Composites: Part B 52 (2013) 323–333
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
An ABAQUS toolbox for multiscale finite element computation Adjovi Tchalla a,b, Salim Belouettar a,⇑, Ahmed Makradi a, Hamid Zahrouni b a b
Centre de Recherche Public Henri Tudor, 29, Avenue John F. Kennedy, L-1855 Luxembourg, Luxembourg Laboratoire d’Étude des Microstructures et de Mécanique des Matériaux, LEM3, UMR CNRS 7239, Université de Lorraine, Ile du Saulcy, 57045 Metz, France
a r t i c l e
i n f o
Article history: Received 31 October 2012 Received in revised form 6 March 2013 Accepted 7 April 2013 Available online 19 April 2013 Keywords: C. Computational modeling C. Finite Element Analysis C. Micro-Mechanics
a b s t r a c t In this paper, we propose to implement, in the framework of a commercial finite element software, a computational multilevel finite element method for the modeling of composite materials and structures. In the present approach, the unknown constitutive relationship at the macroscale is obtained by solving a local finite element problem at the microscale. The main advantages of the proposed computational approach are that it can greatly save computer memory and CPU time, and it has good accuracy at the same time while it allows to easily building nonlinear behavior for high order mechanical theories to deal with problems which cannot be handled by classical multiscale or homogenization theories. The linear and the non-linear cases are introduced and implemented in ABAQUS. A Python script and user-defined FORTRAN subroutines have been developed for this purpose. Finally numerical results show that the method presented in this paper is effective and reliable. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Mechanical properties and response of composite materials and structures have been topics of intense research for several decades. In framework of elasticity, homogenization techniques are extensively developed to provide predictions of the effective mechanical properties of composite materials [1–6]. Several methods for the homogenization of heterogeneous materials exist in the literature (see [7] for an overview). The pioneering work, using this framework is the one based on Eshelby’s theory of inclusion embedded in an equivalent continuum matrix [8]. A detailed review on these approaches could be found in [9]. These mean field theories treat the particulate material with its local mechanical behavior as a whole material, which is defined by the overall (effective) material parameters. Therefore, the model does not directly describe the local response and its interaction within the material but describes the overall mechanical behavior on a macroscale. Indeed, the homogenization of heterogeneous materials leads to the ‘‘calculated moduli or properties’’. Basically, all the existing mean field methods are based on the following elementary assumptions: the macrostructure is formed by a spatial periodicity of the representative volume element (RVE), the macroscale fields are constant within the representative volume element and that the solution is locally periodic in the statistical sense. Basically, all these assumptions are only valid away from the boundary and as long as the material heterogeneity is significantly smaller than the typical dimension of the macrostructure. In the meanwhile, it is com⇑ Corresponding author. Tel.: +352 42 59 91 530; fax.: +352 42 59 91 333. E-mail address:
[email protected] (S. Belouettar). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.04.028
monly observed that the overall behavior of composite materials is affected by the mechanical properties of the individual constituents as well as by geometrical characteristics, such as volume fraction, shape, size, spatial orientation and distribution of particles. An alternative approach to answer this serious drawback is to develop a kind of superposition methods as proposed by [10–14]. These approaches are based on a hierarchical decomposition of the solution space into a local solution and a global one and by enforcement of the solution compatibility conditions. Regarding to the modeling and simulation of composite mechanical response, finite element methods are generally used for two-scale analysis. Moreover, if periodic microstructures have nonlinear behavior, the two-scale analysis requires that nonlinear finite element analysis of a unit cell be performed at all integration points in each element in the macrostructure at every incremental step. Subsequently, computational schemes have therefore been developed to perform nonlinear two-scale analysis (e.g., [15–18]). In the effort, basically the majority of the