Advanced Methods in Materials Processing Defects (Studies in Applied Mechanics)

STUDIES IN APPLIED MECHANICS 45

Advanced Methods in M a t e r i a l s Processing Defects

STUDIES IN APPLIED M E C H A N I C S 20. 21. 22. 23. 24. 25. 28. 29. 31.

Micromechanics of Granular Materials (Satake and Jenkins, Editors) Plasticity. Theory and Engineering Applications (Kaliszky) Stability in the Dynamics of Metal Cutting (Chiriacescu) Stress Analysis by Boundary Element Methods (Balas, Sl&dek and Sl~dek) Advances in the Theory of Plates and Shells (Voyiadjis and Karamanlidis, Editors) Convex Models of Uncertainty in Applied Mechanics (Ben-Haim and Elishakoff) Foundations of Mechanics (Zorski, Editor) Mechanics of Composite Materials - A Unified Micromechanical Approach (Aboudi) Advances in Micromechanics of Granular Meterials (Shen, Satake, Mehrabadi, Chang and Campbell, Editors) 32. New Advances in Computational Structural Mechanics (Ladev~ze and Zienkiewicz, Editors 33. Numerical Methods for Problems in Infinite Domains (Givoli) 34. Damage in Composite Materials (Voyiadjis, Editor) 35. Mechanics of Materials and Structures (Voyiadjis, Bank and Jacobs, Editors) 36. Advanced Theories of Hypoid Gears (Wang and Ghosh) 37A. Constitutive Equations for Engineering Materials Volume 1: Elasticity and Modeling (Chen and Saleeb) 37B. Constitutive Equations for Engineering Materials Volume 2: Plasticity and Modeling (Chen) 38. Problems of Technological Plasticity (Druyanov and Nepershin) 39. Probabilistic and Convex Modelling of Acoustically Excited Structures (Elishakoff, Lin and Zhu) 40. Stability of Structures by Finite Element Methods (Waszczyszyn, Cicho5 and Radwar~ska) 41. Inelasticity and Micromechanics of Metal Matrix Composites (Voyiadjis and Ju, Editors) 42. Mechanics of Geomaterial Interfaces (Selvadurai and Boulon, Editors) 43. Materials Processing Defects (Ghosh and Predeleanu, Editors) 44. Damage and Interfacial Debonding in Composites (Voyiadjis and Allen, Editors) 45. Advanced Methods in Materials Processing Defects (Predeleanu and Gilormini, Editors) General Advisory Editor to this Series: Professor Isaac Elishakoff, Center for Applied Stochastics Research, Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.

STUDIES IN APPLIED M E C H A N I C S 45

Advanced Methods in M a t e r i a l s Processing Defects

Edited by

M. Predeleanu

a n d P. G i l o r m i n i

Laboratoire de M~canique et Technologie, ENS de Cachan, CNRS, Universit6 de Paris 6, 61 avenue de President Wilson, 94235 Cachan Cedex, France

1997 ELSEVIER Amsterdam - Lausanne - New York- Oxford - Shannon - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 RO. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN 0-444-82670-x 91997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, RO. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher f9r any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.

Third International Conference on Materials Processing Defects J u l y 1-3, 1997, C a c h a n , F r a n c e O r g a n i z e d by: Laboratoire de Mdcanique et Technologie (ENS de Cachan, CNRS, Universitd Paris VI) Conference Chairmen: M. PREDELEANU and P. GILORMINI International Advisory Committee: L. ANAND D. BESDO P.R. DAWSON I.S. DOLTSINIS S.K. GHOSH P. HARTLEY M. HASHMI T. INOUE J. KIHARA K.W. NEALE A. NEEDLEMAN E. OI~ATE O. RICHMOND E. STEIN C.L. TUCKER III

USA Germany USA Germany Germany UK Ireland Japan Japan Canada USA Spain USA Germany USA

R.H. W A G O N E R

USA

O r g a n i z i n g C o m m i t t e e (France): J.F. AGASSANT B. BAUDELET M. BRUNET J.L. CHENOT A~ COMBESCURE J.P. CORDEBOIS J.C. GELIN

P. LADEVI~ZE G. MAEDER J. OUDIN A~ POITOU J. GIUSTI C. TEODOSIU

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vii

This volume includes the contributions to the Third International Conference on Materials Processing Defects, which follows its two predecessors held in Cachan, France (1987), and Siegburg, Germany (1992). The Conference focused on advanced methods for predicting and avoiding the occurrence of defects in manufactured products. A new feature was included, namely, the influence of the processing-induced defects on the integrity of structures. The following topics were developed: Damage modeling Damage evaluation and rupture Strain localization and instability analysis Formability characterization Prediction of shape inaccuracies Influence of defects on structural integrity The main manufacturing operations have been covered and various materials were considered: new and conventional metal alloys, ceramics, polymers and composites. It is worth noting that damage theory is used increasingly both for estimating the soundness of the formed product and for defining the workability of a given material. Moreover, new damage models were proposed. High-rate loading conditions, which occur for instance in machining or explosive forming, were also considered. Finally, numerical simulations were used extensively to predict and avoid the shape inaccuracies arising in most forming processes. We believe that this series of conferences stimulates innovative approaches in this important field with obvious economic implications and m u s t be continued. July 1997 M. Predeleanu and P. Gilormini

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CONTENTS Preface ..............................................................................................................................................

vii

DAMAGE MODELING On the dynamic cavitation in solids L. Badea and M. Predeleanu ...........................................................................................................

3

A ductile damage model including shear stress effect J.C. Boyer and C. Staub ...................................................................................................................

13

A mesoscopic approach of ductile damage during cold forming processes G. Brethenoux, P. Mazataud, E. Bourgain, M. Muzzi and J. Giusti ...............................................

23

A fully coupled elasto-plastic damage damage theory for anisotropic materials J.E Charles, Y.Y. Zhu, A.M. Habraken, S. Cescotto and M. Traversin ..........................................

33

Mathematical modelling of dynamical deforming and combined microfracture of damageable thermoelastoviscoplastic medium A.B. Kiselev .....................................................................................................................................

43

A mathematical model for the formation and development of defects in metals V.L. Kolmogorov, V.P. Fedotov and L.F. Spevak ............................................................................

51

Healing of metal microdefects after cold deformation V.L. Kolmogorov and S.V. Smirnov ................................................................................................

61

Definition of the form for kinetic equation of damage during the plastic deformation S.V. Smirnov, T.V. Domilovskaya and A.A. Bogatov .....................................................................

71

DAMAGE EVALUATION AND RUPTURE The influence of critical defect size in a ceramic of alumina elaborated by process sol-gel route N.H. Almeida Camargo, M. Murat and E. Bittencourt ...................................................................

83

Defect evolution during machining of brittle materials A. Chandra, K.E Wang, Y. Huang and G. Subhash ........................................................................

89

Modeling the influence of gradients in strength on the evolution of damage in metals P.R. Dawson, D.J. Bammann and D.A. Mosher .............................................................................

99

On the fracturing of brittle solids with microstructure I. St. Doltsinis ..................................................................................................................................

111

Fracture prediction of sheet-metal blanking process R. Hambli, A. Potiron, S. Boude and M. Reszka ............................................................................

125

Elastic-plastic finite-element modelling of metal forming with damage evolution P. Hartley, F.R. Hall, J.M. Chiou and I. PiUinger .....................................................................

135

Processing of zinc oxide varistors:sources of defects and possible measures for their elimination A.N.M. Karim, S. Begum and M.S.J. Hashmi ................................................................................

143

Analysis of metallic solid fractures by quasimolecular dynamics Y.S. Kim and J.Y. Park ....................................................................................................................

155

Damage framework for the prediction of material defects: identification of the damage material parameters by inverse technique E Lauro, T. Barfi~re, B. Bennani, P. Drazetic and J. Oudin ...........................................................

165

Damage influence in the finite element computations for large strains elastoplastic mechanical structures P. Picart, G. Piechel and J. Oudin ....................................................................................................

175

Microplasticity and tensile damage in Ti-15V-3Cr-3A1-3Sn alloy and Ti-15V-3Cr-3A1-3Sn/SiC composite W.O. Soboyejo, B. Rabeeh, Y. Li, A.B.O. Soboyejo and S.I. Rokhlin ...........................................

185

STRAIN LOCALIZATION AND INSTABILITY ANALYSIS Defects in hydraulic bulge forming of tubular components and their implication for design and control of the process M. Ahmed and M.S.J. Hashmi ........................................................................................................

197

Nimerical and experimental analysis of necking in 3D sheet forming processes using damage variable M. Brunet, S. Mguil-Touchal and F. Morestin ................................................................................

205

Localization of deformation in thin shells with application to the analysis of necking in sheet metal forming J.C. Gelin and N. Boudeau ..............................................................................................................

215

Microcrack induced bifurcation of stress-strain relations for sintered materials D.G Karr and S.A. Wimmer ............................................................................................................

225

Instability analysis for ellipsoidal bulging of sheet metal D.W.A. Rees .....................................................................................................................................

235

FORMABILITY CHARACTERIZATION Compression of a block between cylindrical dies and its application to the workability diagram S. Alexandrov, N. Chikanova and D. Vilotic ...................................................................................

247

Sheet metal formability predicted by using the new (1993) Hill's yield criterion D. Banabic ........................................................................................................................................

257

Characterization of the formability for aluminum alloy and steel sheets S. Barlat, J.C. Brem, D.J. Lege and K. Chung ................................................................................

265

Material plastic properties defects and the formability of sheet metal J.D. Bressan .....................................................................................................................................

273

xi Formability analysis based on the anisotropically extended Gurson model E. Doege, A. Bagaviev and H. Dohrmann ................................................................................

281

Rupture criteria during deep drawing of aluminum alloys J. Proubet and B. Baudelet ..............................................................................................................

289

PREDICTION OF SHAPE INACCURACIES Prediction of flange wrinkles in deep drawing J. Cao, A. Karafillis and M. Ostrowski .......................................................................................

301

Filling defects in ceramics forming process F. Chinesta, R. Torres, I. Mont6n, A. Poitou and F. Olmos ......................................................

311

Localisation of debinding zone for fluid-particle flows in metal injection molding M. Dutilly and J.C. Gelin ................................................................................................................

321

Simplified approaches for the prediction of deep-drawing ears P. Gilormini and B. Bacroix ............................................................................................................

331

Springback and 'rebound' phenomenon analysis with the software PLIAGE F. Morestin, M. Boivin, F. Bublex, X. Deng and M. E1 Mouatassim ... ..................................

341

Prediction of elastic springback defects in sheet stamping processes using finite element methods E. Ofiate and J. Rojek ......................................................................................................................

349

Creep deformation in heat treated components T.C. Tszeng and W.T. Wu ................................................................................................................

361

A study of sharskin defects of linear low density polyethylene C. Venet, J.E Agassant and B. Vergnes .............. .............................................................................

373

INFLUENCE OF DEFECTS ON STRUCTURAL INTEGRITY Influence of initial imperfection on the collapse of thin walled structures A. Combescure .................................................................................................................................

385

On modeling of laminated composite structures featuring interlaminae imperfections M. Di Sciuva, U. Icardi and L. Librescu .....................................................................................

395

Delamination, instability and failure of multilayered composites E. Stein and J. Tel3mer .....................................................................................................................

405

Statistical damage tolerance for cast iron under fatigue loadings H. Yaacoub Agha, A.S. Brranger, R. Billardon and E Hild ...........................................................

415

Author index ....................................................................................................................................

425

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DAMAGE MODELING

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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

O n t h e d y n a m i c c a v i t a t i o n in solids L. Badea a and M. Predeleanu b aInstitute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-70700, Bucharest, Romania bLaboratoire de M6canique et Technologie/E.N.S. de Cachan / UnivSrsit6 Paris 6/61, Avenue du Pr6sident Wilson, 94235 Cachan, France

1

INTRODUCTION

The nucleation of voids or cavities represents one of the main aspects of the ductile microfracture process in solids, specially in metals and alloys, subjected to high straining as in forming or impact problems. It is established that the ductility of the materials is increased if the formation of new microcavities can be stopped or diminished. The influence of the void nucleation on the various coalescence mechanisms (as direct impingement of the voids, microstrain localisation, etc.) was determined quantitatively in recent studies [1-3]. The void nucleation sites were extensively observed, the most attention being focussed on the nucleation attached to second phase particles : void formation by particle fracture or by decohesion of the particle - matrix interface or even at grain boundaries in polycrystals [4]. This kind of nucleation was classified as heterogeneous in contrast with homogeneous nucleation observed in particle free regions [5]. In [6], it was reported, for instance, that in AISI 304 stainless steel the void density was greater than the particle density by a factor of one hundred. Many microcavities were observed in areas of high dislocation densities. The threshold stress for the homogeneous nucleation of voids is of course greater than the one for heterogeneous nucleation and is generally very high. It is worth noting that in shock-loaded materials subjected to explosive forming or high rate impact loadings the tensile mean stress can exceed 10-20 GPa. The sudden formation of voids in solids (a phenomenon named "cavitation" as in fluid mechanics) has been observed in various kinds of materials (rubbers, elastomers, metals, composites) submitted to tension loadings. The paper of Gent and Lindley [7] has drawn special attention because the experimental critical load for the occurrence of a void in a short rubber cylinder pulled in tension agrees with the theoretically deduced one. Analysing the behaviour of spherical cavities in an infinite elastic (neo-Hookean) body loaded by a hydrostatic tension, they found that there is a critical value of the load at which the void grows rapidly without bound (" cavitation instability"). Consequently, Gent and Lindley explained the cavitation phenomenon by the rapid growth (without bound) of a pre-existing microscopic void. An alternative approach for cavitation problems was proposed by Ball [8] which used the discontinuous radially symmetric solutions for the equilibrium equations of an

elastic solid sphere submitted to traction loadings on the boundary. He has shown that for a certain class of elastic materials there exists a critical value of the loading at which a non-homogeneous solution describing the formation of a central cavity bifurcates from the trivial homogeneous solution, which becomes unstable. The cavitation solution is energetically favorable. Due to scaling of the two above problems in finite elasticity, the critical load obtained by Gent and Lindley for cavitation instability is that obtained by Ball for the bifurcation of the discontinuous solutions. Equivalent results were reached by Sivaloganathan [9] and Horgan and Abeyaratne [10], who have treated the cavitation problem for the sphere by considering at its center an infinitesimal pre-existing void (the initial radius approaches zero at critical load). The study of the solutions that allows the cavitation for the full three-dimensional problem in nonlinear elasticity is given in [11]. The cavitation phenomenon has been examined also for elastic-plastic solids for symmetric loadings in [12-17] and for non-symmetric loadings in [lS]. The radially symmetric cavitation problem for rate-dependent materials has been treated in [19]. A comprehensive review of the literature on cavitation may be found in [20]. Very few studies have been concerned with the dynamical problem of cavitation and that in the context of finite elasticity, [21-22], and of viscoplasticity, [23]. This paper is concerned with the continuum micromechanics analysis of homogeneous nucleation and its objective is to deduce analytically the conditions for the sudden cavitation in elasto-plastic and viscoplastic solids subjected to high dynamic loadings. In the next section we state the formulation of this mechanical problem for a hollow sphere submitted to a symmetric traction loading in the dynamical case, for both viscoplastic and elasto-plastic materials. By considering the incompressibility of the material our problem reduces to solving some second order nonlinear differential equations. These equations proceed from the equation of motion and have as unknown the current void radius. In Section 3 we shall consider the initial void radius to be infinitesimal, and consequently, we shall take this radius as vanishing. In what follows we find an expression for the critical load and we give some theoretical results concerning the dependence of the solution on this critical load and various initial conditions. We also make some remarks concerning the theoretical results and we shall draw some conclusions for each material type, viscoplastic and elasto-plastic. 2

MECHANICAL

PROBLEM FORMULATION

Consider a hollow sphere with its center situated at the origin of a spherical coordinate system (R, O, r and denote by A and B its inner and outer radii in the undeformed configuration. The sphere is supposed to have in time only a spherically symmetric motion and therefore, the coordinates in the current deformed configuration (r, 0, ~o) are of the form

o=o,

(:)

for the time interval 0 < t < oo. The inner and outer radii in the current deformed configuration will be denoted by a and b, respectively. m

Taking into account (1), the velocity field is 0r = ~

= ~,

(2)

and consequently, the Eulerian strain rate tensor has the form Ovr

D = -b-;~ o ~ + ~(~0~ o ~ + ~ ~ ~ ) .

(3)

The Cauchy stress tensor can be written as (4)

er = a f t e r | er + aoo(eo | eo + e~ | e~).

As usual, we shall assume an additive decomposition of the strain rate tensor, D = D" + D p,

(5)

in its elastic, D ", and plastic, D p, parts. We shall suppose that the sphere is an isotropic and homogeneous material. The thermoelastic response is defined by v

(6)

D~ = 1 + E v& _ -~tr&l + ar

where E and v are the elastic constants, I is the unit tensor, T the absolute temperature and a the coefficient of linear thermal expansion. The dot denotes the material time derivative. We note that in our case, the corotational (Jaumann) time rate of the Cauchy stress tensor coincides with its material time derivative. The viscoplastic strain rate is written in the overstress form, widely used in dynamic problems, [27], F

DP=

r

~-

OF

1 >)0er,

(7)

where "r is a temperature-dependent viscosity function, r is a control function, r is the viscoplastic overstress function, Ey the material isotropic hardening-softening temperature-dependent function, and F the quasi-static yield function. We shall consider a von Mises material, F(er) = ~/3erd. era, where erd is the deviatoric part of the stress tensor. For the elastic-plastic material, we shall take Dp

. OF

= P0~'

(8)

where i5 = ~ / ] D p " D p is the equivalent plastic strain rate. Also, the yield criterion for the plastic zone will be written as, F - ay - kp n = 0,

(9)

where k > 0 and 0 < n _~ 1 are some material constants and ay is the elastic limit. Supposing the material incompressible, t r D = O, from (3) we get OUr Vr 0--7 + 2--r = O.

(10)

As consequence, we deduce that r 3 = a 3 + R 3 - A 3. The motion equations are reduced in our case to a single one, (%Trr

T

Or

2-,r

p6,=

(11)

where the density p is constant and r = a00 - a ~ . As we mentioned above, the inner surface of the hollow sphere is stress free, a~(a) = O, and a uniform nominal tensile stress is applied on the its exterior surface, arr(b) = po(-~) 2. We shall multiply the above equation by v~ and by the integration on the whole volume of the hollow sphere, taking into account the boundary conditions, we get, d [a3( 1

b)h2 ] = 2a2h(..b_) [p0

_

2

b

(12)

For the viscoplastic solid, we have (elastic strain neglected), 7 = sign(v.)Ev[1

+

r162

3'

I v. IH(I~ I

~))1.

_

r

(13)

and for the elastic-plastic solid

r = ay Jr kp" = 0,

ln~=

(14)

3E r + ~

where H the Heaviside step function, i.e. H(x) = 1 if z > 0 and it vanishes for x < 0. In the above equations we shall take r = x 1/m, with 0 < m < 1, E~ = a~p", 0 < n < 1 and r = 1. The temperature-dependent coefficients are taken as,

uy(T)

t

0

= =,(T)

ifT>T,,,

(16)

~(T), ~(T) = ~ exp V ( 1 / T - 1/V~), where r/is viscosity, Y, V > 0 are some material constants, and Tm and r/m are the melting temperature and viscosity at the melting temperature, respectively, [25]. The simplified heat conduction equation may be written as

pcoT - - d i v q + X~" D ,

(17)

where co is the specific heat, q the heat flux and X the Taylor-Quinney coefficient that takes into account the stored elastic energy (X ~- 0.9). In our spherical case the heat 0T flux, using the Fourier's law, is of the form, q - - k o g r a d T - -ko~Ter. Therefore, we have,

i)T a2h tOT " 02T a2a pco(-~-~ + --~--~-~r ) = ko-~r2 + 2X7 r 3

(18) 9

For this equation we shall consider the initial condition T(0) = To. The heat flux vanishing on both surfaces will be considered as a boundary condition, q(a) = q(b) = O.

3

CRITICAL LOAD, CAVITATION AND BIFURCATION

In this section, in order to study the cavitation phenomenon, we shall consider the initial void radius is vanishing, A = 0. 3.1

Viscoplastic solid

We shall substitute the temperature dependent coefficients (16) in the formula of r, (13), and then the motion equation (12) becomes

2 B)2 [po(t)_ sign(h)Po(t)],

pd[a3(1 - b)h 2] + Q(t) - 2a h ( ~

(19)

where Q(t) = 4a2lhl

~(

r3

)'~H(]~-]- Zu)(21n~) r '

b )2 f b r ). dr P0(t) = 2 ( ~ au(2 In 5 -r--.

(20)

(21)

For the equation (19) we shall first consider the following homogeneous initial conditions, a(0) -- 0,

h(0) -- 0.

(22)

The convergence of the integrals in (20) and (21) is better seen if we change the inte{1 we get gration variable, s = 7n3" Writing x = ~, [ 3

2

1,,

J0

ds

(23)

In this way, we see that the above integral is convergent. Also, we remark that Po(t) > 0 and Q(t) > 0 for 0 < T < Tin. On letting t -+ 0 in (23), the critical load is obtained as

2 fl

2

1

ds

Per = P0(0)= 3 ]o au(To)(-~ ln-s )" 1 - s "

(24)

R e m a r k 3.1 In the following, the temperature T will be considered as a continuous function of a and h, that is, the solution T of equation (18) depends continuously on the coefficients. By a solution a of equation (19) on the interval [0, co) we shall consider the classical solutions, that is, a E C2(0, cr f3 Ca[0, co). In the following we shall give some theoretical results. Theirs proofs can be found in [23]. The next lemma gives the behaviour of a non zero solution on a time interval.

L e m m a 8.1 /f on a time interval (tx, t2), 0 < tl < t2, a solution a of equation (19) satisfies

a(t) > O, a(t) # 0 on (t~, t2), a(tx) = 0 or a(t~) = 0, a(t2) > 0 and a(t2) # 0, sign(po(t) - sign(h)Po(t)) = sp ~ O, constant on (ta, t2),

(25)

then sign(a) = sp, and consequently, on (tl, t2) we have the following equivalences i)po(t) > Po(t) ~ a> 0 ii)po(t) < -Po(t) ~ a < O,

the case IPo(t)l < Po(t) being impossible. Concerning the solution of the problem corresponding to applied loads smaller than the critical load P~, we have P r o p o s i t i o n 3.1 If the applied load po(t) is continuous and 0 < po(t) < P~ on any closed finite time interval, then problem (19), (22} has a unique solution, that is, it has only the zero solution. R e m a r k 3.2 The above proposition also states that the problem (19), (22) can have more than one solution only if po(t) >_ P~, and consequently, for po(t) = P~ we may have a bifurcation of the solution. Let us consider now nonhomogeneous initial values for our problem, that is in the place of the conditions (22) we shall consider the following initial conditions a = ao, ei = v0,

(26)

where a0 > 0 or v0 > 0 and aovo = 0. The following proposition proves that for the above initial conditions, the problem has a solution only if the applied load exceeds the critical load, or, in other words, we can get cavitation only with applied loads greater than the critical load. P r o p o s i t i o n 3.2 Supposing t > O, we then have / } / f 0 < po(0) < P~ then i/}/fpo(0) > P~ and a(t) exists a t2 > 0 such that a(t)

that the applied load po(t) is a continuous function for problem (19) and (e6) has no classical solution, is a non-nut solution of problem (19) and (Ca), then there > 0 for t ~ (0, t2).

R e m a r k 3.3 The result of item i) in the above proposition is natural taking into account that in our model the initial radius of the hole is vanishing. The following theorem describes the local behaviour of a solution for a time tl > 0. T h e o r e m 3.1 Let 0 < tl < t2 and a(t), t E [0, t2] be a solution of problem (19) with initial conditions (22) or (26). Then i} if 0 < po(tl) < P0(tl) then there exists a neighbourhood of tl on which either

a(t) = a(t) = 0 or a(t)a(t) # O, ii} if po(ta) > Po(ta) and a(tl) > 0 or a(t~) # 0 then there is a tz, tx < t3 < t2 such that a(t) > 0 on (t~, t3), or there exists a neighbourhood of tl on which a(t)a(t) ~ O.

3.1.1

Conclusions

We remark that in the above results the connection between the applied load po(t) and Po(t) is essential. In what follows we shall make some remarks concerning the behaviour of a nonzero solution and the critical load at which the cavitation take place, plotting in a plane P - x the curves P ( z , T) =_ Po(t), in which the temperature T is considered as a parameter. In Fig. 1 we have considered a specific example to point out the shape of these curves. This example corresponds to n = 0.24 and Y = 2206 M P a and the initial and melting value of the temperature were To = 293 K and T,~ = 1673 K, respectively.

\

~\

1

I

0.8

\ , \ \ \ \ \ \ \

0.6

\

0.4

\

I

I

I

I

I

T1 = 293 K T2 = 493 K T3=693K T4 = 893 S T5 =1093 K

~

T6=1293K =1493 K

-

-

t

0"2

t~, 0

500

Ts

T4

T3

T2

T1

-

P (MPa) 1000 1500 2000 2500 3000 3500 4000 4500 5000

Fig. 1. Post-cavitation for n = 0.24 and different temperatures First, we remark that P(x, T) is a decreasing function depending on T. Also, the curves are left turning for n < 1.0 (see also Abeyaratne and Hou, [19], for the case of rate-dependent materials), and we shall assume that the temperature is an increasing function during the deformation. Suppose now that the load po(t) > 0 will remain greater than P~ if it exceeds this value at a certain time (even if it starts from a value p0(0) < Pc~), and that the problem has homogeneous initial conditions. In this case, using Proposition 3.1, the item i) of Lemma 3.1 and the item ii) of Theorem 3.1, we can prove that for po(t) > P~ a non-zero solution is an increasing function in time. Consequently, for such loads, the point (P, x) will be situated in our figures, either on the segment [0, P~], if po < P~, or in the right region of all curves, if po > P~. On the other hand, if the load po(t) > 0 assumes values less than P~ move than once, then the point (P, x) can theoveticaly be situated in the left region of the curve corresponding to the temperature T(t). In this case, from item i) of Theorem I it results that for a time to, the solution will be discontinuous in two situations: either the point is not on the segment [0, P~r] and h(to) = O, or the point lies in [0, P~] and h(to) # O. As for the points on the curves P - z, we remark from equation (19) that there are no static states (in which the sphere remains undeformed while stressed) if p(t) > O. Indeed, if the point lies on a curve P - x, replacing h(to) = 0 in (19), we get/i(t0) > 0, and consequently, in a time interval (to, t l) the solution will pass into the region to the

10

right of the curves. Suppose now that, for the problem with homogeneous initial conditions, until the time to we have only the zero-solution and at this time we perturb the solution taking a(to) = ~ > 0. Taking into account Proposition 3.2, we can conclude that: ifpo(to) < P~ then the solution will go back to zero (having a jump) and if po(to) > P~, then a(t) for t > to will be an increasing function. The obtained critical load P~, for our example, calculated from (24), was of 4767.6 MPa. Also, we remark from Fig. 1 that P~ is an decreasing function of temperature T. Other aspects of the thermal effects on the void growth are reported in [26-27]. 3.2

E l a s t o - p l a s t i c solid

In this subsection the temperature influence will not be considered. For simplicity, we shall also assume the imposed load p0 to be constant in time. In the case of po depending on time and temperature influence is taken into account, we can derive similar conclusions to those in the previous subsection. The term in (12) which contains the integral will be denoted as in the previous subsection by P0 = 2(~-) 2 f~ ~dr, and we shall write, P = (-~)2(p0 -/90). In order to study the behavior of this term we shall R3 change the variable in the integral, s = 7 , and also, we shall write x = ~-. We obtain in this way

Po(x) =

2

_ fo ~ 1 - s ds'

2 P(x) = po x2 _ -~ rio

(27)

r(s) 1- s

(28)

and we shall denote 2

P= = Po(1)= 5 fo x T(S) d3,

(29)

which will be named in the following critical load. We mention that ~" in the above formulas is obtained as the solution of (14) and (15). We also remark that 0 _< x _ 1. We shall give in the following some theoretical results. Concerning P(x) we have L e m m a 3.2 For 0 < x < 1 the integral in (PT) makes sense. Concerning the behaviour of P(x) we have 9 (i) if po > _2 x/3~--~-- then P(x) is an increasing function on [0,1], E (ii} if po > i-4";, then there ezists 0 < Xo < 1 such that P(z) is increasing on [x0,1], E (iii} if po < i-4-;, then there exists 0 < zo < 1 such that P(z) is decreasing on [z0,1]. From the item (i) of the above lemma we shall immediately get the following consequence. C o r o l l a r y 3.1 An upper bound of the critical load is given by P~ < -

21~3E--E-

X-l-v"

11 For the second order differential equation (12) we shall consider the following initial conditions a(O) = O,

a(O) =

(30)

where vo > 0 is given. We have, P r o p o s i t i o n 3.3 If po < P~, then the problem (1P) and (30) has only the null solution. If po > P~, then this problem should have without the null solution another one nonnull. P r o p o s i t i o n 3.4 Assume po = P~. Then the problem (12) and (30) has only the null E If P~ < Tu E then this problem should have without the null solution if P~ > T-4"~" solution another one non-null. 3.2.1

Conclusions

An upper bound of the critical loads is given in Corollary 3.1. Evidently, we expect the values of the real critical load to be lesser than this enough large upper bound which is valid in the case of the linear elasticity, too. Proposition 3.3 shows that cavitation is possible only if the imposed load is greater than the critical load, p0 > Per. For the imposed load less than the critical,/90 < P~, cavitation is impossible. Finally, Proposition 3.4 shows that the solution bifurcation at the critical load, p0 = P~r, is possible only if this value does not exceed a certain upper bound, Per < l+v" E E cavitation at the critical load is impossible. This fact does In the case when P~r > 1-~' not contradict the previous remarks, where po ~ P~ and we assume that p0 is constant in time.

4

REFERENCES

1. V. Tvergaard, Adv. Appl. Mech., 27 (1990), 83. 2. G. Perrin and J. B. Leblond, Int. J. Plasticity, 6 (1990), 677. 3. X.P. Xu and A. Needelman, Int. J. Plasticity, 8 (1992), 315. 4. S.H. Goods and L.M. Brown, Acta Metallurgica, 27 (1979), 1. 5. M.A. Meyers and C. Taylor Aimone, Progress in Material Science, 28 (1983), 1. 6. H.G.E. Wilsdorf, Mat. Sci. Engn, 59 (1983), 1 7. A. N. Gent and P. B. Lindley, Proc. R. Soc. Lond., A, 249 (1958), 195. 8. J. M. Ball, Phil. Trans. R. Soc. Lond., A, 306 (1982), 557. 9. J. Sivaloganathan, Arch. Rational Mech. Anal., 96 (1986), 97. 10. C. O. Horgan and R. Abeyaratne, J. Elasticity, 16 (1986), 189. 11. S. M/iller and S. J. Spector, Arch. Rational Mech. Anal., 131 (1995), 1. 12. R. F. Bishop, R. Hill and N. F. Molt, Proc. Phys. Sco. 57 (1945), 147. 13. R. Hill, The mathematical theory of plasticity, Clarendon Press, Oxford (1950). 14. D. Durban and M. Baruch, J. Appl. Mech., 46 (1976), 633.

12 15. D.-T. Chung, C. O. Horgan and R. Abeyaratne, Int. J. Solids Structures 23 (1987), 983. 16. Y. Huang, J. W. Hutchinson and V. Tvergaard, J. Mech. Phys. Solids, 39 (1991), 223. 17. V. Wvergaard, V. Huang and J. W. Hutchinson, Eur. J. Mech., A/Solids 11 (1992), 215. 18. H.-S. Hou and R. Abeyaratne, J. Mech. Phys. Solids, 40 (1992), 571. 19. R. Abeyaratne and H.-S. Hou, J. Appl. Mech., 56 (1989), 40. 20. C. O. Horgan and D. A. Polignone, Applied Mechanics Reviews, 48 (1995), 471. 21. K. A. Pericak-Spector and S. J. Spector, Arch. Rational Mech. Anal., 101 (1988), 293. 22. M.-S. Chou-Wang and C. O. Horgan, Int. J. Engn Sci., 27 (1989), 967. 23. L. Badea and M. Predeleanu, Mechanics Research Comm. 23 (1996), 461. 24. P. Perzyna, Adv. Appl. Mech., 9 (1966), 243. 25. M. M. Carroll, K. T. Kim and V. F. Nesterenko, J. Appl. Phys., 59, 6 (1986), 1962. 26. D.R. Curran and L. Seaman, Proc. Int. Seminar on high temperature fracture mechanisms and mechanics, Dourdan, 13- 15 October, 1987. 27. A. Neme and M. Predeleanu, in Structures under Shock and Impact, eds N. Jones et al., CMP (1996).

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

13

A ductile d a m a g e m o d e l including shear stress effects J.C.BOYER a and C.STAUB b a Laboratoire de M6canique des Solides, INSA Lyon. 20, av A.Einstein. 69 621 Villeurbanne Cedex France b Centre de Recherche et d'Etudes Technologiques, GIAT Industries. 2, rue Alsace-Lorraine. B.P. 1450 - 65014 Tarbes Cedex France

1. INTRODUCTION Since the early work of Gurson [ 1], ductile damage macroscopic constitutive laws based on microvoid growth have been improved, the well-known Tveggaard [2] plastic potential included some coalescence effects, Rousselier [3] formulated relations for finite deformations of plastically dilatant materials, more recently Brunhs and Schiesse [4] have developed a continuum model of elastic-plastic materials with anisotropic damage by oriented microvoids. Most of the existing plasticity-damage theories consider the mean normal stress as the main parameter controlling the microvoid growth even though the deviatoric part of the stress tensor acts on the phenomena at the microscopic or mesoscopic levels. In the first part of this work, numerical modellings of the mechanical behaviour of a microvoid with or without a free inclusion in an elastic-plastic material under several stress states are presented. Then, these predictions of the void volume fraction evolution are compared to a modified Rice and Tracey model intended for any stress states including pure shear loadings. In the second part, a yield function for a stress dependent density material is discussed and identified with the modified Rice and Tracey model.

2. FINITE ELEMENT MODELLING OF THE VOID GROWTH

2.1. Uniaxial loading In their last paper the authors [5] started to simulate the growth evolution of a spherical void in a cylindrical unit cell under uniaxial loading. Using axisymmetry only one quarter of a vertical section is analysed. A spherical void is located centrally in the cylinder such that typical void fractions and distributions are in agreement with experimental data [6]. On the outer surface normal to the cylindrical axis, the displacement is imposed as a remote uniaxial tensile or compressive loading. On the cylindrical surface, the radial components are constant. The behaviour of the matrix is supposed elastic-plastic with linear piece-wise hardening. The particle material is elastic with a ratio of 10 or 0.1 to the matrix's Young modulus. The matrixparticle interface is supposed debonded since the start of the loading. The behaviours of a cell with an inclusion and a cell without an inclusion are completely different. Under tensile loading the void volume fraction increases with effective strain (Figure 1). The phenomenon is less emphasised for a void without inclusion. The more the

14 inclusion Young modulus is high, the more the closing of the void is constraint, but the void volume fraction evolution is quite similar, that's the reason why in the next simulations, the inclusion will be considered as a rigid body. The void volume change predicted with the Rice and Tracey modified model is in very good agreement with the finite element modelling of the filled void. Under compressive loading (Figure 2), the void volume fraction for a cell without an inclusion reduces, whereas it grows for the void with an inclusion.

Figure 1 9Void volume fraction evolution under tensile loading

Figure 2 9Void volume fraction evolution under compressive loading

2.2. Void volume fraction evolution under zero mean stress loading 2.2.1 Mesh

The model used for the finite element simulation is quite similar to the one used for uniaxial loading, but to take into account local plasticity with more accuracy, the mesh is improved (Figure 3). The void volume will be integrated numerically at each step of the loading. When the loading is symmetric (Figure 4) only one quarter of the unit cell is analysed.

Figure 3 9Generalised plane strain Mesh of the unit cell 96962 nodes

Figure 4 9Axisymmetric loading Mesh of a quarter : 1886 nodes

The behaviour of the matrix and the boundary conditions were presented in section 2.1. Such a model is useful to analyse the behaviour of a cylindrical cell with a spherical void as well as to observe the growth evolution of a cylindrical void in a planar cell.

15 2.2.2 Axisymmetric loading" cylindrical shear stress state Using the axisymmetric model, the displacements dz = 2a and dr = - a are imposed.

Figure 5 : Cylindrical shear Unit cell without an inclusion

(see Figure 4)

Figure 6 : Cylindrical shear Unit cell with an inclusion

As shown in Figure 5 and Figure 6 the void volume fraction changes with the plastic strain. The rate is positive for a filled void and negative for an empty void. The behaviour of a cell under cylindrical shear strain is quite similar to the one of a cell under compressive loading. 2.2.3 Generalised plane strain : pure deviatoric stress state

Figure 7 : Pure shear Unit cell without an inclusion

Figure 8 : Pure shear Unit cell with an inclusion

16 In the principal axis, a pure deviatoric strain state can be easily simulated by imposing a displacement dr = a and a displacement dz = - a in the perpendicular direction. As for the cylindrical shear strain state, the void volume fraction decreases with effective strain for a void without an inclusion (Figure 7) whereas it grows for a cell with an inclusion (Figure 8). Shear stress induces cavity closing for empty voids, whereas for a filled void, the void volume fraction increases whatever the nature of the loading.

2.3 Principal direction of the ellipsoid Using the generalised plane strain model without an inclusion, the aim of this section is to find the evolution of the principal direction of the ellipsoid versus a representative parameter which could be used in a constitutive law.

Figure 9 : Loading history

Figure 10 : Evolution of the principal direction of the elliosoid

Figure 11 : Loading history

Figure 12 9Ellipsoid principal direction

The strain rate principal direction ranging from 45 degrees to 31 degrees was gradually decreased and increased (see Figure 9). The principal direction of the ellipsoid is quite

17 identical to the strain rate principal direction (Figure 10). Even if the principal strain state direction is changed rapidly (Figure 11), the principal direction of the ellipsoid follows it. The more a loading direction is maintained constant, the more the ellipsoid direction is closed to the imposed principal strain rate direction (Figure 12).

3.A MODIFIED RICE AND TRACEY MODEL

For the case of an empty spherical void of radius R in a remote uniform plastic strainrate field D ], Rice and Tracey [7] have developed a growth model of microvoids under arbitrary remote stress states ~j = sij +OmS~j, where s~j is the deviatoric stress tensor and O' m the mean normal stress. An approximate estimate of the rates of change in the radii of the void in the directions of the principal strain-rates for the case of isotropic linear hardening materials has the form" 5

+ 3 O"m DP]R

(1)

1L= ?D~ 4~M n

the equivalent plastic strain-rate and ~M the yield stress of the matrix material considered as incompressible. with D p

The void growth rate equation (1) can be used to give useful estimates of the change in shape and volume of an initial spherical void by means of an approximate or numerical integration for proportional loading but it has to be extended to pure shear stress states. 3.1. Void without an inclusion

Thomason [8] gives general expressions for the three integrated principal radii of the resulting 'ellipsoidal' void but the predicted apparent plastic dilational strain does not change under zero mean normal stress. Once the void is no longer spherical, the rates of change of the three radii are different as the 'ellipsoidal' void has different stiffness in the three directions of the principal plastic strain rates, some curvature correction linked to the equivalent spherical void must be introduced in the original model :

+2 (~m Dp

Rk

with the 'spherical' void radius R

4 NM

= ( R , R 2 R 3 ) '/3

(2)

If Vc is the void volume, its logarithmic rate is given as 9

v

r 3ool

(3)

For an empty void, this equation leads to a good fit with volume void fraction rates predicted by numerical modelling for tensile, compressive and shear stress-state, see Fig. n~ 2, 5 and 7 respectively. For large effective plastic strain, the real shape of the void is far from the assumed ellipsoid, these geometric differences are the main reasons of the discrepancies

18 between the values of the void volume fraction predicted with this modified Rice and Tracey model and by the finite element analysis. Nevertheless, the initial rates are in good agreement and the first stages of the plastic deformation of an initial spherical void under non-isotropic plastic strain rates induce volumetric change even under zero mean normal stress. 3.2. V o i d with an inclusion

For a void filled by an inclusion without interface strength, the void growth rate equation (1) is still a versatile basis with unilateral kinematic boundary conditions on the direction of the principal plastic strain-rates. a. Tensile stress state D p = D p and (Yl

= ~ M = 3t3m9

} E ] ~

DPei = - D p / 2 ,1~3 = 1~2 -

d

v(y)

(2)

From internal energy U we pass to the free energy F = U - T s and then inequality (2) gets us, with the help of equation (1), to the next inequality 1

.

1 (gradT

_ Ts - F > 0

(3)

Because F = F ( e e j , e i ~ , w , ~ , T ) , we get next inequality from (3): OF

OF

1

.p

OF.

OF

OF .p Or

1 ~ gradT - - ~ > 0 p T -

As it's shown in [5], it follows from (4) that

(4)

45

OF

O'ij -- P ~ e ' Oeij

OF

.S =

(5)

OT

Using (5), we lead (4) to the next inequality

(6)

d = dM + dF + dT >__0,

d = PT, 7 is the production of entropy and the next disignations are introduced here

OF

OF.

d M -- ( a i j -- p ~ _ p j )~i

OF.

dR = - - p ( - ~ w w + - ' ~ a ) ,

dT = --

~ gradT T

'

(7)

dM is mechanical dissipation, dF is dissipation of continuum fracture, dT is thermal dissipation. Turn again to the equation (1) which is written over with the help of (5) in the next form Th =

OF

.p

(o'ij - p ~ 6 ~ j ) e i j

OF.

