A Projection Method for an Elasto-plasticity Model with Linear Kinematic Hardening

Abstract

We consider a dynamical elasto-plasticity system with Kelvin--Voigt viscosity and linear kinematic hardening of Melan--Prager type. The model is formulated in a variational framework in which a constraint set for the stress evolves in time and is translated by an internal (backstress) variable. As a consequence, the flow rule is coupled with an equation of motion through a quasi-variational structure, since the constraint set depends on the unknown internal variable. To construct solutions, we employ Rothe's method and introduce a projection-based time discretization. Each time step consists of solving a linear viscous-elastic subproblem to obtain a trial stress, followed by a projection onto the translated constraint set. We establish stability of the resulting discrete solutions under suitable norms. By compactness and passage to the limit as the time step tends to zero, we prove existence of a weak solution in the variational sense, and uniqueness follows from an energy argument. The results cover time-dependent yield bounds without assuming spatial continuity or a strictly positive lower bound, and the discretization provides a constructive basis for numerical approximation.

Figures (1)

• Figure 1: The geometric configuration of $\alpha_{n-1}^D$, $\alpha_n^D$, $\sigma_n^D$, and $(\sigma_n^*)^D$ in ${\mathcal{S}}_d$. Points $\alpha_{n-1}^D, \alpha_n^D, \sigma_n^D, (\sigma_n^*)^D$ lie on a line in this order. The distances satisfy $|\sigma_n^D - \alpha_n^D| = g_n$ and $|\alpha_n^D - \alpha_{n-1}^D| = b|(\sigma_n^*)^D - \sigma_n^D|$.

Theorems & Definitions (11)

• Definition 1.2
• Theorem 4.1
• Theorem 4.2
• Lemma 5.1: Trangenstein13
• Lemma 5.2: HR90
• Lemma 5.3: AM25
• Proposition 5.4: AM25
• Lemma 5.5: The positive definiteness and continuity of $S$
• proof
• Lemma 5.6