Numerical Modeling of Concrete Cracking

iil SpringerWienNewYork

CISM COURSES AND LECTURES

Series Editors: The Rectors Friedrich Pfeiffer - Munich Franz G. Rammerstorfer - Wien Jean Salençon - Palaiseau

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Executive Editor 3DROR6HUDÀQL8GLQH

The series presents lecture notes, monographs, edited works and SURFHHGLQJVLQWKHÀHOGRI0HFKDQLFV(QJLQHHULQJ&RPSXWHU6FLHQFH and Applied Mathematics. 3XUSRVHRIWKHVHULHVLVWRPDNHNQRZQLQWKHLQWHUQDWLRQDOVFLHQWLÀF and technical community results obtained in some of the activities RUJDQL]HGE\&,60WKH,QWHUQDWLRQDO&HQWUHIRU0HFKDQLFDO6FLHQFHV

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 532

NUMERICAL MODELING OF CONCRETE CRACKING

EDITED BY GÜNTER HOFSTETTER UNIVERSITY OF INNSBRUCK, AUSTRIA GÜNTHER MESCHKE RUHR UNIVERSITY BOCHUM, GERMANY

This volume contains 185 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned VSHFLÀFDOO\WKRVHRIWUDQVODWLRQUHSULQWLQJUHXVHRILOOXVWUDWLRQV broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2011 by CISM, Udine SPIN 80073533

All contributions have been typeset by the authors.

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WKH VWDELOLW\ FRQGLWLRQ  LV DVVXUHG &RQGLWLRQ  LV NQRZQ DV WKH +VWDELOLW\ condition. We remark that there are alternative stability criteria. 7KRVH PDWHULDOV YHULI\LQJ  PXVW KDYH D VWUHVVVWUDLQ FXUve (σ, ε ZLWK SRVL tive slope for the complete range of strains. This condition precludes any softening response. Thus, it is an excessively strong requirement for those continuum constitutive models we are interested in.

104

A. Huespe and J. Oliver

Uniqueness conditions. 8QLTXHQHVV RI WKH UHODWHG %93 VROXWLRQ FDQ EH JXDUDQ WHHG XQGHU JHQHUDO FRQGLWLRQV +LOO   +RZHYHU LQ WKe present work, we are interested in analyzing a more restricted type of solution uniqueness. &RQVLGHULQJ D ERG\ Ω VXEMHFWHG WR D ORDGLQJ SURFHVV ZLWK D VPRRWK VWUDLQ ÀHOG we analyze the possible existence of solutions displaying bifurcation points, see )LJXUH F ZLWK VHFRQGDU\ EUDQFKHV EHLQJ FKDUDFWHUL]HG E\ weak discontinuity VROXWLRQV ,W PHDQV WKDW WKH VHFRQG VROXWLRQ GLVSOD\V D FRQWinuous displacement ÀHOG EXW VWUDLQV DQG VWUHVVHV DUH GLVFRQWLQXRXV DFURVV D JLven surface S. The analysis follows the classical methodology proposed by 5LFH   WKH 0D[ZHOO·V FRPSDWLELOLW\ UHODWLRQV GHÀQH WKH VWUXFWXUH RI Vtrain tensors displaying jumps or discontinuities. Then, a perturbed strain with this structure is introduced LQ WKH %93 JRYHUQLQJ HTXDWLRQV IURP ZKHUH WKH FRQGLWLRQV Rn the material model admitting a non-unique solution can be found. Maxwell’s compatibility conditions. /HW XV FRQVLGHU D FRQWLQXRXV VFDODU IXQFWLRQ φ GHÀQHG LQ WKH ERG\ Ω with a discontinuous gradient across a surface S, as shown LQ )LJXUH D 7KH VXUIDFH KDV DQ RUWKRJRQDO XQLW YHFWRU n and divides the domain Ω in two parts that we denote as Ω + and Ω −  DV VKRZQ LQ )LJXUH D 7KH YHFWRU n points toward Ω + .

f

W+ y W+

x (a)

n

S

W+ t-

n

S

W

W

W-

(b)

W-

t+

S

W-

(c)

Figure 3. D 6FDODU IXQFWLRQ φ with a weak discontinuity across the surface S EF ERG\ GLVSOD\LQJ D ZHDN GLVFRQWLQXLW\ DFURVV WKH VXUIDce S. The jump of the gradient of φ, i.e. the gradient difference on each side of S: ∇φ+ − ∇φ− , where (·)+ and (·)− denote the values on each side of S, is: ∇φ+ − ∇φ− = [[∇φ]] = ξn



being ξ a proportionality scalar factor. The projection of [[∇φ]] on S is zero because the continuity of φ along S. ,I D YHFWRULDO FRQWLQXRXV IXQFWLRQ u, has a discontinuous gradient on S, the gradient of each components of u VDWLVÀHV HTXDWLRQ   7KXV WKH MXPS RI WKH gradient is:

Crack Models with Embedded Discontinuities ∇u+ − ∇u− = [[∇u]] = β ⊗ n

105 

where, β LV D YHFWRU ,Q WKH SDUWLFXODU VLWXDWLRQ WKDW u is a continuous displacement ÀHOG ZLWK D GLVFRQWLQXRXV JUDGLHQW WKH VWUDLQ MXPS RQ S is: [[ε]] = [[∇u]]sym = ε+ − ε− = (β ⊗ n)sym sym

the notation (·)



means the symmetric part of (·).

Bifurcation analysis. /HW XV FRQVLGHU RQ WKH ERG\ Ω, subjected to a loading process, the existence of a weak discontinuity solution, with a strain jump across the surface S DV VKRZQ LQ )LJXUH E 7KH FRQWLQXLW\ RI WKH UDWH RI WUDFWion vector on S LV D FRQGLWLRQ UHTXLUHG E\ WKH ERG\ HTXLOLEULXP VHH )LJXUH-b:c: ˙ ·n=0 (t˙+ + t˙− ) = σ˙ + · n − σ˙ − · n = [[σ]]



˙ where DT Replacing the stress rate σ˙ by the constitutive relation: σ˙ = DT : ε, LV WKH WDQJHQW WHQVRU RI WKH FRQVWLWXWLYH UHODWLRQ ZH FDQ UHZULWH HTXDWLRQ  DV follows: σ˙ + · n − σ˙ − · n = [(DT )+ : ε˙ + − (DT )− : ε˙ − ] · n = 0



The smooth fundamental solution ε˙+ = ε˙ − YHULÀHV HTXDWLRQ   +RZHYHU ZH search for a second solution with strains having a jump across S. Thus, the strain ÀHOG RI WKH VHFRQG VROXWLRQ IROORZV HTXDWLRQ   ε˙+ = ε˙− + (β˙ ⊗ n)sym , which, UHSODFHG LQ HTXDWLRQ   JLYHV [(DT )+ : (ε˙ − + (β˙ ⊗ n)sym ) − (DT )− : ε˙ − ] · n = 0 +

and assuming that (DT ) FRQVLGHUHG  WKHQ

= (DT )

−



= DT (this assumption should be carefully

[DT : (β˙ ⊗ n)sym ] · n = [n · DT · n] β˙ = 0   



Q

where the tensor Q is the so called acoustic tensor, or localization tensor. Existence ˙ different from zero, means that β˙ = 0 for some vector n. Which of a solution [[ε]] is only possible if: det Q(n) = 0  &RQWUDULO\ WKH FRQGLWLRQ det Q(n) > 0

∀n



called strong ellipticity condition, precludes any type of discontinuous bifurcation. Remark: IRU D UDWHLQGHSHQGHQW PDWHULDO +VWDELOLW\ ε˙ : DT : ε˙ > 0 ∀ ε˙ implies strong ellipticity.

106

A. Huespe and J. Oliver

1.3

Example of a Material Model Subjected to Stability Loss and Bifurcation: Isotropic continuum damage model for concrete

7DEOH  GHVFULEHV D VSHFLÀF LVRWURSLF FRQWLQXXP GDPDJH PRGHl, with a scalar internal variable D, the damage varaible, describing the elastic stiffness degradation due to micro cracking: D = 0, for the undamaged material, and D = 1 for the fully damaged material (for more generic continuum damage models, see for H[DPSOH /HPDLWUH DQG 'HVPRUDW   ,Q 7DEOH  DQG IROORZLQJ 2OLYHU   WKH GDPDJH YDULDEOH D depends on an internal strain-like variable r and its stress-like conjugate internal variable, q which depends on r. The free energy, denoted W , depends on the strain tensor ε and the damage variable D. The elastic strain energy for the undamaged material is denoted W0 , and E LV WKH +RRNH·V HODVWLF WHQVRU μ and λ DUH WKH /DP`H·V SDUDPHWHUV and I and 1 are the fourth and second order identity tensors respectively. ¯ LV WKH HIIHFWLYH VWUHVV ,WV SRVLWLYH FRXQWHUSDUW LV WKHQ GH,Q HTXDWLRQ   σ ÀQHG DV ¯ + = ¯ σi  pi ⊗ pi  σ where ¯ σi  VWDQGV IRU WKH SRVLWLYH SDUW 0F$XOH\ EUDFNHWV RI WKH LWK Srincipal σi  = σ ¯i for σ ¯i > 0 and ¯ σi  = 0 for σ ¯i ≤ 0 DQG pi stands effective stress σ ¯i ( ¯ for the i-th principal stress direction. (TXDWLRQ  GHÀQHV WKH GDPDJH IXQFWLRQ f and the initial elastic domain ¯ + = 0  as: f < 0. This domain is unbounded for compressive stress states (σ Therefore, damage evolution is only possible with tensile stress states, as it is usually observed in concrete crack phenomenon. √ AlternativHO\ WKH GHÀQLWLRQ RI ¯ : ε, in which case, the elastic ¯ , such as: τε = σ τε can be performed with σ damage, in the stress space, is a symmetric domain with respect to the origin (null VWUHVV SRLQW The stresses σ and the stress-like variable q are determined from equations  DQG   (TXDWLRQ  GHÀQHV WKH VRIWHQLQJ ODZ LQ WHUPs of the softening parameter H ,Q HTXDWLRQ   ft and E are, respectively, the tensile strength and WKH 0 and q > 0, as well as: det Qe = det(n · E · n) = μ2 (2μ + λ) > 0



7KHUHIRUH HTXDWLRQ  FDQ EH UHZULWWHQ DV  1−(

 q − Hr  Z(n) ¯ · n] · (Qe )−1 · [σ ¯ · n] = [1 − ]=0 ) [σ qr 2 ξ(H)



(TXDWLRQ  LPSOLFLWO\ GHÀQHV WKH YDOXH RI WKH VRIWHQLQJ Podulus H as a function of the directions n 'XULQJ D ORDGLQJ SURFHVV WKH FULWLFDO VRIWHQLQJ Hcrit and normal ncrit  DUH LGHQWLÀHG DV WKRVH YDOXHV WKDW ÀUVW YHULI\ HTXDWLRQ   7KH\ FDQ EH HYDOXDWHG E\ PHDQV RI D FORVHG IRUPXOD XVHG LQ 2OLYHU DQG +XHVSH   Additional details of this procedure can be obtained there.

2 Material Failure Analysis Using the Continuum-Strong Discontinuity Approach (CSDA) 2.1

Motivation

The formation of weak discontinuities, characterized by continuous displacements but discontinuous strains across a surface S )LJXUH D FRXOG JLYH ULVH WR concentration of strains whenever two discontinuity surfaces, S and S  , bound a EDQG RI ÀQLWH ZLGWK VXFK DV WKDW VKRZQ LQ )LJXUH E 7KH FRQFept of strong discontinuity is recovered when the width of the localization band tends to zero, and WKH YDOXH RI WKH VWUDLQ MXPS WHQGV WR LQÀQLW\ 7KXV WKH VWURQg discontinuity problem can be regarded as a limit case of the strain localization ZHDN GLVFRQWLQXLW\ one. The previous Section presented the necessary conditions that a mechanical problem must verify, in order to show bifurcation points with secondary equilibrium branches characterized by non-smooth, discontinuous, strain solutions. ,Q WKH SUHVHQW 6HFWLRQ DGGLWLRQDO GHWDLOV RI WKH PHFKDQLFDl problem displaying a non-smooth kinematics, with strong discontinuities, are addressed. Particularly,

Crack Models with Embedded Discontinuities

109 loading unloading

e

eu

e S

weak discontinuity (a)

P

unloading

eu

e W

W

el

S

S'

Two weak discontinuity (b) surfaces bounding a finite bandwidth with a strain jump

s

bifurcation point

s=P/A

(c)

eu

el

e

Figure 4. )RUPDWLRQ RI D ÀQLWH ZLGWK UHJLRQ ERXQGLQJ D VWUDLQ MXPS DE DV D UHVXOW RI D ELIXUFDWLRQ SRLQW F 7KH ' SUREOHP ZLWK D ZHDN GLVFRQWinuity.

we show a methodological framework that makes possible the consistency between strong discontinuity kinematics and continuum constitutiYH PRGHOV ,Q WKH LQWHU face of a strong discontinuity, where displacement jumps arise, this methodology provides a discrete traction-separation law similar to that proposed by classical cohesive models. The so derived traction-separation law is compatible and consistent with the continuum constitutive relation of the bulk, while, the classical discrete DSSURDFK GHÀQHV FRKHVLYH IRUFHV ZKLFK DUH LQGHSHQGHQW RI WKe bulk constitutive relation. Conventional one-dimensional (1D) damage model ,Q 7DEOH  WKH GDPDJH model presented in the previous section is particularized fRU WKH ' FDVH ZKLFK LV XVHG WR VROYH D ' EDU SUREOHP :LWK WKLV VROXWLRQ ZH VKRZ Whe inconsistent and nonphysical response provided by the conventional approach when it is used for modeling structural problems characterized by materials with strain softening. 6\PEROV DQG HTXDWLRQV LQ 7DEOH  DUH WKRVH H[SODLQHG LQ 7DEOH 1. :LWK WKH LQLWLDO FRQGLWLRQV JLYHQ E\   LW UHVXOWV D(t = 0) = 0 and the GDPDJH FULWHULRQ LV YHULÀHG IRU σ = ft , where ft is the material tensile strength. 'XULQJ D PRQRWRQLF ORDGLQJ SURFHVV√ZLWK ε˙ ≥ 0 and r˙ > 0, and after reaching the condition f = 0, the identity r˙ = E ε˙ follows from the loading condition: f˙ = 0 7KHQ WKH UDWH HTXDWLRQ  LV REWDLQHG IURP WKLV LGHQWLW\ which gives a constitutive modulus DT = HE. A plot of the stress-strain response can be observed in FigurH D ZKLOH WKH softening law q(r) LV SORWWHG LQ )LJXUH E The 1D bar problem /HW XV FRQVLGHU WKH ' EDU SUREOHP RI )LJXUH F ZLWK D PDWHULDO GHVFULEHG E\ WKH GDPDJH PRGHO RI 7DEOH  $IWHU WKH Oimit load is reached, in the softening regime, one possible solution is obtained by considering that a

110

A. Huespe and J. Oliver Table 2. 2QHGLPHQVLRQDO FRQWLQXXP GDPDJH PRGHO Free energy and stress-strain relation 1 q(r) q(r) ; σ= E ε2 Eε 2 r r Damage function √ f (ε, r) = τε (ε) − r ; τε (ε) = E ε2

W (ε, r) =





Initial conditions ft r|t=0 = r0 = √ E

;

ft q|t=0 = r0 = √ E



Loading-unloading conditions r˙ ≥ 0

;

f ≤0

;

rf ˙ =0



Softening law q˙ = H r˙ ;

H = constant < 0



Incremental constitutive law σ˙ =

q E ε˙ r

;

XQORDGLQJ FDVH



ORDGLQJ FDVH



H

 q,r r − q q σ˙ = E ε˙ + Eεr˙ = HE ε˙ r r2

;

fraction β, (0 ≤ β ≤ 1  RI WKH EDU OHQJWK , is in loading, while the remaining part, with a length (1 − β), is elastically unloading. 'HQRWLQJ WKH YDULDEOHV LQ WKH ORDGLQJ ]RQH ZLWK VXELQGH[ l and those of the unloading zone with subindex u WKH UHODWLRQ EHWZHHQ WKH ORDG P and the total displacement at the end of the bar ΔuT , can be obtained through the following set of equations: i) Compatibility equation ΔuT = Δul + Δuu = (β)εl + (1 − β)εu



where Δul and Δuu are the length increment of the bar zones corresponding to loading and unloading, respectively. ii) Equilibrium equation and traction boundary condition σ l = σu =

P A



Crack Models with Embedded Discontinuities

111

iii) Constitutive relation, see Figure 5-d ft ) ; σu = E εu E $IWHU UHSODFLQJ HTXDWLRQV  LQ HTXDWLRQ   LW FDQ Ee found:  AE β P = (1 − H)ft ΔuT + EH [1 − (1 − H1 )β] σl = ft + EH(εl −

or, in rates: P˙ = kΔu˙ T

;

k=

AE 1 [1 − (1 − H )β]







