Implicit integration scheme and its consistent linearization for an elastoplastic-damage model with application to concrete

Computers and Structures 75 (2000) 135±143

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Implicit integration scheme and its consistent linearization for an elastoplastic-damage model with application to concrete R. Mahnken*, D. Tikhomirov, E. Stein Institut fuÈr Baumechanik und Numerische Mechanik, UniversitaÈt Hannover Appelstraûe 9A, 30167, Hannover Germany Received 30 April 1998; accepted 3 March 1999

Abstract A two-phase coupled elastoplastic-damage model proposed by Ortiz (Ortiz M. Mechanics of Materials 1985;4:67± 93) for modeling inelastic behavior of concrete is considered. For integration of anisotropic damage evolution equations an implicit integration scheme is proposed and the resulting nonlinear equations, both on the constitutive level and the ®nite element equilibrium level, are solved based on consistent linearization. The algorithm is implemented on a parallel computer. A numerical test for a concrete console demonstrates the eciency of the developed algorithm. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Implicit integration; Consistent linearization; Anisotropic damage; Concrete modeling

1. Introduction The inelastic behavior of engineering materials such as concrete is generally associated with irreversible thermodynamical processes involving elastic degradation and plastic ¯ow. Both mechanisms play a crucial role in the change of the elastic sti€ness and formation of the residual strains in the material, respectively. Concrete usually shows both types of responses. A detailed comparison of di€erent plasticity models for the simulation of the concrete response was made by many authors, for example, Meschke [9], Feenstra [4], etc. For modeling of brittle response, which is often dominant in concrete structures, many authors use continuum damage mechanics as an e€ec-

* Corresponding author. Tel.: +49-511-762-2220; fax: +49511-762-5496.

tive tool for the description of inelastic deformation processes [2,3,14,15]. A detailed analysis of damage phenomena and an appropriate mathematical representation of damage was made by Krajcinovic [7]. Usually damage is combined with other physical phenomena for the formulation of constitutive equations such as creep, fatigue, elasticity and plasticity. A literature review on this aspect is given by Ju [6]. In this contribution constitutive relations originally proposed by Ortiz [12] are considered which are able to model inelastic behavior and elastic degradation typically observed in concrete. The characteristic features of the model are summarized as follows: . From the outset concrete is considered as a composite material comprising two phasesÐmortar and aggregate ([11]). The theory of interacting continua is adopted in order to quantify the induced phase stresses as a function of applied ones. . A Drucker±Prager model is used for modeling

0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 8 9 - 9

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R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

inelastic behavior of the aggregate. . In order to simulate elastic degradation of the mortar, an anisotropic damage evolution is postulated. In the original work of Ortiz [12], an explicit Euler method is applied for the integration of the evolution equations for anisotropic fourth-order damage tensors. It is well known that this strategy may lead to unstable results [13] unless the load steps are reasonably chosen. In this work an implicit Euler method is suggested, thus preventing the above-mentioned diculties. For solution of the resulting nonlinear equations on the constitutive level a Newton method is applied. Analogously the global equilibrium problem is solved with a Newton method based on consistent linearization of the resulting equilibrium problem [16].

F ˆ am I s1 ‡ ksk

…3†

Here I s1 is the ®rst invariant of the stress tensor s , s is the deviatoric stress tensor, s0 is the yield stress, k
k indicates the norm of the argument, and ak and am are coecients which depend on the angle of internal friction f and dilatation angle c, respectively ak ˆ

6sin f 3 ÿ sin f

am ˆ

6sin c 3 ÿ sin c

The non-associative ¯ow rule takes the form ! n‡1 @ F s p EÇ ˆ g_ ˆ g_ am 1 ‡ n‡1 @s k sk

…4†

…5†

where 1 is the second-order unit tensor and g_ is obtained from the loading/unloading conditions

2. Review of a two-phase theory for concrete

g_ r0,

2.1. Main considerations

The resulting algorithm for integration of the constitutive equations is Section 3.