authors, who used a two-scale analysis, have developed their own code. For example, Feyel and Chaboche [16] have developed an object-oriented code called ZéBuLon which is based on a Newton–Raphson procedure to solve nonlinear problems and more recently, Nezamabadi et al. [19,20] developed a MATLAB code, from scratch, to build up a multilevel nonlinear finite element analysis coupled with an asymptotic numerical method (ANM) to solve nonlinear problems. Even though, the two-scale finite element based homogenization methods are well established and very efficient to handle nonlinear problems, they remain generally unused because of the important implementation effort needed. To overcome this limitation, one worthwhile alternative solution is the implementation of a
324
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
computation approach within a conventional finite element code like ABAQUS as proposed by [21]. In the present paper, we propose a comprehensive procedure to implement such computational approach in ABAQUS for both linear and nonlinear solid mechanics problems. A Python script and user-defined FORTRAN subroutines have been developed for this purpose. The details on the implementation procedure and the corresponding Python code are presented. Finally numerical results show that the method presented in this paper is effective and reliable. The layout of this paper is as follows: in Section 2, the formulation of micro–macro problem is described. In Section 3, details on the computation procedure in ABAQUS using a Python scripting and user-defined subroutines are given for linear and nonlinear problems. In Section 4, the implemented technique is validated on a heterogeneous composite structure. 2. Problem formulation Most of constitutive material models in ABAQUS are path dependent and usually the constitutive relationships are defined in rate form. In addition, the lagrangian viewpoint is used for the mechanical modeling capabilities and codes like ABAQUS. 2.1. Problem at the macroscale We consider a domain X in RD, D being the domain dimension, with an external boundary @ X, describing the structure at the macroscale level in its current configuration. We assume that the material is heterogeneous and characterized by a periodic microstructure. The structure is subjected to prescribed displacements and forces on the disjoint complementary part of the boundary @ Xu (the Dirichlet boundaries) and @ Xq (the Neumann boundaries). Large displacements are considered and the so-called updated lagrangian formulation is adopted. In the following, the notation ðÞ will be used to denote macroscale quantities. The problem to solve is defined, in the absence of body forces, as follows:
ðxÞ ¼ 0; rr
ð1Þ
ð where r xÞ is the Cauchy stress tensor associated with a point x of the macroscale structure in the current configuration. The appropriate stress measure should be the Kirchhoff stress defined with respect to the reference configuration (t = 0). since an updated Lagrangian formulation is used, the computation at the current configuration (t + dt) is based on the configuration at (t), which is the reference configuration. The assumption that this reference configuration and the current configuration are only infinitesimally different makes the Kirchhoff and Cauchy stress measures almost the same. The conjugate strain rate to Cauchy stress is the rate of deformation D given as:
D¼
1 ðrv þ rv > Þ; 2
ð2Þ
where v is the velocity field. The boundary conditions are defined as:
ðxÞ ¼ u ^ ðxÞ on @ Xu ; u r N ¼ F on @ Xq :
ð3Þ
¼ x X denotes the macroscale displacement field, X and x where u being the coordinates of a point in the initial and deformed config^ is a prescribed displacement. In Eq. (3), N is uration respectively. u the outward unit normal vector to @ X, F is a prescribed load. The weak form associated with Eqs. (1) and (3) is given by:
Z X
r : dDdX ¼
Z @ Xq
F : dv dC:
ð4Þ
and D is unAt this scale, the constitutive relation between r known. In the context of a multilevel finite element analysis, the stresses at the macroscale are computed by solving a local nonlinear finite element problem. The associated mathematical formulation is presented in the next section. 2.2. Problem at the microscale We assume that the material is heterogeneous with a periodic microstructure and characterized by a unit cell that occupies a domain x in RD with external boundary @ x in the current configuration. The equilibrium equation is given, in the absence of body forces, as:
rrðxÞ ¼ 0;
ð5Þ