The set of values β ∈ [0, 1] represents admissible solutions to the mathematical problem. Therefore, in the softening regime, the bar stiffness k depends on the length β RI WKH LQHODVWLF ]RQH VHH )LJXUH H Balance of energy in the bar problem. &RQVLGHULQJ WKDW WKH UHGXFHG GLVVLSDWLRQ D of an inelastic process is the difference between the stress power and the internal HQHUJ\ UDWH VHH 6LPR DQG +XJKHV   WKH GDPDJH PRGHO GLVsipation is given by: ∂r q

D

=

r2     ˙ = σ ε˙ − [ 1 H r − q (Eε2 ) r˙ + ( q Eε) ε] ˙ σ ε˙ − W r 2   r   2 ∂r W

∂ε W =σ

1 1 = (q − Hr)r˙ = (1 − H)ro r˙  2 2 where loading conditions are considered (f = 0  $OVR FRQVLGHULQJ WKDW IURP HTXDWLRQ   r 2 = Eε2  D OLQHDU FRQVWDQW VRIWHQLQJ PRGHO H < 0, such as WKDW VKRZQ LQ )LJXUH E ZLWK q = qo + H(r − ro ) and by the initial conditions: qo = ro  WKH GLVVLSDWHG HQHUJ\ XQWLO UHDFKLQJ WKH FRPSOHWH PDWHULDl degradation (r = rd LQ )LJXUH E FDQ EH GHWHUPLQHG DV IROORZV  r=rd  t=∞ 1 Ddt = Y = (1 − H)ro dr t=0 r=ro 2 1 1 1  = (1 − H)ro (rd − ro ) = (1 − )ro2    2 2 H −ro H

which, once multiplied by the volume of the damaged zone, gives the total structural dissipated energy as: YΩ =

1 1 (1 − )ro2 βA 2 H



112

A. Huespe and J. Oliver

2Q WKH RWKHU VLGH ZH HYDOXDWH WKH H[WHUQDO ZRUN SURGXFHG E\ Whe force P , as the DUHD EHORZ WKH HTXLOLEULXP FXUYH LQ )LJXUH H ,W LV HTXDO WR:  Δuend 1 1 Area = P du = (1 − )ro2 βA  2 H 0 which, as required by the energy balance, coincides with the structural dissipated energy at rupture YΩ  ,W LV QRWHG WKDW WKH WRWDO VWUXFWXUDO GLVVLSDWHG HQHUJ\ UXpture HQHUJ\ WHQGV WR ]HUR DV WKH VROXWLRQ WHQGV WR WKH VWURQJ GLVFontinuity limit, with β → 0, which is a nonphysical structural response. Remark: )URP HTXDWLRQ   LW FDQ EH REVHUYHG WKDW WKH UXSWXUH ZRUN Gepends on the damaged zone length: β. This length is not provided by the constitutive relation or by any other equation of the model. Thus, it is arbitrary. Multiplicity of solutions and the nonphysical response in the strong discontinuity limit are charDFWHULVWLFV RI DQ LOOSRVHG PDWKHPDWLFDO SUREOHP ,Q IDFW %93 ZLWK FRQYHQWLRQDO material models having strain softening provide ill-posed mathematical problems. Well-posed property is recovered if the constitutive relation introduces a characteristic length, such as the width of the strain localizatioQ ]RQH ,Q WKLV FDVH ZH will refer to them as regularized constitutive models. s

q unloading loading

qo

ft HE E

H

unloading

sl ,el bl

loading

q rE

unloading

su , eu

loading

P

su , eu

Du l

Area of the bar: "A"

e

(a)

ro

s

rd

(b)

r

(c)

P

ft

Pmax = f t A

loading

AHE/l (b=1)

unloading HE

su = sl

b

E

eu

ft /E (d)

el

e

AE/l (b=0)

0

all these curves are solutions of the 1D bar problem

Dumax = ft l/E

AE [1-(1- 1 )b] l H

(e)

Du end

Du

Figure 5. DE ' GDPDJH PRGHO FG %DU SUREOHP H ORDG YV GLVSODcement solution of the bar problem. Remark: As a consequence, numerical simulations of this type of problems present

Crack Models with Embedded Discontinuities

113

VHULRXV GHÀFLHQFLHV ZKHQHYHU QRQUHJXODUL]HG FRQVWLWXWLYH UHODWLRQV DUH XVHG ,W LV QRWHG WKDW LQ GLVFUHWH PRGHOV GHULYHG IURP ÀQLWH HOHPHQW IRrmulations, a charDFWHULVWLF OHQJWK LV LQGXFHG E\ WKH ÀQLWH HOHPHQW PHVK VL]H Thus, in some well GHÀQHG FDVHV WKLV GLVFUHWL]DWLRQ SURSHUW\ FRXOG EH XVHG DV a valid regularization procedure. Alternatively, it is possible to assume that the total rupture work of the bar, per unit of area A, is a material parameter. According to the concept introduced in the previous section, we call this parameter the fracture energy: Gf  ,WV DGRSWLRQ LV an important methodological assumption that can be interpreted in two different ways: i) considering that the softening modulus, H, is a given parameter of the mateULDO FRQWLQXXP GDPDJH PRGHO 7KHQ IURP HTXDWLRQ  DQG WKH GHÀQLWLRQ of Gf : Gf = YΩ /A, a characteristic length is induced as follows: β =

2Gf H ro2 H − 1



and represent the size of the strain localization zone, i.e. the width of the VWUDLQ ORFDOL]DWLRQ ]RQH LV GHÀQHG RQFH H and Gf DUH JLYHQ ii) alternatively, it can be assumed that the rupture process zone is concentrated in a very thin band (β → 0  7KHQ IRU FDSWXULQJ D FRUUHFW UXSWXUH ZRUN LW is necessary that the continuum softening parameter H → 0 such that, the H is bounded and takes the value: ratio β

2 2 H ¯ = − 1 ro = − 1 ft =H H→0;β →0 β 2 Gf 2 Gf E

lim



¯ the ,Q WKH ODVW LGHQWLW\ IURP   LW LV UHSODFHG ro2 = ft2 /E. We call H −1 LQWULQVLF VRIWHQLQJ PRGXOXV ZKLFK KDV D GLPHQVLRQ RI /HQJWK . Replacing, ¯ it introduces automatin the constitutive equations, the modulus H by H, ically a length dimension in the problem. Therefore, considering a strong discontinuity solution, the material parameter H is determined by Gf . 2.2 The 1D Continuum-Strong Discontinuity Approach (CSDA) We analyze the strong discontinuity solution of the bar problem that happens when strains localize in the rupture surface S (β → 0 DQG WKH GLVSODFHPHQW ÀHOG GLVSOD\V D ÀQLWH MXPS DFURVV WKLV VXUIDFH ,Q WKLV FDVH ZH VHarch for a continuum approach that describes the rupture process. ,Q RUGHU WR SURFHHG ZLWK WKLV DSSURDFK VRPH DGGLWLRQDO LQJUedients must be considered: i) the assumption that the material constitutive relation of the continuum model describes the mechanical response of the dissipative phenomena taking place in the discontinuity zone S

114

A. Huespe and J. Oliver

ii) it is noted that, as the width of the strain localization band goes to zero, the strains become unbounded. Thus, a regularization procedure must be introduced such that the kinematic singular terms, due to the unbounded growth of strains, become compatible with the continuum material model. The fracture problem approach adopting these assumptions has been called the &RQWLQXXP6WURQJ 'LVFRQWLQXLW\ $SSURDFK &6'$ 2OLYHU D 2OLYHU HW DO  2OLYHU  DQG 2OLYHU HW DO   Kinematics Regularization. /HW XV FRQVLGHU WKH EDU UXSWXUH SUREOHP RI )LJXUH D ZLWK D GDPDJH ]RQH KDYLQJ D ZLGWK WKDW JRHV WR ]HUR 7KHQ in the limit, this zone tends to S DQG WKH GLVSODFHPHQW ÀHOG u(x) displays a discontinuity across S. By denoting the displacement jump across this surface: [[u]] = u+ − u− , were u+ is the displacement on Ω evaluated on the right part of S (Ω +  DQG u− evaluated on the left part of S (Ω −  ZH FDQ ZULWH WKH GLVSODFHPHQW ÀHOG DV IROORZV ! u(x) = u ¯(x) + H(x)[[u]]

;

H(x) =

0 1

∀x ∈ Ω − ∀x ∈ Ω +



where u ¯ LV D VPRRWK ÀHOG DQG H(x) WKH +HDYLVLGH·V VWHS IXQFWLRQ ,QWURGXFLQJ WKH JHQHUDOL]HG GHULYDWLYH FRQFHSW ∂x H(x) = δs (x), where δs (x) LV WKH 'LUDF·V GHOWD IXQFWLRQ RQ S WKH VWUDLQ ÀHOG ε, compatible with the displacePHQW   LV JLYHQ E\ ε(x) = ∂x u ¯(x) + ∂x H(x)[[u]] =

ε¯(x)  regular term

+

δs (x)   

[[u]]



singular term

For the subsequent mathematical treatment, we will approacK WKH 'LUDF·V IXQF WLRQ E\ D UHJXODUL]HG VHTXHQFH RI IXQFWLRQV /HW Ωsk EH WKH EDU ÀQLWH WKLFNQHVV ]RQH of width k, which includes the discontinuity surface S and such that limk→0 Ωsk = S VHH )LJXUH E 7KHQ GHÀQLQJ ! 1 0 ∀x ∈ Ω\Ωsk k  δs (x) = μ(x) ; μ(x) = 1 ∀x ∈ Ωsk k WKH 'LUDF·V 'HOWD IXQFWLRQ FDQ EH DSSURDFKHG E\ WKH NUHJXODrized sequence lim δsk (x) = δs (x)

k→0



5HSODFLQJ  LQ   ZH REWDLQ D NUHJXODUL]HG VHTXHQFH RI VWUDLQV 2QH WHUP RI this sequence can be interpreted as a weak discontinuity solution, where, through a ÀQLWH EDQGZLGWK k WKH VWUDLQ KDV D MXPS VHH )LJXUH E 6WURQJ GLVFRQWLQXLWLes are recovered when k → 0. Thus, with this regularized kinematics, we characterize two families of discontinuities:

Crack Models with Embedded Discontinuities Strong discontinuity kinematics

l S

W-

k-regularized kinematics (weak discontinuity) Du T

W+

k

l

Du T f t

W ks

W-

P

H(x)

115

S

W+

tS HE

P

Gf

Hk

unloading

u

u [[ u[[

[[ u[[

ft

(c)

u

u

HE

P

A ft e

ds

oo

e

e

k

ds

1/k Wk

(a)

(b)

e

Gf f t l/E

A HE (1+Hl)

Du T (d)

Figure 6. Strong discontinuity approach for the bar problem.

Discontinuous kinematics ε(x) = ε¯(x) +

1 μ(x)[[u]] h



h = 0 : weak discontinuity h = k → 0 : strong discontinuity

Constitutive Model Regularization: Traction-Separation Law as a Projection of the Continuum Constitutive Relation. Problems involving strong discontinuities can be analyzed by assuming that Ω + and Ω − are two independent bodies interacting mechanically through cohesive forces applied along the surface S, such as it has been assumed in the classical discrete cohesive modHOV ,Q WKLV W\SH RI PH chanical approach, the continuum is described by means of a constitutive relation WKDW LV LQGHSHQGHQW RI WKH WUDFWLRQVHSDUDWLRQ ODZ GHÀQHG Ln the fracture process zone. Some issues of the discrete cohesive methodology can be questioned, as it happens for example when cohesive forces depend on the crack tip stress state, W\SLFDOO\ WKH WULD[LDOLW\ UDWLR ,Q WKLV FDVH WKH FRKHVLYH discrete law must depend on the stress state, and thus, the traction-separation law parameters must be stress GHSHQGHQW VHH IRU H[DPSOH WKH ZRUN RI 6LHJPXQG DQG %URFNV  ZKHUH DQ application of this approach is used in order to simulate ductile fracture problems. $V DQ DOWHUQDWLYH WR GLVFUHWH FRKHVLYH DSSURDFKHV WKH &6'$ provides cohesive laws that are consistent with the continuum constitutive relation. The tractionVHSDUDWLRQ ODZ LV GHÀQHG YLD D FRQWLQXXP PRGHO SURMHFWLRQ Ln the discontinuity

116

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surface, that is obtained after introducing the discontinuous kinematics in the conWLQXXP FRQVWLWXWLYH UHODWLRQ ,W LV QHFHVVDU\ WR LPSRVH VRPe constraints such that the continuum model provides bounded stresses when unbounded strains arise in Ωsk . These constraints are determined as follows: i) The equilibrium condition requires that: σsk = σΩ\Ωsk =

P A



where σsk is understood as the stress in Ωsk and σΩ\Ωsk is the stress outside Ωsk . ii) 6XEVWLWXWLRQ RI WKH GLVFRQWLQXRXV NLQHPDWLFV HTXDWLRQ   LQ WKH FRQVWLWX WLYH PRGHO RI 7DEOH  JLYHV σs = lim σsk = lim k→0

k→0

1 qs 1 E(¯ ε + [[u]]) = lim [qs E([[u]])] k→0 krs    rs k



bounded

√

where the bounded character of : qs ∈ [0, ft / E], E and [[u]] is emphasized. (TXDWLRQ  FRQVWUDLQWV WKH k−sequence of stresses, σsk , to be bounded. 7KHUHIRUH HTXDWLRQ  UHTXLUHV WKDW lim krs = α ¯

k→0



must be bounded. iii) 7KH UHJXODUL]DWLRQ RI WKH FRQVWLWXWLYH PRGHO UHGHÀQHV WKH Voftening modulus H IROORZLQJ WKH FRQFHSW LQWURGXFHG LQ HTXDWLRQ  WKURXJh the intrinsic ¯ which is given by: softening modulus H, 2 ¯ = H = − 1 ft H k 2 Gf E



¯ LQ WKH VRIWHQLQJ ODZ HTXDWLRQ   WKH UDWH RI WKH and replacing H by H internal variable q˙s LV GHÀQHG E\   H ¯α ¯˙  q˙s = H r˙s = limk→0 k r˙s = H k which provides a bounded term qs during the complete damage process, even when r˙s is not bounded. Thus, the stress σs (traction ts  HTXDWLRQ  FDQ EH UHZULWWHQ DV (σs =)ts =

qs (E[[u]]) α ¯



Crack Models with Embedded Discontinuities

117

Remark: a discrete traction-separation model (ts , [[u]]  VHH 7DEOH  FRQVLVWHQW with the damage continuum model, is obtained from equations  MRLQWO\ ZLWK WKH UHJXODUL]HG GDPDJH IXQFWLRQ VHH HTXDWLRQ   

fs = lim k( k→0

(Eε2 ) − r) =

 (E[[u]]2 ) − α ¯



and the loading-unloading conditions: fs ≤ 0 ; α ¯˙ ≥ 0 ; α ¯˙ fs = 0. Table 3. 'LVFUHWH RQHGLPHQVLRQDO WUDFWLRQVHSDUDWLRQ ODZ UHVXOWing from the continuum damage model projection (tb is the activation time of the strong discontiQXLW\ PRGH  Traction-separation law qs E [[u]] α ¯ Discrete Damage function

fs ([[u]], α) ¯ = E[[u]]2 − α ¯



ts =



Loading-unloading conditions α ¯˙

≥

0

fs ≤ 0

;

;

αf ¯˙ s = 0



Softening law ¯α q˙s = H ¯˙ ;

2 ¯ = 1 ft = constant < 0 H 2 Gf E



initial conditions α| ¯ t=tb = 0

;

qs |t=tb = q(tb )



Remark: 7KH UHVSRQVH RI WKH ODZ GHVFULEHG LQ 7DEOH  LV SORWWHG LQ )LJXUH F √ ,QLWLDOO\ LW VKRZV D ULJLG UHVSRQVH ZLWK lim[[u]]→0 ts = qs E = ft (observe that lim[[u]]→0 α ¯ = 0  $IWHU WKH DFWLYDWLRQ RI WKH GLVSODFHPHQW MXPS WKH VWLIIQess ¯ becomes HE. Solution of the Bar Problem Provided by the CSDA :H VROYH WKH ' EDU SURE OHP RI WKH SUHYLRXV VHFWLRQ E\ PHDQV RI WKH &6'$ ,Q WKH VRIWHQing regime (post FULWLFDO UHJLPH  WKH VROXWLRQ REWDLQHG XVLQJ WKLV PHWKRGRlogy assumes a loading condition in S and elastic unloading in Ω\S. Then, by denoting with subindex s and Ω\S the bar domain where variables are evaluated, the following equations PXVW EH YHULÀHG

