Numerical implementation of the Murnaghan material model in ABAQUS/Standard

Abstract

The paper presents a numerical implementation of the Murnaghan material model (M) [1] in the finite element method software ABAQUS / Standard v. 6.14 [2]. The UHYPER user subroutine is employed, which is suitable for the class of isotropic hyperelastic models [3]. As a special case of the M model, the Saint Venant-Kirchhoff (SVK) model is considered [4]. Formal verification on the basis of elementary tests is performed. Among others, a special attention is paid to a simple shear deformation. In all tested types of deformation, analytical values confirms results based on the finite element procedure within assumed numerical precision and accuracy. It should be noted that the stored-energy function of the M and SVK models do not meet any requirements of the mathematical theory of non-linear elasticity [4, 5]. Therefore, these models are suitable for relatively small deformations, while there are no restrictions on finite rotations. As an example of applications, a tube under axial compression is considered in two cases. Various starting parameters for the Riks procedure [6, 7] are adopted to obtain different solutions of corresponding boundary value problem. Material parameters of steel are considered according to Lurie [8].

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“…In order to implement Murnaghan’s model, the elastic potential function, W , needs to be expressed as a function of invariants to eliminate the volume change [ 33 ]. The invariants, Green deformation tensors and stretch tensors can be decomposed into their dilatational and distortional parts, leading Equation (1) to be expressed as [ 34 ]: where is the left Cauchy–Green deformation tensor with the first and second distortional invariants and , i.e., , . is the total volume ratio for the spatial description of equilibrium requiring the Cauchy stress tensor, and are material constants, which can be expressed as a linear combination of Lamé constants and Murnaghan’s TOECs [ 34 ].…”

“…The elastic potential function for Murnaghan’s model of hyperelasticity in terms of invariants of Left-Cauchy Green deformation tensor is given by [ 34 ]: …”