where r(x) is the Cauchy stress tensor at a microscale point x. The RVE is subjected to boundary conditions (b.c.) depending on macroscale deformation tensor defined here as the integral of the deformation rate. Notice that the computation of this integral is nontrivial, particularly for the general case where the principal axes of the strain rotate during the deformation. In ABAQUS the total strain is computed by approximating the integral of the strain rate over an increment using a central difference algorithm [22]. Notice that geometrical nonlinearity is not taken into account at the microscale. Periodic boundary conditions (PBCs) are adopted on the limits of the RVE such as:
uþ u ¼ ðxþ x Þ;
ð6Þ
where u is the displacement at the microscale. In Eq. (6), the exponents + and are associated with node indices on opposite sides of the RVE. Regarding the constitutive relation, we can consider different laws (elastic or inelastic) available in the ABAQUS library. In the case of linear constitutive law for each phase, this relation can be written as follows:
rðxÞ ¼ CðrÞ : ðxÞ;
ð7Þ
ðrÞ
where C refers to fourth-order elastic tensor associated with phase (r), and is the strain tensor. The weak form associated with the microscale problem (5) is written as:
Z
r : ddx ¼ 0:
ð8Þ
x
The two problems at the micro and the macro scales are coupled through the following relation, that enables to evaluate the at a particular point effective stress r x of the macroscale domain X
r ¼ hrðxÞi ¼
1
Z
j xj
rðxÞdx:
ð9Þ
x
Here jxj denotes the volume of the RVE. The relationship be and the macroscopic strain tween the macroscopic stress r , in case of linear constitutive law, can be written as follows:
r ¼ hrðxÞi ¼ C :
ð10Þ
where C is the homogenized macroscale constitutive tensor depending on the elastic properties of the microscale constituents. In the case of a nonlinear constitutive law at the microscale, this relationship can be written in an incremental form:
¼ Ct : D Dr
ð11Þ
where Ct is the homogenized macroscale tangent modulus depending on the nonlinear behavior of the microscopic constituents. Computation of the operators C and Ct is detailed in the next section.
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
325
the mechanical constitutive model. The UMAT is used in conjunction with FORTRAN environment and manage interactions with external data files that are used in conjunction with user subroutines. The output quantities that are accumulated over multiple elements in COMMON block variables within user subroutines and written to external files at the end of a converged increment for postprocessing. The local scale problems are summarized as follows:
3. Computation procedures in ABAQUS and instructions Based on the above formulations, the implementation of a twoscale homogenization strategy in ABAQUS is presented, at this point, for two cases of solid mechanics problems: the linear elasticity analysis and the more general nonlinear case which accounts for material and geometrical nonlinearities. 3.1. Linear computational homogenization The implementation of the two-level linear analysis is detailed below:
8R > < x r : ddx ¼ 0 in x; rðxÞ ¼ CðrÞ : ðxÞ; > : þ u u ¼ ðxþ x Þ on @ x:
Step 1: The analysis in ABAQUS is processed first and once at the microscale in the linear case. The Jacobian, defined in Eq. (10), is then computed and used in the user-defined material subroutine (UMAT) at each integration point of the macroscale level. The user-defined mechanical material behavior in ABAQUS is provided by means of an interface whereby any mechanical constitutive model can be added to the library. It requires that a constitutive model (or a library of models) is programmed in User subroutine Material UMAT (ABAQUS/Standard). User subroutine UMAT updates the stresses and solution-dependent state variables to their values at the end of the increment for which it must provide the material Jacobian matrix, for
ð12Þ
Notice that, the main difficulty when solving this specific problem is that, the boundary conditions depend on the un . With this in mind and considknown macroscopic strain ering the coupling conditions between the two-scales, we primarily solve partially the microscale problem by considering a macroscale unitary strain imposed at the boundary of the RVE in order to obtain the macroscopic constitutive matrix. Since the problem (12) is linear, the superposition principle for the loading applies and hence the macroscale strain can be decomposed as follows:
Table 1 The two-scale solution scheme for the linear computational homogenization in ABAQUS. MICRO
MACRO
unit 1. Solve the RVE problem using an overall unit strain . Python script for the microscale model . Apply periodic boundary conditions on the RVE 2. Compute the constitutive modulus . Compute the Jacobian matrix ½C
½C
!