118

A. Huespe and J. Oliver

i) displacement compatibility condition: ΔuT = ΔuΩ\S + [[u]]



where ΔuT is the total displacement of the load application point, [[u]] is the displacement jump in S, and ΔuΩ\S the difference between both displacePHQWV ii) equilibrium condition: P ts = σΩ\S =  A iii) constitutive relations: The discrete damage model in S HTXDWLRQV LQ 7DEOH  SURYLGHV D WUDFWLRQ ts versus jump displacement [[u]] JLYHQ E\ VHH )LJXUH F  ¯ + ft ts = HE[[u]]



while in Ω\S, the elastic stress is: σΩ\S = EΔuΩ\S /. Solution of these equations provides the load P as a unique function of the displacement ΔuT  VHH )LJXUH G   ¯ HEA ft ft  Δu ; for ΔuT ≥  P = + T ¯ ¯ E 1 + H HE This solution gives a post critical structural response which depends on a charac¯ teristic length determined by the fracture energy Gf through the parameter H. 2.3

The Continuum-Strong Discontinuity Approach in 3D Problems

/HW XV FRQVLGHU QRZ D %93 FRUUHVSRQGLQJ WR D ERG\ Ω ∈ R3 , displaying a displacement solution with a jump across the surface S. This surface S, with its normal vector n, divides the body in two disjoint parts Ω + and Ω − , as shown in )LJXUH D 7KH GLVSODFHPHQW MXPS YHFWRU LH WKH GLIIHUHnce of displacements on both sides of the surface S, is denoted by [[u]] = u+ − u− . Thus, the displacement ÀHOG FDQ EH ZULWWHQ DV IROORZV ¯ u(x) = u(x) + Hs (x)[[u]]



¯ where u(x) LV D VPRRWK ÀHOG DQG Hs (x) WKH +HDYLVLGH·V VWHS IXQFWLRQ RQ S (Hs (x) = 0 in x ∈ Ω − and Hs (x) = 1 in x ∈ Ω +  &RQVLGHULQJ WKH JHQHUDOL]HG gradient of the function Hs : ∂x Hs = δs (x)n, where δs (x) is the generalized 'LUDF·V GHOWD IXQFWLRQ RQ S WKH VWUDLQ ÀHOG LV JLYHQ E\ ¯ ε(x) = ε(x) + δs (x)(n ⊗ [[u]])sym



Crack Models with Embedded Discontinuities

119

¯ ¯ while ZKHUH WKH ÀUVW WHUP LV WKH VPRRWK SDUW ε¯ and is given by: ε(x) = ∇sym u, the second term is singular. $ UHJXODUL]HG YHUVLRQ RI WKH VWUDLQ ÀHOG   LV FRQYHQLHQW for its subsequent PDWKHPDWLFDO DQG QXPHULFDO WUHDWPHQW /HW D EDQG Ωsk RI ÀQLWH ZLGWK k including the surface S be introduced, and such that limk→0 Ωsk = S, as shown in Figure E 6LPLODU WR WKH ' FDVH LW LV SRVVLEOH WR GHÀQH WKH NUHJularized sequence of functions 1  δsk = μ(x), k where: μ(x) = 1 in x ∈ Ωsk DQG  RWKHUZLVH DQG VXFK WKDW limk→0 δsk = δs . 5HSODFLQJ  LQ   ZH REWDLQ WKH KVHTXHQFH RI UHJXODULzed strains: ¯ ε(x) = ε(x) +

1 μ(x)(n ⊗ [[u]])sym h



Thus, this kinematics setting captures either a weak (h = 0 RU D VWURQJ GLV continuities (h = k → 0  UHVSHFWLYHO\ W

W W+

n S

(a)

W+

W(b)

n

S

k

WSk W-

Figure 7. D 6WURQJ GLVFRQWLQXLW\ NLQHPDWLFV ' FDVH E UHJXODULzation of the VWURQJ GLVFRQWLQXLW\ ZHDN GLVFRQWLQXLW\ LQ Ω k .

Traction-Separation Law in 3D Problems 8WLOL]LQJ DQ LGHQWLFDO SURFHGXUH WR WKDW RI WKH ' FDVH ZH FDQ GHWHUPLQH WKH WUDFWLRQVHSDUDWLRn law projected by the FRQWLQXXP GDPDJH PRGHO GHÀQHG LQ 7DEOH  9DULDEOHV HYDOXDWHG LQ WKH GRPDLQV Ωsk (limk→0 Ωsk = S RU Ω\Ωsk are denoted with subindex s or Ω\Ωsk respectively. The procedure is as follows: i) the equilibrium condition requires the traction vector continuity across the surface bounding Ωsk and Ω\Ωsk : σs · n = σΩ\Ωsk · n



From this equation, and given the bounded character of σΩ\Ωsk , the stresses σΩsk should be bounded, even when k → 0.

120

A. Huespe and J. Oliver

ii) replacing the strain εΩk  IURP HTXDWLRQ   LQWR WKH GDPDJH FRQVWLWXWLYH s UHODWLRQ RI WKH FRQWLQXXP PRGHO HTXDWLRQ  LQ 7DEOH  DQd taking the limit for k going to zero:    qs 1 sym ¯ σs · n = lim E : ε(x) ·n= + (n ⊗ [[u]]) k→0 rs k qs qs (n · E · n)[[u]] = Qe [[u]]  = lim k→0 krs α ¯ ,Q WKH ODVW LGHQWLW\ Qe LV WKH HODVWLF DFRXVWLF WHQVRU HTXDWLRQ   8VLQJ D VLPLODU DUJXPHQW DV LQ WKH ' FDVH DQG FRQVLGHULQJ WKDW qs , Qe and [[u]] are bounded, as also the traction vector, then, α ¯ = limk→0 krs must be a bounded term. ¯ = H , the evolution of the internal iii) E\ UHGHÀQLQJ WKH VRIWHQLQJ PRGXOXV H k variable q˙s is bounded and given by: ¯α ¯˙ q˙s = lim H r˙s = H k→0



iv) introducing the strain ε JLYHQ E\ HTXDWLRQ   LQWR WKH GDPDJH IXQFWLRQ IURP HTXDWLRQ   DQG WDNLQJ WKH OLPLW RI H[SUHVVLRQ lim kf (ε, r) = 0 ;

k→0

a discrete damage criterion can be determined by:  ¯=0 ; fα ([[u]]) = [[u]] · (Qe ) · [[u]] fα ([[u]]) − α





Remark: the traction-separation law is automatically induced once the strong discontinuity kinematics and the softening regularization are introduced in the model. Thus, the cohesive traction is determined by a degeneration of the continuum constitutive relation.

3

Finite Elements with Embedded Discontinuities

Finding good numerical solutions of concrete fracture probOHPV XVLQJ WKH &6'$ depends on the discretization technique and its capacity for adequately capturing one of the most salient aspects of this approach: the strong discontinuity kinePDWLFV $ ZHOO DGDSWHG WHFKQLTXH WR UHDFK WKLV REMHFWLYH LV Ànite elements with embedded discontinuities. ,Q UHFHQW \HDUV WKHVH ÀQLWH HOHPHQWV KDYH EHHQ WKH REMHFW RI increasing study DQG GHYHORSPHQW ,WV ULVLQJ SRSXODULW\ FRPHV IURP WKH IDFW What, a displacement discontinuity model can be introduced in the bulk of the element (having an arbitrary direction regardless of the orientation of the elemenW PHVK  LQ FRPELQDWLRQ

Crack Models with Embedded Discontinuities

121

Table 4. 'LVFUHWH ' WUDFWLRQVHSDUDWLRQ ODZ DV D FRQWLQXXP GDPDJH Podel projection (tb LV WKH DFWLYDWLRQ WLPH RI WKH VWURQJ GLVFRQWLQXLW\ PRGH  Traction-separation law qs e Q · [[u]] α ¯ Discrete Damage function  fα ([[u]], α) ¯ = [[u]] · Qe · [[u]] − α ¯ ts = σs · n =





Loading-unloading conditions α ¯˙

≥

0

fα ≤ 0

;

;

αf ¯˙ α = 0



Softening law ¯α q˙s = H ¯˙ ;

2 ¯ = 1 ft = constant < 0 H 2 Gf E



initial conditions α| ¯ t=tb = 0

;

qs |t=tb = q(tb )



with appropriate propagation mechanisms. Moreover, it has been shown that, in strain localization scenarios, some of those elements joinWO\ ZLWK WKH &6'$ RU GLV crete cohesive laws, completely overcome the well-known problem of the spurious mesh size and mesh orientation dependence of the results. $ TXLWH ODUJH IDPLO\ RI ÀQLWH HOHPHQWV ZLWK VWURQJ GLVFRQWLQuities has been presented in the literature. We are addressing only two particular classes of them, as also, their numerical performances related with accuracy and convergence of solutions. 3.1

Strong Discontinuities: The Local Form of the BVP Governing Equations

/HW XV FRQVLGHU WKH ERG\ Ω, of Figure 8-a, that is subjected to displacements u on the boundary, Γu , and imposed traction loads t on Γσ . Without loss of generality, it is considered that there are no volume forces /HW XV DOVR FRQVLGHU WKDW the body is undergoing a displacement discontinuity [[u]](x; t) across the surface S, and that this surface splits the body into Ω + (pointed by the unit normal n to S DQG Ω − . 7KH FRUUHVSRQGLQJ TXDVLVWDWLF ERXQGDU\ YDOXH SUREOHP %93 FDQ EH GHVFULEHG in rate form, through the following set of equilibrium equations:

122

A. Huespe and J. Oliver ∇ · σ˙ = 0

in Ω\S

internal equilibrium

σ˙ · ν = t˙

on Γσ

external equilibrium

σ˙ Ω + · n − σ˙ Ω − · n = 0

on S

outer traction continuity

σ˙ Ω + · n − σ˙ S · n = 0

on S

inner traction continuity



(TXDWLRQV  DUH WKH FODVVLFDO HTXLOLEULXP HTXDWLRQV 7KH HTXDWLRQV FG are the equilibrium conditions on the surface S. σΩ+ and σΩ − are the stresses evaluated on Ω + and Ω − , respectively, while σs stands for the stresses arising in the strain localization zone, S ,W ZDV SUHYLRXVO\ VKRZQ WKDW WKH HYDOXDWLRQ RI σs using the continuum constitutive relation is a basic assumpWLRQ RI WKH &6'$ $GGLWLRQDO HTXDWLRQV RI WKH %93 DUH WKH IROORZLQJ

ε˙ − ∇sym u˙ = 0

in Ω

kinematical compatibility

u˙ = u˙

on Γu

kinematical boundary conditions

˙ σ˙ = Σ(ε, α)

in Ω

constitutive relation



ZKHUH WKH FRQVWLWXWLYH UHODWLRQ F GHSHQGV RQ WKH VWUDLn ε, as also, on a set of internal variables α. 3.2

A Variational Consistent Formulation of the BVP with Strong Discontinuities

For the variational formulation of a problem displaying strong discontinuities, LW LV FRQYHQLHQW WR GHVFULEH WKH YHORFLW\ ÀHOG DV IROORZV ˙ ˙ ¯˙ u(x) = u(x) + Ms (x)[[u]]



¯˙ LV D VPRRWK YHORFLW\ ÀHOG DQG Ms (x) LV D FRQYHQLHQWO\ GHÀQHG IXQFWLRQ where u that presents a jump in S ([[Ms ]] = 1 DQG LWV VXSSRUW LV Ω k , being Ω k a domain like a band covering the discontinuity line S, see Figure 8-b and c. The function Ms (x) is built as follows: Ms (x) = Hs (x) − ϕ(x)



Crack Models with Embedded Discontinuities t* = s.u

Gs

u

W+ n

W+

S

Gu

HS

W W

Wk

1

x

j

n

W

S x

MS x

1

(b)

(a)

x

1

S

W

123

Wk

(c)

Figure 8. A mechanical problem with strong discontinuity.

where Hs LV WKH SUHYLRXVO\ GHÀQHG +HDYLVLGH·V VWHS IXQFWLRQ RQ S, and ϕ(x) is a smooth function such that ϕ = 0 in Ω − \Ω k and ϕ = 1 in Ω + \Ω k . 7KH UHDVRQ RI GHÀQLQJ D QHZ IXQFWLRQ Ms for describing the velocity jump, OLHV RQ WKH UHGXFHG VXSSRUW RI WKLV IXQFWLRQ ,Q IDFW E\ DVVXming that Ω k does not intersect a boundary Γu (Ms ≡ 0 outside Ω k  WKH YHORFLW\ ERXQGDU\ FRQGL ¯ as it is done in classical tions on Γu are only imposed on the smooth function u, approaches. The generalized gradient of Ms is: ∇Ms = δs n − ∇ϕ



The BVP Weak Form. /HW XV FRQVLGHU WKH IROORZLQJ YDULDWLRQDO IRUPXODWLRQ 

 t˙ · η dΓσ

σ˙ : ∇η dΩ = Ω

;

∀η ∈ Vo = {η = η¯ + Ms β}



Γσ

where Vo is the space of the test functions, η¯ LV GHÀQHG DV D VPRRWK IXQFWLRQ LQ Ω verifying the homogeneous essential boundary conditions (η¯ = 0 in Γu  DQG β LV GHÀQHG LQ S 7KH YDULDWLRQDO HTXDWLRQ  LV HTXLYDOHQW WR WKH ORFDO IRrm RI WKH HTXLOLEULXP HTXDWLRQV   7KLV VWDWHPHQW LV EULHÁ\ demonstrated in the IROORZLQJ DGGLWLRQDO GHWDLOV FDQ EH VHHQ LQ 6LPR DQG 2OLYHU   i) &KRRVLQJ VPRRWK WHVW IXQFWLRQV η (with β = 0  WKH YDULDWLRQDO IRUPXODWLRQ  \LHOGV YLD WKH *UHHQ·V WKHRUHP LQ WKH UHJXODU SDUWV Ω + and Ω − , plus VWDQGDUG DUJXPHQWV WKH HTXDWLRQV  D E DQG F ii) &KRRVLQJ WHVW IXQFWLRQV η, such that η¯ = 0, then: ∇η = [β⊗(δs n−∇ϕ)]+ Ms ∇β and η = 0 in Γσ (due to the assumption that the Ms -function has

124

A. Huespe and J. Oliver D UHGXFHG VXSSRUW DQG VXEVWLWXWLQJ η LQ HTXDWLRQ  \LHOGV    δs (σ˙ · n) · β dΩ + (σ˙ · ∇ϕ) · β dΩ + σ˙ : ∇η dΩ = Ω Ω Ω + Ms (σ˙ : ∇β) dΩ = 0  Ω

which results in the following equation:   (σ˙ · n) · β dS − (σ˙ Ω + · n) · β dS = 0 S



S

7KH ÀUVW WHUP LQ  UHVXOWV IURP WKH ÀUVW WHUP LQ WKH OHIW KDQd side in HTXDWLRQ   ZKLOH WKH ODVW WHUP LV GHULYHG IURP WKH VHFRQG and third terms, considering: a) WKH *UHHQ·V WKHRUHP RQ WKH GRPDLQ Ω k , b) the strong form of the equilibrium equation in the regular slice Ω k \Ω + and the jump condition across S obtained in the item (i), c) ϕ = 0 in S − ; ϕ = 1 in S + and that d) the edges of the slice Ω k intersecting the body boundary are free of tractions. (TXDWLRQ  ÀQDOO\ SURYHV WKH HTXLYDOHQFH EHWZHHQ WKH ZHDN HTXDWLRQ   DQG WKH VWURQJ IRUP HTXDWLRQ   RI WKH HTXLOLEULXP RI ERGLHV with strong discontinuities. 3.3

Finite Elements with Embedded Discontinuities

/HW XV FRQVLGHU WKH ERG\ Ω GLVFUHWL]HG ZLWK D OLQHDU &67 WULDQJXODU ÀQLWH element mesh. Also, let us assume that an available discontinuity path detection algorithm (the so called tracking algorithm, to be explaineG LQ WKH QH[W VHFWLRQ determines the subset J of the elements that are crossed by S at the considered WLPH DV VKRZQ LQ )LJXUHD )RU HYHU\ HOHPHQW RI J , the tracking algorithm also provides the position of the element discontinuity interface Se VHH )LJXUHE RI length le ZKLFK GHÀQHV WKH GRPDLQV Ωe− and Ωe+ and the nodes j 1 , j 2 and j sol , with j sol being the node that lies in Ωe+ . ˙ inside a given Kinematics. 7KH IROORZLQJ LQWHUSRODWLRQ RI WKH YHORFLW\ ÀHOG u, element e is considered: ˙ u(x) Mes (x)

= =

3  i=1 e

Ni (x)d˙i + Mes (x)β˙

[H (x) − N sol (x)]



