A Mode I Crack Problem for a Thermoelastic Fibre-Reinforced Anisotropic Material Using Finite Element Method

In this article, the theory of generalized thermoelasticity with one relaxation time is used to investigate the thermoelastic fiber-reinforced anisotropic material with a finite linear crack. The crack boundary is due to a prescribed temperature and stress distribution. By using the finite element method, the numerical solutions of the components of displacement, temperature and the stress components have been obtained. Comparisons of the results in the absence and presence of reinforcement have been presented.


INTRODUCTION
.ibers are assumed to be an inherent property of matter, rather than some form of inclusion in such models as Spencer [1]. .iber-reinforced composite materials are widely used in technical structures. These materials have considerably high strength with respect to their weight, even at high temperatures they keep their stiffness. Continuum models are commonly used to explain the mechanical properties of these materials. There are some researches in the field of thermoelastic behavior of fibrous composites, among which two of them are well-known. .irstly, Lord and Shulman [2] presented the generalized thermoelastic theory with one relaxation time by postulating a new law of thermal conduction instead of the classical .ourier law. Secondly, Green and Lindsay [3] presented two relaxation times effects on the generalized thermoelastic theory. Dhaliwal and Sherief [4] extended the generalized thermoelastic theories for the anisotropic medium. The material strength in the presence of cracks is an attracting problem of fracture mechanics and the knowledge of the elastic stress fields is potentially useful for strength estimation based on the available theories for brittle fracture [57]. Several researches have been published which treated the stress distributions in an unbounded solid due to the application of normal pressure or temperature on the faces of a circular internal flat crack [8,9].
The exact solution of the basic equations of generalized thermoelastic models for linear/nonlinear coupled system exists for initial and boundary issues which are very specific and simple cases. Therefore one can choose the finite element method. Basically there are three steps to apply the finite element method. The first step is to take the overall behavior of the variables so as to satisfy the differential equations given unknown field. The second step is temporal integration. The temporal derivatives of the unknown variables must be determined by the previous results. The last step is to solve the resulting equations from the first and second steps by the algorithm of the finite element method [10].
The present paper investigates a Lord and Sherman model in a two-dimensional thermoelastic medium containing a mode I crack. The nondimensional equa-PHYSICAL MESOMECHANICS Vol. 21 No. 2 2018 tions have been solved numerically using the finite element method.

BASIC EQUATIONS AND .ORMULATION O. THE PROBLEM
An infinite space ∞ < y < ∞, ∞ < x < ∞, containing a crack on the y axis, , 0 x b x ≤ =± was considered for the problem. The crack surface is subjected to a prescribed temperature and normal stress distribution. The preferred direction of the x axis was considered for the fiber direction as a ≡ (1, 0, 0). All the considered functions depend on x and y with the time t. Thus, the components of displacement vector are u(x, y, t) and L(x, y, t). In this case, the governing equations have the following form [11]: xy yy , α α are the linear thermal expansion coefficients, 0 T is the reference uniform temperature, T is the incremental temperature, 11 K and 22 K are the thermal conductivity components, ρ is the mass density, e c is the specific heat at constant strain, λ and T µ are the elastic constants, , T m = λ + α + µ − µ + β With respect to the nondimensional quantities in Eq. (7), after neglecting the primes for convenience, the previous equations reduced to 2 2 2 2 To solve this problem, the boundary and initial conditions must be considered. The initial conditions are the following:

T x y T x y t u x y u x y t
x y x y t At x = 0, the boundary conditions are assumed as where 1 T and 1 P are constants and H is the Heaviside unit step function. This means that mechanical and heat loading are applied on the crack surface (.ig. 1).

.INITE ELEMENT SOLUTION
The formulation of finite element method for the thermoelastic problem can easily be obtained using standard procedure. In the finite element method, the temperature {T} and the components of displacement  can be written in the following form: .ig. 1. .iber-reinforced plate.
In the domain V and the boundary A, the principle of virtual displacement can be given by  are the vectors of external force, C is the damping matrix, M is the mass matrix and K is the stiffness matrix. Based on an implicit time integration method, the time derivatives of the unknown variables have been determined.

NUMERICAL RESULTS AND DISCUSSION
We assume that the plate is fiber-reinforced. The physical parameters are listed below [12]: .igure 3 show two curves predicted by generalized thermoelastic interaction on the fiber-reinforced medium (with fiber) and on the isotropic medium (without fiber), the solid lines refer to the solution obtained for the isotropic medium (i.e. α = 0, β = 0 and L µ −

0)
T µ = while the dotted lines refer to the solution obtained for the fiber-reinforced medium. The reinforcement has a great effect on field quantities as expected.
.igure 3a shows the temperature variation along the y direction and it indicates that temperature field has its maximum value along the crack line (1 ≤ y ≤ 1), and it starts to decrease just near the crack edges (y = ±1) where it smoothly decreases and finally reaches zero. .igure 3b displays the variation of horizontal displacement with respect to y and it indicates that the magnitude of the displacement has maximum value along the crack line (1 ≤ y ≤ 1), and it starts to decrease just near the crack edges (y = ±1), and then decreases to zero to obey the boundary conditions. .igure 3c displays the variation of vertical displacement along the y axis. The vertical displacement starts decreasing at both ends of the crack, and has a minimum value at the middle of the crack, after which it starts increasing and reaches a maximum value just near the crack edges (y = ±1) and then it decreases to become zero. The , xx xy σ σ and yy σ stress components are shown in .igs. 3d3f, respectively.

CONCLUSION
In this work, the solution of a two-dimensional problem on fiber-reinforced thermoelastic plate with a finite linear crack was studied. The differences of the predicted field quantities were remarkable in the presence and absence of fiber reinforcement. The properties of the fiber-reinforced material tend to increase the temperature variation and reduce the magnitudes of the other considered variables, which may be significant in some practical applications.