3. Input file for the macroscale problem . Define ½C in UMAT associated to the macrostructure . Solving of the macroscopic problem
4. End the microscale problem . Updated displacements, stresses, strains in the RVE
. Extract the macroscale strain
Table 2 The two-scale solution scheme for the nonlinear computational homogenization in ABAQUS. MICRO 1. Initialization . Python script for the microscale model . Apply periodic boundary condition on the RVE . Solve the RVE problem with unit using a StaticLinearpertubation Step object . Compute the Jacobian matrix ½Ct
MACRO
½C
!
. Input file for the macroscale problem . Define ½Ct in the UMAT . Solving of the macroscopic problem
2. Updating . Restart the analysis with a StaticLinear Step . Compute macroscopic stress
D
r
!
. Get the macroscale strain increment D 3. Convergence analysis Macrostructure analysis . Update stresses . Check for the convergence if not converged ) Next iteration: Steps 2 and 3 else ) Next load increment: Step 4 4. Microscale RESTART analysis . Invoke in the UMAT the RVE python script with the command: CALL SYSTEM (‘‘ABAQUS cae noGUI’’) . Restart the analysis on the RVE ) Execute Steps 1, 2 and 3
326
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
Fig. 1. Homogenization plug-in interface.
Fig. 2. Geometry of the bending problem and the cells considered to compare the stresses.
¼ 11
1 0
0 0 |fflfflfflfflffl{zfflfflfflfflffl} unit XX
þ12
0 1
0 0 |fflfflfflfflffl{zfflfflfflfflffl} unit XY
þ21
0
0
1 0 |fflfflfflfflffl{zfflfflfflfflffl} unit YX
þ22
0 0
0 1 |fflfflfflfflffl{zfflfflfflfflffl} unit YY
ð13Þ
ij are the components of the macroscale strain and where unit unit unit unit ; are the unit strain tensors. These unit XX YY ; XY ; YX strains constitute the three independent loading cases (considering the symmetry 21 ¼ 12 ) on the RVE in the context of the periodic boundary conditions (PBCs) Eq. (6). These
327
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
800
4000 Multiscale−ANM 3500
Multiscale−ANM
700
Fully meshed model
Fully meshed model
Multiscale with Abaqus
500
Load (N)
Load (N)
2500
E2/E1=0.1 2000
Horizontal elliptical inclusion
400 300
1500 1000
200
E2/E1=10
100
500 0
Multiscale with Abaqus
600
3000
0
20
40
60
80
0
100
Vertical elliptical inclusion 0
20
Fig. 3. The load–displacement diagrams of the bending problem for the biphasic RVE with circular inclusion and different rigidity ratios, displacement of point C.
independent loading are performed in ABAQUS using the Static Linear Perturbation Step object which do not impact the subsequent steps (for more detail see Appendix A). It is worth to mention, at this point, that the proposed formulation is developed for a 2D case but the methodology is still valid and easily extendable for a 3D case. The solution, u(x) of the RVE boundary value problem employing a local finite element procedure, can be expressed as a linear combination of the solution modes obtained by imposing independently the unit strain field through the boundary conditions Eq. (6):
~ ð11Þ ðxÞ þ 212 u ~ ð12Þ ðxÞ þ 22 u ~ ð22Þ ðxÞ; uðxÞ ¼ 11 u
ð14Þ
~ ðijÞ ðxÞ is the solution of the following system of where u equations:
(
40
60
80
Displacement (mm)
Displacement (mm)
~ ðijÞ ; duÞ ¼ 0 in x Lðu þ ~ ðijÞ u ~ ðijÞ ¼ xþðijÞ xðijÞ on @ x; u
Fig. 5. The load–displacement diagrams of the bending problem for the biphasic RVE with vertical and horizontal elliptical inclusion, and rigidity ratio of 10, displacement of point C.