∀e ∈ J

where Ni (x) DUH WKH OLQHDU VWDQGDUG VKDSH IXQFWLRQV RI WKH WULDQJXODU ÀQite element and d˙i WKH QRGDO YHORFLWLHV 7KH VHFRQG WHUP LQ D FDQ EH FRQVLGered an enrichment velocity mode that captures the discontinuous parW RI WKH YHORFLW\ ÀHOG

Crack Models with Embedded Discontinuities

125

The unit jump shape function Ms KDV D VXSSRUW RI RQH ÀQLWH HOHPHQW ZLWK He EHLQJ WKH +HDYLVLGH·V IXQFWLRQ LQ WKH HOHPHQW e, and N sol the linear shape function that corresponds to the node j sol (N sol = Nj sol  )LJXUHF GLVSOD\V WKH IXQFWLRQ Mes . The parameter β˙ ∈ Rndim , with ndim the space dimension, is a vectorial elemental parameter, which represents the veORFLW\ MXPS RI RQH ÀQLWH HOHPHQW 7KLV LV DQ HOHPHQWDO SDUDPHter, not shared with RWKHU ÀQLWH HOHPHQWV )URP HTXDWLRQ   WKH VWUDLQ UDWH UHDGV ε˙ =

3 

1 ˙ sym ˙ sym − (∇N sol ⊗ β) (∇Ni (x) ⊗ d˙i )sym + μ(n ⊗ β) k i=1



where, in order to compute the singular term arising from the generalized derivaWLYH RI WKH GLVFRQWLQXRXV YHORFLW\ ZH KDYH UHSODFHG WKH 'LUDF·V GHOWD IXQFWLRQ E\ the k-regularized sequence: δse = (1/k)μ, where: μ = 1 in Ω k DQG  RWKHUZLVH VHH )LJXUHE DQG G j2

j2

J ={e1 ,...,em , ..., ep } ep

Se

le

W-e

j1

em

e1

(a)

1/k

1

n

Wk

W +e

k

e

MS

j1

jsol

(c)

jsol

(b)

wPG1 = W e - k le wPG2 = k l e

d

h S

S (d)

PG2

PG1

(e)

Figure 9. Finite element approach. ,Q RUGHU WR LQWHJUDWH WKH WHUPV GHÀQHG LQ Ω k , we add a second sampling point, DV VKRZQ LQ )LJXUHH WKDW UHSUHVHQWV WKH GRPDLQ Ω k , and whose weight is the area of the domain Ω k : area Ω k = kle , and the regularization parameter k is an arbitrarily small factor. a) Symmetric FE: Variationally Consistent Formulation. The variationally FRQVLVWHQW )( IRUPXODWLRQ XWLOL]HV D VLPLODU YHORFLW\ ÀHOG interpolation to that pro-

126

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SRVHG LQ HTXDWLRQ   ZKLOH WKH WHVW IXQFWLRQ VSDFH LV FKRVen as follows: Vo = {η | η = Ni (x)η¯i + Mes (x)δβ e }

η = 0 on Γu

where η¯i and δβ e are nodal and element parameters of the interpolation functions, UHVSHFWLYHO\ ,QWURGXFLQJ WKLV GLVFUHWH IXQFWLRQDO VSDFH in the variational formulaWLRQ   DQG DIWHU SHUIRUPLQJ WKH UHVSHFWLYH YDULDWLRQV Rf parameters, it yields the following system of equations:   t Ni dΓσ = 0  σ˙ · ∇Ni dΩ − Ω Γσ   1  σ˙ · ∇N sol dΩ e = 0 μ (σ˙ · n) dΩ e − Ωe k Ωe (TXDWLRQV  DUH WKH VWDQGDUG YDULDWLRQDO ÀQLWH HOHPHQW Hquation arising in classical problems without strong discontinuities. Thus, there are ndim equations of WKLV W\SH IRU HYHU\ QRGH ´Lµ RI WKH PHVK (TXDWLRQV  LPSRVes a weak equilibrium across the discontinuity line S. There are ndim equations of this type for every element “e” in J . 2D implementation. )RU WKH WZRGLPHQVLRQDO FDVH LQ D &DUWHVLDQ FRRUGLQDWH system (x, y) XVLQJ WKH 9RLJW·V QRWDWLRQ IRU WKH VWUDLQV {ε} = [εxx , εyy, 2εxy ]T and the stresses {σ} = [σxx , σyy, σxy ]T (where (·)T stands for the transpose of (·)  FRQVLGHULQJ WKH OLQHDU WULDQJOH DV XQGHUO\LQJ HOHPHQW DQd using the standard ÀQLWH HOHPHQW B-matrix IRUPDW =LHQNLHZLF] DQG 7D\ORU   HTXDWLRQV    FDQ EH ZULWWHQ DV IROORZV  n" elem T ext(e) (e) ˙ =0  B (e) · σ({ε} ) dΩ −F˙ e=1

Ωe

where F˙ ext(e) stands for the classical element external forces vector and: ˙ (e) = B (e) · d˙(e) {ε} # $T d˙(e) = d˙1 , d˙2 , d˙3 , d˙4 , β˙ e & % (e) B (e) = B1(e) , B2(e) , B3(e) , G ⎤ ⎡ (e) 0 ∂x Ni ⎢ (e) (e) ⎥ Bi = ⎣ 0 ∂ y Ni ⎦ (e) (e) ∂y Ni ∂ x Ni ⎤ ⎤ ⎡ ⎡ (e) ∂x N sol 0 nx 0 ⎢ (e) ⎥ ny ⎦ − ⎣ 0 G(e) = k1 μ ⎣ 0 ⎦ ∂y N sol (e) ny nx sol sol (e) ∂y N ∂x N





Crack Models with Embedded Discontinuities

127

where stands for the classical assembling operator and n = [nx , ny ]T . Remark: 7KH VWUXFWXUH RI HTXDWLRQV  VXJJHVWV WKH LQWURGXFWLon of an internal additional fourth node for each element e, that is activated only for the elements crossed by the discontinuity interface (e ∈ J DQG ZKRVH FRUUHVSRQGLQJ GHJUHHV of freedom and associated shape function are, respectively, the displacement jumps (e) (e) βe and MS LQ HTXDWLRQ   6LQFH WKH VXSSRUW RI MS is only Ωe , those internal degrees of freedom can be eventually condensed at the element level and removed from the global system of equations. Remark: 7KH LQWHJUDWLRQ RI HTXDWLRQ  LV HYDOXDWHG XVLQJ DQ DGGLWional integration point that accounts for those terms in Ω k . The weight of the additional integration point is the area assigned to Ω k : kle . Furthermore, at the additional integration point μ = 1 DQG ]HUR RWKHUZLVH VHH )LJXUH H b) Non-symmetric FE. An alternative formulation to the symmetric FE approach of the previous subsection, can be deduced by replacing the weak equilibrium condition on S HTXDWLRQ   E\ RQH GHWHUPLQHG WKURXJK WKH DYHUDJH WHUPV of the identity: σ˙ Ω + · n = σ˙ s · n, as follows: 1 Ωe 



1 (σ˙ · n) dΩ = le Ωe   

mean value on Ω\S



e

s

(σ˙ s · n) dS  



mean value on S

Convergence Test for the Symmetric FE Formulation 7KH HTXLYDOHQFH RI WKH %93 ZHDN IRUP  ZLWK WKH VWURQJ IRUP equations  FDQ EH ULJRURXVO\ SURYHQ VHH IRU LQVWDQFH 6LPR DQG 2OLYHU   7KHUH fore, using the symmetric formulation, it is expected that mHVK UHÀQHPHQW ZLOO GHWHUPLQH D FRUUHFW WUHQG FRQYHUJHQFH WR WKH IXOÀOOPHQW Rf those equations. Taking the linear triangle as the underlying element, the teVW LQ )LJXUH  FRQ stitutes a simple corroboration of this fact and provides an assessment of the order RI FRQYHUJHQFH RI WKH V\PPHWULF ÀQLWH HOHPHQW ZLWK HPEHGGHG discontinuities. The test consists of a homogeneous rectangular strip pulled from the right end with a force P , imposing a displacement Δ, up to the formation of an inclined failure line and, then, continued to the total failure and reOHDVH RI WKH VWUHVVHV 'XH WR WKH LQGXFHG FRQVWDQW VWUHVV ÀHOG WKH FRQVLGHUHG ELOLQHDr stress-strain law for the constitutive model should translate into a bilinear force-displacement (P − Δ) curve. To check the convergence to the right slope of the descending branch and, therefore, to the right energy dissipation Gf  D KRPRJHQHRXV PHVK UHÀQHPHQW parameterized in terms of the element size h LV SHUIRUPHG )LJXUH D VKRZV WKH UHVXOWV IRU LQFUHDVLQJO\ ÀQH PHVKHV FRQYHUJLQJ WR WKH H[Dct solution (the curve OLPLWLQJ WKH JUD\ ]RQH  7KH HUURUV LQ WKH GLVVLSDWHG HQHUJ\ IUDFWXUH HQHUJ\ IRU

128

A. Huespe and J. Oliver

the different levels of discretization are displayed in FigXUH E ZKHUH WKH OLQHDU convergence of the element is observed. ,W LV ZRUWK PHQWLRQLQJ WKDW WKH V\PPHWULF YDULDWLRQDO IRUPXlation, although convergent, exhibits accuracy smaller than the non-symmetric formulation of subVHFWLRQ E ZKLFK IRU WKLV W\SH RI KRPRJHQHRXVFRQVWDQt stress problem, provides the exact solution with only one element. 1.2

P 2.m

1

1.m

log(error Gf)

-5

D P

-6

0.8 h

0.6

-7

1

h

h=1/13 h=1/9

Gf

0.2

-9

h=1/3 h=1/2

-10

0 0

(a)

0.004

D[m]

1

-8

S

0.4

0.008

0.012

-3

(b)

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0

Figure 10. 6\PPHWULF ÀQLWH HOHPHQW IRUPXODWLRQ FRQYHUJHQFH DQDO\VLs.

3.4

An Embedded Strong Discontinuity FE not Needing the Crack Path Continuity Enforcement

)RU WKH FRUUHFW LPSOHPHQWDWLRQ RI ERWK ÀQLWH HOHPHQW WHFKQRlogies described SUHYLRXVO\ WKH FUDFN SDWK DFURVV WKH ÀQLWH HOHPHQW PHVK PXVt be known, and furthermore, it should be continuous. The crack path continuity condition is necesVDU\ WR GLVFULPLQDWH WKH UHODWLYH SRVLWLRQ RI WKH ÀQLWH HOHPent nodes with respect to the discontinuity line S, and to know whether they stay on Ω + or Ω − . A wrong selection of their relative position produces severe numerical locking. Furthermore, in the non-symmetric formulation, for capturing a correct dissipation of energy it is required an exact evaluation of the length le (as shown in )LJXUH E  ZKLFK LV RQO\ SRVVLEOH LI WKH LQWHUVHFWLRQ SRLQts of the discontinuity line with the element sides, are known. The symmetric formulation does not need this value. ,Q WKLV VHFWLRQ ZH SUHVHQW D ÀQLWH HOHPHQW WHFKQLTXH ZLWK HPEedded strong discontinuity not needing the crack path evaluation. The formulation was proposed E\ 6DQFKR HW DO  DQG LWV LPSOHPHQWDWLRQ LV YHU\ VLPSOH, providing good results in concrete fracture problem simulations.

Crack Models with Embedded Discontinuities j sol

n

N

j1

129

f(b)

S sol

Gf j2

(a)

b

(b)

Figure 11. Finite element approach without enforcing crack path continuity.

Kinematics. ,W LV EDVHG RQ WKH VWURQJ GLVFRQWLQXLW\ NLQHPDWLFV ZLWK D YHOocity ÀHOG LQWHUSRODWLRQ VLPLODU WR WKDW GHVFULEHG LQ HTXDWLRQ  XVLQJ &67 OLQHDU WUL DQJOHV SDUHQW HOHPHQWV 7KXV XVLQJ DQ LGHQWLFDO QRWDWLRQ WKH YHORFLW\ ÀHOG LV given by: ˙ u(x) =

3 

Ni (x)d˙i + [He (x) − N sol (x)]β˙

;

∀e ∈ J



i=1

An important issue of this technique is referred to the way that cohesive forces, DW WKH GLVFRQWLQXLW\ LQWHUIDFH DUH GHÀQHG DQG KDQGOHG 7KHse cohesive forces are provided by a discrete law, independent of the continuum conVWLWXWLYH PRGHO ,Q this case, the stresses and strains in S are meaningless and the strains in Ω\S are computed as follows: ε˙ =

˙ sym ; −(∇N sol ⊗ β)

¯˙ ε 



(∇Ni ⊗d˙i )sym

WKXV RQO\ UHJXODU WHUPV GHÀQH   6LPLODU WR WKH SUHYLRXV FE techniques, N sol is WKH VWDQGDUG VKDSH IXQFWLRQ RI WKH &67 ÀQLWH HOHPHQW UHODWHG with the node lying in Ω +  VHH )LJXUH D +RZHYHU ZH ZLOO VHH LQ WKH QH[W SDUDJUDSK What the choice of the node lying on Ω + does not require an algorithm to trace the crack path. Cohesive Model. 7KH FRKHVLYH PRGHO LV GHÀQHG XVLQJ D GLVFUHWH ODZ RI FHQWUDO forces. Thus, considering two bodies Ω + and Ω − , the interaction forces t, are proportional to the displacement jumps β, and its magnitude depends on the (hisWRULFDO PD[LPXP VHSDUDWLRQ EHWZHHQ WKHP ˜ β t = f (β) β˜

;

β˜ = max  β(t) t>0



130

A. Huespe and J. Oliver

˜ is a monotonic decreasing function of its argument, and must be where f (|β|) GHÀQHG VXFK WKDW WKH DUHD HQFORVHG E\ WKH FXUYH f (β) vs. β, is the fracture energy Gf , see Figure 11-b. Kinematically Consistent Variational Formulation. The equilibrium of the body is imposed through the following discrete variational approach:    σ : δε dΩ + t · δβ dS = t · δu dΓσ  Ω

S

Γσ

∀ δu ∈ Vo ; δε = ∇sym δu Vo = {δu | δu = Ni (x)δdi + (Hs − N sol )δβ} ; i = 1, 2, 3 by performing the variations of both parameters, (δdi , δβ  GHÀQLQJ WKH LQWHUSROD WLRQ RI YLUWXDO GLVSODFHPHQWV ÀHOG WKH IROORZLQJ V\VWHP RI equations is obtained:   (σ · ∇Ni ) dΩ − Ni t dΓσ = 0  Ω Γσ   t dS = 0  − (σ · ∇N sol ) dΩ + Ω

S

(TXDWLRQ  LV LPSOHPHQWHG WKURXJK WKH ORFDO IRUP t=n·σ



Replacing the material bulk constitutive relation, which is assumed to be linear elastic, the stress can be written: σ = E : ε = E : (ε¯ − (∇N sol ⊗ β)sym )



DQG ÀQDOO\ LQWURGXFLQJ  DQG WKH WUDFWLRQGLVSODFHPHQW MXPS UHODWLRQ  LQ   WKH UHODWLRQ EHWZHHQ WKH VWUDLQ ε and the displacement jump β results: 

 ˜   f (|β|) sol 1 + (n · E · ∇N ) β = n · E : ε¯ − (∇N sol ⊗ β)sym ) ˜ |β|



which must be solved in conjunction with the equilibrium equDWLRQ  DQG WKH FRQVWLWXWLYH UHODWLRQ   Remark: This approach is derived from a non-symmetric variational formulation WULDO DQG WHVW IXQFWLRQV DUH GLIIHUHQW  7KHUHIRUH WKH GLscrete stiffness matrix is non-symmetric. Selection of N sol . ,Q RUGHU WR FKRRVH WKH VKDSH IXQFWLRQ N sol , in the context of WKH &67 ÀQLWH HOHPHQW 6DQFKR HW DO  KDYH SURSRVHG D YHry simple methodRORJ\ ,W FRQVLVWV RI WDNLQJ WKH JUDGLHQW YHFWRU ∇Ni , (i = 1, 2, 3) which is as parallel as the normal vector n to the crack:

Crack Models with Embedded Discontinuities |∇Ni · n| i=1,2,3 |∇Ni |

∇N sol = max

131 

An important ingredient of the methodology is that, after the onset of the crack activation, the normal vector to the crack, n, is allowed to rotate for a small anJOH EHIRUH WKH ÀQDO FUDFN GLUHFWLRQ EHFRPHV À[HG 7KLV DUWLIact is called crack DGDSWDELOLW\ E\ 6DQFKR HW DO  

4

Algorithmic Aspects of the CSDA

6RPH ÀQLWH HOHPHQWV ZLWK HPEHGGHG VWURQJ GLVFRQWLQXLWLHV explained in the previous Section, require the use of procedures for computing the position of the disFRQWLQXLW\ VXUIDFH RU FUDFN SDWK LQ IUDFWXUH SUREOHPV E\ Dssuming its geometrical FRQWLQXLW\ DV LW FURVVHV WKH ÀQLWH HOHPHQW PHVK ,Q JHQHUDO WKH VHOHFWLRQ RI WKH GHIRUPDWLRQ PRGHV FDSWXULQg the displacement jumps crucially depends on the way that the element is crossed by the discontinuity. The strategies devoted to predict and capture the geometrical position of the discontinuity surface, are termed tracking strategies. There are two basic tracking strategies: i) WKRVH EDVHG RQ D ORFDO RU SURSDJDWLQJ SURFHGXUH ZKHUH WKH crack path is WUDFNHG HOHPHQW E\ HOHPHQW VHH IRU H[DPSOH 2OLYHU E  2OLYHU HW DO   2OLYHU HW DO   +DQGOLQJ WKLV DOJRULWKP ZKHQ several cracks VKRXOG EH FRPSXWHG LV FRPSOH[ ii) alternatively, global tracking algorithms are based on the information proYLGHG E\ WKH SURSDJDWLRQ ÀHOG RI WKH FRPSOHWH ÀQLWH HOHPHQW Pesh. All crack tracking algorithms are based on two types of data that must be determined, in every point of the body,using a convenient material failure criterion. They are: 1) a condition for detecting the onset of crack propagation, and 2) a GLUHFWLRQ IRU WKH FUDFN SURSDJDWLRQ ,Q WKH IROORZLQJ VXEVHction, we introduce a summary of these criteria. After that, a global tracking strDWHJ\ WDNHQ IURP 2OLYHU HW DO   LV SUHVHQWHG Selection of a Local Material Failure Criterion. ,Q WKH VLPSOHVW FDVHV WKH crack propagation onset condition is marked by the end of the elastic regime, WKURXJK WKH IXOÀOOPHQW RI VRPH \LHOG RU GDPDJH FULWHULRQ $Oso, it could provide a crack propagation direction by assuming, for example, that the crack is orthogonal to the maximum tensile stress. A criterion like this one, works reasonably well in some quasi-brittle fracture problems, such as concrete fracture.

