Here, we present the basic equations of the theory of magneto-thermoviscoelasticity, which will be used for the solution of the problem described above. With reference to a Cartesian frame denoted by 0xyz, consider an initially stressed anisotropic solid is placed in a constant primary magnetic field H0, acting in the direction of the z-axis. Here we address the generalized two-dimensional deformation problem in xy-plane only, therefore all the variables are constant along the z-axis. In the 0xy plane, the solid occupies the region
which is bounded by a simple closed curve C.
In non-dimensional form the governing equations of magneto-thermo-viscoelasticity for an anisotropic solid can be written as follows (see Fahmy and El-Shahat [
The initial and boundary conditions for the current problem are assumed to be written as
where 
and 
are the same expressions 
and 
respectively, 
is the mechanical stress tensor, 
Maxwell’s electromagnetic stress tensor, 
is the displacement, T is the temperature, 
is the initial stress in the solid, 
and 
are respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium, 
is the viscoelastic material constant, 
is the magnetic permeability, 
is the perturbed magnetic field, 
are the thermal conductivity coefficients satisfying the symmetry relation 
and the strict inequality 
holds at all points in the medium, 
is the density, 
is the specific heat capacity of the solid and 
is the dimensionless time, 
is the heat source density and 
is the Euclidean distance between the field point 
and the load point
. Also, 
, 
and 
are suitably prescribed functions of
, 
is suitably prescribed function of
, 
and 
are nonintersecting curves such that
, 
and 
are non-intersecting curves such that 
and 
are the tractions defined by
.
A superposed dot denotes differentiation with respect to the time and a comma followed by a subscript denotes partial differentiation with respect to the corresponding coordinates.
We focus our attention to the case of inhomogeneity along x-axis, so we characterize the elastic constants
, magnetic permeability
, initial stress
, thermal conductivity coefficients 
and density 
of non-homogeneous material by
where
, 
, 
, 
and 
are constants (the values of
, 
, 
, 
and 
in homogeneous matter) and m is a rational numbers.
In terms of the definitions (12) and (13), Equations (1) and (5) can be written as follows
With the heat flux vector 
given by Fourier’s law
where 
and 
are taken to be constant in
.
It is usually difficult to find out the analytical solutions of such state equations except for some special cases.
yields
collocation points on the boundary 
and another 
in the interior of 
so that the total number of interpolation points is
.
is a continuous function in the 
-plane and 
is a polynomial of degree 
is called a radial basis function (RBF). The importance of splines is that they can provide interpolatory approximations to 
for very general sets of interpolation points 
in the xyplane.
-plane a spline is of the form
is the Euclidean distance between the field point 
and the load point
. and 
is a polynomial of degree
. For 
these are the thin plate splines (TPS)
and 
and 
are then approximated by means of shape functions 
and nodal values 
and 
as follows
, 
and 
are matrices We can also approximate the particular solutions 
and 
on the boundary as the unknown field variables by means of nodal values 
and 
as follows
and 
are matrices Using (22) and (23) and applying the point collocation procedure to (21), we have the following system of equations
and 
contain the particular solutions.
and unknown coefficients 
as follows
and 
are obtained from the first row of matrix equation (32) are expressed as follows
, equation (34) can be written in the following form
as the time step, and use the subscripts 
to denote the 
discrete time. A mesh is defined by
, 
being the time step Using finite difference, we can approximate the temperature as follows
determined, the remaining task is to solve (14) subject to (6), (7) and (8).
, the magnetic term
, the viscosity term
, the temperature gradient 
and the initial stress term 
are treated as the body forces.
is a weighting function and 
is the approximate solution.
Therefore, it follows that
by the relation
in (44) by
, then we have
is the Bessel function of the third kind of order zero and
, which is known as Euler’s constant.
and source function 
can be obtained by evaluating
lies inside the domain
. To obtain an expression that contains only boundary variables, the load point 
has to be moved to the boundary. Therefore, the boundary is deformed by a small circular region with radius 
around the load point 
as shown in 
and shape functions 
and 
are matrices.
can be calculated from (48) using the point collocation procedure, which yields
and 
denote, respectively, the known and unknown parts Hence we can write the system (63) in the following form
can be obtained from the first row of (66) as follows
and accelerations 
at time step 
are approximated as follows
the values of 
are established. Once these values have been obtained, the unknown 
and 
can be obtained from (70) and (71), respectively. For the case in which
, the procedure described in Bathe [
and
. Lastly, we compute the traction vector 
using (67).
are plotted in 
and heat capacity
, 
Oersted, 
Gauss/Oersted, 
, 
,
. The boundary 
is
, the boundary 
is
, the boundary 
is
, while 
is
. The numerical values of the temperature and displacement are obtained by discretizing the boundary into 120 elements 
and choosing 60 well spaced out collocation points 
in the interior of the solution domain.
and for the homogeneous solid, we assume that
.
, the dashed line denoted by “C” represents the solution for non-homogeneous solid in the absence of initial stress, the dotted line denoted by “B” represents the solution for homogeneous solid in the presence of initial stress 
and the dashed-dotted line denoted by “D” represents the solution for non-homogeneous solid in the presence of initial stress.
-axis at various values of the time
. It is noticed that the temperature increases with the increase of 
and
.
and 
and thermal stresses
, 
and
. Also, they show the difference among the four cases “A”, “C”, “B” and “D”.
increases and then decreases with the increase of
. It is noticed that the maximum value happens in the homogeneous solid in the presence of initial stress.
decreases and then increases for all cases. It is clear from this figure that the homogeneous and non-homogeneous curves
increases with increasing 
for all cases. It is apparent from this figure that the increasing rate is more pronounced in non-homogeneous solid in the presence of initial stress.
increases with the increase of 
for all cases. It will be observed from this figure that the increasing rate is more pronounced in the presence of initial stress.
decreases with the increase of 
for the non-homogeneous solid, but for homogeneous solid it increases with the increase of
. It is clear from this figure that the increasing rate is more pronounced in the presence of initial stress.
and 
and thermal stresses
,
and 
are closer in the absence of initial stress than in the presence of initial stress. We may also observe from these figures that the initial stress has an important effect on the magneto-thermo-visco-elastic stresses along 
-axis through the material constants, thermal constants, magnetic constants and viscosity factor, which are essential parameters to be considered in the design of various devices. Furthermore, while there is no limitation in the solution procedure, all boundary and initial conditions are strongly satisfied.
-axis in a non-homogeneous anisotropic initially stressed viscoelastic solid placed in a constant primary magnetic field, we can design various viscoelastic solids under magnetothermal load to meet specific engineering requirements and utilize it in measurement techniques of thermoviscoelasticity.