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