New Trends in Thin Structures: Formulation, Optimization and Coupled Problems

CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau

The Secretary General Bernhard Schrefler - Padua

Executive Editor Paolo Serafini - Udine

The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 519

NEW TRENDS IN THIN STRUCTURES: FORMULATION, OPTIMIZATION AND COUPLED PROBLEMS

EDITED BY PAULO DE MATTOS PIMENTA UNIVERSITY OF SÃO PAULO, SÃO PAULO, BRAZIL PETER WRIGGERS LEIBNIZ UNIVERSITY OF HANOVER, HANOVER, GERMANY

This volume contains 67 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2010 by CISM, Udine Printed in Italy SPIN 80013683

All contributions have been typeset by the authors.

ISBN 978-3-7091-0230-5 SpringerWienNewYork




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A Plate Theory as a Mean to Compute 3D Solutions

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18

O. Allix and C. Dupleix-Courdec

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A Plate Theory as a Mean to Compute 3D Solutions

19

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20

O. Allix and C. Dupleix-Courdec

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A Plate Theory as a Mean to Compute 3D Solutions

21

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A Plate Theory as a Mean to Compute 3D Solutions 34 nby 34

25     





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26

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28

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A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type Paulo M. Pimenta, Edgard S. Almeida Neto and Eduardo M.B. Campello Department of Structural and Geotechnical Engineering, Polytechnic School at the University of São Paulo, São Paulo, Brazil {ppimenta, almeidae, campello}@usp.br Abstract This work presents a fully nonlinear Kirchhoff-Love shell model. In contrast with shear flexible models, our approach is based on the Kirchhoff-Love theory for thin shells, so that transversal shear deformation is not accounted for. We define energetically conjugated cross-sectional generalized stresses and strains. The fact that both the first Piola-Kirchhoff stress tensor and the deformation gradient appear as primary variables is also appealing. The weak form of the equilibrium equations and their boundary conditions of the model are consistently derived. Elastic constitutive equations are obtained from fully threedimensional finite strain constitutive models in a consistent way. A genuine plane-stress condition is enforced by the vanishing of the mid-surface normal nominal stress (first Piola-Kirchhoff stress), yet rendering a symmetric linearized weak form. A plane reference configuration is assumed for the shell midsurface, but, initially curved shells can be accomplished, if one regards the initial configuration as a stress-free deformed state from the plane position. As a consequence, the use of convective nonCartesian coordinate systems is not necessary and only components on orthogonal frames are employed.

1 Introduction The main objective of this work is to present a fully nonlinear finite deformation thin shell formulation, which can be employed by numerical methods. To this end, the geometrically exact shell formulation developed in Pimenta (1993) and Campello et al. (2003), which represents an alternative to the work initiated by Simo et al. (1990), is now constrained to obey the Kirchhoff-Love assumption. As in Pimenta (1993) and Campello et al. (2003), our approach defines energetically conjugated generalized cross-sectional generalized stress and strains based on the concept of a shell director. Besides their practical importance, cross section quantities

30

P.M. Pimenta et al.

make easy the derivation of equilibrium equations in weak and strong sense, as well as the achievement of the tangent weak form, which is always symmetric for hyper-elastic materials and conservative loadings, even far from an equilibrium state. On the other hand, the concept of director allows the introduction of a parameter that describes thickness variation, as done in Campello et al. (2003). This is useful for the derivation of shell constitutive equations from 3-D finite strain constitutive equations by applying a simple consistent plane stress condition, which does not destroy the symmetry of the tangent operator. A plane reference configuration was assumed for the shell. Initially curved shells can then be regarded as a stress-free deformation from this configuration. This approach was already employed for rods and shells in Pimenta (1996) and Pimenta et al. (2009). It precludes the use of convective non-Cartesian coordinate systems and other complicate entities like Christoffel symbols and the second fundamental form. It simplifies the comprehension of tensor quantities as well, since only components on orthogonal systems are employed. Throughout the text, italic Greek or Latin lowercase letters a , b, !, B, C, ! denote scalars, bold italic Greek or Latin lowercase letters a, b, !, B, C, ! denote vectors and bold italic Greek or Latin capital letters A, B, ! denote second-order tensors, as well as linear operators built with them. Summation convention over repeated indices (subscripts) is adopted in the entire text, whereby Greek indices range from 1 to 2, while Latin indices range from 1 to 3.

2 Nonlinear Kirchhoff-Love shell theory 2.1

Kinematics

It is assumed at the outset that the shell is plane at the initial configuration, which is used as reference. This formulation can be straightforwardly used for plane finite elements. Let E 8  \e1r , e2r , e3r ^ be an orthogonal system. The vectors r eB , B  1,2 , are placed on the reference middle plane of the shell, as shown in Figure 1. Thus, e3r is orthogonal to this plane. The position of the shell material points in the reference configuration can be described by Y  [ ar ,

(1)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

31

where [  YBeBr

and

a r  [e3r .

(2)

[ is contained in the middle plane and a r is the director.



e3



e2 a

e1



x z

e r3



ar

Y



[

e 2r e1r

O

Figure 1. Shell description and basic kinematical quantities In (2)2, [ is the across the thickness coordinate, with [ ‰ H  h i , h s . Note that h  h i h s is the shell thickness at the reference configuration. The boundary of H is here denoted by C  \ h i , h s ^ . Coordinates YB are such that \ Y1, Y2 ^ ‰ 8 , whereby 8 ‡ \ 2 is the shell domain. The boundary of the domain 8 is denoted by ( , as usual. Thus, Y1, Y2 and [ build a Cartesian coordinate system. In the current configuration the position of the material points is given by

x  z YB a ,

(3)

where z describes the position of the middle surface at the current configuration and a is the current director given by

32

P.M. Pimenta et al.

a  Qa r ,

(4)

where Q is the rotation tensor. Note that no thickness change is assumed. This issue will be better discussed in section 3, where a plane stress condition is stated. A general Kirchhoff-Love theory that incorporates the thickness change will be presented in a coming work under preparation. Remark 1: Back-rotated or material vectors. As in (4), the notation x  Qx r ” x r  QT x for vectors in \ 3 is used. The vector x r is said to be the back-rotated or material counterpart of x and is not affected by superimposed rigid body motions. Notice that the vector x has the same components on system \ei  Qeir , i  1, 2, 3 ^ as the vector x r has on the system E 8 .

2.2

Strains

Let us introduce the following local orthogonal system in the current configuration (see Figure 1) e1  z,1

1

z,1 ,

e2  e3 q e1

and

e3  z,1 q z,2

1

z,1 q z,2 ,

(5)

where the usual notation for the cross product was used and the following notation for partial derivatives has been defined

¸ ,i



s ¸

sYi

.

(6)

Note that eB are tangent to the shell middle surface in the current configuration, while e3 is orthogonal to the shell middle surface. Note also that only e1 and e3 are material, i.e. permanently tangent to same material fibers, while e2 is not. The displacements of the points on the middle plane are defined by

u YB  z YB  [ .

(7)

Note also that

z,B  eBr u,B

and

z,BC  u,BC .

(8)

The Kirchhoff-Love assumption states that the director a remains orthogonal to the middle surface of the shell. Therefore, with the aid of (5), the rotation tensor can be expressed by

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

33

Q  ei  eir .

(9)

ei  Qeir .

(10)

Note that

If (7) is smooth, the bottom and top shell surfaces are continuously and smoothly described by (3). Let us define the following skew-symmetric spin tensor

 T , 8  QQ

(11)

where the superposed dot denotes the differentiation with respect to time. The corresponding spin vector is the axial vector of (11) and is denoted by

X  axial 8 .

(12)

The spin tensor in (11) is given by 8  ei  ei ,

(13)

where ei are obtained by time differentiation of (5). The result is

e2  e2 e3  e3 u,1 , 1   e2   e1 ¸ z,1 ¡ e1  e2 u ,1 e1 ¸ z,2 e2 ¸ z,2 e3 ¢¡ 1

e1  e1 ¸ z,1

1

1

¯  e3 ° u,1 ±°

e2 ¸ z,2 e3  e3 u ,2 and 1 1   ¯ e3  e1 ¸ z,1 ¡ e1 ¸ z,2 e2 ¸ z,2 e2  e3  e1  e3 ° u ,1 °± ¢¡ 1  e2 ¸ z,2 e2  e3 u,2 .

(14)

The axial vector (12) is consequently given by X  e 3  e2 e1 e1  e3 e2 e2  e1 e3 .

(15)

Introducing (14) in (15), we arrive at

X  ( Bu ,B , where

(16)

34

P.M. Pimenta et al. 1

( 1  e1 ¸ z,1

1

( 2  e2 ¸ z,2

1   ¯ ¡ Skew e1  e1 ¸ z,2 e2 ¸ z,2 e1  e3 ° ¢¡ ±°

e1

 e3 .

and

(17)

With ( B we may compute 1

( 1e3   e1 ¸ z,1

1

( 2e3  e2 ¸ z,2

1   ¯ ¡ e1 ¸ z,2 e2 ¸ z,2 e1 e2 ° ¡¢ °±

and

(18)

e1 ,

as well as 1

( T1 e1   e1 ¸ z,1

1

e1 ¸ z,2 e2 ¸ z,2

1

( T1 e2   e1 ¸ z,1 e3 , ( T2 e2

e3 , 1

( T2 e1  e2 ¸ z,2 e3

and

(19)

o.

The deformation gradient sx  x,i  eir , sY

(20)

F  QF r ,

(21)

F r  I H Br  eBr

(22)

F 

can be expressed by

where

is the back-rotated deformation gradient and H Br  IBr LBr q a r

(23)

are back-rotated strains. In (23) the following cross-sectional generalized strains have been introduced IBr  QT z,B  eBr

and

LBr  axial QTQ,B .

(24)

We observe that HBr ¸ e3r  IBr ¸ e3r  0 .

(25)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

35

due to the Kirchhoff-Love assumption. (23) and (24) are the back-rotated counterparts of H B  I B LB q a ,

IB  z,B  eB

and

LB  axial Q,BQT .

(26)

A consequence of (25) is

e3 ¸ IB  e3 ¸ z,B  0 .

(27)

Similarly to (12), we have from (26)3 the following curvature vectors .

LB  ( C u,CB .

(28)

Since Q ,B  Q,B , we have

X,B  L B  X q LB .

(29)

From LBr  QT LB (see Remark 1) and (29), we arrive at the important relation displayed below L Br  QT X,B .

(30)

The velocity gradient is given by time differentiation of (21) F  8F Q H Br  eBr ,

(31)

H Br  I Br L Br q a r .

(32)

where

Time differentiation of (24) yields I Br  QT u ,B Q T z,B  QT u ,B z,B q X .

(33)

Hence, with the aid of (16), we get I Br  QT EBC I Z,B ( C u ,C ,

(34)

where the following skew-symmetric tensors have been introduced

Z,B  Skew z,B

(35)

and the traditional notation for the Kronecker symbol has been adopted EBC  eBr ¸ eCr . From (30) and (16), we also have

36

P.M. Pimenta et al.

L Br  QT ( C,Bu ,C ( C u ,CB ,

(36)

where the derivatives ( C,B are given in the Appendix A. We can collect the cross section generalized strains in two vectors, as displayed below   Ir ¯ FBr  ¡¡ Br °° ¡¢ LB °±

and

 I ¯ FB  ¡¡ B °° . ¡¢ LB °±

(37)

 Q O ¯ °. -  ¡¡ ° O Q °± ¢¡

(38)

Note that

FB  -FBr ,

where

We introduce now the vectors T d   ¡ u u,1 u,2 ¯° and ¢ ± d   ¡ u u,1 u,2 u,11 u,12 ¢

T u,22 ¯° , ±

(39)

as well as the differential operators     ¡I % ¡ ¢   %  ¡¡ I ¢¡

s I sY1

T s ¯° I sY2 °±

s I sY1

s I sY2

and I

s2 sY12

s2 I sY1sY2

T s2 ¯° I ° , sY22 ±°

(40)

such that

u d  %

d  %u .

(41)

F Br  -T :B d  -T :B %u ,

(42)

and

Note that we may write

whereby the following operators have been introduced

 O E1B I Z,B (1 E2B I Z,B ( 2 O O O ¯ ° . (43) :B  ¡¡ ° ( ( ( ( ( ( O E E E E 1,B 2,B 1B 1 1B 2 2B 1 2B 2 ±° ¢¡ Remark 2: Variance of the rotation. We remark that Q , and consequently X in (12) as well, depends on the material fiber, to which e1r is chosen

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

37

to be tangent. Nevertheless, the relation e3  Qe3r remains valid for any choice of that fiber.

2.3

Stresses

Let the 1st Piola-Kirchhoff stress tensor be expressed by P  U i  eir .

(44)

We can now introduce the back-rotated 1st Piola-Kirchhoff stress tensor by P r  QT P  U ir  eir ,

(45)

U ir  QT U i ,

(46)

where

i  1,2, 3 ,

are the back-rotated (material) nominal stress vectors. The following generalized cross sectional forces are obtained by integration of the stresses U B along the thickness nB 

¨H U BdH

and m B 

¨H a

q U B dH .

(47)

n B are the true forces and m B are the true moments, both per unit length at reference configuration, acting on a cross section that is normal to eBr . Their back-rotated counterparts are

n Br  QT n B and m Br  QT m B .

(48)

Hence, we may also write n Br 

¨H U Br dH

and m Br 

¨H a r q U Br dH .

(49)

n Br are the back-rotated cross section forces and m Br are the backrotated cross section moments. The membrane and transversal (or shear due to bending) part of the forces nB are defined by

n Bm  I  e3  e3 n B  N BCeC

and

nbB  VBe3 ,

(50)

respectively, where

N BC  nB ¸ eC

and

VB  nB ¸ e3 .

(51)

38

P.M. Pimenta et al.

Notice that N 11 and N 22 are the membrane normal forces per unit length at reference configuration, as well as N 12 and N 21 are the membrane shear forces per unit length at reference configuration. Notice that N12 v N 21 . V1 and V2 are the transversal shear forces per unit length at reference configuration. The back-rotated counterparts of (50) are n Bmr  N BCeCr

and

r nbr B  VBe 3 ,

(52)

For the moments, we can define

M BC  mB ¸ eC .

(53)

In contrast to the usual notation of the classical plate theory, M11 and M 22 are the torsion moments per unit length at reference configuration, whilst M 12 and M 21 are the bending moments per unit length at reference configuration. Note also that mB ¸ e3  mBr ¸ e3r  0 .

(54)

The local moment balance states Skew PF T  O ,

(55)

where O is the null tensor. (55) enforces PFT : 8  0 .

2.4

(56)

Power

From (44), (31) and (56), we get P : F  U Br ¸ H Br ,

(57)

where : denotes the scalar product of two tensors. (57) is the stress power per unit reference volume. Introducing (32) in (57), we get P : F  U Br ¸ I B a r q U Br ¸ L B .

(58)

Note that U 3r and U Br ¸ e3r are powerless in this model. With the aid of the definitions (49) the integration of (58) over the thickness furnishes

¨H P : F dH

 n Bmr ¸ I Br m Br ¸ L Br .

(59)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

39

(59) is the stress power per unit area of the reference middle surface. It is important to remark that U ir , Hir , n Br , m Br , IBr and LBr are not affected by superimposed rigid body motions. We can collect the cross-sectional resultants that do work and their back-rotated counterparts in two vectors, respectively, as displayed below   nm ¯ TB  ¡¡ B °° ¡¢ m B °±

  n mr TBr  ¡¡ Br ¡¢ m B

and

¯ ° °. °±

(60)

With the aid of (38)2, one may write TB  -TBr .

(61)

With the aid of (60), we can write (59) as follows

¨H P : F dH

 TBr ¸ F Br .

(62)

The internal power on the domain 8 is then given by 8 Pint 

¨8 TBr ¸ F Br d8 .

(63)

Thus, with the aid of (42) and (61), one has 8 Pint 

¨8 TBr ¸ -T :B d d8  ¨8 TB ¸ :B d d8.

(64)

On the other hand, the external power on the same domain can be expressed by 8 Pext 

¨8  ¡¢ t

¸ x C

¨H b ¸ x d [ ¯°± d8 ,

(65)

where t is the surface traction per unit reference area prescribed at the surfaces of the shell and b is the body force per unit reference volume. The notation ¸ C  ¸ [ h i ¸ [ hs has been introduced in (65). The time differentiation of (3) yields x  u X q a .

(66)

Introducing (66) in (65), we can write 8 Pext 

where



¨8 q ¸ d d8 ,

(67)

40

P.M. Pimenta et al.   n ¯ ¡ ° q  ¡¡ ( 15 m °° . ¡ T ° ¡ (2 m ° ¢ ±

(68)

In (68) the following generalized external forces have been introduced

n  t

C

¨H bd[ and m  a q t C ¨ a q b d [ . H

(69)

n is the applied external force per unit reference area and m is the applied external moment per unit reference area. Remark 3: Invariance of the stress power. We remark that either (57) or (59) are invariant with respect to the fiber chosen to be tangent to e1r in the Cartesian reference system.

2.5

Weak form of the local equilibrium equation

The internal virtual work on the shell domain 8 ‡ \ 2 is given by 8 EWint 

¨8 TBr ¸ EFBr d8 ,

(70)

where

EFBr  -T :B Ed  -T :B %Eu .

(71)

Thus, we have 8 EWint 

¨8 TBr ¸ -T :B %Eu d8  ¨8 :BT TB ¸ %Eu d8 .

(72)

The external virtual work on a domain 8 ‡ \ 2 is similarly given by 8 EWext 

¨8 q ¸ Ed d8  ¨8 q ¸ % Eu d8.

(73)

The local equilibrium equations of the shell are obtained by applying the Virtual Work Theorem as follows 8 8 EW 8  EWint  EWext  0,

Ed in 8,

with Ed  o on ( . After introducing (70) and (73) in (74), we get

(74)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

¨8 ¡¢  nC

m

41

 ( TC z,B q nBm m ¸ Eu,C m B ¸ ( C Eu,C °¯ d8 ,B ±  ¨ n ¸ Eu d8  0 .

(75)

8

Performing integration by parts on the integrand m B ¸ ( C Eu,C , we ,B obtain

¨8 nC  ( C mB,B z,B q nB m ¸ Eu,C  n ¸ Eu d8 (76) ¨ mB O Br ¸ ( C Eu,C d (  0 , ( m

T

m

where O Br  O r ¸ eBr are the components of O r , the outward unitary normal to ( . We introduce now the vectors nB  nBm  (TB m H,H z,H q n Hm m ,

(77)

such that (76) can be written as

¨8 nB ¸ Eu,B  n ¸ Eu d8 ¨( mB OBr ¸ ( C Eu,C d (

 0.

(78)

The identity e3 ¸ ( TB z,H q n Hm  ( Be3 ¸ z,H q n Hm  0

(79)

can be easily demonstrated, if one notes that, according to (18), ( Be3 are on the tangent plane. Since z,H and n Hm are also on the same plane, z,H q n Hm is normal to tangent plane and orthogonal to ( Be3 . Because of (79), we may write definition (77) as nB  n Bm  ( TB m H,H m ,

B  1,2,

(80)

Applying integration by parts in (76) on the term with Eu,B , we obtain ¨

8

nB,B n ¸ Eud8 ¨( mB OBr ¸ ( C Eu,C nB OBr ¸ Eu d (

 0 . (81)

By standard arguments of Calculus of Variation, the first integral of (81) delivers the following local equilibrium equation in 8 nB,B n  o .

(82)

It remains the boundary term of (81), given by

¨(  ¢¡ m B OBr ¸ ( C Eu,C nB O Br ¸ Eu ¯±° d ( If we define the tensors

 0.

