Phase‐shift convergence method of homogenization in viscoplasticity
Sergiy Nesenenko, Hans‐Dieter Alber · PAMM · 2008
Setting of the problemIn this work we justify the homogenized problem for the quasistatic initial boundary value problem with internal variables, which models the deformation behavior of viscoplastic materials with a periodic microstructure.We formulate the problem: Suppose that Ω is an open bounded set, the set of material points of the body, with the Lipschitz boundary ∂Ω.T e denotes a positive number (time of existence).Let S 3 denote the set of symmetric 3 × 3-matrices, and let u(x, t) ∈ IR 3 be the unknown displacement of the material point, T (x, t) ∈ S 3 be the unknown Cauchy stress tensor and z(x, t) ∈ IR N be the unknown vector of internal variables.The model equations of the problem are