As suggested by Ortiz and Popov [11], we treat concrete as a two-phase material consisting of mortar and aggregate with volume fractions a1 and a2 , respectively. The thermodynamic consistent representation is based on the theory of interacting continua with the stress tensor s ˆ a1 s 1 ‡ a2 s 2

…1†

Here s 1 is the stress tensor in the mortar, and s 2 is the stress tensor representing the contact forces, which are acting among aggregate particles and surrounding mortar. Thus, the notion of an aggregate particle as a ®nite volume is lost in the context of the mixture theory. This is a plausible assumption, if the representative volume element of the concrete contains both phases in ®xed proportions and the processes to be described have macroscopic representation. The following compatibility condition for the strains follows from the absence of di€usion between both phases E1 ˆ E2 ˆ E

…2†

2.2. Aggregate For the constitutive representation of the aggregate as a granular material, a perfect Drucker±Prager type plasticity model is chosen. The non-associated plasticity criterion represents the aggregate behavior and is characterized by a yield function F and a plastic potential F F ˆ ak I s1 ‡ ksk ÿ s0

FR0,

g_ F ˆ 0:

…6†

2.3. Mortar In order to describe elastic degradation of mortar, Ortiz [12] suggested fourth-order damage tensors in the framework of continuum damage mechanics. In this way the full anisotropic e€ects of sti€ness degradation in the concrete are taken into account. Based on the representation E ˆC:s

Ç :s 9 EÇ ˆ C : sÇ ‡ C

…7†

for the strain and rate of strain tensor, respectively, an additive decomposition of the compliance tensor C ˆ C0 ‡ Cc

…8†

is assumed. Here C0 is the elastic compliance tensor and c

c

Å : P‡ ‡ Pÿ : C Å : Pÿ Cc ˆ P‡ : C I II

…9†

is the added compliance tensor due to microcracks obtained from a constrained minimization problem for the Gibbs free energy function. In order to de®ne the projection operators P‡ and Pÿ in Eq. (9), a spectral decomposition of the stress tensor is performed s ˆ s‡ ‡ sÿ ˆ

X poss^ i

‡ s^ ‡ i mi ‡

X negs^ i

ÿ s^ ÿ i mi

…10†

^ÿ where s^ ‡ i and s i are positive and negative eigenvalues, ÿ respectively, and m‡ i and mi are corresponding eigenvalue bases. Subsequently, s ‡ and s ÿ denote positive and negative parts, respectively, of the stress tensor s

R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

137

based on its spectral decomposition. Then, according to Ortiz [12], the following projection operators are proposed P‡ ˆ

X poss^ i

‡ m‡ i mi ,

Pÿ ˆ

X negs^ i

ÿ mÿ i mi

…11†

Å c and C Å c are added ¯exibility Furthermore in Eq. (9), C I II tensors due to the response of microcracks in the splitting and compressive modes, respectively, if all microcracks are regarded to be active. They are obtained from the evolution rules c

ÇÅ ˆ mR C _ I …s s †, I

c

ÇÅ ˆ mR _ II …s C s† II

…12†

where m is a cumulative variable and RI and RII are damage direction tensors RI ˆ

s‡ s‡ , s‡ : s‡

RII ˆ c

sÿ sÿ sÿ : sÿ

…13† 3. Numerical implementation

The cumulative variable m in Eq. (12) is de®ned such that the loading/unloading conditions _ mr0,

Fd …s ,m †R0,

_ dˆ0 mF

…14†

with damage function 1 ‡ 1 ÿ 1 s : s ‡ ‡ cs s : s ÿ ÿ t2 …m †, 2 2 2 ÿ
0 ln 1 ‡ E m where t…m † ˆ ft e 1 ‡ E 0m