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A. Huespe and J. Oliver

+RZHYHU ULJRURXV SURFHGXUHV WR GHWHUPLQH WKH RQVHW RI ORFDl material failure should be based on the material stability concept, as it was explained in Section 1, associated to the singularity of the so called localization tensor Q(n, H) = n·DT ·n. The critical value of the softening modulus, H crit , signaling the onset of failure, and the corresponding propagation directions, ncrit , are then determined IURP WKH SUREOHP det Qtg = 0. 4.1

A Global Tracking Algorithm

&RQVLGHU WKH YHFWRU ÀHOG n(x; t) DQG WKH RUWKRJRQDO YHFWRU ÀHOG T (x; t), which indicates the direction of propagation of a discontinuity line at the point x ,Q WKH FRQWH[W RI D ÀQLWH HOHPHQW DQDO\VLV ZH DVVXPH WKDW WKHUH LV RQe available vector T (e) for every sampling point in the element e VHH ÀJXUH  :H GHWHUPLQH D VFDODU ÀHOG θ(x; t), such that, its iso-level contour lines are the HQYHORSHV RI WKH YHFWRU ÀHOG T (x; t). Therefore, these lines represent the set of possible crack paths across the body. Evaluation of the Vector Field Envelopes. Heat conduction-like problem. /HW us now focus on a procedure to compute the envelopes of the vecWRU ÀHOG T (x; t) in a two-dimensional domain Ω ,WV H[WHQVLRQ WR WKH JHQHUDO ' VLWXDWLRQ LV VWUDLJKW forward. We shall assume that T LV D XQLW YHFWRU ÀHOG T  = 1  ZKRVH VHQVH LV QRW UHOHYDQW $ VFDODU ÀHOG θ(x) whose iso-level contour lines are the envelopes of WKH YHFWRU ÀHOG T , must verify: T · ∇θ = 0

in Ω

;



0XOWLSO\LQJ HTXDWLRQ  E\ T , it results: (T · ∇θ)T = K · ∇θ = 0

;

K = (T ⊗ T )

where we introduce the rank-one tensor K 'HÀQLQJ WKH YHFWRU q by: q = −K · ∇θ

in Ω,



WKHQ LWV GLYHUJHQFH LV ]HUR HYHU\ZKHUH (TXDWLRQ  FDQ be written in terms of q DV D %93 LQ Ω VHH )LJXUH  ∇·q q·ν θ

= 0 = 0 = θ∗

in Ω in Γq in Γθ



where ν is the outward normal to the boundary ΓΩ of the body Ω 1HZPDQ boundary conditions are prescribed in the boundary Γq DQG 'LULFKOHW ERXQGDU\ conditions in Γθ .

Crack Models with Embedded Discontinuities

133

u

W

q= cte

n

T

Figure 12. The pseudo steady-state heat conduction problem.

7KLV %93 GHÀQHG LQ Ω is similar to a steady-state heat conduction problem ZLWK QR LQWHUQDO KHDW VRXUFHV DQG QXOO KHDW ÁX[ LQSXW LQ WKH ERundary Γq (adiabatic ERXQGDU\  ,Q WKLV FDVH θ SOD\V WKH UROH RI D SVHXGRWHPSHUDWXUH ÀHOG q is the KHDW ÁX[OLNH YHFWRU DQG K is a point dependent rank-one anisotropic thermal conductivity tensor. Finite element formulation of the heat conduction-like BVP. 7KH SUREOHP  can be numerically solved by means of well-known, very simple and computationDOO\ FKHDS ÀQLWH HOHPHQW SURFHGXUHV 2OLYHU HW DO   These features and the fact that all possible crack paths could be determined once tKH ÀHOG θ is known, are the most important advantages of the present tracking algorithm method. Remark: at every point, the global tracking algorithm requieres an adequte material failure criterion providing the local crack propagation direction. This criterion must predict the propagation direction even in those points where an elastic proFHVV LV WDNLQJ SODFH 2QFH WKLV FRQGLWLRQ LV IXUQLVKHG WKH Dlgorithm can be used to simulate different kind of fracture problems, such as ductiOH RU EULWWOH IUDFWXUH QR matter the material model type. Representative Numerical Simulations. )LJXUH  GLVSOD\V WKH DSSOLFDWLRQ RI this algorithm to a concrete fracture problem: the four-point beam test. Also, in the same Figure, an application to a slope stability analysis is VKRZQ ,Q WKLV FDVH WKH PDWHULDO UHVSRQVH LV VLPXODWHG XVLQJ D - HODVWRSODVWLF FRQstitutive relation. The material failure mode corresponds to a shear band formation, which is typical for this type of constitutive response. ,Q ERWK H[DPSOHV LW FDQ EH REVHUYHG WKH GLVWULEXWLRQ RI WKH YHFWRU ÀHOG T through the complete loading process, in every body point, even in regions which are far away from the fracture process or shear band zones. 7KLV PHWKRGRORJ\ LV HDVLO\ H[WHQVLEOH WR ' SUREOHPV DV VKRZQ LQ 2OLYHU HW DO  

134

A. Huespe and J. Oliver



)DFWRUV ,QÁXHQFLQJ WKH 6WDELOLW\ DQG $FFXUDF\ RI WKH 1XPerical Method

9HU\ RIWHQ GXH WR D GHÀFLHQW QXPHULFDO VWUDWHJ\ WKH QXPHULcal procedures for solving fracture problems provide non-physical responses. Spurious solutions are intimately related with algorithms and discrete formulations which are unstable. Furthermore, this type of algorithms is an important source of numerical troubles for the evaluation of solutions. Concrete fracture problem

crack path

T(x) vector field

iso-level curves of q field

J2 - plasticity

T(x) vector field

iso-level curves of q field

shear-band solution

Figure 13. The pseudo steady-state heat conduction problem. There are a number of algorithmic issues that have a strong inÁXHQFH RQ WKH numerical stability. Two of them are: i) the FE technology and ii) the numerical time-integration scheme of the constitutive material model: i) WKH FRPSDULVRQ SUHVHQWHG LQ 6HFWLRQ  VKRZV WKURXJK D VLPSOe example, that the non-symmetric FE formulation is more accurate than the symmetULF RQH ,W LV PDLQO\ GXH WR WKH IDFW WKDW WKH QRQV\PPHWULF DSproach, in linear triangular or tetrahedral FE, imposes exactly (in stURQJ IRUP WKH WUDF tion continuity condition on the discontinuity line, while, in the symmetric DSSURDFK LW LV ZHDNO\ LPSRVHG +RZHYHU LI FRQVLGHULQJ WKH numerical stability property, the second approach is more stable than the non-symmetric approach. The reason of this behavior has been explained in the papers of -LUDVHN  DQG 2OLYHU HW DO   ii) the strain softening of the constitutive relation is the responsible for the appearance of negative eigenvalues in the material constitutive modulus DT .

Crack Models with Embedded Discontinuities

135

,Q WKHVH FDVHV KRZHYHU WKH DOJRULWKPLF FRQVWLWXWLYH PRGXlus Dalg does not necessarily will display negative eigenvalues (observe the difference be˙ tween the tensors DT and Dalg  DT is the tensor that associates σ˙ with ε while, Dalg LV GHÀQHG DV Dalg = ∂εn+1 σn+1 , where σn+1 is the stress value at time n + 1 obtained by an time-integration scheme as a function of the strains εn+1 . The stability of the numerical approach depends on Dalg . For example, the use of classical implicit time-integration algorithms %DFNZDUG(XOHU VFKHPH IRU HYDOXDWLQJ FRQVWLWXWLYH PRGHls equipped with strain softening, leads to accurate results, even for large time step increPHQWV +RZHYHU LW DOVR GHWHUPLQHV DOJRULWKPLF FRQVWLWXWLve tangent tensors, Dalg , with negative eigenvalues provoking ill-conditioned stiffness matrices in the associated discrete problem. $OWHUQDWLYHO\ LI WKH SRVLWLYH GHÀQLWH FKDUDFWHU RI WKH DOJorithmic constitutive modulus Dalg is ensured by the time-integration scheme at any point of the body, this source of numerical instability would be removed. This observation motivates the implicit-explicit integration procedure presented in the following. The IMPL-EX (Implicit–Explicit) Time-Integration Scheme. /H XV UHFDOO WKH GDPDJH PRGHO GHSLFWHG LQ 7DEOH  ,Q WKH SUHVHQW VHFWLRQ ZH Vhow a time integration scheme such that, given the strain εn+1 in the time step n + 1, determines the stresses σn+1 and the internal variables rn+1 and qn+1 . This scheme, called ,03/(; LQWHJUDWLRQ DOJRULWKP 2OLYHU HW DO   GHWHUPLQHV D SRVLWLYH GHÀ nite algorithmic tangent tensor, and therefore, displays a more robust behavior for solving problems where the material constitutive relation is equipped with strain softening. 7KH ,03/(; LQWHJUDWLRQ DOJRULWKP LV EDVHG RQ WKH IROORZLQJ two stages performed in the time step (n + 1  ˜ n+1 , and the stressi) ,Q D ÀUVW VWDJH DQ H[SOLFLW HYDOXDWLRQ RI WKH VWUHVVHV σ like internal variable, q˜n+1 , is performed in terms of the implicit values at the previous time step n and extrapolated values of the strain-like internal variable, r˜n+1  'HWDLOV RI WKH RSHUDWLRQV SHUIRUPHG LQ WKLV VWDJH DUH JLYHQ in next subsection. i) ,Q D VHFRQG VWDJH D VWDQGDUG LPSOLFLW %DFNZDUG(XOHU LQWHJration of the constitutive model is performed and the implicitly integrated stresses, σn+1 , are REWDLQHG 'HWDLOV RI WKH HYDOXDWLRQV SHUIRUPHG LQ WKLV VWDJe for the damage PRGHO DUH JLYHQ LQ 7DEOH  Remark: LQ DGGLWLRQ IXOÀOOPHQW RI WKH PRPHQWXP EDODQFH HTXDWLRQ Hquations ˜ n+1 . ²  DW WLPH VWHS n + 1 LV LPSRVHG LQ WHUPV RI WKH ,03/(; VWUHVVHV σ

136

A. Huespe and J. Oliver

Table 5. ,PSOLFLW EDFNZDUG(XOHU LQWHJUDWLRQ VFKHPH IRU WKH LVRWURpic continuum damage model. DATA: given εn+1 , rn , qn Compute effective stresses and trial values:  ¯ n+1 ¯ n+1 : (E)−1 : σ ; τn+1 = σ

¯ n+1 = E : εn+1 σ r trial = rn

q trial = qn



Compute loading-unloading condition: IF τn+1 < r trial

THEN

Elastic unloading: rn+1 = r trial σn+1 =

;

qn+1 = q trial

;

qn+1 = qn + H(rn+1 − rn )

qn+1 ¯ σ rn+1 n+1

ELSE Damage evolution: rn+1 = τn+1 σn+1 =

qn+1 ¯ σ rn+1 n+1

ENDIF Compute algorithmic modulus (used for determining the material bifurcation condition): Dalg =

Unloading condition: Loading condition:

Dalg =

qn+1 E rn+1

qn+1 E rn+1

−

qn+1 −Hrn+1 ¯ n+1 (σ (rn+1 )3

¯ n+1 ) ⊗σ

The explicit stage. For the isotropic damage model, the explicit stage is summaUL]HG LQ 7DEOH  /HW XV FRQVLGHU WKH VWUDLQOLNH LQWHUQDO YDULDEOH ÁRZ  r ˙ GHÀQLQJ the evolution of the plastic damage, as shown in Table 1. Also, consider that at the beginning of the time step computations, n + 1, the implicit integration results of r, at the previous time steps, (n, n − 1, ... DUH DYDLODEOH 7KXV WKH 7D\ORU VHULHV expansion of r in the time reads: rn+1 = rn +

Δtn+1 (rn − rn−1 ) + O(Δt2n+1 ) Δtn



n−1 ) where we have approximated r| ˙ tn by the expression r(t ˙ n ) ≈ (rn −r , being Δtn Δtn = tn − tn−1 ; Δtn+1 = tn+1 − tn the time step increment at the steps n and n + 1, respectively. &RQVLGHULQJ WKH WUXQFDWLRQ RI H[SDQVLRQ  WR WKH ÀUVW RUder term, it provides WKH VHFRQG IRUPXOD VWHS  LQ 7DEOH  6WHSV  DQG  LQ 7DEOH  DUH REWDLQHG LQ WHUPV RI WKH H[WUDSROated value of r: r˜n+1  \LHOGLQJ WKH ,03/(; LQWHJUDWHG YDOXHV RI WKH UHPDLQLQJ YDriables q˜n+1 ˜ n+1 . and σ

Crack Models with Embedded Discontinuities

137

Table 6. ([SOLFLW H[WUDSRODWLRQ ÀUVW VWDJH RI WKH ,PSOH[ DOJRULWKP  LVRWURSLF FRQ tinuum damage model. DATA: given εn+1 , and the implicit values of the previous time step: (rn , qn , rn−1 , qn−1 ) Compute explicit extrapolation: r˜n+1 = rn +

Δtn+1 (rn Δtn

− rn−1 )

step 1

Update internal and damage variables: VWHS 

q˜n+1 = qn + HΔ¯ rn ˜ n+1 = σ

q˜n+1 (E r ˜n+1

: εn+1 )

VWHS 

Compute the Impl-Ex algorithmic tangent operator: ˜ alg = D

q˜n+1 E r ˜n+1

r ~

rn +1

;

q˜n+1 r ˜n+1

VWHS 

>0

implicit explicit

O (D tn2+1 )

rn rn -1

t n -1

tn

tn +1

t

Figure 14. ,03/(; LQWHJUDWLRQ DOJRULWKP

˜ n+1 DUH WKHQ 7KH YDOXHV REWDLQHG ZLWK WKH ,03/(; VFKHPH ˜ rn+1 , q˜n+1 , σ VXEVWLWXWHG LQ HTXDWLRQV ² WR IXOÀOO WKH PRPHQWXP EDOance equation. Remark: as the equilibrium equations, at step n + 1 DUH HYDOXDWHG ZLWK WKH ,03/ ˜ n+1 , the structural stiffness matrix, which is necessary to compute (; VWUHVVHV σ WKH 1HZWRQ5DSKVRQ LWHUDWLRQV IRU WKH FRQYHUJHQFH RI WKH UHsidual forces, must be ˜ alg . ˜ n+1 = D HYDOXDWHG ZLWK WKH ,03/(; FRQVWLWXWLYH WDQJHQW WHQVRU ∂εn+1 σ Remark: unlike in the standard implicit integration, the values r˜n+1 and q˜n+1 are independent of the current value of the strains, εn+1 , and therefore, they are known at the beginning of time step n + 1 ,W PHDQV WKDW r˜n+1 and q˜n+1 remains constant during the current time step. This furnishes relevant properties to the constitutive ˜ alg : tangent tensor D i) ,Q DOO LW LV V\PPHWULF DQG SRVLWLYH GHÀQLWH 7KH DUJXPHQWV Dre trivial for the LVRWURSLF GDPDJH PRGHO $V VKRZQ LQ VWHS  RI 7DEOH  E is symmetric and