OF

- -~ww - -'~&

1 pdiv(

(8)

1.1. T h e m a i n a s s u m p t i o n s Next simplifical assumptions were done: e e 1. Elastic deformations are small: eijeij 0 (6) may be written as sum of three nonnegative addends

(7): d M >_ 0,

dF >_ 0,

d T >_ 0

(9)

Note that for Fourier low of heat ~' = -e;gradT termal dissipation has the form of Fourier inequality dT = a(gradT) 2 / T >_ 0; hypothesis dM >_ 0 is Plank inequality 9 The inequalities (9) will used for formulation of kinetic equations for internal parameters of state ~iPj, w and a. 4. For dissipation of continuum fracture we assume that

OF -p~-~- = A&,

OF -Pffw-w = A&,

(10)

where A >_ 0, A >_ 0 - parameters of material. Note that equations (10) are the consequenses of Onzager theory under the condition A = const and A = const. 5. Volume modulus K, shear modulus # and dynamical viscosity 77 are the next function of parameter of damage w for damaged material: K

= K o ( l - w),

# = #o(I - w),

~ = ~/o(I - w)

46 Here K0, #0 and 770 - parameters for nondamaged material (they can be the functions of temperature, pressure and other parameters [7]). 6. Kinetic equations for parameters w and a have next form: =

=

Here a -- akk/3 - the first invariant of the stresses tensor O'ij and r second invariant of the deviator of stresses tensor 7"ij = aij - aSij. Introduce next thermodynamic potential

(TijT"ij) 112 - the

=

e

G = F-

~O'ijCij

(11)

Derivation of (11) on the time with regard of (5) gives 1

= ----O'ij~i~ -- T s "4-

OF.

"4-

OF

(12)

&

Because G = G(o'ij,w, a , T ) , we get from (12) r

OG = - P oaij,

s =

OG OT'

OG Ow

OF Ow'

OG Oa

OF Oa

(13)

Using the hypothesis 1 - 5 we get (M

1

1

- 3 K a ~ k + -'4~ O'ij O'ij + -3 a V O ' k k ( T - To) + A - p G = 2#36K-------~

I

O~

~odw+ A

0

I

Cda+

0

+Go(T),

(14)

where a v - the coefficient of volume extension. If we introduce specific heat under constant stresses c~, then we get next equation of heat conduction from equation (8) [7]" pc,,~' + a v & T

= ,,:ls "v -3t- Ad,,2 + A& 2 - d i v ~ "

(15)

"

Using the equations (13), (14) and the hypothesis 5 - 6 we get {M

o, ~kk = ~ o + a v ( T - T o ) + -~

Ot .

~adw,

eij = 2#---~ -

.+

~a~j da

0

(16)

0 !

Here ei~ - deviator of elastic strain, a ' = a/(1 - w ) , rij = vij/(1 - w ) . Let us assume that viscoplastic deformations are described by kinetic equations of Perzina type [12]"

2,/0

r'

H(r'-

~

Yo)

(17)

47 Here ~-' = r / ( 1 - w ) . The statical yield strength Iio depends on tempetature, pressure and other variables of state [7] , Y = Yo + 277oV~C.,ij,.,ij /3 ;v ~v is dynamical yield strength, H(x) is function of Heaviside. Now we define concreteness kinetic equations for damaged parameters w and c~: :

:

-

&= r

-

= C(r'-

~-.)H(v' - 7.)

(18)

Here B, a . , C, r. - parameters of material. 1.2. The system of constitutive equations of the m o d e l Thus full system of constitutive equations for model of damageable thermoelastoviscoplastic medium consists of the next equations: Od

a' e k k = -~o +

~ v

A / O~ ( T - To) + ~ --~adw ,

r'j + A ] eiJ

=

0

27/0

r'

d.; = c2(w,o ) = B(a' - a.) H(a' - a.),

0r

2#0

0-~ij d~' 0

'

6 = r

pc~,7' + o~v&T = 7"ijii~ + A&2 + A&2 -- dive,

c~, r) = C ( ' r ' - 7.) H ( T ' - r.),

~=-x

gradT

(19)

This model generalizes model of elastoviscoplastic flow [12] and takes into account the accumulation of damage in area of intensive tension and in area of intensive shear, the effects of the processes of deformation and accumulation of micro-structural damage, the thermal effects.

1.3. Criterion of beginning of destruction Criterion of beginnig of destruction (the origin of cracks - new free surface in material) - is the entropy criterion of breaking specific dissipation (as in models [7, 8])" t.

D = f _1(dM + dF + dT)dt = D , P

(20)

o

Here t, is fracture time, D, is parameter of material (maximum of dissipation), which is defined experementally [7, 8]. Criterion (20) may be referred to the class of entropy criteria of failure. Such a criterion makes it possible to describe, in principle, the process of failure, using the mechanism of cumulative mierostructural damages occurring, for instance, at break-off failure in tension waves (in this case a decisive contribution to (6) and (20) is made by the term Ad; 2 the power of continual failure dissipation along with the power of ...v Use can also be made of the mechanism of shear mechanical dissipation dM = "r,;r

48

which is the case, for example, in problems of punching plate targets of a finite thickness with a flat-face striker. In his particular case, narrow zones of intensive abiabatic shear are known to develope in target in places of stress concentration. The work of plastic deformations converts almost completely to the heat that, because of high local deformation velocities, has no time to extend over any significant distance from the zones of developed plastic deformations. As a result, the temperature in the zones rises and great thermal gradients occur which cause an additional plastic flow and a further concentration of local plastic deformations and eventually forces a "plug" out of the target. At shear failure, a decisive contribution to (6) and (20) is made by the terms dM= rij~i~, dT = and A&2.

--qgrTdT

2. E X A M P L E S Now we consider examples of formulation of special problems of deforming and fracture of solids which make possible to estimate capabilities of the model. 2.1. F l a t c o l l i s i o n o f t w o p l a t e s w i t h s p a l l a t i o n d e s t r a c t i o n in a plate t a r g e t Consider the problem of flat collision of two thin plates which is studied experimentally very well. This problem is most often used for calculation of constants of the solids models by means of comparison of results of physical and numerical experiments of the problem of flat collision of two plates with spallation destraction in a plate target [7, 8]. Since the thickness of the plates are small as compared with the size and the characteristic time of the process is the time of several runs of elastic wave across the whole thickness of the plate target, the problem may be solved using a one-dimensional mathematical formulation (a uniaxial deformed state) and an adiabatic approximation (div~ = 0). In this case, the equations of mass, momentum and internal energy are written on the Cartesian coordinate system Oxyz (the x - axis is perpendicular to the plate surface) as follows:

/5

- = -i,

p

1 a ( f + a)

7) = -

p

pc~T + a~&T = 23--'rgl~ + AdJ2 + A&2

Ox

Here v = v. is the velocity, i = i~. = 0--7, o. i p = i ~ , r = Txx the rest of designations is identical to the ones introduced above. Besides, we take into account of the following

eyy'' = L-P:: = -~/2,

ryy = rzz = -r~=/2, since rkk = 0 and gkk "p

5-'= K 0 ( i r'

arT2 Y0

e' - ~-~0 (1 - ~7[)

=

0. Constitutive equations are given by

3 ( 1 - ~ 1 co) '

H(I

r'

2

I- 5Y0),

=

+

i _

i~

~"

+ iv,

+

~

AC 1-w

& sign T,

49

~ "- C ( ~

& = B(a~ - a,) H(a~ - a,),

[TI[ -- T,) H ( ~

[Tll -- T,)

Spallation destraction take place in the cross-section x = x*, where is realized criterion of destruction (20).

2.2. O n e - d i m e n s i o n a l s p h e r i c a l l y s y m m e t r i c p r o b l e m Now consider one-dimensional spherically symmetric problem in adiabatic approximation. In view of the spherical symmetry, the equations of the laws of conservation of mass, change of momentum and heat influx are written as PP -- -g~ - 2go ,

piJ = ~Oa~ + 2 ~a~, r - ao

pc~T + av&T = r~g~ + 2rogPO+ A&2 + A&2

Here r is the distance from the center of symmetry; v is the velocity of radial motion; ar and a0 = ar are the components of the stress tensor decomposed into the spherical and deviator parts a = (a~ + 2a0)/3 and a~ = a + r~, a0 = a + r0, respectively; gr =~'7 ov and go = vr__are the deformation rates that are representable as sums of elastic and inelastic deformation rates, the inelastic part consisting of the deformation rates due to viscous and plastic effects: g~ = e~ + g~, g0 = e~ + g~; it is also assumed that g~ + 2g~ = 0. The system of constitutive equations are given by ~

~

BA + 2go 2-~r~ o9I &t = Ko(g~+2go-avT-3(l~_w----------~&) ' g; = -----~---+

3 +-~o1-w T{ Yo ~ = ~ o ( 1 - i T~.;--------~) ;_ H(lv'-r;l-go),

T; ~ = ~o(1-i,r

u

o

1-wAC & sign(r;_r~O )

~

Yo H(lr:-r;l-Yo),

u

o

2.3. O n e - d i m e n s i o n a l cylindrically s y m m e t r i c p r o b l e m Consider one-dimensional cylindrically symmetric problem in adiabatic approximation. In this case the equations of the laws of conservation of mass, change of momentum and heat influx are written as

[)

- = -g,- - go p '

piJ =

O~Tr

-~r

+

6rr - - ~ 0

r

pc~T + a , & T = 2T~g~ + 2rOgPo + T~g~ + 7og~ + A& 2 + A& 2 In one-dimensional cylindrically simmetric case e = (a~ + a o + e z ) / 3 and a~ = a + r~, "

r. -t- ra -l- r. = O.

9

9

50 The system of constitutive equations are given by

BA

~' = go ( ~ + ~o - . , T - 3(1 - ~ )

~ = ~ + ~ +~ 3 +5-~,o + =

TIr (1 -

'

~ = ---Y-

+ ~foo + ~ - ~

r,

AC ~ 2~-'~+ r" 1~-~ ~-'

/-2 Y0 ~ H(T ' -

5.,=B(a'-a.)H(a'-a.),

5.,)

i 2 Y0)

~;

/~y0

~ = ~-~0( 1 - y ~ - ~ - 7 ) H ( r ' -

& = C(v' - v.)H(v' - r.),

~_ Y0),

v' = V/2(r~2 + ,'r;rb + r; 2)

REFERENCES

1. L.M.Kachanov, Izv. Akad. Nauk SSSR. Section of Eng. Sciences, 8 (1958) 26 (in Russian). 2. Yu.N.Rabotnov, Fatigue of Structural Components, Moscow, 1966 (in Russian). 3. A.A.II'yushin, Mechanics of Solids, 3 (1967) 21. 4. S.Murakami and Yu.N.Radaev, Mechanics of Solids, 4 (1996) 93. 5. B.D. Coleman and H.E. Gurtin, J. Chem. Phys., 2 (1967) 597. 6. V.I.Kondaurov and L.V.Nikitin, Theoretical Bases of Geomaterials' Rheology, Moscow, 1990 (in Russian). 7. A.B. Kiselev and M.V. Yumashev, J. Appl. Mech. Tech. Phys., 5 (1990) 116. 8. A.B. Kiselev and M.V. Yumashev, J. Appl. Mech. Tech. Phys., 6 (1992) 126. 9. A.B. Kiselev and M.V. Yumashev, Moscow Univ. Mechanics Bulletin, 1 (1994) 14. 10. A.B.Kiselev, Fourth Int. Conf. of Biaxial/Multiaxial Fatigue (St. Germain en Laye, France, May 31 - June 3, 1994), 2 (1994) 183. 11. A.B.Kiselev, M.V.Yumashev and A.S.Zelensky, Advances in Fracture Resistance in Materials, Eduted by V.V.Panasyuk etc., New Delhi, India, 2 (1996) 281. 12. P.Perzina, Quart. Appl. Math., 3 (1963) 321.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

51

A Mathematical Model for the Formation and Development of Defects in Metals V.L.Kolmogorov, V.P.Fedotov, L.F.Spevak Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91 Pervomaiskaya St., GSP-207, 620219, Ekaterinburg, Russia SUMMARY To simulate metal forming processes, the formation and development of defects in metals, one has to solve relevant boundary value problems. The progress in the theory of plasticity is obvious (for example, the slip-line method, the finite element method, etc.), yet it retains too many unsolved problems to be applied to attain these ends. A mathematical model for the formation and development of continuity defects in metals under deformation cannot be constructed within the theory of plasticity alone (or any other section of continuum mechanics) because of the fundamental axiom of continuity. The proposed mathematical model of continuity defect formation deals with a new boundary value problem, in which the classical problem of plasticity is supplemented with a kinetic ordinary differential equation for a scalar functional depending on the stress-strain state and temperature histories. This kinetic ordinary differential equation is written for each material particle. The functional is called "metal damage", ~', caused by microdiscontinuities. Here we present a new technique for solving rather general boundary value problems, which can be characterized by the following: microdamage and macrofragmentation; the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; nonisothermal flow; rapid flow with inertial forces; nonstationary state; movable boundaries; changeable and nonclassic boundary conditions etc. 1. INTRODUCTION The proposed mathematical model for the formation and development of continuity defects in metals deals with the corresponding general boundary value problem, which consists of the classic problem of the plasticity theory and the condition of fracture for each material particle. This fracture condition (theory) was published in English in [ 1], then in [2-4] and in [5]. The solution for the corresponding general boundary value problem is presented here. Recent papers and books, for example [6 - 9], speak of the state of affairs in the calculation of the stress-strain state in metal forming with the application of the finite element method (FEM). Naturally, the calculation is approximate, and, notwithstanding the progress in computer hardware, the matter of topical importance is to find a better approximate solution method. To calculate the stress-strain state in metal forming (and, on the whole, in deformation mechanics) is to solve the boundary value problems of mechanics. Continuum mechanics

52 equations may be schematically divided into three types: kinematic equations, dynamic equations and constitutive relations. The characteristic feature of the majority of the works [69] is the fact that they satisfy kinematic equations exactly, satisfy constitutive relations, but, strictly speaking, they do not satisfy dynamic equations. In fact, the solution is constructed in velocities and displacements, e.g., by Lagrange's, Jourdain's, Markov's principles or Galerkin's method. Then, by the found flow kinematics (for of Markov's principle, by kinematics and mean normal stress), by means of canstitutive relations, the stress tensor field is calculated. Since direct variational methods are used for the solution, the obtained stress fields do not satisfy continuum dynamic equations, namely, the differential equations of equilibrium (or motion if the flow has mass inertial forces), the stresses do not satisfy the boundary conditions in stresses exactly ("softer" satisfiability). Certainly, as the number of variation parameters grows and/or if more suitable coordinate functions are used, the discrepancy in the satisfaction must decrease. Even solutions for very complicated problems by the FEM (as in [6-9]) often seem verisimilar. Man has accumulated some experience in estimating body forming, and the visualisation of kinematic solution results (displacements, velocity fields and even strain distribution) creates the impression of safety. However, there is little physical notion of stress fields, and paper authors seldom bring their results to the analysis of the stress state obtained. We do not state that this approximate solution method is worse than the one described in the present paper, but we are of the opinion that the latter method deserves discussion. The solution in the form of kinematic fields (of velocities, displacements etc.) satisfying all the kinematic equations and in the form of stress fields satisfying all the dynamic equations is viewed as an alternative method. Since the solution is still approximate, the "softening" falls on the constitutive relations. It should be noted that the constitutive relations are always approximate and found f~om experiments, where there are experimental errors, therefore this way of "softening" seems more preferable. The idea of simultaneous variation of the stress and strain states with the "softening" of constitutive relations alone is not new. It was proposed independently in [10,11] and developed in [ 12-17]. 2. THE CORRECT FORMULATION OF THE GENERAL BOUNDARY VALUE PROBLEM

The boundary value problem consists in the integration of an extended set of continuum mechanics equations with respect to the variables describing flow kinematics, stress, microdamage and macrol~agmentation for certain boundary and initial conditions. Some of these equations (constitutive relations) and the boundary conditions are defined fiom experiments for a specific class of problems, particularly, those of metal forming mechanics; however, any formulation ought to be correct. The correct formulation is as follows. Let a material body of volume V undergo finite plastic deformation. Let the constitutive relations be given V A / e V . They can be given in any form (e.g., they describe plastic deformation in volume Vp, which is part of V, and elastic deformation in volume V e, which is the rest of the volume etc.). We suppose that, under conditions of developed forming, the material possesses rheonom properties, the constitutive relations are known functionals of the deformation history, temperature, 0 and density, ,o etc. However, at every

53 specified time, t , including the one under study, they turn into some known tensor functions, together with their inverse functions s o. = s o. ( e k l ) , e o. = e o. ( s kl ); (1) cr = o'(9~),

r = ~(o').

(2)

..

Here, s ~J and e/j are stress deviator components and strain rate deviator components; cr and are mean normal stress and the rate of relative volume change respectively; ekl and s kl form a set of deviator components which appear in functions (1) as arguments. Among the arguments in functions (1) and (2) there may be any characteristics of the stress and strain e..

9

states (e.g., derivatives s v , o', microdamage, ~ etc.), but in (1) and (2) there are arguments that are principal ones for the following reasoning. The adopted coordinates are Lagrangian ones. Let functions (1) and (2) satisfy the conditions

3

~"/o e~l~_~ > 0,

(3)

C70-/C7~ > 0. (4) SO constitutive relations at any specified time, t (functions (1) and (2)) ought to be differentiable with respect to the mentioned arguments and have inverse functions; conditions (3) and (4) ought to be fiflfilled for the functions, as they express the known properties of metal viscosity. Let the solution be sought in the flow velocity fields, o i and temperature fields, O, which are continuous in coordinates, and in the stress fields, o" 0, for which the surface stresses, fi

_ cr iJnj are continuous on any surface inside V. Here n is a unit normal to the surface

S. Suppose the body of volume V under deformation is bounded by an external surface, S, where the boundary conditions are generalized at any specified time, t as follows

VM eS

f i _ fi(vj,...) ' vi - v ; ( f J,...),

(5)

o7'/ jl;:j > 0; VM e S

(6)

00 - 2 - ~ - - (p,(M, O).

(7)

Here f.;, v?, ~. are known functions, wita f.;, V,.* be~-g reversible; 2 ~ heat ~o.au~ti~ity coefficient. Relation (6) shows the viscous properties o f th e environment. Finally, suppose that V M ~V distributed mass forces, g/* are given. Suppose the following initial conditions are given (at t = 0) for time integration:

V M e V,

vi- v ~

o ' 0 " - o "0", P = Po,

g / - ~o,

0-0o.

(8)

On the fight, marked by zero, there are known coordinate functions. Thus we have formulated the boundary value problem of mechanics of solids under deformation. Let us solve it.

54 3. A N A P P R O X I M A T E S O L U T I O N F O R THE G E N E R A L B O U N D A R Y V A L U E PROBLEM

Consider an arbitrary, but specified of time, t . The space integration of the boundary value problem can be replaced by the equivalent task of solving the following variational equation for the principle of virtual velocities and stresses

eo'..

d"

~"

o-"

0

+ Ir 0

6{f[ I s'J (e)ae + I eij(s)ds + I o-(~)ar z 0

0

:/" -I[I f/av + I v ? a f ] a s }

+ p(W i

-

gi* )Vi" ]dV

-

(9)

vl

SO

= o.

0

l

The variation is isochronous only with respect to the virtual quantities cr 0'', v i marked in (9) by a prime. The summation is made over the indices i and j appearing in the upper limits of the integrals and in the expressions under the integral sign. Naturally, the constitutive relations ought to be such that the functional J in Eq. (9) (curly brackets) were differentiable. Besides !

the continuity in V and on S, the virtual o i ought to satisfy all the kinematic conditions. For example, if the material is incompressible, velocity vector is given on S v and its normal component is given on S S ( Sv k.) S S L) S f - S ), then VM

~ V dive'=

O; V M

~ Sv h = h ,9" V M ~ S s v ~

(10)

' -- V*v.

!

Besides the continuity of the surface stresses f i ' = cr 0 n7 in V and on S, the virtual o-0" ought to satisfy the following conditions V M ~ V, V;cr#" + p (g.i _ w J ) = 0 , V M e ST, cr #"nj = f.~.

cr 0 ' - cr J;';} (11) !

Here w j is the acceleration of material particles. Note that the virtual o-0" and o i satisfy all the equations of continuum mechanics (which are linear in this case, and this simplifies the practical implementation of the principle of virtual velocities and messes), except for the constitutive relations. The functional J calculated for any virtual stress-strain state, even entirely different from the actual one, all other factors being equal (i.e., invariable quantities), is not negative and becomes zero on reaching the absolute minimum at the actual state being the space solution for the boundary value problem with specified time. The value of J calculated for some virtual state described by the fields o i and cr/j expresses a discrepancy in their satisfying the constitutive relations. The temperature part of the problem is also solved on the basis of the variational principle. The solution of the heat conduction differential equation is equivalent to the finding of the extremum of a functional t

!

55

)2 oO'~o'dO+ IcP 0 tdO]dV +--I I~o,(M,O) doriS} - 0 (12) v z o o ' 2s0 on the virtual temperature fields (continuous in V and on S), with the variation with respect to

/9'. Here V is Laplacian operator, C is mass heat capacity factor. The approximate solution at an arbitrary time, t will be sought by using the principles (9) and (12) in the following form n m l o~cr~O"( y ) ; 0 ' = ~-'~CkOk(y). (13) vi' - Y'~agivki(Y); frO"- Z'-O" k=l k=l k=l Here, y are Lagrangian coordinates; aki , b~ and Ck are variable coefficients (with specified t and, generally speaking, functions of time);

Oki(Y), cr~'(y) and Ok(y ) are known

suitable coordinate functions (in the fight hand part there is no ~mmation over the repetitive indices

i, j). The suitable functions are selected so that o i,! ~

On v - 67x

account

that

P-Po

t where p 0 and p

..

and

0 !

are virtual.

detllOX~o/dyJll/detllCTXk/dYtll , /l il/l I[ / II

are the initial and current material density,

co-

t,

x i~ and x k are

the initial and current Eulefian coordinates of the particle, yJ and yl are their Lagrangian coordinates, the variational equation (9) and (12) are reduced to the integration of a set of

or .a y

equa o,

respectto ak, - ak, tt), t g =

(t), ck - ck(t)

view of the initial conditions (8). The question of the unique existence of the solution is discussed for each stage of the proposed solution algorithm separately. According to the theorem of functional analysis, conditions have been found that suffice to provide the unique existence of the extreme problem (9) in the Odich-Sobolev space

W lLM [18]. When the problem is solved

numerically, the properly chosen virtual state allows one to obtain the closest possible solution to the variational problem at any time. In the second stage, the problem solution is reduced to a set of ordinary differential equations where the conditions for the unique existence of the solution are known. Note that, for different times (different stages of elasto-plastic deformation), the chosen virtual states give different deviations from the exact solution. Therefore one can speak only of the unique existence of the approximate solution found with an accuracy up to the virtual state, with the convergence of the general problem solution being established numerically. There is one more variable, ~ which is still to be defined, and it is discussed in the next section.

4. ON THE FRAGMENTATION OF DEFORMABLE BODIES In the above-described solution for the boundary value problem, it was assumed (as usual) that the material volume V remains continuous in the process of deformation, that it does not become divided into parts, and that no macroscopic holes or macrocracks appear in it, i.e.,

56 that there is no considerations and mechanics is valid) time tp). This time

macroscopic fragmentation of the body under deformation. The the boundary value problem solution are valid (in as far as continuum until macroscopic fragmentation begins, i.e., discontinuity appears (at the can also be viewed as the be~nning of a new stage, i.e., a new solution for

a new boundary value problem, because on the newly-formed surfaces there appear additional boundary conditions requiring a new boundary value problem statement. The second stage continues until the next new surfaces appear and so on. The time tp when fragmentation starts and the time of fil_rther macrodiscontintfities can be determined by means of the fracture theory [8], supplied with some new statements. According to this theory, in every material particle of the body under deformation, accumulation of damage, ~ takes place. By the time t , ~t(t) is calculated by some kinematic relations. To this end, firstly, we solve the appropriate boundary value problem. Secondly, we specify (or find from special experiments) plastic characteristics of the body under deformation. Damage, ~r is calculated for every material particle. For this purpose, on the particle motion path, separate sections of monotonic deformation, are specified where the components of the particle strain rate tensor do not change the sign. We indicate the time when the tensor components change the sign as tl, t2,... , tn_1 (transition through zero of at least one component). In the first section (t o _< t < t 1), damage is determined as

H(t)

d ~br1

~/(t) = ~l(t),

d t - A? [kl(t) , k2(t)] ' ~rl(t~ = 0;

in the second section (t 1 ~ t < t 2 ), [ / / ( t ) - [ ~ l ( t l ) ] al + [ ~ 2 ( t ) ] a2,

d ~2

d---T =

H(t) A p [k,(t), ko(t)] ' ~ 2 ( t l ) = 0; K

x

-

-

--

in the n-th section (t~_ 1 _< t

< t),

d ~'n

/~(t)

d t - Ap [Klt/r--~tx,K2t)J--:t ~1' ~n(tn-1)= 0.

~ t ) = ~~'~",__~

(14)

Firstly, here H - H ( t ) i s shear strain rate intensity; k 1 = kl(t) and k 2 - k 2 ( t ) are dimensionless independent invariants of the stress tensor [k 1 = O'/T, k 2 = 2 ( 0 - 2 2 - o " 3 3 ) / ( O - l l - o " 3 3 ) - 1 where o" is the mean normal stress and T is tangential stress intensity; o'11 >_ 0"22 > 0-33 are principal normal messes]. Secondly, we have plastic characteristics of the body under deformation, which have been found from oxpo mo ts:

- A

valuos of~o ~ ~ o n

Ikl,

rofors to

va uos

~ = ~ (g~, ge) m ~o i - ~ so~on of monotom~ Ooformauo~

aro

57

By e eof ao e('--'p), (t) = ~ (tp) = 1,

(15)

and the body is saturated with microdamage (not appearing in the boundary value problem solution), the material becomes brittle and ready to form a macrocrack (body fragmentation onset). Condition (15) is the end of solving the boundary value problem within the adopted statement and the be~nning of a new stage, i. e., a new solution. How can we find tp, the macrompture spot and formulate boundary conditions on the new surface? Damage is calculated by the given algorithm simultaneously with the time integration of ordinary differential equations, allowing one (as is described above) to obtain an approximate solution to the boundary value problem. At every time t , we can solve the problem of seeking the x coordinates of the point where there is maximum damage

max [ ~/(t)]. x~V

(16)

The time t = tp will be determined when, according to (24), the maximum value of IF reaches unity, i.e.,

max [ g/(tp)] =1. x~V

(17)

At the same time, the point (or points) where a macrocrack appears can be found. How will the unlace of the macrocrack be oriented?::We can assume that, if plastic deformation precedes fracture, then, at the time t = tp, the crack will be oriented along the sites of maximum tangential stresses. The crack will have finite dimensions owing to the continuous change of ~ in the volume V. The dimensions can be calculated by solving a new boundary value problem and calculating the stress-strain state around the crack. On the spots of maximum tangential mess, the tangential and normal messes will, respectively, be as follows

l

~'n = 2 (O'l l -- 0"33 ); 1

1

0_33). j

If, when t = tp, at the point with ~ = ~max = 1

(18) , o-n > 0, then there appears a crack with

"edges" free from surface stresses, i.e., ~'n = ~ = 0. Impact unloading takes place at the edges by the value A r n = rn, Ao-n = o n . (19) If, when t - t p, at the point with ~ - ~'max- 1, we have O"n < 0 , then there appears a shear crack with edges not free from surface stresses. Impact unloading takes place at the edges by the value

I

(20)

if between the shear crack edges there is assumed to be Coulomb friction (fl is friction factor). In the subsequent solution, the shear crack edges should be viewed as surfaces with sliding friction.

58 CONCLUSION The problem solution by the proposed method can be divided into two stages: (i) space integration for any specified time up to some parameters and (ii) time integration of a set of ordinary differential equations (including the kinetic one) for these parameters. To solve the problem in stage (i), a variational principle is proposed, which develops the well-known classical principles of Jourdain, Castigliano and Bio. The field of velocities of material particles, the stress tensor field and the temperature field are varied independently at a specified time. This method allows one to find the solution satisfying all the equations of continuum mechanics exactly, with the constitutive relations being fulfilled approximately. The "soitening" of the constitutive relations, which are always approximate, is more preferable as compared with the most familiar methods when the Newtonian mechanics equations are "soitened", whereas the constitutive relations are fulfilled exactly. To substantiate the solution method, we have proved the equivalence of the application of the variational method to differential equations of continuum mechanics and the unique existence of the solution, etc. The use of the functional minimum condition results in a set of algebraic and ordinary differential equations for variable parameters depending on time (stage ii). The discussion and the solution to the boundary value problem in terms of classical continuum mechanics are valid up to the time of discontinuity tp, i.e., the onset of macroscopic fragmentation. At this time a defect appears. Impulse relief takes place on its boundary. A boundary condition is formulated according to a certain rule, and one can start solving the boundary value problem in the next stage of deformation up to the formation of the next defect etc. The instant of fragmentation onset, tp and the instants of further macroruptures can be determined by the proposed kinetic equation for the fimctional ~ . Some applications of the mathematical model for the formation and development of metal continuity defects under deformation are given. For dynamic processes, impact problems are discussed for materials with different properties under different kinematic conditions, some of the problems having known solutions. This has offered appropriate correlations. For simplicity, the application of the mathematical model for the appearance and development of continuity defects in metal under deformation is illustrated by test examples of thin elastic and elasto-plastic bars impacting a rigid obstacle. The elastic problem was solved at the following mechanical characteristics of the material: /9 = 8 0 0 0 k g / m 3,

E = 200000MPa,

Ap = 0.2 exp(-2cr / T), a - 1.2exp(l + 0.24cr / T), bar

length l = 0.1m. The virtual state was given in the form of Fourier series, the solution of the variational problem resulted in a series segment coinciding with the exact solution segment. Fig. 1 shows the values of damage, ~ cumulated along the bar at the instant when the first fracture takes place at an impact velocity of 250m/s. Fig.2,3 show fracture time tp and fracture point Xp in the bar as dependent on impact velocity ~. Note that the origin of coordinate x = 0 is in the point of contact between the bar and the obstacle. Fig.4 shows fracture point displacement before and after fragmentation at an impact velocity of 140m/s, rip. ~XOM tp = 0 . 0 0 0 0 4 8 S, Xp -- 0 . 0 3 m. As is seen from the figure, as a result of

59 unloading, the rupture edges move along different paths after fracture. The elastoplastic problem was solved with the following constraint equation for the plastic region cr = 2000x/-~, the rest parameters being the same as in the elastic problem. The virtual state was given in the difference form based on the representation of the bar as 10 linear finite elements. Tab. 1 shows the average values of damage, Ip' for each element at the instant when the first fracture takes place at an impact velocity of 300m/s, t p - - 0 . 0 0 0 0 1 5 s . The fracture is seen to occur in Element 1. 6 tp• 103 5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 100

0.8 0.6 0.4 0.2 0

0.02

0.04

0.06

0.08

0.1

x ----~

Figure 1. Damage along the elastic bar.

u(t,x)•

, 150

200

-10 \ -

250

V~

v -----

250

~

3 2 1 0

0"07f~

0.011 100

200

Figure 2. The instant of the first fracture.

0.09~Xp

t I

150

Figure 3. The point of the first fracture.

1

2

3

/txl05-t 4

5

6

7

Figure 4. The displacement of the fracture point.

Table 1 Damage along the elastoplastic bar dement no. damage,

1

2

3

4

5

6

7

8

9

10

1.012 0.213 0.027 0.006 0.002 0.001 0.001 0.001 0.002 0.004

The proposed method for solving boundary value problems provides an approach to solving the problem of motion stability, which can be described by continuum mechanics equations. The method presented here within the mathematical model for defect formation is a matter of great siL-mificance in itself. It can be successfi~y applied to calculate stress-strain states and temperature fields in metal forming.

60

Acknowledgement We are grateful to Mrs E.E.Verstakova for assistance in the preparation of this paper. REFERENCES 1. V.L.Kolmogorov. Model of metal fracture in cold deformation and ductility restoration by annealing. Materials Processing Defects, S.K.Ghosh and M.Predeleanu (Editors), 1995, Elsevier Science B.V. 2. V.G.Burdukovsk% V.L.Kolmogorov, B.A.Migachev. Prediction of resources of materials of machine and construction elements in the process of manufacture and exploitation. I.J. of Materials Processing Technology, 55 (1995), 292-295. 3. V.L.Kolmogorov. Friction and wear model for a heavily loaded sliding pair. Part I. Metal damage and fracture model. I.J. Wear 194 (1996) 71-79. 4. V.L.Kolmogorov, V.V.Kharlamov, A.M.Kurilov. Friction and wear model for a heavily loaded sliding pair. Part II. Application to an unlubricated journal bearing. The Journal of Wear 197 (1996) 9-16. 5. V.L.Kolmogorov, S.V.Smirnov Healing of metal microdefects after cold deformation (an article in the present volume). 6. J.-L.Chenot, T.Coupez, L.Fourment. Recent progresses in finite element simulation of the forging process. Proceedings of the Fourth International Conference on Computational Plasticity: fundamentals and applications, Barcelona, 3-6 April 1995, pp. 1321-1342. 7. Zhi-Hua Zhong. Finite element procedures for contact-impact problems. Oxford University Press, 1993. 8. Jurgen Gerhardt. Numerische Simulation dreidimensionaler Umformvorgange mit Einbezug des Temperatmverhaltnes.Springer-Verhg, 1989 (German). 9. A.A.Pozdeev, P.V.Trusov, Yu.I.Nyashin. Large plastic deformations: theory, algorithms, addenda. M.: Nauka, USSIL 1986 (Russian). 10. A.Baltov. Variational theorems in the dynamic theory of viscoplasficity. Bull. de l'Acad. Pol. SoL set. SoL techn. 17, No5, 1969. 11. V.L.Kolmogorov. The Principle of possible variation of stress and strain. Mekhanika tverdogo tela. N2. 1967 (Russian). 12. V.L.Kolmogorov. Stresses, strains, fracture. Metallurgiya. USS1L 1970 (Russian). 13. E.P.Unksov. W.Johnson. V.L.Kolmogorov et al. Theory of plastic strains in metals. Mashinostroyenie. USS1L 1983 (Russian). 14. V.L.Kolmogorov. Mechanics of metal forming. Metallurgiya. USSR, 1986 (Russian). 15. V.L.Kolmogorov and 1LE.Lapovok. The calculation of stress-deformed state under nonisothermic plastic flow - the example of parallelepiped setting. Computers and structures. Vol. 44 No. 1/2. 1992 (English). 16. E.P.Unksov. W.Johnson. V.L.Kolmogorov et al, edited by E.P.Unksov, A.G.Ovchinnikov. Theory of forging and stamping. Mashinostroyenie. Russia. 1992 (Russian). 17. V.P.Fedotov. Variational solutions for elastic-plastic problems. Conf. Modem Problems of Plastic Metal Forming. Bulgaria, Vama. 1990 (Russian). 18. 1LA.Adams. Sobolev spaces. Academicpress, New-York - San-Francisko - London. 1975.-315p.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

61

Healing of Metal Microdefects after Cold D e f o r m a t i o n V.L.Kolmogorov, S.V.Smirnov Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91 Pervomaiskaya st., GSP-207, 620219, Ekaterinburg, Russia

I. Introduction As is known, cold plastic deformation of metal (rolling, die-forging etc.) from the first stages is accompanied by microscopic defects in continuity (we call these defects and the phenomenon "microdamage" or simply "damage"). As deformation accumulates, the development of damage can result in the appearance of macroscopic defects or even in the division of the body under deformation into parts, i.e., in defective products. Definitely, this is inadmissible. Macroscopic defects can be easily revealed (external ones by visual observation, internal ones by introscopy), but correction either is impossible or requires that the defective bulk of the metal should be removed. One of the methods for avoiding macrodamage is multistage deformation with intermediate annealing at the end of every stage. This provides metal dishardening and, above all, the restoration of metal plasticity (i.e., the ability of metal to be deformed without fracture). The amount of deformation in a separate stage can be established intuitively, from one's practical experience. This paper gives specific rules for it. As distinct from macrodefects, microdiscontinuities are harder to detect under service conditions. Now industry lacks means for checking microflaws, therefore all metal products must have microflaws, which can lower the efficiency of machine parts. They have been ascertained to influence fatigue life [2]. Therefore it is urgent to study the mechanisms of eliminating (or healing) microflaws, i.e., the mechanisms of metal plasticity margin restoration by heat treatment and the ways of making it more efficient. This is the subject matter of the present paper. Together with their helpers, the authors have done the work complemented with the phenomenological theory of metal fracture developed by them. The principal features of the theory were briefly discussed in [ 1-3] published in English. The reader can familiarize himself with the details of the theory by reading [4-6] in Russian. In [2] it was shown that the abovementioned theory of metal fracture proved to be general enough and applicable to the description of the fatigue life of machine parts. In brief, the ideas of the fracture theory are as follows. The physical picture phenomenologically described by the fracture theory consists in the fact that plastic deformation is always accompanied by microfracture, i.e., by the formation of flaws (point, linear, plane and three-dimensional). As deformation accumulates this damage develops to fracture by way of the appearance of new defects and the growth of those previously formed. This can be presented mathematically as follows.

62 For a material particle of a body being deformed, the increment of this damage (A~) is assumed to be proportional to the increment of deformation and inversely proportional to plasticity margin. The increment of deformation HAt (H is shear strain rate magnitude, At is the increment of time) results from the calculation of the stress-strain state; plasticity margin, Ap is a constitutive relation (function) determined by experiment. Plasticity margin (function Ap = Ap (| of temperature | and two dimensionless invariants of the stress tensor kl,k2) is determined by the method worked out within this theory. If the increment of time approaches zero, this assumption brings us to the differential equation d~

H(t) m

dt

Ap [0(t),kl (t),k2 (t)]

(1)

Here the arguments of the function Ap (| result from the calculation of the stressstrain state and temperature. Equation (1) gives good results (at the time of macrofractureq~=l) if deformation develops monotonically. It develops monotonically if no component of the strain rate tensor changes its sign. If a material particle undergoes nonmonotonic deformation (and it is a more common case), then the damage accumulated by the time t must be calculated in a different way: n

V(t) = 2 V ai

(2)

1

where n is the number of monotonic deformation stages the particle has gone through by the time t; ~Pi is the value calculated for the i-th stage of monotonic deformation (i -- 1...n) in accordance with relation (1); ai is an exponent that is taken as a mean value for the conditions corresponding to the i-the stage of monotonic deformation (it has been found that a i > 1, and this formally reflects damage development retardation upon the change in the deformation direction). It has been found that a - a(kl,k2) , and it is the second (afterAp) constitutive relation in the fracture theory presented here. This version of the fracture theory gives good results, whereas some familiar theories of fatigue can only be viewed as particular cases. (In some cases, the mathematical model (1) can be made more accurate, but it has to be done at the cost of introducing some nonlinearity [6].) It was shown in [2] that the above-mentioned model of fracture can be viewed as generalized known conditions of fracture from low-cycle fatigue, creep, etc. There is an unconventional way of "healing" steel (metal) "tired" of deformation. The restoration of metal plasticity is based on diffusion or on the transfer of the substance into pores and microcracks and the transfer of these discontinuities to the surface of the body. This idea of using heating to maintain the serviceability of machine parts was discussed in [3] with a heavily loaded sliding bearing as an example. In this paper the problem of metal plasticity restoration by heating is discussed in detail within the above-mentioned theory of fracture described in [1-3,6].

63

2. A model for damage history in annealing As was mentioned above, damage accumulates in metals under loading of all kinds (plastic forming, machine part operation, etc.). In the experiments described below, to initiate the onset of damage, active plastic deformation was used. This proved to be an easier method, besides, flaws from plastic deformation are large, as compared with those from other kinds of loading, and therefore they are harder to heal. To work out the technology of manufacturing cold-deformed products with annealing, it is necessary to give a mathematical description (within the above-mentioned model) of how the restoration of plasticity margin (or the reduction of microdamage) proceeds under annealing. The following technique has been developed for solving this problem. Tests are performed on metal, with the plasticity Ap(kl,k2) being already known. Test specimens undergo different amounts of plastic deformation, each specimen being deformed to different amounts of damage, Wi (for example, by tension not to fracture, but to different amounts of shear strain A 1 = 2~f3 ln(d 0 / dl) where d o and d 1 are diameters of specimens before and after tension; W1 = A1 / A p is determined by a plasticity diagram for every specimen, provided that the process of tension is monotonic). Thereafter all the specimens undergo annealing in accordance with the chosen regime (0 is temperature and t is annealing duration). Plasticity restoration (or damage decrease by the value AV) takes place in annealing. After annealing, once again, all the specimens undergo plastic deformation in the same direction, but this time to fracture. Then A 2 is determined and calculated W2 = A 2 / A p . The second plastic deformation plays an auxiliary part in the determination of AW. Evidently, the total value for the specimens is ~Ij = ~IJl - A~t j + t ~ 2 =

1,

since the specimens have been brought to fracture by the second deformation. This allows one to determine the unknown decrease in damage resulting from annealing in accordance with the chosen regime AW - ~1 + ~2 - 1,

(3)

and to predict permanent damage Wp = W1 - AW which has not been healed by heat treatment. The results on damage decrease by annealing of some steels and a titanium alloy are presented in Fig.1 as an example. The annealing regimes: for steels, Oranges from 500 to 750~ and t ranges from 5 to 300 minutes; for the titanium alloy, | = 680~ and t = 60 min. The dependence presented here was found in other investigations to apply to various other metals and alloys, therefore we can draw general conclusions [6].

64 The restoration of plastic properties (or decrease in microdamage AW) under annealing by recrystallization depends on W~ to a great extent. Separated by two critical values of damage W, and ~**, there are three intervals of W~with different rates of restoration. If damage resulting from deformation is 0 < W1 < W,, then complete healing takes place on annealing, since Aq~=q~, in Fig.1. If q~, < qJ~ < q~**, then damage is not completely removed by normal annealing. AW continues to increase as W~ grows, but at a lower rate. When qJ~ > q~,, AW decreases, and it goes to zero when q~l = 1. The damage - time history for the steel of grade 20 at a temperature of 600~ is shown in Fig.2 as an example. The lower curve at ~1=0 illustrates that the annealing of a blank to be deformed, for example, of hot rolled metal, can lead to higher plasticity due to the healing of microdamages that appear in the stage of hot rolling. The curves have three distinctive parts: AB is rapid exponential decrease in damage; BC is considerable deceleration (and even stopping) of metal plasticity restoration; CD is further acceleration of the process.

