A Result regarding the Seismic Dislocations in Microstretch Thermoelastic Bodies

The aim of our study is to derive a relation of De Hoop-Knopoff type for displacement fields within context of thermoelastic microstretch bodies. Then, as a consequence, an explicit expression of the body loadings equivalent to a seismic dislocation is obtained. The results are extensions of those from the classical theory of elastic bodies.


Introduction
The theory of thermomicrostretch elastic solids was first elaborated by Eringen [1], and, in short, this is a theory of thermoelasticity with microstructure that includes intrinsic rotations and microstructural expansion and contractions.
The purpose of this theory is to eliminate discrepancies between classical elasticity and experiments, since the classical elasticity failed to present acceptable results when the effects of material microstructure were known to contribute significantly to the body's overall deformations, for example, in the case of granular bodies with large molecules (e.g., polymers), graphite, or human bones.
These cases are becoming increasingly important in the design and manufacture of modern day advanced materials, as small-scale effects become paramount in the prediction of the overall mechanical behaviour of these materials.
Other intended applications of this theory are to composite materials reinforced with chopped fibers and various porous materials.
This theory can be useful in the applications which deal with porous materials as geological materials, solid packed granular materials, and many others.
On the other hand, materials which operate at elevated temperatures will invariably be subjected to heat flow at some time during normal use.Such heat flow will involve a nonlinear temperature distribution which will inevitably give rise to thermal stresses.For these reasons, the development, design, and selection of materials for high temperature applications require a great deal of care.The role of the pertinent material properties and other variables which can affect the magnitude of thermal stress must be considered.
The main difficulty of the thermomicrostretch materials is the large number of the thermoelastic coefficients and, as such, the problem of their determination in the laboratory.Yet many authors consider that this problem will be solved in the future.
Already, in the isotropic case, when the number of coefficients decreases a lot, they are calculated as can be seen in many works due to Eringen or Iesan.
The present paper must be considered a first step to a better understanding of microstretch and thermal stress in the study of above enumerated materials.
The reciprocity and representation relations that appear in our study constitute powerful theoretical tools in the assessment of the theory of seismic-sources mechanism, in the studies connected with seismic wave propagation.
Also, we think that this paper is a good help to understand the application of microstretch mechanism to earthquake problems.

Mathematical Problems in Engineering
For instance, in [6], the authors establish a reciprocity relation which forms the basis for a uniqueness result, a continuous dependence of solutions upon initial data and body loads and a variational characterization of solutions.The effect of a concentrated heat supply and of a concentrated heat volume charge density in an unbounded homogeneous and isotropic electromagnetic body is investigated.
We find other results regarding thermoelasticity of nonclassical materials, as in [1,[9][10][11][12].The main results of our study are extensions of some similar results in the classical elasticity in order to cover the thermoelasticity of microstretch bodies (see the results due to Saccomandi in [13]).

Basic Equations
For convenience the notations and terminology chosen are almost identical to those of our study [11].Our paper is concerned with an anisotropic and homogeneous material.
Let the body occupy, at time  = 0, a properly regular region  of the three-dimensional Euclidian space, bounded by the piece-wise smooth surface  and we denote the closure of  by .We refer the motion of the body to a fix system of rectangular Cartesian axes   ,  = 1, 2, 3 and adopt the Cartesian tensor notation.Points in  are denoted by   and  ∈ [0, ∞) is temporal variable.Throughout this work the Einstein summation convention over repeated indices is used.The subscript  after comma indicates partial differentiation with respect to the spatial argument   .All Latin subscripts are understood to range over the integers (1, 2, 3), while the Greek indices have the range (1,2).A superposed dot denotes the derivatives with respect to the -time variable.Also, the spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion.
Let us denote by   the components of the displacement vector and by   the components of the microrotation vector.Also, we denote by  the microstretch function and by  the temperature measured from the constant absolute temperature  0 of the body in its reference state.
As usual, we denote by   the components of the stress tensor and by   the components of the couple stress tensor over .Also, we denote by   the components of the microstress vector.
For clarity and simplicity in presentation, the regularity hypotheses on the considered functions will be ommited.
On these grounds, the field equations in the dynamic theory of thermoelasticity of microstretch bodies are as follows (see [1,11,14]): (i) the equations of motion (ii) the balance of the equilibrated forces The equation of energy is given by In the above equations we have used the following notations: (i)   are the components of body force; (ii)   are the components of body couple; (iii)  is the generalized external body load; (iv)  is the generalized internal body load; (v)  is the reference constant mass density; (vi)  and   =   are the coefficients of microinertia; (vii)  is the entropy per unit mass; (viii)  is the heat supply per unit mass; (ix)   are the components of heat flux vector.
For an anisotropic and homogeneous microstretch thermoelastic material, the constitutive equations have the form: where   ,   ,   ,   ,   ,   ,   ,   ,   ,   ,   ,   , , , , and   are the characteristic constitutive coefficients.The components of the strain tensors   ,   , and   are defined by means of the geometric equations: where   is the alternating symbol.The constitutive coefficients obey the following symmetry relations: One can assume that a positive constant  0 exists such that Also, the Second Law of Thermodynamics implies that We denote by   the components of surface traction,   the components of surface couple,  the microsurface traction, and  the heat flux.These quantities are defined by at regular points of the surface .
Here,   are the components of the outward unit normal of the surface .