Fd …s ,m † ˆ

…15†

are satis®ed. Here E 0 denotes Young's modulus of the uncracked mortar, ft is the uniaxial tensile strength, c is the cross e€ect coecient which takes into account the relative extent of the damage under tensile and compressive stresses, respectively, and e ˆ 2:71828 is a constant. A graphical representation of the damage surface Fd in the principal stress space is given in Fig. 1. As it can be seen from Fig. 1, parameter c enables to move the damage surface in the principal stress space so that the ratio of the compressive strength to p the tensile strength is 1=c ˆ 10, constant for any class of concrete. For subsequent developments we take the time derivative of Cc in Eq. (9) ÇÅ c ‡ 2PÇ ÿ : C ÇÅ c Ç c ˆ 2PÇ ‡ : C Å c : P‡ ‡ C Å c : Pÿ ‡ C C I II I II

Fig. 1. Damage surface in principal stress space.

…16†

Å c and where we have exploited major symmetries of C c c cI c Ç ‡ ÇÅ ‡ ÿ ÇÅ Å Å CII cand the relations P : CI : P ˆ CI and P : CII : Pÿ ÇÅ , which hold for the chosen form of projection opˆC II erators, Eq. (11).

3.1. Aggregate The implicit integration scheme for Drucker±Prager plasticity is described elsewhere in the literature, see for example [1,5]. Here we give only the resulting expressions for stresses and the consistent elastoplastic tangent moduli. We consider a typical load step within a strain-driven algorithm with given strain increment Dn‡1 E and given initial condition n s : The stress tensor at the new state is obtained from the relation ÿ
n‡1 s ˆn‡1 s tr ÿ Dgn‡1 3am k1 ‡ 2mn‡1 n , where Dgn‡1 ˆ

n‡1 tr tr 3an‡1 s k ÿ s0 k p ‡k c

…17†

Here, n‡1 s tr ˆ n s ‡ De : Dn‡1 E is the trial stress tensor with elastic sti€ness of aggregate De , and k and m are the bulk and shear modulus, respectively. Furthermore, n‡1 n ˆ n‡1 str =kn‡1 str k is the normal to the yield surface, n‡1 tr p is the hydrostatical pressure and c ˆ 9ak am k ‡ 2m is a coecient. From now on, if confusion is out of danger, the index referring to the state n ‡ 1 is neglected. A linearization of Eq. (17) yields the consistent tangent operator as Dˆ

@s ˆ k1 1 ‡ 2m1dev @E

ÿ

9ak am k2 6km ‰am 1 n ‡ ak n 1Š 1 1ÿ c c

ÿ


4m2 4m2 Dg ÿ dev n nÿ 1 ÿn n tr ks k c

…18†

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R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

where 1dev ˆ 1 ÿ 1=31 1 is the deviatoric unit tensor and 1 is a fourth-order unit tensor.

3.2.2. Algorithmic tangent modulus The starting point for derivation of the algorithmic tangent modulus is the following representation of the residual equation (20) in its most general form

3.2. Mortar

r…EE,s s,Dm† ˆ 0

3.2.1. Integration of the constitutive equations As in the previous section we consider a typical load step within a strain-driven algorithm with given strain increment Dn‡1 E and given initial conditions n P‡ , n Pÿ , n Å c, nC Å c , n m and the index n ‡ 1 will be s , n C, n C I II omitted. Then, discretization of the rate constitutive equation (7) leads to

s,Dm† ˆ 0 Fd …EE ,s Then the total di€erentials of r and Fd are dr ˆ

@r @r @r dEE ‡ ds s‡ dDm ˆ 0 @E @s @ Dm

E ˆ C : Ds s ‡ DC : s ˆ …C0 ‡n C ‡ DC† : …s ÿn s † n

n

ˆ …C0 ‡ C† : Ds s ‡ 2DC : s ÿ DC : s

dFd ˆ …19†

Upon substitution of the discretized form for Eq. (16) in Eq. (19), using DC ˆ DCc and the damage function Eq. (15), we obtain the following system of equations which determines the new state fs s,mg

"