/'x

f

Aq~

i

0.4

0.2

0

0.2

0.4

0.6

0.8

qJ 1

Figure. 1. Decrease in metal microdamage owing to recrystallization annealing (for fixed time and fixed temperature) for the steels 12X18H10T (1) and CT3Kn (2) and the titanium alloy BT1-0 (3) with damageW1 after plastic deformation.

Assume that the material damage, ~I-/1 resulting from plastic deformation changes exponentially (in Fig.2 within 0.5 h.), i. e., it may be written as follows q~(t) = ~Jl exp(-[3t). Here 13> 0 is the index of the exponent steepness.

(4)

65 3. I n v e s t i g a t i o n

results,

discussion,

recommendations

The restoration of plasticity margin in metals after cold deformation has been studied for a number of years in co-operation with researchers from some institutes, mostly with Prof. A.A.Bogatov (Ural State Technical University, Ekaterinburg). Fig.3 shows diagrams illustrating the degree of damage healing for some alloys in coordinates (~[,~p). By the amount of permanent damage, ~gp (caused by cold deformation resulting in calculated damage Wp and further recrystallization annealing), one can estimate the completeness of the healing of flaws appearing in the preloading stage. (Mind that, if q~p= 0, the healing is complete, whereas if q-'p=W1, all the flaws remain in the metal.) Healability is seen to be different for different alloys. However, the values of q-',,W** are similar for different alloys. This fact enables one to state that, if machine part damage (in the most unsafe place) < q~, = 0.2-0.3 in cold plastic deformation or in service, then the margin of plasticity can be restored by conventional recrystallization annealing. P 0.7, A

B

D

C

0.5 0.3 "-------~

G

0

~

e

I

0.1 ' ~ , ~ :'

2 i '~ " ~ ~ ' ~ D . . . . . . . . ~ ,

'

I

I

I

-0.1

~-,

0

:40

u

:80

~

~

120

.~

.

T

: 160

a ,w

t, mln

Figure.2. Damage history W during heat tretment (at 600 0 C). The grade of steel is 20.

Generally speaking, in this equation, the exponent is a function of heat treatment and the third constitutive relation of the fracture theory: In Fig.3a, the behaviour of q~p for steels is noteworthy. Thus, for example, carbon steels tend to show lower permanent damage as carbon content increases. Electrone microscopy has made it possible to reveal that this can be attributed to a smaller size of the flaws and, consequently, to more intensive healing by annealing. The healing of deformation damage by means of heating is connected with the change in the dislocation structure, decrease or disappearance of microdiscontinuities. Depending on the temperature of heating and the amount of deformation, healing can follow the pattern observed in polygonization, recrystallization and other phenomena. Investigations have shown that recrystallization annealing results in the healing of subgrain-size microdiscontinuities (i.e., under 2-5 ~t) by

66 intensive surface diffusion of voids when they are crossed by the moving intergrain boundary of the grain being recrystallized.

q., P

0.8

0.8

0.6

0.6

0.4

0.4

1//

0.2 O0

0.2

0.4

0.6

bo

4

0.2 0 0

0'.8

0.2

0.4

0.6

0.8

q-'I

Figure.3. Permanent damage after plastic deformation of alloys (with damage W1) and recrystallization annealing. 1-3 - carbon steels with 0.2%, 0.5%, 0.7% carbon content respectively; 4 - titanium cx-alloy of the system Ti-A1-Mn; 5 - W-Ni-Fe alloy; 6 - aluminium alloy with 7% of rare-earth metals.

The rate of damage healing can be increased by the activation of other healing mechanisms, for example, by the repeated passing of the grain boundary through a microflaw during cyclic heat treatment with phase recrystallization. This enables one to increase the value of W, limiting the range of damage, which can be completely removed by recrystallization annealing (Fig.4). Naturally, by increasing W,, heat cycling in annealing allows for fewer cycles in making colddeformed products due to a greater amount of deformation between annealings. However, the initialization of the mechanism of metal leaking into flaws (Laplace flow) due to local plastic creep seems to be more promising. This mechanism can be realized by treating materials with discontinuities in a gasostat (HIP process). q., t9

0.8 0.6 0.4 0.2 0

~ 0

0.2

2 0.4

0.6

0.8

q~l

Figure.4. Diagrams of permanent damage for the steel 40X after single annealing with phase recrystallization (1) and after heat-cycling annealing with phase recrystallization (2).

67 Fig.5 (line 1) shows the result of studying permanent damage of the nickel alloy 3H698 after treatment in a gasostat for four hours at 1100 o C in the argon environment under a pressure of 180 MPa. This kind of treatment is seen to result in the complete healing of damage caused by preceding deformation. Triaxial uniform compression does not cause any alteration of the specimen's geometry after treatment, and yet it initiates the development of metal microcreep in internal discontinuities. ~.J.,

8325

P

0.8

o

xx)'

0.6

%

8315

4 o

0.4

3

o

o

o

8295

0.2

. 9o

0

o n

8305

^

x x X'~x

•

x x

x

x

1,2

a

a ,,,,,,,~x ox

o

9

9g o

~

8285

u

A

m

-0.2

0

0.2

0.4

0.6

0.8

~

8275 1

0

0.2

0.4

0.6

0.8

I

Figure.5. Diagrams of permanent damage for the nickel alloy 3 H 6 9 8 : 1 - annealing in a gasostat; 2 - annealing under uniaxial compression; 3 - conventional annealing Figure.6. Density of nickel alloy samples: 1 -after deformation (without annealing); 2 - after deformation and conventional Complete damage healing also proved to be attainable by heating specimens under uniaxial compressive stress. To this end, samples of the same nickel alloy having undergone different amounts of cold tensile deformation, were heated up to 1100~ and subjected to initial compression with a force running 80 per cent of the yield stress at the above-mentioned temperature (30MPa). They were held at constant temperature to complete relaxation (2.5min). As distinct from HIP, uniaxial compression proved to result in the rise of undesirable permanent creep. Although the amount of permanent strain was small, it proved to be sufficient for complete damage healing (Fig.5, line 2). For comparison, Fig.5 (line 3) shows a diagram of permanent damage after conventional annealing (1100 ~ C, 1-hour holding, air cooling). The validity of the above-mentioned results, which have proved high effectiveness of heat treament after deformation under compressive stresses, has been proved. To this end, we have studied the change in the density of the speciments after deformation to different amounts of damage and after conventional heat tretment in a gasostat with uniaxial compression has been studied. It follows from Fig.6 that heat treatment under compressive stresses leads to complete healing of metal damaged because of deformation up to strains preceding fracture (with ~P, = 1). Note that, if conventional annealing is used, the area of damage to be healed must not exceed 0.3. The initiation of the gasostat healing of microdamaged metal that has undergone cold plastic deformation or reached the limit of its fatigue life in a machine or a construction is now applicable only in particular cases. Gasostatic treatment is expensive and underproductive. Besides, gasostats are unsafe because of high potential energy saved in

68 compressed gas. Instead of gasostats, fluid-metal hydrostats can be used. They can be fairly efficient and safe owing to low compressibility of metals. Pressure can be forced electromagnetically. Therefore the development of such apparatuses seems to have prospects. They are being designed by the Institute of Engineering Science, Russian Academy of Sciences (Urals Branch). Timely heat treatment offers better deformability of metal in multipass processes of metal forming. Fig.7 shows an example of a successful technological solution to the problem of how to reduce cracking of drawn tungsten wire for filament lamps. Predicted damage (q~ = 0.6) supported by metallographic examination has shown that in 0.8mm finished wire there are microcracks that cannot be healed by annealing and cause delamination of heating elements under deformation. In Fig.7 the intermediate annealing of 1.4mm wire, with damage not exceeding W = 0.6, is seen to heal the deformation defects. The practical application of this innovation has resulted in significantly reduced wire scrappage caused by delamination in heating element coiling. As far as we know, the restoration of the in-use fatigue life of machine parts by heat treatment is not adopted in practice in Russia. However, this phenomenon has been known for a long time. Thus, in [7] there is a description of a favourable result concerning the restoration of the mechanical and physical properties of rails after service by heat treatment. There is no doubt that it entails some technological difficulties. For example, it is necessary to maintain the condition of the surface during heat treatment. However, for some machine parts, including those used in forging and stamping, this way of prolonging service life seems feasible. Thus, the operational lifetime of dies can be increased after 25-30 per cent exhaustion by tempering causing the dispersal of dislocation clusters and the healing of submicroscopic flaws.

~3

P,

0

0.4

microcracks ,_7

_.

0.2

tm

t~ t" o ~

rc -

o f

0

"278

o

~

"2?4

~ .6

176

,

2 ,.~ "'J/ d, mm

Figure.7. The change in the damage of tungsten wire for electric lamps (1-conventional technology, 2-proposed technology).

Acknowledgement We are grateful to Mrs E.E.Verstakova for assistance in the preparation of this paper.

69

References 1. V.L.Kolmogorov. Model of metal fracture in cold deformation and ductility restoration by annealing. In: Materials Processing Defects, S.K.Ghosh and M.Predeleanu (eds.), Elsevier Science B.V, 1995 2. V.G.Burdukovsky, V.L.Kolmogorov, B.A.Migachev. Prediction of resources of materials of machine and construction elements in the process of manufacture and exploitation. I.J. of Materials Processing Technology, 55 (1995), 292-295. 3. V.L.Kolmogorov. Friction and wear model for a heavily loaded sliding pair. Part I. Metal damage and fracture model. I.J. Wear 194 (1996), 71-79. 4. V.L.Kolmogorov. Stresses, strains, fracture. M: Metallurgiya, 1970, 232 p. (in Russian). 5.V.L.Kolmogorov, A.A.Bogatov, B.A.Migachev et al. Plasticity and fracture. M: Metallurgiya, 1977, 336 p. (in Russian). 6. A.A.Bogatov, O.I.Mizhiritsky, S.V.Smirnov. Metal plasticity margin in metal forming. M.: Metallurgiya, 1984. (in Russian). 7. V.S.Ivanova, S.E.Gurevich, I.M.Kopiev et al.Fatigue and brittleness of metallic matrials. I: Nauka, 1968, 216 p. (in Russian).

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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

71

Defimtion of the form for kinetic equation of damage during the plastic deformation S.V.Smimov ~, T.V.Domilovskaya" and A.A.Bogatovb "Institute of Engineering Science of the Russian Academy of Sciences (Ural Branch), 91, Pervomaiskaya st., GSP-207, 620219, Ekaterinburg, Russia bUral State Technical University, 19, Mira st., 620002, Ekaterinburg, Russia

1.1ntroduction According to contemporary ideas the damage of metals is not a simply catastrophic phenomenon, but a multi-stage process of appearing and development of microdefects, which is called in mechanics a processes of damage accumulation (plastic loosening, fracture, cracking, etc.) Purely brittle fracture is possible only in non-metallic materials with a great quota of covalent part in inter-nuclear link. Historically, the problem of fracture during plastic deformation was first considered from the position of technological interests based on empirical criteria. This approach allowed to solve some easy applied tasks and held back investigations of general problems of metal fracture during the complex stress strain state. Progress has been made in ideas of mechanics of fracture of metals during plastic deformation connected with appearance of dissipated fracture kinetic theories. As it was shown in the Novozlfilov work [ 1], the plastic deformation should be accompanied by a residual change of deformed metal volume (plastic loosening): ~ = tx].,,

(1)

where a - plastic loosening module, L - loading path which, for the large plastic deformations could be taken as a shear strain AA= ~/(2e~j e~j), where e~j - components of deformations increment tensor. Another expression of fracture processes during the plastic deformation within theories of dissipated fracture is the kinetic equations of damage in a form, where the accumulated damage is defined by kinetic equation: do/dA = f(sl,s 2),

(2)

where r represents damage of material, sl,s2 - thermomachanic parameters of deformation, depending on the loading conditions. Relations between the value of dissipated fracture with

72 plastic deformation[ 1] allow to connect the damage accumulation (cracking) with shear strain A and to insert a term ~F named the reserve of plasticity expenditure stage [2]. This value has been initiated by linear summarization principle and became a base for the majority of phenomenology theories of metals fracture Ap dA ~F= oJ" Ap ' where Ap - plasticity, could be defined as a shear strain rate, accumulated in material till the fracture point, under the monotonic deformation processes with the given constant values of thermomeehanieal parameters. Damage is a scalar parameter, equal to 0 for undeformed material and corresponding to 1 in a point of macro-fracture. Later Bogatov [3] offered to describe accumulation of plastic loosening by the more common power function e = bA a and considered it's increment as de = a(A)dA,

(4)

where ct(A) = baA(al) - intensity of plastic loosening coefficient, b,a - empirical coefficients. The limit value of plastic loosening e" in a point of macro fracture cracking appearance depends on physical-mechanical nature of the deformed material and thermomechanical conditions of loading processes. The parameter e* for constant conditions of loading in a point of fracture, estimated by value of accumulated shear strain Ap, could be defined by a

e* = b.A/p. Relation de/s* = do was identified as a damage increment. Then kinetic equation (2) could be described as dto/dA = e,A/e"

(5)

Consideration of equation(5) allows to assume that defining the form of kinetic equation of damage under constant deformation conditions can be reduced to investigation of dependence e,A and 6" of loading conditions and the value of accumulated deformation. It should be emphasized, that in literature there is no agreement in opinions on the form of kinetic equation and different authors base their choices on hypotheses fragments of published data on metal-physical investigations. As postulated in linear models of damage accumulation, the e,A depends only on momentary loading conditions [2]. Some investigators assume, that even during the monotone simple deformation, with constant external loading parameters, relationship e,A has a power [3, 4, 5] or exponential form. It is accepted in [6] that change of damage depends on accumulated damage. There are some inconsistent data in literature on the dependence of limit level of plastic loosening of loading conditions [3,5,7]. It is usually accepted in theoretical consideration of damage accumulation, that the parameter e* depends only on the nature of deformed metal.

73 The purpose of the present work is to carry out experimental investigations for basing the choice of the form for kinetic equation of deformation and cheek out hypothesis of dependence of the limit parameter r* on stress state.

2. Materials and experimental methods Relative change of density p during the deformation was considered as plastic loosening g=Ap/p0. Density definition was carried out by triple hydrostatic method. Every specimen was subjected to five times weighing to reduce a statistical error. A resulting value was defined as an arithmetic mean. Specimens were made of 6 different kinds of steel, brass and molybdenum. The surface of all specimens was treated by grinding and polishing before experimental tests. A ZDMU-30 universal testing machine was used for testing specimens of steel 45; an original testing machine equipped with a controlled hydrostatic pressure chamber [3] was used for the test specimens. The liquid used in the chamber is castor oil, which is neutral to deformed metal. The dimensions of the specimens before and after tests were measured with instrumental microscope. Five general series of experiments were carried out. Shear strain rate A calculated by formulas [2,3] was used as a measure of plastic deformation: - for tension of cylinder specimens in a minimal cross section A = 2~/31n(d0/d~); - for tension of flat specimens with a width and thickness ratio >_0.6 A = 2In(to/h); - for torsion of cylinder specimens

A = tg(p, where do, d~ - diameter of specimen in a minimal cross section before and after the test; to, t~ thickness of flat specimen in its minimal section before and after the test; q) - twisting angle of line, brought on a surface of specimen parallel to its longitudinal axis. Stress state coefficient k and coefficient of Lode-Nadai go were used to characterize a stress state of the deformed metal: k = o/T; go = (2022 - 011 - 033)/( O l l

- 033),

where 0 = (0~ + 022 + o33)/3 - average normal stress; T = ~/~S~S~ - tangent stress intensity; 0~1,022, o33 - principal normal stresses; saj - components of stress deviator. The above mentioned values are invariant parameters, so they allow to compare stress state of deformed materials with different levels of strength properties.

74 The separate effect of stress state coefficients k, go was investigated by conducting the tests in a chamber with controlled hydrostatic pressure. In this case the coefficient !~ does not change and the current value of coefficient k could be determined from the decision of Davidenkov and Spiridonova k=(1 + 3/4 d/R)q3 - pq3/a,

(6)

where d - diameter of specimen in its minimal cross section; R - radius of longitudinal profile of deformed specimen in area of deformation localization, as - intensity stress. It is obvious from equation (6), that when p is not equal to 0, as a result of changes os due to deformational hardening of material a coefficient k is not constant even within the uniform deformation stage. To define density the areas with length 2d0 containing the localized deformation zone in the center were cut off from the tested specimens. It is obvious that the density that was defined during hydrostatic weighting is the average within the volume of an investigated specimen. So changes of density correspond to shear strain rate and coefficient of stress state that are also average in deformation path and volume V 1 vj'k(V)dV, kv =-V

A v = V1V~ A(V)dV

(7)

Deformation resistance as used for calculation of coefficient k was investigated by inverse extrapolation of Lode method for multi-stage tension of specimens. For analytical description of deformation resistance Ludwick model was used a. = a,o + aAb,

(8)

where a~0 ,a ,b - has been defined by statistic treatment of experimental results. Tests were carried out using a universal testing machine ZDMU-10t with velocity of loading 6 ram/rain.

3. Simple loading The purpose of the first one was to establish the form of equation (1). Obviously, to define correctly whether the S,A value depends on accumulated deformation, it is necessary to carry out experiments with changing conditions of stress state during the loading process. So during the first series of experiments a preliminary loading was carried out by tension of cylinder or flat specimens in a range of uniform deformation. In this case it is possible to cut some area of effective part of specimens, where stress state coefficients are constant during deformation (k-0.58, p~ = -1 - for tension of cylinder specimen; k=- +1, !~ = 0 - for tension of flat specimen), and use them to define density. Comparing the results of specimens tested and the value of strain rate accumulated during the process of uniform tension allows to determine a form of equation (1) for constant quantities of stress state conditions. Experimental results of test performed are shown in fig. 1. It is obvious, that under constant parameters of stress state during the deformation process, relation between increment of

75 density and accumulated shear strain is linear. This behaviour is characteristic for tension of cylinder ( ~ . = -1) as well as for fiat shaped specimens ( ~ . = 0). As it was mentioned above, the experimental data obtained by a number of scientists suggests non-linear relation between deformation and physical characteristics of damage. These published data were subjected to analysis and the following were observed. Most of the investigated specimens had the areas with non-uniform stress-strain state (for example, tested in a non-uniform range of deformation or during drawing of wire). It is necessary to point out, that in the first case during the deformation a change of stress state coefficient according to relationship is observed (6). This causes a growth of plastic loosening intensity proportionally with variations of stress state coefficient or by the power law. In the second case non-liner relation "s-A" could be connected with signveriable type of deformation which reduces an intensity of damage accumulation. An explanation of non-linearity during the torsion of cylinder specimen is not so obvious. Similar data were observed in [5]. But in our opinion these results could not serve as a basis for definition of the form of equation (1) because of tension stress appearing in deformation hardening material, made specimens under the conditions of the big angles of torsion. The level of this stress grows with deformation development and results in the growth of coefficient of stress state (investigation results of A.V.Konovalov) and damage accumulation intensity which show themselves as non-linear decrease of density. E;

.

104 I. ~

(a)

4;

-8

,.6

-16

~

3

(b)

~ o

-8 "100

0.1

0.2

"240

0.3 A

0.05

0.1

0.15 '

0.2

A

|

i

lO~ ' ~ -2

o

-10

~ ~ o

4

(C)

' { ~ - ~ , ~

(d)

-4

o

i

.201

-6

5

xX'x

-10

4% . . . . 0'.1'

0'.2

0'.3

o'.,~

'

0

0.1

0.2

0.3

0.4 A

Figure 1. The plastic loosening e under simple loading: 1 - carbon steel 0.2%C , ~ = -1; low-alloyed structural steel (0.12%C, l%Cr, 0.3%Mo, 0.2%V), ~ = - 1 , 3 - brass molybdenum, ~ = -1; 4 - leaded brass, ~ = -1; 5 - carbon steel 0.45%C, !~ = 0; 6 - carbon steel 0.45%C, !~ =-1 _

76 4. E f f e c t o f s t r e s s s t a t e

The purpose of the second one was to define a functional relation of value S,A and limit quantity s* of the parameters of stress state. For definition of form for the relation between plastic loosening intensity and the stress state coefficient k, some specimens made of carbon steel 0.20%C were subjected to tension tests in uniform deformation range in a chamber with a controlled hydrostatic pressure. There is no reason to average k and A within the volume, which decrease statistic error. Some deviation of relation from the linear one shown at fig.2 connected with the variation of coefficient k because of the deformational hardening according to formula (6). So the average value of the plastic loosening intensity s was defined as ds/dA = (Ap/po)/AA,

(9)

and associated with the value of integrated mean coefficient of stress state k, calculated by formula (7).

in (S,A)

.

.

.

.

.

.

.

.

.

1

-7 -8

-3

-2

-1

0

k

Figure 2. Changes of plastic loosing intensity under the stress state coefficient k, p~ = -1 1 - carbon steel 0.45%C, 2 - carbon steel 0.10%C, 3 - carbon steel 0.20%C; 4 - carbon steel 0.24%C; 5 - alloyed structural steel (0.4%C, 0.8%Cr, 0.8%Ni, 0.2%Mo, 0.3%Si) Results were obtained in semi-logarithmic coordinates and that allowed to describe the required relationship in exponential manner: ds/dA = am exp(azk), where a~ and a2 - empirical coefficients, to be defined by least square method.

(10)

77 Specimens of the test materials were pulled or twisted till failure. The parts of specimens with length L=do, cut from the deformed specimens were subjected to hydrostatic weighting. An average in volume coefficient of stress state was calculated from formula (7). As it is obvious from fig.2, for all the tested materials a relationship between the plastic loosening intensity and the coefficient of stress state described correctly by the exponential form (10). It should be pointed out that the growth of tension stress during the deformation calls an increasing of plastic loosening while the growth of compressive ones are decreasing it. The direct experimental determination of the limiting value of plastic loosening ~* corresponding to the moment of micro-crack appearing is very complicated since failure starts in locally undetermined volume dimensions of which do not allow to carry out the density determining. That is why in this experiments the value e* has been calculated using an above mentioned relationship (10). It is known from the experimental tests, that in tension tests the crack of plastic metals appears in a middle area of minimal cross section of specimen neck. So the value 8 in the moment of failure, accounted for this moment taking in consideration the history of stress state coefficient was taken as Ap

(11)

e*= ~ k(h)dA, 0

where relationship k(A) could be determined by formula (7). In

s,

, , ~

-3.5

1

-4

-5.5

-'

-0.5

0

0.5

k

Figure 3. Changes of limited value of plastic loosening e* under the different stress state coefficient, Ix~ = -1:1 - carbon steel 0.45%C; 2 - carbon steel 0.10%C; 3 - carbon steel 0.20%C; 4 - carbon steel 0.24%C; 5 - alloyed structural steel (0.4%C, 0.8%Cr, 0.8%Ni, 0.2%Mo, 0.3%Si) Calculated value of plastic loosening accumulation in a central area of specimen neck till the failure moment is shown in fig. 3. Results of e* calculations by formula (11) are also shown there. Obviously, the value ~, is not constant for specimens pulled apart under different hydrostatic pressure. Variation of e* in comparison with integral mean value of stress state

78 coefficient during the tension process is shown at fig.3 in semi-logarithm coordinates and has an exponential character s* = bleXp(bzk),

(12)

where b~ and b2 - empirical coefficients. The result of investigations of other materials shown at fig. 3 are also reinforce a validity of relationship (12). Decreasing of limiting plastic loosening with increasing of tension stress contribution (k is increasing) corresponds to existing ideas about the fracture of metals. Actually, the rate of viscous fracture associated with body defects of continuity is decreasing with a concentration of tension stress and brittle fracture rate associated with appearing and spreading of the increasing fiat cracks. Obviously in limiting ease of the brittle fracture value s* will tend to zero.

5. Two-stage loading In third series of tests the cylindrical specimens made of 0.20%C carbon steel were subjected to tensile tests under atmospheric pressure ( pl = 0.1 Mpa) till different shear strain rates, then specimens were placed to the controlled hydrostatic pressure chamber with a pressure of working liquid p2 = 200, 500, 800 MPa and subjected to following tension. In the fourth series of tests the specimens were subjected to tension i n t o the chamber under hydrostatic pressure pl = 200, 500, 800 MPa till shear strain rate A~ = 0.14 and 0.35, then tension was continued under atmspheric pressure p2 = 0.1 MPa. Changes of specimens density under this conditions goesnon monotonic (see riga and fig.5). t; 10 -4

",,,,.

-2

........

1

~..,...

"~

-4

"%-.% '%

X "~.%

~ 4

'-,.. -6

~176 ~ ~ %,, "%~163

-8 -10

(a)

0

.

.

.

. 0.2

.

.

.

.

. . 0.4

.

0.6 A

Figure 4. Changes of plastic loosening intenity under two-stage tension: 1 - p = O. 1 MPa; 2 - p = 500 MPa; 3 - pl = 0.1 MPa, A1 = 0.14, P2 = 500 MPa; 4 - Pl = 0.1 MPa, A1 = 0.3 5, P2 = 500 MPa

79 It was observed that for two-stage variation of stress state the intensity of plastic loosening changes not immediatly but within certain period of adaptation of metal behaviors to the changed conditions. The rate of adaptation depends of the magnitude of damage accumulated at the first stage of deformation. Similar phenomena were observed under the conditions of changing direction of deformation. 10 "4

,~- ,-,,~,...~..

'

%

-2

(b)

"",,t.,.. .

i--?,~t h~'',,a

=

-4 -6

-8 -10

0

0.2

0.4

0.6 A

Figure 5. Changes of plastic loosening intenity under two-stage tension: 1 - p = 0.1 MPa; 2 - p = 500 MPa; 3 - pl =500 MPa, A~ = 0.14, p2 = 0.1 MPa

6. Practical calculations Comparison of the calculated results shows that non of the known theories of metal fracture could describe the observed experimental relationships correctly, but the best agreement reaches by using the formulas of Bogatov [3], which could be recommended for practical calculations under the multi-stage loading: 1/ai AAi co i = (o~i_l + )ai,

Api

where i - number of loading stage, AAi - increment of the shear strain rate at loading stage, Api- plasticity of metal under k and la, at i-stage of loading, ai - empiric index of damage accumulation intensity. Values of k and la~ could be defined by solving the plasticity test. Relationship between a, Ap and k and !~ to be defined experimentally [2,3]. Calculation and analysis of damage accumulated in metal allow to optimize technology processes of plastic treatment. One of number of practical examples may be given. At the Pervouralsk Pipe-Making Plant in Pervouralsk (Russia) pipes of carbon steel 0.45%C for poles has been produced by cold rolling. Existing equipment did not allow to satisfy the demand for this type of product. To increase a volume of production it was offered to be produce at

80

automated triple drawing line. One of the major questions stated for engineers was a question of damage of pipes during drawing process because the manufacturing line design did not suppose the intermediary annealing. Theoretical calculations allowed to choose an optimal drawing parameters, when the level of residual damage was not dangerous (fig.6). Experimental and then industrial tests of theoretical results show a validity of prediction.

O" 0.2

-"

-4.0

0.15 0.1

f -8.0

0.05

II

III

IY

Figure 6. Changes of calculated damage co (1) and plactic loosening ~ (2)under drawing of 0.45%C carbon steel pipe: I - III - number of drawing pass; IY- annealing

REFERENCES 1. V.V. Novozl~ov. Applied Mechanic and Mathamatic, v.29, No 4 (1965), 681-690 (in Russia) 2. V.L.Kolmogorov. Stress, Strain, Fracture, M., Metallurgy, 1970, 230 p.(in Russia) 3. A.A.Bogatov, O.I.Mizhiritsky, S.V.Smimov. Reserve of metal plasticity during the treatment of metals by pressure, M., Metallurgy, 1984, 144 p (in Russia) 4. V.A.Ogorudnikov. Estimation of the deformability of metals during the treatment by pressure, Kiev, Vishya Shkola, 1983, 175 p (in Russia) 5. Z.J.Luo,W.H.Ji, N.G.Guo at al. Journal of Materials Processing Technology, N_o 30 (1992), 31-43. 6. B.A. Migachiev. Metals (1994), 3, 52-55 (in Russia) 7. H.Sekiguchi, K.Osakada, H.Hayashi. Journal of Institute of Metals v.101 (1973), 167-174

DAMAGE EVALUATION AND R U P T U R E

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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

83

The influence of critical defect size in a ceramic of alumina elaborated by process sol-gel route. N.H.Almeida Camargoa; M. Murat b and E. Bittencourt a. aFaculdade de Engenharia de Joinville - UDESC, Campus Universithrio-Bairro Born Retiro 89.203-100 - Joinville - S.C.- Brasil blNSA de Lyon, Laboratoire GEMPPM, URA CNRS n ~ 341, 20, Av. A. Einstein 69621 Villeurbanne. Abstract: Optimizing the conditions of growth of a-alumina during the thermal transformation of a transition alumina prepared by the sol-gel process allows us to obtain a hot-pressed aalumina ceramic with low critical defect size, which presents mechanical characteristics significantly better than those of alumina ceramics obtained from conventional powders.

1. INTRODUCTION The research program has elaborated a ceramic by the process sol-gel route using as precursor organo - mineral a blend of pseudo-boehmite and acetate of aluminum (powder DISPERAL P3 from CONDEA CHEMIE, RFA). The filiation sol-gel obtained was dryed resulting a xerogel, then alumina of transition by thermal processing and a-alumina at T~ and final ceramics by hot-pressing under mechanical charge at T2. The temperature T~ was varied from 1100~ until 1500~ The ceramic resulting hot-pressing at T: = 1550~ during 30rain presented low critical defect size, and mechanical properties significantly superior than of the ceramics of alumina obtained from conventional powders. Observations in scanning electron microscopy (SEM) on the powder processed thermicaUy at T~ = 1300~ put in obviousness a elementary crystals growth constituted of aggregates of t~-alumina and the hot pressed product presented an homogeneous micro structure, with dimensions of elementary grain of alumina not exceeding some microns. The micro structure of the product transformed into corundum at T1 = 1500~ and then hot pressed at T2 = 1550~ put clearly in obviousness the grains growth detected by a significant decreasing mechanical properties, related by an increase of critical defects size.

2. EXPERIMENTAL PROCEDURE

The presem research works on the elaboration of a ceramic of pure alumina prepared by the process sol-gel route, the alumina was optimized characteristic. A blend of pseudo-boehmite and acetate of aluminum (powder Disperal P3 of Condea Chemistry., RFA),

84

transition, which considered in this case a mixture of phases eta and theta. This powder presents itself in the form of aggregates of elementary crystals with size, d T] to realize the hot-pressing. In the present case T2 = 1550~ with isothermal maintenance of the charge during 30 minutes to avoid a too pronounced growth of grains of a-alumina, what would be deleterious to the obtaition of good characteristic and mechanical properties. A preliminary study of alumina of transition behavior prepared according to previously described protocol had leaned to choose l l00~ as temperature T~, after isothermal processing under void for three hours at this temperature, the pure alumina of transition was totally transformed into corundum [1-2] The surface grains of corundum showed then a very strong increasing, the classic micro structure microporous vermicular (the elementary crystals having a transverse diameter in the order 100 to 200nm) observed by others authors having used others precursor types [3-7]. The mechanical properties of ceramic obtained after hot-pressing at 1550~ ffigures, la and lb) are similar to those of ceramic of alumina prepared by conventional hot-pressing powders (flexural strength generally in order to 350-400 MPa although recently have been obtained values in the order to 560 MPa for samples prepared by conventional process with very fine commercial powders of alpha alumina. In the present case, the microstructure fracture of the material after hot pressed comprises very fine particles (some hundred of nm) dispersed around crystals aggregates of alumina with well superior dimension (some microns)

700 600 E

~ ' 500

g

-4

-

o. r~

400

-

300

-

etO

-3

O b-

200 I00

'

I000

I

1100

'

i

'

I ......

' .......

I

1200 1300 1400 Temperature T1 (Oc)

'

I

1500

2

'

1600

Figure 1" Variation, according to the temperature TI, the 4 flexural point strength (curve a) and toughness ( curve b).

85 We tried to improve this micro structure varying the parameter T, from 1100~ until 1500~ This had put in obviousness the existence of a temperature Tt (in the occurrence 1300~ where the ceramic resulted of hot pressing at 1550~ present case maximum value of the flexural 4 point strength fracture (op = 684 MPa, fig. l a) although the value of the report d/dth (apparent density and theoretical density) is only 96%. For Tl = 1300~ was observed equally (fig. 1b) a maximum value of the toughness Ktc (4,12 MPa re'a), and the hardness Hv (15,1 GPa ) but especially a minimum value of the dimension of the critical defect size (x = 1 l~tm, fig. 2) calculated from the relation K,c Y'.CrR.(Tt.X)]/2 with Y chosen arbitrarily equal to unit. =

100 ..,90

8o ~'70 ~

60

N 5o "= r.) 4 0 30 ~ 20 100 ~

~

1000

'

I

1100

'

I

'

I

'

I

1200 1300 1400 Temperature T 1(~

J

I

1500

'"

1600

Figure 2 9Variation, according to the temperature Tt in relation of critical defect size. The observation of the powder obtained at T, = 1300~ (and even that already obtained at 1200~ put in obviousness a growth of elementary crystals constituting alpha alumina aggregates (figure 3a), and the product hot pressed presented a homogeneous micro structure (figure 3b), the elementary grain dimension of alumina not exceeded some jam. For values of T, over 1300~ an incontest process of pre-densification (natural sintering) of powders, associated with the transformation of the alumina of transition. This phenomenon is perfectly put in obviousness by thermal dilatation curves (figs. 4 and 5), powders after processed at Tt ( notable decreasing of the retraction at high temperature when T, was increased). This natural sintering, associate to a exagered growth of alumina, limits the previous action of the hot-pressing. The micro structure of the product transformed into corundum at T, = 1500~ then hot-pressed at T2 = 1550~ put clearly in obviousness this increasing of grains, but the aspect presented by crystals is not favorable to characteristics of the material that presents an important porosity (d/dth = 80,5%) with lair value of the flexural strength fracture (173 MPa) and the toughness (2,83 MPa.m ~'2, figures l a and 1b). The value of critical defect size tbr this material was raised (90~tm).

86

Figure 3a: Microstructure of the powder of alumina sol-gel Tt = 1300~ (SEM)

Figure 3b: Microstructure sol-gel pure alumina strength fracture by a couple of temperature T~/T2= 1300~176 (SEM)

9,

A

Temp6rature (~

0 /

200

400

600

800

1

1400

1600

-5 -10 -15 ~ 0 O Q9 99 O--

9

9 1 4 9 9 -O.O O 9

- - ~-"'1e"o o Oeo..o.-u__._,. _ _

-25 ~ -30

9 1 4 99 O ' O - 0 9

__

.... 9

........

E

re 9149149 9oo-ei,.'..i" 'l~.~,dMl~:~dp~oQ

a:- o e w , o

.1_

Figure 4: Thermal dilatation curve of the x6rogel powder of pure alumina treated at T~ = 1100~ (curve OABC: linear increasing of the temperature; curve OF: isothermal return at 1500~ A direct thermal processing at 1550~ to realize the hot-pressing without using a constant sintering temperature T~ conducted to the obtainment of a material with large grains (10~tm), high K[c (4,1 MPa.m ~/2) but of small C~R(134 MPa), what results the very high value of the critical defect size (309~tm, figure 2).

87 0 t

A d(~

-5 .1

Temperature (~

o o~0~ o o6~0 o o o 8 ~ o o d ~ 0 ~ ) o o g g ~ a o o ~ 4 ~ d ~ C 1600

~176176 ~

fie O0

E

-10 -15 -20 --25 --30 --

Figure 5" Thermal dilatation curve of the xerogel powder of pure alumina treated to T1 = 1300~ (curve OAC: linear increasing of temperature; curve OF: isothermal return at 1500~

3. CONCLUSION The optimization of the temperature formation of corundum at T~ = 1300~ from a alumina sol-gel behaved, after hot-pressing of this corundum, to a ceramics presenting mechanical characteristic clearly superior than of ceramic prepared from powders of conventional alumina. This result can be interpreted as a better control condition of formation of the alpha alumina, from a powder of alumina of transition prepared itself by thermal processing of a xerogel of alumina [10], and the homogenization of the micro structure (figure 3a and 3b), and diminution of the critical defect size (figure 2).

REFERENCES 1. M. Murat, F. Mignard, F. Hue et G. Fantozzi, Elaboration des poudres c~ramiques composites "Alumine/Whiskers SiC par proc6d6 sol-gel. L'ind. C6rm., n ~ 892, (1994), p. 236-238. 2. M. Murat, N.H.A. Camargo et F. Sorrentino, Stabilit6 thermique d'une alumine de transition: effet de l'incorporation pr6alable d'une poudre nanom6trique de carbure de silicium. C.R. Acad. Sci., Pads, 318, s6rie II, (1994), p. 611-614. 3. G. Varhegyi, J. Fekete et M. Gemessi, Reaction kinetics and mechanism of ot-A1203 formation. Compte Rendu du 36me Congr6s International de I'I.C.S.O.B.A., Nice. (1973), Sedal Editeur, p. 575-584. 4. F.W. Dynys, J.M. Halloran, Alpha Alumina formation in alum-derived Gamma alumina. J. Amer. Ceram. Soc., 65, (1982), p. 442-448.