138

A. Huespe and J. Oliver

SRVLWLYH GHÀQLWH ZKLOH WKH IDFWRU qr˜˜n+1 is positive. n+1 ii) For the present damage model, the algorithmic tangent operator is constant (independent of the current strain, εn+1  Property (i) could be easily extended to other constitutive models equipped ZLWK VWUDLQ VRIWHQLQJ VXFK DV SODVWLFLW\ VHH IRU H[DPSOH 2OLYHU HW DO   $OVR property (ii) FDQ EH YHULÀHG LQ PRUH JHQHUDO FDVHV +RZHYHU LW LV QRWHG WKDt, ZKLOH WKH ÀUVW SURSHUW\ LV DQ LPSRUWDQW UHTXLUHPHQW WR UHDFK numerical stability, the second one is not. Stable integration schemes could be developed without preserving the property (ii). Time-Integration Accuracy Assessment of the IMPL-EX Method. 7KH 'RX EOH &DQWLOHYHU %HDP '&% WHVW RI )LJXUH D ZLWK PDWHULDO parameters shown LQ WKH VDPH )LJXUH LV VROYHG LQ RUGHU WR HYDOXDWH WKH LQÁXHQFH RI WKH ,03/(; LQ cremental time step length Δs on the accuracy of the obtained numerical solution. :H DQDO\]H WKH VROXWLRQ FRQYHUJHQFH LQ )LJXUH E WKURXJK Gifferent equilibrium solutions when Δs → 0 (Gf = 50N/m  $OVR )LJXUH F VKRZV WKH convergence rate of computed fracture energy, Gf , with Δs. 1.00

Error (Gf) 1

P

δ

0.3 L

L = 0.4 m Thickness = 0.4 m

P

0.25 L

2.0 P [kN]

8D s 4Ds

0.01 2

1.5

f t = 3.0 MPa 1.0 E = 36.5 MPa 0.5 n = 0.18 Gf = 50 N/m (a)

1.6

0.10

Ds 5

10

(c) 2 Ds

Ds

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Displacement δ (b)

Figure 15. $FFXUDF\ DVVHVVPHQW RI WKH ,03/(; PHWKRG '&% WHVW D *H RPHWU\ DQG ÀQLWH HOHPHQW PHVK E /RDG 3 YV YHUWLFDO GLVSODcement δ curves using different integration-time steps that are proportional to Δs F (UURU RI WKH computed Gf as a function of the integration-time step length.

5 Applications of the CSDA Methodology to Concrete Fracture Problems ,Q WKLV 6HFWLRQ WKUHH DSSOLFDWLRQV RI WKH &6'$ PHWKRGRORJ\ for solving typical concrete fracture problems are shown. They are: a) the fracture process zone

Crack Models with Embedded Discontinuities

139

analysis of two-well known concrete structure tests (SubseFWLRQ   b) the capture of the size effect observed in fracture of concrete structurHV 6XEVHFWLRQ   DQG c) WKH VLPXODWLRQ RI D G\QDPLF IUDFWXUH SUREOHP 6XEVHFWLRQ   Emphasis is given to show the capability of the computational tool for analyzing such problems, and not to the physical aspects involved in every case. Thus, detailed explanations of the size effect phenomena or the phenomenology involved in dynamics fracture mechanics are not provided here, The interested reader is adGUHVVHG WR WKH VSHFLÀF ELEOLRJUDSK\ LQ WKH UHIHUHQFHG ZRUNs, for additional details. 5.1

Modeling the Fracture Process Zone

$ VDOLHQW IHDWXUH RI WKH &6'$ IRU PRGHOLQJ FUDFN SUREOHPV LV What it can simulate the complete fracturing process phenomena, from the continuum to the disFUHWH RQHV XVLQJ D XQLÀHG PHWKRGRORJ\ ,Q WKLV VHQVH LQHODstic deformations with no macro crack formation in the material bulk, a typical phenomenon welldescribed by continuum models, can be represented using the same methodological framework as that utilized in the simulation of macro-crack propagation processes. This feature is remarked in this section: an accurate description of the FPZ in two standard problems is shown. As it was explained in Section 1, during the structural loading process in fracture problems, a material point subjected to cracking displays successive stages of material behavior that can be idealized as follows: i) LQLWLDOO\ WKH PDWHULDO VKRZV DQ HODVWLF UHVSRQVH ii) at the time td , the material starts displaying an inelastic response, that, in the particular case of concrete fracture problems, could be described by a GDPDJH PRGHO iii) in a subsequent time, tloc , the singularity of the localization tensor, Qtg , GHÀQHG LQ WHUPV RI WKH WDQJHQW FRQVWLWXWLYH RSHUDWRU DT as: Qtg = n · DT · n

;

DT =

∂σ ∂ε



signals the onset of material instability, strain localization, and eventually, WKH GHYHORSPHQW RI GLVFRQWLQXLWLHV LQ WKH GLVSODFHPHQW ÀHOd that characterizes the macro-crack formation. For a given material model, the delay between td and tloc depends on the material properties and the stress state. For a given stress state, the interval [td , tloc ] EHFRPHV ODUJHU ZLWK ODUJHU OHVV QHJDWLYH VRIWHQLQJ PRGXOus, H. Therefore, from a phenomenological approach to crack modeling, one can take advantage of this fact in order to consider the amount of volumetric energy dissipation in front of the FUDFN WLS DV DOVR WKH VL]H DQG LQÁXHQFH RI WKH IUDFWXUH SURFess zone. The involved SKHQRPHQRORJ\ WKDW FDQ EH VLPXODWHG ZLWK WKH &6'$ LV VNHWFKHG LQ )LJXUH  WKH initial value of the continuum softening modulus, H VHH )LJXUH D  UXOHV WKH

140

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stable damage production at the considered point. Material points in this stage, in front of the crack tip, are displayed in the gray zone in FigurH E EHWZHHQ SRLQWV D VWDQGLQJ IRU RQVHW RI GDPDJH DQG L VWDQGLQJ IRU RQVHW RI ORFDOL]DWLRQ  7KH length d of this stable damaged zone is governed by the value H as said before. As, at time tloc , material instability is detected, the continuum softening modulus is no longer determined as a material property but regularized from the intrin¯ LQ WXUQ GHSHQGLQJ RQ WKH IUDFWXUH HQHUJ\ DFFRUGLQJ WR sic softening modulus H HTXDWLRQ  LQ 7DEOH  VHH )LJXUH D $W WKH VDPH WLPH WKe strong discontiQXLW\ NLQHPDWLFV HTXDWLRQV ²  LV DFWLYDWHG WKLV HQVXULQJ IXOÀOOPHQW RI WKH degenerated traction/separation law and the subsequent correct dissipation of the fracture energy Gf up to the complete stress release. Points in this stage are sigQDOHG LQ )LJXUH E DV WKH FRKHVLYH ]RQH EHWZHHQ SRLQWV L (standing for onset RI ORFDOL]DWLRQ DQG R VWDQGLQJ IRU VWUHVVIUHH ]RQH OLPLW  q qd

lFPZ

H q loc kH rd

rloc

ld lloc

D stable damage

L

cohesive zone

R (a)

stress-free zone

r (b)

Figure 16. The fracture process zone. Both, the stable damage and the cohesive zone, constitute the generalization, LQ D &6'$ VHWWLQJ RI WKH IUDFWXUH SURFHVV ]RQH )3=  FODVVLcally considered in QRQOLQHDU IUDFWXUH PHFKDQLFV %D]DQW DQG 3ODQDV   Ds the locus of material points where the fracture is processed by dissipative mechanisms. As it will be shown in the following, the total size of this fracture process zone, FPZ, basically depends on the amount of fracture energy Gf , and the length of the stable damage zone, d DQG WKH FRUUHVSRQGLQJ GLVVLSDWLRQ LV UXOHG E\ WKH FRQWLQXum softening modulus H. Therefore, the size of the cohesive zone, loc , can be expressed as: loc (Gf , H) = F P Z (Gf ) − d (H) with the conditions:

dF P Z ≥ 0; dGf



dd ≤0 d |H|

5.1.1 The Three-Point Bending Test. The crack propagation problem in the notched concrete beam supported on three points, shown in FiJXUH D LV XVHG

Crack Models with Embedded Discontinuities

141

to assess several aspects of the fracture process zone analy]HG ZLWK WKH &6'$ methodology, additional details of this analysis have been UHSRUWHG LQ LQ +XHVSH HW DO D  The continuum constitutive relation adopted for this problem, is the damage PRGHO SUHVHQWHG LQ 7DEOH  )URP HTXDWLRQ  LQ WKLV 7DEOH note that damage evolution is inhibited in compressive stress states. The daPDJH PRGHO LV VSHFLÀHG for the plane stress case. a) Cohesive Force Induced by the Continuum Damage Model. )LJXUHV  c and d show the evolution, in terms of the pseudo-time uy (the monotonically increasing vertical displacement of the load application pRLQW  RI WKH QRUPDO DQG tangential components of the traction vector (tS = σS · n DW IRXU GLIIHUHQW SRLQWV RI WKH FUDFN SDWK SRLQWV $ % & DQG ' DW D GLVWDQFH ζ from the notch tip, Figure E  7KHUH LW FDQ EH REVHUYHG WKDW ERWK FRPSRQHQWV RI tS are continuous along the time, irrespective of the transition between different fracture process stages. z

y

uy

E=20000[MPa] n=0.2 f t =2.4[MPa] Gf =113N/m

20mm

240mm

P

450.mm

z= 34mm

Point D Point C Point B Point A

z= 22 z= 0

z= 12 z= 3

(b)

(a)

x Traction normal component t n [MPa]

0.1

3

Traction shear component t s [MPa] Activation of cracks

point A

2.5

0

point B

2

point C

1.5

point B

-0.1

point D

point D

1

-0.2

Activation of cracks

0.5

point C

-0.3

point A

0 0

0.05

0.1

0.15

0.2

0.25

(c)

uy [mm]

0

0.05

0.1

0.15 uy [mm]

0.2

0.25

(d)

Figure 17. The Three-Point Bending Test. b) Study of the Fracture Process Zone Characteristic Lengths. /HW XV IRFXV RQ the role played by the initial continuum softening modulus, H, the softening rule LQ )LJXUH D DQG LWV LQÁXHQFH RQ WKH IUDFWXUH SURFHVV ]RQH structure. &XUYHV LQ )LJXUH D VKRZ WKH GHSHQGHQFH RI WKH IUDFWXUH SURcess zone length, F P Z , on the fracture energy, Gf , as a function of the material characteristic EG length parameter (ch = f 2f FKDUDFWHUL]LQJ WKH PDWHULDO EULWWOHQHVV VHH (OLFHV t

142

A. Huespe and J. Oliver

HW DO   ,Q WKH ÀJXUH LW FDQ EH REVHUYHG WKDW F P Z depends clearly on Gf but it barely changes with substantial changes of H 1RWH WKDW WKH VROXWLRQ FDOOHG “with elastic bulk” means that H is a very large negative parameter, such that, strain localization conditions are immediately detected at onset of the inelastic deformation. Therefore, no stable damage is induced in the FPZ. Thus, the experiment assesses the dependence of the total FPZ length with the fracture energy, Gf  ,Q )LJXUH E LW LV VKRZQ WKH GHSHQGHQFH RI WKH VWDEOH damage zone size, d , with the H and Gf values. Results are parameterized in terms of the normalized length, χ, (χ = −Hch  3ORWV RI FXUYHV d (χ), for different values of ch (Gf ) DUH WKHQ SUHVHQWHG ,Q DJUHHPHQW WR WKH DVVHUWLRQ LQ %D]DQW   DERXW WKH FKDQJLQJ OHQJWK RI WKH PLFUR FUDFNLng zone in front of the crack tip, the numerical simulations show a similar behavior. Therefore, all lengths in Figure 18-b have been measured at the time that stresses are completely UHOHDVHG DW WKH QRWFK WLS FRPSOHWH VWUHVV UHOHDVH  0.20

Response with elastic bulk

Response with elastic bulk

0.16

ld lloc

lFPZ

l FPZ(G f )[m]

Response with damaged bulk

0.12

z

0.08 Range of parameters for concrete

0.04 0.00

0.04

0

ld (H)[m]

0.2

0.4

lch =EG f /(s u) 2 [m]

0.6

H = -c / l ch

lch =0.6 lch =0.4 l ch=0.2 lch =0.1

0.01

(c)

lch =0.1m c=0.002m ld =0.008m

0.03

0.02

(a)

ld lFPZ

lloc

(d) (e)

l ch=0.6m

c=0.001m

ld =0.030m

0 0

1

2

3

c x10 -3 [m]

4

(b)

Figure 18. The Three-Point Bending Test, fracture process zone analysis. ,W FDQ EH REVHUYHG WKDW IRU D JLYHQ YDOXH RI Gf , increasing values of |H|, diminish the stable damage zone size d (the lower bound is, approximately, the VL]H RI RQH ÀQLWH HOHPHQW 0.004²0.007 P  7\SLFDO GHYHORSPHQWV RI WKH )3= IRU different material parameter values, are displayed in FiguUH FH ÀQLWH HOHPHQWV XQGHUJRLQJ GLVVLSDWLRQ DUH SORWWHG LQ OLJKW JUD\ FRORU  7Kere, the stable damage ]RQH URXQG VKDSHG LQ IURQW RI WKH FUDFN WLS FDQ EH FOHDUO\ GLfferentiated from the

Crack Models with Embedded Discontinuities

143

remaining domains of the FPZ. 5.1.2 Double Cantilever Beam (DCB Test). ,Q WKLV VXEVHFWLRQ ZH VKRZ DGGL tional details of the fracture process zone through the classical double cantilever EHDP WHVW DQDO\VLV '&% 7HVW RI )LJXUH D a)Predicted Peak Load Values. This test emphasizes the effects that the stable dissipation zone, as a function of the continuum softening modulus H, induces on the predicted peak loads. 7KH SUREOHP LV VROYHG E\ XVLQJ WZR GLIIHUHQW SURFHGXUHV ,Q 3rocedure A, we impose an elastic response of the bulk material, like in standard cohesive models. ,W LV GHÀQHG D YHU\ ODUJH QHJDWLYH YDOXH RI WKH LQLWLDO VRIWening modulus, H (the VWDEOH GLVVLSDWLRQ ]RQH GRHV QRW H[LVW  ,Q 3URFHGXUH % EXOk dissipation is admitted by specifying a softening modulus H ≈ 0 ODUJH VWDEOH GLVVLSDWLRQ ]RQH  ,Q both cases the same value for the fracture energy Gf LV FRQVLGHUHG )LJXUH E GLVSOD\V WKH FRUUHVSRQGLQJ IRUFHGLVSODFHPHQW &02' FXUYHV ,W FDQ EH FKHFNHG that allowing dissipation on the bulk has a non-negligible effect on the predicted peak load. 80.

P 0.2m

u

z

0.12m

P[kN] Procedure A

E = 36500[MPa] 60. f t = 3[MPa] n = 0.18 Gf = 50N/m 40.

t1

Procedure B

t2 t3

0.4m 20.