(83)

42

P.M. Pimenta et al.

N  n B  eBr ,

  n  e r N B B

and

M  m B  eBr ,

(84)

as well as the moments and forces per unit reference length on the boundary ( m  M Or

and

 Or , n  N

(85)

we can write (83) as

¨(  ¢¡ m ¸ ( C Eu,C n ¸ Eu ¯±° d (

 0.

(86)

From (86) we get also the following boundary term

¨( NC ¸ Eu,C

n ¸ Eu d (  0 ,

(87)

where

NC  ( TC m

(88)

are pseudo-moments that are energetically conjugated with u,C .

2.6

Statics

The shell local equilibrium equations can be directly derived by Statics, as for example in Pimenta et al. (2004). They are displayed below n B,B n  o

and

m B,B z,B q n B m  o .

(89)

From (89)2, we get z,B q nB   m H,H m , which implies e3 q z,B q n B  e3 q m H,H m .

(90)

Considering the double cross product formula and making use of (27) and (51)2, from (90) we arrive at

VBz,B  e3 q m H,H m .

(91)

The scalar product of (91) by eC delivers

z,B ¸ eC VB from which we get

 e 3 q eC ¸ m H , H m ,

(92)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type 1

  e ¸ m m  e ¸ z V ¯ H, H 1 ,2 2 ±° ¢¡ 2 1   ¯ V2   e2 ¸ z,2 ¡¢ e1 ¸ m H,H m °± . V1  e1 ¸ z,1

43 and

(93)

In view of (19), we may write (93) as VB   ( Be3 ¸ m H,H m

(94)

Hence, by means of (94), the transversal shear forces can be directly recovered from the bending moments, as in the linear theory. From (89)2, we get z,H q nbH   m H,H z,H q n Hm m ,

(95)

which can be introduced in (77), leading to nB  n Bm ( TB z,H q nbH .

(96)

With the help from (50)2, z,1  e1 ¸ z,1 e1 and z,2  eB ¸ z,2 eB , we get

z,H q nbH  z,2 ¸ e2 V2e1   ¢ z,2 ¸ e1 V2  e1 ¸ z,1 V1 ¯± e2 . Now, with the aid of (19) and (97), we may write ( TB z,H q nbH  VBe3  n Bb ,

(97)

(98)

which, together with (96), furnishes nB  n B in 8

and

n  n on ( .

(99)

The scalar product of (98) by e3 leads to VB   ( Be3 ¸ m H,H z,H q n Hm m ,

(100)

with ( Be3 given by (18). Introducing (79) in the equation above, one arrives also at (94).

2.7

Boundary conditions

We define a local orthogonal system at the boundary ( in the reference configuration, expressed by E (  \ U r , O r , e3r ^ , where O r is the outward unitary normal to the boundary ( and U r  O r q e3r

(101)

44

P.M. Pimenta et al.

is tangent to ( . The directional derivative of u along the tangent and the normal to ( can be calculated through u,U  ‹u U r

and

u,O  ‹u O r ,

(102)

respectively, where ‹u  u,C  eCr .

(103)

Hence, one has z,U  U r u,U

and

z,O  O r u,O .

(104)

Associated with the local system E ( at the boundary in the reference configuration, there is a local orthogonal system at the boundary in the current configuration, denoted by \ U, O,e3 ^ , where U  z,U

1

O  e3 q U

z,U ,

and

e3  z,U q z,O

1

z,U q z,O . (105)

The underlined subscripts remind the reader that the summation convention does not apply for subscripts U and O . Note that U  Q ( U r , O  Q ( O r and e3  Q ( e3r , where Q (  U  U r O  O r e3  e3r

(106)

is the corresponding rotation tensor (see Remark 2). Now the unitary vector U is material, i.e. it is permanently tangential to the material fiber around the boundary. In analogy with (17), we also have 1   ¯ ¡ Skew U  U ¸ z,O O ¸ z,O U  e3 ° ¢ ± 1  O ¸ z ,O U  e 3 . 1

( U(  U ¸ z,U

( O(

and

(107)

If X(  axial Q (Q ( T , then one may write

X(  ( U( u ,U ( O( u ,O .

(108)

Because of the invariance of the stress power (see Remark 3) and (108), the boundary term (86) can be transformed to

¨(  ¡¢ m ¸ ( U( Eu,U

( O( Eu,O n ¸ Eu ¯° d (  0 , ±

where (99) was already taken into account. (109) leads to

(109)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

¨( NU(

45

¸ Eu,U NO( ¸ Eu,O n ¸ Eu d (  0 ,

(110)

where NU(  ( U(T m

and

NO(  ( O( T m

(111)

are boundary pseudo-moments that are truly conjugated with u,U and u,O . They are analogous to the pseudo-moment at the boundary in Campello et al. (2003) and Pimenta et al. (2004). Note that   ¡ U q m U ¸ z,O ¢ 1 (  O ¸ z,O mb e3 , 1

NU(   U ¸ z,U

NO(

1

O ¸ z,O

¯ mb( e3 ° ±

and

(112)

where mb(  m ¸ U

(113)

is the bending moment per unit reference length at the boundary. Observe also that NU( ¸ U  NU( ¸ O  0

1

NU( ¸ e3   U ¸ z,U

and 1   ( ( ¯ ¡ mt U ¸ z,O O ¸ z,O mb ° , ¢ ±

(114)

where mt(  m ¸ O

(115)

is the torsion moment per unit reference length at the boundary. Hence, we can also write (112) as NU(  NU( e3

and

NO(  NO( e3 ,

(116)

where 1   ( ( ¯ ¡ mt U ¸ z,O O ¸ z,O mb ° ¢ ± 1  O ¸ z,O mb( . 1

NU(   U ¸ z,U

NO(

and

(117)

If ( is smooth, then the unitary outward normal vector O r is uniquely defined around the boundary. Consequently, after integration by parts of the first integrand in (110) along the boundary, it delivers

46

P.M. Pimenta et al.

¨( r ¸ Eu NO(

¸ Eu,O d (  0 ,

(118)

whereby the following vector has been introduced r  n  NU( ,U .

(119)

r is the reaction per unit reference length on the boundary ( . In (119)

NU( ,U  ‹NU( U r

(120)

is the directional derivative of NU( along the boundary. Observe that the reaction on the boundary is not equal to the cross-sectional force on the boundary, a fact that is well known in the linear theory of plates (see Timoshenko, 1940). From (112)2, we get NO( ¸ Eu,O  NO( EB ,

(121)

EB  e3 ¸ Eu,O .

(122)

where

Thus, (118) can be written as

¨( r ¸ Eu NO(

¸ EB d (  0 ,

(123)

The boundary terms emanating from (123) are r ¸ Eu  0

and

NO( ¸ EB  0 .

(124)

From (124)1, with ( r ƒ ( u  ( , ( r ‚ ( u   , one gets the following natural and essential boundary conditions

r  r on (r

or

u  u on (u .

(125)

From (124)2, with ( O ƒ ( B  ( , ( O ‚ ( B   , the following natural boundary condition arises

NO(  NO(

on ( O .

(126)

The essential boundary condition emerging from (124)2 is such that

EB  e3 ¸ Eu,O  0 .

(127)

If e3 is constant in time, i.e. fixed, (127) means that B  e3 ¸ u,O has a prescribed value, as follows

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

BB

on ( B .

47

(128)

For non-fixed e3 , (127) leads to a more involved analysis, which will be treated on a forthcoming paper. When the boundary ( is not smooth, the integration by parts of (110) furnishes

¨(

nc

r ¸ Eu NO( ¸ Eu,O d ( œ rc ¸ Euc  0 ,

(129)

c 1

where rc  ced NU( fhg  NU(  NU(  is a concentrated force on the corner c and nc is the number of corners on the boundary ( . In view of (114), one may conclude that

rc  rce3

(130)

at the corners, i.e. the concentrated force on the corners are normal to the shell in the current configuration. This is also a generalization of a well known fact in the linear theory.

2.7.1 Free and unloaded edges For free and unloaded edges, the boundary conditions, according to (125) and (126), are r o

and

NO(  0 .

(131)

Note that (131)2 leads to mb(  0 , i.e. to a null bending moment at the edge. If there are corners at a free edge, one has additionally rc  o on them.

2.7.2 Fixed but free to rotate edges For edges that are fixed but free to rotate around the boundary, one has, according to (125) and (126), the following boundary conditions apply u o

and

NO(  0 .

(132)

Notice that (132)2 also leads to mb(  0 .

2.7.2 Clamped edges On the other hand, the boundary conditions for clamped edges are, as expected,

48

P.M. Pimenta et al. u o

2.8

and

B  0.

(133)

Complete weak form of the equilibrium equations

In view of the discussion on the boundary conditions above, we define now the complete weak form of the equilibrium equations that also includes the natural boundary conditions. The virtual work of the external forces and moments on the boundary is defined by ( EWext 

¨(

nr

r

r ¸ Eud ( œ rc ¸ Euc ¨ NO( EBd ( , c 1

(O

(134)

whereby the corners c  1, !nr are assumed to belong to ( r . Thus, with the aid of (72), (73) and (134), the complete weak form is then given by 8 8 ( EW  EWint  EWext  EWext  0, Ed in 8,

(135)

with Eu  o on ( u and EB  0 on ( B .

2.9

Tangent weak form of the equilibrium equations

8 8 The Gâteaux derivative of EW 8  EWint  EWext with respect to u leads to the following bilinear form

E EW 8 

T r T 8 ¨8 %Eu ¸  ¢¡ :B -DBC - :C G ¯±° %Eu d8  Eu ¸ L8 %  Eu d8 ,  ¨ % 8

(136)

which is very important for the solution of (135) by the Newton Method. r In (136) DBC is the constitutive tangent operator defined by

r DBC

  sn r B ¡ ¡ r r sTB ¡ sIC  ¡ r ¡ sm Br sFC ¡ ¡ sIr ¡¢ C

sn Br ¯° ° sLCr ° °, sm Br ° ° sLCr °°±

whilst G 8 and L8 are geometric operators defined as follows

(137)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

G8 

s

:BT -TBr

s %u



TrB

 O ¡ ¡O ¡ ¡ ¡O ¡ ¡O ¡ ¡O ¡ ¡ ¡¢O

O

O

O

O

G22 G23 G24 G25 G32 G33 G34 G35 G42 G43

O

O

G52 G53

O

O

G62 G63

O

O

49

O ¯ ° G26 °° ° G36 ° ° O ° ° O °° ° O ° ±

(138)

and

L8 

sq . sd

(139)

The matrix G 8 is always symmetric, even far from an equilibrium state, and its sub-matrices are given in the appendix B. Bilinear form (136) is symmetric when r r DBC  DCB

T

L8  L8T .

and

(140)

Condition (140)1 is true when the material is hyper-elastic, while condition (140)2 is satisfied when the external loading is locally conservative. We introduce now the following tensors with constitutive tangent moduli sU Br sH Cr

r .  C BC

(141)

The sub-matrices of (137) can be computed by the chain rule. With the aid of (141), sH rH sICr

 EHC I

sH rH

and

sLCr

 EHC Ar ,

(142)

where EBC is the Kronecker symbol and Ar  Skew a r , they result as follows sn Br sIrC sm Br sICr

 

r d[ , ¨t C BC

¨t A

r d[ C BC

r

sn Br sLCr

r  ¨ C BC Ar d [ ,

and

t

sm Br sLCr

(143)  ¨ A

r C BC Ar d [

r

t

.

50

P.M. Pimenta et al.

Therefore, (140)1 is true if r r T . C BC  C BC

(144)

3. Elastic constitutive equations 3.1 Plane stress condition The issue of the plane stress condition within this shell formulation can be better discussed if the following expression for F r is assumed F r  I HBr  eBr H33e3r  e3r  fir  eir ,

(145)

whereby fir stands for fir  eir Hir .

(146)

The element H33 was introduced in (145) to allow for transversal normal strains. It can be regarded as an additional degree of freedom that should be eliminated at constitutive level by a shell plane stress condition. With (45) in mind, we state here the following plane stress condition r U 33  U r3 ¸ e3r  0 ,

(147)

which follows from Pe3r ¸ e3  P r e3r ¸ e3r  0 . This means that the r projection of the traction Pe3r on the director a  [e3 is zero. So, U33 is powerless, as in (58). (147) yields then the following equation r U 33 H33  0 .

(148)

The Newton Method can iteratively solve equation (148) for H33 , as follows k 1 H 33



k H 33

  sU r ¯ 1 33 k ° ¡ ¡ H33 ° U33r H33k , s H ¢¡ 33 ±°

k  0,1, !,

0  0, H 33

(149)

Remark 4: Workless transversal stresses. The just developed consistent plane-stress condition supposes that following condition holds for the material equation

U Br ¸ e3r  0 .

(150)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

51

If this is not the case, a more general approach must be developed. This consists in assuming, in place of (145), the expression for F r below F r  I H Br  eBr Hi 3e3r  eir  fir  eir .

(151)

The elements HB 3 , B  1,2 have been introduced in (151) to allow for transversal shear strains. Hi 3 , i  1, 2, 3 can be regarded as three additional degrees of freedom that should be eliminated at constitutive level by the conditions given by

Uir3  U ir ¸ e3r  0 .

(152)

Thus, Ui 3 , i  1, 2, 3 , are powerless, as in (58). (152) lead to the system

Uir3 H13 , H23 , H 33  0 ,

i  1,2, 3 .

(153)

3.2. General elastic isotropic material A general elastic isotropic material may be described by a strain energy function Z J , I 1, I 2 with the following strain invariants

J  det F r ,

I 1  F rT : F r

and

I2 

1 rT r F F : F rT F r , (154) 2

which, with the aid of (145), can be computed through J  f3r ¸ f1r q f2r  1 H 33 J ,

with

J  e3r ¸ f1r q f2r ,

I 1  fir ¸ fir  fBr ¸ fBr 1 H33

and 4 1 1 1 I 2  fir ¸ f jr fir ¸ f jr  fBr ¸ fCr fBr ¸ fCr 1 H33 . 2 2 2 2

(155)

It is not difficult to show that the derivatives of Z J , I 1, I 2 with respect to Hir furnish the first Piola-Kirchhoff stress vectors U ir as below U ir 

sZ sHi

r



sZ sJ sZ sI 1 sZ sI 2 , r r sJ sfi sI 1 sfi sI 2 sfir

so that after some algebra one has

(156)

52

P.M. Pimenta et al. sZ sZ sZ 1 H33 FBC fCr q e3r 2 fBr 2 fCr ¸ fBr fCr and sJ sI 1 sI 2 (157) 3 sZ sZ sZ  J 2 1 H33 2 1 H33 , sJ sI 1 sI 2

U Br  r U 33

with FBC  e3r ¸ eBr q eCr as a permutation symbol. H33 is computed by r r given by (157)2 . The stress resultants TB may be solving (148) with U33 then computed via integration of (159)1 across the thickness. We can observe in (157) that, for elastic isotropic materials, (150) always holds. Remark 5: 2nd Piola-Kirchhoff stress and Cauchy-Green strain tensors. We remark that neither the 2nd Piola-Kirchhoff stress tensor S  F r 1P r nor the right Cauchy-Green strain tensor C  F rT F r is necessary in our formulation.

3.3. Neo-Hookean material A simple poly-convex neo-Hookean material can be represented by the following strain energy function (see Simo and Hughes, 1998) Z J , I 1 

¬ 1 1 ž 1 2 M ž J  1  ln J ­­­ N I 1  3  2 ln J , ž ­® 2 2 Ÿ2

(158)

Hence, from (157), we get  1 ¯1 U Br  ¡ M J 2  1  N ° FBC fCr q e3r NfBr ¡¢ 2 °± J  1 ¯J r U 33  ¡ M J 2  1  N ° N 1 H33 . ¡¢ 2 °± J

and

(159)

Introducing (159)2 in (147), we arrive at following solution for (148) H33 

M 2N MJ 2 2N

1,.

(160)

i.e., H33 may be consistently eliminated. Introducing (160) back into (159), we get U Br  K J

where

FBC fCr q e3r NfBr

,

(161)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type K J

 N

M 2N MJ 3 2NJ

53

(162)

.

r as follows Substitution of (161) into (141) yields C BC r C BC  K a J

FBH fHr q e3r  FCE fEr q e3r  K J FBC E3r

NEBC I , (163)

where E3r  Skew e3r and

K a J  K J



3MJ 2 2N MJ 3 2NJ

.

(164)

3.4. Incompressible isotropic elastic material For incompressible isotropic elastic materials we state the strain energy function in the form Z  Z I 1, I 2 , with J  1 . From J  1 , we have H 33 

1 1. J

(165)

The back-rotated first Piola-Kirchhoff stresses are then given by 1 F f r q e3r 2 ssIZ fBr 2 ssIZ fBr ¸ fCr fCr and J BC C 1 2 sZ 1 sZ 1  pJ 2 2 , sI 1 J sI 2 J 3

U Br  p r U 33

(166)

where p is a hydrostatic pressure. Although it represents an additional degree-of-freedom, p can be eliminated with the aid of the plane stress condition (147), what renders

 sZ sZ ¬­ J 2 p  2J 2 žžž ­­ . žŸ sI 1 sI 2 ­®

(167)

For instance, the Neo-Hookean model of Treloar (1943), given by 1 Z I1  N I1  3 , (168) 2 yields U Br  N  ¡ J 3 e3r q FBC fCr fBr ¯° and ¢ ± r  2N  ¡ J 3 FBH fHr q e3r  FCE fEr q e3r J 2 FBC E3r EBC I ¯° C BC ¢ ±

(169) .

54

P.M. Pimenta et al.

4. Conclusions The geometrically-exact shell formulation presented in Pimenta (1993) and Campello et al. (2003) was extended to a Kirchhoff-Love type shell theory. As in Pimenta (1993) and Campello et al. (2003), our approach has defined energetically conjugated cross-sectional generalized stress and strains based on the concept of a shell director. Besides their practical importance, cross section quantities make easy the derivation of equilibrium equations, as well as the achievement of the corresponding tangent bilinear form, which is always symmetric for hyper-elastic materials and conservative loadings, even far from an equilibrium state. The consistent derivation of the boundary conditions of this theory was also accomplished within this work. The concept of director allows the introduction of a parameter that describes thickness variation. This was useful for the derivation of shell constitutive equations from 3-D finite strain constitutive equations by applying a simple consistent plane stress condition that do not destroy the symmetry of the tangent operator. The developed ideas are so general that can be easily extended to inelastic shells, once a stress integration scheme within a time step is at hand. A plane reference configuration was assumed for the shell. Initially curved shells can be regarded as a stress-free deformation from this configuration. This approach was already employed for rods in Pimenta (1996) and for shells in Pimenta et al. (2009). It precludes the use of convective non-Cartesian coordinate systems and simplifies the comprehension of tensor quantities, since only components on orthogonal systems are employed. The same idea can be used in a co-rotational formulation, which is necessary for incremental rotations greater than Q / 2 . This is object of a future work.

Bibliography E.M.B. Campello, P.M. Pimenta & P. Wriggers, A triangular finite shell element based on a fully nonlinear shell formulation. Computational Mechanics, 31 (6), 505-518, 2003. P.M. Pimenta. On a geometrically-exact finite-strain shell model. In Proceedings of the 3rd Pan-American Congress on Applied Mechanics, III PACAM, São Paulo, 1993. P.M. Pimenta. Geometrically-Exact Analysis of Initially Curved Rods. In Advances in Computational Techniques for Structural Engineering, Edinburgh, U.K., v.1, 99-108, Civil-Comp Press, Edinburgh,1996.