88 5. F.W. Dynys, M. Ljungberg et J.W. Halloran, Micro structural transformations in alumina gels. in Bitter Ceramics trough Chemistry, Materials Research Society, vol. 32, (1994), p. 321-326. 6. J.E. Blendell, H.K. Bowen et R.L. Coble, High purity alumina by controlled precipitation from Aluminum sulfate solutions. Ceram. Bull., 63, (1984), p. 797-801. 7. S. Rajendran, Production of ultrafine alpha alumina powders and fabrication of fine grained strong ceramics. J. of Mat. Sic., 29, (1994), p. 5664-5672. 8. L.M. Sheppard, Enhancing performance of ceramic composites. Ceram. Bull., 71, (1992), p. 617-631. 9. J. Zhao, L.C. Stearns, M.P. Hamer, H.M. Chart and G.A Miller, Mechanical behavior of almnina-silicon carbide nanocomposites. J. Amer. Ceram. Soc., 76, (1993), p. 503-510. 10. N. H. Almeida Camargo et M. Murat Caract6ristiques m6caniques d'une c6ramique d'alumine prdparde par le procdd~ sol-gel: influence des conditions de formation de l'almnine alpha. C.R. Acad. Sci., 319, s6rie II, (1994), p. 893-897.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

89

Defect evolution during machining of brittle materials A. Chandra, K. E Wang, Y. Huang and G. Subhash Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University, Houghton, M149931 ABSTRACT A simple stress based defect evolution model is developed to assess the influence of various process paramters on material removal rate (MRR) and induced damage during ceramic grinding processes. Model predictions for normal and lateral damage zones under normal indentations are first compared to fracture models as well as experimental observations on pyrex glass. The proposed model is then extended to simulate oblique indentation events depicting abrasive gritworkpiece interactions during ceramic grinding. It is also easily extendable to real grinding situations involving multiple interacting abrasive grits. Process design options for reducing induced damage in the finished part, and increasing MRR are considered next. In particular, the potential of a new design avenue involving intermittent unloading is investigated. For pyrex glass, it is observed that intermittent unloading can facilitate significant increase in force per abrasive grit without increasing the associated surface and sub-surface fragmentation in the finished part. This design feature may enable significant increase in MRR, while maintaining a very low level of process induced damage in the finished product. 1. INTRODUCTION Currently, precision ceramic components are being used ever increasingly in various engineering applications. Compared to metals, however, most ceramics (e.g., glass, SiC, Si3N4, ZrO 2, CBN, sapphire, spinel, etc.) are also much harder and much more susceptible to brittle fracture. These advanced ceramic components, in most cases, require finishing operations to achieve surfaces of required geometry, tolerance and finish, while maintaining the desired level of strength. Finishing operations, e.g., grinding, typically induces surface and sub-surface damages in the finished part, that can severely degrade its strength and useful life under service conditions. Hence, very expensive free abrasive polishing is necessitated for such precision ceramic parts. Accordingly, finishing operations typically account for 25-80% of the total cost of many precision ceramic components. It is observed (e.g., Eckert and Weatherall 1990, Office of Technology Assessment 1988) that process induced damage and high finishing cost pose two significant barriers against commercial success of precision structural ceramic components. Jahanmir et al (1992) provide a review and assessment of current practices and research needs in precision finishing of ceramic components. Brittle grinding makes the finished surface susceptible to induced surface and sub-surface damages and the strength of finished parts have been observed to be reduced by 30-60% (Jahanmir et al 1993). Accordingly, the present work focuses on first developing a simple stress based model that will be capable of depicting defect evolution at high strain rates (103 - 10S/s) that are representative of a machining process. The abrasive grit - workpiece interaction in grinding is modeled as an indentation event subject to normal and tangential loading. Model simulations for

90 normal and lateral damage under normal indentations are compared against fracture models, and verified against experimental observations on pyrex glass. The validated model is then utilized to further explore the "process design space", and identify potential design parameters for process improvements. Intermittent unloading during chip formation is identified as a keydesign variable, and its effects are investigated in detail. Following introduction, existing literature on scratch tests and indentation fracture mechanics models for brittle materials are briefly reviewed in the background section. This is followed by development of the proposed model, and its validation against experimental observations from static and high strain rate normal indentations on pyrex glass specimens. The proposed model is then utilized, in particular, to investigate the effects of intermittent unloading on defect evolution and material removal mechanisms in ceramic machining processes. 2. DEFECT EVOLUTION MODEL AND VALIDATION A single chip, in a ceramic grinding process, is typically formed in 10-100 ~ts. This time gets even shorter for modem high speed grinding at 60,000-150,000 rpm (Kovach et al 1996, Ashley 1995, O'Connor 1995). As a result, strain rates approach 103-105/s. Accordingly, the present work focuses on depicting defect evolution in brittle ceramics under high loading and strain rates. It is important to note here, that most process induced damage in ceramic grinding is observed (e.g., Jahanmir et al 1992, 1993, Allor et al 1993, Xu and Jahanmir 1994, 1995a, b, Xu et al 1995) to be less than 1 mm in size. Typical dilational wave speeds are about 104 m/s in ceramics and 23x103 rn/s in glass. In view of this, the time needed for the wave to propagate 1 mm is expected to be

(1)

is proposed for estimations of different damage zones induced by a single grit indenting on a workpiece. In Eq. (1), I~1 is a suitable norm of the induced stress field due to the action of the abrasive grit on the workpiece. As warranted, I~1 may be identified with magnitudes of appropriate stress components, principal stresses, or other invariants. 6f(g) is the rate dependent failure strength of the workpiece material, and admits a failure strength of the material that has been observed experimentally at the apprpriate strain rate regime. For a completely elastic solution under a concentrated load, a stress singularity exists at the point of contact between the abrasive grit and the workpiece. Hence, the damage zone will initiate there, and propagate outward as long as 16[ is greater than the intrinsic rate dependent failure strength 6f(~) of the material. This finite zone, is then identified as the corresponding damage zone. In the present work, we focus on damage evolutions normal to the finished surface (median and radial damage), and parallel to the finished surface (lateral damage). The normal damage zone represents the effects of median and radial cracking (and induced secondary cracking), while the lateral damage zone represents the extent of lateral cracking. It is interesting to note, that normal

91 damage is typically left behind in the finished part, and is responsible for its strength degradation. The lateral damage, on the other hand, facilitates material removal (Cook and Pharr 1990). Thus, they may be benevolent, and aid in enhancing the material removal rate (MRR). Details of the damage model is presented in Chandra et al (1997). The experimental scheme aims at delineating the effects of various material and process parameters influencing the material removal and damage evolution mechanisms during ceramic grinding processes. A set of static normal indentation tests at peak loads of 5-100 N is carried out first~ A modified split Hopkinson pressure bar (also called Kolsky bar) facility with a momentum trap is used for indentation experiments involving high loading rates (Subhash and Nemat-Nasser 1993, Koeppel and Subhash 1997, Koeppel et al 1997). 2.2. Model Predictions and Validations for Normal Indentations: For the present work, tensile failure strength of pyrex glass is assumed to be 20% of its compressive failure strength. For model simulations, the tensile failure strength is taken to be 200

MPa for nominal strain rates less than 103/s, and is assumed to increase linearly to 400 MPa for a strain rate of 104/s. For simulations using a fracture model (e.g., Lawn et al 1980, Evans and Marshall 1981, Chiang et al 1982a,b), Klc of pyrex glass (typically in the range of 2.7 - 3.3 MPam 1/2) is assumed to be 3 MPa-m 1/2. Fig. 3 shows comparisons of simulated results with experimental observations for fully unloaded pyrex glass specimens that have experienced different peak indentation loads. Predictions of normal damage zone depths (shown in Fig. 2a) based on the proposed simple stress-based model agree quite well with the median crack size estimates obtained from a fracture model (e.g., Lawn et al 1980, Evans and Marshall 1981, Chiang et al 1982a,b, Marshall 1984, Ritter et al 1984, 1985, Hu and Chandra 1993). The normal damage depths were measured experimentally by polishing a window on the side of the glass specimens. Two sets of experimental data are shown in Fig. 2a. It may be observed that model predictions also agree well with the experimental observations. Fig. 2b shows the comparisons of simulated surface traces of normal damage with experimental observations. In this case, the model seems to over-predict the surface trace extensions. The model assumes the surface traces of median cracks. Experimentally observed surface traces may be due to median or radial cracks, and this may explain the discrepancy between model predictions and experimental observations. Lateral damage occurs due to unloading, and Fig. 2c shows the comparisons of lateral damage zone predictions with experimental observations under fully unloaded configurations. It may be observed that model predictions for lateral damage zone sizes agree very well with the experimental observations at fully unloaded configurations. Upon indentation loading, normal damage is initiated. The normal damage zone grows bigger in size and penetrates deeper with increasing load, finally reaching its maximum depth at the peak load. At this instant, no lateral damage is observed, and the surface trace of normal damage is also usually very small. Thus, the normal damage zone evolves as a series of elongated ellipses (with major axis oriented along the loading direction) during the loading phase. Upon unloading from the peak indentation load, the depth of normal damage remains almost constant, however, its surface trace gradually extends, approaching a circular shape under complete unloading. The evolution of normal damage in soda-lime glass during a complete loading-unloading cycle is shown in Fig. 3. The model predictions are then compared to experimental observations of

92 Marshall and Lawn (1979) on the evolution of a median crack in annealed soda-lime glass slabs (H = 5.5 GPa, Kie = 0.75MPa-m 1/2) during a complete loading-unloading cycle. For the damage model, tensile failure strength of soda-lime glass is taken to be 50 MPa. At the peak load of 90N, the damage model predicts a normal damage zone depth of 0.3 mm, while the fracture model predicts 0.34 mm. The experimental data (Marshall and Lawn 1979) ranges from 0.214 mm to 0.363 mm. Thus, the model predictions are within the range of experimental observations. It is particularly interesting to note the non-self-similar nature of normal damage growth under unloading. Typically, fracture models assume a half-penny shaped crack front, and its predictions are restricted to self-similar circular crack-fronts. A damage, model, however, can fully capture such effects. Upon unloading, the simulation results predict extensions in surface traces of the normal damage zone, while its depth of penetration is held constant. Such a trend is also evidenced by crack arrest marks observed experimentally by Marshall and Lawn (1979). Their experimental data (Fig.6 in Marshall and Lawn 1979) shows a fully unloaded aspect ratio (depth/ surface trace) of about 0.5, while the present simulations predict 0.52. Similar to experimental observations, the proposed stress-based model also predicts lateral damage initiation only upon unloading beyond a certain threshold (that depends on the tensile failure strength of the material). With continued unloading, the lateral damage zone propagates parallel to the free surface, reaching its maximum size at the fully unloaded configuration. 3. DESIGN SPACE EXPLORATION Utilizing the validated model, several issues pertaining to process design of a ceramic grinding operation are investigated in this section. In Figs. 2 and 3, the normal force on the indenter is correlated with various damage zone estimates. In view of these results, it may be observed that Peak Force/Abrasive Grit is a key parameter effecting defect evolution during a ceramic grinding process (Bifano et al 1991, Subramanian et al 1996, Kovach et al 1996). In the present work, we focus on identifying additional variables, that may significantly effect the product quality and economy (through Material Removal Rate) in ceramic grinding processes. 3.1. Potential of Intermittent Unloading: It is particularly interesting to note from the above investigations, that detrimental normal damage nucleates and propagates in depth under loading, while benevolent lateral damage is primarily facilitated by unloading. Upon unloading, the normal damage zone has also been observed to propagate in the transverse direection with extending surface traces, while maintaining constant depth of penetration. This indicates that process induced damage (left behind in the finished part) in a ceramic grinding process is typically induced as the abrasive grit penetrates the workpiece and reaches its full indentation load. By contrast, the material removal mechanism (of lateral damage) is initiated only upon unloading from the peak load, and material is removed as the indenter gradually leaves contact with the workpiece. Thus, the loading

produces the undesired product damage and strength degradation, while unloading provides the desired material removal. In a conventional grinding process, however, loading always precedes unloading. As Force/Grit is increased gradually from zero, normal damage is initiated first, and propagates to its maximum depth at peak load. Only later, upon unloading, lateral damage responsible for material removal is initiated. Thus, in a traditional grinding operations on brittle materials, detrimental normal damage is usually fully developed, even before material removal mechanism of lateral damage has had a chance to initiate. It is important to note here, that one does not necessarily need to wait for unloading from the

93 peak pulse load to initiate lateral damage. Unloading, from a threshold load (dependent on the failure strength of the material) is only required to induce lateral damage in the material. Accordingly, the proposed intermittent unloading technique attempts to achieve the following: 1. Initiate the material removal mechanism of lateral damage early in the process, before normal damage is fully developed~ 2. Utilize the shielding effects of lateral damge to retard and potentially eliminate normal damage from the finished product. This shielding effect is shown schematically in Fig. 4. It is interesting to note that the lateral crack (or damage), initiated by the intermittent unloading, acts as a barrier against further penetration of normal crack (or damage) upon reloading. Given a fixed load Pint from which unloading has occured, this shielding effect is effective until a threshold Pshield is exceeded in reloading. The effect of intermittent unloading from an indenter load of Pint, is shown in Fig. 5. In order to reduce process induced damage in the finished part, an additional intermittent unloading step from Pint is introduced within a typical load-pulse during normal indentation prior to reaching the peak indenter load of Pmax. The load pulse is shown as inset in Fig. 5. It is observed that intermittent unloading establizes a lateral damage zone before normal damage becomes fully developed. The size of this lateral zone depends on the magnitude of Pint- Upon reloading beyond Pint, the normal damage zone would normally attempt to grow in size, and penetrate to greater depth. Unlike a traditional case, however, the pre-existing lateral crack or damage (generated from prior unloading) now provides a shielding effect, and essentially annihilates any singularilty associated with median/radial crack or normal damage. As a result, the normal damage zone cannot penetrate to greater depth until (under continued increase in load) it's lateral size has grown bigger than the size of the lateral damage zone induced by Pint- The corresponding indenter load at this instant is denoted as Pshield. Thus, upon reloading followed by intermittent unloading from Pint, the normal damage zone is inhibited from penetrating to greater depth so long as the indenter load does not exceed Pshield" Fig. 5 shows a typical case, where unloading from a Pint = 75N resulted in a Pshield exceeding 250N. Fig. 6 graphically presents the effects of intermittent unloading, and resulting interactions of lateral and evolving normal damage zones in the load range Pint < P < Pshield- Defining AP = Pshield--Pint' the workpiece can carry an additional indenter load AP beyond Pint without any further increase in normal damage depth. Upon exceeding Pshield, the normal damage penetrates to higher (as predicted for monotonous loading) depth, and the shielding effect of intermittent unloading is lost. The present investigation shows that the effect of intermittent unloading remains the same over a wide range of indentation loads (0.1N to 300N), and results in a Pshield that is about 3*Pint. This offers a potential avenue for increasing Force/Grit (and associated MRR) in grinding of ceramics, without any associated increase in process induced damage in the finished product.

3.2. Oblique Indentation Events: Both normal and tangential loads are significant in real life grinding processes. Accordingly, implications of intermittent unloading under more realistic oblique indentation events are investigated in this section. Single- and multi-grit scratch tests show that normal load (Pn) / tangential load (Pt) ratios typically vary from about 1 for sharp well dressed wheels to about 2 for

94 worn and loaded wheels (e.g., Hockey and Rice 1979, Jahanmir 1993). Accordingly, oblique indentation simulations are conducted with Pn/Pt = 0.5, 1.0 and 2.0, at fixed resultant loads (R). As a first approximation, it is assumed that Pn/Pt remains fixed throughout the chip formation cycle. Fig. 7 shows the contours of normal damage zones for oblique indentations for different ratios of Pn/Pt. As expected, the normal damage zone is skewed. It is interesting to note, that the fundamental nature of defect evolution remains the same as that for normal indentation. Normal damage is initiated and propagated under loading, while lateral damage is initiated and propagated under unloading. The qualitative nature of the normal damage evolution also remains the same. Fig. 8 shows the effects of intermittent unloading for oblique indentation events. Again, the qualitative nature of the shielding characteristics remains the same as that observed under normal indentation. For a given Rint, the actual value of AR = Rshield- R int is now dependent on the Pn/Pt ratio. It is observed that (for 0.1N < Rint < 300N), AR varies from 0.26*Rin t to 1.05*Rint as Pn/Pt is varied from 0.5 (corresponding to single point cutting) to 2.0 (worn grinding wheel). For a sharp grinding wheel (Pn/Pt = 1.0), a AR of 0.55*Rin t is observed. Thus, the benevolent effects of intermittent unloading, observed under normal indentations, seems to carry forward to realistic grinding situations. 4. DISCUSSION AND CONCLUSION A very simple stress based model is proposed in the present work to represent defect evolution during grinding of brittle materials such as glasses and ceramics. For grinding, the zone of interest is usually restricted to a depth of less than 1 mm from the free surface. A strain rate dependent static model is a reasonable approximation of the essentially dynamic event, as long as the loadpulse time is much larger than the time taken by the dilatational wave to travel the depth of interest (usually A)i

and

eo = V ~-~(Nt < b > A)i

(33)

i=1

Ai

requiring evaluation of the integral only over the area of the crack. The last expression applies to flat cracks with unique normal ni, and < bi > denotes the average crack displacement vector over the crack area Ai. For a single isolated crack, the approximate relation < bi > = Bi < ti >

(34)

is proposed in [3] between the average crack opening displacement < bi > and < ti >, the average of a nonuniform traction, respectively a traction vector ti uniformly applied to the crack faces. The matrix Bi, the crack compliance, depends on crack shape and size, the geometry of the sample and on the properties of the elastic medium. In the anisotropic case, orientation is also of relevance. Expressions for Bi were given in [31, [41 for representative crack geometries. As an elementary example we consider a rectilinear crack in the plane, and with reference to the local n, s-system (normal and tangential to the crack line), eq. (34) assumes the form

b, > i

B,,B,,

i < t, >

For the isotropic material, normal and shear modes are uncoupled, and the off-diagonal terms in the matrix Bi vanish. Otherwise B,n = B,, for the elastic medium, and Bi is a symmetric matrix. In the infinite plane, B,n = B,, = 7rli/2E" (crack length Ii, elastic modulus Eo, Poisson's ratio to, E'o = Eo for plane stress, E'o = E o / ( 1 - U2o) for plane strain). In this case, Bi = (Trli/2E~o)I is an isotropic matrix invariant to coordinate rotations, I denoting the identity operator. The average tractions along the crack faces defined by the local stresses, are specified here as to compensate the action of the applied loading at the same location in a crack-free solid. Thus,

1 f(NadA)i

< ti > = ~iAi

= Ni < ai >

and

ti = Nia

(ai = const. = a)

(36)

where ai denotes the stress along the plane of the virtual crack defined by the normal vector ni respectively the matrix Ni. It accounts for the presence of any other defects but for the cracks. For a uniformly distributed stress we have along each single crack ai = a, the macroscopically applied stress.

118 If interactions can be neglected, relation (34) for the single isolated crack can be applied to each member of a crack system. With (36), (34) and (33), the contribution (32) of the cracks to the strain then becomes ec' =

NtBNA)i a = N t l 3 N a

"~

,

N = { N i } , B =-v

1 [AiBiJ

(37)

In the last expression in (37), the definition of the matrix arrays N and 13 substitutes the summation sign. We notice that expression (37) for the strain ee establishes a symmetric constitutive relation as by (31). It reads e = n / i + N t I ] N ] = n-1 a

and

g-1

=

go- 1 _[_N t l ] N

(38)

Representation of the crack system in (38) is based on the orientation Ni and the compliance Bi of the individual cracks, which thus define the parameters D in (1). 3.4.

Interacting

crack

system

For interacting cracks, we assign a stiffness to the crack system introduced by the relations < t~ > = CII < bl > +C12 < b2 > + . . . + C~. < b . > < t2 > = C21 < bl > -~-C22 < b2 > + . . . + C2n < b . > (39) < t, >=

Cnl


+Cn2 < b2 > + . . . + C , , < b , >

A short-hand form of (39) reads t = Cb

with

t = {< ti > } , b = {< bi >} and C = [Vii]

(40)

where the symbols t, b and C denote hypervectors and hypermatrix arrays respectively. and the hypermatrix. In (39), each coefficient matrix Cij determines the average traction < ti > induced Mong the plane of the i th crack by an average displacement discontinuity < bj > imposed on the j th crack, whilst all other cracks are kept closed. Interpreting (36), determination of the above contribution to < ti > requires the stress distribution resulting from opening the j th crack. For this purpose, we assume that the stress Vrj can be represented as o'j = a~tj where a~ denotes a 6 x 3 array formed from unit solutions for the isolated j th crack and t] is the uniform, respectively average traction along the crack face. In the two-dimensional case for instance, the crack-induced stress aj may be represented as

crj = pjapj + qj%j

(41)

where pj = n j tt j ,

,

q j - - s jtt j ,

(42)

119 denote the traction components along the normal direction n and the tangential direction sj of the i th crack. The stress fields apj, aqj are those arising in the elastic plane from unit normal and shear tractions pj = 1 and qj = 1 respectively at the j th isolated crack. Combining (41) with (42) yields aj - [apnt-t - aqS t] J tj' -- ajtj''

(43)

which defines a~. Then, with (34) aj =

a~t] = a]B~"1 < bj >

(44)

and from (36) the single contribution to < ti > follows 1

< ti >j---- Aii [Ni f a;dAi]B~" < bj >

(45)

Ai

~,From (45) the coefficient matrix Cij is deduced as, Cij = ~i1 [Ni ] a~dAi] B~-' = (~ijS~-1

(Cii = B~-x , Cii - = I)

(46)

Ai

and essentially requires averaging over the i th crack of the unit solution for the j th crack constituting a~. The stiffness matrix of the crack system may be written as C = (~B -1 with (3

=

[Cij], B---- [BiJ

(47)

The particular constitution of the diagonal elements of the hypermatrices in (47) is shown in (46). Ultimately, the tractions applied to the crack faces must compensate the action of the external loading on the specimen. For a uniformly distributed stress a they axe given by t = N a and solving (40) for the crack opening displacements yields b = C - 1 N a = B(~ - 1 N a

(48)

With b = { < bi > } from (48), the contribution of the crack system to the overall strain is determined by (32), respectively (33). In matrix notation, e.c =

Nt]3(~-SNa

(49)

which replaces the uncoupled expression (37). The constitutive relation following from

(31) i~ e -- [too1 + N t I 3 ( ~ - 1 N

]a

= n-xa

with

~-1

= ~o'+ N~t]e-XN

(50)

_

and is now non-symmetric because of the interaction matrix C. Thereby, mutual positions of the cracks appearing in the interaction matrix enter as additional parameters in the array D in (1).

120 4. E V O L U T I O N OF M I C R O C R A C K I N G

4.1. Stability of crack system The question arises now, whether at a given state of loading the crack system can undergo changes by the appearance of additional cracks. In this connection, the following considerations do not deal with continuous crack formation and extension, but rather with an increasing number of discrete crack entering the system instantaneously with the full dimension. A modification of the crack system with nl to one with n2 cracks (n$ > nl ) at constant applied stress a causes a change in strain e. The strain difference emanates exclusively from the part ev, the contribution of the crack system, cf. (31), and determines the energy release in the material volume, Vvrt [ec$ - eel ] = ~ E t[ < b~ - b, >i Ai 2 i=1

(51)

In (51), the energy release rate has been alternatively expressed for the discrete system in terms of the tractions ti = Nia along the n$ crack areas and the change in average crack opening displacement < b2 - bl >i. For the additionally introduced (n2 - n l ) cracks < b~ - b ~ > i = < b~ >i. Comparison of the energy release from (51) to the surface energy required for the creation of the new cracks states that the change will not appear as long as, n2

y~ 2

[~o~ _ ~o,] _ 2 ~ ~jAj < 0 ~

(52)

where the second term represents the surface energy for the extension of the system from n l to n2 cracks. If crack interaction can be neglected, the strain cv is determined by (37). The difference for the two configurations of the crack system reads, f-C$- f-C1 ~ ' ~

L nl

J

the summation extending over the trial cracks (n2 - n l ) only. This is a consequence of the non-interaction assumption stating that the new cracks do not affect the displacement discontinuity along the existing cracks. Using (53) and for 7j = 7 = const., the stability criterion (52) becomes, Aj

(NtBNA)j a - 2~' _< 0

(54)

L nI

which is the counterpart of the continuous criterion (8) for a certain virtual extension of the crack system. Alternatively, (53) could be derived from (8) in conjunction with (31), (37) for the present virtual crack extension. The same result is obtained directly in terms of microscopic quantitites, using the energy expression on the right-hand side of (51). The contribution of interacting cracks to the strain is given by (49), and yields

Aj

N ~ [B~C~ ~ - B, C; ~] N ~ - 2~ < 0

(55)

121 instead of (54). Interaction necessitates consideration of the complete system in the two configurations, the existing and the one virtually extended by the (ng - n l ) cracks.

4.2. Progressive fracturing The above theory relies entirely on deterministic crack systems requiring specification of orientation, shape, size and mutual position of cracks. Nevertheless, the data may be generated at random within the sample. Accordingly, examination for possible appearance of microcracks can be performed at random, in principle. If the structural morphology of the material favours any particular crack formation, the task of exploring the evolution of fracturing is simplified. As an example, if fracturing preferably occurs along grain boundaries, a network of potential cracks is defined by the grain structure of the material. Expressions (54) and (55) introduce microscopic morphological parameters in the respective continuum forms. For an algorithmic treatment of progressive fracturing we preferably refer to the microscopic variables. We consider a state of fracturing (n cracks) stable under the given conditions, and modify the loading. It is assumed that a new single crack is thereby formed if the elastic energy release balances the surface energy. Using the expression on the right-hand side of (51), (56)

t i < b, - b ~ >i Ai = 27jAj 2 i=l

and 2vjAj denotes the surface energy for the j th trial crack. In the non-interacting case, the introduction of the new crack does not affect the response of the existing system. With reference to (34) for constant loading iCj = 0 ,

j==Bj

(57)

It follows from (56) that at a given state of uniform macroscopic stress a the j th crack is considered formed if, tjt < bj > = tjtBjtj > 47j

(tj -- Nja)

(58)

and variation of the index j -- n + 1, m supplies the cracks forming by the performed modification of the loading. Since, however, the criterion for crack formation does not depend here on the state of fracturing, the crack pattern associated to a given macroscopic stress a can be obtained at once. In the case of radial loading (a = tat) the equality sign in (58) specifies the instant tj when the j th crack forms 4-yj t~ = att(NtBN)jat

(59)

The above equation may be deduced also directly from (54). Within a defined network of potential cracks, (59) determines a priori the sequence of fracturing patterns during the course of the radial loading. If the interactions in the system can not be neglected, the determination of the energy release in (56) involves all cracks, the existing and the trial one, and requires a complete solution of the matrix equation (40) for the displacement discontinuities in the presence

122 of the surplus crack at constant applied load. For the system of n + 1 cracks, equation (30) can be presented in the form (60) In (60), the index n refers to the existing array, the index j to the new crack. Depending on the number (m - n) of the trials to be undertaken, an exploration of the extended crack pattern may become cumbersome. A simplified treatment disregards the influence of the single new crack on the system. Then the upper row in (60) supplies the average crack opening displacements bn = {< bl > } , ( i = 1, n), as

(6:)

bn = C n-In t n = C n-1n N n a

Consequently, the difference < b~ - bl >i vanishes for all i = 1, n. The lower row in (60) yields, bj = C~ 1 [tj - Cjnbn] = Bj [Nj - ajnbn]

and

< b~ - bl >j--< bj > - bj

(62)

which completely accounts for the influence of the existing system on the trial crack. It follows from (56) that at a given state of uniform macroscopic stress a the j th crack is considered formed if tj> < =bj t

ttBj [tj - Cjnbn] >_ 47j

(63)

The criterion (63) differs from (58) by the interaction term Cj~b~. If the transition from n to n + 1 implies a radial stress variation (a~+x = fa~), the equality sign in (63) determines the multiplier

f~ = min (47j(a~NjBj[Nj- Cj.C:=:N.]a.)-')

(64)

for the formation of the next among the (m -- n) candidates in the network. Expression (64) helps to indicate explicitly the reference stress an prior to the progress of fracturing. Its evaluation is conveniently performed with the quantities in (63) determined for an and scaled by the factor f to the actual ones. We remark that the assumption of a negligible influence of the extension on the existing system at least necessitates a small number of additional cracks, preferably a single one in each incremental modification of the loading. 4.3. M i c r o c r a c k i n g n e t w o r k For a simulation of microcracking in the material it is proposed to consider a sample representative of the structure defining a network of potential cracks. Such a network may be provided by the grain boundaries. Information on the morphology of the structure, respectively the grains must be available in a form suitable for numerical processing. For this purpose, artificial microstructures with specific characteristics are generated by a computer algorithm [5]. In two dimensions, the procedure starts from a regular hexagonal lattice. The cells, representing grains, are subjected to statistical distortions by displacing triple points. This usually leads to only moderate fluctuations in grain size, and in order to obtMn larger variations, several adjacent grains are merged together. Thereby, some triple points are eliminated at random and new grain boundaries are created there.

123 Apart from grains, other elements may appear in the microstructure as well. Plasmasprayed ceramic coatings, for instance, exhibit a lamellar structure as a result of the manufacturing process, and pores. Lamellae are introduced in the computer generated microstructure by the definition of polynomial curves attracting triple points in the vicinity. Pores are randomly placed by activating a virtual void growth process at triple points in the sample accounting for the fact that they preferably appear at lamella~ interfaces. The resulting microstructure is specified by a record containing the topological and geometrical data of the facets. An algorithm pursuing crack formation in the defined network in accordance to Section 4.2 can be given by the following instructions:

Loading loop Advance state of applied stress a. Determine tractions tj = Nja and average crack opening displacements < bj > an by eq. (62) for each of the (m - n) unbroken facets.

Facet loop Calculate "energy release" expression t~ < bj >. Allocate a crack if tjt < bj > >_ 47j.

End facets Update cracks n and facets ( m - n). Determine strain state e and elasticity matrix n for actual crack pattern.

End loading REFERENCES

1. J.R. Rice, Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in A.S. Argon (ed.), Constitutive Equations in Plasticity, The MIT Press, Cambridge, Massachusetts, and London, England (1974). 2. J.W. Dougill, On stable progressively fracturing solids, Journal of Applied Mathematics and Physics (ZAMP) 27 (1976) 423-437. 3. M. Kachanov, Elastic solids with many cracks: a simple method of analysis, Int. J. Solids Structures, 23/1 (1987) 23-43. 4. M. Kachanov, Effective elastic properties of cracked solids: critical review of some basic concepts, in V.C.L.Li (ed.), Micromechanical Modelling of Quasi-brittle Behaviour, Applied Mech. Rev., 45 (1992) 304-355. 5. I.St. Doltsinis and R. Handel, Modelling the behaviour and failure analysis of brittle microcracking materials, Proceedings, ECCOMAS 96, John Wiley, 1996.

This Page Intentionally Left Blank

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

Fracture

prediction

125

of sheet-metal blanking process

Ridha HAMBLI 9Alain POTIRON 9Serge BOUDE and Marian RESZKA ENSAM CER-ANGERS; Laboratoire de G6nie M6canique et C.A.O 2 Boulevard du Ronceray - B.P. 3525 49035 ANGERS CEDEX

1. I N T R O D U C T I O N In industrial processes, the making of thin mechanical parts which requires costly tools and machines, is widely used. A modern way to decrease the development's costs is to implement a numerical simulation. In the case of sheet-metal forming, the process involves complex sollicitations of the material and many physical phenomena, such as hardening and damaging, may occur leading to modifications of the material's behaviour. In some processes as blanking, shearing and punching, the rupture of the sheet is wanted. Consequently, during the numerical simulation, a mechanical behaviour model will necessarily account for damaging and will include several failure criteria. This allows for a more realistic outlining of the industrial process from its starting point up to the final breaking of the part. Moreover, prediction of the spreading damage defaults inside the material and the final geometrical shape of the blanking part, will be well predicted. The aim of this work is to provide a quite general finite element model allowing for the numerical studies of structures, subjected to damage and ductile fracture. In order to meet this goal, the best suited models describing the whole blanking process, will finally be used. 2. NUMERICAL MODELLIZATION OF DAMAGE AND DUCTILE FRACTURE Inspection of the last studies in the field of industrial process simulation and despite the increasing progress reached in numerical computing simulation, there appears to be a lack in the prediction of the blanking process including damage and ductile failure phenomena. Recently, Clift and al (1990) [1] carded out a comparative study by means of finite element models including failure of the material, in order to investigate the metal forming problems. The fracture has been numerically predicted by means of several criteria described in literature. Comparing the numerical and experimental results they concluded that some

126 literature. Comparing the numerical and experimental results they concluded that some criterion which are well suited to simulating the behaviour of the material for some particular process and geometry, cannot efficiently be used in other technological cases. In conclusion, there is not a universal failure criterion which allows for the description of any industrial metal forming process.

2.1 Ductile fracture criteria In order to predict the possibility of a structure to undergoing rupture, numerous authors have proposed their own criterion. In the isotropic case, these failure criteria are scalar functions invol'~ing stresses and/or strains, depending on some physical and mechanical parameters. These are usually identified by performing rheological tests on specimens. Particularly the failure is represented by a mathemetical function which is supposed to represent the physical behaviour of the material, and occurs when it reaches a critical value Cc. The mathematical formulations are often written in the form of: If

fO f ( o, eeq ) deeq - Cc < O, there is no failure.

(l-a)

If

f ~ f ( O, eeq ) deeq- Cc >_O, the failure occurs.

(l-b)

In the above expressions, e is the total strain, eeq is the equivalent strain defined by means of plastic part eeq

-

V3-

Different failure criteria which can be applied to predict the ultimate elasto-plastic behaviour of structures subjected to external forces, have been detailed in [2]. It is shown that these criteria are not suitable for describing the gradual material-degradation initiating the first cracks. Consequently, theoretical formulations have been developed by means of elastoplastic laws coupled with continuous damage, overcoming the aforementioned inefficiencies. With this in mind, we are reminded of the work of Lemailre and Chaboche [3].

2.2. Lemaitre behaviour law coupled with damage [3] As a matter of simplification, the damage will be considered as isotropic and depending on a scalar value D. In the unidimensional case and for an elemental domain, D is defined as the ratio between the default and apparent areas. As a result, the material damage state corresponds to the following situations: a- If: D = 0 then there is no damage. b- If 0 _ D ___0 the predicted stress value falls into a non permissible region and a plastic correction is needed. The corresponding algorithm is the two-step scheme elastic-prediction plastic-correction, involving the orthogonal projection P(oT+I)

on the yield surface.

This has been implemented into the F.E. code ABAQUS [4] by means of the user's routine UMAT. 2.4. C rack initiation and propagation During the structural analysis, it is supposed that cracks initiates at some points of the material where the damage value reaches a critical value Dc. Consequently, the damaging parameter D is set to a value DR near 1. Therefore, it can be seen from (3) that the material

stiffness value falls to zero and in the finite element modellization, the stiffness of the elements belonging to that region is negligible. In this way, the failure-of the finite element occurs in the vicinity of the critical points and the crack grows and follows the damaged material domain, characterized in the mesh by the elements with the lower values of the stiffness. 3. N U M E R I C A L

TEST

A fiat plate subjected to axial loading has been studied as a testing model for the numerical algorithm which has been developed. The EE. model includes just one rectangular eightnodes element. Clamped on one side, it is subjected to a prescribed displacement U on the other side. Geometry and boundary conditions are depicted on figure 1. The material's elastic parameters are taken to be the Young'modulus E = 210000 Mpa and the Poisson's ratio v = 0.3. For isotropic hardening, the equation is:

eq~ o0 = oel + K (epl

with the values of:

oel =200 Mpa ; K = 480 Mpa ; n = 0.406. Accounting for Lemaitre damaging law (6), the material parameters take the following values: e D = 0 . - e R = 1. - O c = 0 . 3 5

129

I

~

U 9 "1

t

/

b

~-H--~ Figure 1- Tensile test of a quadrangular finite element with eight nodes. Performing all the aforementionned steps of the numerical algorithm leads to the following conclusions, depicted on figures 2a-b. Accounting for the material damage D, or not, results in two different curves drawn on figure 2a. These curves correspond to the equivalent stress evolution Oeq versus equivalent total strain et- If the damage is taken into account, the failure of the structure can be predicted. Otherwise, the structure would never fail. 800

r~va~ aress(lVl~

700

~7

0,8

J

J

(:J30

j f

500

0,6

300

0,4

200 o2 1130 .

o

.

.

.

i

.

0,2

.

.

.

i

0,4

.

.

.

.

,

0,6

-a-

.

.

.

.

;

0,8

.

.

.

.

,

1

,

9

,

9

i

1,2

0

,

0

,

02

,

,

,

0,4

0,6

,

I

,

i

0,8

-b-

Figure-2- Equivalent stress and damage evolution versus equivalent strain. As it is shown on figure 2b, when the equivalent strain exceeds the value eR, the damage increases rapidly inducing a decreasing stress due to the reduced stiffness of the material, as it can be seen on figure 2a. Taking these results into account, we can conclude that the algorithms we have implemented, allow for a good prediction of the material behaviour and damage, leading to the complete rupture of the structure without computing divergence. In the following, the blanking process simulation of a XC 60 sheet-metal will be described by employing the previous numerical method.

130 4. S H E E T - M E T A L B L A N K I N G S I M U L A T I O N

Among the industrial processes dealing with large elasto-plastic deformations, the sheetmetal blanking simulation is one of the most hard to perform, due to the difficulties arising from a right description for the damage evolution, the crack initiation and its propagation throughout the material. In former works and especially [5], the complete material failure of the sheet could not be attained and consequently, the numerical results don't match experiments. In order to overcome these difficulties, we propose the following approach. 4.1. Modellization of damage and failure mechanisms The sheet-metal blanking process on press, has been investigated by many researchers. Recently, we have pointed out by means of an analytical stud)" and experimental tests, that the physical mechanisms leading to the complete failure of the sheet material can be described as lbllows: Firstly, a crack initiates at the cutting edges A and B of the t ~ l s (figure 3) due to the

penetration of the punch into the sheet. Secondly, this crack goes on the region where eeq overpass a critical value and cut progressively the material fibres one after another.

~~n~h

-

fibres

~/~~

~ Sheet -a- initial

-b- crack initiation

-e- crack propagation (fibres cutting)

Figure-3- Rupture of the sheet from [2]. Experiments on technical devices equipped with electrical gauges and a force transducer, were performed on a 4000 KN hydraulic press and we have implemented different failure modellizations in the algorithm. Having analysed the results, it was found that the model of LemMtre was the more appropriated to describe the progressive degradation of the material, the complete rupture of the sheet being reached in a realistic manner.

131

4.2- Numerical blanking simulation of a XC 60 steel sheet The axisymetric blanking operation of circular parts has been choosen as experimental work. All the geometrical parameters are shown on figure 4-a. Accurate experiments performed on the sheet material have given the following values: Young's modulus E = 200000 M p a , Poisson's ratio v =0.3 The hardening law is characterized by the values of: Oel = 250 Mpa , K = 1045 Mpa, n = 0.194. The damage coefficients a~sociated with Lemai'tre's model are: eD = 0. - eR = 0.8. - Dc = 0.37 The meshing of the model involving 1400 quadrangular four nodes axisymetric element, is depicted on figure 4-b.

4.3.

Results

The computations corresponding to different steps of the punch penetration, figures 5-a and 5-c, show the crack propagation inside the mesh.

132

ABA@L0 Punch

! -a- punch penetration = 40%

,2

-b- punch penetration = 42%

numerical profi!

i ~ -V ~/i

I

,-~,1

!

/experimental profil ~

I\

I ! I

r

-~- punch penetration = 45%

-d- Complete rupture of the sheet (punch penetration = 65%)

Figure-5- Numerical prediction of the rupture in the sheet Despite all non-linearities arising from contacts between the sheet and the tools and the elasto-plastic behaviour of the material, the results enlighten the robustness and the fiability of the algorithm. A Coulomb's law has been choosen in the contacts. It can be seen that the distorsion of the mesh has no influence on the results accuracy because the distorted dements vanish during computation. From the several failure criteria which have been implemented in the F.E. code, particularly the Gurson's criterion, it has been found that the better results are deduced from the Lemaitre and Chaboche approach. Consequently the sheet-metal blanking process simulation would be best predicted using a constitutive damaging law coupled wiyh elasto-plasticity.

133 As the optimal choice of the press and corresponding tools is always an industrial goal, we have computed the punch force vs. the punch displacement, occuring all along the blanking process. On figure 6, in the case of an optimal clearance - 10% of the sheet thickness, two CUl-Ves are drawn. They are respectively, the numerical prediction depending on all aforementionned models and algorithm, and the experimental results. 350

Fax

lii

s ~

300I j~,

250-

a-

Ip--=,ql-,,-o ~

200.

r

Punch

Die 50-

an~rmmme/~ O~ 0

,

I

I

I

I

I

I

I

i

10

20

30

40

50

03

70

80

Figure 6 Punching forces vs. punch travel

Figure 7 D a m a g i n g map

It can be viewed that the more realistic description corresponds to a damaging material model and consequently it can be concluded that the blanking process would necessary account for damage.The failure of the sheet material is obtained for a punch penetration of about 65% of the sheet thickness. The resulting computed damaging field is given on figure 7, on which it can be observed that the damage is restricted in the neighborhood of the crack line and mainly apparent in the gap between punch and die. 5. C O N C L U S I O N The numerical studies which have been presented in this paper, are based on robust and efficient algorithms. The experiments have shown the computed results' accuracy. It has been shown that the damaging and ductile failure phenomenon occuring during a sheet-metal blanking operation must be accounted for. Then the correct description of the general elastoplastic behaviour of the material is achieved. The crack initiation and propagation can accurately be predicted without computational divergence. From the moment of the crack initiation to the complete rupture of the sheet-part, all along the process simulation, experimental and numerical results are always in good agreement. These algorithms and constitutive laws seem also to be siuted in the case of extrusion and plastic forming.

134

REFERENCES [1]- C L I F F S.E., H A R T L E Y P., S T U R G E S S C.E.N. and R O W E G.W. " Fracture prediction in plastic deformation process" Int. J. Mech. Sci. Vol. 32, n ~ - 1990 - p: 1-17 [2] H A M B L I Ridha " Etude exp6rimentale, num6rique et th6orique du d6coupage des t61es en vue de l'optimisation du proc6d6 " - Th~zsc de Doctorat - E N S A M d ' A N G E R S - 15 Oct. 1996 [3]- L E M A I T R E J . , C H A B O C H E J.L. " M6canique des mat6riaux solides " - Dunod - 2/~me 6dition - 1988 [4]- A B A Q U S

- HKS

9 Manuel d'utilisation - version 5 . 4 - 1

[5]- H O M S I M., W R O N S K I M. et R O E L A N D T J.M. " Mod61isation num6rique de la coupe " - S T R U C O M E - 1994 - p: 677 - 690 [6]- P O P A T P.B., G H O S H A. and K I S H O R E N . N " Finite element analysis of the blanking process" Jour. of Mech~W-6rking Techn. p" 2 6 9 - 282

- 1989.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

135

Elastic-plastic f i n i t e - e l e m e n t modelling of metal forming with d a m a g e evolution E Hartley, ER. Hall, J.M. Chiou and I. Pillinger Solid Mechanics & Process Modelling Research Group, School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, Birmingham, B15 2TI', UK.

ABSTRACT A model for damage accumulation proposed by Lemaitre has been incorporated in an elastic-plastic finite-element simulation of the plane-strain side pressing of an aluminium rod. Predictions of failure site agree closely with experimental observations. The level of deformation at which failure is predicted shows a small difference to experiment. Further predictions of failure were undertaken with variations on the basic Lemaitre model, (i) with no damage accumulation permitted for compressive triaxiality, and (ii) with an exponential dependence on triaxiality. When combined with a suitable fracture criterion the latter model showed very close correlation with experiment. 1. INTRODUCTION The recent advances in computer simulation techniques combined with models for damage evolution has provided the opportunity to analyse in detail material behaviour in forming processes and how the evolution of damage affects the integrity of a formed product. There is however considerable debate on the most effective damage model to use. In this paper the Lemaitre damage model is used [1,2] together with modifications proposed by Chiou [3]. Each of three different models are incorporated in simulations of the plane-strain side pressing of a circular section workpiece, see figure 1. This type of test, investigated initially by Jain and Kobayashi [4], has proved very useful in assessing finite--element models incorporating ductile fracture criteria [5] or damage models [6]. This is largely due to the consistency with which fracture initiation sites can be identified and related to the initial geometry of the workpiece. The circular section workpiece used here will always develop a fracture at the centre of the cross-section. 2. FINITE ELEMENT FORMULATION The elastic-plastic thermo-mechanical finite-element code (epfep3) used for the modelling employs an updated-Lagrangian approach with a large displacement finite-strain formulation [7,8]. The material is assumed to be isotropic and the constitutive expression is derived from the Prandtl-Reuss flow rule and von Mises' yield criterion. A radial return algorithm [3,9] is used for return mapping to the yield surface.