(a) (b) 4

Traction vector component tn [MPa]

Procedure A

ft

0.1

4

Traction vector component tn [MPa]

t1

t2

t3

t1

2

1

1

0

0

0.2

0.3

0.4

u[mm]

Procedure B

ft

3

3 2

0 0

t2

t3

-1

-1

(c)

-2 0

0.1

0.2

z[m]

(d)

-2 0

0.1

0.2

z[m]

Figure 19. 7KH '&% 7HVW IUDFWXUH SURFHVV ]RQH DQDO\VLV )LJXUH F SORWV WKH GLVWULEXWLRQ RI WKH FUDFN QRUPDO VWUHVs component, tn ,

144

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DORQJ WKH FUDFN SDWK VWUHVV SURÀOH  DW WKUHH GLIIHUHQW WLPes, t1 , t2 and t3 , specLÀHG LQ WKH HTXLOLEULXP FXUYHV RI )LJXUH E 7KRVH FXUYHV Fomputed with the 3URFHGXUH $ VWDQGDUG FRKHVLYH PRGHO VKRZ SHDN YDOXHV JUHDter than the material tensile strength (ft = 3M P a  7KLV LQFRQVLVWHQF\ KDV DOVR EHHQ GHWHFWHG E\ RWKHU authors using a similar procedure, where it is claimed that this unwelcome effect GLVDSSHDUV RU GLPLQLVKHV ZLWK PHVK UHÀQLQJ 5HVXOWV REWDined with Procedure % EXON GLVVLSDWLRQ LV DOORZHG VKRZ RQ WKH FRQWUDU\ FRQVLVWHQW VWUHVV SURÀOHV They are not spike-shaped, as in the previous case, and the maximum stress level remains always below ft , even for the relatively coarse mesh used in the analysis. b) Study of the Fracture Process Zone Characteristic Lengths. 8VLQJ D VLPLODU WHFKQLTXH RI WKDW SUHVHQWHG LQ VXEVHFWLRQ  ZH HYDOXDWH Whe sensitivity of the FPZ length with the fracture energy Gf parameter. Thus, we determine the size F P Z for three different values of Gf : (= 25, 50 and 75N/m  7KH UHVXOWV DUH compared with the material characteristic length ch = EGf /ft2 LQ )LJXUH  IRU ERWK WHVWV WKH WKUHHSRLQW EHQGLQJ WHVW RI VHFWLRQ  DQG WKH SUHVHQW '&% WHVW lFPZ (Gf ) [ m ]

Range of concrete Three-point Beam Bending test

l

EGf f t

Figure 20. &RPSDULVRQ RI WKH IUDFWXUH SURFHVV ]RQH OHQJWKV YV WKH PDWHrial charDFWHULVWLF OHQJWK IRU ERWK WHVWV 7KUHHSRLQW EHQGLQJ DQG '&% WHVW

5.2

Size Effect Analysis

7KH LQÁXHQFH RI WKH VWUXFWXUH VL]H RQ WKH QRPLQDO VWUHQJWK Zhen proportional RU JHRPHWULFDOO\ VLPLODU VWUXFWXUHV DUH FRPSDUHG LV FDOled the size effect, see %D]DQW DQG 3ODQDV   7KH QRPLQDO VWUHQJWK LV WKH UDWLR Eetween the ultimate structural load and a structure representative area. 7KH VL]H HIIHFW LV RQO\ VLJQLÀFDQW LQ WKRVH VSHFLPHQV ZLWK D Fharacteristic size not being so small, if compared with FPZ size. There are several laws describing

Crack Models with Embedded Discontinuities

145

this effect in concrete structures subjected to fracture problems. Some of them DUH EDVHG RQ 1RQ/LQHDU )UDFWXUH 0HFKDQLFV SULQFLSOHV ZKLOe others are empirical ODZV +RZHYHU WKH\ DUH QRW JHQHUDO ODZV LQ WKH VHQVH WKDW WKHy cannot be applied to a wide range of geometries or loading conditions. &RPSXWDWLRQDO )DLOXUH 0HFKDQLFV DQG SDUWLFXODUO\ WKH &6'A approach, is an alternative tool for evaluating the size effect phenomenon in structural analysis. We show in the following examples an application to this problem extracted from %ODQFR   5.2.1 The Brazilian Test. The Brazilian test has been included in several building standards and design codes as a valid procedure to determine the tensile strength of quasi-brittle materials. 2QH YHUVLRQ RI WKLV WHVW FRUUHVSRQGV WR D F\OLQGULFDO VSHFLPen that is compressed by two diametrically opposed loads, as it is shown in )LJXUH D 7KH tensile stresses produced by these loads induce the formation of a vertical planar FUDFN DORQJ WKH SODQH GHÀQHG E\ WKH RSSRVLWH ORDGV ZKLFK HYentually, produces the splitting of the specimen. 7KH PDLQ VSHFLPHQ GLPHQVLRQV DUH VSHFLÀHG LQ )LJXUH D 7Ke test provides the splitting tensile strength value, fst , given by: fst =

2Pu πBD



where B and D are specimen dimensions and Pu the ultimate load. The value fst decreases with larger specimen sizes and with smaller bearing strip sizes b. +RZHYHU fst tends, asymptotically, to the material tensile strength. We have reproduced the fst values obtained with different specimen sizes. The experimental values of these test were reported by Rocco HW DO   ZKR have analyzed granite cylindrical specimens with the following dimensions: B = 30mm and diameters: D = 30, 60, 120 and 240mm )LJXUH E SUHVHQWV WKH material parameters adopted for the numerical model. )LJXUH F GLVSOD\V VHYHUDO H[SHULPHQWDO UHVXOWV WDNHQ Irom Rocco et al.   IRU WKH FDVH b/D = 0.16 . The plots correspond to the structural response: load P vs. diametric opening wd . Three characteristic points are displayed in each curve: Point 1 is the maximum load coinciding with the vertical diametric crack onset. The crack formation causes a marked structural load carrying capacity loss. 7KH 3RLQW  LV WKH PLQLPXP YDOXH DWWDLQHG DIWHU  $IWHU WKH SRLQW  VHYHUDO FUDFNV show up below the load-bearing strips and evolve until reaching a second maxiPXP LQ WKH 3RLQW  The performed nine numerical tests were run until reaching tKH SRLQW  7KXV the formation of the second pattern of cracks, below the load-bearing strips, were QRW FDSWXUHG )LJXUH G GLVSOD\V WKH ÀQLWH HOHPHQW PHVK IRU FDVHV   DQG 

146

A. Huespe and J. Oliver t

ft = 10.1 MPa E =33.9 GPa Gf =167N/m w1 = 19mm

ft

Gf

Wd

b

w1

(a)

(b)

(c)

P/Pmax1

1.2 b/D=0.16

1

D=302

0.8

numeric

2 experimental

D=60 2

Case 1: D=240mm ; b/D=0.16

experimental 2 numeric

Case 7:

Case 4:

D=120mm ; b/D=0.04D=60mm ; b/D=0.0.04

(d)

D=120

0.4 D=240

(e)

2

2 experimental 2 numeric numeric

2 experimental

wd/wd1

0.0 0

(f)

1

2

3

4

5

Experimental Numeric

Figure 21. Size effect analysis: the Brazilian test.

WKDW ZH KDYH PRGHOHG DQG )LJXUH HI WKH QXPHULFDO VROXWLons obtained with the &6'$ PHWKRGRORJ\ WKDW DUH FRPSDUHG LQ DOO FDVHV ZLWK WKH H[perimental results. We remark the capacity of the numerical model for capturing the effect observed in the experimental test: the larger is the specimen the greater is the brittleness. 5.3

Dynamic Fracture Simulation

$Q DSSOLFDWLRQ RI WKH &6'$ DSSURDFK WR VLPXODWH G\QDPLFV IUDcture problems, UHSRUWHG LQ +XHVSH HW DO E  LV KHUH SUHVHQWHG ,W FDSWures the most intriguing aspects of the problem.

Crack Models with Embedded Discontinuities lch= (Gf E)/ f t =68.mm 2

u=3000m/sec

(a)

u=3000m/sec

f t =129MPa E =3.24GPa n =0.35 Gf =0.352N/m r =1.19g/cm3

Crack tip velocity [m/sec]

3.mm

1000

0.25mm

147

Mesh 1 * Mesh 2 Mesh 3

800

t=5.6msec

600 400

Falk et al.

200 0

Potential law, from Sharoin et al.

Begining of branching (mesh 3)

4

5

(b)

6 7 8 Time t [x1e-6 sec]

9

(c)

1.mm t=5.6msec

(d)

t=6.5msec

t=5.6msec t=6.5msec

Mesh 1

t=5.6msec

(e)

(c)

t=6.3msec

t=5.6msec

t=6.3msec

Mesh 2

t=5.6msec

t=5.6msec

(f)

Mesh 3

Figure 22. 3UHGLFWLRQ RIQ G\QDPLF IUDFWXUH D VSHFLPHQ JHRPHWU\ E FUDFN SURSDJDWLRQ YHORFLW\ G FUDFN SDWWHUQ PRUSKRORJ\ GHI &UDFN SDWKV IRU 0HVK  IRU 0HVK  DQG IRU 0HVK 

Numerical Methodology. The FE technology with embedded strong discontinuities which does not need enforcing the crack path continuity, reported by Sancho HW DO  DQG H[SODLQHG LQ 6HFWLRQ  LV DGRSWHG $V LW LV Pentioned in that Section, an important advantage of this procedure is that it does not need a crack path tracking algorithm. Therefore, this technique results more convenient in order to capture possible crack bifurcations, which are common in dynamic fracture problems. Additional details of the implementation aspects, that were specially addressed WR FDSWXUH FUDFN EUDQFKLQJ FRXOG EH IRXQG LQ +XHVSH HW DO E  a) Prediction of Dynamic Fracture in PMMA. ,W LV DQDO\]HG WKH FUDFN SURSD gation problem in a square specimen with 3.mm sides and a notch 0.25mm long VHH )LJXUH D  ,Q WKH DQDO\VLV LW LV LPSRVHG D YHUWLFDO Fonstant velocity of

148

A. Huespe and J. Oliver

3000.m/sec on the top and bottom edges. The material is characterized with parameters depicted in the same Figure. With these values, the dilatational wave speed is: cd = 2090m/sec, the shear wave speed: cs = 1004.m/sec, and the Rayleigh wave speed: cR = 938, m/sec. A numerical simulation, using cohesive LQWHUIDFH ÀQLWH HOHPHQWV ZDV SUHVHQWHG E\ )DON HW DO   a.1) Crack Tip Velocity. ,Q )LJXUH E LW LV SORWWHG WKH FUDFN WLS YHORFLW\ ZKLFK has been evaluated as the numerical derivative of the crack tip position as a funcWLRQ RI WLPH $OVR LQ WKLV ÀJXUH LW LV SUHVHQWHG WKH UHVXOWV Ueported by Falk et al. where two different implementations of a discrete cohesive model have been used. a.2) Crack Pattern Morphology. Also, with this methodology, it is possible to FDSWXUH WKH FUDFN EUDQFKLQJ HIIHFW DV LW LV VKRZQ LQ )LJXUH GI :H GHSLFW WKH distribution of cracks at the end of the simulation process fRU WKH WKUHH PHVKHV  DQG   7KH EODFN ]RQH FRUUHVSRQGV WR DFWLYH HOHPHQWV RU FUDcks in opening mode, while the gray zone represents elements that previously have been activated but that at the end of the simulation process are arrested. 2WKHU FUDFN GLVWULEXWLRQ SDWWHUQ PRUSKRORJLFDO IHDWXUHV Zere reported in the literature, and it was concluded that the branches of the cracks are distributed IROORZLQJ D SRWHQWLDO ODZ OLNH WKDW SORWWHG LQ )LJXUH F ,t can be observed that &6'$ VLPXODWLRQ DJUHHV UHDVRQDEO\ ZHOO ZLWK WKLV H[SHULPHQtal fact.

6 A Model for Reinforced Concrete Fracture via CSDA and Mixture Theory 5HLQIRUFHG FRQFUHWH FRQVWLWXWHG E\ FRQFUHWH ZLWK ORQJ ÀEHUV UHLQIRUFHPHQWV RUL ented in different directions embedded in it, could be analyzed following two different conceptual models implying different length scales: i) a mesoscopic scale model describing the response of every composite material component (matrix DQG UHLQIRUFHPHQW DV DQ LQGHSHQGHQW VXEV\VWHP WKDW LV PHFKanically interacting with the neighbor components. The numerical simulation of mesoscopic models UHTXLUHV KLJK FRPSXWDWLRQDO FRVWV ii) a macroscopic scale model describing the response of the composite material via a homogenized constiWXWLYH PRGHO ,Q WKLV case, the success of the model relies on the homogenization procedure, which becomes the key issue in this type of conceptual approach. Following the second approach, it is possible to develop a rather simple hoPRJHQL]DWLRQ SURFHGXUH EDVHG RQ FRPELQLQJ &6'$ ZLWK PL[WXUe theory, which provides a computational model that captures the most salient phenomena governing the failure of reinforced concrete structures. An important feature of this model is that it requires a reduced computational effort. The resulting model was SUHVHQWHG LQ /LQHUR   ,Q WKH SUHVHQW 6HFWLRQ LW LV H[SODLQHG WKH )( PRGHO ZKLFK IROlows this proposal.

Crack Models with Embedded Discontinuities 6.1

149

A Mixture Theory for Reinforced Concrete

Reinforced concrete is assumed to be a composite material made of a matrix FRQFUHWH DQG ORQJ ÀEHUV VWHHO EDUV DUUDQJHG LQ GLIIHUHQt directions, as shown in )LJXUH  According to the basic hypothesis of mixture theory, the composite is a conWLQXXP LQ ZKLFK HDFK LQÀQLWHVLPDO YROXPH LV RFFXSLHG E\ DOO Whe constituents with volumetric fractions given by the factor ki ≤ 1 (for the iWK FRQVWLWXHQW  $VVXP ing a parallel layout, all constituents are subjected to the composite deformation ε. The composite stresses, σ, are obtained by adding the stresses of each constituent, weighted according to their corresponding volumetric participation. Thus, the matrix strains, εm , coincide with the composite strains, ε: εm = ε



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,Q RUGHU WR WDNH WKH GRZHO DFWLRQ LQWR FRQVLGHUDWLRQ WKH ÀEHr shear strains, γ, are obtained as the shear components of the composite strain ÀHOG ,Q D ORFDO ' orthogonal reference system (r, s  WKH ÀEHU VKHDU FRPSRQHQW εfrs is given by: εfrs =

f γrs =r·ε·s 2



The stresses of a composite with nf ÀEHUV RU ÀEHU EXQGOHV RULHQWHG LQ GLIIHU ent directions rf (f = 1, 2, .., nf FDQ EH REWDLQHG XVLQJ WKH IROORZLQJ ZHLJKWHG sum of each contribution:

σ = k m σ m (εm ) +

nf 

f f k f [σ f (εf )(r f ⊗ rf ) + 2τrs (γrs )(r f ⊗ sf )sym ] 

f =1

where km and k f DUH WKH PDWUL[ DQG WKH ÀEHU f volumetric fraction, respectively, f σ m is the matrix stress tensor, σ f LV WKH ÀEHU QRUPDO D[LDO VWUHVV DQG τrs is the ÀEHU VKHDU VWUHVV FRPSRQHQW )RU WKH VDNH RI VLPSOLFLW\ LQ HTXDWLRQ   LW LV DVVXPHG What the normal and WDQJHQWLDO VWUHVV FRPSRQHQWV RI WKH ÀEHUV DUH UHODWHG WR WKH corresponding strains E\ PHDQV RI VSHFLÀF FRQVWLWXWLYH HTXDWLRQV LQ D FRPSOHWHO\ Gecoupled behavior of the matrix response.

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S

n

W+

s y x x

matrix r

x

fiber r W-

discontinuity composite material

Figure 23. Reinforced concrete model.

The incremental form of the composite constitutive equation can be written as: σ˙ DT

=

˙ DT : ε;



=

k m Dm T +

+

4Gfrs (r f

nf 

k f [ETf (r f ⊗ r f ) ⊗ (r f ⊗ r f ) +

f =1

⊗ sf )sym ⊗ (r f ⊗ sf )sym ]

m m f  ETf = ∂σ f /∂εf and Gfrs = ∂τrs /∂(γ f /2) are where Dm T = ∂σ /∂ε the tangent operators for the involved constitutive relatiRQV RI FRQFUHWH DQG ÀEHU respectively.

6.2 Constitutive Model for the Composite: Regularization Procedure based on the CSDA &UDFN RQVHW DQG JURZWK LQ WKH FRPSRVLWH FDQ EH PRGHOHG LQ WKH Fontext of the &6'$ 7KHUHIRUH D VWURQJ GLVFRQWLQXLW\ NLQHPDWLFV OLNH What presented in Section  HTXDWLRQV ²  LV DVVXPHG a) Constitutive Model for the Concrete. The concrete shows very different reVSRQVHV HLWKHU LQ WHQVLOH RU FRPSUHVVLYH VWUHVV UHJLPHV ,Q each case, the most VWULNLQJ GLIIHUHQFH LV REVHUYHG LQ WKH IDLOXUH PRGH 8QGHU Wensile stresses, the concrete displays a much lower strength than in compressive states. Also, it shows a KLJKHU EULWWOHQHVV LQ WHQVLOH VWUHVV FRQGLWLRQV ,Q IDFW Iormation of cracks are expected only if tensile states are observed while in compressive states, the concrete behaves like a plastic material, sometimes displaying a failure mechanisms like shear bands.