8 þ þ x xð11Þ ¼ unit > XX ðx x Þ; > < ð11Þ þ xþð12Þ xð12Þ ¼ unit XY ðx x Þ; > > : xþ x ¼ unit ðxþ x Þ: YY ð22Þ ð22Þ
ð16Þ
Step 2: With the solution of the microscale problem in hand, the associated Jacobian is computed for the user-defined material subroutine (UMAT) associated to the macroscopic analysis. The solution of the problem (15) using finite element discretization within ABAQUS, provides the stress distribution outputs at the microstructural cell which allows computing the macroscopic elastic modulus C through the averaging relation defined in (10). Therefore, ~ ðr ~ XX ; r ~ YY ; r ~ XY Þ, using the resulting stress field functions r obtained from the aforementioned three loading cases, the macroscopic modulus can be computed as expressed below:
ð15Þ ~i ¼ C ¼ hr
1 jxj
Z
r~ ðxÞdx:
ð17Þ
x
with L the stiffness operator and The resulting outputs from the various steps are stored in ABAQUS in a dedicated output database. A Python script (see ABAQUS documentation for an overview) is used to extract the stress at each integration point n in the RVE and compute the integral defined in Eq. (6) n
800 Multiscale−ANM Fully meshed model Multiscale with Abaqus
700 600
½C ¼
1 X ~ X ðnÞ; r ~ Y ðnÞ; r ~ XY ðnÞ IVOLðnÞ: ½r jxj n¼1
ð18Þ
Load (N)
500 400
Inclusion volume fraction of 28%
300 200 100 0
Inclusion volume fraction of 7% 0
20
40
60
80
Displacement (mm) Fig. 4. The load–displacement diagrams for the biphasic RVE with circular inclusion, different volume fractions and rigidity ratio of 10, displacement of point C.
IVOL(n) = J W(n) is an integration point output variable which represents the integration point volume or section point volume in case of beams and shells. n represents the number of integration points. Step 3: Once the macroscopic modulus C is computed, the constitutive law is thus known for the macroscopic problem. Macroscopic displacements and therefore macroscopic can be obtain at each integration point. strain Step 4: At this stage, the macroscale strain is available and the postprocessing of the displacements as defined in Eq. (14) can be performed. The updating of the stress and strain fields is performed, subsequently, in the RVE as follows:
328
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
rðxÞ ¼ 11 r~ ð11Þ ðxÞ þ 212 r~ ð12Þ ðxÞ þ 22 r~ ð22Þ ðxÞ; ðxÞ ¼ 11 ~ð11Þ ðxÞ þ 212 ~ð12Þ ðxÞ þ 22 ~ð22Þ ðxÞ:
ð19Þ
is computed at each integration point, by solving a finite element problem at the RVE level. The nonlinear computational steps are detailed here below:
The implementation scheme is summarized in Table 1. It is remarked that when the problem is linear at the microscale, the homogenized macroscopic constitutive modulus C can be computed once and used for all calculation steps at the macroscale.
Step 1: Initialization Considering the equations system (12), in which the constitutive law is replaced by a nonlinear law, the PBC is rewritten in the following form:
Duþ Du ¼ Dni ðxþ x Þ
3.2. Nonlinear computational modeling
ð20Þ
where n and i are respectively the number of macroscale load increment and the number of equilibrium iterations. Then the microscale tangent problem can be written as follows:
The two-scale procedure presented in Section 3.1 can be extended to a more general nonlinear case that accounts for material nonlinearity. This requires first an appropriate two-scale bridging procedure environment. In this context, the solution of the macroscale nonlinear system is performed using a standard iterative technique quite similar to a modified Newton–Raphson procedure. Indeed, at each macroscale load increment, the tangent modulus Ct
8R > < x Dr : ddx ¼ 0 in x; Dr ¼ CtðrÞ : D; > : þ Du Du ¼ Dni ðxþ x Þ on @ x:
ð21Þ
S, Mises (Avg: 75%) +5.636e+02 +5.169e+02 +4.702e+02 +4.234e+02 +3.767e+02 +3.299e+02 +2.832e+02 +2.365e+02 +1.897e+02 +1.430e+02 +9.625e+01 +4.951e+01 +2.770e+00
Y
Z
X
(a) S, Mises (Avg: 75%) +4.240e+02 +3.953e+02 +3.666e+02 +3.378e+02 +3.091e+02 +2.803e+02 +2.516e+02 +2.228e+02 +1.941e+02 +1.654e+02 +1.366e+02 +1.079e+02 +7.912e+01
S, Mises (Avg: 75%) +4.127e+01 +3.924e+01 +3.722e+01 +3.519e+01 +3.316e+01 +3.113e+01 +2.910e+01 +2.707e+01 +2.504e+01 +2.301e+01 +2.098e+01 +1.895e+01 +1.692e+01
Y Z
Y X
Z
(b)
X
(c)
Fig. 6. Two-scale solution with ABAQUS: von Mises stresses. (a) Deformed shape of the macroscopic structure loaded at F = 100 N. (b) Deformed microstructure at point A. (c) Deformed microstructure at point B.