136

Figure 1. Plane strain side pressing of a circular section workpiece. 3. DAMAGE MODELS Damage models are used to represent the effect of plastic deformation and progressive deterioration of the material. Often damage models may be used in conjunction with some local criterion to predict ductile fracture. Numerous damage models have been proposed, for example those of Rice and Tracey [10], Gurson [11], Tvergaard [12], Rousselier [13] and Lemaitre [1,2]. In this paper the focus is on the incorporation of the Lemaitre model in an elastic-plastic finite-element formulation. The damage evolution equation is shown below (this is referred to here as the L model). /)=

D~ 2 e,~ - e~ [-5(1 + v)+ 3(1- 2v)(

)Z]pZ/Mp

(11

where v is Poisson's ratio, M is the Ramberg-Osgood hardening exponent, ER and ED are the uniaxial rupture strain and uniaxial damage threshold, p is the generalised plastic strain and Dc is the critical damage value Two variations on Lemaitre's model are also investigated. In the first case the effect of hydrostatic compression is neglected (referred to as the LS model), giving the conditions below. 15 = e,~D~ 2 + v) + 3(1 - 2V)(O~q ot~ )2 ]p2/M p - e~9 [3(1

when on O,q

>0

/)

when~

1 we may neglect the two t e r m s in ( ~ e ) 2 to give the following closed solution to ~ v . e~.e = (z/2w){C)w[(1 + l/to) - Q] + 0[(1 + 1/r~9) - 11- 2nr (1/r o + Q/rgo)} (1/ro + Q/rgo) + 90 + l/to) - (2] + 010(1 + 1/rgo) - 1]}

(27)

In calculating ~ v from Eq.(27), the input data required are: r o , rg0 and r45, the Hollomon contants n, A, the initial sheet thickness, t o, and the ratio between the lengths of the minor and major ellipse axes; a . = B / A < 1. For a given orientation, the radii of c u r v a t u r e (9b,c) and the stress ratio (10) are calculated from an a s s u m e d dimensionless height (h = H / A ) . Then, w, r and z follow from Eqs (19a,b,c) and (23). The equivalent strain is also found from Eqs(16a) and (21b) as: H

~v= z f dH/R~ = z 0

In (1 + h 2

~at2 )

(28)

If e~e from Eqs (27) and (28) do not agree then h is incremented and w, r and z are re-calculated until the difference in ~i v becomes negligible. The equivalent stress at instability t h e n follows from Eq.(5). Eqs (24b,c) allow ~ v to be converted to surface strains, e.l~e ' a n d e22v ' aligned with the ellipse axes. There also exists a shear s t r a i n y12P ' w h e n 0 ~ < 0 < 90" which is found from combining Eq.(18c) with Eq.(24b): Y12 v ' = g e ~ v ' = ( g / z ) ~ v. The three strain components: e l f ', %2v ' and ~q2v ', are sufficient to d e t e r m i n e the principal, in-plane pole strains. The t h r o u g h - t h i c k n e s s strain follows from the incompressibility condition: e3v = - (e I f '+ e22v '). 3.4. P r e s s u r e v e r s u s b u l g e h e i g h t The analysis of instability has been associated with m a x i m u m pressure being reached. This leads to a theoretical prediction to the p versus H curve which m a y be compared to a test recording. Using h = H / A and a r = B / A , to normalise Eqs (9a,b,c), the radii of c u r v a t u r e become: r = Rr

(h : + arz)/2h and r o = R o / A = (h 2 + 1)/2h

p = R / A = r 0 % [%+(.1 + 1/r90)r 0 ]/ro[r~+ (1 + llr9o~r 0 ]+re [to+(1 + 1/ro)r~ ]

(29a,b) (3o)

241 Eq.(28) supplies the equivalent pole strain from which ~ is found from Eq.(5). The relationship between p and h requires the current pole thickness, t to be found. Noting that s3P = ln(t/t) and using Eqs(24a) and (28): (31)

t= to exp [(w/rz)g el = to(1 + tiZ/a~.Z)~'~ Combining Eqs (31) and (20a) with Eq.(8a) gives:

(32)

p = t a l A p x = (~tolApx)(1 + h21~2) ~

where w, r and x follow from Eqs (19a, b), and (22) respectively, given Eq.(lO).

Q from

4. E X P E R I M E N T A L Table 1 gives details of the test materials: annealed 18/8 stainless steel, as-rolled 60/40 brass sheet and soft CR1 steel sheets with and without zinc cladding. Bulge tests were restricted to orientations of 0 = 0 ~ and 90 ~ in dies of aspect ratios; a r = 1.0, 0.89, 0.78, 0.63, 0.50 and 0.42. The length of the major axis of all die apertures was 100 mm. The bulge height H was measured by positioning an lvdt transducer vertically above the pole. The lvdt was supported on two legs giving 2A = 50.8 m m i n contact with the major axis. The voltage outputs from the lvdt and a pressure transducer were plotted together during each test to establish a continuous p versus H curve through to failure. We have seen t h a t the calculations require the K and n - values and the three r - values for a material. These were determined from tension testpieces machined at orientations of 0 ~ 45" and 90 ~ to the rolling direction. Dimensional changes to the parallel length and width dimensions were measured following unloading from the plastic range using 10 increments of increasing plastic strain. The r - values and the Hollomon constants as found from true stress and n a t u r a l strain plots are given in Table 1. Table 1 Tensile properties of four sheet materials Material

Stainless Brass C-Steel Zn-plated

toirrm

steel

steel

0.50 0.40 0.78 0.81

r o

1.00 0.89 2.17 1.70

r45

rgo

K

1.00 0.73 1.47 1.86

1.00 0.59 2.20 2.02

1260 650 600 600

n

0.42 0.20 0.21 0.25

Note that for the present predictions average Holloman constants K and n in Table 1 were used to described both bulge flow and tensile flow where e x a c t equivalence was not achieved.

242 The pressure versus height curves for dies of various aspect ratios are given in Figs 2 - 6 . Stainless steel (Fig.2) shows a maximum pressure particularly for dies of lower aspect ratios. 200 -

/

bor

4.9 4-~ § +~+'~

150-

-

,oo-

,r

/2

S~ )I

so-

,,~ ~_.~*-

, ~+ i~ x , X

.fi(. ~ .'~

_/- /_.o~.-

-x-

..ISy~-bo~

. /I~,o

xh

./

l9

X

I~

+

~

o

I---1

8 ,I o.~Io,~ o.~, o.~o o.~i,~~

f

22~" 0

I

I

2

i+

h/ram

I

1

6

8

Figure 2. Pressure versus H for ellipsoidal bulging 18/8 stainless steel. In contrast, brass (Fig.3) failed on a rising pressure curve. A pressure plateau appears for bulging CR steel in all dies (Figs 4 - 6). Theoretical predictions from Eq.(32) for extreme aspect ratios, contain all the observed curves, reproducing their shape but yielding some error in the maximum pressure attained.

/

80

P bar

60 9

.§ .. ,a'~ §

-~

~+/+ 4.,,§

r

+I + ~

_.qyx~

20

15 2.

hlm m

5SO

Figure 3. Pressure versus H for ellipsoidal bulging 60/40 brass.

243 We see from Eqs (20) and (21) that the theory depends upon equivalent pole flow being achieved when bulging sheet through dies of all aspect ratios. The K and expressions for @ = 0 ~ and 90 ~ orientations in the 100 mm die tests did not produce a unique flow curve for each material investigated [1]. ~ 200 -

P

100

I

I

2

~

h/mm

I

I

6

8

Figure 4. Pressure versus H for ellipsoidal bulging CR1 steel. Of these, isotropic stainless steel exhibited the least divergence in its bulge flow behaviour at large strain. It appeared for all materials t h a t pole flow was restricted by the near die wall when bulging through the dies of low a .

200 ~

P bor

- - - - - - - - - -

o

P

100

I

0

2

~

I

h/ram

6

Figure 5. Pressure versus H for bulging zinc-plated CR1 steel r = 0~

244 Also, as the radii of curvature decrease with increasing pressure, ultrasonic thickness measurement taken at the sharply curved pole become unreliable.

200 0

~

P bar ..+."+

§ +

"~§

"-§ ~ ' §

~W

+-

100

0

4

hlmm

6

8

Figure 6. Pressure versus H for bulging zinc-plated CR1 steel (0= 90"). A preliminary investigation of bulging Zn-plated steels through larger 180 mm dies has achieved the desired equivalence and these will be employed for a future appraisal of the theory presented here.

REFERENCES ~

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

D.W.A. Rees, Int. J. Mech. Sci., 37 (1_995)373. H.W. Swift, J. Mech. Phys. Solids 1 (1952) 1. R. Hill, Phil. Mag., 41 (1_950) 1138. D.W.A. Rees, J. Mats Process Tech., 40 (1994) 173. J.H. Hollomon, Trans AIME, 162, (1945) 268. N.A. Weil and N.M. Newmark, J. Appl. Mech., 22 (1955) 533. A.B. Haberfield and M.W. Boyles, Sheet Met. Ind., 50 (1973) 400. W.F. Brown and G. Sachs, Trans ASME, 70 (1948) 241. P.B. Mellor, J. Mech. Phys Solids, 5 (1956) 41. A.N. Gleyzal, Trans ASME 70 (1948) 288. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 10 (1968) 157. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 10 (1968) 143. J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 9 (1967) 681. M.I. Yousif, J.L. Duncan and W. Johnson, Int. J. Mech. Sci., 12 (1970) 959. D.W.A. Rees, J. Mats Process Tech., 55 (1995) 146. D.W.A.Rees J. Strain Anal., 30, (1995) 305.

FORMABILITY CHARACTERIZATION

This Page Intentionally Left Blank

Advanced Methods in MaterialsProcessingDefects M. Predeleanuand P. Gilormini(Editors) 9 1997 Elsevier Science B.V. All rights reserved.

247

C o m p r e s s i o n o f a b l o c k b e t w e e n cylindrical dies and its a p p l i c a t i o n to the workability diagram S. Alexandrova, N. Chikanovab and D. Viloticc alnstitute for Problems in Mechanics RAS, 101 Prospect Vernadskogo, Moscow 117526, Russia* bDepartment of Applied Mechanics, Bauman Moscow State Technical University, Baumanskaya 5, Moscow 107005, Russia

2 nd

clnstitution for Production Engineering, University of Novi Sad, V.Perica-Valtera 2, Novi Sad 21000, Yugoslavia 1. INTRODUCTION In engineering practice two methods are commonly used for modeling technological processes of metal forming, the slab method and the upper bound method (see, for example, [1, 2]). Despite the recent development of the upper bound method [3, 4], its applications are restricted to the special friction laws and material models for which the necessary variation principle is proven. On the other hand, the slab method often gives reasonably good predictions for average stresses. However, this method does not permit the description of the kinematic variables, which are of great importance for initiation and growth of microdefects during technological processes. Therefore, in the present paper Hill's method [5] is used to determine the stress-strain components in the deforming region during upsetting performed with two identical cylindrical dies. The material is assumed to be rigidplastic, hardening. The friction law proposed by Carter [6] is applied on the die surface. This problem was investigated by Vilotic and Shabaik [7] using the slab method. Brovman [8] found a slip line solution for a rigid perfectly plastic material. The results of calculations and experimental data are used to find a point on the workability diagram introduced by Vujovic and Shabaik [9]. The experimental determination of stress distribution on the die surface was performed by the method described by Vilotic [10] and Plancak et al. [11]. Steel specimens (0.35 percent C) with initial height 18 mm and length of 40 mm are used in this study. The stress - strain curve for the material is determined experimentally using tension and compression tests according to the Rastegaev method, the results are approximated in the form of the Ludwick relation. The upsetting experiments with cylindrical dies were conducted on a Sack and Kieselback hydraulic press with a capacity of 6.3 MN. *Present address: Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069-0001. The main part of this work was performed at GKSS Research Centre, Geesthacht, Germany.

248 2. VELOCITY FIELD Due to Hill's method it is necessary to choose a kinematically admissible velocity field satisfying the kinematic boundary conditions. Moreover, this velocity field must satisfy the incompressibility condition. For the process under consideration (Fig.l) it is natural to adopt a bipolar coordinate system. The transformation equations for this coordinate system (see,

rigid-plasticd~=~obOUndary]D[ U E rigid zone

vo

vo

v=v0 die surface

Figure 1. Geometric representation for upsetting. for example, Flugge [12]) are:

x = Asinhg,/(coshg, +cos~);

y = Asin(~/(coshg, +cos~)

(1)

and the components of the metric tensor and the Christofel symbols of the second kind are given by

g ~ = gr162= A2/(cosh V/+ cos r ~, F~,~ = F~ = - F # _

_

sinhg, cosh~ +cos~'

(2) ~, ~ = F~ = sin~ F~,~=-F~,~, # cosh~, + c o s ~

(3)

where A is the distance between the die centers, a function of time, t. It follows from the geometrical considerations (Fig. 1) that

A = Rsinh 9'o,

H = R(cosh 9'o - 1)

(4)

249 where R is the die radius and W0 is the W-coordinate of a line coinciding with the die surface. Let us assume that the rigid-plastic boundary coincides with a coordinate line of the ~b-family, ~ = ~0, which passes through a point N (Fig.l). This coordinate line is determined by the equation

cOSr

= Asinh ~o I H

o

(s)

-cosh~ o

where H0 is the height of the undeformed specimen. The outward normal to the tool surface is given by

(cosh ~'o + fi = -

cos

~)~

A

1 + cosh ~o cos ~_ -

A sin

cosh ~o + c o s ~ / + R(cosh ~o + c o s ~) }

(6)

where v0 is the given magnitude of the tool velocity, v0 > 0. The velocity vector of the tool in terms of Cartesian basis is 9 = -Voi from which it follows that v . n = v n = -Von x

(7)

On the other hand, in bipolar coordinates we have v

9

n

=

Vn

:

v~n ~ + V2 n 2

(8)

Here and in what follows, 1 corresponds to the w-direction and 2 to the d~-direction. Combining (7) and (8) and taking into account (6) one can obtain the boundary condition at W = W0

v~ = - voA(1 + cosh ~o cos ~)//(cosh ~o + cos ~)2

(9)

Analogously, on the rigid-plastic boundary at ~ = ~o

u : v~(cos,,,,,§247

(10)

where u is the velocity of rigid material moving along the y-axis (Fig.l). The incompressibility condition in bipolar coordinates has the form A ' , / v ~ + A'2/d~ = 0

(11)

To satisfy boundary condition (9) and the condition on the symmetry axis, Vl = 0 at W = 0, we assume

v~

=

- ~(~, ~o)V0~0 § oo~. ~o oo~ ~)/[~(~o, ~o)(CO~. ~o § oos~) ~]

~1~

250 where F(W,~0) is an arbitrary function of ~ and ~0 satisfying the condition F = 0 at ~ = 0. Substitution of (12) into (11 ) gives

v2=

F'(~,~o)voAsin~/[F(~o,

~o)(cosh ~'o + cos ~) 2] + C(~, ~o)

(13)

where F' = ct=/c~ and C(~,W0) is an arbitrary function which is determined by virtue of (10)

ruo+cos,

v~

c~ COS ~0

~'~

~~

}

(14)

- F(~o

The magnitude of u can be obtained from the incompressibility condition in integral form

~vknkdl =0

(15)

L

where L is a contour bounding the plastic zone, CNBO (Fig. 1), dl is the length element, n k is the outward unit normal to L. From symmetry, vl = 0 at t~ = 0 and v2 = 0 at t~ = 0. Therefore, substitution of (10) and (12) into (15) leads to u = v o sin ~o/sinh ~/o

(16)

Then, C(~,~o) can be rewritten as

C(~',~'o) = Avo sinr

1+ cosh ~cOSr )2 sinh~,o(cosh~, +cOSr

'o)(c~

+ cos #o)

(17)

Thus, expressions (12), (13) and (17) determine a kinematically admissible velocity field involving one arbitrary function F(~,~0). The components of the strain-rate tensor can be found by means of tensor analysis sinh ~,

sin~

~ =v~,~ + cosh~ +cos~ v~ + cosh~ +cos~ v2;

oc'22 -- V2, 2

-

-

sinh~ sin~ cosh ~, + cos ~ v~ - cosh ~, + cos ~ v2;

Vl, 2 -I- V2,1 ~12 =

sin~

sinh ~,

cosh ~ + cos ~ v~ + cosh V/+ cos ~ v2

(18)

251 3. THIN PLATES If the plate is sufficiently thin that ~o 2 (( 1 then analysis is essentially simplified. We put sinh~0 ~ ~g0, sinh~g ~ ~g, cosh~g0.~ 1, cosh~g ~ 1

(19)

Then the geometrical relationships (4) take the form

A = R~' o, H = Rgo 2//2

(20)

As F(~,~g0) must be an odd function of ~g, we take F(~g,~g0)=V Ago

(21)

Using (2), (19) and (21) the expressions for velocities (12) and (13) and for strain-rates (18) in terms of physical components become v~ = - V o g / / ~

=-.,

o,

v~ = v o

sin~/~,o

(22)

cos )cos /(A o)

v~ [3sin~+ sin ~~(c~

e~,~ - 2A go

~~(1+ - c~ cos~~ ~o)?)(1+ cos ~) 2

1

(23)

4. SISTEM OF EQUATIONS Due to Hill's method [5] it is necessary to choose orthogonalizing motion. Since the kinematically admissible velocity field (22) does not involve any arbitrary functions and, on the other hand, this field is a solenoidal one, orthogonalizing motion must contain one arbitrary function corresponding to the hydrostatic stress, or. This stress is an even function of ~g therefore we assume c = cr(~,H)

(24)

to the leading order. Hence, a comparable orthogonalizing motion in the deforming zone may be chosen in the form

v; = o. v; = v(o..)

(25

where v is an arbitrary continuous function of ~ and H satisfying v(0,H) = 0. According to (20) the independent variable H may be replaced by ~o and vice versa. Applying the

252 Hill's method with the orthogonalizing motion (25) and the boundary conditions on the surface ~ = 0 cry,, = 0

at

~ =0

(26)

and on the surface ~ = 0 ~v, = 0

at

~ = 0

(27)

and on the surface ~ = ~0 (friction surface) where the friction law proposed by Carter [6] is adopted i

i

rt =-o" w =/~Is~,I

(28)

we obtain the approximate equilibrium equation in the form

2S~ sin cosh~, + cosr

d~ ~'o + ~L or

=0

'

(29)

_1_~=~, 0

where the upper sign "~" denotes the stresses calculated from the velocity field (22) and from the constitutive equations; S~, and Sr are the deviatoric portions of the normal stresses. To determine the boundary condition for o, we integrate the equilibrium equation over the rigid zone BNED (Fig. 1). Then, ~r

cr=-~

d~

at

r162

(30)

~o The yield criterion under plane strain condition is \/ 2

2

S~ - S~ } + 4cr;,~ = 4k

2

(31)

where k is the shear yield stress. Its dependence of the equivalent strain % has been determined experimentally using tension and compression tests according to the Rastegaev method. The results can be approximated by the equation k = 213.45 + 385.87 eeq 0"38 MPa

(32)

The equivalent strain rate is given by

[

Oeq = 2(6~,G~, + ~6~ + 2 % %

(33)

253 Applying the associated flow rule we obtain from equation (31) that (34) Eliminating ~ from (34) we find (35) Then, taking into account (23) and (19) we obtain from (35)

S~,=-S~=-k,

cos~)2cos~ sinr176176 ~~162162176 o'~,~=k~,(l+

2) + 3sin~

]

(36)

and from (33) eeq -" Vo COS~l

4- COS ~)/(%~-")

(37)

Due to (36) equation (29) takes the form

~,0rolk

2k sin r 7_,

(38

to the leading order. Taking into account (32), equations (37) and (38) constitute the closed form set with respect to o and eeq. Since eeq should be an even function of ~, it does not depend on W to the leading order. In this case integration with respect to ~ in (30) and (38) may be done. Using (36) it leads to

dcr cik 2k~o sin r ~'O-d--~-+~-~o+ l+cos~ -/zk=O

(39)

o = -k

(40)

at

~ = ~b0

5. SOLUTION We begin with equation (37) which does not contain o. This equation can be transformed to ~eq

--~--

(1 q- COS ~)sin ~ ~e,eq 2H

---~- +

(1 -!- COS ~)sin 4r3H

=0

(41)

The exact initial condition to this equation is eeq = 0 at H = H0. However, the adopted approach does not allow one to consider the initial stage of the process when the plastic

254 deformations are localized near the die surface. Therefore, neglecting this stage we assume the initial condition in the form eeq = 0

at

h

(42)

= cz

where cz = const and h = H/H0. The general solution of equation (41) can be obtained by the characteristics method in the following form

# 1 ;)]sin eeq = ----~ln{OIhtan ~exp(~tan2

r

(43)

where 9is an arbitrary function of its argumentand htan[~/2)exp[O.5tan2[~b/2)]=C / _ \

r

/ _ x l

defines the characteristics of equation (41). The characteristic curves are shown in Fig.2 (except the line AB which shows the coordinate of the moving rigid-plastic boundary found from (4) and (5)). The function 9 is determined by the different analytical expressions on the left and on the right sides from the characteristic passing through the point B (Fig. 2). On the left side, the condition (42) must be satisfied, then

~[a tan(#/2)exp(0.5tan2(#/2))]sin # =1

(44)

On the right side, the condition on the rigid-plastic boundary, where

eeq-" O,

leads to

010.5(1+ cos #)tan(#/2)exp(0.5tan2(#/2))]sin #: 1 h

(45) q)

A

10

0.7 0.6

/

4

Be 0.5

I

1

1.5

Figure 2. Characteristic curves and position of rigid-plastic boundary (AB line).

2 0.2

0.4

0.6

0.8

Figure 3. 9 as function of its argument (solid line from eq. (44) and dashed line from eq.(45)).

Functions tg(z), where z is their argument, are shown in Fig. 3. From the structure of equation (41) it follows that it should be treated near point ~ = O. If ~ is sufficiently small

255 quantity then the solution (43) may be rewritten as (46) Then the condition (42) leads to eeq = 2/~/31n(~/h)

(47)

in the vicinity ~ = 0. Thus, the analytical solution of the equation (41) is found. Hence, the magnitude of k can be calculated from (32). After that, equation (39) can be solved by the characteristics method. It is clear that H = const are the characteristics of equation (39) which is the relation along the characteristics. 6. A P P L I C A T I O N TO T H E W O R K A B I L I T Y D I A G R A M We apply our results to the workability diagram proposed in [9]. This workability diagram is defined by effective strain at fracture to be a function of the stress ratio 13, where [3 is given by (48)

[~ = 3~/~eq

For a Mises material

(Yeq ---

~/3 k. Therefore, we may rewrite (48) in the form

13= ~/3~/k

(49)

The upsetting experiments have shown that fracture initiates at the point ~ = 0 at h ~ 0.5. For this value of h equation (39) has been solved numerically at R = 50 mm, ~t = 0.3 and cz = 0.75 (this value of cz has been taken because the experimental observations showed that the plastic zone reaches the symmetry axis, x = 0, at this value of cz). After this the value of [3 has been determined from (49). At ~b= 0 we have obtained 13= -0.46

(50)

As we have said above, at the beginning ot~the process the analysis is not correct. However, the local deformations at ~ = 0 on this stage influence fracture conditions. We may take them into account approximately since the average effective strain at ~ = 0 is still given by (47) even if a = 1. In this case eeq ~ 0.8

at

h = 0.5

(51)

The values of [3 and eeq given by (50) and (51) determine a point on the workability diagram.

256 7. CONCLUSION Using Hill's method a semi-analytical solution for compression of a block between cylindrical dies is found. This solution combined with the experimental data determines a point on the workability diagram. ACKNOLEDGMENT The authors would like to thank Dr. P.A.Hollinjhead for his help with English. REFERENCES

1. W.F. Hosford and R.M. Caddel, Metal Forming: Mechanics and Metallurgy, Prentice-Hall, New York, 1983. 2. B. Avitzur, Metal Forming: The Applications of Limit Analysis, Dekker, New York and Basel, 1980. 3. A. Azarkhin and O. Richmond, Trans. ASME J. Appl. Mech., 58 (1991) 493. 4. B.A. Druyanov, Technological Mechanics of Porous Bodies, Clarendon Press, Oxford, 1993. 5. R. Hill, J. Mech. Plys. Solids, 11 (1963) 305. 6. W.T. Carter, Trans. ASME J. Eng. Mater. Technol., 116 (1994) 8. 7. D. Vilotic and A.H. Shabaik, Trans ASME J. Eng. Mater. Technol., 107 (1985) 261. 8. M.Y. Brovman, Kuznechno Shtampovochnoy Proizvodstvo, 9 (1966) 1. (in Russian) 9. V. Vujovic and A.H. Shabaik, Trans ASME J. Eng. Mater. Technol., 108 (1986) 245. 10. D. Vilotic, Ponasanje Celicnih Materijala u Razlicitim Obradnim Sistemima Hladnog Zapreminskog Deformisanja, Naucno-Obrazovni, Novi Sad, 1987. (in Serbian) 11. M. Plancak, A.N. Bramley and F.H. Osman, J. Mater. Processing Technol., 60 (1996) 339. 12. W. Flugge, Tensor Analysis and Continuum Mechanics, Springer-Verlag, Berlin, 1972.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

257

Sheet metal formability predicted by using the new (1993) Hill's yield criterion

D. Banabic a,b "Technical University of Cluj-Napoca, 15 C. Daicoviciu St., 3400 Cluj-Napoca, Romania b present address: Institut of Metal Forming, University of Stuttgart, Holzgartenstrasse 17, 70174 Stuttgart, Germany

The paper attempts to settle a mathematical model in order to predict theoretically Forming Limit Diagrams (FLD's) by using the Marciniak-Kuczynski analysis with the new orthotropic yield criterion (transversaly isotropic version) proposed by Hill in 1993. The model is solved numerically. The algorithm is very flexible, so that it allows to study the influence of various parameters upon FLD's: strain hardening coefficient (n), strain-rate sensitivity index (m), normal anisotropy coefficient (r), geometrical non-homogeneity coefficient (f), ratio between uniaxial and biaxial yield stress (o JOb) (a).

1. INTRODUCTION After the introduction of the Forming Limit Diagram concept by Keeler [1] and Goodwin [2], the research in the field of sheet-metal formability has focused mainly on the development of some mathematical models for the theoretical determination of FLD's. The first realistic mathematical model has been proposed by Marciniak and Kuczynski (M-K model) [3]. They have made the assumption that the strain localization, in case of the biaxial straining, appears in the region of a geometric non-homogeneity of the sheet-metal. Marciniak and Kuczynski have considered this non-homogeneity as a variation of the thickness (a notch) directed along the minimum principal stress axis. The M-K model has been used for analysing the influence of some material parameters (such as the strain hardening coefficient n, normal anisotropy coefficient r, strain-rate sensitivity index m etc.) or process parameters (strain rate, strain path, punch curvature, temperature, vibrations etc.) upon the shape and position of the FLD's. A survey of the research performed in this field has been presented in [4]. The limit strains computed on the basis of the M-K model (using the Mises yield criterion) are overestimated in the domain of biaxial straining and underestimated in the domain of plane straining. The shape and position of the FLD's are strongly influenced by the expression of the yield criterion used in the model. As a consequence, the above-mentioned drawback may be attenuated by considering an appropriate yield function. Several authors have tested this possibility by introducing different yield criteria in the M-K model: Parmar and Mellor [5], Marciniak and Ike [6], Lian [7] have used the criterion proposed

258 by Hill in 1979 [8]; Bassani et al. [9], Neale and Chater [ 10] have used the criterion proposed by Bassani [11]; Gotoh [12] has used the criterion proposed by himself [13]; Graf and Hosford [ 14,15], Padwal and Chaturvedi [16] have used the criterion proposed by Hosford [ 17]; Lian et al. [18] have used the Barlat - Richmond criterion [ 19] extended by Barlat and Lian [20]; Ferron and Touati [21] have used the criterion proposed by Budiansky [22]. Further information concerning these yield criteria and their introduction into the M-K model can be found in [23-25]. The above-mentioned yield criteria has been extended recently in order to make them more flexible. Thus, Weixian [26] has extended the Hill's 1979 yield criterion, Tourki et al. [27] have extended the Budianski's yield criterion, Chu [28] has extended the Barlat-Lian yield criterion.

2. HILL'S 1993 YIELD CRITERION Hill has proposed in 1993 a new yield criterion [29] for orthotropic sheets, valid in the first quadrant. This criterion is expressed by a function containing only quadratic and cubic terms. The principal stresses o~, 02 does not appear in the cubic terms. Thus, it is possible to express one of the principal stresses with respect to the other by means of a quadratic relationship. The expression of the Hill's 1993 yield criterion is O12 _ r 002

+ 022 + [(p+q) - (po,+qo2) ] 0,02

00090

0902

Ob

- 1,

(1)

00090

where: p and q are non-dimensional parameters that may be obtained from the following relationships: (___1+ I -- ___])p = 2ro(Ob-090) -- 2rgOOb + ~C, ~ 090 Ob (1 +r0)o~ (1 +rg0)o~0 ~

(1 + 1 _l)q

. 2rgo(Ob-Oo) . . . (1 +r90)o~0

O0 090 Ob

2roOb , C ; (1 +r0)o ~ 090

(2)

(3)

c may be obtained from the relationship c 00090

1 2

00

1

1

090

0b

(4)

01 and 02 are the principal stresses; 0 0 and 09o are the yield stresses determined by uniaxial straining along the rolling and transverse directions, respectively; Ob is the yield stress obtained by biaxial straining; r 0 and rgo are the anisotropy coefficients corresponding to the rolling and transverse directions, respectively. By rewriting En. (1) for the two cases of uniaxial straining (01 = o . o2 = 0 and o 2 = %0, o~ = 0), one may obtain different values for the anisotropy coefficients

259 ro * rgo even ifoo = 09o. The yield criteria mentioned in the previous section cannot describe such a situation. Assuming that oo = 09o = o= is the yield stress in uniaxial straining and ro = rgo = r is the normal anisotropy coefficient, the yield criterion (in a transversaly isotropic version) proposed by Hill may be rewritten as ol 2 -

where

p

2 2 = o~, o----;)oio2+ 022 + [(p+q) - (POl+q02)]OlO o

(2 - ~

and

q

(5)

may be obtained from the relationship 2

P

= q =

2 (1 +-"-~

-~)/(2 - ~ Ob 0b

-

(6)

Relation (5) is valid only in the case o~ > 0, 0 2 > 0. Its extension to other stress states imposes the cubic terms to be rewritten as

(plo~ I + qlo~l)o~o~

(7)

The following notations are useful when introducing the Hill's yield criterion in the M-K model" OI

t-

Ou

; o

a-

.

(8)

o

With this notations, Eqn. (5) becomes

012 + [p(1-t)+q-(2-a2)]olo2

02

+ (1-qt)o2 2 = o2,

1"I 0.6

0.4

0.2

o'o" o:.:/;' oo.4. 4 016" 11 0.2 O.6 ola' o.a

Eqn. (9) will be used for obtaining the relations between stress and strain increments in the flow theory (the so-called Levy - Mises equations) and the expression of the equivalent stress increment. Figure 1 shows a graphic representation of the yield locus (in a transversaly isotropic version) corresponding to the first quadrant (normal anisotropy coefficient is assumed r=l). One may notice that the yield curve moves towards the origin when the value of a is raised. For a = 1, the yield curve becomes the locus described by the von ~1.2 Mises yield criterion.

0.90

0.8

oI

(9)

Figure 1. The influence of the a parameter on the yield locus

260 3. THE MARCINIAK-KUCZYNSKI MODEL The M-K model assumes that the strain localization appears in the region of a geometrical or structural non-homogeneity. The model presented in this paper assumes the existence of a geometric non-homogeneity in the form of a notch (zone b) perpendicular to the direction of the maximum principal stress at. The initial thickness of the sheet-metal t'0 is greater than the initial thickness in the region of the notch tbo (see Figure 2). The sheet-metal is stretched by the principal stresses o I and o~. The current value of the nonhomogeneity coefficient is expressed by the relationship f-

Figure 2. Geometrical model of the Marciniak-Kuczynski approach

tt, to,

(10)

where t. and t b are the current values of the thickness in the regions a and b, respectively~

The basic equations of the model The following equations are valid for each of the two regions of the sheet: a) The yield criterion:

012 + [p(l_t)+q_(2_a2)]ol02 + (l_qt)022

2

(11)

b) The Levy- Mises equations: de I 2 0 l+[p(1

-t)+q-(2-a2)]o2

de 2

de e

2(1-qt)o 2+[p(1-t)+q-(2-a2)]Ol

2o~

(12)

The increment of the equivalent strain may be expressed from the equality of the incremental work done by the principal stresses and the incremental work done by the equivalent stress. c) The strain - hardening law:

o e = k (e0+e,)" ~m, where: K is a material parameter; e0 is a pre-strain; [~is the equivalent strain-rate.

(13)

261

d) Volume constancy condition: dl~ 1 + d ~ 2 + d ~ 3 = 0.

(14)

The model is completed with two equations expressing the link between regions a and b: - equation expressing the equilibrium of the forces acting along the interface of the two regions: o1~ to = Olb tb;

(15)

- equation expressing the fact that the strains parallel to the notch are equal in both regions: de2a = de2b-

(16)

In addition, the model assumes that the strain ratio in zone a is constant during the whole process:

de2a = O de1,,.

(17)

The model has been solved by using the numerical procedure presented in [30-32]. The determination of the strain and stress state in zone b has been made by using a double predictorcorrector algorithm (a detailed presentation of this alghoritm is in the paper [33 ]). 4. RESULTS AND DISCUSSIONS The mathematical model presented in this paper has a broad generality, which allows the analysis of the influence of several material parameters upon FLD's. Figure 3 shows the influence of the strain-hardening coefficient upon the FLD's. One may notice an increase of the limit strains when the this coefficient is raised. This fact is in good agreement with the experimental and theoretical results presented in [ 14, 34] etc. Figure 4 shows the influence of the normal anisotropy coefficient upon FLD's. According to this figure, the influence of the parameter r is very strong for the case ofbiaxial straining. One may see on Figure 4 that an increase ofr causes a decrease of the limit strains. This fact contradicts the experimental and theoretical results presented in [ 14-16] (Hosford's yield criterion is used in these papers). The curves shown in Figure 4 agree with the theoretical results presented in [34-36] etc. (Hill's 1948 yield criterion [37] is used in these papers). Figure 5 shows the influence of the strain-rate sensitivity index m upon FLD's. One may notice an increase of the limit strains when m is raised. A suggestive explanation of this phenomenon is presented by Marciniak and Duncan in an excellent monograph published recently [38]. The raise of the non-homogeneity factor f (which means the reduction of the thickness variations)causes an increase of the limit strains, as may be seen on Figure 6. This conclusion is in agreement with the conclusions of other papers dealing with the problem of FLD calculation [3, 14, 34, 35] etc. The parameter a defined by Eqn. (8) influences the FLD's shape and position in the manner presented in Figure 7. One may notice that the FLD moves downwards when a is raised. The phenomenon is more significant for the case of equi-biaxial tension. Figure 8 shows the influence

262

0.5 eps 1

e~sl

0.4 ............................................................................................................................ ~ i

!

0.3 ...........................................i..................................... ,',~

n-

0.6 .............................................................~............................................-.................~..........................................................................................

........................

i

i

......................................................................................................

0.4

............................................................. i ........................................................; ............................................................... i...............................

0.2 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.1

~ 1 , 5

0

0

02

0.l

01

0.3 eps

0.4

0.5

.

.

.

0

.

02

.

.

.

.

.

.

0.4

0.6

et~

2

Figure 3. Influence of the strain-hardening coefficient, n, on the FLD (r=l, m=0.01,f=-0.95, a=l)

Figure 4. Influence of the anisotropy coefficient, r, on the FLD (n=0.2, m=0.01,f=-0.95,a=l) 0.6. epsl

0"4i~1 0 . 3 t ............

: ................................... !

o.4t. ..................................~....................................!................................

0.2

"

.

"

0 . 3 ......................................-.................................................................................................. :..................................~...................................

...............................i . . . .

i....................................i ...................................i....................................i ..................................

o.1 ............................................

i....................................i ...................................i....................................i....................................

o.z 0.1

O

0.2

0.I

0.3

0

0.4

eps2 Figure 5. Influence of the strain-rate sensitivity index, m, on the FLD (n=0.2, r=l, f=0.95, a=l)

0.30e~l

0

0.I

02

0.3

0.4

0.5

0.6

eps2 Figure 6. Influence of the non-homoge-neity coefficient, f, on the FLD (n=0.2, r=l, m=0.01, a=l) 0.30

epsl

0.25

i

i

ia--

i

i

io. iio

0.20

0.10 0.05 0

.................... .........................................................................................................

0.05

0.10

~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.15

0.20

0.25

0.10

0.05

0.30

epsa

Figure 7. Influence of the parameter a on the FLD (n=0.2, ~ 1 , m=O.O1, f=0.95)

i]

84 .....................................................................................................................

...........................! .......................................................................................................................................................................................

0

0.80

'

'

'

'

0.85

0.90

0.95

1.00

1.05

1.10

1.15

a

Figure 8. Dependence of computed limit strains on the parameter a (n=0.2, r=l, m=O.O1, f=0.95)

263 of the parameter a upon the limit strain epsl for the case of equi-biaxial tension. One may notice that the variation is approximately linear. 5. CONCLUSION The paper presents the use of a new yield criterion (in a transversaly isotropic version) introduced by Hill in 1993 in the M-K model. The influence of different material parameters upon FLD's is studied. The influence of the yield curve shape upon FLD's is analyzed by using the parameter a (expressed as the ratio of the uniaxial yield stress and biaxial yield stress). The increase of a causes an approximately linear increase of the limit strain for the case of equi-biaxial tension. ACKNOWLEDGEMENTS This research was supported by the Humboldt Foundation at the Institut of Metal Forming, University of Stuttgart. The author are grateful to Professor Klaus Siegert (Director of this Institut) and Mr. Eckart Dannenmann for their help.

REFERENCES S. P. Keeler and W.A. Backofen, Trans. of the A.S.M., 56 (1963) 25. G. M. Goodwin, La Metallurgia Italiana, (1968) 767. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9 (1967) 609. R. M. Wagoner, K. S. Chan and S. P. Keeler, Forming Limit Diagrams: Concepts, Methods and Applications, TMS, Warendale, 1989. A. Parmar and P. B. MeUor, Int. J. Mech. Sci., 20 (1978) 385. 6. Z. Marciniak and H. Ike, Proceedings of the IDDRG Meeting, Helsinki, 1983. 7. Lian, J., Instabilite plastique, surface de plasticite et endommagement au cours de la defomation plastique et superplastique (These de doctorat), Inst. Nat. Politehnique Grenoble. 1987. R. Hill, Math. Proc. Cambr. Phil. Sot., 85 (1979) 179. 9. J. L. Bassani, J. W. Hutchinson and K. W. Neale, In: Metal Forming Plasticity, Edited by H. Lippmann, Springer Verlag, Berlin, 1989, 1. 10. K. W. Neale and E. Chater, Int. J. Mech. Sci., 22 (1980) 563. 11. J. L. Bassani, Int. J. Mech. Sci., 19 (1977) 651. 12. M. Gotoh, Int. J. Solids and Struct., 21 (1985) 1131. 13. M. Gotoh, Int. J. Mech. Sci., 19 (1977) 505. 14. A. Graf and W. F. Hosford, In: Forming Limit Diagrams: Concepts, Methods and Applications, Edited by R.M. Wagoner, K.S. Chan, S. P. Keeler, TMS, Warendale, 1989, 153. 15. A. Graf and W. E. Hosford, Met. Trans., 21A (1990) 87. 16. S. B. Padwal and R. C. Chaturvedi, Int. J. Mech. Sci., 34 (1992) 541. 17. W. F. Hosford, Proc. 7th NAMRC, Dearborn, 1979, 191. 18. J. Lian, F. Barlat and B. Baudelet, Int. J. of Plasticity, 5 (1989) 131. 19 F. Barlat and O. Richmond, Mat. Sci. Eng., 95 (1987) 15. 20. F. Barlat and J. Lian., Int. J. of Plasticity, 5 (1989) 51. 21. G. Ferron and M. Mliha Touati, Int. J. Mech. Sci., 27 (1985) 121. 22. B. Budiansky, In: Mechanics of Material Behaviour, Edited by: G. J. Dvorak and R. T. Shield, .

.

264 Elsevier, Amsterdam, 1984, 15. 23. F. Barlat, Mat. Sci. Eng., 91 (1987) 55. 24. D. Banabic and I. R. D6rr, Sheet Metal Formability, O.I.D.I.C.M. Publishing House, Bucharest, 1992, (in Romanian). 25. D. Banabic and I. R. Dorr, Mathematical Modelling of the Sheet Metals Processes, Transilvania Press Publishing House, ClujoNapoca, 1995, (in Romanian). 26. Z. Weixian, Int. J. Mech. Sci., 32 (1990) 513. 27. Z. Tourki, R. Makkouk, A. Zeghloul andG. Ferron, J. Mater. Process. Technol., 45 (1994) 453. 28. E. Chu, J. Mater. Process. Technol., 50 (1995) 207. 29. R. Hill, Int. J. Mech. Sci.,35 (1993) 19. 30. D. Banabic and S. Valasutean, J. Mater. Process. Technol., 34 (1992) 431. 31. D. Banabic, Research on the Thin Sheet Metal Formability, Ph.D. Thesis, Technical University of Cluj-Napoca, 1993, (in Romanian). 32. D. Banabic and I. R. Dorr, J. Mater. Process. Technol., 45 (1994) 551. 33. D. Banabic, Proc. Int. Conf. NUMISHEET'96, Dearborn, 1996, 240. 34. Q. Q. Nie and D. Lee, J. Mater. Shaping Technol., 9 (1991) 233. 35. Z. Marciniak, K. Kuczynski and T. Pokora, Int.J.Mech.Sci., 15 (1973) 789. 36. S. N. Rasmussen, Limit Strains in Sheet Metal Forming, (Master Thesis), Technical University of Denmark, Lyngby, 1981. 37. R. Hill, Proc. Royal Society of London, 193A (1948) 281. 38. Z. Marciniak and J. Duncan, Mechanics of Sheet Metal Forming, E.Arnold, LondonMelbourne-Auckland, 1992.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

265

Characterization of the formability for aluminum alloy and steel sheets F. Barlat a, J.C. Brem a, D.J. Lege a and K. Chung b aAlcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15060-0001, USA bDepartment of Fiber and Polymer Science, College of Engineering, Seoul National University, 56-1 Shinlim-Dong, Kwanak-Ku, Seoul, 151-742, Korea

In this work, the formabilities of an aluminum and steel sheet samples were assessed both experimentally and theoretically. The forming limit diagram (FLD) and the limiting dome height (LDH) tests were simulated using mathematical models incorporating suitable constitutive equations for the steel and aluminum alloys. Experimental and predicted results were compared and discussed. It was suggested that this type of mathematical modeling can be used to design and optimize forming processes, especially for aluminum alloy sheets.

1. INTRODUCTION The automotive industry, which has relied primarily on steel as a material of choice for structure and body applications, is beginning to use alternate materials for lightweighting purposes. This trend is driven by environmental concerns and energy saving needs. Because aluminum alloys exhibit a lower density with a relatively high strength, they can be used as good substitutes for steel in autobody sheet applications. However, steel has some economical and technological advantages over aluminum alloys. In particular, drawing quality steels have better formability. Moreover, sheet forming practitioners have used drawing quality steels for over a century in stamping operations and they have empirically optimized processing conditions for such materials. Comparatively, little experience has been gained in optimizing forming processes for aluminum alloys. However, numerical simulations using finite element methods (FEM) can be used to reduce the number of experimental trials. As an important input, these simulations require information regarding the material properties, i.e. constitutive equations. Because steel and aluminum sheets exhibit different formabilities, the constitutive equations must be specific for each material. The purpose of this work is to use constitutive equations which differentiate between steel and aluminum sheets, to characterize the formabilities of these materials with numerical models, and to validate the results with experiments.