‡2…P‡ ÿn P‡ † : ‡2…Pÿ ÿn Pÿ † :

n

Å c ‡ DC Å c : …2s s ‡ ÿ P‡ : n s † C I I Å c ‡ DC Å c : …2s C s ÿ ÿ Pÿ : n s † II II

1 ‡ 1 ÿ ÿ 1 2 s : s ‡ ‡ cs s :s s ÿ t …m † ˆ 0 2 2 2

…20†

Here also the notation Ds s ˆ s ÿn s has been used. Upon de®ning a vector of unknowns X ˆ ‰s s,DmŠt and a residual vector R ˆ ‰r,Fd Št , the solution of the problem (20) is obtained with a Newton scheme according to Xk‡1 ˆ Xk ÿ Jÿ1 k Rk ,

k ˆ 0,1,2, . . .

…24†

#ÿ1   @r @ Fd ÿ1 @ r @ Fd ÿ

@s @ Dm @ Dm @s

…25†

The above result allows the interpretation, that the inverse of D is obtained by static condensation of the Jacobi matrix J in Eq. (22).



  Å c ‡ DC Å c : n s ÿ DEE ˆ 0 ‡2Dm…s ‡ ‡ cs s ÿ † ÿ DC I II Fd ˆ

Dˆ



@ Fd @ Fd @ Fd dEE ‡ ds s‡ dDm ˆ 0 @E @s @ Dm

After some algebraic operations and taking into account that @ Fd =@ E ˆ 0 and @ r=@ E ˆ ÿ1, the above equations can be solved for the consistent tangent operator D ˆ ds s=dEE as

r ˆ …C0 ‡n C† : Ds s n

…23†

…21†

where k refers to the iteration number and J is the Jacobi matrix 2 3 @r @r 6 @s @ Dm 7 6 7 Jˆ6 …22† 7 4 @ Fd @ Fd 5 @ s @ Dm Detailed expressions for the partial derivatives in J are given in Appendix A.

Remark. An alternative way for derivation of the consistent tangent operator exploits the fact, that sti€ness and compliance tensors are inverse to each other, i.e. D : C ˆ 1 yields the fourth-order unit tensor. Then, upon using the representations E ˆ E …s s,Dm…s s†† and Fd …s s,Dm…s s†† ˆ 0, straightforward di€erentiation by the chain rule yields for the compliance tensor dEE @E @E @ Dm ˆ ‡

, ds s @ s @ Dm @s   @ Dm @ Fd ÿ1 @ Fd ˆÿ @s @ Dm @s

Cˆ

where …26†

Taking the inverse of C and using the relation @ E =@ s ˆ @ r=@ s yields the result for D as given in Eq. (25).

The detailed representation of the consistent tangent operator by taking into account its minor symmetries is

R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

(

Expressions for @ mi =@ s are derived by Miehe [10] as

D ˆ Cÿ1 ˆ C0 ‡n C ‡

h i s^ i @ mi ÿ1 1 ÿ I s3 …s^ i †ÿ1 1s ˆ @s Di

 @ P‡ n Å c Å c : …2s s‡ ÿ P‡ : n s † : CI ‡ DC I @s

‡ 2…P‡ ÿn P‡ † :

Åc @ DC I

‡

: …2s s‡ ÿ P‡ : n s †

@s    @ s ‡ @ P‡ n c c ‡ n ‡ n Å Å ‡ 2…P ÿ P † : CI ‡ DCI : 2 ÿ : s @s @s   @ Pÿ n Å c Å c : …2s ‡2 s ÿ ÿ Pÿ : n s † : CII ‡ DC II @s Å c @ s : …2s ‡ 2…Pÿ ÿn Pÿ † : @ DC s ÿ ÿ Pÿ : n s † II    @ s ÿ @ Pÿ c c n Å Å CII ‡ DCII : 2 ÿ : s @s @s   ‡   Å c @ DC Åc @s @ sÿ @ DC I II ‡ 2Dm ‡c ÿ ‡ : ns @s @s @s @s   " Åc @ t ÿ1 @ DC I … ÿ t…m † 2…P‡ ÿn P‡ † : : 2s s‡ @ Dm @ Dm ‡ 2…Pÿ ÿn Pÿ † :

n

ÿ P‡ : n s † ‡ 2…Pÿ ÿn Pÿ † :