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

55

P.M. Pimenta & E.M.B. Campello. Geometrically nonlinear analysis of thin-walled space frames. In Proceedings of the Second European Conference on Computational Mechanics, II ECCM, Cracow, Poland, 2001. P.M. Pimenta, E.M.B. Campello & P. Wriggers. A fully nonlinear multiparameter shell model with thickness variation and a triangular shell finite element. Computational Mechanics, 34 (3), 181-193, 2004. P.M. Pimenta & E.M.B. Campello. Shell curvature as an initial deformation: geometrically exact finite element approach, International Journal for Numerical Methods in Engineering, 78, 1094-1112, 2009. J.C. Simo, D.D. Fox & M.S. Rifai. On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory, Computer Methods in Applied mechanics and Engineering, 79, 21-70, 1990. J.C. Simo & T.R.J. Hughes. Computational Inelasticity. Springer, Berlin, 1998. S.P. Timoshenko. Theory of plates and shells, McGraw-Hill, New York, 1940. L.R.G. Treloar. The elasticity of a network of long chain molecules-I. Transactions of the Faraday Society, v.39,36-41,1943. L.R.G. Treloar, The elasticity of a network of long chain molecules-II, Transactions of the Faraday Society, v.39,241-246,1943.

Appendix A: Derivatives of ( B Similarly to (14), one may write e1,B  B e2  e2 e3  e3 u,1B ,

e2,B  B e1  e2 cEe3  e3 u,1B E e3  e3 u,2B

and

(170)

e3,B  B cEe2  e3  e1  e3 u,1B  E e2  e3 u,2B .

where the following coefficients have been introduced a  z,1 , B  z,1

1

, c  e1 ¸ z,2 , d  e2 ¸ z,2 and E  d 1 .

(171)

With the aid of the coefficients (171), one may write (17) as follows ( 1  B e3  e2  e2  e3  cEe1  e3

and

The derivatives of (172) are displayed below

( 2  E e1  e3 . (172)

56

P.M. Pimenta et al.

( 1,B  B,B e3  e2  e2  e3  cEe1  e3

B e3,B  e2 e3  e2,B  e2,B  e3  e2  e3,B

 B Ec,Be1  e3  cE,Be1  e3  cEe1,B  e3  cEe1  e3,B and

(173)

( 2,B  E,B e1  e3 E e1,B  e3 e1  e3,B ,

where ei,B are given in (170) and the derivatives of (171) are as follows

a,B  e1 ¸ u,1B , E,B  E d,B 2

B,B  B2a,B , and

c,B  Bd e2 ¸ u,1B e1 ¸ u,2B ,

d,B  Bc e2 ¸ u,1B e2 ¸ u,2B .

(174)

Appendix B: Operator G 8 Let t be an arbitrary constant vector. Then, with the aid of (172), one gets (T 1 t  BTe1  BcE e1 ¸ t e3

(T2 t  E e1 ¸ t e3 ,

and

(175)

where T  Skew t . Thus, time differentiation of (175) leads to ( TC t  VCH t u ,H ,

(176)

where the tensors VCH t are given by

V11  B2 T  2 t q e1  e1  BcW1  BdW2  BcW3  cW4 , V12  cW5  W3 ,

V21  W1 aW4

and

V22  aW5

(177)

with W1  C e 3  t , W2  C e1 ¸ t e3  e2 , W3  C e1 ¸ t e3  e1 ,

W4  C e1 ¸ t cC e2  e3 e3  e2  C e1 ¸ t Be1  e 3 d Ce 3  e1

W5  a C

2

e1 ¸ t e3  e2 e2

(178) and

 e3 .

(176) allows for

E(TC t  VCH t Eu,H . Differentiation of (179) with respect to YB leads to

(179)

A Fully Nonlinear Thin Shell Model of Kirchhoff-Love Type

57

E( TC,Bt  VCH,B t Eu,H VCH t Eu,HB .

(180)

Thus, with the aid of (179) and (180), from (136) follows

G8 %Ed ¸ %Ed  ( CT N H  N C ( H Eu,H ¸ Eu,C ( CT Z,B N B ( H  VCH z,B q nB E u,H ¸ Eu,C  ( C,BT M B ( H E u,H ¸ Eu,C VCH mB  ( CT M B ( H E u,H ¸ Eu,CB VCH,B mB E u,H VCH mB E u,HB ¸ Eu,C .

(181)

and In (181) the skew-symmetric tensors N B  Skew nB

M B  Skew mB have been introduced. The sub-matrices of G in (138) can then be expressed by GC 1,H 1  ( CT N H  N C ( H ( CT Z,B N B ( H  VCH z,B q n B VCH ,1 m1 VCH ,2 m 2  ( C,BT M B ( H ,

GC 1,4  VC 1 m1 ,

GC 1,5  VC 1 m 2 VC 2 m1 ,

GC 1,6  VC 2 m 2 ,

(182)

G4,H 1  V1H m1  ( 1T M 1 ( H ,

G5,H 1  V1H m 2 V2H m1  ( 1T M 2 ( H  ( 2T M 1 ( H G6,H 1  V2H m 2 

( 2T M 2 ( H

and

.

The symmetry of G can be established with the aid of the following properties Skew z,B q n B  Z,B N B  N BZ,B ,

T VCH t  ( TC T ( H  VCH t

VCH,B t 

( TC,BT ( H

T  VCH ,B

(183)

and

t

( TC T ( H,B

.

(183)1 is a known property of the cross product. (183)2 can be verified after some manipulations and is analogous to the property obtained in Campello et al. (2003) for the Euler rotation vector. (183)3 follows directly from (183)2 by differentiation. The tensors VCH,B t in (180) are obtained by differentiation of (177) with respect to YB . The result is displayed in the following page

58

P.M. Pimenta et al. V11,B  2BB,B  ¡T  2 t q e1  e1 ¯° ¢ ±  2B2 t q e1 ,B  e1  cW4,B  c,BW4  2B2 t q e1  e1,B  B,B cW1  dW2  cW3

 B c,BW1  d,BW2  c,BW3

(184)

 B cW1,B  dW2,B  cW3,B ,

V12,B  c,BW5 cW5,B  W3,B , V21,B  W1,B a,BW4 aW4,B

and

V22,B  a,BW5  aW5,B ,

with W1,B  C,B e3  t C e3,B  t ,

W2,B  C,B e1 ¸ t e3  e2 C e1 ,B ¸ t e3  e2 C e1 ¸ t e3,B  e2 e3  e2,B ,

W3,B  C,B e1 ¸ t e3  e1 C e1 ,B ¸ t e3  e1 C e1 ¸ t e3,B  e1 e3  e1 ,B ,

W4,B  c C 2 e1 ,B ¸ t e2  e3 e3  e2

 e1 ,B ¸ t BCe1  e3 d C 2e3  e1

c,B C 2 2c CC,B e1 ¸ t e2  e3 e3  e2

 B,B C BC,B e1 ¸ t e1  e3

 d,B C 2 2d CC,B e1 ¸ t e3  e1

c C 2 e1 ¸ t e2,B  e3 e3  e2,B e 3,B  e2 e2  e3,B

 BC e1 ¸ t e1 ,B  e3 e1  e 3,B  d C 2 e1 ¸ t e3,B  e1 e3  e1,B

and

W5,B  a,B C 2 2a CC,B e1 ¸ t e3  e2 e2  e3 a C 2 e1 ,B ¸ t e3  e2 e2  e3

a C 2 e1 ¸ t e3,B  e2 e2  e3,B e3  e2,B e2,B  e3 .

A Beam Finite Element for Nonlinear Analysis of Shape Memory Alloy Devices $@K=N@K

NPEKHE 




=J@ %AN@EJ=J@K

QNE??DEK 



=J@ 1K>ANP + 3=UHKN

1

4

Department of Civil Engineering, University of Rome ‘Tor Vergata’, Rome, Italy 2 Department of Structural Mechanics, University of Pavia, Pavia, Italy 3 IMATI, Institute of Applied Mathematics and Information Technology, National Research Council, Pavia, Italy Department of Civil and Environmental Engineering, University of California, Berkeley, CA, USA Abstract A large displacement finite rotation beam finite element formulation for shape memory alloy structural analysis is proposed. The Reissner-Mindlin beam model is considered in the total Lagrangian form. A reference configuration macroscopic constitutive model with internal variables is adopted for the evaluation of the stress components acting on the beam cross section. The computation of stress resultants and couples is performed iteratively using an algorithm that grants cross section equilibrium given material strain measures.

1

Introduction

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The Author acknowledges the ESF S3T EUROCORES Programme ‘SMARTeR: Shape Memory Alloys to Regulate Transient Responses in civil engineering’ for partial financial support during a research stay in 2008 at the University of California, Berkeley, where he initiated this work.  The Author acknowledges the financial support of the ESF S3T EUROCORES Programme ‘SMARTeR: Shape Memory Alloys to Regulate Transient Responses in civil engineering’.

60

E. Artioli et al.

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P PDEO LKEJP SA =NA NA=@U PK ?KILQPA = JAS OKHQPEKJ PD=P LNKRE@AO = JAS R=HQA BKN   3DEO EO =??KILHEODA@ >U ?KILQPEJC NA@Q?A@ IK@QHE BNKI PDA ?KJRANCA@ OP=PA =>KRA =O        





6



Numerical tests

3DA >A=I JEPA AHAIAJP IK@AH LNAOAJPA@ EJ OA?PEKJ  PDA OD=LA IAIKNU =HHKU EJPACN=PEKJ =HCKNEPDI KB OA?PEKJ  =J@ PDA ?NKOO OA?PEKJ EPAN=PERA O?DAIA LNKLKOA@ EJ OA?PEKJ  =NA ?K@A@ =J@ EILHAIAJPA@ EJ PDA JEPA AHA IAJP OUOPAJ %$ / 3=UHKN  =O = QOAN OEIQH=PEKJ PKKH>KT 3K =OOAOO =??QN=?U =J@ NK>QOPJAOO KB PDA LNKLKOA@ IAPDK@ = OAP KB OEILHA JQIANE?=H PAOPO =NA LNAOAJPA@ EJ PDA BKHHKSEJC %ENOP SA ?=NNU KQP = ?=HE>N=PEKJ EJ KN@AN PK E@AJPEBU PDA OAJOEPEREPU KB PDA OKHQPEKJ SEPD NAOLA?P PK PDA ?NKOO OA?PEKJ EJPACN=PEKJ NQHA KN@AN =J@ PDA IAOD OEVA 2A?KJ@HU SA OPQ@U PDA OPNQ?PQN=H NAOLKJOA KB OEILHA PDNAA@EIAJOEKJ=H OPNQ?PQNAO SDE?D ATLHKEP PDA OQLANA H=OPE? AA?P =J@ =NA OQ>FA?PA@ PK ?U?HE? HK=@EJC EJ H=NCA @EOLH=?AIAJP =J@ JEPA NKP=PEKJ NACEIA 3DA =??QN=?U KB PDA IAPDK@ EO IA=OQNA@ >U ?KI L=NEOKJ SEPD =??QN=PA PDNAA@EIAJOEKJ=H JEPA AHAIAJP OKHQPEKJO QOEJC >NE?G AHAIAJPO +=OPHU PDA OEIQH=PEKJ KB = OLNEJC =?PQ=PKN @ARE?A SDE?D ATLHKEPO PDA OD=LA IAIKNU AA?P EO LNAOAJPA@ 4JHAOO KPDANSEOA OLA?EA@ SA =@KLP PDA BKHHKSEJC I=PANE=H LNKLANPEAO

QNE??DEK =J@ /APNEJE = $   ,/= ,   * $   *    ,/=

D   ,/= ,   *
  *    ,/=

     *    ,/= *

      *   

=OOQIEJC =HOK = I=PANE=H EJ PDA L=NAJP LD=OA EA &    =P PDA >A CEJJEJC KB A=?D HK=@EJC DEOPKNU (P EO EILHE?EPHU =OOQIA@ PD=P PDA =J=HUOEO ?KJOE@ANO QJEBKNI PAILAN=PQNA @EOPNE>QPEKJ EJ PDA I=PANE=H >K@U =J@ JK PDANIKIA?D=JE?=H ?KQLHEJC AA?P EO P=GAJ EJPK =??KQJP 

",*#2"4*/. /' 4)& 02/0/3&% 3$)&-&

3DA ?=HE>N=PEKJ KB PDA IAPDK@ EO =OOAOOA@ ?KJOE@ANEJC = OEILHA ?=JPEHARAN >A=I QJ@ANCKEJC EILKOA@ H=NCA @EOLH=?AIAJP %KN OEILHE?EPU PDA EJEPE=H ?=HE>N=PEKJ EO LANBKNIA@ QOEJC PDA ?H=OOE?=H 5KJ,EOAO AH=OPKLH=OPE? IK@AH

QNE??DEK =J@ 3=UHKN  3DA I=PANE=H EO ?D=N=?PANEVA@ >U =J AH=OPE?

A Beam FE for Shape Memory Alloy Devices 10

3500

85

−1

3000

Relative error [−]

Force Reaction

2500

2000

1500

3D 3x3 4x4 5x5 6x6 7x7 8x8

1000

500

0 0

1

2

3

4

Displacement

5

6

10

10 7

−2

−3

10

0

10

1

10

2

10

3

# elmt [−]

*(52&  "=HE>N=PEKJ KB PDA LNKLKOA@ JQIANE?=H O?DAIA +ABP AH=OPKLH=OPE? NA=?PEKJ@EOLH=?AIAJP NAOLKJOA KB PDA ?=JPEHARAN >A=I 1ECDP NAH=PERA ANNKN KJ PEL I=TEIQI @EOLH=?AIAJP KB PDA ?=JPEHARAN >A=I I=@A KB OD=LA IAIKNU =HHKU 52 JQI>AN KB JEPA AHAIAJPO

IK@QHQO $   ,/= = /KEOOKJ ?KA?EAJP   =J@ = UEAH@ OPNAOO   $
  3DA >A=I D=O = NA?P=JCQH=N ?NKOO OA?PEKJ SEPD OE@AO (  +   I   I =J@ HAJCPD -   I 3DA ?=JPEHARAN EO OQ>FA?PA@ PK =J EJ?NAIAJP=H PN=RANOA @EOLH=?AIAJP =P PDA PEL SDE?D NA=?DAO = I=TEIQI KB    I SEPD = N=PA KB  I LAN QJEP PEIA (J %ECQNA  +ABP = ?KIL=NEOKJ EO @N=SJ NALKNPEJC PDA BKN?A NA=?PEKJ =P PDA ?H=ILA@ AJ@ RANOQO PEIA K>P=EJA@ SEPD @EANAJP ?NKOO OA?PEKJ MQ=@N= PQNA KN@ANO =J@ = TA@ IAOD KB  AMQ=HHU OL=?A@ >A=I AHAIAJPO 3DA NABANAJ?A OKHQPEKJ EO ?KILQPA@ QOEJC      # JK@A 0/ IETA@ JEPA AHAIAJPO 9EAJGEASE?V =J@ 3=UHKN  NA=OKJ=>HA =CNAAIAJP >A PSAAJ PAOPA@ =J@ NABANAJ?A OKHQPEKJ ?=J >A K>P=EJA@ SEPD = IEJEIQI KN@AN KB    MQ=@N=PQNA LKEJPO KN DECDAN (J %ECQNA  1ECDP SA NALKNP PDA NAH= PERA ANNKN KB PDA OEILHA ?=JPEHARAN >A=I LNK>HAI DANAEJ ?KJOE@ANA@ =@KLPEJC PDA OD=LA IAIKNU =HHKU I=PANE=H L=N=IAPANO NALKNPA@ =>KRA 3DA NABANAJ?A OKHQPEKJ EO K>P=EJA@ SEPD = IAOD KB      JK@A IETA@ 0/ AHA IAJPO O EP ?=J >A K>OANRA@ PDA OKHQPEKJ >A?KIAO OP=>HA =P = B=ENHU ?K=NOA IAOD OEVA EJ@E?=PEJC PDA NK>QOPJAOO KB PDA LNKLKOA@ JEPA AHAIAJP O?DAIA ARAJ BKN ATPNAIA HK=@EJC ?KJ@EPEKJO

86

E. Artioli et al. 1

6

0.8 4

0.4

Displacement [m]

Loading function [−]

0.6

0.2 0 −0.2

2

0

−2

−0.4 −0.6 −4 −0.8 −1 0

0.5

1

1.5

2

2.5

Time [−]

3

3.5

4

−6 0

0.5

1

1.5

2

2.5

3

3.5

4

Time [−]

*(52&  "=JPEHARAN >A=I +ABP +K=@EJC DEOPKNU BKN PDA PEL ODA=N BKN?A 1ECDP TE=H @EOLH=?AIAJP 52 PEIA ?KJPEJQKQO HEJA 3N=JORANOA @EOLH=?A IAJP 52 PEIA @=OD@KPPA@ HEJA !HQA HEJA NABANAJ?A OKHQPEKJ NA@ HEJA LNAOAJP OKHQPEKJ



".4*,&6&2 #&"-

6A JKS ?KJOE@AN = ?=JPEHARAN >A=I QJ@AN = ?U?HE? HK=@EJC DEOPKNU 3DA >A=I D=O = NA?P=JCQH=N ?NKOO OA?PEKJ SEPD OE@AO (  +   I   I =J@ HAJCPD -   I PN=JORANOA ODA=N HK=@ SEPD PEIA R=NUEJC EJPAJOEPU =J@ = I=TEIQI R=HQA !    EO =LLHEA@ KJ PDA >A=I BNAA AJ@ 3DA LEA?ASEOA HEJA=N HK=@EJC BQJ?PEKJ SDE?D @NERAO PDA =LLHEA@ HK=@ EO NALKNPA@ EJ %ECQNA  +ABP PEIA OPAL
3   EO =@KLPA@ BKN PDA EJPACN=PEKJ KB PDA ?KJOPEPQPERA N=PA AMQ=PEKJ 3AILAN=PQNA EO GALP ?KJOP=JP =P $  * 3DA ?KIL=NEOKJ EJ PANIO KB PDA PEL @EOLH=?AIAJP =J@ NKP=PEKJ EJ PDA HK=@EJC LH=JA EO ?=NNEA@ KQP QOEJC = QJEBKNIHU OL=?A@ IAOD KB  >A=I AHAIAJPO =J@ =J    ?NKOO OA?PEKJ MQ=@N=PQNA NQHA SDEHA PDA NABANAJ?A OKHQPEKJ EO K>P=EJA@ SEPD = IAOD KB      # JK@A // IETA@ AHAIAJPO O EP ?=J >A =LLNA?E=PA@ EJ %ECQNA  1ECDP = O=PEOB=?PKNU =CNAAIAJP KJ PDA KRAN=HH OPNQ?PQN=H >AD=REKN EO K>P=EJA@ SEPD = I=TEIQI NAH=PERA ANNKN KB   (J %ECQNA  PDA @ABKNIA@ IAOD =P 3   EO NALKNPA@ EJ@E?=PEJC PDA OPNQ?PQN=H I=TEIQI @AA?PEKJ =J@ ODKNPAJEJC SDE?D =NA NAOLA?PERAHU  =J@ 

KB PDA EJEPE=H >A=I HAJCPD 3DEO AHAIAJP=NU PAOP @AIKJOPN=PAO PDA LNKLKOA@ O?DAIA A?EAJ?U EJ PANIO KB ?KILQP=PEKJ PEIA SDE?D EO @N=OPE?=HHU NA@Q?A@ SEPD NAOLA?P PK = # IK@AHEJC OPN=PACU UAP PDA KRAN=HH OPNQ?PQN=H NAOLKJOA EO ?=LPQNA@ SEPD CKK@ =??QN=?U