266 2. EXPERIMENTAL

An aluminum AA2008-T4 sheet sample, a drawing quality aluminum killed (DQAK) steel and a drawing quality interstitial free (DQIF) steel were investigated in this study. The chemical compositions of these materials are listed in Table 1. For each material, the stress strain curves were measured in uniaxial tension at 0 ~ 45 ~ and 90 ~ to the rolling direction (RD). The r values, the width to thickness plastic strain ratio in uniaxial tension, were obtained in these three directions. The stress-strain curves were also measured in balanced biaxial tension with the bulge test at two different constant strain rates. This material information is necessary to calculate the different coefficients of the constitutive equations. The initial portion of the shear stress-strain curve (up to about 5% shear strain) was measured with the Iosipescu shear test [ 1] in the 0 ~ and 45 ~ orientations.

Table 1 Chemical composition in weight% of AA2008-T4, DQIF and DQAK steels Si Cu Mg A1 Fe Mn Cr AA2008-T4 0.68 0.96 0.41 Bal 0.17 0.06 0.01 A1 Fe Mn C DQAK steel 0.051 Bal 0.21 0.06 DQIF steel 0.042 Bal 0.17 0.01

(remelt analysis) Zn Ti 0.01 0.02 P S 0.005 0.02 0.008 0.02

Many processes and material parameters affect formability and, consequently, no one single parameter can be used to characterize it. However, specific formability tests are conducted in material processing or stamping plants to assess formability. The forming limit diagram (FLD), the limiting dome height (LDH), the stretch-bend and guided-bend tests are performed rather routinely to evaluate sheet formability. For autobody applications these tests correlate generally well with field performance. The FLD represents the limit strains above which localized necking occurs. Rectangular specimens are stretched using a hemispherical punch and different strain paths are achieved by varying the aspect ratio of the specimen and the lubrication conditions. Grids are applied onto the sheet surface prior to deformation and are measured after deformation, near necked or fractured regions. These measurements provide strains which are reported on a diagram whose axes correspond to the in-plane strains. This information is used by metallurgists to characterize formability, by sheet forming engineers to design processes, and by stamping engineers to assess safety margins during product manufacturing. The FLD gives information about the strains that a material element (about the size of the grid spacing which is on the order of the sheet thickness) can sustain, but it does not give information about how the material distributes strain in a larger area. For this purpose, the LDH is used. Again, a rectangular specimen is stretched using a hemispherical punch and the height at which the specimen fails is recorded. Several specimen widths are tested and the minimum height corresponds to the LDH. This minimum is associated to limit strains in a

267 state of plane strain tension, where one of the in-plane strains is zero. It is well known that plane strain is the critical state in practical stamping operations. Other tests, such as the stretch-bend and guided bend tests are conducted because they are used to assess the material performance in drawing or hemming operations. These tests characterize the ability of the material to resist fracture under bending in various conditions. However, these tests are not discussed further in this paper

1.5

'0

800

I

700

i

I

I

I

I

I

I

0.1 0.2 0.3

I

-

).5

ID

51111 ~9 ~ -

a~.

~

.400

2

f

300

T ~

f

100

x5r

0 0.0

_

D

.

.

'

\~ !

.

.

.

!

!

,o-=

)Jj

-

- ..... 0 ~ - - 45 ~ t e n s i o n

D Q A K steel --o- Bulge D Q I F steel --U-- Bulge

- - - 90 ~ t e n s i o n ~ 0~ shear --o-- 45 ~ s h e a r -

,I

I

I

I

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

iiiiii

-

2008-T~1 (S.N. 7 1 4 8 6 4 ) 1.5 I . J -1.5 -1.0 -0.5 0.0

1.0

Effective strain

i I 0.5

0

sh~r

I 1.0

1.5

x

Figure 1. Stress- strain cuves for AA2008T4, DQIF and DQAK steels.

3. CONSTITUTIVE

0.4 0.5

i [ '

__~._ Bu!g e

J ~ " r/f.."

200

4

._, I I I Gla i xy i i contours ! ! i 0.0 ............. i ............. . . . . . . . . . . . . . . . . . . . .~ "

Figure 2. Yield surface for AA2008-T4 predicted with Yld91 and Yld94.

MODELING

For the modeling aspect, the material behavior was assumed to be adequately described with a yield surface and its associated flow rule, to be rate sensitive, and to harden isotropically. Yield surface, as well as strain and strain rate hardening have been identified as the macroscopic parameters that influence the FLD the most [2-3]. Under these conditions, it is necessary to define a yield function and a stress-strain curve giving the rate of isotropic expansion of the yield surface with the strain and strain rate. The yield surface was described with an anisotropic non-quadratic yield function which contains a parameter that characterizes the crystal structure of the materials (BCC for steel and FCC for aluminum) [4,

5]. , = c~llsz -s3l a + ~zls3 -Sl[ a + ~31s1 -s2l a

= 2~ a

(1)

In Equation (1), the components S 1 , S 2 and S 3 a r e the principal values of a traceless tensor, s, obtained by linear transformation of the Euler stress components. This tensor reduces to the stress deviator in the isotropic case. The exponent a (6 for the steels and 8 for AA2008-T4)

268 characterize the crystal structure of the materials. The terms ~ k are functions of the orientation of the principal directions of s with respect to the anisotropy axes of the materials, rolling (RD), transverse (TD) and normal (ND) directions. ~ was selected to represent the flow stress in balanced biaxial tension (bulge test) because the hardening curve is evaluated over a larger strain range in this deformation mode (see Figure 1). The yield function defined by Equation (1) and subsequently referred to as Yld94, generalizes a previous version (referred to as Yld91, [6]) where the terms 0~kare constant (0t 1 = ~ 2 = 0 t 3 = l ) The coefficients of the yield function were derived from the flow curves obtained in balanced biaxial tension (bulge test) and uniaxial tension in the 0 ~ 45 ~ and 90 ~ orientations (Figure 1). The shear flow curves were just used to assess the validity of the yield function. The yield surface for AA2008-T4 is shown in Figure 2. The balanced biaxial stress-strain curves were represented by the Swift and the Voce equations for steel and aluminum alloy sheets, respectively. In general, these equations lead to the best fit with experimental data for these respective materials. Finally, the strain rate effect was assumed to be represented by a reference strain rate, ~o and a strain rate sensitivity coefficient, m. This coefficient is usually different for steel and aluminum alloys. The mathematical representation for strain and strain rate hardening is given by the following equations and the corresponding material coefficients are listed in Table 2. = [ A - B exp(-CE)](~/to}m : K(Eo + ~n){~/~o }m

(AA2008-T4)

(2)

(DQIF and DQAK steels)

(3)

Table 2 Strain and strain rate hardening parameters A (MPa) B (MPa) C 2008-T4 396.0 230.6 6.024 K (MPa) eo n DQIF steel 686.5 0.0037 0.274 DQAK steel 694.1 0.0032 0.261 R

..1%

Eo

m

0.005 eo 0.005 0.005

0.00 m 0.(12 0.02

4. FORMABILITY MODELING Using the constitutive equations, the forming limit diagram (FLD) and the limiting dome height (LDH) were predicted for the aluminum alloy and the two steel sheet samples. The FLD was calculated using a model in which a local defect is growing as plastic deformation increases until all the deformation localizes in this imperfection [7]. The FLD was computed for each of the three sheet materials using the Yld94 yield function and the respective hardening function given above. This model had been successfully used in the past to predict the FLD for many aluminum alloy sheets based on crystallographic texture measurements [8].

269 The LDH tests were simulated using the finite element method (FEM) code ABAQUS/EXPLICIT 5.3. In this model, the geometry of the blank and tools were given as input. The friction was accounted for by a single coefficient, ~, typical for the kind of material/tool interfaces employed in the experiments (It = 0.1). For numerical efficiency, the draw bead was not modeled. Rather, a large friction coefficient was used to characterize the interface between the blank holder and the blank. As with the physical experiments, dome heights were computed for several specimen widths. For a given specimen width, the dome height was defined as the height at which the largest strain in the sample reached the predicted forming limit. The width leading to plane strain deformation in the critical area of the specimen was reported and the corresponding dome height was used to define the LDH.

0.40 IC

t\

r~

]

~

/ ~~176

i

~ :

~

o.8o,,

t

oo-'oF ~7~-~,W,/ ~ 1 7 6 ~lJ / :: o.oo7~i ~:~:~ -0.10

0.00

0.10 0.20 TD strain

,

,

,

,

.,

~ r oo,~.,~176

\'~

~I~ ~176_ _ J

-0.20

0.00

to I :: o.oo ~176 ~lJ / [~_~:,o~ , }~~..~,o,.R

,

-0.20

,

~\o~

, o,o F

0.30

Figure 3. Experimental and predicted FLDs for AA2008-T4.

0.4

-0.40

0.20 0.40 TD strain

0.60

0.8

Figure 4. Experimental and predicted FLDs for DQIF steel.

The dependence of strain and strain rate hardening on the strain distribution in formed parts is well-known. The influence of the yield surface shape on the strain distribution has been shown using FEM codes [9]. The LDH test has been modeled with FEM codes using Yld94 [ 10] or a more sophisticated material model using a polycrystal approach [ 11]. All of these models indicated that the crystal structure and the crystallographic texture that affect the shape of the yield surface have a strong influence on the strain distribution during forming processes. In this work, because the yield function Yld94 was not implemented into ABAQUS when these computations were performed, Yld91 was used for the LDH calculations. This yield function is not as accurate as Yld94 over the entire stress range. However, the stresses and strains involved in the LDH test are close to the plane strain mode and the coefficients of Yld91 were determined so that Yld91 and Yld94 lead to almost identical yield surface shapes near plane strain tension (indistinguishable in Figure 2).

270 As a summary of this section, the FLD and LDH tests can be assessed theoretically if a minimal amount of information is known about materials. A coefficient of friction between the punch and blank, and three uniaxial tension and two balanced biaxial stress-strain curves are needed to adequately describe the friction and plasticity aspects of the forming process. Everything else is accounted for by mathematical modeling and numerical simulations.

5. DISCUSSION

The predicted FLDs for AA2008-T4 alloy, DQAK and DQIF steels are generally in good agreement with experimental FLDs reported in the literature [8, 12]. Figures 3 and 4 show the predicted FLD curves and experimental data obtained for AA2008-T4 and DQIF steel, respectively. The experimental FLDs cannot be drawn accurately as a curve because of the scatter in the data. The experimental FLD which lies somewhere below the experimental data points labeled "neck affected," agrees reasonably well with the predictions. Figure 5 shows the predicted FLDs for AA2008-T4, DQAK and DQIF steels. This plot indicates that the FLD is lower for the aluminum alloy and that the DQIF steel performs slightly better than the DQAK steel, which is in agreement with current knowledge.

0.80

i

I

i

i

/

"7:

0.60

0.40

~176 f

=

e-

m

0.20

FLo

:t

0.35

o

0.30

9 108.0 m m , 6.0 o 108.0 m m , 6.5

.,

120.7 m m , 6.0 s

0.25 0.20

0.00

0.10 211t18-1"4

-0.20 -

-0.40 -0.40

_

DQAK steel DOIF steel I -0.20

I 0.00

0,05 N 714864 ).

I

i

I

0.20

0.40

0.60

TD strain

Figure 5. Predicted FLDs for AA2008-T4, DQIF and DQAK steels.

0.80

0.00 -0.15 -0.10 -0.05 0.00 0.05

0.10

0.15

0.20

0.25

TD strain

Figure 6. Predicted strain distributions during LDH test for AA2008-T4.

Figure 6 shows how the minimum (plane strain) punch height in the LDH test was theoretically determined. The test was simulated with a constant punch displacement rate (250 mm/min) for specimens that were 177.8 mm long, while different widths were used to determine the plane strain conditions, as in experimental trials. In this figure, the predicted FLD is plotted with the strain profiles taken along the specimen length for three combinations of specimen width and displacement duration. For the sample with a width of 120.7 mm, the

271 maximum strain does not occur in plane strain condition, where one principal surface strain is zero (y-intercept). Figure 6 shows that for this width, the material is subjected to biaxial expansion. However, the maximum strain is clearly in the plane strain condition for a width of 108.0 mm. The LDH is obtained for this width when the maximum strain reaches the forming limit, i.e. after a punch displacement of 6.5 sec which produces a height of 27.8 mm. 0.50

I

I

1.60

I

DQIF (S.N. 714875)

.

0.40

i

L

I 1.50 I'-

"

I

"

I

"

I

"

I

"

I

o 2oo8-T4 [] DQIF steel

"

I

"

I

/

"/3 /

/ j

J

]

1.40 1.30

0.30

1.20

=

0.20

1.10 I

0.10

0.00 -0.10

iI ~ / ii H ~ i ~ ~/ ,

~

DQIF steel / o 120.7mm, 8.Os-1 " 120.7mrn, 9.Os| o 120.Tmm, 9.5s] I

0.00

0.10 0.20 TD strain

I

0.30

0.40

Figure 7. Predicted strain distributions during LDH test for DQIF steel.

1.00 0.90 0.80 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 Predicted LDH

Figure 8. Experimental vs. predicted LDH for AA2008-T4, DQIF and DQAK steels.

Figure 7 shows the same kind of information for the DQIF steel. For this material, the predicted height was found to be 39.6 mm with a specimen width of 120.7 mm. The predicted LDH for DQAK steel was determined with the same method. Figure 8 shows the predicted versus experimental LDH values. This figure indicates that the absolute LDH is not perfectly predicted with the model. This is not surprising because many assumptions have been made concerning the material properties, friction and boundary conditions. However, the trends given by the modeling work are excellent. The predicted LDH is higher for steel than for aluminum and slightly higher for DQIF than for DQAK steel, in total agreement with the experiments. The specimen width corresponding to the LDH (plane strain) is smaller for aluminum than for steel, again in full agreement with experiments (Table 3). These results concerning the FLD and the LDH clearly validate the theoretical approach. The modeling work is able to accurately describe the formability of the steels and the aluminum alloy with minimal mechanical information needed as an input (2 bulge tests, 3 uniaxial tension tests).

Table 3 Experimental and predicted widths (mm) for LDH specimens 2008-T4 DQIF steel DQAK steel Experimental 114.3 133.4 133.4 Predicted 108.0 120.7 120.7

272 The experimental and predicted FLD and LDH results were found to be in good agreement for the three materials studied. In particular, it was found that the model was able to rank the relative value of the LDH very well. This result is particularly interesting because it shows that this kind of relatively simple constitutive modeling can be used in FEM simulations of forming processes to differentiate material forming performance according to plastic behavior. Thus, it is likely that this type of modeling can be successfully used in FEM codes to design and optimize forming processes for specific materials, particularly for aluminum alloy sheets.

ACKNOWLEDGMENTS

The authors would like to thank Shawn Murtha for his many helpful discussions, Greg Fata, Harry Zonker and Jerry Morson for their help in obtaining materials and conducting/analyzing experiments, and Rich Becker for implementing the constitutive equations into ABAQUS.

REFERENCES

1. 2. 3.

N. Iosipescu, J. Materials 2 (1967) 537. F. Barlat, Mat. Sci. Eng. 91 (1987) 55. F. Barlat, Forming Limit Diagrams - Concepts, Methods, and Applications', ed. R. H. Wagoner et al., The Metallurgical Society, 1989, p. 275. 4. F. Barlat, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, Proc. Fourth International Conference on Computational Plasticity, (Complas IV), Barcelona, Spain, April 1995, ed. D.R.J. Owen, E. Ofiate and E. Hinton, Pineridge Press, Swansea (UK), p. 879. 5. F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, D.J. Lege, K. Matsui, S.J. Murtha, S. and Hattori, Submitted to Int. J. Plasticity (1997). 6. F. Barlat, D.J. Lege and J.C. Brem, Int. J. Plasticity, 7 (1991) 693. 7. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci. 9 (1967) 609. 8. H.R. Zonker, J.C. Brem and J.L. Morson SAE Technical Paper 950701 (1995). 9. K.N. Shah and R.E. Dick, SAE Technical Paper 950925 (1995). 10. Y. Hayashida, Y. Maeda, K. Matsui, N. Hashimoto, S. Hattori, M. Yanagawa, K. Chung, F. Barlat, J.C. Brem, D.J. Lege, S.J. Murtha, Simulation of Materials Processing: Theory, Methods and Applications, ed. Shen and Dawson, Balkema, Rotterdam, 1995, p. 722. 11. J. Bryant, A.J. Beaudoin and Van Dyke, SAE Technical Paper 940161 (1994). 12. A.K. Ghosh, S.S. Hecker and S.P. Keeler, Workability Testing Techniques, Ed. G.E. Dieter, ASM, 1984.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

273

Material Plastic Properties Defects And The Formability O f Sheet Metal J. D. Bressan Departamento de Engenharia Mec~.nica- Faculdade de Engenharia de Joinville Campus Universithrio - 89.223-100 - Joinville - SC - Brasil. Abstract

A main thecnological concern of sheet metal forming is the analysis of the influence of material imperfections on its formability. The imperfections can be defined as local variations in thickness and in plastic properties which can affect the plastic flow and, therefore, influence the neck formation phenomena. From the micro structural point of view this could be due to the variations in grain size and orientation, inclusions and second phase contents, porosity, as well as residual cold work, originating from the melting, solidification and rolling processes. As consequence of these imperfections within the material, the thickness, strain hardening ,strain rate sensitivity and strength coefficients and initial prestrain may exhibit local variations and thereby initiation of the neck and limit strains be influenced. Using the concept of strain gradient at the neck, the limiting strains are investigated with respect to the material parameters mentioned above. It is discussed the role of these defects on the development of the local neck and the forming limit diagram FLD. The predicted results are compared with Marciniak Theory and wtih the experimental results obtained by Azfin.

Keywords :forming limit diagram FLD, defects, sheet metal. I. INTRODUCTION The forming of sheet metal is a widespread technological process which requires a deep understanding of the mechanics of materials deformation and failure in order to optimise this process and to develop new high formability materials. The Forming Limit Diagram FLD is a well established aid in the development of sheet metal. It displays the combination of major and minor strains, e 1 and g2 respectively, which represent the limit of performance for a sheet material in principal strain space; or it thus display, in principal strain space, the combinations of useful strains that can be achieved before the onset of local necking. FLDs obtained from laboratory sheet stretching and punch forming agree reasonably well with those from pressshop experience. Although the FLD was originally empirical [ 1] a number of theoretical treatments have been developed aimed at predicting the forming limit curve in terms of readily measured material parameters and the conditions of the forming operation itself. For a material with a strain hardening eponent n, within the biaxial stretching region the limiting strain is approximately n when theory predicts the formation of a dif~se neck [2]. Within the drawing region of the FLD the Hill's theory [3] predicts an increase in the limiting strain el from n to 2n in uniaxial tension. In general, for constant strain path, the agreement between Hill's theory

274 and experimental points is reasonably good whereas in the positive quadrant the diffuse necking prediction is a lower bound limit for the limiting strains which may increase to 1.Sn in the balanced biaxial stretching case. Therefore, most theoretical effort has been concentrated on the biaxial stretching region of the FLD. Within the biaxial stretching region experimental observations [1] on sheet metal deformation clearly show that failure is generally preceded by local necking. The first theorectical approach which has overcome the problem of no direction of zero extension in the plane of the sheet under biaxial stretching was that presented by Marciniak and Kuczynski [4]. They argued that under the actual conditions of press forming local necking or fracture can occur either before or after the classical instability conditions or the applied load has attained its maximum. The authors postulate that local necking originates at initial inhomogeneities of thickness and/or material properties within the sheet which lead to conditions at the failure site moving towards those of plane strain and thus local necking could occur. These initial inhomogeneities or imperfections are characterised by a conceptual parameter f which might include thickness variations, non-uniform distribution of impurities, varying texture, different size and orientation of grains, porosity and others. Although this concept provides a powerful and fruitful model there are discrepancies not yet fully explained. A different approach to the onset of local necking in sheet metal forming by the development of strain gradients was presented by Bressan and Williams [5]. The approach describes neck initiation and growth as a continous process of flow or strain localisation owing to initial inhomogeneities in thickness or porosity. These initial inhomogeneities are characterised by the parameter ~t which is the initial gradient in area. The limit strain or local necking occurs when the strain gradient ~ attains a critical value or when the ratio M~t is approximately 100. Later the analysis is extended to variations in the plastic properties in a similar approach. 2. THEORETICAL ANALYSIS In this work an attempt is made to analyse the influence of the defects in thickness and material on the limit strains and, thus, affecting the FLD. The approach analysis consider strain hardening and strain rate hardening materials having a constitutive equation of the form,

~=

k ( 6 o + ~ ) " ~ "M

(1)

where k is the strength coefficient, s o is the prestrain, n is strain hardening coefficient, M is the strain rate sensitivity coefficient, ~ is the equivalent strain, ~ is the strain rate and ~- is the equivalent or flow stress. Assuming that the phenomena of local necking is a flow localisation process it is straightforward to conclude from eqn.(1) that necking is likely to be highly influenced by both the n-value and M-value as the flow stress ~" has strong dependence on these plastic flow parameters [5]. However, other parameters of eqn.(1) can also influence the limit strain. The imperfections that may influence the necking formation phenomena can be defined as local variations in thickness and in the flow parameters above. From the micro structural point of view these could be due to variations in grain size and orientation, porosity, inclusions

275 and second phase content as well as residual cold work, originating from the production processes. As a consequence of these imperfections within the material the plastic flow parameters n, M, k, R and e o may exhibit local variations and thereby neck initiation be influenced. A straightforward way to investigate the influence of these inhomogeneities on the neck growth behaviour is to consider the plastic flow parameters as a function of the coordinate x perpendicular to the neck, i.e., n(x), M(x), k(x), and eo(X). From eqn.(1) the equivalent stress gradient is given by,

.

.

-~ dx

.

.

k dx

~

6o+-~ dx

,,xl

+

+ -.-~

~" dx

/

~" /

/

--v-g '

/

+ en(Co + a ) ' - ~

n dx

~/

~

,

+ en ~

(2)

M dx

A

U / . ~ ~l (/

~SF1

/ / / / / / s2

F2

I s,~_A ~s,+.s, "A+dA b----'-~-- ~~176- - ~ 2Ax

I I

Fig. 1 - Sheet element with a local neck From the analysis presented by the author [5], the governing equation for the necking formation in sheet metal forming is,

d-~- M

~

(l+a)Z

.}

(60+?) 2

(3)

where 2 = c~/6x is the equivalent strain gradient at the neck, a = 661 is the strain path and

&2

Z is the critical value of the subtangent for necking. Equation (3) was derived from the analysis of an element of the sheet exhibiting a current neck. See Fig.1. This analysis lead to the following equation,

276 ld-~ dx

a =

c7~

(4)

+12

(1+ a)Z dx

Equating equation (2) to equation (4) we arrive at,

-

mE o

n]

d2 =__p + ~ ~ _ LY~ M M (1+ a)Z

2

(5)

6o+~

where the equivalent defect parameter ~ is,

12 =12 + 12k + (6 '0%+ -~)12 6 o +

1 OA o

,u

Ao Ox

12 , I l k , 12n,

12n +

1 Ok

Pk

12e~

_n

e,,( ~o + e)

,12M

k Ox

gn(~)M

12M

(6)

1 8~ o

tUeo

eo d x

1 On

'Un

ndx

1

t'tM

dM

MOx

are the material imperfections or defects. Equation (5) is similar to

equation (3). Therefore, it is quite clear that the K-type of imperfection, 12k , give a similar effect as thickness imperfections bt and both play the most decisive role on the development of the strain gradients and, thus, on the FLD. For a drawing quality steel, we can assume n = 0.25, M - 0.012, 60= 0.01, ~ - 0.001, and at the onset of the diffuse necking strain ~ is approximately 0.5. Thus, equation (6) can be reduced to,

the

(7)

= 12 + 12k +0"005126o-0"16812n- 0"08212M

The effect of the initial defects can be better explained by considering the point of instability or at the onset of diffuse necking, that is at ~= zdn -- Co . Thus, from equation (5), at this point, the strain gradient development is,

c~

=

(8)

M

If the initial imperfections 12k =126~ = ,Un = 12M =0

then, ~ = p

, and, for thickness

imperfection only, equation (8) yields, 32

=

12 M

(9)

277 But, if this imperfections are not zero and have the same size, then the strain gradient development intensity at the begining of diffuse necking is, for #k =#60

=,Un = t i M = t t ,

d2 _ 1.755~ M

(10)

Therefore, the velocity of localization of the strain gradient is almost double the value for thickness imperfection only, That is, the velocity of the necking process considering all types of imperfections is 1.755 times greater than that for a thickness defect only. However, if we have k-type and thickness type of imperfections only, the velocity of the necking process at instability point is, for #k = P equation (8) becomes, dA

_

2.0/t

(11)

M

Thus, the k-type and thickness-type of defects are the most severe in sheet metal forming. 3. RESULTS

The influence of the defect type on the limit strain for a = 3.6 is presented in Fig.2 bellow. 0.00 0.60 I

0.04 '

Defect size f 0.08

I

'

I

n =0.24" .X..~ 0~

0.12

'

!

0.16

I

'

M =0.017"

I

'

R =1.5

~o = 0.01

.= 0.40

~ 0.20 -

gM

~

gn

_

kt or ktk

1 0.00

i

0.00

/

5.00

i'

I

i

I

10.00 15.00 Defect size

I

j

I

I

20.00

25.00

Fig.2 - Influence of Defect type on the limit strain.

278 The present theory predicts that n-type and M-type of defects are quite similar. The worst defect are the thickness-type and k-type as mentioned above. In order to validate the present theory, comparisons were made with experimental results and with the Marciniak and Kuczinski theory (M-K model). The experimental results obtained by Azrin [6] using sheets of aluminium-killed steel with pre-machined grooves are compared with both theories. The co-relation between the defect parameters f and IX is,

where h is the thickness and 2 A x is the width of the machined groove and f is the M-K defect parameter. For the strain path 62/61 = 2.8 , Azrin used constant groove width of 0.015in. Therefore IX = 133.3 f/in. In Fig.3 bellow comparisons of the present theory are made with the experimental results of aluminium-killed steel from Azrin [6]. It is assumed a critical value for ~, at the limit strains, that is, 2=, = 20 in the present analysis. The predicted curve for the limit strains decrease with the defect size and is greatly affected by the strain rate sensitivity parameter M. For M = 0.017 the predicted curve agree closely with the experimental points. It is clear that M is beneficial to increase the limit strain in the presence of defects up to approximately f-- 5%.. For defect size greater than this the benefit vanishes.

0.00

0~0

0.02

I

Defect Size f 0 . 0 4 0.06 0 . 0 8 0 . 1 0 0 . 1 2 0.14

I

I

I

0.40

I

I

l

PresentTheory n=0.24"R= 1.5"~=0

r.x)

01

~

o.oo I 0.00

i I0.00

,' 20.00

Defect size ~ or Ix Fig.3 - Influence of the thickness defect size ix and strain rate sensitivity M on the limit strain 6~* compared with the experimental results [6] for aluminium-killed steel.

279 Defect Size f 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

I

0.50

0.40

I

I

I

\

9

-

I

Experimental

_ _

Present Theory

....

- -

\

0.20

I

n = 0 . 2 4 , R = 1.5 ; ~ = 0

--

0.30

I

M-K model

\

-

0.10 -

0.00

i

I

5.00

0.00

i 15.00

10.00

20.00

Defect size Fig. 4 - Comparisons between theories and the experimental results for aluminium-killed steel [7].

In Fig.4 the present theory is compared with the M-k theory for 62/6~ 2.8 .The correlation is good and shows the same trend with the defect size. The predicted curve for the M-K model was obtained from the paper by Ghosh [7]. In the paper by Azrin the parameter M was not measured but it is reasonable to suppose M = 0.017 for aluminium-killed steel. The comparisons between both theories and the experimental results for aluminium-killed steel are made assuming M = 0.017 and the critical =

strain gradient value I=, = 20 for the Present Theory. For the M-K model it is adopted M = 0 [7]. The correlations are good except that for the M-K predicted curve M = 0. The discrepancies between the theories should increase with the parameter M.

4. Conclusions

The present theory defines an equivalent defect size ~ that takes into account the imperfections in thickness, strength coefficient k , workhardening coefficient n , strain rate sensitivity parameter M and pre-strain. The limit strains decrease with the equivalent deflect size. The worst defects are imperfections in thickness and in the strength coefficient k and the least important are small variations in M and pre-strain. Thickness type and k-type have equivalent contribution to decrease the limit strains in sheet metal forming. For an equivalent defect size of 7.5/in or 5%/in the limit strains can decreased by approximately 100%. On the

280 other hand, positive variations in n are beneficial to balance the imperfections in thickness and coefficient k. The present approach takes into account the width of the local defect besides its thickness size f. It is clear that both have large influence on the limit strains of sheet metals. For aluminium-killed steel a realistic critical strain gradient is 2or, = 20. The random distribution of defects in the directions of the sheet plane should produce scatter in limit strains as seen from the experimental FLD curves.

5. Acknowledgements The author would like to thanks the University of CNPq for the financial support.

Santa Catarina State - UDESC and

6. References [1] S. P. Keeler, Sheet Metal Industries, Sept., 633 (1968). [2] H. W. Swirl, J. Mech. Phys. Solids, 1 (1952) 1. [3] R. Hill, J. Mech. Phys. Solids, 1 (1952) 19. [4] Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9 (1967) 609. [5] J. D. Bressan and J. A. Williams, J. Mech. Working Tech., 11 (1985) 291. [6] M. Azrin and W. A. Backofen, Met. Trans., 1 (1970) 2857. [7] A. K. Ghosh, The Mechanics of Sheet Metal Forming, Ed. Koistinen and Wang, Plenum Press, New York, (1978).

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

281

Formability analysis based on the anisotropically extended Gurson model E. Doege, A. Bagaviev and H. Dohrmann Institute for Metal Forming and Metal Forming Machine Tools, University of Hannover, Welfengarten 1A, 30167 Hannover, Germany A formability analysis of anisotropic sheet metal is carried out based on the anisotropically extended rate independent constutive equations suggested by Gurson and subsequently modified by Tvergaard and Needleman. The advanced model takes into account the influence of the porosity as well as and anisotropic properties of the matrix material. The integration scheme was implemented in the ABAQUS general purpose finite element program. The validity of the anisotropic extended model is proved by simulating special cases of the implemented laws as e.g. yon Mises, Hill and original Gurson plasticity. These constitutive models can be obtained from the extended model as particular cases. 1. I N T R O D U C T I O N The numerical simulation of complex sheet metal forming processes by means of the finite element method (FEM) is of great relevance for the industry. Redundant trial and error steps can be avoided during the die design period and thus manufacturing costs can be lowered. For an accurate computational prediction of the sheet metal behaviour, it is indispensable to choose an adequate constitutive model. Indeed it might be necessary to take into account the complex nature of the problem including the onset of instability phenomena, because its formation is an important precursor to failure. Workability of a sheet metal can be defined as the ability of the material to be deformed without failure. There is no consensus among researchers on the best criteria to be used. The approach advocated in the present paper utilises a porosity (damage) as such a parameter. The various ways of the occurrence of the instability phenomena are the reason to take into account the combination of anisotropy and damage as the factors affecting a sheet metal forming process.

2. C O S T I T U T I V E B E H A V I O U R OF P O R O U S D U C T I L E M A T E R I A L The main instability phenomena in a deep drawing process are wrinkling caused by the influence of circumferential compressive stresses and necking as a localised plastic deformation due to internal material inhomogenities. The computational prediction of instability phenomena in plastic solids under deformation depends substantially on the

282 assumed material response. In the following the Gurson's plasticity model and its possible anisotropic extension are briefly outlined. 2.1. G u r s o n ' s plasticity m o d e l The Gurson-type yield function for a plastic material containing randomly distributed voids gets popular for considering the ductile damage development during sheet metal forming. Based on an upper-bound solution for deformations around a single spherical void under the assumption of rigid-plastic isotropic material, Gurson proposed yield function [1], which has the following form after modification [5]:

42 --

q

-~1

+ 2qlfcosh

(

2k] ] - (1 + q3f2) ,

(1)

where k / i s the equivalent tensile flow stress of the matrix material and f is the current void volume fraction, which is small for real sheet metals. The parameters ql, q2, and q3 were introduced by Tvergaard [5] to agree the predictions of the Gurson model with his numerical studies of materials containing periodically distributed circular cylindrical and spherical voids, q is the root of the second stress deviator, p is the hydrostatic pressure. This model takes into account the progressive damage due to void nucleation and growth of an initially dense material [7,8]. The limit case with zero porosity for the isotropic Gurson model is the Levy-Mises plasticity. 2.2. F o r m a l anisotropic extension In this work it is adopted [4] that the deviatoric part of the model can be extended to an anisotropic plasticity model with the fourth rank anisotropy tensor Ai/kz, i.e.

q= i3a.TAer.

(2)

The constants in the Hill's stress function [2,3], representing planar anisotropy, are calculated based on the r-values obtained by tests of the material under different orientations c~ to the rolling direction [6]:

r~

S 9

In so

(3)

"

Therefore in the limit case with zero porosity the modified model should turn into the Hill quadratic yield condition. Additionally, when Aijkl = Iijkl, i.e., an isotropic fourth rank tensor, the conventional isotropic Gurson model is obtained. 3. C O M P U T A T I O N A L

PROCEDURE

The constitutive library provided in the ABAQUS general purpose finite element program enables the description of the behaviour of porous (mildly voided) metals [7,8]. An implicit and unconditionally stable method for the numerical integration of a class of isotropic pressure dependent plasticity models has been developed by Aravas. The procedure

283 has been applied to the material model for void-containing ductile solids with isotropic hardening developed by Gurson and modified by Tvergaard and Needleman. Following Aravas, the same Euler-backward integration method was applied. As it is characteristical for metals, undergoing small elastic and large plastic deformations, the additive decomposition of the strain increment into an elastic and an inelastic part is assumed. de = de ~; + ds pt.

(4)

For the case of linear isotropic elasticity it is assumed (r = C d ~ d

,

(5)

where C ~t is the fourth order elasticity tensor C~]kt -

K - 5G

(6)

5ijSkt + 2GSikSit.

G and K are the elastic shear and bulk moduli respectively, and 5ij is the Kronecker delta. The equation (5) gives cr~+~ = cr d -

C ~ I A e p.

(7)

The use of the Euler-backward integration scheme yields the expression for the plastic strain increment

0+] t+~t"

(8)

As p t - A,~ ~

The combination of the equations (5), (7) and (8) results in the following expression for the stress increment

The numerical algorithm proposed by Aravas is based on an assumption that is only valid for isotropic plasticity. Thus, for the integration of (1), using (2), the continuum tangent modulus C EP is used. The continuum tangent modulus is derived from the continuum rate equations by the enforcement of the consistency condition that upon yielding the stress point must remain on the yield surface if no unloading occurs. The stress increment can be calculated implicitly by using the continuum tangent moduli for any given strain increment. The procedure is straightforward and well documented thus only the expression for C EP is presented:

cel oo (cel O(~~T cEP = C~l_

'~

(10)

oo'J

Ggur wherein

(OI~I T Ga,.,,. --

~

2 ~

- ( 1 - f)~-~-~-~ + (1 - f ) k ) \

-~]

"

(11)

284 To solve the non-linear system of constitutive equations for the new approach (12)

O't+At --- ~ t -~- c E P t A ~ ,

an integration procedure based on the method of successive approximation was used. The implemented routine is verified on an one-element model for various load cases. Very good consistency with the particular cases such as isotropic Gurson's plasticity, anisotropic Hill's plasicity and isotropic Mises plasticity models, was achieved, comparising the results directly with those gained with ABAQUS. 4. S H E E T T H I C K N E S S

I N C A S E OF " P L A N E S T R E S S "

For the simulations of complex sheet metal forming processes shell elements were employed. These elements use the "plane stress" assumption 0.33 = 0.

(13)

The out-of-plane strain component is not defined kinematically. In ABAQUS the component A:33 is treated as one of the unknowns to be calculated. In the present paper another approach was employed. On the one side, for the pore-free metals the following physically based equation is adopted:

/~g33- --(/~Cll "JI-Z~C22).

(14)

On the other side, under the assumption (8) one gets

t A : o = AAa~j, where

(15)

0.~j is the deviator of the stress tensor. Consequently 1

/~Cll + /~C22- 5i)~(0.11 -1t- 0"22)

(16)

and 1

A:33 -- -~AA(0":: -t- 0"22).

(17)

Therefore, it is possible to calculate the out-of plane strain component for the porous metal formally as 0r A:33 - A ~ - -

(18)

00"33

and then to take 0"33= 0 in the above equation. 5. P R A C T I C A L

APPLICATION

The simulations were carried out with ABAQUS/Explicit. The flange of a trapezoidal cup after 40 mm punch-travel can be seen in Figure 1. This is in very good agreement with practical results. Figure 2 shows the distribution of the sheet thickness of the deformed cup. The considered material is a St1503 with an initial sheet thickness of 1 mm. The initial porosity

285 is assumed as f = 0.001. As the sheet thickness is connected with the porosity, Figures 2 and 3 have to be discussed together. The sharper corner of the cup shows the more critical value as the thinning of the sheet reaches approximately 27 % and the porosity is increasing upon 0.117 what allows the assumption that here a fracture will occure with great possibility. Thus, the porosity (damage variable as a material parameter) distribution allows to indicate localisation of the failure onset. The second example is the simulation of the S-rail deep drawing process according to the NUMISHEET'96 Benchmark geometry. In the Figures 4 and 5 are shown the sheet thickness and porosity distributions. One can see that the porosity defines precisely localisation of the damage occurrence and possible failure, while the sheet thickness might lead to ambiguously indicated place of necking. 6. C O N C L U S I O N S The numerical simulation of the deep drawing process of the anisotropic sheet metal was carried out taking into account the influence of anisotropy on the damage during the forming process. From the results it may be concluded that as a precursor of the area endangered by ductile failure, porosity (damage variable), predicted from the anisotropic extended Gurson's model, is a material-dependent refined criteria for the formability estimation of a metal sheet and the process reliability. The results emphasise that the anisotropic extension ~of the Gurson model is a good method for failure assessment in the design of metal forming processes of metal sheets with anisotropic properties. REFERENCES

1. GURSON, A.L.: Plastic Flow and Fracture Behaviour of Ductile Materials Incorporating Void Nucleation, Growth, and Interaction. Ph.D.-thesis, Division of Engineering, Brown University, 1975 2. HILL, R.: A theory of the yielding and plastic flow of anisotropic metals. Proc. of the Royal Society, London, 1948 3. HILL, R.: Theoretical plasticity of textured aggregates. Math. Proc. Camb. Phil. Soc. 85 (1979), S. 179-191 4. SEIBERT,D.: Untersuchung des duktilen Versagens von Feinblechen beim Tiefziehen. Dissertation, Universit/it Hannover, VDI-Verlag, Dfisseldorf, 1994 5. TVERGAARD,V.: MechanicM Modelling of Ductile Fracture. Mechanica 26 (1991), S. 11-16 6. IDDRG: Ermittlung der senkrechten Anisotropie (r-Wert) yon Feinblech im Zugversuch. Stahl-Eisen-Prfifblatt 1126-84, Nov. 1984 7. ABAQUS/Explicit. User manual. Hibbitt, Karlsson'& Sorensen, Inc., 1994 8. ABAQUS. Theory manuM. Hibbitt, Karlsson & Sorensen, Inc., 1994

286

Figure l. Flange of a trapeziodal cup after 40 mm punch-travel.

Figure 2. Sheet thickness distribution after 40 mm punch-travel.

287

Figure 3. Porosity distribution after 40 mm punch-travel.

Figure 4. Deep drawing of the S-rail. Sheet thickness distribution at the end of deformation.

288

Figure 5. l)eep drawing of the S-rail. Porosity distribution at the end of deformation.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

289

R u p t u r e c r i t e r i a d u r i n g d e e p d r a w i n g of a l u m i n u m a l l o y s J. Proubet a and B. Baudelet b apechiney Centre de Recherches de Voreppe, BP 27, 38340 Voreppe, France bGPM2 Department, INPG, BP 46, 38402 St-Martin d'H~res, France

1. INTRODUCTION The drawing operation is a key process for the packaging industry as well as for the automotive i n d u s t r y : every day, tens of millions of food and beverage cans are m a n u f a c t u r e d by Pechiney which also promotes the use of aluminium for drawing car body panels. Hence, this industry often faces the problem of designing new series of tools for making cans or car body panels either newly designed, or with decreased thickness, or out of a different alloy. Another recurrent problem is choosing the best alloy for a given drawing operation. For all these reasons, research is going on at Pechiney to try and find out failure criteria during deep drawing of aluminium alloys. Aims of this research are diverse : have a better understanding of why failures occur, reduce the lengthy and costly procedures necessary to characterize the formability of aluminium alloys, and reduce the trial and error experiments necessary to design new series of tools each time a different type of can or car body panel has to be manufactured. Two approaches came out of this research : numerical simulations and a simplified failure criterion for deep-drawing. These two approaches are presented here. Theoretical results obtained are compared with experimental results. 2. NUMERICAL SIMULATIONS OF R U F r U R E IN D E E P DRAWING 2.1. Application of existing theories on rupture to a l u m i n i u m alloys A lot of work has been published in literature on forming limits of metal during drawing. Most of the work available, however, is concerned with local failure criteria : the aim is to understand why a flat sheet of metal will break under given stress and strain histories. Well-known theories on the subject include those of Hill [1] and Marciniak-Kuczinski [2]. Hill theory predicts that failure can only occur for states of strains varying from plane strain to pure shear.

