Thermo-mechanical coupling of transversely isotropic materials using high-order finite elements
Constitutive modeling and numerical analysis of the behavior of anisotropic materials, particularly transversely isotropic and orthotropic materials, attained increasing attention in the last few years. The attention is motivated by the wide range of applications of these materials in engineering industries and biomedical technologies. This work aims to develop a constitutive model for transversely isotropic materials undergoing thermo-mechanically coupled finite deformations. The model is based on the idea of multiplicative decomposition of the deformation gradient. Furthermore, making use of high-order finite elements, the capability of the model to simulate the behavior of transversely isotropic material under isothermal and thermo-mechanically coupled loadings is demonstrated by performing some numerical experiments. First of all, a constitutive model for the case of isothermal transversal isotropy is formulated. The proposed model is an extension of the volumetric/isochoric decoupling of the deformation gradient, where the isochoric part is decomposed into two parts, one part containing only the deformation along the preferred direction, while all remaining deformations are included in the other part. This formulation has the advantage that it leads to a clear split of the stress-state, i.e., the stress along the preferred direction is splitted from the remaining stresses. Additionally, the proposed model overcomes the obstacle related to the application of volumetric/isochoric decomposition to anisotropy. The formulation is, then, extended to the case of thermo-mechanically coupled problem, where a thermodynamically consistent constitutive model for transversal isotropy is developed. Moreover, a directionally dependent, i.e. transversely isotropic heat flux vector is derived, which takes into consideration the anisotropy in heat conductivity. The proposed model is implemented into a high-order finite element code, in which the p-version finite element method (p-FEM) and the high-order diagonally implicit Runge-Kutta (DIRK) methods are used for the spatial and time discretizations, respectively. In p-FEM the accuracy of the solution is improved by increasing the polynomial degree of the elements, and this makes p- FEM more convenient for the analysis of thin structures, like in the case of laminated composites. Thus, computations are carried out in order to investigate the behavior of the proposed model with different numerical examples. To this end, the influence of different factor, namely, existence of anisotropy, orientation of the preferred direction, anisotropic thermal expansion as well as anisotropic heat conductivity, on the response of transversely isotropic material under isothermal and/or thermo-mechanical loadings is discussed. Furthermore, the efficiency of the p-version implementations is demonstrated by comparing it with two different h-version finite element implementations.