Abstract
Based on the complementary potential energy variational principle, in this work, we proposed a stress-driven homogenization procedure to compute overall effective material properties for elastic composites with locally heterogenous micro-structures. We have developed a novel incremental variational formulation for homogenization problems of both infinitesimal and finite deformations where the macro-stress-based complementary potential energy is obtained for hyperelastic materials for a global minimization problem with respect to fine-scale displacement fluctuation field. The point of departure of our approach is a general complementary variational principle formulation that can determine material responses of elastic composites with heterogeneous micro structures. We have implemented the proposed stress-driven computational homogenization procedure with the finite element method. By comparing the numerical results with the analytical method and the strain-driven homogenization method, we find that the stress-driven homogenization offers the lower bound estimate of materials properties for elastic composites.
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References
Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357
Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A Math Phys Sci 326(1565):131
Willis JR (1981) Variational and related methods for the overall properties of composites. In: Advances in applied mechanics, vol 21, Elsevier, pp 1–78
Castaneda PP, Suquet P (1997) Nonlinear composites. In: Advances in applied mechanics, vol 34, Elsevier, pp 171–302
Li S, Wang G (2008) Introduction to micromechanics, introduction to micromechanics. World Scientific, Singapore
Sánchez-Palencia E (1980) Non-homogeneous media and vibration theory. Lecture notes in physics 127
Bensoussan A, Lions JL, Papanicolaou G ( 2011) Asymptotic analysis for periodic structures, Asymptotic analysis for periodic structures, vol 374 (American Mathematical Soc.)
Terada K, Hori M, Kyoya T, Kikuchi N (2000) Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct 37(16):2285
Geers M, Kouznetsova V, Brekelmans W (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175
Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3–4):387
Miehe C (2002) Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Methods Eng 55(11):1285
Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A Math Phys Sci 241(1226):376
Li S, Wang G, Sauer R (2007) The Eshelby tensors in a finite spherical domain Part II: applications to homogenization. ASME J Appl Mech 74:784
Suquet P (1985) Elements of homogenization for inelastic solid mechanics, homogenization techniques for composite media. In: Lecture notes in physics, vol 272, Springer, p 193
Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157(1–2):69
Miehe C (2003) Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput Methods Appl Mech Eng 192(5–6):559
van Dijk N (2016) Formulation and implementation of stress-driven and/or strain-driven computational homogenization for finite strain. Int J Numer Methods Eng 107(12):1009
Javili A, Saeb S, Steinmann P (2017) Aspects of implementing constant traction boundary conditions in computational homogenization via semi-Dirichlet boundary conditions. Comput Mech 59(1):21
Hashin Z, Shtrikman S (1962) A variational approach to the theory of the elastic behaviour of polycrystals. J Mech Phys Solids 10(4):343
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127
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**e, Y., Li, S. A stress-driven computational homogenization method based on complementary potential energy variational principle for elastic composites. Comput Mech 67, 637–652 (2021). https://doi.org/10.1007/s00466-020-01953-8
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DOI: https://doi.org/10.1007/s00466-020-01953-8