290 This theory was extended to all states of strains by Marciniak et al. who introduced an imperfection in the sheet and simply applied to it equations from the mechanics of continuous media. Questions were raised on why such a defect is necessary and what magnitude should be taken for it. Another difficulty with the Marciniak approach is that results depend also on the strain rate sensitivity of the material. Many authors tried to link the defect with metallurgical damage [3]. In our opinion, the introduction of a defect is indeed necessary for explaining the rupture of a fiat sheet of metal submitted to stress and strain histories but m a y not be necessary any more when dealing with ruptures during deep drawing : it that case, ruptures occur in areas which do show defects but these defects may be created by complex stress and strain histories leading to thickness heterogeneities and not necessarily by metallurgical damage. This means that for alloys with little metallurgical defects, mechanics of continuous media should be able to predict rupture. Indeed, most of the aluminium alloys used for deep drawing have a very small initial damage compared to imperfections in thickness created during deep drawing. Typical damage in an AA 3004 H19 aluminium alloy was measured to about 5.10 -4 [4] while thinnings of 10% are frequently observed in deep drawing. Thus, it should not be necessary to introduce artificial defects in a finite element simulation of deep drawing for predicting rupture of most aluminium alloys : thickness heterogeneities will be naturally generated by the complex stress and strain histories. Since finite elements simulations use the same equations of the mechanics of continuous media as Marciniak et al., running such simulations is indeed equivalent to solving the complex equations of the M-K theory. Also, aluminium alloys are not much sensitive to deformation rate, which simplifies further their simulation. On the contrary, rupture of alloys which do have some large imperfections will not be correctly simulated without introducing into finite element codes an appropriate failure criterion. Such theories, not based on the mechanics of continuous media, exist and try to link metallurgical damage with failure [3,5].

2.2. Experimental results A common way of investigating the drawing ability of a given alloy is to determine the failure curve giving the maximum blank holder force (BHF) allowable for drawing without breaking a blank with a given diameter. Pechiney has determined such curves for many of its alloys. Problem with these curves is that they not only depend on the material and geometry used but also on the lubrication. Fig.1 below shows two curves obtained for an AA 5754-0 alloy used at Pechiney. One curve ('high lub') was obtained by lubricating carefully the blanks, while for the other ('low lub') blanks were wiped. These curves themselves are approximate due to high dispersion in the experimental results.

291 Geometry used []

H i g h lub lub

punch diameter" Dp = 69.9 mm

3

punch radius 9 rp=3mm 2 ~J

die diameter 9 Dm = 72.5 mm 1

die radius 9 rm = 5 mm 0

. 138

I

"

140

I

142

'

I

'

144

I

146

'

148

blank thickness 9 t = I mm

B l a n k d i a m e t e r (mm) F i g u r e 1. E x p e r i m e n t a l failure curves for AA 5 7 5 4 - 0 P r e s s u r e s P s h o w n on t h e c u r v e s a r e h y d r a u l i c p r e s s u r e s a p p l i e d in press. A p r e s s u r e P of 1 b a r c o r r e s p o n d s to a force F of 15.7 k N a p p l i e d on blank holder. T h e slopes of t h e two curves are respectively a b o u t - 0 . 7 b a r / m m for 'high a n d - 0 . 3 b a r / m m for 'low lub'. Rheology of t h e AA 5754 O u s e d is t h e following : - y i e l d s t r e n g t h " Ro,2 = 103 M P a - L a n k f o r d coefficient : r = 0.61 - h a r d e n i n g d e s c r i b e d by Voce law : (~ = A - B exp (-C e) w h e r e (~ is

the the lub'

the

e q u i v a l e n t s t r e s s in MPa, e t h e e q u i v a l e n t d e f o r m a t i o n a n d A = 285.4 M P a , B = 194.3 MPa, C = 12.9

2.3. Description of numerical simulations F i n i t e e l e m e n t s i m u l a t i o n s w e r e c a r r i e d o u t to t r y a n d r e p r o d u c e e x p e r i m e n t a l f a i l u r e curves. T h e explicit code O p t r i s TM w a s used. R h e o l o g y d e s c r i b e d above w a s m o d e l l e d u s i n g t h e available Hill model. Since i n t e r e s t w a s not in m o d e l l i n g t h e effect of p l a n a r a n i s o t r o p y , a n d to m i n i m i z e C P U t i m e , only 4 ~ of t h e a x i s y m m e t r i c g e o m e t r y w e r e m e s h e d w i t h a p p r o p r i a t e b o u n d a r y conditions. P u n c h a n d die w e r e a s s u m e d u n d e f o r m a b l e ; t h e i r r a d i i m e s h e d w i t h a b o u t 10 e l e m e n t s . V a r y i n g forces w e r e a p p l i e d on a d e f o r m a b l e , elastic b l a n k holder. P r e l i m i n a r y r u n s s h o w e d t h a t b e s t r e s u l t s a r e o b t a i n e d w h e n t h e b l a n k is m e s h e d as t h i n as possible u s i n g available shell e l e m e n t s : t h e m i n i m a l l e n g t h of a n e l e m e n t m u s t be l a r g e r t h a n its t h i c k n e s s to p r e v e n t self-contact a n d to k e e p t h e a s s u m p t i o n of 'shell' e l e m e n t s (whose t h i c k n e s s is s u p p o s e d to be m u c h s m a l l e r t h a n t h e i r span). Only one e l e m e n t w a s u s e d in t h e o r t h o r a d i a l d i r e c t i o n to describe t h e 4 ~ s p a n (4 ~ w e r e chosen so t h a t t h e o r t h o r a d i a l l e n g t h

292 be always larger t h a n the blank thickness). The part of the blank u n d e r the punch was meshed with a single 3-node shell element. P u n c h speed was accelerated to 10 m/s (this speed was found optimal to minimize CPU time without introducing perturbing inertial forces). Average CPU time for such a drawing simulation was about 10 min on a Silicon Graphic R8000 processor. 2.4. Influence of friction coefficients

E x p e r i m e n t a l results (see Fig.l) clearly showed t h a t friction has a large influence on failure curves. Three different friction coefficients (Coulomb's law) are defined in the simulations : IXfor the part of the blank under the blank holder, Ixm on the die radius and ~tp on the punch radius. P r e l i m i n a r y trials showed that in order to have a correct t h i n n i n g of the blank under the punch (0.9 mm in the experiments), ~tp should be higher t h a n 12 %. Otherwise, simulations predict too much thinning. Other simulations also showed t h a t in order to prevent too much t h i n n i n g u n d e r the punch radius, a 'good' value for ~tp is 20 %. This value may seem quite high, especially for lubricated blanks, but may be explained by the very high local pressures exerted by the blank on the punch radius. Then, simulations were run to determine the influence of Ix and Ixm on failure curves. To compute such a curve, simulations with v a r y i n g b l a n k holder pressures are run for two or three blank diameters D (typically close to d r a w i n g ratios of 2) until pressures at which r u p t u r e occurs are reached. Ruptures do occur when pressures get too large and a r e r e v e a l e d in numerical simulation by the localisation of deformation on a given set of finite elements which get thinner and thinner. All ruptures simulated occur on the part of the b l a n k located between the punch and die radii. Failure curves are a s s u m e d s t r a i g h t (this a s s u m p t i o n is justified for drawing ratios close to the limit drawing ratio usually situated between 1.9 and 2.1 for aluminium alloys). The following table shows the slopes and elevation of the failure curves obtained for varying friction coefficients : Table 1 : slope and elevation of failure curves (alloy AA 5754-0) ~m Ixp pressure (bar) slope for D= 142 m m (bar/mm) 0.02 0.02 0.2 11.5 -1.25 0.04 0.04 0.2 4.2 -0.6 varying 0.06 0.06 0.2 2.2 -0.5 0.08 0.08 0.2 0.5 -0.25 0.06 0.04 0.2 3.5 -0.5 l.tm varying 0.06 0.08 0.2 0.9 -0.5 0.04 0.06 0.2 3.2 -0.75 ~t varying 0.08 0.06 0.2 1.2 -0.25 .

_

_

It is seen t h a t " - the slope of the failure curve depends on IXbut not (much) on lXm,

293 - the height of the failure curve depends on both }~ and ~tm. By c o m p a r i n g all these curves with e x p e r i m e n t a l curves (Fig.l), it is seen t h a t the 'low lub' curve is best fitted using (~, ~tm)=(8%, 6%) and t h a t the 'high lub' curve is best fitted using (~, pro)=(4%, 6%). These curves are s h o w n in Fig.2 below.

9

4

Low lub (NS) (~,~m)=(8%,6%)

%

~-

High lub (NS) 6%)

~2

~2 r~

Low lub (exp) 0

'

138

I

140

'

I

142

'

I

144

~''

I

146

0

'

148

'

138

B l a n k d i a m e t e r (mm)

High lub (exp) I

140

'

I

142

'

I

"~ '

144

I

146

'

148

B l a n k d i a m e t e r (mm)

Figure 2. F a i l u r e curves for AA 5754-0 9experiments vs. simulations 2.5. C o n c l u s i o n

It can t h u s be concluded t h a t n u m e r i c a l s i m u l a t i o n s are able to reproduce failure curves as long a s " - the alloy s i m u l a t e d does not have large metallurgical defects, - the finite e l e m e n t m e s h is fine enough, - friction coefficients are adapted. The modelling of the rheology of the m a t e r i a l should be accurate b u t is less critical. 3. S I M P L I F I E D A N A L Y T I C A L C R I T E R I O N 3.1. P r i n c i p l e

Most r u p t u r e s d u r i n g deep drawing occur in the cup wall (between the die a n d p u n c h radii) : only this p a r t is u n c o n s t r a i n e d . A r u p t u r e criterion can t h e n be found if computed stresses in this p a r t of the cup are h i g h e r t h a n limit s t r e s s e s allowable. This criterion will only be valid if, as for n u m e r i c a l s i m u l a t i o n s , no large m e t a l l u r g i c a l defects are p r e s e n t in the initial flange. It will not be able to predict failures occurring u n d e r the blank holder but it can be t h o u g h t t h a t if a failure occurs there, t h e n the criterion should detect a failure in the cup wall for n e a r b y b l a n k holder pressures.

294 3.2. C o m p u t a t i o n o f s t r e s s i n t h e w a l l

The r a d i a l s t r e s s gr due to s h e a r in the flange above the wall can be computed as [6] 9 c

r

Rm = malog~ Ri

(1)

where ~ is the equivalent flow stress, 'm' is a coefficient enabling adjusting the Tresca criterion (used for establishing the formula) to the von Mises criterion (m = 1.1), Rm is the flange radius (at the given draw depth) and Ri the die i n t e r n a l radius. To this stress m u s t be added the stress asfdue to the blank holder pressure 9 ~t-F Csf - 2. ft. Ri. t m

(2)

where F is the blank holder force and t the flange thickness (it can be t a k e n to the initial flange thickness as an approximation). W h e n the flange slides over the die radius, these stresses are t r a n s f o r m e d by the so-called rope formula, so t h a t the stress ~ in the can wall becomes 9 ~ = (Gr + Osf)" el'tm"0

(3)

w h e r e 0 is the angle over which the flange is sliding on the die radius. This angle can be c o m p u t e d as a function of the d r a w i n g d e p t h (0 at d r a w beginning, n/2 at the end). An a d d i t i o n a l s t r e s s gbub m u s t be added to this s t r e s s due to the b e n d i n g / u n b e n d i n g of the flange :

(~LULou= ~" 2" rm --

(4)

where rm is the die radius and E is the Young's modulus. The stress (~t in the flange wall is thus [7] 9 crt = (~r + ~ s f ) "el'tm'0 +~bub

(5)

S t r a i n h a r d e n i n g can be t a k e n into account by varying the equivalent flow stress (~ as a function of an equivalent strain. This equivalent s t r a i n can, for example, be calculated in the middle of the flange (at radius (Rm+Ri)/2).

295 3.3. F a i l u r e

criterion

The s t a t e of stress at failure is necessarily located on t h e yield curve. The m a x i m u m positive s t r e s s is w h e r e the t a n g e n t to the yield curve is vertical. W i t h an anisotropic Hill criterion, this stress is computed as [8] 9

max

=o

-

z

l+r

=~ ~ ~/2r + 1

(6)

w h e r e r is the L a n k f o r d coefficient. At failure, the criterion is 9 ot = (~max

(7)

And the corresponding orthoradial stress o0 is" r o 0 = ~ o

r+l

(8)

z

It is not null, as frequently a s s u m e d but positive (tensile). Therefore, close to r u p t u r e , compressive o r t h o r a d i a l stresses in the flange become tensile after the flange h a s gone over the die radius. This fact w a s confirmed by n u m e r i c a l simulations. Also, the stress Crmax is an increasing function of r, w h i c h p r o v i d e s a n e x p l a n a t i o n to the fact t h a t alloys with a high L a n k f o r d coefficient are best for drawing. 3.4. F a i l u r e

curves

A p p l y i n g the above criterion, the following failure curves are found (blank holder force F at r u p t u r e for varying b l a n k radii Rm) 9

F - 2 gRi t(( (~max - Gbub )e -]'tm0 - m~ log -~1 Rm )

(9)

Since failure curves are u s u a l l y d r a w n for d r a w i n g ratios b e t w e e n 1.5 a n d 2.5, the l o g a r i t h m can be a p p r o x i m a t e d by" mlog Rm -- 0.5 R m - 0.35 Ri

(with m=l.1)

(10)

Ri

so t h a t failure curves are about s t r a i g h t with slopes K a p p r o x i m a t e d by 9 g

2.rt.tx.t.~ K -- -0.5 g -- - ~ g g Results found experimentally explained by Eqs.9 and 11 9

(11) a n d in n u m e r i c a l

simulation

are

thus

296 - the lower the friction under the blank holder, the more vertical the failure curve get. In a perfect case with no friction, the curve would be vertical. - the height of the curve depends on both friction coefficients ~t and ~trn. The higher the friction ~tm on the die radius, the lower the failure curve.

3.5. Comparison between analytical criterion and numerical simulations First comparison is made on the slopes of the failure curves. Table 2 below compares slopes found in numerical simulation for alloy AA 5754-0 (see Table 1) and slopes K' found from Eqs.9 and 10 (t = 1.mm, ~ = A = 0.285 GPa). Slopes K' are expressed in bars per mm of blank diameter so that E q . l l becomes : P (bar) = K'. Dm (mm) with

K ' = K / 2 / 15.74

(12)

Table 2 9slope of failure curves (alloy AA 5754-0) slope K' (bar/mm) simulation analytical 0.02 -1.25 -1.4 0.04 -0.6/-0.75 -0.7 0.06 -0.5 -0.5 0.08 -0.25 -0.35 It is concluded t h a t given uncertainties on slopes computed n u m e r i c a l l y (due to small n u m b e r of cases computed), a g r e e m e n t b e t w e e n a n a l y t i c a l criterion and numerical simulations is very good. A F o r t r a n program was written to compute failure curves according to Eq.9. F a i l u r e condition Eq.7 is computed for each blank diameter, p r e s s u r e and drawing depth. It was found t h a t in order to match finite element curves, the friction coefficient ~tm t a k e n for the simulations has to be increased by about 0.08 when applied in Eq.3. Some authors [8] pointed out that the so-called rope formula is not well adapted to deep drawing due to bending stiffness of the sheet. Fig.3 below shows a comparison between failure curves o b t a i n e d in simulation and analytically in the case of alloy AA 5754-0 with low and high lubrication 9

297

-

Low lub (Anal)

(p,pm)=(4%,6%)

Low lub (NS) 3 ~2 2 00

1 ~

Hi High lub (NS)

(~t,pm)=(8%,6%) 0

'

138

I

140

'

!

142

'

I

144

'

I

146

Blank diameter (mm)

0

'

148

'

138

I

140

'

I

142

'

I

144

"

(iPI

146

'

148

Blank diameter (mm)

Figure 3. Alloy AA 5754-0. Numerical simulation vs. analytical criterion Here again, the agreement is quite good. It should be noted that slopes found with the Fortran program do not match exactly slopes computed according to Eq.12 since this equation is a simplified version of Eq.9. 4. CONCLUSION This research has shown that deep-drawing failure curves can be predicted both by numerical simulation and an analytical criterion as long as the metal studied does not have large metallurgical defects - as in the case of commercial aluminium alloys used for drawing cans or car body panels. Results of the simulations are influenced by mesh density, which should be as fine as possible, by rheology but mainly by the friction coefficients defined. The analytical criterion confirms and explains the importance of the friction coefficients. In particular, the slope of failure curves strongly depends on the friction under the blank holder. As a first approach, the analytical criterion provides accurate enough failure curves but problem remains to enter adequate friction coefficients to use this model in a predictive way. REFELtENCES 1. R. Hill, J. Mech. Phys. Solids, Vol.1 (1952), 19-30. 2. Z. Marciniak & K. Kuczinski, Int. J. Mech. Science, Vol.15 (1973), 789. 3. I.L. Dillamore, J.G. Roberts, A.C. Bush, Metal Science, 2 (1979), 73-77. 4. B. Baudelet & B. Grange, Scripta Met. et Mat., Vol.26 (1992), 375-379. 5. L. Felg~res, Mem. Sci. Rev. Met., 77 (1980), 327-342. 6. W. Johnson & P.B. Mellor, Engineering Plasticity, Ellis Horwood Ltd (eds), England, 1983. 7. K. Manabe & H. Nishimura, Proc. of the 2nd Inter. Conf. on Techn. of Plasticity, Ed. K.Lange, Stuttgart, 24-28 Aug., 1987, Vol.II, pp. 1297-1304. 8. J.A.H. Ramaekers et al., Proc. IDDRG'94, Lisbon, 403.

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PREDICTION OF SHAPE INACCURACIES

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Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

301

Prediction of flange w r i n k l e s in deep d r a w i n g Jian Cao a, Apostolos Karafillis b and Michael Ostrowski b aDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, U.S.A. bResearch and Development Center, General Electric Company, Schenectady, NY 12301, U.S.A.

ABSTRACT A numerical method for predicting the onset of flange wrinkles of small wavelength during deep drawing process is presented. The method is based on the approach developed by Cao and Boyce (1997) for predicting the buckling behavior of sheet metal under lateral constraint using a combination of energy conservation and finite element method. Continuum elements are used in a simple Finite Element Analysis model to study wrinkles with a maximum wavelength of ten times the sheet thickness. The analysis provides the critical buckling stress and the resulting buckling wavelength as functions of normal pressure. Such relationships are then implemented in a Finite Element Method (FEM) package that uses membrane elements to simulate the workpiece deformation during a forming process. The use of membrane elements significantly reduces the amount of computation time required in comparison to using structural shell elements with multiple integration points through the thickness. The stress histories calculated from the FEM membrane analysis are used to predict the onset of buckling during the forming process. The application of the forming of a rectangular pan is examined. The comparison between numerical simulation and the experimental results is presented. Our approach predicts the onset of buckling in excellent agreement with the experimental observations.

1. INTRODUCTION Wrinkles during a sheet metal forming process is a major consideration due to the fact that it alters the ability to impose stretching during processing and also adversely affects final part appearance, assembling and function. Computational prediction of the onset and growth of the wrinkles has significant ramification for optimizing the design of parts and tooling, selecting materials and improving part formability. The problem that we are particularly interested in is the buckling in the flange (Fig. 1) which is held between a blankholder and a die. This buckling can be combined with subsequent draw-in of the already wrinkled part of the flange, thereby causing the appearance of wrinkles at the sidewalls of the part by the end of the process. The methodologies for predicting the

302 onset of buckling can be mainly divided into two categories. One is the bifurcation analysis initiated from Hill's (1958) general theory of bifurcation and uniqueness, and later detailed by Hutchinson (1974) in the plastic buckling range. Triantafyllidis and Needleman (1980) studied the problem by assuming sheet metal resting on an elastic foundation whose stiffness relates to the binder pressure. Although their results were found to compare favorably with some previous empirical models for the cases where binder stiffness K=0 (no binder constraint), no comparison between the numerical results and experiments was given when K:~0. However, they calculated the effect of binder stiffness on the critical buckling stress and the wave number. A similar approach for elastic rectangular plates on a non-linear uni-laterial elastic foundation can be found in recent literature such as Elisakoff et. al (1994) and Shahwan et. al (1994) whose work was directed towards understanding the buckling of films bonded to a substrate. Overall, the bifurcation analysis is essentially an eigenvalue approach. The major obstacle of this method is that numerically it becomes extremely complicated if large deformation theory and anisotropic material constitutive law are involved. Nevertheless, the analytical method using shell theory is limited to shallow buckling, i.e., the buckling wavelength is large compared to the sheet thickness.

Fig. 1 A rectangular cup (left) having buckling and tearing failure (right shows a corner). The other methodology is Finite Element Method (FEM) with either implicit or explicit integration method, which becomes a prime tool to predict buckling behavior for complicated geometry and boundary conditions. Implicit method is essentially an eigenvalue approach and the post-buckling behavior has to be traced by initial imperfections built in the original mesh which usually is a specific modal shape. Unlike the implicit method, the nature of the explicit integration as a dynamic code and the numerical error accumulated in the analysis, which acts as imperfections, can automatically generate wrinkled deformed shape. However, the predicted onset and post-buckling behavior is affected by variations in the FEM model, such as, material density, simulation speed and mesh density, etc. Therefore, the robustness of the FEM simulations is not quite reliable. On top of these, an accurate prediction of buckling in the order of sheet thickness requires a very fine mesh which is considered impossible or unrealistic to have for a complicated three-dimensional forming simulation. Recently, a different approach which promises the ability to predict sheet buckling under lateral constraint was developed by Cao and Boyce (1997) using a combination of energy conservation and the implicit finite element method. They simplified the flange buckling

303 behavior in 3-D sheet metal forming process into a rectangular plate under lateral constraint. The approach was tested for shallow flange buckling using shell elements and the predicted results, which are critical buckling stress and the resulting buckling wavelength, match the experimental conical cup forming results extremely well. In this paper, the same analytical method is adopted to study short-wavelength flange buckling. The results are then compared to the forming of a rectangular pan conducted at GE.

2. N U M E R I C A L S O L U T I O N

2.1 Review of the Buckling Criterion As known, for plates of length L under in-plane compression (Ux) with no lateral constraint, the strain energy in a perfect flat plate (Go) is greater than that in a buckled plate (ew). Such difference can be interpreted as the work (gZn) executed by external lateral force (F) to suppress the buckling. Specifically, (1)

Wn-- G o - Ew n = f n ( U x ) -- I0 2~n~ w F d u z

where n represents the buckling mode (mode 1 is a half sinusoidal wave), Ux is the in-plane edge displacement, w is the width of the plate, Uz is the out-of-plane displacement, and ~max is the maximum buckled height at a displacement of Ux. By assuming F = a (Uz - ~max) 2 -F b , where a and b are fitting parameters to a parabola, the maximum pressure required to suppress the buckling of the n th mode at a given ux/L can be calculated as Pmax n =

3 (Co -

(2)

Ew n ) / 4 ( ~ m a x n L W )

m

t~ O

Mode 1 ....

/

Mode 2

f

E b E LU x~ E 9

.'/

.,

**./ .,/1

p**

4

0~

~

~. 2

O

.

/

z

/ ' ~ " d " I" " "" "

,-

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

p*

,

0.0000 0.0010 0.0020 0.0030 Normalized edge displacement Ux/L Fig. 2 Maximum binder pressures as functions of the edge displacement for the first and second mode buckling to illustrate the concept of the buckling criterion.

304 Figure 2 shows the calculated Pmax] and pmax2 as a function of ux/L. The two curves are found to cross over each other at a specific point, the transition pressure p]2, which is the binder pressure where the favored mode of buckling transits from mode 1 to mode 2. For example, consider the case of a binder pressure p* applied on the plate: the plate would buckle in mode 1 and mode 2 at displacements of u]* and u2*, respectively. For the case where p* < P12, we have Ul* < u2* (Fig.2), i.e., the critical buckling stress for mode 1 is lower than that for mode 2, and therefore, the plate would favor mode 1. On the other hand, for the case of p** > p12, we have u2** < Ul**, and therefore, the plate would buckle in mode 2. This criterion thus quantitatively defines the critical buckling stress (the associated stress at the critical or transit u/L) and also the mode of buckling which will occur in the presence of a given binder pressure. The criterion was tested for the drawing of aluminum conical cup and excellent predictions were obtained as shown in Fig. 3.

40

.-=

~o

7

m

O

t.-

30

Conical Cup Forming Data

o = ..Q

Numerical Results

i n

6 TL-O

8~5

, m

"E O

"o ~ 2 0 --~

~

O~ t_

O

.-- ~ ~ a

~_ ~ 10 E o z 0

Conical Cup Forming Data Numerical Results

0

z

0

20

40

60

80

2

0

100

Normalized binder pressure P/Oyo (10 4)

20

40

60

80

100

Normalized binder pressure P/~yo (10 4)

Fig. 3 Calculated critical buckling parameters compared to experimental results.

2.2 Study of short-wavelength buckling under constraint

The buckling criterion reviewed in the previous section provides a general principal for characterizing sheet buckling under lateral constraint. The methodology itself has no limitation on the applications of using various material constitutive laws or considering small or large deformation. However, the results presented in Fig. 3 are valid only for shallow buckling, since shell elements were used in a simple FEM analysis to obtain eo and ewn. Here, we will present an approach to study short-wavelength buckling under lateral constraint for a forming quality steel sheet. 2.2.1 Finite Element Model Using the commercial finite element code ABAQUS, a plate of length L is modeled with 8-node reduced integration plane stress continuum elements (ABAQUS CPS8R) when considering a uniaxial in-plane stress state, and 20-node reduced integration brick elements (ABAQUS type C3D20R) when considering a biaxial in-plane stress state. The reduced

305 integration element CPS8R has four integration points whereas C3D20R has 8 integration points. The finite element code uses an implicit solver where equilibrium is converged upon in each increment using a Newton-Raphson method. For the uniaxial stress state case, which corresponds to the stress condition at the outer edge of the flange in a sheet metal forming operation, the plate, having unit width, is modeled with 12 CPS8R elements along the length (x direction) and 3 elements through the thickness (y direction). The loading condition consists of a monotonically increasing compressive displacement Ux of edges at x=0 and x=2L. Two cases are simulated for each 2L: a perfect plate and a plate buckled to mode 2. Notice that mode 2 buckling of a length 2L is equivalent to mode 1 buckling of a length L. In Cao and Boyce (1997), mode 1 was used in the simulations to perform the analysis. Here, mode 2 is used due to the simplicity of setting the boundary conditions associated with continuum elements. As known, buckling initiates from some form of imperfection in the structure which can be a geometric imperfection (such as lack of flatness), a material imperfection (e.g., non-uniform thickness), or loading imperfection (e.g., off-center loading). Cao and Boyce (1997) have shown that the effect of geometric imperfection on the initiation of buckling is more significant than that of material imperfection. Here to obtain the buckled plates, the form of imperfection is chosen to be a geometric imperfection which positions nodes to a sinusoidal mode shape (Zo = Ato (1-cos (2rex/L))) of a very small amplitude A=0.005, where to is the plate thickness. Incorporation of the imperfection in this manner acts to predefine the buckling mode obtained where A=0 for obtaining a perfect flat plate under in-plane compression. The deformed mesh of a buckled plate with the above imperfection form (A~:0) is shown in Fig. 4. The strain energies of a flat plate (eo)and a buckled plate (ew), and the buckled amplitude are recorded as functions of edge compression ux/L.

z u z[mx Fig.4 Deformed mesh in uniaxial stress state.

Fig.5 Deformed mesh in biaxial stress state.

For studying the buckling behavior under biaxial loading (compression along the length direction and tension in the width direction), which represents the stress condition near the binder radius in sheet metal forming, the plate is modeled with twelve C3D20R elements along the length direction (x direction), one element in the width (y) direction and three elements through the thickness (z direction). The width is defined as one-twelfth of the total length in x direction. The loading condition consists of a constant tensile stress in y direction and a

306 monotonically increasing compressive displacement Ux of nodes at x=0 and x=2L. A transverse tensile stress (-- 0.5 (~yo) is used to examine the effect of tensile stress on the onset of sheet buckling. Again, two simulations of a perfect and a wrinkled plate are calculated for each length 2L where the same imperfection form as described before is adopted. The deformed shape of a wrinkled plate using three dimensional brick element is shown in Fig. 5.

2.2.2 Buckling stress prediction Following the calculation procedure reviewed in Section 2.2.1, the critical buckling wavelength normalized to the sheet thickness as a function of applied normal pressure normalized to the initial yield stress is plotted in Fig. 6a for both uniaxial and biaxial loading cases. Fig. 6b shows the critical buckling stress normalized to the initial yield stress versus normalized pressure. Two curves are plotted in this diagram. The solid lines correspond to a uniaxial compression stress state, whereas the dashed lone corresponds to the biaxial stress state described in Section 2.2.1. By comparing the two curves in the diagram of Fig. 6, we see that a transverse tension in the plane of the sheet metal will initiate buckling earlier and will result in a longer buckling wavelength. A buckling severity index is then defined a s (]applied/(~crit as a function of binder pressure. A value higher than 1.0 means that buckling will occur. We call the diagram of Fig. 6b the Buckling Limit Diagram (BLD). The BLD of a material can be used for the prediction of the initiation of small wavelength buckling, once the stress state in the material is known. 12

._=

"| .

.

10

~l

0:3 t'-

8

~.

-o ~9 N

6

~

o Z

~

.

.

.

.

.

uniaxial ......

~

biaxial

N~

m"

. B m

--.

4

~f

"-----uniaxial ......

z.~

2

i

0

J

0

~

0.1

Normalized binder pressure p/O'y o

0.2

0

biaxial

I

0.1

i

0.2

Normalized binder pressure P/~yo

Fig. 6 Calculated buckling parameters as functions of binder pressure

3. A P P L I C A T I O N AND C O M P A R I S O N W I T H E X P E R I M E N T A L R E S U L T S The proposed buckling prediction method was implemented in a finite element analysis code used for sheet metal forming process simulation. The code is based on a dynamic explicit analysis of the forming process and uses three node membrane elements to model the sheet metal workpiece. The contact calculations area effected by using a cubic polynomial based description of the forming tool surfaces in combination with a predictor-corrector algorithm for

307 contact boundary conditions enforcement. During the dynamic explicit calculations, strains and stresses are calculated in every element using the combination of an explicit integration scheme, a plasticity flow theory, and an elastic-plastic material model.

3.1 Defect prediction The forming process simulation is used in order to predict the deformation behavior of the workpiece during a sheet metal forming operation. One of the major uses of our computer simulations of sheet metal forming processes is the prediction of two types of forming defects: tearing, due to excessive stretching of the workpiece, and buckling, due to in-plane compressive stresses. The methods used for the prediction of the occurrence of these defects are explained below. 9 Tearing prediction: To predict tearing, we use the concept of the Forming Limit Diagram (FLD), already introduced by Keeler (1964). The FLD of a material provides the magnitude of the major strain allowed prior to failure by localized thinning for a given value of the minor strain. A schematic representation of an FLD is shown in Fig. 7. Using the FLD of a material, and the calculated strains at an integration point of our finite element analysis process simulation, we can define the Forming Severity (FS) as shown in Fig. 7, i.e. (FS) = Emax/E* where Emax is the calculated maximum strain at a material point, and e* is the maximum allowed strain prior to failure by localized thinning. Therefore, a value of the forming severity higher than 1 indicates failure by tearing. During our finite element analysis simulation of the process, the highest value of the forming severity obtained in every material point is recorded as a state variable. Contours of forming severity can then be displayed upon completion of the simulation.

~max

o,

t

J

S~

~max

0 Minimumstrain, Fig. 7: A schematic representation of the Forming Limit Diagram.

308

Buckling severity: The measure of the buckling severity for the prediction of buckling initiation has been already introduced in Section 2.2.2. During the finite element analysis simulation of a sheet metal forming process, we calculate the principal stresses at every material integration point. Then, based on the Buckling Limit Diagram of Fig. 6b, we calculate the forming severity as ~applied/~crit, where ~applied is the absolute value of the minimum principal stress, and CYcdt is the critical stress for the initiation of small wavelength for a given normal blankholder pressure. The buckling severity is updated at every time increment of the finite element analysis only in the elements that are in the blankholder area and in which 0< ]~trans/(Yappliedl0, and ~appliedO.Ol~yo, where 6t~ans is the major principal stress. This range of stresses corresponds to a typical stress state in the blankholder area of a sheet metal forming operation. The maximum value of the buckling severity during the forming process simulation is recorded. Contours of the value of the buckling severity distribution in the workpiece can be plotted upon completion of the forming simulation. Note here that an element that is initially in the blankholder area can register high values of the buckling severity during the initial steps of the process and then draw in the forming area. In this case, the highest value Of the buckling severity will be retained throughout the process. This feature can be used to predict side wall wrinkles that initiate in the flange area and then draw in the forming area. However, secondary effects such as contact pressure changes and localized stress and strain gradients due to post-buckling behavior are not modeled.

3.2 Process simulation results

The finite element analysis simulation and the process defect prediction techniques introduced and discussed in this paper were used in the modeling of the forming of the square pan of Fig. 1. The contour of the buckling severity, as obtained from our finite element analysis, is shown in Fig. 8, where only a quarter of the actual part is presented. Only the values of buckling severity which are higher than 1.0 are displayed in this case, in order to identify the areas where buckling of small wavelength had initiated. Therefore all areas with a buckling severity contour level higher than 1.0 will incur small wavelength buckling during the forming process. By comparing Figs. 1 and 8, it can be seen that our prediction of buckling initiation is in excellent agreement with the experimental observation. We were able to successfully predict that all the flange will undergo the development of small wavelength wrinkles, as the buckling severity exceeds the value of 1.0 in the whole flange. Although not very clear in Fig. 1, wrinkles from the flange area were drawn in the forming area during the deep drawing process. This phenomenon was also predicted (compare Fig. 1 and Fig. 8). In general our model predicted the areas at which buckling of small wavelength occurred in very good agreement with the experimental observations on the test part of Fig. 1. Additional forming simulations and experiments need to be performed in order to further validate our buckling initiation modeling and process simulation capability used to predict the development of small wavelength wrinkles for different blankholder pressures and different forming geometry.

309

Fig. 8: A contour of the buckling severity of the part of Fig. 1 It is worth noting here that substantial tearing was also developed in the part of Fig. 1. In Fig. 9 we plot the forming severity of the part of Fig. 1 upon completion of forming. The maximum predicted value of severity was equal to 0.92, developed in the area of the vertical comer of the box, see Fig. 9. As it can be also seen in Fig. 1 this is an area where cracking has occurred. The maximum predicted forming severity was slightly lower than 1.0 (=0.92) indicating a high likelihood of failure by localized thinning. The area of high forming severity was also an area of buckling severity higher than 1.0 implying that both buckling and tearing were developed in the same area, compare Figs. 8 and 9. This prediction was verified by our experimental observations, see Fig. 1.

Fig. 9: A contour of the forming severity of the part of Fig. 1.

310 4. CONCLUSIONS A numerical method for predicting the onset of wrinkles of small wavelength was developed. This method uses an energy based approach to predict the contact pressure required to suppress small wavelength buckling in the flange area of a sheet metal formed part. Based on this approach, a stress based Buckling Limit Diagram (BLD) for a drawing quality steel was created. The Buckling Limit Diagram was then used in conjunction with a Finite Element Analysis code to predict the initiation of small wavelength buckling during the forming of a square box test part. Based on the BLD used and the stress calculations of the Finite Element Analysis code, we were able to successfully predict the initiation of small wavelength buckling. Our method was also capable of predicting side wall wrinkles that initiated in the flange area demonstrating the effectiveness of our approach.

ACKNOWLEDGEMENTS The authors would like to acknowledge the efforts of William Carter and Michael Graham of GE Corporate Research and Development in the development of the Finite Element Analysis code used in this work. Philip Stine of GE Appliances provided the square box test part and all the data for its forming.

REFERENCES Cao, J. and Boyce, M. (1997) Wrinkling behavior of rectangular plates under lateral constraint, Int. J. Solids Structures, 34 (2), 153-176. Elishakoff, I. and Cai, G.Q. (1994) Non-linear buckling of a column with initial imperfectio via stochastic and non-stochastic convex models, Int. J. Non-linear Mech., 29(1), 71-82. Keeler, S. and Backofen W.A., (1964), Plastic instability and fracture in sheet stretched over rigid punches, Trans. ASM, 56, 25-48. Hill, R. (1958) A general theory of uniqueness and stability in elastic/plastic solids, J. Mech. Phys. Solids, 6, 236-249. Hutchinson, J. W. (1974) Plastic Buckling, Adv. in Appl. Mech., 14, 67-144. Shahwan K. W., and Waas A. M. (1994) A mechanical model for the buckling of unilaterally constrained rectangular plates, Int. J. Solids Structures, 31(1), 75-87. Triantafyllidis, N. and Needleman A. (1980) An analysis of wrinkling in the Swift cup test, J. Eng. Mater. and Tech., 102, 241-248.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

311

F i l l i n g d e f e c t s in a c e r a m i c s f o r m i n g p r o c e s s F. Chinesta, R. Torres, I. Mont6n ~ A. Poitou b and F. Olmos c Universidad Polit~cnica de Valencia. Camino de Vera s/n. 46071 Valencia. Spain. bLMT, ENS de Cachan, 61 Avenue du President Wilson, 94235 Cachan Cedex. France. Universitat de Valencia. Campus de los Naranjos. 46071 Valencia. Spain. This paper whishes to present a simple semi-heuristic method to predict mold filling defects in a porcelain forming process. It involves both a mechanical modelling of this process as a squeezing flow, and a postprocessing calculus which unables a defects prediction as soon as the shape and the size of artistic reliefs are known. This shape analysis is carried out with a multiresolution wavelet analysis. 1. I N T R O D U C T I O N Traditionnal porcelain processing is mainly achieved in two steps. During the first one, a paste is formed in a mold with a process similar to "turning". The difficulty of this stage is to fill every superimposed drawing correctly (Figure 1). A second step consists in drying the molded part so that the schrinkage allows for its demolding. The aim of this paper is to model the filling stage and to propose a criterium to avoid any filling defect which could waste the quality of the drawings. In the following, section 2 deals with a simplified modelling of the process during which the "turning" stage is approximated by the deformation under pressure of a Norton Hoff ceramics paste squeezed between the mandrel and the averaged surface of the mold. The problem is solved with the Hele Shaw equations generalized to the case of a Norton Hoff's materials on an axisymetrical shell with appropriate boundary conditions. Section 3 proposes a criterium for a possible mold filling defect which can be applied as a postprocessing procedure as soon as the average velocity of the flow front has been computed and as the shape and location of the artistic reliefs are quantified. The technic of localisation and analysis of the reliefs involves a wavelet decomposition of the exact shape of the mold which is assumed here to have been previously numerized for its machining. 2. A V E R A G E M O L D F I L L I N G The mold filling operation is threedimensionnal. An excentered mandrel rotates both around its axis and around the axis of symetry of the bell. Its spacing from the midsurface of the mold is reduced progressively in order to control the width of the part (Figure 2). This stage has been modelled as follows.

312

.j i i ! |

I Z+ Z--

i |

|

I !

Figure 1" Porcelain bell

Figure 2: The forming process

2.1. G e o m e t r y E denotes the average mid-surface of the mold. It is a surface of revolution which is generated by rotation around the z axis of a curve parametrized by its curvilinear coordinate s. A point _x~ of this surface is parametrized by two orthogonal coordinates s and ~o 9 (1) We note 9 g-* =

g-v =

dx_r. ds =

dr(s) ds

cos~pi +

dr(s) ds

sin ~ j + d g - l ( r 2 ( s ) ) k -

= - r ( s ) sin~p i + r ( s ) cos~ j

d~

(2)

ds

(3)

Thus g_,.g_, = 1, g_ .g_~ = r 2 et g_,.g_~ = 0. The domain occupied by the material is = E x [-h, hi, so that a point in 1~ is denoted by _~ = ~

+ o~

(4)

In equation 4 _zr. is the projection of _z on E, 0 lies in [-h, h] and g-0 is the unit vector normal to E, defined by"

x~

(5)

= I1~ x ~ l l The equations are classically written in the orthogonal coordonate system (s, ~, 0) associated to the basis ( ~ , g_~,go) with use of the metric tensor g" 10

O)

g_ =

0

r2

0

-

0

0

1

(6)

313

2.2. T h e t h r e e d i m e n s i o n n a l m o d e l l i n g The material is assumed to be incompressible and to exhibit a Norton Hoff's behavior. Moreover, the contact between the mandrel and the paste is assumed to be perfect (i.e. a normal velocity as well as a zero shear stress are assumed to be prescribed), the contact between the paste and the mold is described by a Norton's law F:

ak"'lk = 0

# = k

with v.n_=0

(r

diJdij

)n-1

(flow front)

(n_.=a.n__)= F(Vg)

on E_

a,-k = _pg,.k + 2 #g mr g ksd m8

where

on E_

z . n = 0 on Eut a.n-

vkl k = 0

and

1

(7)

and

dij = ~ (viii + viii)

(8)

v.n=U

on E+

(9)

(n.=a.n) = 0 on E+

(10)

a.n-

where V_g is the slipping velocity on E_

(11)

2.3. H e l e S h a w e q u a t i o n s The threedimensionnal equations are solved with approximations similar as those described in [4]. These approximations are an extension of Hele Shaw equations and consists in searching, with use of the virtual work theorem, a solution for which the 0 dependance would be that of a simple shearing motion. Thus we search a velocity field v whose projection w on the tangent plane writes:

w__ =

v,p

= f(O) u + V_g = f(O)

u~(s, qp)

+

(12)

Va~(s,~)

with

f(O) = l + 2n { 1 _ ( O - h ) ~--~""} 1+ n

(2h) ~-t~-. "

(13)

w and Vg can then be written as functions of the pressure p which is assumed to depend on s and T only:

1

_w-

(2h) Itpl~ll~pl~ -~

with

2hplk =-F(V__g)

and

a=

h

21o1 o /14)

p(s, r is then calculated in solving the generalised Reynolds equation: (2h giJuj + 2h giJVgj) I` = U(s,~)

(15)

Equation 15 is solved numerically with a finite element method. At each time step, the computation provides the averaged flow front velocity _u_u,which is used simultaneously to actualise the flow front location and to predict with the following results an eventual mold filling defect.