A Beam FE for Shape Memory Alloy Devices

87 _________________ DISPLACEMENT 2 0.00E+00 -4.81E-01 -9.61E-01

2

3

-1.44E+00 1

-1.92E+00 -2.40E+00 -2.88E+00 -3.36E+00 -3.84E+00 -4.32E+00 -4.81E+00 -5.29E+00 -5.77E+00

Time = 1.00E+00

*(52&  "=JPEHARAN >A=I "=NPAOE=J REAS KB PDA @ABKNIA@ ?KJCQN=PEKJ =P 3  



,#/7 #&"-

 AH>KS ?H=ILA@ >A=I EO ?KJOE@ANA@ JKS EJ KN@AN PK AOP=>HEOD PDA LANBKNI=J?A KB PDA O?DAIA EJ LNA@E?PEJC PDA OPNQ?PQN=H >AD=REKN KB ?KILHAT =TE=H ATQN=H =J@ PKNOEKJ=H @ABKNI=PEKJO 3DA >A=I HEAO EJ PDA '  '  LH=JA D=O = NA?P=JCQH=N ?NKOO OA?PEKJ SEPD OE@AO (  +   I   I =J@ A=?D =NI D=O HAJCPD -   I 3DA LNK>HAI OAP QL EO =O BKHHKSO 3DA OUOPAI EO OQ>FA?PA@ PK =J EJLH=JA PN=JORANOA BKN?A =HKJC ' @ENA?PEKJ =?PEJC KJ PDA AH>KS ?H=ILA@ >N=J?D AJ@ =J@ EO OQ>FA?PA@ PK =J KQPKB LH=JA PN=JORANOA BKN?A =HKJC ' @ENA?PEKJ =LLHEA@ KJ PDA PEL 3DA HK=@EJC DEOPKNU EO CKRANJA@ >U PDA @NEREJC BQJ?PEKJ LKNPN=UA@ EJ %ECQNA  +ABP PEIA OPAL
3   EO =@KLPA@ BKN PDA EJPACN=PEKJ KB PDA ?KJOPEPQPERA N=PA AMQ=PEKJ 3AILAN=PQNA EO GALP ?KJOP=JP =P $  * 3DA NABANAJ?A OKHQPEKJ EO ?KILQPA@ SEPD = IAOD KB      # JK@A // IETA@ AHAIAJPO BKN A=?D =NI KB PDA OPNQ?PQNA O EP ?=J >A K>OANRA@ EJ %ECQNA 

+ABP = O=PEOB=?PKNU =CNAAIAJP KJ PDA PEL @EOLH=?AIAJP DEOPKNU EO K>P=EJA@ SEPD = I=TEIQI NAH=PERA ANNKN KB   (J %ECQNA  PDA @ABKNIA@ IAOD =P 3   EO NALKNPA@ EJ@E?=PEJC PDA OPNQ?PQN=H I=TEIQI @AA?PEKJ 

02*.( "$45"4/2

PPAJPEKJ EO JKS BK?QOA@ KJ PDA PULE?=H OLNEJC =?PQ=PKN SDE?D ATLHKEPO PDA OD=LA IAIKNU AA?P ?KJOEOPEJC KB =J DAHE?=H OLNEJC HK=@A@ SEPD =J =TE=H BKN?A #QANEC AP =H ? +E=JC =J@ 1KCANO  (B PDA OPNAOO =LLHEA@ >U PDA BKN?A =P $   EO CNA=PAN PD=J PDA I=NPAJOEPE? ?NEPE?=H OPNAOO PDA LD=OA

E. Artioli et al. 15

1.5

10

1

5

0.5

Rotation [−]

Displacement [m]

88

0

0

−5

−0.5

−10

−1

−15 0

0.5

1

1.5

2

2.5

3

3.5

4

−1.5 0

0.5

1

1.5

Time [−]

2

2.5

3

3.5

4

Time [−]

*(52&  $H>KS >A=I +ABP 3EL @EOLH=?AIAJP 52 PEIA #=OD@KPPA@ HEJA @EOLH=?AIAJP EJ ' @ENA?PEKJ @=ODA@ HEJA @EOLH=?AIAJP EJ ' @ENA?PEKJ ?KJPEJQKQO HEJA @EOLH=?AIAJP EJ ' @ENA?PEKJ !HQA HEJA NABANAJ?A OKHQPEKJ NA@ HEJA LNAOAJP OKHQPEKJ 1ECDP 3EL NKP=PEKJ 52 PEIA 1A@ @=OD@KPPA@ HEJA NKP=PEKJ SNP ' =TEO NA@ @=ODA@ HEJA NKP=PEKJ SNP ' =TEO NA@ ?KJ PEJQKQO HEJA NKP=PEKJ SNP ' =TEO

3 1

3 2

1

_________________ DISPLACEMENT 3

2

_________________ DISPLACEMENT 3

0.00E+00

0.00E+00

-1.10E+00 -2.20E+00

-1.10E+00 -2.20E+00

-3.30E+00

-3.30E+00

-4.40E+00 -5.50E+00

-4.40E+00 -5.50E+00

-6.60E+00

-6.60E+00

-7.70E+00 -8.80E+00

-7.70E+00 -8.80E+00

-9.90E+00

-9.90E+00

-1.10E+01 -1.21E+01

-1.10E+01 -1.21E+01

-1.32E+01

-1.32E+01

Time = 1.00E+00

Time = 1.00E+00

*(52&  $H>KS >A=I +ABP "=NPAOE=J REAS KB PDA @ABKNIA@ ?KJCQN=PEKJ =P 3   1ECDP LANOLA?PERA REAS KB PDA @ABKNIA@ ?KJCQN=PEKJ =P 3  

A Beam FE for Shape Memory Alloy Devices

89

PN=JOBKNI=PEKJ BNKI IQHPER=NE=JP PK OEJCHAR=NE=JP I=NPAJOEPA EO EJ@Q?A@ =J@ PDA =LLHEA@ BKN?A LNK@Q?AO = OECJE?=JP AHKJC=PEKJ KB PDA OLNEJC (B PDA =LLHEA@ OPNAOO NAOQHPO HKSAN PD=J PDA =HHKU NA?KRANU OPNAOO DA=PEJC PDA I=PANE=H =>KRA  EJ@Q?AO PDA EJRANOA #%   PN=JOBKNI=PEKJ =J@ PDA OLNEJC ODKNPAJO >=?G ATANPEJC = BKN?A KJ PDA HK=@A@ AJ@ ,KNAKRAN ?KKHEJC >AHKS   PDA PN=JOBKNI=PEKJ BNKI =QOPAJEPA PK OEJCHAR=NE=JP I=NPAJOEPA P=GAO LH=?A =J@ PDA BKN?A OPNAP?DAO =C=EJ PDA OLNEJC EJ OQ?D = S=U PD=P CHK>=HHU = ?U?HE? PSKS=U IKPEKJ ?=J >A K>P=EJA@ 3K =J=HUVA PDEO @ARE?A SA ?KJOE@AN = OLNEJC ?KJOEOPEJC KB  =J@  ?KEHO LEJJA@ =P KJA AJ@ SEPD =J EJJAN N=@EQOPKPKP=HHAJCPD N=PEK KB  3DA OLNEJC EO EJ?NAIAJP=HHU HK=@A@ >U =J =TE=H BKN?A =P KJA AJ@ =J@ KJ?A EPO I=TEIQI R=HQA EO NA=?DA@ GAALEJC PDA HK=@ ?KJOP=JP EP EO OQ>FA?PA@ PK PAILAN=PQNA ?U?HAO 3DA HK=@EJC DEOPKNU EO =OOQIA@ =O = LEA?ASEOA HEJA=N BQJ?PEKJ KB PDA PEIA L=N=IAPAN 3DA HK=@EJC DEOPKNU H=OPO  PEIA QJEPO =J@ ?KJOEOPO KB RA @EANAJP OP=CAO @AO?NE>A@ EJ PDA BKHHKSEJC P=>HA Time [-] 0KQJ@=NU R=HQA LNK>HAIO 0"'  2 "' , *  W > )' NCUNEO J AT?QNOEKJ EJPK H=NCA NKP=PEKJO -+.32 $2' ..* $"' ,&0& W  % QNE??DEK =J@ + /APNEJE (ILNKRAIAJPO =J@ =HCKNEPDIE?=H ?KJOE@AN=PEKJO KJ = NA?AJP PDNAA@EIAJOEKJ=H IK@AH @AO?NE>EJC OPNAOOEJ@Q?A@ OKHE@ LD=OA PN=JOBKNI=PEKJO ,2  3+ $2' ,&0& W  % QNE??DEK =J@ + /APNEJE PDNAA@EIAJOEKJ=H IK@AH @AO?NE>EJC OPNAOO PAILAN=PQNA EJ@Q?A@ OKHE@ LD=OA PN=JOBKNI=PEKJO L=NP E OKHQPEKJ =HCK NEPDI =J@ >KQJ@=NU R=HQA LNK>HAIO ,2  3+ $2' ,&0&  W = % QNE??DEK =J@ + /APNEJE PDNAA@EIAJOEKJ=H IK@AH @AO?NE>EJC OPNAOO PAILAN=PQNA EJ@Q?A@ OKHE@ LD=OA PN=JOBKNI=PEKJO L=NP EE PDANIKIA ?D=JE?=H ?KQLHEJC =J@ DU>NE@ ?KILKOEPA =LLHE?=PEKJO ,2  3+ $2' ,&0& W > % QNE??DEK =J@ 1+ 3=UHKN 3SK I=PANE=H IK@AHO BKN ?U?HE? LH=OPE?EPU JKJHEJA=N GEJAI=PE? D=N@AJEJC =J@ CAJAN=HEVA@ LH=OPE?EPU ,2  * 1 2("(27 W  % QNE??DEK / "=NKPAJQPK =J@  1A=HE .J PDA CAKIAPNE?=HHU AT=?P >A=I IK@AH ?KJOEOPAJP AA?PERA =J@ OEILHA @ANER=PEKJ BNKI PDNAA @EIAJOEKJ=H JEPAAH=OPE?EPU ,2$0, 2(-, * -30, * -% -*(#1 ,# 203" 230$1 W  8 !AHHKQ=N@ .OD=LA IAIKNU =HHKUO BKN IE?NKOUOPAIO NAREAS BNKI = I=PANE=H NAOA=N?D LANOLA?PERA  2$0( *1 "($,"$ ,# ,&(,$$0(,&  WW  ,+ !KQ>=G=N 2 ,KUJA " +AT?AHHAJP =J@ /D !KEOOA 2, LOAQ@KAH=OPE? JEPA OPN=EJO 3DAKNU =J@ JQIANE?=H =LLHE?=PEKJO -30, * -% ,&(,$$0(,&  2$0( *1 ,# $"',-*-&7 W  & !KQN>KJ " +AT?AHHAJP =J@ 2 +A?HAN?M ,K@AHHEJC KB PDA JKJ EOKPDANI=H ?U?HE? >AD=REKQN KB = LKHU?NEOP=HHEJA "Q9J H OD=LA IAIKNU =HHKU -30, * #$ '71(/3$ "W  " !KQRAP 2 "=HHK?D =J@ " +AT?AHHAJP ,A?D=JE?=H !AD=REKN KB = "Q H !A 2D=LA ,AIKNU HHKU 4J@AN ,QHPE=TE=H /NKLKNPEKJ=H =J@ -KJLNKLKN PEKJ=H +K=@EJCO 0 ,1 "2(-, -%  W   "=N@KJ= =J@ , &AN=@EJ >A=I JEPA AHAIAJP JKJHEJA=N PDAKNU SEPD JEPA NKP=PEKJO ,2  3+ $2' ,&0& W  # "DNEOP =J@ 2 1AAOA %EJEPA @ABKNI=PEKJ LOAQ@KAH=OPE?EPU KB OD=LA IAI KNU =HHKUO ?KJOPEPQPERA IK@AHHEJC =J@ JEPA AHAIAJP EILHAIAJP=PEKJ ,2$0, 2(-, * -30, * -% -*(#1 ,# 203"230$1 W  !# "KHAI=J =J@ ,$ &QNPEJ  '$+ '71  

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A Unified Approach for the Nonlinear Dynamics of Rods and Shells Using an Exact Conserving Integration Algorithm Paulo M. Pimenta and Eduardo M. B. Campello Department of Structural and Geotechnical Engineering Polytechnic School at the University of São Paulo, São Paulo, Brazil {ppimenta, campello}@usp.br Abstract A unified formulation is presented in this work for the nonlinear dynamics analysis of rods and shells undergoing arbitrarily large deformations and rigid body motions. Based on our previous works, we develop a special notation and describe both rod and shell kinematics with the same set of expressions. Differences are observed only at the constitutive equation. Important aspects of the above-mentioned works are preserved, such as the special parameterization of the rotation field, the concept of stress resultants and the ability to handle nonlinear hyperelastic materials in a totally conserving way. The time integration algorithm developed for the equations of motion follows an energy-momentum approach and results in a fully conserving scheme. The formulation is well-suited for (but not restricted to) finite element approximations and its unified character leads to a straightforward simultaneous implementation of both rod and shell dynamics models within a finite element code. Assessment is made by means of numerical simulations.

1

Introduction

Based on our previous papers (Pimenta et al., 2008, and Campello et al., 2009), in this work we present a unified formulation for the nonlinear dynamics analysis of rods and shells in 3-D space. By “unified” we mean that both rod and shell models are described by the same set of expressions, with differences only at the constitutive equation. This is achieved solely due the development of a special, yet simple, notation. Important aspects of the above mentioned works are entirely preserved, namely: (i) a special parameterization is adopted for the rotation field,

100

P.M. Pimenta and E.M.B. Campello

with which update of the rotational degrees-of-freedom is made extremely simple, (ii) energetically conjugated cross-sectional stresses and strains are defined based upon the first Piola-Kirchhoff stress tensor and the deformation gradient and (iii) nonlinear hyperelastic materials are permitted in a totally consistent way. The equations of motion are derived in strong and weak forms, the time-collocation of which (following an energymomentum approach) ensures exact conservation of both momentum and mechanical energy in the absence of external forces. We believe the unified character of the formulation leads to a straightforward simultaneous implementation of both rod and shell dynamics models within a finite element code. Computational aspects are discussed and assessment of the scheme is made by means of several numerical simulations. Throughout the text, italic Greek or Latin lowercase letters a, b, !, B, C, ! denote scalar quantities, bold italic Greek or Latin lowercase letters a, b, !, B, C, ! denote vectors and bold italic Greek or Latin capital letters A, B, ! denote second-order tensors in a threedimensional Euclidean space. The same bold fonts are used for general tensor operators. Subscripts play a special role in the notation developed herein. For this reason, we stress our assumptions as follows. Latin indices range from 1 to 3, with the summation convention over repeated indices implied. Greek indices, however, take the values of 1 for rods and 1 and 2 (with summation implied) for shells. This is adopted all over the text (unless where clearly stated) and is summarized below: £¦ 1 for rods; (1) B, C, H, !  ¦¤ ¦¦¥ 1, 2 for shells (with summation over repeated indices).

At last, vectors and matrices built of tensor components on orthogonal frames (e.g. for computational purposes) are expressed by boldface Greek or Latin upright letters a, b, ! A, B, ! .

2

Parameterization of the rotation field

Following the parameterization we have developed for the rotation field in Pimenta (1993a) and Campello et al. (2003), let R be the classical Euler rotation vector representing an arbitrary finite rotation on 3-D space, with R  R as its magnitude and

A Unified Approach for the Dynamics of Rods and Shells

Q I

101

2 sin R 1  sin R / 2 ¬­ 2 ­­ 2 2 žž R 2 žŸ R / 2 ­®

(2)

the associated rotation tensor, in which 2  Skew R . We define the Rodrigues rotation vector B by means of B 

tan R / 2

R, R /2

(3)

and opt to describe the rotation field with B instead of R . The rotation tensor may be expressed in terms of B as follows (see e.g. Pimenta and Campello, 2005, and Campello et al. 2003, 2009)





Q  I h B " 2 "2 , 1

h B 

with

4 4 B2

,

(4)

where B  B and "  Skew B . In this case, the Cayley transform



1



1

Q  I 2" I  2"

1





1

 I  2"

1

I

1 " 2



(5)

holds for Q , and the following relations may be derived: 1 2

I



Q  I  2 "



Q I  A I 

1

1 " 2

1



1



and



 I 

1 " 2

1



(6) A.

The skew-symmetric spin tensor associated to the rotation Q is defined  T , with its axial vector X  axial 8 being called the spin by 8  QQ vector or angular velocity vector. One can show that

X  Ȅ B ,

where





Ȅ  h B I 2 " . 1

(7)

Tensor Ȅ relates X to the time derivative of B and has the remarkable property QT ȄQ  Ȅ , from which follows ȄT  QT Ȅ  ȄQT . From these identities, the back-rotated counterpart of X is given by Xr  QT X  ȄT B ,

(8)

where the notation with a superscript was introduced to define backrotated quantities as follows: ¸ r  QT ¸ . Let now t be a generic vector and T a second-order tensor such that T  Skew t . The following result (useful subsequently in the text) may be obtained by differentiation

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P.M. Pimenta and E.M.B. Campello

s ȄT t

sB

 W B, t ,

where

W B, t  2 h B T  ȄT t  B . (9) 1

Other useful properties are 1 det  ¡ 2 I Q ¯°  h B , ¢ ± T T 1 1 1 Q  det  ¡ 2 I Q ¯°  ¡ 2 I Q ¯°  h B  ¡ 2 I Q ¯° and ¢ ±¢ ± ¢ ±   1 I Q a ¯ q   1 I Q b ¯  ¢¡ 2 ±° ¢¡ 2 ±°  Q a q b , a,b ‰  3 Nanson's rule .

det Q  h 2 B ,

(10)

The parameterization with the Rodrigues rotation vector leads to simpler expressions when compared to the Euler representation, and allows for a very simple update scheme as we shall see later on in the text. It should be mentioned, however, that due to definition (3) we must have 0  R  Q . This restriction does not affect the formulation since we adopt an updated description for the dynamical problem. Hence, the rotations may not exceed Q only within a single time increment, what is not considered to be a practical limitation.

3

Rod and Shell Dynamics

3.1 Kinematics Based on the geometrically-exact quasi-static rod and shell models of Pimenta (1993a, 1993b), Pimenta and Yojo (1993) and Campello et al. (2003), a straight reference configuration is assumed for the rod axis whereas a flat reference configuration is assumed for the shell mid-surface. A local unitary orthogonal system \e1r , e2r , e3r ^ with corresponding coordinates \ Y1 , Y2 , Y3 ^ is defined at each of these configurations, as shown in Figure 1. Points of the rod or shell in these configurations are described by Y  [ rr ,

(11)

where [ describes the position of points on the rod axis or on the shell mid-surface, and r r is the reference director. For the rod description, [  Y1 e1r , Y1 ‰ 8   ¢ 0, A ¯± (with A being the rod reference length), and r r  Y2e2r Y3e3r . For the shell description, on its turn, [  Y1e1r Y2e2r and r r  Y3e3r , with Y3 ‰ H   ¢ h 2 , h 2 ¯± ( h being the shell reference thickness). In a compact manner, one may write

A Unified Approach for the Dynamics of Rods and Shells

103

e3

current configuration

x

e2

e2

r current configuration

e1 r

e1

e3 x

x

z u

z u

Y

8 0 reference configuration

x

r [ e3r

e2r e 3r

r

Y

e1r

[ O

rr

e2r

8 e1r reference configuration

Figure 1. Rod and shell geometries and basic kinematical quantities. [  YBeBr

(12)

for both rod and shell descriptions, with index B as displayed in (1). Let now the unitary vectors \e1, e2 , e3 ^ define a local orthogonal system in the current configurations as depicted in Figure 1. We describe the rod and shell motions by a vector field x  xˆ Y , so that in these configurations the position of the material points is given by x  z r ,

(13)

z  [ u

(14)

where

describes the position of points in the deformed rod axis or in the deformed shell mid-surface (with u as the displacement vector) and

r  Qr r

(15)

is the current director at these points, with Q as the rotation tensor given by equation (4). Notice that expression (15) embeds our basic kinemat-

104

P.M. Pimenta and E.M.B. Campello

ical assumption, i.e. the director remains rigid during the motion (no cross-sectional deformations or thickness changes occur) and may only rotate as a rigid body, with first-order shear deformations being accounted for. Relation ei  Qeir holds for the local systems. Time differentiation of expression (13) furnishes the velocity and acceleration vectors of a material point:

x  u X q r

and

x  u X q r X q X q r .