Crack Models with Embedded Discontinuities

151

Additionally, in reinforced concrete structures, due to the reinforcement, the FRQFUHWH LV JHQHUDOO\ VXEMHFWHG WR KLJK FRQÀQHPHQW VWUHVV Uegimes, which plays a very important role in the structural strength, suggesting that this effect must be considered in the concrete model. Therefore, it is advisable to use a concrete constitutive relation having the ability to capture the phenomenology observed under both tensile and compressive stress conditions. This motivates the concrete model presented in S´anchez et al.   ZKLFK LV EDVHG RQ WKH GDPDJH FRQVWLWXWLYH UHODWLRQ depicted in Table 1 and UHJXODUL]HG YLD &6'$ IRU SRVLWLYH PHDQ VWUHVV VWDWHV DQG D plastic response for negative mean stress states. Additional details of this model can be found in the mentioned reference. b) Constitutive Model for the Steel Fibers. 6WHHO ÀEHUV DUH UHJDUGHG DV RQH dimensional elements embedded in the matrix. They can contribute to the composite mechanical behavior introducing axial or shear strength and stiffness. 7KH D[LDO FRQWULEXWLRQ RI HDFK ÀEHU EXQGOH GHSHQGV RQ LWV PHFhanical properWLHV DQG WKH PDWUL[ ÀEHU ERQGVOLS EHKDYLRU 7KH FRPELQDWLRn of both mechanisms LV PRGHOHG E\ WKH VOLSSLQJÀEHU PRGHO GHVFULEHG EHORZ ,Q WKis framework, the GRZHO DFWLRQ FDQ EH SURYLGHG E\ WKH ÀEHU VKHDU VWLIIQHVV FRQWribution in the crack zone. 6OLSSLQJÀEHU 0RGHO 7KH ÀEHU D[LDO FRQWULEXWLRQ FDQ EH PRGHOHG WKURXJK RQH dimensional constitutive relations, relating extensional strains with normal stresses. 7KH DVVXPHG FRPSDWLELOLW\ EHWZHHQ PDWUL[ DQG ÀEHU VWUDLQV Dllows capturing the VOLS HIIHFW GXH WR WKH ERQG GHJUDGDWLRQ E\ PHDQV RI D VSHFLÀF Vtrain component DVVRFLDWHG ZLWK WKH VOLS 7KXV WKH ÀEHU H[WHQVLRQDO VWUDLQ, εf , can be assumed as D FRPSRVLWLRQ RI WZR SDUWV RQH GXH WR ÀEHU PHFKDQLFDO GHIRUPation, εd , and the other related to the equivalent relaxation due to the bond-sOLS LQ WKH PDWUL[ÀEHU interface, εi : εf = εd + εi



$VVXPLQJ D VHULDO FRPSRVLWLRQ EHWZHHQ ÀEHU DQG LQWHUIDFH Ds illustrated in Figure  WKH QRUPDO VWUHVV RI WKH VOLSSLQJÀEHU PRGHO σ f , is equal to each component stress: σf = σd = σi



7KH VWUHVV DVVRFLDWHG WR WKH ÀEHU HORQJDWLRQ DV ZHOO DV WKH Rne associated with WKH PDWUL[ÀEHU VOLS HIIHFW FDQ EH UHODWHG WR WKH FRUUHVSRQding strain component by means of a uniaxial linearly elastic/perfectly plastic constitutive model

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sd ed sd syd

ei

ef

(d)

fiber mechanical elongation

+ ed

(a)

sf s yf

si i s adh E dfr

sf

E E

iir

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=

(i)

fiber matrix bond-slip

ei

E f fr E slipping-fiber

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(b)

Figure 24. 6OLSSLQJÀEHU PRGHO WKH FRPSRVLWLRQ ⊕ RI ERWK HOHPHQWV G DQG L must be understood as a serial mechanical system, in the sense that, deformations are additive (εf = εd + εi DQG VWUHVVHV DUH QRW σ f = σ d = σ i 

6.3

Representative numerical simulations

The capability of the model to predict reinforced concrete structure failure modes is shown through two typical problems which solutions are contrasted with H[SHULPHQWDO UHVXOWV 7KH ÀUVW RQH LV WKH VLPXODWLRQ RI D UHLnforced concrete corEHO D W\SLFDO ' WHVW DQG WKH VHFRQG RQH LV D UHLQIRUFHG FRQFrete beam assumed a plane strain case. a) Reinforced Concrete Corbel. 7KH FRQFUHWH FRUEHO RI )LJXUH  LV VLPXODWHG and results are contrasted with the experimental test published by Mehmel and )UHLWDJ  DQG WKH QXPHULFDO VROXWLRQ RI +DUWO   7he test description, geometry and reinforcing bar distribution, are supplied in )LJXUH DE VHH DOVR 0DQ]ROL HW DO  ZKHUH DGGLWLRQDO GHWDLOV RI WKH QXPHULcal simulation are presented. 6\PPHWU\ FRQGLWLRQV DUH DVVXPHG WKHUHIRUH RQH IRXUWK RI Whe structure is disFUHWL]HG 7KH ÀQLWH HOHPHQW PHVK LV VKRZQ LQ )LJXUH G (DFh rebar and the surrounding concrete were modeled as a composite material with equivalent mechanical properties. The corbel was then divided into sub-regions involving the VWHHO EDUV ,Q HDFK VXEUHJLRQ WKH ÀEHU GLUHFWLRQ DQG YROXPetric fraction corresponds to the embedded steel bars in the real problem. The assumed mechanical properties for the comonent materials are depicted in the Figure.

Crack Models with Embedded Discontinuities

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(b)

Concrete f t =2.26MPa E =21.9GPa n =0.2 Gf =100.N/m

Steel

(d)

s y =430MPa Ed =206.GPa

Bond s fy =300MPa Ef =E

(c)

Figure 25. 5HLQIRUFHG FRQFUHWH FRUEHO DE PRGHO GHVFULSWLRQ F Ooad vs. vertical displacement of the load application point curves, insert fracture crack SDWWHUQ VLPXODWHG OHIW DQG H[SHULPHQWDO ULJKW  G ÀQite element mesh.

)LJXUH F FRPSDUHV WKH VWUXFWXUDO DFWLRQUHVSRQVH FXUYHs (load P vs. verWLFDO GLVSODFHPHQW RI WKH ORDG DSSOLFDWLRQ SRLQW REWDLQHG numerically, with the proposed methodology, and the ones obtained using a model based on the smeared crack model with embedded representation of the rebar, presHQWHG E\ +DUWO   Both methodologies provide a reasonable prediction of the experimental ultimate ORDG FDSDFLW\ ,Q WKH LQVHUW RI )LJXUH F OHIW  LW LV VKRZn the simulated crack pattern that are represented by the iso-displacement contours at the end of the analysis. The predicted crack pattern is in good correspondence with that observed in the experimental test, as shown in the right half symmetric part of the same picture.

154

A. Huespe and J. Oliver

b) Application to Corroded Reinforcement Concrete Beams. The corrosion SKHQRPHQRQ REVHUYHG LQ 5HLQIRUFHG &RQFUHWH VWUXFWXUHV 5& VWUXFWXUHV LV D YHU\ important failure mode that limits the service life of these VWUXFWXUHV ,Q WKH SUHVHQW example, taken from S´DQFKH] HW DO   LW LV VKRZQ DQ DSSOLFDWLRQ RI WKH DERYH presented numerical approach to simulate the mechanical consequences, typically the structural load carrying capacity loss, due to the reinforcement corrosion. The mechanism by which the steel corrosion causes a loss of the structural strength is due to an expansion of the corroded rebar, which induces cracking in the concrete cover, loss of steel-concrete bond, as also net area reductiRQ RI WKH VWHHO ÀEHU 7KH effects of the above mentioned mechanisms on the structural load carrying capacity can be analyzed as a function of the reinforcement corrosion degree. Therefore, WKH PRGHO PDNHV SRVVLEOH WR GHWHUPLQH WKH LQÁXHQFH DQG VHQVLtivity of this key variable, the reinforcement corrosion level, in the structural deterioration problem. The proposed numerical strategy can be applied to beams, columns, slabs, etc., WKURXJK WZR VXFFHVVLYH DQG FRXSOHG ' PHVRVFRSLF PHFKDQLFal analyzes, as follows: X+DR

X

Interface Delamination

Reinforcement Concrete y

Ri

Rf=Ri+DR

x

Reinforcement

Inclined cracking +

Interface

Ri : initial (uncorroded) bar radius Rf : final (expanded) bar radius X: corrosion attack depth DR: bar radius increment

(a)

(b)

(c)

Figure 26. Analysis of the rebar corrosion: cross section analysis of the structural member. (i) At the structural member cross section level, we simulate the reinforcement expansion due to the volume increase of the steel bars as a consequence RI FRUURVLRQ SURGXFW DFFXPXODWLRQ 'DPDJH DQG FUDFN SDWWHUn distribution, in the concrete bulk and cover, is evaluated, which (indirecWO\ GHÀQHV WKH concrete net section loss in the structural member. (ii) $ VHFRQG PDFURVFRSLF KRPRJHQL]HG PRGHO DW WKH VWUXFWXUDO level, considering the results of the previous analysis, evaluates the mechanical response of the structural member subjected to an external loading system. This evaluation determines the global response and the macroscopic mechanisms of failure.

Crack Models with Embedded Discontinuities

155

b.1) Cross Section Analysis of the Structural Member (expansion mode). /HW XV FRQVLGHU WKH FURVV VHFWLRQ RI DQ DUELWUDU\ 5& VWUXFWXUDO Pember, as displayed in )LJXUH E ZKRVH UHLQIRUFHPHQW LV H[SHULHQFLQJ D FRUURVLon process. The products derived from the steel bar corrosion, such as ferric oxide rust, reduce the net steel area and accumulates, causing volumetric expansion of the bars (see Figure D  ZKDW LQGXFHV D KLJK KRRS WHQVLOH VWUHVV VWDWH LQ WKH VXrrounding concrete. As a consequence the cover concrete undergoes a damage and a degradation process GLVSOD\LQJ WZR W\SLFDO IUDFWXUH SDWWHUQV L LQFOLQHG FUDFNV DQG LL GHODPLQDWLRQ FUDFNV DV REVHUYHG LQ )LJXUH F 2EYLRXVO\ WKHVH LQGXFHd cracks can increase the rate of corrosion process in the structural member. The two-dimensional plane strain mesoscopic model, as idealized in Figure E FRQVLGHUV WKUHH GLIIHUHQW GRPDLQV RI DQDO\VLV L WKH FRQFUHWH PDWUL[ LL WKH VWHHO UHLQIRUFHPHQW EDUV DQG LLL WKH VWHHOFRQFUHWH LQWHrface. Each of them are characterized by a different constitutive response, and FE technology, that takes into account the main mechanisms involved in the corrosion process. A steel-concrete interface model is considered in order to capture the possible slipping movement between both components once the concrete fractures. ,Q )LJXUH  ZH GLVSOD\ WZR GLIIHUHQW QXPHULFDO VROXWLRQV Uelated to the exSDQVLRQ PHFKDQLVP RI VWHHO EDUV IRU D SUHGHÀQHG FRUURVLRQ DWtack depth, and the degradation induced in concrete cover at the cross section level. From a qualitative point of view, it can be observed that the proposed mesoscopic plane strain numerical model captures physically admissible failure mechanisms. The introGXFWLRQ RI IULFWLRQFRQWDFW LQWHUIDFH ÀQLWH HOHPHQWV LQ the simulations is a key aspect in order to obtain consistent crack patterns, which match very well with the semi-analytical predictions published in the literature.

1.00 0.80 0.60 0.40 0.20 0.00

(a)

(b)

(c)

Figure 27. Analysis of the rebar corrosion problem for two different reinforcement EDU GLVWULEXWLRQV 3ODQH VWUDLQ H[SDQVLRQ DQDO\VLV D LVo-displacement contour OLQHV FUDFN SDWWHUQ  E VFDOHG GHIRUPHG FRQÀJXUDWLRQ F GDPDJH FRQWRXU ÀOO b.2) Macroscopic Model to Simulate the Structural Load Carrying Capac-

156

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ity (Flexure Mode). The model presented in this subsection provides qualitative information related to the concrete degradation mechanisms due to the steel exSDQVLRQ +RZHYHU LW GRHV QRW JLYH DGGLWLRQDO LQIRUPDWLRQ Dbout the mechanical EHKDYLRU RI D GHWHULRUDWHG 5& VWUXFWXUH VXEMHFWHG WR H[WHUQal loads. 7KHUHIRUH D ' PDFURVFRSLF KRPRJHQL]HG FRPSRVLWH PRGHO as it was shown in the previous sections, could be proposed in order to evaluate the residual load FDUU\LQJ FDSDFLW\ RI WKH FRUURGHG 5& PHPEHU 7KH LGHDOL]HG Vcheme of the disFUHWH PRGHO LV VKRZQ LQ )LJXUH  ,W LV PRGHOHG WKH SODLQ FRQFrete by means of WKH &6'$ DSSURDFK ZLWK WKH FRQVWLWXWLYH UHODWLRQ GHVFULEHG LQ 6HFWLRQ  b.3) Coupling Strategy Between the Cross Section and the Structural Member Analysis. Finally, a coupling strategy must be provided in order that the results of WKH FURVVVHFWLRQ PRGHO 6XEVHFWLRQ  FRXOG EH WUDQVIHUred, as a data set, to the PDFURVFRSLF DQDO\VLV RI WKH VWUXFWXUDO PHPEHU ,W FRQVLVWV of projecting, from one domain of analysis to the other, the average value of the damage variable d across horizontal slices of the cross section model. This projection is consistent because both analyses use the same continuum isotropic damage model for simulating the FRQFUHWH GRPDLQ 7KXV WKH ÀQDO GHJUDGDWLRQ VWDWH RI FRQFUHte, induced by the steel bar volumetric deformation process, is considered to be the initial damage condition for the subsequent structural analysis. This means that we are assuming that the two models are coupled in only one direction. Macroscopic model

Homogeneized composite material

concrete

Figure 28. Macroscopic FE model to simulate the structural load carrying capacity RI D VWUXFWXUDO PHPEHU ÁH[XUH PRGH  b.4) Numerical results of the corroded structural member. The structural anal\VLV RI WKH 5& EHDP ZLWK WKH PRGHO RI 6XEVHFWLRQ  ZKHUH DQ Lnitial damage distribution was adopted following the results of the analyVLV LQ 6XEVHFWLRQ  provides structural responses which change with the rebar corrosion attack depth. For a typical beam subjected to different levels of corrosion attack, these responses DUH SORWWHG LQ )LJXUH D GHSLFWLQJ VHYHUDO ORDGV YV PLGspan vertical displacePHQW SORWV ,W FDQ EH REVHUYHG WKH VWUXFWXUDO VWUHQJWK VHQVLtivity with the rebar corrosion level. The same structural analysis provides the damage distribution that is displayed LQ )LJXUH E IRU RQH FDVH RI FRUURVLRQ OHYHO $OVR )LJXUH F VKRZV WKH LVR

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GLVSODFHPHQW FRQWRXU OLQHV ZKLFK H[KLELW WKH ÀQDO IDLOXUH Pechanism and the active cracks at the end of analysis. P

y 1.00

(b)

(a)

0.00

z y

Increase of the (rebar) corrosion attack depth (c)

(b)

z

(a)

Figure 29. 1XPHULFDO UHVXOWV RI WKH PDFURVFRSLF )( PRGHO WR VLPXODWH WKe strucWXUDO ORDG FDUU\LQJ FDSDFLW\ D ORDG YV PLGVSDQ YHUWLFDl displacement plots, FXUYHV FRUUHVSRQG WR WKH VDPH 5& EHDP VXEMHFWHG WR GLIIHUHQW corrosion attack GHSWKV E ÀQDO FRQWRXU ÀOO RI GDPDJH F LVRGLVSODFHPHQt contour lines in z-direction.

Bibliography Z.P. Bazant. Mechanics of distributed cracking. Appl. Mech. Rev. ²  Z.P. Bazant. Analysis of work of fracture method for measuring fracture energy of concrete. J. Eng. Mech. Div. ASCE SDJHV ²  Z.P. Bazant and J. Planas. Fracture and size effect in concrete and other quasibrittle materials &5& 3UHVV %RFD 5DWRQ )/  S. Blanco. Contribuciones a la simulaci´on num´erica del fallo material en medios tridimensionales mediante la metodolog´×D GH GLVFRQWLQXLGDGHV IXHUWHV GH FRQ tinuo 3K' WKHVLV (76 (QJLQ\HUV GH &DPLQV &DQDOV L 3RUWV 7HFKQLFDO 8QL YHUVLW\ RI &DWDORQLD 83&  %DUFHORQD  ,Q VSDQLVK 0 (OLFHV *9 *XLQHD - *´omez, and J. Planas. The cohesive zone model: advantages, limitations and challenges. Engineering Fracture Mechanics  ²  0/ )DON $ 1HHGOHPDQQ DQG -5 5LFH $ FULWLFDO HYDOXDWLon of dynamic fracture simulation using cohesive surfaces. J. de Physique IV SDJHV 3U²² WR 3U²²   + +DUWO 'HYHORSPHQW RI D FRQWLQXXPPHFKDQLFV EDVHG WROO IRU ' ÀQLWe element analysis of reinforced concrete structures and application to problems of

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soil-structure interaction 3K' WKHVLV *UD] 8QLYHUVLW\ RI 7HFKQRORJ\ *UD] $XVWULD  . +HOODQ Introduction to Fracture Mechanics 0F*UDZ+LOO 1HZ