314

Figure 4: Local variance pattern

Figure 3: Criterium for a mold filling defect 3. M O L D F I L L I N G D E F E C T S

3.1. C r i t e r i u m for a d e f e c t The above computation is achieved in assuming that the mold wall do not exhibit any artistic drawing. Thus it is impossible through this calculation to predict directly if the reliefs of the mold are correctly filled in. To answer this question, we propose here a simplified semi-empirical criterium. Let's assume that the averaged front velocity has been calculated in solving the Reynolds equation (equation 15). Let's assume moreover t h a t the material arrives at the edge of a relief (Figure 3). If to denotes the time required for the material, without accounting for the relief, to go from the entry to the exit of the relief in the flow direction (to = a/u). We state that a mold filling defect can possibly occur if the time to is not sufficient to fill in the area ab = S of the relief's section in the flow direction: ~otO

U s(t) dt =

~otO

t~ U u t dt = U u -2-

=

U a2 2 u




(26)

are the inertia resultants, and (-N2n; -m- 2 n ; n--~3) R~In) -" ;

V3--{P3>

(27)

denote the resultants of the applied tractions. Moreover, on F

(.).. = ne(.).e;

(.). = n.(.)o.;

(.).~ = n~(.):. - n:(.)~.

(28)

In Eq.(28), as well as in the forthcoming ones, the repetition of the indices n and t does not imply summations over these indices. Moreover, N [(k)Z+ 2a (~ + E ( ~ . u r,#2a (3) - C ~ u

(~ 3,r~r2a

1 O N;e,.

~U3,'xotU3,. + lZ3,rrU3,.oe) -(33)

401

1B

+~ ~

/ (o)

(o)

(0,0)g.(0) _~_(1)

+(1)rnt~-y ~t~,'v

D(3)u~-y.u(~ ~-~(3,3) U(r3)

---~,3"r3 PLATE

(o) (o) ~

~U3,r~olU3,~ .3ff ~3.~.~~

(0,3)-(3)

(2)

rn.y~ uo. v --

(o) + (N~,~3..)., (0,0)..(0)

rno.v u3,~.y

+ F(3'3) ~.u. - E(3).'v~uu(~ (0)

--

(0,3) ..(0)

rnt~~ ut~ +

MODELS

(0)

WITH

:

-t

__-+

--P3 - P3 +

_~_(0) m(O,O)fi(o)

+21D(3)'~'Y~u I, (0)

(3,3) ..(3) __(I)

m&v ut~

PERFECTLY

(0,3) ..(0)

m~

o

--M~.y,.y~

(34)

(0) + u3.~u3.u (0) O

u3,~ -R~.y,~

BONDED

(35) INTER-

FACES If in the displacement field (7) and in the subsequent derivation the effect due to interlayer slips, (/,~(xj), is neglected, the governing equations of multilayered plates with general layup and perfectly bonded interfaces are obtained. These equations correspond to the plate model developed and discussed in Di Sciuva [10] and Icardi and Di Sciuva [3].

6

NUMERICAL

RESULTS

AND

DISCUSSION

In order to shed light on the implications of interfacial defects and their distribution, the global structural response of solid cross-ply and sandwich beams of length L in the xl-direction and of unit width in the x2-direction is investigated. The beams are symmetrically laminated, simply-supported at both ends and are either loaded by a sinusoidally distributed transverse loading on the top surface, p~ = P3 = _pO sin ~L ' or compressed by inplane loading applied to the ends (for buckling analyses). In the solid cross-ply beams considered in the numerical illustrations, all the layers have equal thickness. In the sandwich beams, the faces are assumed to be made by single unidirectional layers; the core is assumed to be orthotropic. The material properties are displayed in the figures, where EL and ET are the elastic moduli of the individual layer in the directions parallel and normal to the fibres, respectively; GLTand GTT are the shear moduli, and lILT is the Poisson's ratio measuring transverse strain under uniaxial stress parallel to the fibres. Moreover, the indices f and c refer to the face and core of sandwich beam. The plotted results are normalized as follows

W* -- lOOu(~

O)ETsh3/(p~ 4) (36)

N~lc," - N11c,.L2/(ETjh2). Following Cheng et al. [4,5,6], we introduce the dimensionless sliding constant R defined by (k)R~ = 5 ~ (k)Rh/Et. The values attached to R stand for the values taken by R at all interfaces. Figures 2 and 3 refer to a three-layered symmetric cross-ply solid beams. Figure 2 gives the central deflection of the beam in bending, as predicted by small deflection theory, for different values of the length-to-thickness ratio, L / h and of the sliping constant, R. For the same beam, figure 3 gives the compressive buckling load parameter. As expected, the detrimental effect of sliping is, in percentage, more pronounced at lower values of the

402 slenderness ratio, L/h, where the transverse shear effects are more important. The results of figure 2 appear to be quantitatively similar to the ones given in Refs. [4]. The results plotted in figures 4 and 5 pertain the behavior of sandwich beams with stiff faces and soft core. It appears that the previuos conclusions apply also for this case, at least qualitatively. The results plotted in figures 3 to 5 appear to be novel in the open literature. Numerical work is in progress to quantitatively assess the effect of interfacial defects on the behavior of unsymmetric cross-ply beams under large deflections and on the natural frequencies.

7

CONCLUDING

REMARKS

A third-order zig-zag discrete-layer theory of laminated composite plates featuring interlayer slips was presented. The theory incorporates the dynamic and thermal effects as well as the geometric non-linearities. The pertinent equations of motion and consistent boundary conditions are derived by means of the dynamic version of the principle of virtual work. The theory represents a generalization of the first-order theory proposed by Schmidt and Librescu [1] and the third-order theory proposed by Cheng, et alii [4,5,6] for laminated plate featuring interlayer slips. The detrimental implications of interlayer slips on the flexural behavior and buckling loads of plates in cylindrical bending have been numerically emphasized and specific conclusions have been outlined. It is hoped that the present paper, together with the referenced ones by Schmidt and Librescu [1] and Cheng et alii [4,5,6] will stimulate further studies on the modeling of laminated composite structures featuring interfacial defects and will contribute to a better understanding of their implications upon the static and dynamic behavior of structures. A c k n o w l e d g m e n t . M. Di Sciuva and U. Icardi would like to thank the Consiglio Nazionale delle Ricerche for the partial support of this research by Grant CNR 96.01765.CTll. Liviu Librescu acknowledges partial support of this research by NATO Grant, CRG 960118.

References [1] Schmidt, R.; Librescu, L. "Geometric Nonlinear Theory of Laminated Anisotropic Composite Plates Featuring Interlayer Slips," Nova Journal of Mathematics, Game Theory and Algebra, 5(2), 131-147, (1996). [2] Di Sciuva, M. "A Generalization of the Zig-Zag Plate Models to Account for General Lamination Configurations.", Atti Accademia delle Scienze di Torino-Classe di Scienze Fisiche, Matematiche e Naturali, 128(3-4), 81-103 (1994). [3] Icardi, U.; Di Sciuva, M. "Large-deflection and Stress Analysis of Multilayered Plates with Induced-Strain Actuators.", Smart Materials and Structures, 5, 140-164 (1996).

403

[4] Cheng, Z.Q.; Jemah, A.K.; Williams, F.W. "Theory for Multilayered Anisotropic Plates with Weakened Interfaces," paper to appear in the J. Appl. Mechanics, Trans. ASME (1996).

[5] Cheng, Z.Q.; Kennedy, D.; Williams, F.W. "Effect of Interfacial Imperfection on Buckling and Bending Behavior of Composite Laminates," AIAA J., 33(12), 2590-

2595,

996).

[6] Cheng, Z.Q.; Howson, W.P. Williams, F.W. "Effect of Interfacial Imperfection on Buckling and Bending Behavior of CompositeLaminates," AIAA J., 33(12), 25902595, (1996).

[7] Librescu, L. Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures, Noordhoff Internt. Publishing, Leyden (1975).

Is] Di Sciuva M. "Bending, vibration and buckling of simply-supported thick multilayered orthotropic plates. An evaluation of a new displacement model," Journal of Sound and Vib, 105(3), 425-442, (1986).

[9] Di Sciuva M. "An Improved Shear-Deformation Theory for Moderately Thick Multilayered Anisotropic Shells and Plates," J. Appl. Mechanics, Trans. ASME, 54, 589596, (1987).

[10] Di Sciuva M. "Multilayered Anisotropic Plate Models with Continuous Interlaminar Stresses," Composite Structures, 22(3), 149-167, (1992).

[11] Di Sciuva M. "A Geometrically Nonlinear Theory of Multilayered Plate with Interlayer Slips," Submitted.

404

layer N

h

(k)h

layer k (k) z+

li

,a,er I

B

Figure1: Plategeometryand numberingof layersand interfaces.

5

18

4.5

Laminaeofequalthickness;Lay.up:0/90/0 MechanicalpropeYdesof unidirectionallamina EUEt=25;GLT/ET=O.5;GTT/ET=0.2;vLT=0.25

.it

4 .,.1... o 3.5

Data as in figure 2.

,~16 Z

R=0.2

,._-14 (3) ,.6.-, (1)

R=0

l::: 12

---~

3

R=0,6

"1= 2.5 ~

2

~

1.5

03 10

R=0.4

R=0.4

03 8 _oo

R=0.6

o) 6 .c::

R=0.2

4

1

::3 CX2

21

0.5 4

6

8

10

12

14

16

18

4

20

6

Length-to-thickness ratio, L/h

8

10

12

14

16

18

20

Length-to-thickness ratio, IJh

Rgure2: Simply-supportedbeamundertransversesinueoidalloading.

R~u'e3: Simply.supportedbeamundercompression..

16 9'

R=0.6

14,

R=0.4

.--12 .~

,

o10 c,

R=0.2

8

Sandwichbeam (h f=0.1h; hc=0.8h) Face lay-up: 0190R Materialproperties: Faces: EUET=25; GLTIET=0.5; GTTIET=0.2; v LT=0.25 Core: EUET=I" GLT/ET=0.4;

R=O

Data as in figure 4.

Z

R=0.2

(1) (1)

E 03 03

GTTIET=1.5;vLT=0.25

R=0.6

ETfacelETcore = 25

03

o

~4 m

R=O

!---2

_

r ::3

0 4

6

8

10

12

14

16

18

20

Length-to-thickness ratio, L/h Figure4: Simply-supportedbeamundertransversesinusoidalloading.

01 4

6

8

10

12

14

16

Length.to-thickness ratio, L/h RgureS:Simply.=pportedbeamunder~ n . .

18

20

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

405

Delamination, instability and failure of multilayered composites E. Stein and J. Tegmer Institute of Structural and Computational Mechanics, University of Hannover, Appelstr. 9A, D-30167 Hannover, Fed. Rep. of Germany A numerical method for the nonlinear analysis of thin-walled composite structures is presented within the finite element method. In addition to the static analysis, onset and propagation of delaminations are considered. The variational formulation is described by a multi-director approach with piecewise polynomial functions of the displacements. Initiation of delamination is controlled by a stress based fracture criterion. For its propagation the energy release rate is calculated as a further criterion. An isoparametric quadrilateral multilayer shell element is used for the numerical calculation. It leads to a comparable number of degrees of freedom as by using brick elements but additionally has some main advantages, as 2D-mesh generation is possible, coupling with conventional 2D-shell elements is easier, bending behavior is better, description of interlaminar interfaces is simple. For the kinematic expansion of delaminated zones, additional degrees of freedom are introduced. They are not adjoined to a distinct interface in advance of the delamination analysis. The theory holds for a complete 3D-stress state. Applications to glulam beams are given. 1. I N T R O D U C T I O N Composites are typically used for light weight structures. They are employed in many engeneering fields. Due to anisotropy and stacking sequences of laminates, i.e. heterogenities, the material shows rather complicated states of strains and stresses. Several models for thin composites have been developed over the past, and the generalisation of the elastic shear deformation theory for thin composites leads to the classic laminate theory which takes into consideration the effects of anisotropy and stacking sequence, e.g. coupling of bending and stretching, [1]. In many cases FE methods were applied, [2,4-7,10-15]. The problems of delamination have been studied in [2,4-7,11,13,15] because they have a major impact on the failure bahavior of the considered structures. Especially at free edges or cutouts, 3D-states of stresses arise, and especially the interlaminar stress components can induce delamination growth between adjacent layers. Typically for thin-walled structures, external loads don't lead to large or critical stresses in thickness direction of the structure. Such the existence of a delamination doesn't directly initiate failure of the system. Yet, under compression stresses a delamination can strongly diminish the critical buckling load of the structure, [2,4,7]. Other problems of delamination result from the brittle behavior of composite structures. Stress singularities, due to the ideal elastic material model or due to incompatibilities between adjacent layers with different fiber orientations, can lead

406 to zipper-like crack propagation. These effects have to be analysed if a further use of a structure is intended under admission of delaminations within certain bounds. Goodminded properties of classic materials like plastification of steel have led to many simplifying calculation rules in engeneering disciplines. These rules have to be reconsidered when composite materials are used. In this paper a multi-director shell element with a deformation mode in thickness direction is used for the nonlinear analysis of composite structures, [4,14]. The interpolation in thickness direction with piecewise polynomial functions over each layer is independent from the bilinear tangential interpolation within the middle surface. The complete 3D-stress tensor is computed by this model. The extended kinematics of delaminated elements are realised by introducing additional degrees of freedom at each node within the delamination zone. The onset of delamination is described by a stress based fracture criterion. Propagation is controlled by energy relase rates. Typically, in most parts of thin composite structures, normal stresses S 33 in thickness direction are much smaller than normal stresses in tangential direction. Such, S 3a is neglected in the classical laminate theory. Furthermore, the Kirchhoff-Love hypothesis holds for undisturbed subdomains which are usually predominant. In those regular parts of a structure a conventional one-director kinematic for the shell is sufficient. Therefore we couple standard 2D-shell elements with multi-director elements by using transition elements, [7]. In the sequal only the multi-director theory for shells is outlined and used. 2. V A R I A T I O N A L F O R M U L A T I O N For the analysis of composite materials we restrict the calculations to thin-walled beams and shells with a layerwise build up. The considered shells consist of n physical layers j with thickness h j and N n u m e r i c a l layers i. Therefore, a subdivision or collection of layers for the numerical calculation is possible. The position vector X0 of the reference surface So is labeled with convected coordinates | An orthonormal basis system tk(| ~) is attached to this surface where ta is a normal vector and | the coordinate in thickness direction. The transformation between the different base systems is given by tk(O~) = Ro(O~) ek

(1)

where R0 is a proper orthogonal tensor. Then the position vectors of the reference and the current configuration of the body are given by x(o

o

=

Xo(O

x(O

o

-

x(o

+ o t (o ,o

h , < 0 a < ho

(2)

+ u(O ,

The kinematics of the shell are based on the assumption of a multiplicative decomposition of the displacement field in shell space with independent shape functions in thickness direction and shape functions in tangential direction, defined on the reference surface of the shell. For the numerical layer i the displacement vector is interpolated through the

407 1ilickIless, [14]

u'(O ~, o ~)

-

~~(o

~) u~ -~(o~) _ ~ ( o ~ ) a ~ ( o

~)

(3)

l=1 f l i ( o a)

__

[ f l ~ , - u2, i

" T ,fi~]

...

(2<m
_ Gc: The energy criterion is fullfilled for some nodes of the trial crack area.

----~T*CT. If

~

A TrK AAK > Gc

set

T* = T

~

proceed with Step 2,

(20)

K

('Condition A' in figures 5 and 6), else close nodes K C { T \ T*} and compute the energy release rate for the remaining trail crack area.

5. E X A M P L E We consider a curved glulam beam under prescribed deflection w of the apex as shown in figure 3. For the numerical analysis one half of the system is discretized due to symmetry. We apply the deflection in 20 load steps. Results are presented in figure 4 to 6. The load-deflection curve F(w) is given in figure 4 and shows the influence of growing cracks which ends at point E.

412

~W

I3,0m

E s= 11000 M N / m**2 Et = 300 M N / m**2 G = 500 MN / m**2 v=0,3 Zo= 0,2 MN / m**2 Ro= 0,9 MN / m**2 Gc= 0,001 MN / m

crack s u ~ ~ ~ ,

3

t,Om

....

,Om

O,lm

lO,Om

Figure 3. System of a curved Glulam Beam with material data, displacement controlled loading at apex. The crack begins in point P and grows in both directions until the points Q and R.

-0.3

GlulamBeam M e s h 6~ w i t h Q u t c r a c k ~ M e s h 6, w l t h crack Mesh 6, energy release rate-m--

0.25

In figure 5 and 6 different finite element meshes are compared. They have one element in y-direction, 10 layers in thickness direction and a variable number of elements in s-direction. Mesh Mesh Mesh Mesh Mesh Mesh

1: 2: 3: 4: 5: 6:

10 elements in s-direction 20 elements in s-direction 40 elements in s-direction 80 elements in s-direction 160 elements in s-direction 320 elements in s-direction

-0.2

0.15

0.05

0

0

i

-0[05 -0'.1 w(s-20.115tm I -o'.2

i

-0 25

-0.3

Figure 4. Load-Deflection curve for Glulam Beam without crack, with prescribed crack and with calculated crack-propagation until the points Q and R in figure 3 which corresponds to point E.

413

Glulam

Beam

-0.2 t

~

., . Mesh Mesh

~

I

$

'

{

~'"

.~

~g

~''~&

-e--~-. -B--

Mesh

-.~.....

Mesh

~--

Mesh

~ ....

-0.15

o" u

§

-0.I

l .....

o

~- ~ -~-..-4-.~

........

.............=.= ......................

o

o

§ ,~

-0.05

.~i o o i

0 0

i

1

i

2

3

crack

lenght

1

i

i

4 [m]

5

6

Figure 5. Energy-Release-Rate-algorithm for Glulam Beam with 'Condition A' in (20)

GlulamBeam

-0.2 ti $

,---, ,..., 4~

'

~

~

~

~

,

.

:

+

~

~

'

?,-0

15

-t

~o

+

T

"~~o -o.1

~" ~....

~ .....

_ ......

iMesh + Mesh Mesh ~ Mesh ," M e s h

~-....

3-s-~ ......... ~--

+

a--

_~ .....

;~i..................+;'.....................~""; ~/ :::::::ii :~...........................:::::::::::: ~,.---

.~

.............• .............x"

..$

'13

-0.05 o o l

0 0

1

i

2 crack

i

i

3 lenght

1

4 [m]

i

5

Figure 6. Energy-Release-Rate-algorithm for Glulam Beam w i t h o u t 'Condition A' in

(20) Acknowledgment The financial support of the Deutsche Forschungsgemeinschaft (DFG) for project Ste 238/25-TP.2 is gratefully acknowledged.

414 REFERENCES

1. ALTENBACH, H., ALTENBACH, J. and RIKARDS, R. Einf~hring in die Mechanik der Laminat- und Sandwichtragwerke, Deutscher Verlag f~r Grundstomndustrie, Stuttgart, 1996. 2. COCHELIN, B. and POTIER-FERRY, M. A numerical model for buckling and growth of delaminations in composite laminates. Comput. Methods Appl. Mech. Engrg. 89, 361-380, 1991. 3. DVORKIN, E. N. and BATHE, K.-J. A continuum mechanics based four-node shell element for general nonlinear analysis. Engineering Computations, 1, 77-88, 1984 4. GRUTTMANN, F. and WAGNER, W. On the numerical analysis of local effects in composite structures. Composite Structures, 29, 1-12, 1994. 5. GRUTTMANN, F., STEIN, E. and WAGNER, W. A Generalized FE-Method for Non-Linear Composite Shells with 2D- and 3D-Modeling, in S.N. Atluri, G. Yagawa, T.A. Cruse (Eds.), Proceedings of the Int. Conf. on Comput. Eng. Science, 2533-2538, Hawaii, 1995. 6. GRUTTMANN, F. Theorie und Numerik dfinnwandiger Faserverbundstrukturen. Habilitationsschrift am Fachbereich Bauingenieur- und Vermessungswesen der Universit/it Hannover, 1996. 7. GRUTTMANN, F. and WAGNER, W. Coupling of 2d- and 3d-composite shell elements in linear and nonlinear applications, Comput. Methods Appl. Mech. Engrg. 129, 271-278, 1996. 8. HAHN, H.G. Bruchmechanik, Teubner Verlag, 1976. 9. HASHIN, Z. Failure criteria for unidirectional composites, Journal for Applied Mechanics,47, 329-334, 1980. 10. KLARMANN, R. Nichtlineare FE-Berechnungen yon Schalentragwerken mit geschichtetem anisotropen Querschnitt, Heft 12 d. Schriftenreihe d. Inst. f. Baustatik, UniversitS~t Karlsruhe, 1991. 11. KRUGER, R. Delaminationswachstum in Faserverbundlaminaten, Bericht 13-96 d. Inst. f. Statik u. Dynamik d. Luft- und Raumfahrtkonstruktionen, Universitgt Stuttgart, 1996. 12. LOGEMANN, M. Abschs der Tragfiihigkeit von Holzbauten m. Ausklinkungen und Durchbrfichen, Fortschritt-Berichte Reihe 4, Nr. 102, VDI, Dfisseldorf, 1991. 13. RIKARDS, R., BUCHHOLZ, F.-G. and WANG, H. Finite element analysis of delamination cracks in bending of cross-ply laminates, Mechanics of composite materials and structures 2, 281-294, 1995. 14. ROBBINS, D. H. and REDDY, J. N. Modeling of thick composites using a layerwise laminate theory. International Journal for Numerical Methods in Engineering, 36, 655-677, 1993. 15. WAGNER, W. and GRUTTMANN, F. A Computational Model for the Delamination Analysis of Composite Shells, in D.R.J. Owen, E. Ofiate (eds), Proceedings of the Fourth Int. Conf. on COMPUTATIONAL PLASTICITY: Fundamentals and Applications, 1191-1202, Barcelona, 1995.

Advanced Methods in Materials Processing Defects M. Predeleanu and P. Gilormini (Editors) 9 1997 Elsevier Science B.V. All rights reserved.

Statistical Damage Tolerance for Cast Iron Under Fatigue Loadings H. Yaacoub Agha a, A.-S. B~ranger b, R. Billardon a and F. Hild a aLaboratoire de M~canique et Technologie E.N.S de Cachan / C.N.R.S / Universit~ Paris 6 61, Avenue du President Wilson, F-94235 Cachan Cedex, France. bRenault - Direction de la Recherche- Service 60152 860 quai Stalingrad, F-92109 Boulogne Billancourt, France.

In this paper, a statistical model accounting for the presence of initial flaws is introduced to study the fatigue failure of Spheroidal Graphite cast iron. An expression of the cumulative failure probability of a structure is proposed for cyclic loading conditions. The proposed model uses a modified Paris' law and is valid when no new flaw nucleates during cycling. An identification procedure is developed to determine the flaw distribution as well as the crack growth law from a series of standard fatigue tests. A post-processing approach is developed to study the failure probability of any complex structure, and applied to analyze a suspension arm.

1. I N T R O D U C T I O N

For competitive reasons the automotive industry tries to reduce the cost of its products without repercussions on the final quality of the cars. The cost may be reduced by using cheap manufacturing processes such as casting, where complex shapes can be obtained with little difficulties and the costs are lower than for any other manufacturing procedure such as machining. Due to its good properties, the Spheroidal Graphite cast iron is widely used now in automotive industry for safety components. For example, it is utilized in ground link elements such as steering knuckle holder, suspension arms. Despite the developments of the production technology, flaws (e.g. pin-holes, shrinkage, cavities) are unavoidable. Flaws occur in solidifying cast due to negative pressures generated during solidification contraction, and pressure developed by gases dissolved in the molten metal. These flaws are usually undesirable and are randomly distributed within the material. The cast components are frequently subjected to high cycle fatigue conditions, and their fatigue strength may be reduced by the presence of these initial casting flaws whereas, conventional design procedures to assess the structural integrity of a component use deterministic crack initiation criteria and ignore micro-inhomogeneities within the material. It is important when studying the fatigue properties of cast material to consider the scatter of experimental

416

results which is observed and develop an evaluation method which accurately calculates the effect of casting flaws. In this paper, it is proposed to model the presence of these inhomogeneities, their possible evolution with the number of cycles, and if needed their statistical distribution. An expression of the cumulative failure probability of a component is proposed on the basis of a modified Paris' law. This failure probability is related to the statistical distribution of flaws. An identification procedure is developed to determine the flaw size distribution as well as the micro crack propagation law. This method is applied to an SG cast iron supplied by Renault car company. The identified results obtained from fatigue tension-tension tests are then compared with independent experimental results. A post-processing approach is developed to study the failure probability of structures, such as a suspension arm.

2. R E L I A B I L I T Y OF S T R U C T U R E S

CONTAINING

FLAWS

It is assumed that initial flaws are randomly distributed within a structure and that the flaw distribution is characterized by a probability density function f. The function f depends upon the flaw size a. Other parameters such as orientation are not considered, since only flaws in pure mode I are considered (i.e. with their orientation perpendicular to the maximum principal stress). In the case of cyclic loading conditions, the evolution of the flaw size leads to the evolution of the flaw size distribution. After N cycles, it is assumed that the flaw distribution is described by a function fN 9 At this stage, it is useful to introduce a function ~ that relates the initial flaw size a0 to the flaw size after N cycles aN

ao = ~(aN)

(1)

If the flaw size evolution is deterministic, the probability of finding a flaw of size aN after N cycles is equal to the probability of finding an initial flaw of size q~(aN). Therefore the cumulative failure probability PRO can be written as [i] PRO =

f~

(ac) fo(a)da

(2)

where 9 (ac) denotes the initial flaw size that, after N cycles of loading, reaches the critical flaw size ac. The flaws are supposed to be described bycracks of size a, whose geometry is taken into account by a dimensionless factor Y such that the energy release rate G is given by y2a2a

G=

E

(3)

where a stands for an equivalent uniaxial stress (for instance the maximum principal stress). It is worth noting that the values of the parameter Y depend upon the geometry of the initial defect and the fact that this flaw intersects or not a free surface. Under monotonic and cyclic loading conditions, local failure can be described by a criterion referring to a critical value of the energy release rate

G _> G

(4)

417 In the case of ductile materials subjected to cyclic loading conditions, stable crack propagation can be described by a generalized Paris' law [2]

da ~_ C (V/~maxg!~)" = -V/-'~thl n .V/ P~cT _ ~g(R) /

(5)

dN

where Gmax (resp. gm~) stands for the maximum (resp. minimum) energy release rate over one cycle, Gth denotes the threshold energy release rate under which ~maxg(R) < Gth) no propagation occurs, and N the number of cycles. The parameters C and n are material dependent, and the function g models the influence of the load ratio R = ~~m/n " An expression for the function g has been proposed by Pellas et al [2] 1-R g(R) = 1 m R

(6)

-

where m is a material parameter. From Eq. (5) the following closed-form solution can be derived [1]

)(

- ~ \ V aM

)n(

V~c -- ~ g(R)

Sth ] NF

(7)

where aM denotes the maximum flaw size in the structure, the dimensionless constant C* is equal to ~--~u' c and Sth the cyclic threshold stress, defined as the lowest value of the stress level below which no failure occurs (i.e. the failure probability is equal to zero). The cyclic threshold stress, Sth, is related to the threshold energy release rate Gth. Its expression, when g(R) = 1, is given by

1 ~/EGth Sth = Y V a----M

(8)

The value of the function ~ depends upon the power n. When n -~ 1 and n -~ 2, the function ~ is given by

~(x) -- 2 ( x - Xth)l-n(Xth --(TL- 1)X)

(n- l)(n- 2)

(9)

where Xth is the normalized threshold defect size given by

Xth =

~

a~h

aM

=

S~h amo~g(R)

(10)

If the interaction between flaws is negligible, an independent events assumption can be made. The expression of the cumulative initiation probability P1 of a structure ft of volume V can be derived in the framework of the weakest link theory. The expression of P1 can be related to the cumulative failure probability PRO of a single link by

418

In the case of global unstable propagation, the structural failure corresponds to the initiation and the expression of the cumulative failure probability PF is given by

PF = PI

(12)

It is worth noting that in the case of high cycle fatigue, the propagation stage tends to become negligible when compared, in terms of number of cycles, with the initiation stage (i.e., local failure). Since the propagation stage is neglected when Eq. (12) is used, this equation corresponds to a lower bound to the cumulative failure probability of the structure. Hence, in the following, 'failure' refers to local failure i.e. macroscopic initiation. 3. A N A L Y S I S

OF A FATIGUE

TESTS

In this section, a series of experiments performed on specimens made of ferritic SG cast iron are analyzed in details. These experiments have been carried out at different stress levels. The ratio between the threshold energy rate and the critical energy rate is on the order of 1/9. The specimens contain 'controlled' initial flaws. These specimens are tested under cyclic tension with two different load ratios (R : -I,R = 0.i). Each curve of a standard S-N plot can be associated with a constant failure probability. When the fatigue limits are known, the identification can be performed in two different stages. The first stage consists in the identification of the flaw size distribution. A minimization scheme is used to determine the minimum error between all the available experimental data on fatigue limits [3]. If we assume that the maximum flaw size is bounded by aM, the flaw size distribution f0 can, for instance, be fitted by a beta distribution

fo(a)=

aa-l(aM_a) ~-1 B , ~ a ~+z-1 , w h e n 0 < a < a M ,

a > 0, /3 > 0

(13)

where a and ~ are the parameters of the beta function, and B~Z is equal to B(a,/3), and where B(., .) is the Euler function of the first kind. The parameters to identify are the powers c~ and fl of the beta distribution, the volume ratio ~, v and the threshold stress Sth. The first step of the identification is applied to the experimental results obtained for the load ratio R = 0.I. It leads to the following values: a : 4.7, /3 = 25, ~V = I, and Sth -- 105 MPa. The second stage of the identification concerns the crack growth law (parameters C, n and m). In tension, this identification is performed by studying one constant cumulative failure probability (e.g. 50~ A constant value of PF is described by a constant cumulative failure probability PRO. The cumulative failure probability 5070 is used to minimize an error function [3]. The following values are obtained: n : 2.0, and C* = : 5.9 x 10 -5. In Figure I predictions of the number of cycles to failure are compared with the experimental observations. Three points were used for the identification and the remaining points are predictions. This figure shows that the identified laws are in good agreement with the experimental results.

419

~" 400 ----350

'~ ~w.

,.

.......

,

9 PF=90 % (Experiments)

........

-'QQ.~.~ " "~$~

...__

~300 "o, " T : ~

9mma

9

PF = 50% (Experiments)

9

PF = 1 0 %

(Experiments)

250 r~

200 o~,,~

o

9 9o 9

- -o-

- PF = 90

%

(Identification)

150

" " 0 " ~ PF = 5 0 %

(Identification)

:~ loo 105 106 107 10 4 Number of cycles to failure, N F

.... o " - PF = 10 %

(Identification)

Figure 1. Predicted failure probabilities compared with experiments (R - 0.1).

4. V A L I D A T I O N O F T H E I D E N T I F I E D

MODEL

The previous results are validated by comparing them with other experimental results obtained independently. The two stages of the numerical identification are compared separately. The flaw distribution is compared with an experimentally identified one and the propagation law is also compared with a measured propagation law. Finally these results are used to predict other experimental results. 4.1. Flaw distribution Experimental investigations are performed to quantify the identified flaw distribution, and also to get more information about the distribution such as the size of the largest defect aM. Systematic microscopic observations of 50 fractured surfaces were performed using a Scanning Electron Microscope. The initial defects on the fractured surfaces can be distinguished with no difficulty since the stable propagation area has different morphological characteristics as compared to those of the initial defects. Pictures of the fractured surfaces were stored in a SUN workstation and an image analysis program was used to determine the defect distribution. Flaws with a diameter less than 80 pm were not considered in order to avoid confusion with graphite nodules (with maximum size on the order of 60 pm in diameter [4]). In Figure 2 the experimental flaw distribution is drawn and compared with the identified distribution. The identified distribution is in good agreement with the experimental one. The experimentally measured value of aM is 400 #m. This result shows that we are dealing with short cracks, and that the threshold energy release rate can be considered as a constant for flaws of this size as shown in [5].

420

Identification

o

7

'

0

t

'

'

'

t

'

'

'

I

Image analysis '

'

'

J

'

'

'

9

6 =

=

9

5 4

m

3

2

o Z

inmmhmmmmmimmmmm-,.m~

0 0

0.2 0.4 0.6 0.8 Normalized flaw size, a/a M

Figure 2. Predicted failure probabilities compared with experiments (R = 0.1). Identification

[]

a o = 6. mm

ao = 0 . 2 4 m m

o

ao = l . m m

10 -7

10_8

-~

10_ 9

~a

Q

10-10

~

) (3

10-1] 10-12

[]

t' o ,.

, A,

A

, ,A

I

2 4 6 8 10 20 Stress intensity range, AK (MPa m 1/2)

Figure 3. Crack growth rate as a function of stress intensity range.

421 4.2. P r o p a g a t i o n

law

Figure 3 shows the crack growth rate as a function of the stress intensity range. The solid curve is the identified one. The other curves represent experimental results obtained on specimens made of SG cast iron. The solid squares concern an artificial short crack of initial length a M - - 240 #m [6]. The open circles correspond to an artificial crack of initial size a0 = 1 mm. The open squares concern an artificial long crack of initial length a0 = 6 mm. [6]. The identified curve is in good agreement with the experimental one for short cracks especially near the threshold regime. The distinction with the curve for long cracks is mainly described by threshold differences. 4.3. P r e d i c t i o n

of experimental

results

The comparison between the threshold values calculated by Eq. (11) for the load ratio R = - 1 and R = 0.1 allows to identify the value of parameter m (m = 0.59). Figure 4 shows the comparison between the experimental results for R = - 1 and the predicted results using the parameters identified previously (R = 0.1) and m. The predictions are

400 350 E

'

'

'

9

....

I

.

.

.

.

.

.

.

.

I

.

.

.

.

.

.

.

9

PF = 90 %

9

PF = 50 %

(Experiments)

.

(Experiments)

o * 9149

300

9

250

--o--PF=

~g

PF= 10% (Experiments) 90 %

Prediction

200 . . . . . . . . . "0. . . . . . . . -0 o~-~

150

0 O001D

----c)--- PF = 50 %

Prediction

9 O00O~ .... 0.--- PF = 10 %

100

' ' ....... 104 105 106 10 N u m b e r of cycles to failure, N F . . . . . . . .

. . . . . . . .

Prediction

Figure 4. Predicted failure probabilities compared with experiments (R = - 1 ) .

in reasonable agreement with the experimental data. This result shows that function g accounts for the influence of load ratio for different cumulative failure probabilities. 5. A P P L I C A T I O N

TO A STRUCTURE

The identification, validation of the present model allows to make an extension of the model to real structures. A post-processing program is developed to compute the failure

422 probability of a cast structure. Figure 5 shows the flow-chart of the program called ASTAR.

Loading

ALTAR I (Stress fieldand volumeof eachelemen9

~

uivalent

stressfield at each integrationpoint)

~umerical computationof ~(ac)) i

~~"u~eprobaUilityofeachelement9 ~ ( [

ii

Flaw ) distribution

(Failureprobabilityof thestructurePF)

Figure 5. Flow-chart of the program ASTAR.

From the results of an elastic computation of a structure through a Finite Element analysis, ASTAR evaluates the equivalent stress at each integration point. A critical flaw size is associated to this stress level and to a given number of cycles, N, which corresponds to the flaw size which becomes critical after N cycles. The failure probability at the integration point is then calculated by numerical integration of Eq. (2). The integration over the total volume of a finite element gives the failure probability of the element. Then the failure probability of the structure can be calculated according to Eq. (II). This procedure has been applied to a suspension arm designed by Renault car company. The mesh consists of 3712 triangular shell elements. The industrial FE package

423

Figure 6. contours of the failure probabilities PROfor a maximum stress level on the order of 300 M P a and a number of cycles N = 107 cycles.

424 ABAQUS [7] is used to perform the elastic analysis of the structure. ASTAR is run by using the results of the FE computation. Figure 6 shows the contours of the failure probabilities PFO for a maximum stress level on the order of 300 MPa and a number of cycles N - 107 cycles. It must be noted that this procedure could take account of different flaw size distribution in different parts of the structure.

6. C O N C L U S I O N S A reliability analysis taking account of flaw size distributions has been developed for components subjected to cyclic loading conditions. Emphasis is put on the initiation stage, which is directly related to the evolution of initial flaws. An expression of the cumulative initiation probability is derived in the framework of the weakest link theory and by assuming that the flaws do not interact. Experimental data on SG cast iron in tension are analyzed within this framework. The predictions of the whole set of data is in good agreement with the experimental number of cycles to failure. This last result shows that the expression of cumulative failure probability proposed herein is able to model fatigue data obtained on SG cast iron and that the model can describe the influence of the load ratio. Typical applications of this approach concern the reliability analysis of cast components. More and more tools are available to predict the different flaw size distributions in the different parts of a cast, whereas the fatigue behavior of the material is in general mainly derived from experiments performed on so-called flawless specimens. These two tools should eventually be coupled. The post-processing approach can predict the reliability of the whole component under cyclic loading conditions, in other words, it enables to predict the number of cycles to macrocrack initiation (local failure), or the probability of reaching a certain number of cycles without failure at any point of the component.

ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of Renault through contract CNRS/109 (H5-24-12) with the Laboratoire de M~canique et Technologie, Cachan. REFERENCES

1. F. Hild and S. Roux, Mech. Res. Comm. 18(6) (1991) 409-414. 2. J. Pellas, G. Baudin and M. Robert, Recherche A~rospatiale 3 191-201 (1977). 3. A.-S. B~ranger, R. Billardon, F. Hild and H. Yaacoub Agha, FATIGUE '96, Berlin (Germany) (1996)1269-1274. 4. P. Clement, J. P. Angeli and A. Pineau, Fatigue Eng. Mat. Struct. 7 (4) (1984) 251265. 5. P. CMment and A. Pineau, Journ~es Internationales de printemps de la SFM, Paris 22-23 May (1984) 203-218. 6. P. Clement, CNAM Report (1984) Paris. 7. H.D. Hibbitt, B. I. Karlsson and P. Sorensen, Abaqus, version 5.5. (1995).

425

AUTHOR INDEX

A g a s s a n t , J . E ..................... 373 A h m e d , M ......................... 197 A l e x a n d r o v , S .................... 247 A l m e i d a C a m a r g o , N . H ...... 83 B a c r o i x , B ......................... 331 B a d e a , L ................................ 3 Bagaviev, L ........................ 281 B a m m a n n , D.J ..................... 99 B a n a b i c , D ......................... 257 Barlat, E ............................ 265 Barri~re, T .......................... 165 B a u d e l e t , B ........................ 289 B e g u m , S ........................... 143 B e n n a n i , B ........................ 165 B6ranger, A.S ................... 415 B i l l a r d o n , B ....................... 415 B i t t e n c o u r t , E ...................... 83 B o g a t o v , A . A ....................... 71 Boivin, M .......................... 341 B o u d e , S ............................ 125 B o u d e a u , N ........................ 215 B o u r g a i n , E ......................... 23 Boyer, J.C ............................ 13 B r e m , J.C ........................... 265 B r e s s a n , J.D ....................... 273 B r e t h e n o u x , G ..................... 23 Brunet, M .......................... 205 B u b l e x , E .......................... 341 C a o , J ................................. 301 C e s c o t t o , S ........................... 33 C h a n d r a , A .......................... 89 C h a r l e s , J . E ......................... 33 C h i k a n o v a , N ..................... 247 C h i n e s t a , E ...................... 311 C h i o u , J . M ......................... 135 C h u n g , K ........................... 265 C o m b e s c u r e , A .................. 385 D a w s o n , P.R ........................ 99

D e n g , X ............................. 341 Di Sciuva, M ..................... 395 D o e g e , E ............................ 281 D o h r m a n n , H ..................... 281 D o l t s i n i s , I.St ..................... 111 D o m i l o v s k a y a , T.V. ............. 71 Drazetic, E ........................ 165 Dutilly, M .......................... 321 E1 M o u a t a s s i m , M ............. 341 F e d o t o v , V.E ............. ........... 51 Gelin, J.C .................. 215, 321 G i l o r m i n i , P. ...................... 331 Giusti, J ............................... 23 H a b r a k e n , A . M .................... 33 Hall, E R ............................ 135 H a m b l i , R .......................... 125 Hartley, E .......................... 135 H a s h m i , M . S . J ............ 143,197 Hild, E ............................... 415 H u a n g , Y. ............................. 89 Icardi, U ............................. 395 Karafillis, A ....................... 301 K a r i m , A . N . M ................... 143 Karr, D . G ........................... 225 K i m , Y.S ............................ 155 Kiselev, A . B ......................... 43 K o l m o g o r o v , V.L ........... 51, 61 L a u r o , E ............................ 165 L e g e , D.J ........................... 265 Li, Y. .................................. 185 L i b r e s c u , L ........................ 395 M a z a t a u d , E ........................ 23 M g u i l - T o u c h a l , S .............. 205 M o n t 6 n , I ........................... 311 M o r e s t i n , E ............... 205, 341 M o s h e r , D . A ........................ 99 M u r a t , M ............................. 83 M u z z i , M ............................. 23

O l m o s , E ........................... 311 Ofiate, E ............................. 349 O s t r o w s k i , M ..................... 301 O u d i n , J ..................... 165, 175 Park, J.Y. ............................ 155 Picart, E ............................. 175 Piechel, G .......................... 175 Pillinger, I .......................... 135 Poitou, A ............................ 3 1 1 Potiron, A .......................... 125 P r e d e l e a n u , M ....................... 3 P r o u b e t , J ........................... 289 R a b e e h , B .......................... 185 R e e s , D . W . A ...................... 235 R e s z k a , M .......................... 125 R o j e k , J ......... ~.................... 349 R o k h l i n , S.I ....................... 185 Smirnov, S.V. ................ 61, 71 S o b o y e j o , A . B . O ............... 185 S o b o y e j o , W . O .................. 185 Spevak, L . E ......................... 51 Staub, C ............................... 13 Stein, E .............................. 405 S u b h a s h , G .......................... 89 TefSmer, J ........................... 4 0 4 Torres, R ............................ 311 Traversin, M ........................ 33 T s z e n g , T.C ....................... 361 Venet, C ............................. 373 Vergnes, B ......................... 373 Vilotic, D ........................... 247 W a n g , K . E ........................... 89 W i m m e r , S . A ..................... 225 Wu, W.T. ........................... 361 Y a a c o u b A g h a , H .............. 415 Z h u , Y.Y. .............................. 33

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