(16)

3.2 Strains The deformation gradient F is computed by differentiation of (13) with respect to Y , and after some algebra can be expressed by F  Q  ¡ I IBr LBr q r r  eBr ¯° , ¢ ±

(17)

where I Br  QT z,B  eBr

and

LBr  QT axial Q,BQT

 ȄT B,B ,

(18)

with the notation ¸ ,B  s ¸ / sYB . In these expressions, IBr and LBr may be regarded as the rod or shell strain vectors. Time differentiation of (18) provides the strain rate vectors as follows and I Br  QT u ,B z,B q X  QT u ,B Z,B Ȅ B

L Br  QT X,B  ȄT B ,B Ȅ T B,B  QT  ¢ Ȅ B ,B QW B, B,B B ¯± ,

(19)

with W B, B,B as in (9). If we place IBr and LBr within a generalized strain vector FBr , i.e.   Ir ¯ (20) FBr  ¡¡ Br °° , ¡¢ LB °± then the generalized strain rate FBr may be written as (in view of (19))

F Br  -T 'B %Bv

where

(no sum on B) ,

(21)

A Unified Approach for the Dynamics of Rods and Shells

 Q O ¯ ° , 'B  -  ¡¡ ° O Q ¢¡ ±°  I s O ¯° ¡ sYB ¡ ° s %B  ¡¡ O I sY °° B ¡ ° ¡ O I °° ¢¡ ±

105

  I O Z,B ¯ ¡ ° ¡O I O ° , °± ¢¡

and

  u ¯ v  ¡ °. ¡X ° ¢ ±

(22)

In addition, if we define an operator YB and a vector d such that

%Bv  6 B %Bd , 6

B

where

 I O ¯ O ¡ ° ¡ °  ¡O Ȅ QW B, B,B ° ¡ ° ¡O O ° Ȅ ¢¡ ±°

and

 u ¯ d  ¡B ° , ¡¢ ±°

(23)

then it is possible to rewrite (21) as F Br  -T 'B 6 B %Bd .

(24)

3.3 Stresses Let the first Piola-Kirchhoff stress tensor be expressed in terms of its column-vectors by P  U i  eir ,

(25)

where U i are nominal stress vectors acting on points of the rod or shell at the current configuration whose unitary normal vectors at the reference configuration are eir . Notice that, since eBr are the normal vectors to the rigid cross-sections in both rod and shell descriptions, U B correspond to the cross-sectional stress vectors. Integration of U B over the rod reference cross-sectional area A or over shell reference thickness domain H furnishes the stress resultants as below nB 

¨S U BdS

and

mB 

¨S r q U BdS ,

(26)

where

S  A for rods (reference cross-sectional area) and S  H for shells (reference thickness domain).

(27)

Vectors n B and m B in (26) stand for the internal forces and internal

106

P.M. Pimenta and E.M.B. Campello

moments acting on the cross-sections, respectively (notice that, in the case of shells, these quantities are both per unit length of the reference configuration). Considering the back-rotated stress vectors U ir  QT U i , the backrotated counterparts of (26) may be written as

¨S U Br dS

n Br  QT n B 

and

m Br  QT m B 

¨S r r q U Br dS .

(28)

For the sake of neatness in notation, we will place these resultants into the vectors below  n ¯ TB  ¡¡ B °° m ¢¡ B ±°

and

  nr ¯ TBr  ¡¡ Br °° . ¡¢ m B °±

(29)

Note that TBr  -T TB and TB  -TBr , with - given in (22)1.

3.4 External forces Let n be the distributed external forces and m the distributed external moments acting on the rod axis by unit length of the reference configuration, or on the shell mid-surface by unit area of the reference configuration. Let n ( be the external forces and m ( the external moments acting on the rod or shell edges (in the former case, they are concentrated forces and moments acting on the rod’s ends; in the latter, both are distributed along the edges, i.e. they are by unit length of the reference configuration). Let also 8  0, A ‡  for rods 8‡

2

and

(30)

for shells

be the rod or shell open domain, with (  s8 as its boundary. The resulting external forces and moments on the rod or shell are then given by

¨8 nd8 n ( ( and  ¨ z q n m d8 z q n ( m ( , ( ( 8

f ext  m ext

(31)

where the following notation has been introduced

¸ ( ¸ (

 ¸ Y  A ¸ Y  0 for rods 1



¨( ¸ d (

1

for shells .

and

(32)

A Unified Approach for the Dynamics of Rods and Shells

107

Notice that the external moments are defined with respect to the origin of the coordinates.

3.5 Linear and angular momentum Let S be the specific mass of the rod or shell at the reference configuration and assume for simplicity that (i) the rod axis coincides with the line of the cross-sectional barycenters and (ii) the shell mid-surface is the medium surface, i.e. H   ¢ h 2 , h 2 ¯± , such that

¨S Sr r dS



¨S SrdS

o

(33)

(recall that S  A for rods and S  H for shells). We then define the following inertia properties for the cross sections:

¨S SdS and  ¨ Skew2 r SdS , S

M  J



J r  ¨ Skew2 r r SdS  QT JQ . S

(34)

Let now V be the volume of the rod or shell at the reference configuration. The linear momentum for both (with the aid of (16)1) is given by M 

¨V SxdV



¨8 Md8,

M  Mu ,

where

(35)

and the angular momentum (with the aid of (13) and (16)1) by N

¨V Sx q xdV

N  J X  QN

r



¨8 z q M N d8 ,

and

where

N J X . r

r

r

(36)

Notice that the angular momentum N is defined here with respect to the origin of the coordinates. Time differentiation of M and N results in





M  ¨ M d8 and N  ¨ z q M N d8 , 8 8 and M  Mu , N  J X X q J X  QN r r r r r r r N  J X X q J X .

where (37)

r We remark that N r  N v Nr ¹ , since Nr  QT N and N r  ¹  Nr X r q Nr .

3.6 Equations of motion The global equations of motion describing the rod or shell’s dynamical equilibrium may be stated by means of the Euler’s laws, i.e.

108

P.M. Pimenta and E.M.B. Campello f ext  M

and

m ext  N .

(38)

Substitution of (31) and (37) into these two expressions renders

¨8 n  M d8 n ( (  o and  ¨8 z q n m  z q M  N d8 z q n ( 

m(

(

o.

(39)

On the other hand, from application of the divergence theorem on the cross-sectional resultants n B and m B , and with the aid of the notation introduced in (32), we may write

¨8 nB,Bd8  n ( ( , ¨8 mB,Bd8  m ( ( and ¨8 z q nB ,B d8  ¨8 z,B q nB z q nB,B d8  z q n ( ( ,

(40)

where n (  OB nB

and

m (  OBmB .

(41)

The vectors above may be regarded as the internal forces and internal moments acting at the boundaries of the rod or shell, with O B as the components of the unit normal O  O Be Br of ( . By introducing these results into (39), we get

¨8 nB,B n  M d8 n (  n ( (  o and  ¨8  ¢¡ z q nB,B n  M mB,B z,B q nB m  N ¯±° d8 

¡  z q n (  n ( m (  m ( °¯ ¢ ±

(

(42) o,

from which it is possible to state n B,B n  M  o

in 8

n (  n ( on ( m B,B z,B q n B m  N  o m(  m(

on (

²¦ ¦¦ ¦ ¦¦ » in 8 ¦¦ ¦¦ ¦¦ ¼

º

£ ext  M ¦¦ f ¤ ext  N . ¦¦¥¦ m

(43)

The first and the third expressions above represent the local equations of motion for the rod and shell models. The second and the fourth expressions constitute the natural boundary conditions.

A Unified Approach for the Dynamics of Rods and Shells

109

3.7 Weak form of the equations of motion One possible strong form for the local equations of motion (43) is

¨8 nB,B n  M ¸ Eu d8  ¢¡ n  n ¸ Eu ¯±° ( ¨ m B,B z,B q n B m  N ¸ EXd 8 8 

(

(

 ¡ m (  m ( ¸ EX ¯° ¢ ±

(

(44) Eu , EX ,

 0,

where Eu and EX are the adopted weighting functions. On the other hand, from the divergence theorem on the cross-sectional resultants nB and m B , it follows

¨8 nB,B ¸ Eu d8  ¨8 nB ¸ Eu ,Bd8 n ( ¸ Eu ( and ¨8 mB,B ¸ EXd8  ¨8 mB ¸ EX,Bd8 m ( ¸ EX ( ,

(45)

with n ( and m ( as in (41). Introduction of (45) into (44) leads to the following weak form, which corresponds to the application of the Virtual Power Theorem: EP 

¨8  ¡¢ nB ¸ Eu,B z,B q EX M  n ¸ Eu ¯°± d8 ¨  ¢ m B ¸ EX,B N  m ¸ EX ¯± d 8 8 

 n ¸ Eu

(

 m ¸ EX

(

(

(

 0,

(46)

Eu , EX .

Expression (46) may be written in a more compact manner as follows EP 

¨8 TB ¸ 'B%B Ev g ¸ Ev  T ¸ Ev d8  T(

¸ Ev

(

 0,

Ev, (47)

where TB , 'B and %B are given by expressions (29) and (22) and where   Eu ¯ Ev  ¡¡ °° , ¡¢ EX ±°

  M ¯ g  ¡¡ °° , ¡¢ N °±

 n¯ T  ¡¡ °° ¡¢ m ±°

and

  n( T(  ¡¡ ( ¡¢ m

¯ °. ° °±

(48)

As already stated in Pimenta (1993a, 1993b), we remark that due to the use of Eu and EX as weighting functions the static part of equation (47) does not correspond to the variation of a functional. For this reason, a symmetric tangent of the weak form is not to be expected. However, this does not represent a drawback, since the presence of rotational degrees-offreedom leads to a nonsymmetrical dynamical problem anyway. Another possible weak form for the equations of motion may be constructed via the Virtual Work Theorem, but demonstration of energy conservation within the time integration scheme becomes extremely complicated – if not im-

110

P.M. Pimenta and E.M.B. Campello

possible. We adopt (47) as the basis in the development of our algorithm and associated finite element approximation.

3.8 Internal and kinetic energy We assume that the rod and the shell are made of a hyperelastic material, with Z  Zˆ FBr as their specific strain energy function per unit volume of the reference configuration. The shell is assumed to be under a plane stress state. The rod or shell internal energy is then written as U int 

¨8 ¨S Z FBr dSd8  ¨8 Z FBr d8 ,

(49)

where Z FBr 

¨S Z FBr dS

(50)

may be regarded as the strain energy of the cross sections. The kinetic energy, on its turn, reads as

T 

1 2

¨8 ¨S Sx ¸ xdSd8  2 ¨8 Td8 , 1

with

T  2 Mu ¸ u 2 J X ¸ X  2 Mu ¸ u 2 J r X r ¸ X r . 1

1

1

1

(51)

The stress resultants T Br and the matrices of material tangent moduli may be derived from Z as follows:

r DBC

T Br 

sZ sFBr

and

r DBC 

sT Br s2 Z  . sFCr sFBr sFCr

(52)

We draw the attention of the reader to the fact that both indices B and C in (52)2 obey the convention (1). Some possible choices for Z (or, equivalently, for Z ) are discussed next. Linear elastic materials. A linear elastic material may be defined by setting a quadratic potential 1

r Z FBr  2 FBr ¸ DBC FCr ,

(53)

r taking DBC as constant so that from definition (52) follows r T Br  DBC FCr .

(54)

For the case of rods with homogeneous cross sections, and recalling that we assume that the rod axis coincides with the line of the cross-sectional r is given in the rod´s local system by the stanbarycenters, matrix DBC

A Unified Approach for the Dynamics of Rods and Shells

111

dard relation

r r DBC  D11 

sT 1r sF1r

0 0   EA 0 ¡ ¡ 0 GAa 0 0 ¡ ¡ 0 a 0 GA 0  ¡¡ 0 0 0 GI T ¡ ¡ 0 0 0 0 ¡ ¡ 0 0 0 ¡¢¡ 0

0 0 0 0 EI 2 0

0 ¯

° ° 0 °° , 0 °° ° 0 ° ° EI 3 °° ± 0 °

(55)

in which A , A a , I 2 , I 3 , IT are the cross-sectional geometrical properties and G , E are the material elastic moduli. The diagonal form (55) assumes in addition that local axes e2r and e3r are principal axes and that the shear center coincides with the barycenter. For the case of shells, recalling that we assume that the shell midr are given in the shell´s local surface is the medium surface, matrices DBC system by

r D11 

r D12 

sT 1r sF2r

sT 1r sF1r

  Eh 0 0 ¡ ¡ 0 Nh 0 ¡ ¡ 0 0 Nh ¡ ¡ ¡ 0 0 0 ¡ ¡ 0 0 0 ¡ ¡ 0 0 0 ¡¢

  0 E Oh ¡ ¡ Nh 0 ¡ ¡ 0 0 ¡ ¡ 0 ¡ 0 ¡ ¡ 0 0 ¡ ¡ 0 0 ¢¡

0

0

0

0

0 1 12

Nh

0

0

1 Eh 3 12

0

0

0

0

0

0

0

0

0

0

0

1  12 Nh 3

1

0  12 E Oh 3

0

0

0

0

° ° 0° ° ° , 0° ° 0 °° ° 0° ± 0°

0 3

0

0

0¯

(56)

0¯

° ° 0° ° rT °  D21 , 0° ° 0 °° 0 °° ± 0°

and

(57)

112

P.M. Pimenta and E.M.B. Campello   Nh 0 0 ¡ ¡ 0 Eh 0 ¡ ¡ 0 0 Nh ¡ ¡ ¡ 0 0 0 ¡ ¡ 0 0 0 ¡ ¡ 0 0 0 ¡¢

0

0

0¯

0

0

0°

° ° 0 0 0° r ° sT 2 r ° , D22  (58) 1 3 Eh 0 0 ° sF2r 12 ° 1 0 Nh 3 0 °° 12 ° 0 0 0° ± in which E is the effective elasticity modulus, N is the transverse shear modulus and O the Poisson´s coefficient. One should recall that E  E / 1  O 2 and N  E / 2 1 O , with E as the standard elasticity modulus. Small strains must be assumed for these constitutive relations to be valid.

General hyperelastic materials. One possible nonlinear hyperelastic material for the case of rods may be defined by the following strain energy function   T œ sgn FBr i Ti ¡¡ FBr i kii i 1 ¢¡ 6

Z FBr 



  ki žžže Ti Ÿž

FBr i

¬¯ ­  1 ­­ °° , ­­® ±°

i  1, 2, !, 6 ,

(59)

which is written in terms of the six components FBr i of vector FBr . In this case, k1  EA , k2  k 3  GAa , k 4  GIT , k5  EI 2 and k6  EI 3 are the cross-sectional stiffness properties and Ti are additional material parameters. From this expression, it follows that the six components TBr i of T Br are given by k   i TBr i  sgn FBr i Ti žžž 1  e Ti Ÿž



FBr i

¬­ ­­­ , ®­

i  1, 2, !, 6 ,

(60)

and the matrix of elastic tangent moduli reads as

\

^

r r DBC  Diag DBC i ,

with

r DBC i  ki e



ki r F Ti B i

,

i  1, 2, !, 6 .

(61)

It is important to mention that this exponential material is a fictitious material that was created here only to render a nonlinear constitutive equation, with which we could assess our integration algorithm. It may used to represent rods that lose stiffness with straining, as if they were made of plastic materials. More general, fully three-dimensional nonlinear hyperelastic materials for rods can be considered in the lines of Pimenta

A Unified Approach for the Dynamics of Rods and Shells

113

and Campello (2003), as already done for statical problems and rectangular cross-sections by Dasambiagio et al. (2007). For the case of shells, if we assume that the material is isotropic, the specific strain energy function Z FBr may be written in terms of strain invariants I 1 and I 2 . By adopting I 1  F : F and I 2  det F  J , a general (neo-Hookean) hyperelastic material may be defined by (see e.g. Ciarlet, 1988) Z FBr  Z I 1, J 

1  1 2 1 M ¡ J  1  ln J ¯° N I 1  3  2 ln J , ± 2 2 ¢2

(62)

where M and N are elastic parameters or generalized Lamé constants. In order to enforce the plane-stress condition in a consistent manner, let us rewrite expression (17) for the deformation gradient as F  Q < I I Br LBr q r r  eBr H 33e 3r  e 3r > 

 Q < I H Br  eBr H 3r  e 3r > ,

(63)

in which H Br  I Br LBr q r r encloses the shell strain vectors and H 3r  H33e3r is the vector of the thickness straining corresponding to the plane-stress state (scalar H33 is here introduced in order to allow for this thickness deformation). Defining fir  eir Hir ,

(64)

expression (63) can be rewritten as F  Q fir  eir ,

(65)

and in this case the strain invariants turn out to be I 1  fir ¸ fir  fBr ¸ fBr 1 H33 2

and

J  f3r ¸ f1r q f2r  1 H33 J ,

with

J  e3r ¸ f1r q f2r .

(66)

It is not difficult to show that the derivatives of Z I 1, J with respect to fir furnish the first Piola-Kirchhoff stress vectors U ir as below U ir 

sZ sZ sI 1 sZ sJ ,  sfir sI 1 sfir sJ sfir

so that after some algebra one has

(67)

114

P.M. Pimenta and E.M.B. Campello sZ r sZ 1 H 33 FBC fCr q e3r

f sI 1 B sJ sZ sZ J,  U 3r ¸ e3r  2 1 H33

sI 1 sJ

U Br  2 U 33

and

(68)

with FBC  e3r ¸ eBr q eCr as a permutation symbol. Now the plane-stress assumption may be invoked for equation (68)2: U 33  U 3r ¸ e 3r  0 ,

(69)

and with the aid of (62) it renders H 33 

M 2N MJ 2 2N

1,

(70)

i.e., H33 may be consistently eliminated. Introducing this result into (68)1, and taking (62) again into account, the following expression is obtained for the first Piola-Kirchhoff stress vectors:

U Br  K J FBC fCr q e3r NfBr ,

(71)

1 M 2N 1 K J   ¡¢ 2 M J 2  1  N ¯°±  N 3 . MJ 2NJ J

(72)

where

The stress resultants T Br may be then computed via integration of (71) across the thickness. Alternatively, one may use expression (52)1 together with (50), (66) and (70) to get T Br . For computation of the matrices of material tangent moduli, from (52)2 one has

r  DBC

sT Br sFCr

  sn Br ¡ ¡ sI Cr ¡ ¡ sm Br ¡ ¡¢ sI Cr

sn Br ¯ ° sLCr ° ° . sm Br ° ° sLCr °±

(73)

Defining the tangent tensors r C BC 

sU Br sH Cr

,

the derivatives in (73) may be written as

(74)

A Unified Approach for the Dynamics of Rods and Shells sn Br sICr sm Br sICr

 

r dH , ¨H C BC

¨H

r dH RrC BC

sn Br sLCr

115

r  ¨ C BC Rr dH , H

and

sm Br sLCr

(75) r  ¨ RrC BC Rr dH , H

r with Rr  Skew r r . Substitution of (71) into (74) yields C BC as follows r C BC  K a J FBC fCr q e3r  FCE fEr q e3r

 K J FBC Skew e3r NEBC I ,

(76)

where EBC is the Kronecker symbol and K a J



M 2N 3MJ 2 2N

sK 3MJ 2 2N N  K J

. 2 3 sJ 3 M N J J 2 MJ 2NJ

(77)

Substitution of (75) and (76) into (73) furnishes the matrices of materir for this neo-Hookean material. It is interesting to al tangent moduli DBC observe that, up to first order in the deformations, this material is entirely equivalent to the linear elastic material, as the above expressions for T Br r collapse to the corresponding ones from the previous sub-section. and DBC More general, fully three-dimensional hyperelastic materials for shells (i.e. without the plane-stress enforcement) can be considered in the lines of Pimenta et al. (2004). Remark. We draw the attention of the reader to the fact that, up to our knowledge, our formulation is the first one in the literature to allow for general hyperelastic materials in the nonlinear dynamics of rods and shells (i.e. models involving rotational degrees-of-freedom at finite rotations) with exact conservation of both momenta and mechanical energy within the time integration scheme.