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
number. ABAQUS performs an (incremental) loading as well as with the time increment Dt, and a new guess D niþ1 for the strain increment until convergence achieved. The user subroutine UMAT supply ABAQUS with new Cauchy stress tensor updated according to the constitutive law as well as with the derivative of stress with respect to the strain increment. With this information, a new guess for the strain increment is calculated and the whole procedure is iterated until convergence. A new tangent modulus is computed from the microscale analysis and used for the macroscale analysis purposes. The procedure is the same in Step 1 except that the microscale analysis will be executed from the previously converged state of the RVE and both analyzes (micro and macro) are performed sequentially:
800 700 600
S11
S12
Load (N)
500 400
S22
300 200 Multiscale−ANM Fully meshed model Multiscale with Abaqus
100 0 −200
0
200
400
600
800
329
1000
Macroscopic stresses (MPa) Fig. 7. The load–stress diagrams of the bending problem for the biphasic RVE with circular inclusion, volume fraction of 7% and rigidity ratio of 10 at the point A.
800
First, the analysis is carried out with StaticLinearPerturbationStep object to compute the macroscale tangent matrix as in Step 1. Second, a restart analysis is performed using StaticStep object as specified in Step 2 in order to update the macroscale stresses according to (9). The solution loop and algorithm is depicted in Table 2. Fig. 1 presents the homogenization plug-in interface developed for this purpose.
700
4. Numerical example: two-scale coupled analysis in bending
S11
600
S22
S12
Load (N)
500 400 300 Multiscale−ANM Fully meshed model Multiscale with Abaqus
200 100 0 −40
−20
0
20
40
60
80
100
Macroscopic stresses (MPa) Fig. 8. The load–stress diagrams of the bending problem for the biphasic RVE with circular inclusion, volume fraction of 7% and rigidity ratio of 10 at the point B.
As in the linear analysis, the microscale analysis is performed first to obtain the initial macroscopic tangent modulus. The local problem is then solved by applying the PBC (20) with a unit strain using the static Linear pertubation Step object as in linear modeling (Section 3.1). Once the initial tangent modulus is computed, the analysis at the macro-level is then performed and the first strain increment D is computed and stored using the developed procedure UMAT. This increment will be used in the subsequent steps via iteration loops through the PBC (20). Step 2: Updating At this stage, the updating of the macroscale stresses is performed. The updating requires that we restart the previous microscale analysis. In parallel, the multiaxial strain loading is performed at the microscale level using StaticStep object which indicates that the step should be analyzed as a static load step according to Eq. (20). The resulting stresses Drni are then averaged according to (9) and used by the developed UMAT for the macroscale stresses updating. Step 3: Convergence analysis At this point, the convergence analysis is performed on the macroscopic stress after an updating of the load increment
The developed two-scale computational homogenization approach is applied to simulate a pure bending of a heterogeneous rectangular beam as in [19]. The beam is subjected to a concentrated load at the top right corner and clamped at its left edge (see Fig. 2). This example was selected to compare and assess the soundness of the proposed multiscale-ABAQUS approach when compared to the multiscale-ANM approach and the fully meshed composite structure as well such as in [19]. To this end, different numerical tests have been performed considering various shapes of the inclusions, volume fractions and rigidity ratios of the inclusion and the matrix as well. The predicted load–displacement curve using multiscale-ABAQUS approach is reported in Fig. 3 and compared to the one using multiscale-ANM [19] and results from a fully meshed model. These simulations are conducted considering two rigidity ratios between the matrix (E1) and the inclusion (E2): the case where the inclusion is more rigid than the matrix E2/E1 = 10 with E2 = 100,000 MPa, and the case where the inclusion is less rigid than the matrix E2/E1 = 0.1 with E2 = 10,000 MPa. Investigation considering different inclusion volume fractions is shown in Fig. 4 and compared to the MultiscaleANM prediction and results from the fully meshed model. Analysis considering different shape of the inclusion is reported in Fig. 5. Therefore, one can easily notice the validity of the implemented