Fiberreinforced materials have widespread applications, which prompt the study of the effect of fiber reinforcement. Research studies have indicated that thermal conductivity cannot be considered as a constant, which is closely related to temperature change. Based on those studies, we investigate the fiberreinforced generalized thermoelasticity problem under thermal stress, with the consideration of the effect of temperaturedependent variable thermal conductivity. The problem is assessed according to the LS theory. A fiberreinforced anisotropic halfspace is selected as the research model, and a region of its surface is subjected to a transient thermal shock. The timedomain finite element method is applied to analyze the nonlinear problem and derives the governing equations. The nondimensional displacement, stress, and temperature of the material are obtained and illustrated graphically. The numerical results reveal that the variable conductivity significantly influences the distribution of the field quantities under the fiberreinforced effect. And also, the boundary point of thermal shock is the most affected. The obtained results in this paper can be applied to design the fiberreinforced anisotropic composites under thermal load to satisfy some particular engineering requirements.
Fiber reinforcement is an inherent property of materials that is considered an effect rather than a form of inclusion in it [
The assumption of infinite propagation speed of the thermal signal in classical thermoelasticity theory is inconsistent with the real phenomenon. Several generalized thermoelasticity theories have been developed to eliminate this paradox [
Thermal conductivity is an important parameter of a material which is typically considered constant. However, several experimental and theoretical studies have indicated that thermal conductivity is closely related to temperature change [
Normal mode analysis is applicable only for solving the steady state problem, whereas integral transformation and timedomain finite element method are suitable for solving dynamic problems. Integral transforms, such as Fourier and Laplace transforms, have been widely used for processing the related generalized thermoelastic problems [
This paper investigates the transient thermal shock problem for fiberreinforced materials with a timedependent variable thermal conductivity according to the LS theory. Timedomain finite element method is applied to derive the nonlinear governing equations. Numerical examples are presented to clarify the transient thermal shock response on a halfspace. Field quantities are obtained for different thermal conductivities and illustrated graphically.
Belfield et al. [
The equation of motion (in the context of LS theory) is
The equation of energy conservation is
The geometrical equation is
The fiberreinforced direction is defined as
From equations (
The equation of heat conduction is
The thermal conductivity
From equations (
The finite element method is an approximate method for solving differential equations. The first step in solving the problem is to establish the governing equations, followed by defining the boundary conditions based on the specific problems and then performing the the structural discrete, unit analysis and overall analysis to obtain the numerical solutions. For nonlinear problems, the finite element expression is obtained using the finite element method, which can eliminate the influence of truncation errors and avoid the tediousness of integral transformation. In addition, the time history of the variables in the constitutive relation can be determined to better reflect the wavefront characteristics. FlexPDE is a useful tool for solving partial differential equations, which can form Galerkin finite element integrals, derivatives, and dependencies aiming at the problem description and then build a coupling matrix and solve it. Therefore, FlexPDE is employed to deal with the related partial differential equations generated by the finite element method. For convenience, the constitutive equations of equations (
The heat conduction equation of equation (
The basic variables in this study include displacement and temperature. After the elements are divided, the variables are represented by shape functions in each element as follows:
According to equation (
Then, the variational forms of equation (
According to virtual displacement principles, the fiberreinforced generalized thermoelasticity problem with variable thermal conductivity can be formulated as
According to equations (
These expressions can be summed as the following matrix form:
To check the validity of the proposed method, reference [
Distribution of temperature along the
Consider the problem of a fiberreinforced anisotropic elastic halfspace (
Initial conditions:
Boundary conditions:
(a) Diagram of a halfspace under thermal shock. (b) Simplified
Under the given conditions, the halfspace model can be simplified as a
Copper material is selected for numerical evaluation, and the parameters are presented in Table
Material parameters of the copper material.











For convenience, the following nondimensional quantities are introduced:
According to equation (
Under the given conditions of equations (
The dimensionless distributions of temperature, displacement, and stress are illustrated graphically in Figures
Case 1:
Case 2:
Case 3:
Distribution of nondimensional temperature along
Distribution of nondimensional temperature along
Distribution of nondimensional horizontal displacement along
Distribution of the nondimensional horizontal displacement along
Distribution of the nondimensional vertical displacement along
Distribution of the nondimensional stress
Figures
Given that the research model is symmetrical about the
Figures
Figure
It is well known that instantaneous changes of temperature can markedly change the thermal conductivity of a material. Therefore, this article investigated the effect of temperaturedependent variable thermal conductivity on a fiberreinforced generalized thermoelastic halfspace. Given the reinforced direction
Based on the simulation, we can draw that the timefinite element method is very effective for analyzing nonlinear problems with given initial and boundary conditions, and by which we can capture a pronounced wavefront effect. In consideration of the fiberreinforced effect, variable thermal conductivity positively affects the distributions of temperature, displacement, and stress. In addition, the boundary point of thermal shock is affected the most.
Components of stress tensor
Kronecker delta
Components of strain tensor
Reinforcement parameters
Lame constants
Coefficient of linear thermal expansion
Reference temperature
Temperature difference
Entropy density
Mass density
Specific heat at constant strain
Displacement vector
Heat flux vector
Relaxation time
Initial thermal conductivity
Small quantity for measuring the influence of temperature on thermal conductivity
Number of nodes in the grid
Heaviside unit step function
Shape functions
Firstorder derivative of
Traction vector.
All the data used to support the findings of this study are included within the article.
There are no conflicts of interest regarding the publication of this paper.