Åc @ DC I … : 2s sÿ @ Dm

ÿ

#

! n

)ÿ1 ‡

s : s ‰s ‡ cs

ÿŠ

3  ÿ
2 ÿ
ÿ2 s^ i X I s3 …s^ i †ÿ1 ÿ s^ j s^ j mj mj Di jˆ1

…29†

Q ÿ1 where Di ˆ 3jˆ1ni …s^ i ÿ s^ j †, 1s ˆ s ÿ1
1
s ÿ1 : From a computational point of view, it is advantageous to reduce the special cases of equal or nearly equal eigenvalues of the stress tensor to the general case of distinct eigenvalues by applying a perturbation technique, ®rstly introduced by Simo and Taylor [17]  if ‰js^ A ÿ s^ B j=max js^ A j,js^ B j,js^ C j < told then 9 s^ A ˆ s^ A …1 ‡ d† = s^ B ˆ s^ B …1 ÿ d† with perturbation d  1: ; s^ C ˆ s^ C =…1 ‡ d†…1 ÿ d†

…30†

Here told is a machine-dependent tolerance.

4. Numerical example

ÿ Pÿ : n s † ‡ 2…s ‡ ‡ cs sÿ † Åc Åc @C @C I II ‡ @ Dm @ Dm

139

…27†

where   ‡ X @ s‡ ‡ @ mi ‡ ‡ ˆ mi mi ‡ s^ i @s @s poss^ i

  X @ m‡ @ P‡ n @ m‡ n ‡ ‡ n i i : sˆ : s mi ‡ m i

: s @s @s @s poss^

The algorithm described for the coupled elastoplastic-damage model has been applied to the numerical simulation of the concrete console (Fig. 2) tested by Franz (Mehmel and Freitag [8]). The schematic crack picture of the experiment for three di€erent loads is depicted in Fig. 3. Material parameters used in the simulation (Table 1) are taken from [12]. The console is discretized with 1024 eight-node isoparametric brick elements and 12 load steps were applied until the ultimate load. The maximum numerical load where the convergence was achieved is 775 kN. From Fig. 4 it can be seen that this numerical failure load exceeds the experimental value for the fracture load by 4.7%. The

i

Table 1 Material parameters

2

@ s‡ n @ s‡ n : s s‡ ‡ s‡

: s Åc 6 @ DC I n @s @s : s ˆ Dm6 X 6 @s 6 sà 2i 4

Mortar

poss^ i

3 ÿ2

s‡ s‡ s‡ : ns 7 7 7 X 2 !2 7 5 sà i

…28†

poss^ i

In analogy to the relations (28) we obtain the exÅ c †=@ s : n s : pressions for @ s ÿ =@ s , @ Pÿ =@ s : n s , @ …DC II

Aggregate

Volume fraction

a1 ˆ 0:3

Young modulus Poisson ratio Uniaxial tensile strength Cross-e€ect coecient Volume fraction Young modulus Poisson ratio Internal friction angle Dilatancy angle Yield stress

E ˆ 2187:0 kN/cm2 n ˆ 0:2 ft ˆ 0:226 kN/cm2 c ˆ 0:01 a2 ˆ 0:7 E ˆ 2187:0 kN/cm2 n ˆ 0:2 f ˆ 208 c ˆ 108 s0 ˆ 1 kN/cm2

140

R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

Fig. 2. Test concrete console: geometry and loading.

distribution of the cumulative damage variable m for the load steps of Fig. 3 and the equivalent plastic strain for the limit load are given in the Figs. 5±8. Fig. 5 shows that damage appears ®rstly in the zone where cracks initiated. In these ®gures there are two zones with local softening representing the highest concentration of damage, which coincide with two zones with multiple cracks. As a result the algorithm described above gives an e€ective tool for numerical modeling of

damage induced by developing microcracks in material. This example was computed on a parallel computer PARASTATION consisting of 16 processors Pentium166 MHz with an iterative cg-solver. The total computation times for di€erent numbers of processors are compared in Table 2. The most important characteristics are the speedup S-ratio of the computation time on one processor to the computation time on n pro-

Fig. 3. Crack distribution for di€erent load from experiment.

