Numerical approximations of problems in plasticity: error analysis and solution algorithms.

*(English)*Zbl 0896.73078The paper starts from the formulation of an initial-boundary value problem of elastoplasticity as a variational inequality with the displacement, plastic strain and internal (hardening) variables as primary unknowns. For this formulation, the existence and uniqueness proofs can be found in the closely related paper of W. Han and B. D. Reddy [Comput. Mech. Adv. 2, 283-400 (1995; Zbl 0847.73078)]. For the numerical solution of the problem, a fully discrete scheme is described which uses finite differences for the quasi-time variable and finite elements in space. The convergence rate of the approximate solution is discussed. It is shown that an improved convergence in quasi-time can be obtained by using the Crank-Nicolson type algorithm, and an improved convergence in space for linear finite elements can be obtained by using discontinuous piecewise linear approximation for the plastic strains. The same convergence rate and stability are proved also for the numerical integration of the functional of plastic strains within the discrete scheme. Finally, an iterative technique of the predictor-corrector type for solving the discrete problem is described. The convergence of this type iterations for the continuous problem is proved in the above reference.

Reviewer: R.Blaheta (Ostrava)

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

74C99 | Plastic materials, materials of stress-rate and internal-variable type |

74S05 | Finite element methods applied to problems in solid mechanics |

74S20 | Finite difference methods applied to problems in solid mechanics |

##### Keywords:

initial-boundary value problem; variational inequality; elastoplasticity; convergence rate; iterative technique; existence; uniqueness; quasi-time variable; Crank-Nicolson type algorithm; linear finite elements; discontinuous piecewise linear approximation for plastic strains; functional of plastic strains; predictor-corrector
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\textit{W. Han} et al., Numer. Linear Algebra Appl. 4, No. 3, 191--204 (1997; Zbl 0896.73078)

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