Consistent numerical implementation of hypoelastic constitutive models
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Consistent numerical implementation of hypoelastic constitutive models Mehrdad Palizi, Salvatore Federico and Samer Adeeb
Abstract. In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi rate for shell and membrane elements. The user subroutine UMAT is the platform to implement a general constitutive model into ABAQUS, but, in order to update the Jacobian matrix DDSDDE in UMAT, the model must be expressed in terms of the Jaumann rate of the Kirchhoff stress tensor. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT subroutine. The developed UMAT subroutine codes are validated using available solutions, and the consequence of using wrong Jacobian matrices is elucidated. The UMAT subroutine codes are provided in the “Electronic Supplementary Material” repository for the user’s consideration. Mathematics Subject Classification. 74A05, 74A10, 74B20, 53A15, 15A69, 53A45. Keywords. Hypoelastic constitutive equations, ABAQUS UMAT subroutine, Stress array update, Jacobian matrix update.
1. Introduction Hypoelasticity is a rate form of elastic material model [1], in which an objective stress rate is linearly related to the rate of deformation by means of a fourth-order elasticity tensor which, in general, is not obtainable from a strain energy density. Originally, Dienes [2] showed that the zero-graded hypoelastic model, i.e. a hypoelastic model with constant isotropic elasticity tensor, exhibits oscillation in simple shear, if it is constructed based on the Jaumann rate of the Cauchy stress. However, zero-graded hypoelastic models which are based on the Truesdell or Green–Naghdi rates do not suffer this problem [3]. We remark that the definition of elements in ABAQUS is based on the Jaumann rate for solid elements [4] and on the Green–Naghdi rate for structural elements (shells, membranes, beams, trusses) [5,6], as mentioned in the ABAQUS Theory Manual, Section 1.5.3 [7]. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT (user material) subroutine. To attain this, it is essential to express the elasticity tensor of the hypoelastic model in terms of the elasticity tensor which relates the Jaumann rate of the Kirchhoff stress tensor to the rate of deformation. According to Pinsky et al. [8], such relations seemed difficult to be constructed for models associated with the Green–Naghdi stress rates; however, the kinematical relations provided in Mehrabadi and Nemat-Nasser [9] enable us to establish such connections. The study starts with a review of some basic definitions, the concept of objective rate and the structure of
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