Modeling of Elastic-Plastic Deformation Based on Updated Initial Configuration of Solid Body

Abstract

In this paper we discuss some modifications of the classical two-step algorithm (elastic-predictor, inelastic-corrector) usually called the radial return-map method. The neo-Hooke rheological model is employed in the analysis of elastic behavior instead of the Hooke model. The von Mises yield criterion $-J_2 (\mathbf{S}) \le \sigma _s^2/3$ is used in the equivalent form $-J_2 (\mathbf{B}_D) \le \sigma _s^2/(3\mu ^2)$. The main difference of the proposed algorithm in comparison with the traditional one is that in the simulations of plastic deformations we switch the emphasis from corrections of the stress tensor to irreversible corrections of the initial configuration of the solid body. The stress tensor is automatically corrected by this procedure. The implicit integration method is suggested for the correction of the initial configuration in the case of plastic flows. While changing the initial configuration, we automatically get plastic (irreversible) deformation at any time step. This algorithm allows us to calculate residual stresses in the elastic-plastic solid after removing the external load as a result of unloading after non-uniform plastic deformation. It is also used for an accurate simulation of deformations of both perfectly plastic and elastic-plastic solids with workhardening including the Bauschinger effect. Numerical examples show some advantages of the algorithm developed in this work for a springback problem.

A Notation

An orthogonal basis for the 3D vector space is a set of orthogonal unit vectors $\mathbf{{\vec {e}}}_i$ ($i=1,2,3$). We use here only fixed rectangular Cartesian coordinate system. The scalar product of any two of these vectors is

$$ \mathbf{{\vec {e}}}_k\cdot \mathbf{{\vec {e}}}_s = \delta _{ks}= \left\{ \begin{array}{rl} 1, &{} \text{ if } k = s, \\ 0, &{} \text{ if } k \ne s, \end{array} \right. $$

where $\delta _{ks}$ is the Kronecker delta symbol. A vector (first-order tensor) $\mathbf{{\vec {a}}}$ can be decomposed in the introduced basis as

$$ \mathbf{{\vec {a}}}= a_k\mathbf{{\vec {e}}}_k. $$

The usual summation convention is assumed over the repeated indices.

The dyadic product of the base vectors is the tensor $\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s$. This tensor serves as a base tensor for the representation of a second-order tensor $\mathbf{A}= A_{ks}\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s$. In particular, $\mathbf{A}\cdot \mathbf{B}=A_{ij}\mathbf{{\vec {e}}}_i\mathbf{{\vec {e}}}_j\cdot B_{ks}\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s =A_{ij}B_{ks}\mathbf{{\vec {e}}}_i \delta _{jk}\mathbf{{\vec {e}}}_s =A_{ik}B_{ks}\mathbf{{\vec {e}}}_i\mathbf{{\vec {e}}}_s$ is the second-order tensor, $\mathbf{A}\mathbf{B}=A_{ij}B_{ks}\mathbf{{\vec {e}}}_i\mathbf{{\vec {e}}}_j\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s$ is the fourth-order tensor, $\mathbf{{\vec {a}}}\cdot \mathbf{A}=a_i\mathbf{{\vec {e}}}_i\cdot A_{ks}\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s = a_i A_{ks} \delta _{ik}\mathbf{{\vec {e}}}_s = a_k A_{ks} \mathbf{{\vec {e}}}_s$ is the vector, $\mathbf{A}\cdot \cdot \mathbf{B}=A_{ij}\mathbf{{\vec {e}}}_i\mathbf{{\vec {e}}}_j\cdot \cdot B_{ks}\mathbf{{\vec {e}}}_k\mathbf{{\vec {e}}}_s =A_{ij}B_{ks}\delta _{jk}\delta _{is} = A_{sk}B_{ks}$ is the scalar.