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In this project, we will study investigate the thermoelastic interactions caused by a continuous point heat source in a homogeneous and isotropic unbounded thermoelastic body, by using the linear theory of thermoelasticity without energy dissipation. We are going to solve the problem using the same boundary conditions using different thermoelastic models, namely Green and Naghdi II, Lord and Shulman and Fractional Order theory of thermoelasticity and compare the results for displacement, temperature, radial stress and transverse stress.
First, let us define the keywords: Generalised thermoelasticity, point heat source.
The generalisation of thermoelasticity arises from a modifying the Fourier's law of heat conduction.
What is thermoelasticity?
Thermoelasticity is the change in size and shape of a solid material as its temperature fluctuates. Objects which are more elastic will expand more and those which are less elastic will expand less. Scientists use their knowledge of thermoelasticity to devise new materials that withstand heat changes better without breaking.
Next, we have a point [heat] source. A point source [of heat] is a single identifiable localized source [of heat].
In mathematical modelling, these sources can be approximated as a mathematical point to simplify analysis. It is good to note that the size of the actual source need not necessarily be small physically. It is relative to the scale where it is being used. For example, in astronomy, stars are treated as point sources.
Now let us see the various thermoelastic models we have.
1.2 COUPLED/CLASSICAL THERMOELASTICITY (CTE)-BIOT 
The classical theory of thermoelasticity (CTE- Biot ) is based on the heat conduction equation of Fourier which assumes that the thermal disturbances propagate at infinite speed. This prediction seems unrealistic physically, especially in situations involving very short transient durations, sudden high heat flux exposures, and very low temperatures near the absolute zero.
These aspects have caused much commotion in the field of heat propagation.
The classical model for propagation of heat is:
which yields the parabolic heat equation:
The governing equations are:
The over-dot represents partial derivatives with respect to time, t.
1.3 EXTENDED THERMOELASTICITY-Lord & Shulman 
Lord and Shulman proposed generalized thermoelasticity equations by modifying the Fourier's heat conduction equation taking into account the time needed for acceleration of the heat flow. That is our energy equation which consists of two coupled partial differential equations, includes a relaxation time and is of the form:
The stress- temperature relations is:
The equation of motion is:
We note that the only way the above equations are different from the governing equations of CTE is the presence of and on removing it we get back our original equations. This constant is the relaxation time which will make sure that the heat conduction equation will predict finite speeds of heat propagation, V,
1.4 TEMPERATURE-RATE-DEPENDENT THERMOELASTICITY
Green & Lindsay 
Green and Lindsay on the other hand, did alternations with the constitutive and the energy equation, without however changing the Fourier heat conduction equation. The thermoelastic equations involve two new relaxation parameters, and this time in the stress- temperature relation and the heat conduction.
The governing equations about velocity, temperature and stresses are given respectively as follows:
We note that by making and , the theory reduces to CTE and L-S respectively.
1.5 Green & Naghdi Models
(a) THERMOELASTICITY WITHOUT ENERGY DISSIPATION- GN II
Green and Naghdi (1992) developed a fully non-linear theory which permits heat transmission at finite speed without any heat loss. In this model thermal-displacement gradient is considered as a constitutive variable and is different from the previous models since it does not accommodate for energy loss.
The GN II model considers undamped thermoelastic waves and is known as the theory of thermoelasticity without energy disspation.
The governing equations are:
The speed of the wave is given by:
Thermoelasticity With Energy Dissipation, G-N III
Based on their previous model, Green and Naghdi developed another thermoelasticity theory. This time, they included enery dissipation. There is an extra to the heat equation.
Another way GN-III differs from GN-II is that it admits damped waves.
The governing equations in vector form are
The finite speed is given by:
1.6. Fractional Order Model
Fractional calculus is a branch of mathematical analysis that investigate the possibility of extending derivatives and integrals to non-integer orders. It is generally known than integer-order derivatives have clear physical and geographical interpretations. However in the fractional order model, which is just a new born in the world of mathematics, is still on the way to evolve. For more than 300 years, there was no clear interpretation of this model.
Only recently, have we found more about its importance and functions. It is used in the study of viscoelastic materials, as well as many fields in science and engineering including fluid flow, in electrical network, electromagnetic theory and probability.
