Study on steady-state thermal conduction with singularities in multi-material composites

Abstract

Increasing demand in material and mechanical properties has led to production of complex composite structures. The composite structures, made of different materials, possess a variety of properties derived from each material. This has brought challenges in both analytical and numerical studies in thermal conduction which is of significant importance for thermoelastic problems. Therefore, a unified and effective approach would be desirable. The present study makes a first attempt to determining the analytical symplectic eigen solution for steady-state thermal conduction problem of multi-material crack. Based on the obtained symplectic eigen solution (including higher order expanding eigen solution terms), a new symplectic analytical singular element (SASE) for numerical modeling is constructed. It is concluded that composite structures composed of multi-material with complex geometric shapes can be modeled by the developed method, and the generalized flux intensity factors (GFIFs) can be solved accurately and efficiently

A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Anisotropic Material

“…where c T is a constant which should be determined by the compatibility condition defined in Eqs. (17) and (18). The solution to the original problem corresponding to the Jordan form eigenvector should be constructed through:…”

“…However, in the analysis of thermal conduction problem of wedge in anisotropic material the equivalent heat conductivity coefficients become functions of  under the polar coordinate system ( , ) r  . Due to this reason, the symplectic approach proposed by some of the authors for the study of steady-state thermal conduction of crack in multiple isotropic materials [17,18] cannot be applied directly to anisotropic materials. Actually, this problem also exists in other analytical methods, and special treatments must be adopted to obtain the analytical solution.…”

“…In the previous studies on the steady-state thermal conduction problem for crack in isotropic materials, a kind of symplectic analytical singular finite element (SASE) was proposed [17,18]. The temperature and thermal flux fields in the SASE were defined by the analytical eigen solution obtained by using the symplectic dual approach [39,40,41], and the finite element formulation was derived based on the variational principle.…”

“…In this way, both of the singularity problem in the vicinity of the crack top and the complex geometry, loading condition and constrain types can be handled very well. It was demonstrated that the proposed numerical method which combines analytical eigen solution and conventional finite element method (FEM) together can bring substantial improvement on solving accuracy and efficiency compared with conventional FEM [17,18]. However, as the equivalent conductivity coefficients are functions of  for anisotropic material, the corresponding eigen solution cannot be solved due to the mathematical complexity.…”

“…With the obtained eigen solution, a new SASE for steadystate thermal conduction of wedge in anisotropic material is constructed in the way similar to the previous studies [17,18]. However, due to the changes of the fundamental equations, the derivation steps for eigenvalue as well as the formulation of the SASE are quite different from those reported in [17,18]. By using the proposed SASE, the numerical prediction of the generalized flux intensity factors (GFIFs) can be solved directly without any post-processing.…”