Main Results
Let  and V be functions defined on ×[0, ∞) and continuous on [0, ∞) with respect to the time variable  for each spatial variable  ∈ .
We denote by  * V the convolution of  and V; that is, Let us introduce the notations Following the same procedure used by Ies ¸an in [5], it is easy to prove the following result that enables us to give an alternative formulation of the initial boundary value problem in which the initial data are incorporated into the field of equations.
Theorem 1.The functions   ,   , , ,   ,   ,   , and   satisfy ( 1), ( 2), (3), and the initial conditions (10) if and only if they satisfy the following system of equations: In our following estimations, we will use formulation (15) of the mixed problem.We wish to find the behavior of the considered medium when embedded in  there is a discontinuity surface Σ for the displacements, the microration vector, the microstretch function, and the temperature.The sides of Σ are denoted by Σ − and Σ + .
Let ]  be the components of the unit normal vector of Σ, directed from the side (−) to the side (+).
Then on surface Σ we have the conditions where  + and  − are the limits of the function () as  approaches a point on the side (+) or (−) of the surface Σ, respectively, and   , Φ  , Ψ, and Θ are prescribed functions.
In this way we can consider (15) in the domain  \ Σ.
It is easy to see that in the absence of the discontinuities we obtain the generalization, in the context of the thermoelasticity of thermoelastic microstretch bodies, of the previous results established in the classical thermoelastodynamics.
Figure 1 shows the 3D variation of displacement with respect to time and space.Because we are interested only in the positive values of time, it is observed that the displacement increases sharply with .The 3D variation of microstretch function with respect to time and space is shown in Figure 2. It is observed that for an increasing of time the microstretch function tends to become constant and the microstretch surface becomes a parallel plane with .In Figure 3 the 3D variation of microrotation function is represented and Figure 4 shows the variation of the temperature with increasing  under the effects of thermoelastic body.
Figure 5 presents the displacement profiles (, ) against  at different times.The displacement achieves a maximum for all chosen times for  ∈ (−2.5; −2.35)The displacement is an increasing function with respect to space fot  ∈ [−5; −2.5] ∪ [3.4; 6.3], it is a decreasing function with respect to space for  ∈ (−2.5; 3.4) and tends to a steady-state.
Figure 6 presents the microstretch profiles (, ) against  at different times.The microstretch is an increasing function with respect to time.It is clear that the microstretch function is not stable.
The profiles of microrotation function (, ) over space are shown in Figure 7 for four values of time.The microrotation function, as it results from Figure 4, is a decreasing function with respect to time.It is observed that for the displacement vector greater values than  = 3.2 the solution is stabilized.The microstretch function increases asymptotically to  = 5.The microrotation vector is stabilized for all chosen moment starting with  in the range (3.6, 3.85).The temperature increases for  ∈ (−5, −3.5), decreases for  ∈ (−3.5, 0), and then increases asymptotically to  = 5.

Conclusions
In the absence of the discontinuities we obtain the generalization, in the context of the thermoelasticity of thermoelastic

Figure 8 Figure 5 :
Figure 8 presents the temperature profiles (, ) against  at different times.The medium temperature is an increasing function with respect to time for  > 0.Based on papers by Eringen[1,14] and Kumar et al.[12] and based on the simulations performed in the laboratory of virtual engineering, we have used the following characteristic constitutive coefficients for the microstretch thermoelastic materials (in the isotropic case):

Figure 6 :
Figure 6: The behavior of (, ) at different values of time.

Figure 7 :Figure 8 :
Figure 7: The behavior of (, ) at different values of time.