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Deformation Driven Homogenization of Fracturing Solids
Ercan Gurses∗, Manuel Birkle, and Christian Miehe
University of Stuttgart, Institute of Applied Mechanics (Chair I), Pfaffenwaldring 7 70550 Stuttgart GERMANY
The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus onmulti–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as wellas matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, weextend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonicalvariational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms fornumerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundaryconditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturingmicrostructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of atraction–type boundary condition in the deformation–driven context.
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Modeling of Fracture
In the context of fi nite elements there are different approaches for the modeling of discrete cracks. Two main approaches canbe identifi ed, interelement discontinuities where cracks can run through the fi nite elements and intraelement discontinuitieswhere cracks can run over the element boundaries. In our work we follow the intraelement approach which can be basedon the insertion of interface elements between standard fi nite elements to model the ductile fracture or the removal of theconnection between fi nite elements to model the brittle fracture. The interface elements may have a softening type materialmodel relating tractions to displacment jumps to model cohesive fracture and in the limit the traction may drop directly to zerowith the initiation of the crack yielding a brittle fracture response.
2 Deformation Driven Homogenization of Nonlinear Composites
The main aspects of the approach are governed by the incremental variational formulation for the local constitutive responseas outlined in [1]. We extend the formulation to the solids with discontinuities by the defi nition of incremental potentials forthe bulk and the crack surface seperately,
Wb(Fn+1) = infI∈G
tn+1∫
tn
[ ψb + φb ] dt and Ws(δn+1) = infJ∈G
tn+1∫
tn
[ ψs + φs ] dt (1)
in terms of free energies ψb, ψs and dissipation potentials φb, φs for the bulk and the crack surface, respectively. I and Jstand for the set of internal variables that are computed by the minimization problem (??), Fn+1 is the deformation gradientand δn+1 is the displacement jump on the crack surface. As the key homogenization condition, we extend the minimizationproblem defi ned in [1] to a more general one considering the discontinuities,
W (Fn+1) = infwn+1∈W
1|V|
∫B[ Wb(Fn+1 + ∇wn+1) + δ(Γ)Ws([[Fn+1X + wn+1]]) ] dV (2)
which is subject to constraints coming from the specifi c boundary conditions. The optimization problem (??) defi nes ahomogenized macro potential W as the minimum volume average of the micro potential W with respect to a fluctuation fi eldwn+1. In (??) Fn+1 is a given macro deformation gradient and δ(Γ) stands for the Dirac function placed on the discontinuitysurface Γ. Having defi ned the homogenized incremental potential the homogenized stresses and the moduli are obtained by
Pn+1 = ∂Fn+1W (Fn+1) and An+1 = ∂2
Fn+1Fn+1W (Fn+1) . (3)
An approximate numerical solution of (??) can be obtained in the context of fi nite element method by discretizing the de-formation gradient F and the displacement jump vector δ. A formulation of the boundary conditions for linear deformation,uniform tractions and periodic deformations can be achieved by a Lagrange or penalty functional. For the details of thenon–trivial implementation of different boundary conditions in the macro–deformation driven context we refer to [1,2].
∗ Corresponding author: e-mail: [email protected], Phone: +49 711 685 6377, Fax: +49 711 685 6347
PAMM · Proc. Appl. Math. Mech. 5, 333–334 (2005) / DOI 10.1002/pamm.200510143
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
displacement b.c. traction b.c.a) displacement b.c. traction b.c.b)
Fig. 1 Final deformed state of the composite for different boundary conditions (a) tension test, (b) biaxial tension test.
3 Numerical Examples
We demonstrate the performance of the above outlined homogenization technique by two examples where a square specimenwith three circular inclusions is subjected to uniaxial and biaxial tensile loading conditions. The inclusions are elastic whilethe matrix is elastoplastic with saturation type isotropic hardening. Two types of boundary conditions are investigated, namelythe displacement and the traction boundary conditions. The value of the normal traction is used as a crack criterion. Thelimiting normal traction is set to a lower value for the segments between inclusions and matrix material in order to trigger thecrack initiation from the interface. In Figure ?? the deformed meshes are plotted where the inclusions are fi lled with yellow.Figure ??(a) and ??(b) are for the uniaxial tensile test and the biaxial test, respectively. In both tests the cracks start from theinterface between the matrix and inclusion and then run through the matrix. In Figure ??(a) for both boundary conditions thecrack paths are mainly perpendicular to tensile axis. The key difference between two boundary conditions becomes obviouswhen the crack reaches to the outer boundary of the RVE. For the case of displacement boundary conditions the crack can notseparate the RVE into pieces completely since the displacement boundary conditions puts the constraint to the nodes on theboundary. On the other hand for the case of traction boundary conditions the crack can run through the RVE and separate itinto two pieces. The outcome of this main difference can be seen also in the plots of homogenized stresses. In Figure ??(a) thehomogenized stresses are plotted for both boundary conditions. The traction boundary condition provides a stress responsewhich can relax to zero level due to complete separation while displacement boundary conditions lock after some softening.The principally same behavior is obtained also for the biaxial loading of the same composite specimen. The crack patternsof the RVE are visualized in Figure ??(b) for both boundary conditions. The traction boundary conditions provide again acomplete relaxation of the homogenized stresses to zero while the displacement boundary conditions show a locking behavioras seen in Figure ??(b). As a key result in the deformation–driven homogenization, we show both the bound character ofdifferent boundary conditions and the signifi cance of the traction–type boundary condition to model a complete separation.
11εnormal strain
11σno
rmal
str
ess
a)
displacement b.c.
traction b.c.
0
0.02
0.04
0.06
0.08
0.1
0 0.005 0.01 0.015 0.02 0.025
11σ
traction b.c.
b)
displacement b.c.
11εnormal strain
norm
al s
tres
s
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.005 0.01 0.015 0.02 0.025
Fig. 2 Homogenized Stress response for different boundary conditions (a) tension test, (b) biaxial tension test.
Acknowledgements Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grant SFB 404 / C11and DFG common project FOR 509.
References
[1] C. Miehe, Int. J. Num. Meth. Engrg. 55, 1285 (2002).[2] C. Miehe, E. Gurses and M. Birkle, Comput. Methods Appl. Mech. Engrg. (submitted to).
© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Section 7 334