The most popular definitions of the fractional derivatives are those of Riemann-Liouville and Grunwald-Letnikov .
The Riemann-Liouville definition is given as:
where is the Gamma function.
The Grunwald-Letnikov definition is
where, is the integer part of x and h is the time step.
The derivation of with respect to is .
Now we want to know the derivative of using a non-integer value. Let us see the steps of the calculation.
The general case is:
where, is the non integer value such that and k is the power of the function
The derivative of with is . Upon differentiating a second time using the derivative of order we get 1.
Thermoelastic Interactions without Energy Dissipation due to a Point Heat Source: The G-N II Model
In this chapter, the Green and Nagdi II (G-N II) model is used to study the thermoelastic interactions caused by a continuous point heat source in a homogeneous and isotropic unbounded thermoelastic body, neglecting any body forces by using the linear theory of thermoelasticity without energy dissipation.
We have to find expressions for the displacement, temperature and stress fields. First, we express our governing equations in a one dimensional spherical system, then we non- dimensionalise them. The Laplace Transform is used to solve the problem and we get an exact solution. We will also be able to observe the behaviours of each component by their discontinuities.
Finally we represent some numerical data for the displacement, temperature and stresses graphically for a copper - like material.
2.2 Formulation of the Problem
In our situation, we are considering a one-dimensional problem of the different components. The thermoelastic interactions caused by the source are spherically symmetric in nature, that is they depend only on and . There are also no body forces present here.
Thus, the displacement component consisting of only the radial component is of the form
where r is the distance from the origin.
For convenience, we apply a non-dimensionalising scheme to the governing equations (1.2.1), (1.2.2) and (1.2.3) using these given transformations:
where is a standard length and is a standard speed.
Applying (2.2.1) to (1.2.1)-(1.2.3) and dropping the primes for convenience, we get the following set of non-dimensional equations:
Note that and the non-dimensional speeds of purely elastic dilatational and shear waves respectively, is the non-dimensional speed of purely thermal wave and that is the coupling parameter.
Suppose initially, the point source is at rest in its undeformed state. The following homogeneous initial conditions hold for :
Now if we substitute
into (2.2.2) and (2.2.3), we get the following equations:
Eliminating from the above two equations we get
This equation serves as the governing equation for . Once we solve it under suitable conditions and find and are obtained from Equations (2.2.10) and (2.2.11) and and are obtained from (2.2.4) and (2.2.5).
2.3 Transform Solution
We suppose that the body is initially stationary in its original state with its temperature change and temperature rate equal to zero. We also assume that the point heat source causing thermoelastic interactions, at time t > 0, is specified by
Here, is a constant,is the Dirac Delta and H(t) is the Heaviside unit step function.
Now we will solve our problem using the Laplace Transform defined as:
Taking Laplace Transform of equation (2.2.13) under homogeneous initial conditions, with given by (2.3.1), we get
Now, Eqn 2.3.2 can be rewritten in the form
Where and are the roots of the biquadratic equation
Then using the Helmholtz equation
and imposing the regularity condition that as we get the following solution for Eqn (2.3.3):
We consider only the positive real roots and of Eqn (2.3.4).
Applying Laplace Transform to (2.2.7) and (2.2.8) we obtain the following solutions for and :
2.4 Exact Solutions
Given that we have obtained the solution for the displacement and temperature, we are now going to find the field variables in time.
Solving the biquadratic Eqn (2.3.4), we find that
Substituting from Eqn (2.4.1) into the transform solutions (2.3.6) and (2.3.7) and applying the inverse Laplace transforms of the expressions obtained, we get the exact expressions for and :
The stresses are now obtained from Eqn (2.2.4) and (2.2.5):
To confirm our calculations, we directly verify that (2.4.5), (2.4.6), (2.4.8) and (2.4.9) satisfy the governing Eqn (2.2.5) and (2.2.6) with given by (2.3.1), the constitutive relations (2.2.7) and (2.2.8) with the homogeneous initial conditions and the regularity conditions. We find that (2.4.5), (2.4.6), (2.4.8) and (2.4.9) are indeed exact solutions, in closed form, for , , and We note that these are made up of two parts, each one corresponding to a wave moving with finite speed, and
Using (2.4.2) and (2.4.3), we find that . and if , and if . The disturbances that we are considering consist of two distinct coupled waves, one after the other, with being faster than .
