Keywords: anisotropy elasto-plasticity, Hill's yield criterion, mixed hardening.

In this work, in the context of finite element procedures [1], an algorithm for small strains orthotropic elasto-plasticity based on Hill's yield criterion [2] with mixed hardening is formulated. In anisotropic elasto-plasticity with mixed hardening, a consistently linearized local implicit Newton-Raphson algorithm based on a single parameter is not possible because both the hardening and the flow direction depend on the consistency parameter, and the final converged parameter depends on those parameters. Approximations using a single scalar parameter are known as the governing parameter method (GPM) [3,4]. Using this method, in the case of anisotropic elasto-plasticity with mixed hardening, approximations are employed and a final scalar nonlinear equation is obtained. As a result of the approximations employed in the method, the most effective Newton-Raphson algorithm for solving the local equations is substituted by the bisection method [4], where a solution is granted but the efficiency of the algorithm is clearly inferior to the former schemes.

A fully implicit Newton stress integration procedure, based on the return mapping algorithm with associated flow and hardening rules is presented. In contrast with the aforementioned procedures, the algorithm is designed as two nested loops so no approximation is made and a local Newton-Raphson procedure is possible. In the inner loop the isotropic and kinematic hardening parameters are frozen and the consistency parameter is obtained. With this converged consistency parameter, the hardening parameters are given and the final solution is obtained.

For the global equilibrium iterations, a consistent elasto-plastic tangent matrix must be obtained in order to preserve the quadratic convergence rate of the Newton type procedures. The consistent tangent is independent of the local specific layout of the stress integration algorithm since it depends only on the converged solution, and may be formulated accordingly.

On the other hand, the stress integration procedure presented in this paper may provide the basis for a formulation in finite deformations. The extension to finite strain kinematics may be reduced to the implementation of a pre-processing and a post-processing from the small radial return algorithm [5,6,7].

A numerical example is presented to illustrate applicability and effectiveness of the computational algorithm. In this example, we compare the results with the alternative algorithm of Kojic et al. [4].

1

K.J. Bathe, "Finite Element Procedures", Prentice-Hall 1996.

2

R. Hill, "A theory of the yielding and plastic flow of anisotropic metals", Proceedings of the Royal Society of London. Series A, 193, 281-297, 1948. doi:10.1098/rspa.1948.0045

3

M. Kojic, K.J. Bathe, "Inelastic Analysis of Solids and Structures", Springer 2005.

4

M. Kojic, N. Grujovic, R. Slavkovic, "A general orthotropic von Mises plasticity material model with mixed hardening: model definition and implicit stress integration procedure", Journal of Applied Mechanics, 63, 376-382, 1996. doi:10.1115/1.2788875

5

F.J. Montáns, K.J. Bathe, "Computational issues in large strain elasto-plasticity: and algorithm for mixed hardening and plastic spin", International Journal for Numerical Methods in Engineering, 63(2), 159-195, 2005. doi:10.1002/nme.1270

6

F.J. Montáns, M.A. Caminero, K.J. Bathe, "Small and large strains algorithms for anisotropic plasticity", Computers & Structures, to appear.

7

M.A. Caminero, F.J. Montáns, K.J. Bathe, "Large strains anisotropic elastoplasticy using logarithmic strains, Lee decomposition and generalized Kirchhoff stress measures", Computers & Structures, to appear.

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