AN EVOLUTIONARY ELASTOPLASTIC PLATE MODEL DERIVED VIA Gamma-CONVERGENCE

作者：Liero Matthias^*; Mielke Alexander

来源：Mathematical Models and Methods in Applied Sciences, 2011, 21(9): 1961-1986.

DOI：10.1142/S0218202511005611

• Linearized elastoplasticity
• rate-independent system
• Gamma-convergence
• Mosco convergence
• hysteresis
• generalized Prandtl-Ishlinskii operator

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff-Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies.

• 出版日期2011-9