4

Time increment

In this section we recast our time-increment notation introduced in Pimenta et al. (2008) and Campello et al. (2009), and show some crucial results needed subsequently in the text. Demonstration of some of the expressions is omitted here for the sake of simplicity, but one can find the detailed derivations in the above-mentioned works. We remark that an updated description of the motion is adopted, and for this reason the con-

116

P.M. Pimenta and E.M.B. Campello

cepts of incremental displacements and rotations will be fully exploited. Consider an arbitrary time increment ti , ti 1 , for which we adopt the notation (¸) ti  (¸)i and (¸) ti 1  (¸)i 1 . Assume that all quantities at time ti are known from the solution at the previous increment and consider the following notation %(¸)  (¸)i 1  (¸)i

and

1

(¸)1/ 2  2 < (¸)i (¸)i 1 > .

(78)

4.1 Incremental displacements and rotations Let u % be the incremental displacement vector, defined by

u %  %u ,

(79)

and let Q% be the incremental rotation tensor, such that

Qi 1  Q%Qi .

(80)

Let B% be the Rodrigues rotation vector associated to the rotation tensor Q% , with magnitude B%  B% , and let us write "%  Skew B% .

(81)

If u % and B% are known, update of the displacement field may be performed by means of (79) and (78)1, i.e

ui 1  ui u % ,

(82)

and update of the rotation field may be performed by using the Rodrigues expression for superposed rotations (see Argyris, 1982, Pimenta and Campello, 2005, and Rodrigues, 1840) Bi 1 





4 1 B% Bi 2 B% q Bi . 4  B% ¸ Bi

(83)

We strongly remark that expression (83) is valid only when the Rodrigues parameterization for the rotation field is adopted, and thus may not be applied if the rotations are described by the Euler vector R or by any other parameters. In these latter cases, update is a much more complicated task and requires additional operations, sometimes with several singularities involved. Let us now define a back-rotated incremental rotation tensor by Q%r  QiT 1Q%Qi 1  QiTQ%Qi ,

(84)

A Unified Approach for the Dynamics of Rods and Shells

with B%r as its Rodrigues rotation vector, "%r  Skew B%r . From (84) and (80), it follows Br%  QiT 1B%  QiT B%

and

117

and

let

B%  Br%  B% .

us

write (85)

Using (5) and (6), the following identities can be derived 1

Q1/ 2  Qi 1 I 2 "%r

1

1

%Q  Qi I  2 "%r

1

1

and

"%r  Q1/ 2"%r  1

 Qi "%r I  2 "%r

1

1

 I 2 "% Qi 1

(86)

 "%Q1/ 2 ,

and here one should notice that Q1/2 above is not a rotation, but has the following important property, obtained with the aid of (85)1: T B%r  Q1/2 B%

and

B%  Q1/2 B%r .

(87)

Other important expressions involving Q1/2 may be derived by using (10): det Q1/2  h B% ,

Q1/2  det Q1/2 Ȅ %TQi ,

1 1 Q1/2  det Q1/2

QiT ȄT%

and

T det Q1/2 Q1/2  Ȅ %Qi ,

(88)

where





1 Ȅ%  Ȅˆ B%  h B% I 2 "% .

(89)

The following Nanson’s rule may be then written:

Q

T a q Q1/2b  det Q1/2 Q1/2 a q b  Ȅ%Qi a q b , a,b ‰ 3 . (90)

1/2

4.2 Incremental strains and strain rates Starting from expression (18) for I Br and LBr , and using (80) and (86), one can show that T %I Br  Q1/ 2 u %,B z 1/ 2,B q B%





and

%LBr  QiT 1axial Qi 1,BQiT 1  LBr i  det Q1/ 2 Q1/21B%,B .

(91)

With these expressions, the incremental strain vectors %FBr may be written after some algebra in a compact manner as

118

P.M. Pimenta and E.M.B. Campello %FBr  -Tm 'B 1/2%Bd% ,

(92)

in which O  Q1/ 2 ¯   I O Z1/ 2,B ¯ °, ' ¡ ° and -m  ¡¡ B 1/ 2  ¡ T ° ° O I O det O Q Q

1/ 2 1/ 2 ° ¢¡ ±° ¢¡ ± (93)   u% ¯ d%  ¡ B ° . ¡¢ % °± For the incremental strain rate %FBr , from time derivative of (91) one may write I Br i 1  QiT 1 ¡  u %,B Z i 1,B Ȅ %B % °¯ and ¢ ± r T L B i 1  Qi 1 < Ȅ %B %,B Q%W B% , B%,B B % > ,

(94)

so that %FBr  FBr i 1  -Ti 1'B i 16 6

B%

 I ¡ ¡  ¡O ¡ ¡O ¢¡

O Ȅ% O

 ,

B % %Bd %

with

¯ ° ° Q%W B% , B%,B ° . ° ° Ȅ% ±° O

(95)

4.3 Increments of momentum, kinetic and strain energy From expressions (35) and (36), increments of linear and angular momenta may be written as %M 

¨8 %Md8

and

%N 

¨8 % z q M N d8 ,

(96)

where %M  Mu % and % z q M N  z1/2 q M %u u % q Mu 1/2 r Q1/2  ¡ J r %Xr B%r q J r X1/2

¯°± . ¢ Increment of kinetic energy is computed by using (51): %T 

¨8 %Td8 ,

where

r r %T  M %u ¸ u 1/2 J r %Xr ¸ X1/2  %M ¸ u 1/2 %Nr ¸ X1/2 ,

(97)

(98)

A Unified Approach for the Dynamics of Rods and Shells

119

and the increment of strain energy (from (49)) is given by %U int 

5

¨8 %Zd8 .

(99)

Time integration Algorithm

5.1 Time collocation of the equations of motion First, we write the global equations of motion (38) at a time instant tm in the midst of the increment:

fmext 



¨8 Mmd8

mmext 

and

¨8 z

1/2



 q Mm N m d8 .

(100)

Then, we assume from (31) that

¨8 nmd8 nm( ( and  ¨ z q nm mm d 8 z 8

fmext  mmext

1/2

1/2

q nm( mm(

(

,

(101)

and from (37) that M m  Mu1/2 and r N m  Q1/2 N m , with

r r r r N m  J r X 1/2 X1/2 q J r X1/2 .

(102)

( ( and mm are mean values of the external loads Vectors nm , mm , nm within the time step that will be defined later. In the same way as in the derivation of (46), the weak form associated to (100) may be constructed, and analogously to (47) it can be written in a compact manner as

EPm 

¨8 TBm ¸ 'B 

( Tm

1/2

¸ Ev



%B Ev g m ¸ Ev  Tm ¸ Ev d8

(

 0 , Ev ,

(103)

where 'B 1/2 is given in (93) and (analogously to (48))   nB m ¯ °, TB m  ¡¡ ° ¡¢ m B m °±

 n ¯ Tm  ¡¡ m °° , ¡¢ mm °±

  n( ¯ ( Tm  ¡¡ m( °° m ¢¡ m ±°

and

  M ¯ gm  ¡¡ m °° . (104) ¡¢ N m °±

In (104)1 n B m and m B m are mean values of the cross-sectional resultants within the time step that will be defined later. Notice that from (103) the algorithmic equations of motion (100) follows, using (101) and (102). One crucial aspect in our algorithm is now introduced: we define the cross-sectional resultants T Bm of (103) by

120

P.M. Pimenta and E.M.B. Campello T B m  -m T Br m ,

(105)

with -m as in (93) and TBr m given by r   ¯ ¡ ¨S U B mdS ° TBr m  ¡ °, ¡ ¨ r r q U Br mdS ° ¢¡ S ±°

(106)

which is based on definitions (29) and (28). Here, we may regard U Br m as mean values of the back-rotated Piola-Kirchhoff stress vectors within the time increment, whose expressions will be defined later.

5.2 Time approximations and algorithmic weak form We adopt the following mid-step approximations for the time-dependent variables: u 1/ 2  u1/ 2  

1 %t 1 %t

u% ,

X1/r 2 

u % 

X 1/r 2 

2 %t 2

u %  %tu i ,



1 %t 1 %t

B%r ,

2 %t 2

B%r  %tXir .

%X r 

(107)

Introduction of these assumptions into (102) yields Mm  N mr 

2 M u %  %tu i

%t 2 2   r J B%r  %tXir %t 2 ¢¡

and



1 r B 2 %

N m  Q1/ 2 N mr ,

with

(108)

q J B% ¯±° . r

r

By introducing (108) into (103), and by taking (105) into account, one arrives at the algorithmic weak form

EPm 

¨8  ¡¢ T B m ¸ -m'B r

T

1/ 2



%B Ev gm  T m ¸ Ev ¯° d8 ±  T m( ¸ Ev (  0 ,

Ev ,

(109)

where gm 

2 %t 2

  ¯ M u %  %tu i

¡ °. ¡ ° ¡Q1/ 2 J r B%r  %tXir 12 B%r q J r B%r ° ¢ ±

(110)

Expression (109) is crucial in the demonstration of energy conservation. However, as already observed in section 3.7, a symmetric tangent of the weak form is not to be expected.

A Unified Approach for the Dynamics of Rods and Shells

121

5.3 Conservation of linear and angular momentum By inserting the time approximations (107) into (97), and by taking (102) and (100) into account, after some algebra one arrives at

¨8 %Md8  %t %M and 1 1 ext mm  %t ¨ % z q M N d8  %t %N . 8 fmext 

1 %t

1

(111)

These expressions ensure that, if the body is isolated (i.e. fmext  mmext  o ), both momenta are exactly conserved within the algorithm. Expression (111) can be regarded as the algorithmic form of the Euler laws of motion.

5.4 Conservation of energy By introducing (107) into (98), one arrives at

%T 

1 %t

%M ¸ u

%

%N ¸ B% 

1 %t

%g ¸ d%  gm ¸ d% .

(112)

On the other hand, for isolated bodies the weak form (109) turns into

EPm 

¨8  ¡¢ T Br m ¸ -Tm'B

1/2



%B Ev gm ¸ Ev ¯° d8  0 , ±

Ev .

(113)

If we set Ev  d % and take (92) and (112) into account, we have

¨8  ¡¢ T Br m ¸ %FBr %T ¯°± d8  0 .

(114)

Therefore, if TBr m is such that %Z  T Br m ¸ %FBr ,

(115)

with Z as the cross-sectional specific strain energy defined in (50), then the increment of mechanical energy vanishes from (114), i.e., the total energy is exactly conserved in a discrete sense. In order to fulfill condition (115), TBr m is defined by setting





T Br m  TˆBr FCr m 

sZ r , F sFBr C m





(116)

where the collocation points FCr m are given by a convex combination of FCr within the increment: FCr m  FCr +  1  + FCr i +FCr i 1  FCr i +%FCr ,

(117)

with + as a local scalar variable yet to be determined. We remark here

122

P.M. Pimenta and E.M.B. Campello

that both indices B and C in (116) follow convention (1). As we have mentioned in Pimenta et al. (2008) and Campello et al. (2009), the idea of using (117) is similar to that proposed by Simo and Tarnow (1992), although in a different framework. Simo, however, did not consider the coupling between + and the deformations (see Laursen and Meng, 2001), leading to an incorrect tangent operator. For this reason, his formulation could deal only with materials of quadratic potentials, for which +  12 and the strain coupling automatically disappears. Here, more general hyperelastic materials can be considered and + is found by solving the energy conservation constraint equation g +  T Br m ¸ %FBr  %Z  0

(118)

by the Newton Method as follows 1

+k 1  +k  < g a +k >

g +k ,



k  1, 2, !,

+0 



1 2

,

(119)

r g a +   ¡ DBC FHr + %FCr °¯ ¸ %FBr , ¢ ±

where

r with DBC as in (52)2. As a consequence of assumption (117) and constraint (118), computation of the tangent of the weak form becomes a much more elaborated task. The following result is obtained after some lengthy algebra: alg DBC 

sT Br m sFCr i 1



sT Br m sFEr m sFEr m sFCr i 1



  ¯ s+ ° r  DBE FIr + ¡¡ +I %FEr   r s%FC i 1 °° ¡¢ ± r  +DBE FIr +









(120)



1 r r DBE FIr + %FEr  ¡  T Cr i 1  T Cr m  +DCH FIr + %FHr °¯ . ga +

¢ ±







Notice that, if Z is quadratic, one deduces +  solution to (118), and then FBr m  FBr 1 / 2 , %Z 

T Br 1 / 2

¸





1

2

r T Br m  T Br 1 / 2  DBC FIr 1 / 2 FCr 1 / 2 ,

%FBr

and

alg DBC



1 Dr 2 BC



as an analytical

(121)

.

We remark again that all Greek indices B , C , H , E and I in (119)– (121) follow convention (1).

A Unified Approach for the Dynamics of Rods and Shells

123

5.5 Tangent of the weak form The Gâteaux derivative of the algorithmic weak form (109) furnishes the tangent of the weak form as given below

E EPm 

¨8 %B Ev ¸ 'BT

1/ 2

alg T -m DBC -i 1'C i 16

¨8  ¢ %B Ev ¸ GB %B Ed

%

C % %C Ed%

d 8

Ev ¸ H Ed%  Ev ¸ LEd% ¯± d 8 (122)

  ¢ Ev ¸ L( Ed% ¯±

,

(

alg where DBC is given in (120), GB is given by

  O ¡ ¡ GB  ¡¡ O ¡1 ¡ Skew Q1/ 2n Br m ¢¡ 2

O O





 Hu  ¡¡ ¡¢ O

O ¯° °, HB ° ±

O





1  2 Skew Qi 1n Br m Ȅ % ¯° ° ° W T B% ,Qi 1m Br m ° ° 1 r Skew Qi 1n B m Ȅ % °° Z 2 1/ 2,B ±







(123)

and H 

sg m sd%

L 

sTm sd %

,

and

L( 

( sTm

sd %

,

(124)

with Hu  H

6

B



2 MI %t 2 2 Q %t 2 1/ 2 1  %t 2

and 1 1 r r r r ¯ T   r ¢¡ J 2 "% J  2 Skew J B% ±° Qi 1 Skew \Qi 1  ¡¢ J r B%r  %tXir 2 B%r q J r B%r ¯°± ^ Ȅ % .

(125)

Finite element implementation and numerical examples

Let p % be the rod or shell element vector that collects the nodal degreesof-freedom, i.e.

  u% ¯ p%  ¡¡ °° , ¡¢ B% ±°

(126)

and let Ep% be the vector of their variations. Let N be the matrix of element shape functions. Finite element approximations of Galerkin type may be then written as

d%  Np% ,

Ed%  NEp%

and

Ev  NEp% .

(127)

124

P.M. Pimenta and E.M.B. Campello

Introducing (127) into (109), we obtain the element residual force vector as follows ¯ ( 'BT1/2 -m TBr m NT g m  NT Tm ° d 8  NT Tm

( , (128) ± and from (127) into (122) we obtain the element stiffness matrix P

 

¨8 ¡¢ %B N

k

T

¨8  ¢¡ %B N 'BT -m DBC -Ti 1'C i 16 C %C N ¯±° d8 ¨ ¢  %B N T GB %B N NT H N  NT L N ±¯ d8 8 T

alg

%

1/ 2

(129)

 NT L( N ( . These expressions are used for assemblage of the global force vector and global stiffness matrix. Within a Newton solution procedure, we must compute, at every iteration and at every integration point, the following quantities at ti 1 from their values at ti and the current interpolated values of u % and B% : u i 1 

2 u  u i , %t % 2 Xi 1  %t B%  Q% Xi LBr i 1  LBr i %LBr ,

ui 1  ,

4

X i 1  Bi 1 

4 %t

u% 

%t 2

4

u i  ui ,

B%  Q%

%t 2 4 4  B ¸B %

i

B

%



4 %t



Xi X i ,

Bi

1 B 2 %

(130)



q Bi .

After convergence, the following updates must be performed at every node

u k u u%

and

u %  B%  o ,

(131)

whereas at each integration point we must set u i k u i 1 ,

X i k X i 1 ,

ui k ui 1 , Xi k Xi 1 ,

LBr i k LBr i 1 , Bi k Bi 1 .

(132)

In the next sections, we assess the performance of the presented formulation by means of some numerical examples. For the rod elements, standard linear shape functions of Lagrangean type are assumed to construct N , and 1-point Gauss quadrature is adopted for the space-domain integration. For the shell elements, the six-node triangular element of Campello et al. (2003) is adopted as the basis for all implementations. The element is

A Unified Approach for the Dynamics of Rods and Shells

125

purely displacement-based and is equipped with quadratic interpolations for u % and linear interpolations for B% , these latter being based on the mid-side nodes only. Gaussian quadrature using 3 integration points is adopted, together with 3 Gauss points for integration across the thickness. No special techniques such as ANS or EAS are employed since the element does not suffer from any locking misbehavior in the thin-shell limit (see Campello et al., 2003). Computation of the scalar parameter + is performed locally at the integration points, so that + is always eliminated at the element level.

P(t)

I2 = I3 = 0.5

0.5P(t)

IT = 0.1

0.8

A = 1.0 200.0

E = G = 1.0×104 0.6

0.1P(t)

 = 1.0 t = various

2.5

5.0

t(s)

P(t) 1000

600

. Internal + kinetic energy

Angular momentum .

400 200 0 -200

X component Y component Z component

-400 -600

800

600 400

Internal plus kinetic energy

200

0 0

100

200

300

Time (s)

t = 2s

t = 3s

400

500

0

100

200

300

400

500

Time (s)

t = 14s

t=0

Figure 2. (a) Large overall motion of an inclined beam, problem data. (b) Time history of angular momentum and energy, and early stages of the motion.





126

P.M. Pimenta and E.M.B. Campello

6.1 Large overall motion of an inclined beam This interesting problem was first proposed by Simo et al. (1995) and deals with the large overall motion of a free flexible beam. The beam is initially at an inclined position as depicted in Figure 2(a), and is subjected to the spatially fixed forces and moments shown. The material is assumed to be linear elastic. Spatial discretization is performed using ten 2-noded elements, along with %t  0.1 s for the time-dependent variables. Graphs of energy and angular momentum are shown in Figure 2(b) for the total analysis time of t  500.0 s, and a side view of the early stages of the motion is also shown in true scale. Notice that the loading produces translational, forward tumbling and out-of-plane displacements. We remark that larger time steps may be employed, and the very same conserving response is attained.

