Saroukhani, S.; Vafadari, R.; Andersson, R.; Larsson, Fredrik; Runesson, K. On statistical strain and stress energy bounds from homogenization and virtual testing. (English) Zbl 1406.74569 Eur. J. Mech., A, Solids 51, 77-95 (2015). Summary: Computational homogenization for quasistatic stress problems is considered, whereby the macroscale stress is obtained via averaging on Statistical Volume Elements (SVE:s). The variational “workhorse” for the subscale problem is derived from the presumption of weak micro-periodicity, which was proposed by F. Larsson et al. [Comput. Methods Appl. Mech. Eng. 200, No. 1–4, 11–26 (2010; Zbl 1225.74069)]. Continuum (visco)plasticity is adopted for the mesoscale constituents, whereby a pseudo-elastic, incremental strain energy serves as the potential for the updated stress in a given time-increment. Strict bounds on the incremental strain energy are derived from imposing Dirichlet and Neumann boundary conditions, which are defined as suitable restrictions of the proposed variational format. For this purpose, both the standard situation of complete macroscale strain control and the (less standard) situation of macroscale stress control are considered. Numerical results are obtained from “virtual testing” of SVE:s in terms of mean values and a given confidence interval, and it is shown how these properties converge with respect to the SVE-size for different prescribed macroscale deformation modes and different statistical properties of the random microstructure. In addition, the upper and lower bounds for a sequence of increasing strain levels, for a fixed SVE-size, are used as “data” for the calibration of a macroscopic elastic-plastic constitutive model. Cited in 4 Documents MSC: 74Q20 Bounds on effective properties in solid mechanics 74Q15 Effective constitutive equations in solid mechanics Keywords:virtual testing; computational homogenization; effective properties Citations:Zbl 1225.74069 PDFBibTeX XMLCite \textit{S. Saroukhani} et al., Eur. J. Mech., A, Solids 51, 77--95 (2015; Zbl 1406.74569) Full Text: DOI References: [1] Brisard, S.; Sab, K.; Dormieux, L., New boundary conditions for the computation of the apparent stiffness of statistical volume elements, J. Mech. Phys. Solids, 61, 12, 2638-2658, (2013) · Zbl 1432.74192 [2] Danielsson, M.; Parks, D. M.; Boyce, M. C., Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers, J. Mech. Phys. Solids, 55, 3, 533-561, (2007) · Zbl 1173.74002 [3] Geers, M. G.D.; Kouznetsova, V. G.; Brekelmans, W. A.M., Multi-scale computational homogenization: trends and challenges, J. Comput. Appl. Math., 234, 2175-2182, (2010) · Zbl 1402.74107 [4] Hazanov, S.; Huet, C., Order relationships for boundary condition effects in heterogeneous bodies smaller than the representative volume, J. Mech. Phys. Solids, 42, 1995-2011, (1995) · Zbl 0821.73005 [5] Huet, C., Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids, 38, 813-841, (1990) [6] Kouznetsova, V.; Geers, M. G.D.; Brekelmans, W. A.M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, Int. J. Numer. Methods Eng., 54, 1235-1260, (2002) · Zbl 1058.74070 [7] Larsson, F.; Runesson, K.; Saroukhani, S.; Vafadari, R., Computational homogenization based on a weak format of micro-periodicity for RVE-problems, Comput. Methods Appl. Mech. Eng., 200, 11-26, (2011) · Zbl 1225.74069 [8] Larsson, F.; Runesson, K.; Su, F., Variationally consistent computational homogenization of transient heat flow, Int. J. Numer. Methods Eng., 81, 1659-1686, (2010) · Zbl 1183.80109 [9] Michel, J. C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Eng., 172, 109-143, (1999) · Zbl 0964.74054 [10] Miehe, C.; Koch, A., Computational micro-to-macro transitions of discretized microstructures undergoing small strains, Arch. Appl. Mech., 72, 300-317, (2002) · Zbl 1032.74010 [11] Miehe, C.; Schröder, J.; Schotte, J., Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials, Comput. Methods Appl. Mech. Eng., 171, 387-418, (1999) · Zbl 0982.74068 [12] Nguyen, Q.-S., On standard dissipative gradient models, Ann. Solid Struct. Mech., 1, 79-86, (2010) [13] Ostoja-Starzewski, M., Material spatial randomness: from statistical to representative volume element, Probab. Eng. Mech., 21, 112-132, (2006) [14] Ostoja-Starzewski, M., Microstructural randomness and scaling in mechanics of materials, (2008), Chapman & Hall/CRC Taylor & Francis Group · Zbl 1148.74002 [15] Quilici, S.; Cailletaud, G., FE simulation of macro-, meso-, and microscales in polycristaline plasticity, Comput. Mater. Sci., 16, 383-390, (1999) [16] Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D. D.; Bieler, T. R.; Raabe, D., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications, Acta Mater., 58, 1152-1211, (2010) [17] Salmi, M.; Auslender, F.; Bornert, M.; Fogli, M., Apparent and effective mechanical properties of linear matrix-inclusion random composites: improved bounds for the effective behavior, Int. J. Solids Struct., 49, 10, 1195-1211, (2012) [18] Schmauder, S.; Wulf, J.; Steinkopff, T.; Fischmeister, H., Micromechanics of plasticity and damage in an al/sic metal matrix composite, (Pineau, A.; Zaoui, A., Micromechanics of Elasticity and Damage of Multiphase Materials, (1996), Kluwer Academic Publishers) [19] Schröder, J.; Balzani, D.; Brands, D., Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions, Arch. Appl. Mech., 81, 975-997, (2011) · Zbl 1271.74056 [20] Suquet, P. M., Effective properties of nonlinear composites, (Suquet, P. M., Continuum Micromechanics, CISM Courses and Lectures, Udine, (1977), Springer Verlag Berlin), 197-264 · Zbl 0883.73051 [21] Suquet, P. M., Overall potentials and extremal surfaces of power law or ideally plastic materials, J. Mech. Phys. Solids, 41, 981-1002, (1993) · Zbl 0773.73063 [22] Temizer, I.; Wriggers, P., On the computation of the macroscopic tangent for multiscale volumetric homogenization problems, Comput. Methods Appl. Mech. Eng., 198, 495-510, (2008) · Zbl 1228.74066 [23] Teply, J. L.; Dvorak, G., Bounds on overall instantaneous properties of elastic-plastic composites, J. Mech. Phys. Solids, 36, 29-58, (1988) · Zbl 0632.73052 [24] Torquato, S., Random heterogeneous materials. microstructure and macroscopic properties, (2002), Springer Verlag · Zbl 0988.74001 [25] Zohdi, T.; Wriggers, P., A model for simulating the deterioration of structural-scale material responses of microheterogeneous solids, Comput. Methods Appl. Mech. Eng., 190, 2803-2823, (2001) · Zbl 1013.74082 [26] Zohdi, T. I.; Wriggers, P., Introduction to computational micromechanics, (2005), Springer · Zbl 1085.74001 This reference list is based on information provided by the publisher or from digital mathematics libraries. 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