multiscale methodology from these comparative analyzes. Indeed, the proposed multiscale-ABAQUS approach reproduce with a very good accuracy the results obtained in [19]. It worth to mention that even though the updated lagrangian viewpoint is adopted here, while in [19] a total lagrangian formulation was considered, the results are still very similar. In addition, the geometric nonlinearity at the microscale [19] does not have any effect on the response at the macroscale due to the fact that displacements on the RVE remains small. In order to show the flexibility and significance of the developed implementation, we present in Fig. 6 the deformed macroscale mesh and the resulting von Mises stress field, on two units cells located at the integration points A and B. In Figs. 7 and 8, the averaged Piola–Kirchhoff stresses S11, S22, S12 versus loading parameter are also depicted at two distinct points A and B corresponding respectively to large deformation and large rotations.
330
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
Listing 1. Example of Python script architecture used for the analysis on the RVE.
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
Listing 1. (continued)
331
332
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
Listing 2. Example of user-defined subroutine associate to the nonlinear analysis.
5. Conclusion A numerical tool fully integrated in ABAQUS and dedicated to a computational multiscale finite element analysis of heterogeneous materials and structures is proposed. The multiscale-ABAQUS approach presented in this work serves twofold purpose: (i) it demonstrates that the computational homogenization can be seamlessly carried out using ABAQUS for both linear and nonlinear solid mechanics problems. Therefore, it constitutes a reference technique for the two-scale analysis implementation, since it provides satisfactory results as shown here. (ii) Using the
proposed methodology allows to bypass some heavy numerical post-processes on the scale bridging relations or on the extraction of the macroscale tangent modulus such as in classical implementation of computational homogenization [23,24]. In the context of a two scale homogenization method, ABAQUS offers a wide outputs database which allows to get many informations for solving mechanical problems without programming any finite element (FE) discretization. Once all input files, user-defined subroutines and Python scripts are setting up, ABAQUS allows to customize itself using a graphical user interface and then with a single ‘‘press button’’ the multiscale analysis can be executed. Those aspects
A. Tchalla et al. / Composites: Part B 52 (2013) 323–333
could be interesting for industrials which prefer to use well-known commercial FE codes which ensure the continuity of use and, at the same time, they do not require frequent maintenance operations with respect to the ‘‘in-house’’ codes. Acknowledgments The authors acknowledge the financial support of the Luxembourgish National Research Fund (FNR). This work was performed as a part of the PhD thesis of the first author under the AFR Grant (882919) and in the framework of SIMUCOMP FP7-Inter Project. Appendix A See Listing 1. Appendix B See Listing 2. References [1] Willis J. Variational and related methods for the overall properties of composites. Adv Appl Mech, Vol. 21. Elsevier; 1981. http://dx.doi.org/ 10.1016/S0065-2156(08)70330-2. [2] Muller S. Homogenization of nonconvex integral functionals and cellular elastic materials. Arch Ration Mech Anal 1987;99:189–212. http://dx.doi.org/ 10.1007/BF00284506. [3] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials, Amsterdam, The Netherlands. Appl Math Mech 1993;37(3):687. [4] Azoti W, Koutsawa Y, Bonfoh N, Lipinski P, Belouettar S. On the capability of micromechanics models to capture the auxetic behavior of fibers/particles reinforced composite materials. Compos Struct 2011;94(1):156–65. http:// dx.doi.org/10.1016/j.compstruct.2011.07.006. [5] Koutsawa Y, Biscani F, Belouettar S, Nasser H, Carrera E. Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials. Compos Struct 2010;92(4):964–72. http://dx.doi.org/10.1016/j.compstruct.2009.09.041. [6] Kari S, Berger H, Rodriguez-Ramos R, Gabbert U. Computational evaluation of effective material properties of composites reinforced by randomly distributed spherical particles. Compos Struct 2007;77(2):223–31. http://dx.doi.org/ 10.1016/j.compstruct.2005.07.003. [7] Kanouté P, Boso D, Chaboche J, Schrefler B. Multiscale methods for composites: a review. Arch Comput Methods Eng 2009;16:31–75. http://dx.doi.org/ 10.1007/s11831-008-9028-8.