R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

141

Fig. 4. Load-displacement curve for console.

Fig. 6. Cumulative damage parameter for P ˆ 700 kN.

Fig. 5. Cumulative damage parameter for P ˆ 300 kN.

Fig. 7. Cumulative damage parameter for P ˆ 740 kN.

Table 2 Time characterics for di€erent number of processors by the computation of console Number of processors

Common solution time

Speedup

Eciency

± 1 2 4 8 16

h:min:sec 1:44:58 1:14:38 51:43 29:13 16:48

± 1 1.41 2.03 3.59 6.34

± 1 0.71 0.51 0.44 0.40

142

R. Mahnken et al. / Computers and Structures 75 (2000) 135±143

follows: @r ˆ C 0 ‡n C @s  @ P‡ n Å c Å c : …2s ‡ s ‡ ÿ P‡ : n s † : CI ‡ D C I @s Åc @ DC I … : 2s s ‡ ÿ P‡ : n s † @s    @ s ‡ @ P‡ n c c ‡ n ‡† n Å Å … ‡ 2 P ÿ P : CI ‡ DCI : 2 ÿ : s @s @s   @ Pÿ n Å c Å c : …2s s ÿ ÿ Pÿ : n s † ‡2 : CII ‡ DC II @s ‡ 2…P‡ ÿn P‡ † :

Åc @ DC II … : 2s s ÿ ÿ Pÿ : n s † @s    n ÿ ÿ Å c : 2 @ s ÿ @ P : ns Å c ‡ DC ‡ 2…Pÿ ÿn Pÿ † : C II II @s @s    ‡  c c Å Å @s @ sÿ @ DC @ DC I II ‡ 2Dm ‡c ÿ ‡ : ns @s @s @s @s ‡ 2…Pÿ ÿn Pÿ † :

Fig. 8. Equivalent plastic strain for P ˆ 740 kN.

cessorsÐand the eciency EÐratio of speedup to the number of processors. We conclude from this table noticeable execution time advantages by the usage of parallel computations.

5. Conclusion In this work, an implicit integration algorithm for a coupled elastoplastic-damage model is suggested. The resulting equations on the local level and on the ®nite element equilibrium level are solved based on consistent linearization. To this end a complete expression for the consistent tangent operator for the anisotropic tensor damage model is derived. The algorithm is applied to the concrete console test and the results of the numerical simulation give a good representation of concentrated damage coinciding with zones of highest crack concentration. The numerical test was performed on a parallel computer thus demonstrating high eciency of parallelized computations.

Acknowledgements The ®nancial support of the German Academic Exchange ServiceÐDAADÐfor the second author of this paper is gratefully acknowledged.

Appendix A The elements of the Jacobi matrix J in Eq. (22) are determined from partial di€erentiation of Eq. (20) as

Åc @r @ DC I … ˆ 2…P‡ ÿn P‡ †: : 2s s ‡ ÿ P‡ : n s † @ Dm @ Dm c

Å @ DC I … : 2s s ÿ ÿ Pÿ : n s † @ Dm ! Åc Åc @C @C ‡ ÿ I II s †ÿ ‡ 2…s ‡ cs ‡ : ns @ Dm @ Dm ‡ 2…Pÿ ÿn Pÿ † :

@ Fd ˆ s ‡ ‡ cs sÿ @s @ Fd @t ˆ t…m † @ Dm @ Dm

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