Stochastic homogenization of plasticity equations

Heida, Martin; Schweizer, Ben

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 153-176

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter $\epsilon >0$ denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit $\epsilon \to 0$. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution $[0,T]\ni t\mapsto \xi \left(t\right)\in$ induces a stress evolution $[0,T]\ni t\mapsto \Sigma \left(\xi \right)\left(t\right)\in$. Once the hysteretic evolution law $\Sigma$ is justified for averages, we obtain that the macroscopic limit equation is given by $-\nabla ·\Sigma \left(\nabla {}^{s}u\right)=f$.

Classification : 74C05, 35R60, 74Q10
Keywords: Small strain plasticity, stochastic homogenization

H.-D. Alber, Initial-boundary value problems for constitutive equations with internal variables. Materials With Memory, Vol. 1682 of Lect. Notes Math. Springer Verlag, Berlin (1998). | MR | Zbl

H.-D. Alber, Evolving microstructure and homogenization. Contin. Mech. Thermodyn. 12 (2000) 235–286. | MR | Zbl | DOI

H.-D. Alber and S. Nesenenko, Justification of homogenization in viscoplasticity: From convergence on two scales to an asymptotic solution in $L^2\left(\Omega \right)$. J. Multiscale Model. 1 (2009) 223–244. | DOI

P.G. Ciarlet, The finite element method for elliptic problems, Reprint of the 1978 original. Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). | MR | Zbl

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization. Annali di Matematica Pura ed Applicata 144 (1986) 347–389. | MR | Zbl | DOI

G. Francfort and A. Giacomini, On periodic homogenization in perfect elasto-plasticity. J. Eur. Math. Soc. 16 (2014) 409–461. | MR | Zbl | DOI

W. Han and B.D. Reddy, Mathematical theory and numerical analysis. Plasticity, Vol. 9 of Interdisciplinary Appl. Math. Springer-Verlag, New York (1999). | MR | Zbl

H. Hanke, Homgenization in gradient plasticity. Math. Models Methods Appl. Sci. 21 (2011) 1651–1684. | MR | Zbl | DOI

M. Heida and B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations. ZAMM Z. Angew. Math. Mech. 96 (2016) 5–23. | MR | Zbl | DOI

V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer (1994). | MR

S.M. Kozlov, The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188–202. | MR | Zbl

A. Mielke, T. Roubicek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31 (2008) 387–416. | MR | Zbl | DOI

A. Mielke and A.M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39 (2007) 642–668. | MR | Zbl | DOI

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials. Archive for Rational Mech. Anal. 99 (1987) 189–212. | MR | Zbl | DOI

F. Murat and L. Tartar, H-convergence. In Topics in the Mathematical Modelling of Composite Materials. Springer (1997) 21–43. | MR | Zbl

S. Nesenenko, Homogenization in viscoplasticity. SIAM J. Math. Anal. 39 (2007) 236–262. | MR | Zbl | DOI

G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients. In {Random fields, Vol. I, II (Esztergom 1979), Vol. 27 of Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam New York (1981) 835--873. | MR | Zbl

R.~Rockafellar and R.-B. Wets, Variational Analysis. Springer (1998). | MR | Zbl

B. Schweizer, Homogenization of the Prager model in one-dimensional plasticity. Contin. Mech. Thermodyn. 20 (2009) 459–477. | MR | Zbl | DOI

B. Schweizer and M. Veneroni. Periodic homogenization of the Prandtl-Reuss model with hardening. J. Multiscale Modell. 2 (2010) 69–106. | DOI

B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization. Netw. Heterog. Media 6 (2011) 755–781. | MR | Zbl | DOI

B. Schweizer and M. Veneroni. Homogenization of plasticity equations with two-scale convergence methods. Appl. Anal. 94 (2015) 376–399. | MR | Zbl | DOI

A. Visintin, On homogenization of elasto-plasticity. J. Phys.: Conf. Ser. 22 (2005) 222–234.

A. Visintin, Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Contin. Mech. Thermodyn. 18 (2006) 223–252. | MR | Zbl | DOI

A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1363–1401. | MR | Zbl | DOI

V. Zhikov and A. Pyatniskii, Homogenization of random singular structures and random measures. Izv. Math. 70 (2006) 19–67. | MR | Zbl | DOI