333
[8] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc London Ser A 1957;241(1226):376–96. http://dx.doi.org/10.1098/rspa.1957.0133. [9] Sanchez-Palencia E. Einstein-like approximation for homogenization with small concentration. I-Elliptic problems. Nonlinear Anal Theory Methods Appl 1985;9(11):1243–54. http://dx.doi.org/10.1016/0362-546X(85)90033-1. [10] Feyel F, Calloch S, Marquis D, Cailletaud G. F.e. computation of a triaxial specimen using a polycrystalline model. Comput Mater Sci 1997;9(12):141–57. [11] Fish J, Markolefas S. Adaptive s-method for linear elastostatics. Comput Methods Appl Mech Eng 1993;104(3):363–96. http://dx.doi.org/10.1016/ 0045-7825(93)90032-S. [12] Zohdi TI, Oden J, Rodin GJ. Hierarchical modeling of heterogeneous bodies. Comput Methods Appl Mech Eng 1996;138(1-4):273–98. http://dx.doi.org/ 10.1016/S0045-7825(96)01106-1. [13] Fish J, Shek K, Pandheeradi M, Shephard MS. Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Comput Methods Appl Mech Eng 1997;148(1-2):53–73. http:// dx.doi.org/10.1016/S0045-7825(97)00030-3. [14] Hughes TJR. Multiscale phenomena: Green’s functions, the dirichlet-toneumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 1995;127(1-4):387–401. http://dx.doi.org/10.1016/0045-7825(95)00844-9. [15] Fish J, Suvorov A, Belsky V. Hierarchical composite grid method for global-local analysis of laminated composite shells. Appl Numer Math 1997;23(2):241–58. http://dx.doi.org/10.1016/S0168-9274(96)00068-2. [16] Feyel F, Chaboche J-L. Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput Methods Appl Mech Eng 2000;183(3-4):309–30. [17] Miehe C. Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput Methods Appl Mech Eng 2002;191(44):4971–5005. [18] Asada T, Ohno N. Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis. Int J Solids Struct 2007;44(22-23):7261–75. http://dx.doi.org/10.1016/j.ijsolstr.2007.04.007. [19] Nezamabadi S, Yvonnet J, Zahrouni H, Potier-Ferry M. A multilevel computational strategy for handling microscopic and macroscopic instabilities. Comput Methods Appl Mech Eng 2009;198(27-29):2099–110. [20] Nezamabadi S, Zahrouni H, Yvonnet J, Potier-Ferry M. A multiscale finite element approach for buckling analysis of elastoplastic long fiber composites. Int J Multiscale Comput Eng 2010;8(3):287–301. [21] Yuan Z, Fish J. Toward realization of computational homogenization in practice. Int J Numer Methods Eng 2008;73(3):361–80. [22] Hughes TJR, Winget J. Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. Int J Numer Methods Eng 1980;15(12):1862–7. [23] Kouznetsova V. Computational homogenization for the multi-scale analysis of multi-phase materials. PhD thesis. Technische Universiteit Eindhoven; 2002. [24] Kouznetsova V, Geers M, Brekelmans W. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 2004;193:5525–50. http://dx.doi.org/10.1016/j.cma.2003.12.073.