6.2 Free vibration of a beam in 3-D space The slender beam with geometrical properties shown in Figure 3 is subjected to self-equilibrated loads whose magnitude follows a hat function in time. P

E = 50×109

L = 3.0

G = 20×109  = 2500

I2 = 0.64×10-6 I3 = 0.15×10-4 A = 0.15×10-2

P /2

IT = 0.64×10-8

(SI units)

P /2

TBr i

P(t)

Ti

t = 0.1 ms, 0.5 ms, 1.0 ms

FBr i

Ti

40×104

0.1

0.2

t(s)

Figure 3. Free vibration of a beam in 3-D space. Problem data.



The beam consists of an exponential elastic material of the type defined in expressions (59)–(61), with density S  2500 and elastic constants E  50 q 109 , G  20 q 109 , T1  T2  3.0 q 107 , T3  7.5 q 107 ,

A Unified Approach for the Dynamics of Rods and Shells

127

T4  3.2 q 104 , T5  7.5 q 105 and T6  1.3 q 102 . At t  0.2 s, the loads are removed and the beam undergoes a finite free vibration with large out-of-plane bending. The problem is analyzed using ten 2-noded elements with no constraints on the 6 nodal DOFs. For the time discretization we adopt (i) %t  0.1 ms, (ii) %t  0.5 ms and (iii) %t  1.0 ms. Time history of the tip displacements obtained with the three different increments %t are shown in Figure 4, together with the energy graphs for the total analyses ending at t  5.0 s (linear and angular momentum are zero by nature and for this reason the corresponding curves are omitted here). A plot of deformed configurations at the beginning of the motion (obtained with %t  1.0 ms) is also shown in true scale, for which typical converged values of + range from 0.488 to 0.511.

6.3 Dynamics of a satellite-like structure Let us now consider the dynamics of a satellite-like structure made up of three intersecting plates. The motion starts with the structure initially at rest and subjected to a set of external loads that is subsequently removed, leading to a force-free motion in which both momentum and energy must be preserved. Problem data are shown in Figure 5, where the material is assumed to be the linear elastic one of equation (53). Also in Figure 5, a plot of selected deformed shapes and time-histories are given. Excellent agreement with the results reported by Simo and Tarnow (1994) is found.

6.4 Free vibration of a hemispherical shell This last example was proposed by Sansour et al. (1997) for the case of linear elastic materials, but here we present a different version of the problem and consider a neo-Hookean hyperelastic material of the type defined in (62). A hemispherical shell with geometric and material properties as shown in Figure 6 is subjected to two pairs of concentrated forces. The forces are applied until t  2.0 ms and then removed, after what the structure undergoes a free vibration motion. We discretize the problem by using four element divisions per quadrant on both radial and circumferential directions. The results obtained for the displacements under the load points are depicted in Figure 7, together with the energy (internal plus kinetic) graph where perfect conservation can be found. Deformed shapes are also shown (no amplification factor is adopted).

128

P.M. Pimenta and E.M.B. Campello

 2.00

1.50

Vertical displacement (m .

Horizontal displacement (m) .

present algorithm (deltaT=0.0001)

1.00 0.50 0.00 -0.50 -1.00 -1.50 0.0

1.0

2.0

3.0

4.0

1.00 0.50 0.00 -0.50 -1.00 0.0

5.0

present algorithm (deltaT = 0.0001)

1.50

1.0

2.0

1.50

Horizontal displacement (m) .

Vertical displacement ( .

0.50 0.00 -0.50 -1.00

1.0

2.0

3.0

4.0

1.00 0.50 0.00 -0.50 -1.00 0.0

5.0

1.0

2.0

3.0

4.0

5.0

Time (s)

2.00

1.50

Horizontal displacement (m) .

present algorithm (deltaT=0.001)

1.00

Vertical displacement (m .

5.0

present algorithm (deltaT = 0.0005)

1.50

Time (s)

0.50 0.00 -0.50 -1.00 -1.50 0.0

4.0

2.00

present algorithm (deltaT=0.0005)

1.00

-1.50 0.0

3.0

Time (s)

Time (s)

1.0

2.0

3.0

4.0

5.0

present algorithm (deltaT = 0.001)

1.50 1.00 0.50 0.00 -0.50 -1.00 0.0

1.0

2.0

3.0

4.0

5.0

Time (s)

Time (s)

Internal + kinetic energy (J) .

100000

t = 0.10 t = 0.02 t = 0.015

All time-steps

80000

t = 0.01

60000

t=0

40000

t = 0.206 20000

t = 0.208 0 0.0

t = 0.210 1.0

2.0

3.0

Time (s)

4.0

5.0



Figure 4. Free vibration of a beam in 3-D space. Time history of the tip displacements, energy graph and selected deformed configurations.

A Unified Approach for the Dynamics of Rods and Shells 9.0 P(t)

5P(t)

P(t) 4.0 14.0

0.25

(SI units)

t(s)

1.0

0.5

t = 1×10-4s, 1×10-3s, 2×10-3s, 5×10-3s

0.5P(t)

t = 0.8s

t = 3.0s

t = 2.0s

100

200

X component Y component Z component

Energy (deltaT = 0.005) Internal + kinetic energy

75

Angular momentum (

P(t)

h = 0.02 E = 2×10-6  = 0.25  = 1.0

t = 0.4s

129

50

all t 25

0

150

100

all t 50

-25

0

-50 0.0

5.0

10.0

15.0

20.0

25.0

0.0

5.0

10.0

15.0

20.0

25.0

Time (s)

Time (s)



Figure 5. Dynamics of a satellite-like structure. Problem data, early stages of the motion and time histories of momentum and energy.

P(t)

P(t) R = 10.0 P(t)

h = 0.4 E = 68250

200

 = 0.3125  = 1×10-7

P(t)

0.001

0.002

t(s)

(SI units) t = 1×10-5s, 2×10-5s, 1×10-4s, 5×10-4s P(t)

Figure 6. Free vibration of a hemispherical shell. Problem data.



130

P.M. Pimenta and E.M.B. Campello

4.00

6.00

2.00

Displacement (m)

Displacement (m)

Horizontal displacement under outward load point (dt=1.0e-5)

4.00

0.00 -2.00 -4.00 -6.00

Horizontal displacement under inward load point (dt=1.0e-5)

-8.00 -10.00 0.000

0.020

0.040

0.060

0.080

2.00 0.00 -2.00 -4.00

0.100

-6.00 0.000

0.020

0.040

0.060

0.080

0.100

time (s)

time (s)

2000

Present algorithm (dt=1.0e-4)

Energy (J)

1600

1200

800

400

0 0.000

0.020

0.040

0.060

0.080

0.100

Time (s)

Figure 7. Free vibration of a hemispherical shell. Time history of displacements under load points, energy graph and deformed shapes.

7

Conclusions

A unified formulation was developed in this work for the nonlinear dynamics of rods and shells undergoing arbitrarily large deformations and rigid body motions. Based on Pimenta et al. (2008) and Campello et al. (2009), we introduced a special notation so that the description of both rod and shell motions was made possible with the same set of expressions. Differences are observed only at the constitutive equation. Time-collocation of the resulting expressions following an energymomentum approach ensured exact conservation of both momentum and mechanical energy in the absence of external forces. We believe this unified description leads to a straightforward simultaneous implementation of both rod and shell dynamics models within a finite element code.

A Unified Approach for the Dynamics of Rods and Shells

131

Bibliography J. H. Argyris, An excursion into large rotations. Comp. Meth. Appl. Mech. Engrg. 32: 85-155, 1982. E. M. B. Campello, P. M. Pimenta and P. Wriggers, A triangular finite shell element based on a fully nonlinear shell formulation. Comput. Mech. 31: 505-518, 2003. E. M. B. Campello, P. M. Pimenta and P. Wriggers, An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: Shells. Submitted to Computational Mechanics, 2009. P. J. Ciarlet, Mathematical Elasticity. Vol.1, North Holland, Amsterdam, 1988. E. R. Dasambiagio, E. M. B. Campello and P. M. Pimenta, Multiparameter analysis of rods considering cross-sectional in-plane changes and out-of-plane warping. In Proceedings of the CMNE 2007 and XXVIII CILAMCE, J.C. de Sá (ed), Oporto, Portugal, 2007. O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comp. Meth. Appl. Mech. Engrg. 190: 1763-1783, 2000. D. Kuhl and E. Ramm, Constraint energy-momentum algorithm and its application to nonlinear dynamics of shells. Comp. Meth. Appl. Mech. Engrg. 136: 293-315, 1996. T. A. Laursen and X. N. Meng, A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics. Comp. Meth. Appl. Mech. Engrg. 190: 63096322, 2001. P. M. Pimenta, On a geometrically-exact finite strain shell model. In Proceedings of the 3rd Pan-American Congress on Applied Mechanics, III PACAM, São Paulo, 1993a. P. M. Pimenta, On a geometrically-exact finite strain rod model. In Proceedings of the 3rd Pan-American Congress on Applied Mechanics, III PACAM, São Paulo, 1993b. P. M. Pimenta and T. Yojo, Geometrically-exact analysis of spatial frames. Applied Mechanics Reviews, ASME, 46 (11): 118-128, 1993. P. M. Pimenta and E. M. B. Campello, A fully nonlinear multi-parameter rod model incorporating general cross-sectional in-plane changes and out-of-plane warping. Lat. Amer. J. Solids Struct. 1: 119-140, 2003. P. M. Pimenta, E. M. B. Campello and P. Wriggers, A fully nonlinear multi-parameter shell model with thickness variation and a triangular shell finite element. Comput. Mech. 34: 181-193, 2004.

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P. M. Pimenta and E. M. B. Campello, Finite rotation parameterizations for the nonlinear static and dynamic analysis of shells. In Proceedings of the 5th International Conference on Computation of Shell and Spatial Structures, E. Ramm, W.A. Wall, K.-U. Bletzinger, M. Bischoff (eds), Salzburg, Austria, 2005. P. M. Pimenta, E. M. B. Campello and P. Wriggers, An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods. Comput. Mech. 42: 715-732, 2008. B. O. Rodrigues, Des lois géométriques qui régissent les déplacements d’un système solid dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire. Journal de Mathématiques Pures et Appliquées, 380-440, 1840. C. Sansour, P. Wriggers and J. Sansour, Nonlinear dynamics of shells: theory, finite element formulation and integration schemes. Nonlin. Dyn. 13: 279-305, 1997. J. C Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z. angew. Math. Phys. 43: 757-792, 1992. J. C Simo and N. Tarnow, A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int. J. Numer. Methods Engrg. 37: 2527-2549, 1994. J. C Simo, N. Tarnow and M. Doblare, Exact energy-momentum algorithms for the dynamics of nonlinear rods. Int. J. Numer. Methods Engrg. 38: 1431-1474, 1995.




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, 4  +, 1J " $  $# $#+   !  9  + / 42+ -1555 9  +  ($+
)+#$ +"  8)/(/


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$# (

##    + ) $ )

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"

 

/0, ""  +  0+ ", "":  + //0, ,  0   , ,,4  /,,+ , 0 "

/,++: $  N , ,+,   "+ "      / ,"+,     ; $ "" ,  0 ,  " , *,4 ,  , ,& "+0"/    + 0 +  4 ,  /&   * "+ +,  " ,0+ *'+   //0,: /,   & ,+ ""4      

134

K.-U. Bletzinger et al.

 *( 2",   , //0, ""

 0+     /   //0, /   + + , 

,* 0 , 4 0"  ,  0 / 0 ,+  / 2   7

": $ "+ ,   +, //0, ""  ,++ , 7 / ,+ "0+"& ",: $ //0, /,,+ +   +  "  +, ,+: $ ,  ,

   , 4 *  , " ,   / 2    "; $ " , ,  0 /  " , *,4 ,  2,+ "7 "     ,/0+ ,+ , ,  , * /,  &  ,  0  0+ *,:   +, .+ ""4  *  , ,++ ,"/  0  "+ ,   ,

" ,,+: $ ,"/   +  "    &+ ,0+   ""  +,&+ + * ,  /,,+ ,  / + ,    , "0+7"& ,4 *   ,   .+ * + * , M: "   + "& "" 0"+ "  /,,+ * ,  , M C:: 5, $7""E4  5"   " ,   &,++ "",+ 0,& "   ,+   +0+; "    &+ ,0+   "0+7"& ""4 ,

 ,+  ,   //0,4 * , 0  " , /,"  ,  ,&  0     , ,+ /,,+     "": %   ++ *4   & ",+    / 2  , " , ,7   ,   ,+4 , *++ , ,  /0,    * 4 * ,++ * , & ,+     ,",+ //0,  ,:

Advanced Numerical Methods for Membrane Structures

135

# ' '              , //0, "" ,++ "   ,0, +, +,/   0"+    ,  +, + , , ,4 ,    7 ""/ /,,+     "" ,   /,     ,+   , ,+ /,,+ / ++ C ! +,+ C@ Cinst mS i

(26)

with the instability constant Cinst . The constant Cinst is known to be 3 for the most basic backward Euler time discretization scheme with a 0th order predictor and even smaller for more sophisticated schemes. See F¨orster et al. (2007) for a more complete discussion of Cinst . The later schemes lead to a recursive instability condition max μi i

mF 12 > mS 8n

for n > 1,

(27)

that depends on the number of time steps n. Therefore, given enough time steps, condition (27) will always be violated. Irrespective of the particular time discretization schemes the sequential staggered coupling scheme itself carries an inherent instability. For every sequentially staggered scheme, a mass ratio mF /mS exists at which the overall algorithm becomes unstable. The above are the results from the investigation reported in F¨orster et al. (2007). This investigation pertains to staggered solution schemes for

184

W.A. Wall et al.

FSI problems with incompressible flows. The case of compressible flows towards the incompressible limit is discussed in F¨orster and Wall (2009). The artificial added mass effect does indeed burden the staggered solution schemes of FSI problems with compressible flows in the incompressible limit, if the incompressible limit is actually reached. As long as there is enough compressibility on the fluid side, the artificial added mass effect is less dominating and weak coupling schemes can usually be employed without severe stability violations.

4 Efficient solver for FSI-block preconditioned Newton-Krylov schemes To solve the nonlinear FSI problem applying a Newton-type method the field residual forces (8)-(11) are combined to obtain the residual of the FSI problem f FSI . The unknowns of the residual are the structural displacement dS,n+1 , the fluid velocity uF,n+1 and pressure pF,n+1 , and the mesh displacement dG,n+1 . f FSI = f FSI (dS,n+1 , uF,n+1 , pF,n+1 , dG,n+1 )

(28)

Assuming the kinematic condition (12) to hold, the residual is

f FSI

f SI  f SΓ + f F  Γ  = F   = 0. fI fG I

(29)

In the following the explicit fluid pressure variable pF is dropped from the notation. Instead, it is understood that uF represents all unknowns of the fluid field, where the fluid pressure actually is submerged in the fluid field’s interior unknowns uF I , since the kinematic coupling condition (13) at the interface couples the structural interface displacement dSΓ with the fluid interface velocity uF Γ. A linearization of the FSI residual (29) is derived to obtain a Jacobian to be used within the Newton scheme. With these field linearizations in place the block Jacobian system can be build df FSI Δxi = −f FSI , i dx

(30)

where x includes all unknowns, JFSI = df FSI /dx is the Jacobian matrix

Advances in Computational Fluid-Thin-Walled-Structure and ( · )i denotes the iteration step. In δSIΓ SII  SΓI δSΓΓ + FΓΓ + δFG FΓI ΓΓ   FIΓ + δFG FII IΓ δAIΓ

detail equation

ΔdSI,i G  FΓI   ΔuF Γ,i   ΔuF FG I,i II AII ΔdG I,i

185

(30) reads

  = −f FSI i . 

(31)

The factor δ = θΔt in (31) converts between displacement and velocity increments (32) ΔdSΓ,i = δΔuF Γ,i according to (12). Equation (31) is solved in each Newton step followed by an update dSi+1 uF i+1 dG i+1

= = =

dSi + ΔdSi F uF i + Δui G dG i + Δdi

(33) (34) (35)

The Newton method is iterated until residual forces (29) satisfy the convergence criteria. 4.1

Block Gauss-Seidel process

In order to solve the linear system (31) using an iterative Krylov method, a preconditioner is required. The blocks in equation (31), however, relate to very different physical fields and need independent field specific preconditioning. To this end a block Gauss-Seidel process is employed. The linear block system (31) can be solved using available individual linear field solvers. This leads to a three step process where each step consists of a linear solution of the respective field SII,i ΔdSI,i,j+1

= −f SI,i − δSIΓ,i ΔuF Γ,i,j

(36)

AII ΔdG I,i,j+1

=

Fext,i ΔuF i,j+1

G G G S = −f F (38) i − FII,i + FΓI,i ΔdI,i,j+1 − SΓI,i ΔdI,i,j+1

−δAIΓ ΔuF Γ,i,j 



(37)

The fluid matrix Fi is augmented with the interface contribution of the G structural field SΓΓ,i and the fluid’s shape derivatives FG IΓ,i and FΓΓ,i G Fext,i = Fi + δSΓΓ,i + δFG ΓΓ,i + δFIΓ,i

(39)

according to (31). Note that the matrix Fext,i has the same dimensions as Fi .

186 4.2

W.A. Wall et al. Preconditioning based on Gauss-Seidel

The Gauss-Seidel process (36)-(38) can be used to precondition an iterative Krylov method that solves the linear system (31). A Newton-Krylov scheme on the coupled FSI problem is an intrinsic monolithic solution scheme. The partitioning of the linear field solvers can be preserved by a block preconditioner as introduced for FSI problems in Heil (2004). A comparison of block preconditioned Newton-Krylov schemes with Dirichlet-Neumann partitioned schemes has been provided in Heil et al. (2008) as well as K¨ uttler et al. (2009). A preconditioner M is introduced to Equation (30) JFSI M−1 M Δxi+1 = −f FSI i

(40)

where M is easily invertible and M ≈ JFSI . Furthermore, it does not necessarily represent a matrix but rather a linear projection for the preconditioning operation (41) z = M−1 y . Equation (41) is performed once for each iteration within a Krylov method. Since the linear field solvers are available, an obvious choice for M is a simplified version of the system matrix (31)

SII 0  SΓI δSΓΓ + FΓΓ + δFG  FΓI FG ΓΓ ΓI  M= (42) G G .  FIΓ + δFIΓ FII FII 0 AII Here, the coupling of the interface velocity increment ΔuF Γ to both structural field and ALE field has been dropped. From a physical point of view the block Gauss-Seidel (BGS) preconditioner (42) models an independent linear structure that influences a linear fluid without any feedback from the fluid to the structure. This is a simplification of the fully coupled linear problem that allows sequential treatment of the linear block equations. Inside the preconditioner the linear field solvers that are applied in the Gauss-Seidel process (36)-(38) can (and should) be approximations. In particular an algebraic multigrid method can be applied for each block solve. 4.3

AMG with block Gauss-Seidel smoothing

If all field preconditioners apply an algebraic multigrid method, these independent multigrid methods can be merged to obtain an algebraic multigrid preconditioner with block Gauss-Seidel smoothing. This idea has been presented in detail in Gee et al. (2009b).

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  Figure 3: Example V-cycle with three levels.

The three operators needed to specify a multigrid method fully are the reˆk , k = 1, . . . , Nlevels , and the grid translaxation (smoothing) procedures, S fers, prolongations Pk and restrictions Rk , k = 1, . . . , Nlevels − 1, where Rk = PTk in the symmetric case and Rk = PTk in the nonsymmetric setting. Note that Pk is an interpolation operator that transfers grid information from level k +1 to level k. The key to fast convergence is the complementary nature of the smoothing and interpolation operators. That is, errors not reˆk must be well interpolated by Pk . The coarse level operator duced by S Ak+1 (k ≥ 1) is specified by the Galerkin product Ak+1 = Rk Ak Pk .

(43)

Together these three operators are used to build a V-cycle of arbitrary depth. The evaluation of such a V-cycle leads to an approximation to the solution. Figure 3 shows an example V-cycle with three levels. At the third level a direct solver is applied to solve the coarse (small) linear system. The construction of the Rk ’s and Pk ’s for each field is outside of this discussion. See Gee et al. (2009b) for a presentation of state of the art multigrid techniques for FSI problems. As soon as the field prolongations and restrictions are available, however, those can be combined to obtain special purpose FSI prolongations and restrictions

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PSk

PF k

PG k





and RFSI = k

RSk

RF k

RG k

.

(44)

The application of these to the linear FSI system (31) leads to the coarsened Jacobian

S SII Δt 2 IΓ G  FSI  SΓI JFSI = RFSI (45) Pk Fext FG k+1 k ΓI + FII Δt A A IΓ II 2

S RSk SIΓ PF Rk SII PSk Δt k 2    S F G G G  (46) =  RF RF RF k SΓI Pk k Fext Pk k FΓI + FII Pk  F Δt G 2 Rk AIΓ Pk

G RG k AII Pk

according to equation (43). Here, one special purpose multigrid method is applied to the linear system (31) instead of three independent multigrid methods combined inside a Gauss-Seidel iteration. The result is an increased coupling between the fields. Figure 4 sketches the Gauss-Seidel process that is based on three independent multigrid methods. In contrast, figure 5 sketches the special purpose multigrid method that uses Gauss-Seidel smoothing to couple the fields on all levels. Numerical examples that evaluate the effectiveness of both preconditioners for real world FSI problems are presented in Gee et al. (2009b).

5

Examples of ALE-based FSI simulations

To illustrate the FSI solution algorithms explained above two distinct numerical examples are shown. These examples demand a throughout explanation. However, to stick to the spirit of this contribution, the present discussion focuses on concise descriptions. A more lengthy discussion of these examples including all subtle details can be found in Gee et al. (2009b). 5.1

Flexible flag behind rigid obstacle

The flexible flag behind rigid obstacle example is a thin-walled FSI benchmark that has been devised by Wall (1999) in the context of partitioned FSI solution schemes. Thus, the flexible flag example is ideally suited for a fixed-point approach with Aitken relaxation, that is discussed in K¨ uttler and Wall (2008). A snapshot of the fluid velocity at time step 330 is shown in figure 6.

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block Gauss-Seidel iteration

Structur

ALE

Fluid

Figure 4: Gauss-Seidel process based on three independent multigrid methods.

Figure 5: Special purpose multigrid method based on Gauss-Seidel smoothing.

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Figure 6: Two dimensional channel flow with flexible flag behind square rigid obstacle. Snapshot of fluid velocity at time step 330.

The artificial added mass effect is of minor importance in this example. The structure in this example is stiff and dense in comparison to the fluid. Even more important, however, the structure is fully submerged into the fluid. There are fluid loads on both sides of the structure. As a result there is artificial added mass on both sides of the structure that cancels itself. The fluid has a damping effect on the structure and the Dirichlet-Neumann partitioning introduces just a slight error and converges rapidly. The flexible flag example poses a challenge to the preconditioning in Newton-Krylov based FSI schemes. The structural field owns far less degrees of freedoms than the fluid field. An AMG preconditioning with block Gauss-Seidel smoothing is no reasonable choice in this case as the structural field is too small to support a sufficient number of grid levels. The preconditioner based on a Gauss-Seidel process can be used and obtains favorable results. Indeed, the Newton-Krylov scheme with block Gauss-Seidel preconditioning does outperform the Dirichlet-Neumann partitioned scheme with Aitken relaxation, if an exact direct solver is used to precondition the small structural block. Furthermore, the convergence of the linear Krylov solver can be improved by exchanging the block preconditioner (42) with the related preconditioning matrix

SII  SΓI M2 =  

δSIΓ δSΓΓ + FΓΓ + δFG ΓΓ FIΓ + δFG IΓ δAIΓ

0 FII

0  .  FG II AII

(47)

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This change does not touch the Gauss-Seidel process (36)-(38). Only the interface equations are moved from the fluid to the structural block. If a stiff and dense structure is used, matrix (47) is a better approximation to the system matrix than (42). For further details see Gee et al. (2009b). 5.2

Abdominal Aortic Aneurysm

The second example is an FSI-simulation of an abdominal aortic aneurysm, an abnormal dilation of the abdominal aorta right above the branch into the two iliac arteries. Here, the geometry obtained from presurgical imaging is prestressed Gee et al. (2009a,c) to account for the deformation due to blood pressure load present at time of imaging. The structure/fluid density ratio is ρs /ρf = 1 and the structure is nearly incompressible isotropic hyperelastic while the fluid is incompressible and Newtonian. Snapshots of von Mises Cauchy stress distribution at t = 3.24s are provided in figure 7. This example has characteristics very different from the flexible flag example discussed above. Here, fluid and structural field contribute a similar number of degrees of freedom. There are 529065 structural, 521885 mesh movement and 769952 fluid degrees of freedom. Furthermore, the field densities are equal. Dirichlet-Neumann partitioned schemes cannot be used for these kinds of problems, as has been shown in K¨ uttler et al. (2009). Newton-Krylov schemes, on the other hand, excel for these problems. Both the Gauss-Seidel and the AMG based preconditioners produce very good results. With this problem the AMG preconditioner with Gauss-Seidel smoothing is up to 30% faster than the Gauss-Seidel preconditioner. The tighter coupling obtained from a Gauss-Seidel process inside all smoothers pays off in biomechanical FSI problems. A conscientious presentation of iteration counts and timings is given in Gee et al. (2009b).

6 Robust formulation for large deformation FSI – an XFEM based fixed-grid approach 6.1

Introduction

Next to an efficient solution, the range of problems that can be handled by the ALE-based FSI formulation can be a limiting factor. The severe fluid-mesh deformation of the ALE approach can be circumvented in an elegant way by the fixed-grid method that is introduced in Gerstenberger and Wall (2008a, 2009) using the eXtended Finite Element Method (XFEM) introduced in Mo¨es et al. (1999); Belytschko and Black (1999). The principle setup and distinctive features are shown in the following. For ease of derivation and, eventually, implementation, the FSI problem

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Figure 7: Spatial (deformed) configuration of aneurysm example at time t = 3.24s. Fluid velocity field and cut through structural domain with von Mises Cauchy stress distribution.

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Figure 8: 3-field FSI approach: Γi separates the computational fluid domain Ω in a physical fluid domain Ω+ and the fictitious fluid domain Ω− . Fixed fluid field and the structure surface are coupled independently to the interface discretization.

is set up as as 3-field problem, where an additional interface discretization between the conjoined fluid and structure surface is introduced. The principle setup is depicted in figure 8. The interface-structure coupling is identical to the methods used in the previous chapters and, consequently, only minor code modification are necessary. Hence, the fixed grid approach only affects the fluid solution process. For clarity of presentation, the following discussion assumes a partitioned approach for the FSI problem. A more explicit presentation of the XFEM based fixed-grid approach can be found in Gerstenberger and Wall (2008b, 2009). For 3D FSI problems, a novel way to weakly prescribe Dirichlet conditions on the fixed 3D fluid grid is described in Gerstenberger and Wall (2009). As can be seen on the left side of figure 8, the interface moves on a fixed Eulerian fluid grid. Assume a domain Ω that contains the fluid domain Ωf completely and extends into the structural domain Ωs . The internal interface separates Ω into two subdomains Ω+ and Ω− , where Ω+ corresponds to the physical fluid domain Ωf and Ω− is the remaining domain filling Ω. Hence, the flow field in Ω− is entirely fictitious with no physical meaning to the FSI problem. The main tasks are to decouple the physical flow in Ω+ and the fictitious flow in Ω− and, using this splitting, to employ the interface conditions between the interface and the physical flow in Ω+ . Therefore, a discontinuous velocity and stress field is needed along Γi and a Lagrange multiplier based velocity coupling is employed.

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6.2

Velocity and stress discontinuity across the fluid-structure interface

The fluid-structure interface is generally not aligned with fluid element edges. Here, the finite element shape functions are extended or enriched by using additional degrees of freedom combined with known solution or special enrichment functions. Applied to the velocity field, an enriched velocity approximation can be defined as   NI (x)˜ uI + NI (x)ψ(x, t)ˆ uI (48) uh (x, t) = I

I

˜ I represent the standard nodal degrees of freedom at node I, while Here, u ˆ I multiplied by a properly chosen enrichment additional degrees of freedom u function ψ(x, t) are used to enhance the solution. The choice of the enrichment function depends on the character of the interface conditions. For the current interface problem, there is a jump from the physical behavior in Ω+ to zero velocity and pressure in Ω− . Likewise, the viscous stress resulting from velocity derivatives are only present in the physical flow field, whereas in Ω− they are zero as no flow occurs. Hence, a jump-like function, also allowing different gradients on both sides of the jump discontinuity, can describe the fields at the interface +1 ∀x ∈ Ω+ ˆ t) = (49) ψ(x, t) = ψ(x, 0 ∀x ∈ Ω− Along Γi , both velocity and pressure are discontinuous and enriched with ψˆ as shown in figure 9. The complete discretization for trial and test functions with equal order ansatz functions for velocity and pressure are given as   ˆ tn+1 )ˆ uh,n+1 (x) = NI (x)˜ un+1 + NJ (x)ψ(x, un+1 (50) I J I

h

v (x)

=



NI (x)˜ vI +

I



J

ˆ tn+1 )ˆ NJ (x)ψ(x, vJ

(51)

J

and ph,n+1 (x)

=

 I

q h (x)

=

 I

NI (x)˜ pn+1 + I NI (x)˜ qI +





J

ˆ tn+1 )ˆ NJ (x)ψ(x, pn+1 J

ˆ tn+1 )ˆ NJ (x)ψ(x, qJ

J

The superscript ‘h’ indicates the discretized field function.

(52) (53)

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Figure 9: Shown are the standard degrees of freedom (◦) with the additional degrees of freedom (
) needed for the jump enrichment. The striped elements are deactivated. The Lagrange multiplier nodes () are constructed on intersection between the interface and fluid element edges.

Spatial derivatives of enriched shape functions with respect to the spatial direction i are done using the product rule as ˆ t)),i = NI,i (x)ψ(x, ˆ t) + NI (x)ψˆ,i (x, t) (NI (x)ψ(x,

(54)

Since ψˆ is a constant function, the derivative of an enriched shape function is simply the derivative of the normal shape function multiplied by the step ˆ Across the interface, such derivatives would result in Dirac funcfunction ψ. tions, which can be identified as traction jumps. These jumps are already accounted for by the traction vector λ along Γi representing the traction jump from Ω+ to Ω− . Higher order shape functions are constructed with no additional effort making this approach capable for consistent low and high order finite element approximations. Along the structural surface there are two possible scenarios: In the first scenario, there can be a thick structure such that only one interface intersects a fluid element as depicted in figure 9. Here, all nodes belonging to an intersected element are enriched. All remaining nodes in Ω+ use the standard degrees of freedom, while the remaining nodes in Ω− carry no degrees of freedom at all. Hence, no additional computation costs for the fictitious fluid domain is generated. The second scenario are several interfaces intersecting one fluid element. In that case, simply more enrichments can be used, using one enrichment for each interface. This feature allows contacting structures and thin structures to be treated within an FSI approach on “large” fluid elements, as can ˆ 1I , u ˆ 2I , ... and enrichment be seen in figure 10. The additional unknowns u

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Figure 10: For thin structures, two overlapping enrichments (
and ) represent the discontinuity, while away from the surface, standard degrees of freedom (◦) are used. Lagrange multiplier meshes () are constructed on each side as usual.

functions ψˆ1 , ψˆ2 , ... are summed as    NI (x)˜ uI + NJ (x)ψˆ1 (x)ˆ u1J + NK (x)ψˆ2 (x)ˆ u2K +... (55) uh (x) = I

J

K

with the time dependence omitted for brevity. The enrichment for a thin structure as depicted in figure 10 consists of 2 overlapping jump enrichments using the step function equation (49). An example for flow around a thin structure is given in figure 11a. Here, different flow fields develop around a thin structure, which thickness is smaller than the fluid element size. On the left side, a horizontal parabolic inflow is applied, while on the right some constant diagonal inflow enters the domain. It can be clearly seen in figure 11a and figure 11b, how the pressure and the velocities of the flow right and left of the structure are completely decoupled from each other and the small fictitious fluid domain between them. The 3 sets of enrichments for the pressure given by    NI (x)˜ pI + NI (x)ψˆ1 (x)ˆ p1I + NI (x)ψˆ2 (x)ˆ p2I (56) ph (x) = I

I

I

are shown in figure 12a-12c (see also equation (55)). The standard degrees of freedom, depicted in figure 12a occupy most of the domain. Near the surface the first and the second enrichment overlap each other (figure 12b and 12c). The sum of these enrichments gives the pressure field as displayed in figure 11b. 1

Note that displayed triangles are not part of the discretization and are only used for proper visualization of the results (and for numerical integration).

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(b) pressure

Figure 11: Flow field around a stationary thin structure: The thickness of the structure is less than the element size of the fluid. No flow is observed within the structural domain and right and left flow field are completely decoupled. The triangular integration cells are shown in the pressure field.

For numerical integration, elements divided by discontinuities are subdivided into subdomains using a 3D constrained Delaunay algorithm as described in Mayer et al. (2009). This integrates the piecewise bi-quadratic functions exact, but requires higher programming effort than alternative integration schemes. 6.3

Fluid-interface coupling and interface discretization

The velocity matching condition between interface and fluid mesh is employed weakly using the Lagrange multiplier field λ as

Γ+


¯ i dΓ+ . δλ u − u

(57)

The Lagrange multiplier field and the corresponding test functions are discretized as   NIi λI and δλh = NIi δλI (58) λh = I

I

The choice of appropriate shape functions NIi can not be made without consideration of the underlying fluid discretization. Inappropriate choices can lead to oscillations or locking due to violation of the inf-sup condition. Here, bi-quadratic ansatz functions for fluid velocity and pressure NI and

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(a) zero-th enrichment:
pI I NI ()˜

(b) first enrichment:
ˆ1 p1 I NI ψ ()ˆ I

(c) second enrichment:
ˆ2 p2 I NI ψ ()ˆ I

Figure 12: Flow field around a stationary thin structure and enrichments used for the flow field around a stationary structure. The dots indicate the enriched nodes used for the particular enrichment. Summation of all enrichment terms results in the complete solution as shown in figure 11b.

linear ansatz function for NIi are used. The construction of the interface mesh is illustrated in figure 9. This is a viable approach for 2D problems but can be quite cumbersome for 3D problems. Hence, an attractive alternative based on a mixed/hybrid approach has been presented in Gerstenberger and Wall (2009). In summary, the presented fixed-grid approach allows to decouple the physical domain from the fictitious domain Ω− and weakly apply Dirichlet boundary conditions along the interface. The interface mesh could be seen as the fluid surface mesh along the fluid-structure interface and can now be coupled to the wet structure surface.

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Interface-structure coupling and FSI formulation

The interface discretization is closely connected to the way the fluid mesh is intersected and, consequently, usually does not match the surface discretization of the structure. The problem of the resulting non-matching grids often occurs in the process of domain decomposition, hence, extensive research has been undertaken. For the interface-structure coupling, the Mortar method for non-matching grids is adopted, see Bernardi et al. (1994). The interface condition in equation (1) is formulated in terms of velocities, however it implies that not only are the velocities equal at all times but also the positions or displacements of the interface and structural surface. In the Mortar method, the kinematic matching condition is enforced weakly by using a Lagrange multiplier field μ, which can be identified with the surface traction μ. The setup for the structure-interface coupling is depicted in the right side of figure 8. With this setup there is a framework available that allows for arbitrary structural movements in an FSI simulation. 2D numerical examples are presented in Gerstenberger and Wall (2008b). Note that it might be favorable to consider also hybrid approaches, where fixed and moving fluid meshes are advantageous. This could be the case, where very thin boundary layer develop around deforming structures. Here, a potentially highly graded, deforming boundary layer mesh should be employed that represents such boundary layers efficiently. The deforming boundary layer mesh and the fixed background grid can be coupled using Chimera-like techniques (see e.g. Wall et al. (2008)) or using the very same XFEM techniques as described above. Such XFEM based fluid-fluid coupling approaches for FSI have been presented in Gerstenberger and Wall (2008a). Figure 13 shows, how the intermediate ALE mesh is coupled to the fixed fluid domain and to the structure surface using the additional Lagrange multiplier field κ. For details on this approach, see Gerstenberger and Wall (2008a). As an outlook, we show preliminary results of a steady-state 3D FSI simulation. In this example, a flexible structure (Poison ratio ν = 0.48, Young’s modulus E = 90 N/m2 ) is deformed due to a flow through a channel with dimensions 0.5 × 1.0 × 3.0m. The fully coupled FSI simulation is performed until steady-state is reached. Top and bottom channel walls have zero (no-slip) velocity prescribed, the inflow from the right is prescribed by a parabolic velocity condition. The Reynolds number based on the mean inflow velocity and the channel height is 16. The stationary equilibrium solution is shown in figure 14. A detailed description of the 3D FSI approach will be the topic of an upcoming paper.

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Figure 13: Hybrid approach: a fixed fluid grid and a deforming ALE fluid grid are coupled using the XFEM/Lagrange Multiplier approach. The moving ALE grid can in turn be coupled to the structure field. Shown are also variables living on each domain and the Lagrange multiplier fields.

Figure 14: Stationary flow field through a channel with a flexible structure. The parabolic inflow (right) and the wall boundaries are standard Dirichlet boundary conditions. The outflow boundary (left) is of Neumann (zero traction) type. The structure is fixed at the lower wall.

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Conclusion

The trends and techniques shown here highlight current developments in FSI solution schemes. Weak coupling partitioned schemes are ruled out when problems with incompressible flows are considered. The quest for fast and reliable FSI solution schemes motivated research of monolithic solution schemes. The resulting schemes are block preconditioned Newton-Krylov methods, that perform surprisingly well, provided efficient preconditioners are employed. Two highly effective preconditioners have been presented. Furthermore, an XFEM based fixed-grid approach has been described, that overcomes the limitations imposed by the ALE approach to mesh movements.

Bibliography Y. Bazilevs, V.M. Calo, Y. Zhang, and T.J.R. Hughes. Isogeometric fluid– structure interaction analysis with applications to arterial blood flow. Comput. Mech., 38(4–5):310–322, 2006. T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engng., 45(5):601–620, 1999. C. Bernardi, Y. Maday, and A.T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. Nonlinear partial differential equations and their applications, 299:13–51, 1994. P. Causin, J.-F. Gerbeau, and F. Nobile. Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Comp. Meth. in Appl. Mech. and Engng., 194:4506–4527, 2005. ´ Fern´ M.A. andez and M. Moubachir. A Newton method using exact Jacobians for solving fluid–structure coupling. Comput. Struct., 83(2–3): 127–142, 2005. Ch. F¨orster and W.A. Wall. A note on the artificial added mass effect for compressible flows interacting with structures. Comp. Meth. in Appl. Mech. and Engng., submitted, 2009. Ch. F¨orster, W.A. Wall, and E. Ramm. On the geometric conservation law in transient flow calculations on deforming domains. Int. J. Numer. Meth. Fluids, 50:1369–1379, 2005. Ch. F¨orster, W.A. Wall, and E. Ramm. Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible

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Advances in Subdivision Finite Elements for Thin Shells

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