From (2.4.5), (2.4.6), (2.4.8) and (2.4.9), we see that , , and are almost zero for . This implies that there exists a certain instant of time where the points of the body which are past the faster wavefront do not experience any disturbance. Thus, the effects of heat source are restricted to a time-dependent bounded region surrounding the source.
2.5 Analysing Discontinuities
After analysing solutions (2.4.5), (2.4.6), (2.4.8) and (2.4.9), we can observe the discontinuities experienced by , , and at the wavefronts , . These are:
Note represents the discontinuity of the function across the wavefront ,.
Expression (2.5.1) shows that displacement is continuous at both wavefronts.
Expressions (2.5.2) - (2.5.4) show us that there are positive jumps in temperature and stress at the slower wavefront , while at the faster wavefront the jump in temperature is positive and in stress is negative. We can also observe that these jumps are inversely proportional to
2.6 Numerical Results
For purposes of numerical evaluation, we consider a copper like material with constants and.
Using expressions (2.4.2) and (2.4.3) the values of and are obtained as:
Graphs are then plotted for .
The graph shows that displacement is continuous for all positive values of including the location wavefront. increases gradually up to of the faster wavefront and becomes zero beyond this point.
Here we see that decreases gradually in two intervals: and undergoing finite jumps at and and it disappears identically for Calculation shows that is approximately equal to at the point found just behind the e-wavefront and 0.031765 at the point just beyond this wavefront. then decreases and comes to the value 0.01003 at the point just before the wavefront and jumps to zero beyond that wavefront.
Fig 2.6.3 shows that is compressive in the interval , experiencing jumps at and and disappears identically for . It increases in the interval and approaches -0.025551 approximately at the point just before the e-wavefront (the slower front). Immediately beyond that point, it jumps down to the value . Then increases again, reaching a maximum value at the point just behind the faster wavefront (-wavefront). Across this wavefront, jumps to become zero.
Using 2.5.2-2.5.4, we calculate the discontinuities in and and store them in the following table:
It can be noted that Fig 2.6.1-2.6.4 show that and jump infinitely at which is the position of the heat source.
While is continuous and experience no jumps, and jump at the position of the -wavefront, that is, at and at the position of the -wavefront, at
In this Chapter, we used the GN-II model to study the one dimensional thermoelastic interctions without energy dissipation due to a point heat source. We used Laplace Transform to solve the problem and got exact solutions for displacement, temperature and stress fields by taking the inverse Laplace transform. Then we plotted the graph of each component to observe their behaviour by keeping constant ( and varying . Thus it was found that was continuous and experience no jumps while and were discontinuous.
In this section, we use the Lord and Shulman model to investigate the thermoelastic interctions without energy dissipation due to a point heat source. The same procedures as before are followed.
Here too, we determine the displacement, temperature and stress distributions. Laplace Transform is again used to solve the problem. However, since the characteristic roots are too complicated, analytical inversion is not possible. Hence, we use small time approximations and using the software Mathematica, we obtain the inverse transforms.
Finally, we illustrate the results graphically for a copper like material.
3.2 Formulation of the Problem
We consider the following transformations to convert the governing equations (1.3.1) - (1.3.3) given in Chapter 1 into non-dimensional form:
Going through the same procedures as before and non dimensionalising our governing equations we obtain the following:
We consider the same initial and boundary conditions as the first chapter.
Applying (2.2.10) to (3.2.2) and (3.2.3), we have:
Eliminating from the above two equations yields:
3.3 Transform Solution
Taking Laplace Transform of (3.2.7) under homogeneous conditions, with given by (2.3.1), we obtain the equation
The above equation may be re-written in the form
and and are the positive roots of the characteristic equation
Using the Helmholtz equation and imposing similar regularity condition that as , we get the following solution for Eqn (3.3.1):
Solution (3.3.2) is used in the Laplace Transform versions of (2.2.10)
and (3.2.5), we get the following solutions for and :
3.4 Inversion of Laplace Transform
Given that the solution of the biquadratic equation (3.3.3) contains square roots, it is difficult to invert directly. So, since thermoelastic interactions are short lived, we can therefore use the short time approximation technique to find the square roots.
Hence we use the Initial Value Theorem
We expand the result using Maclaurin series after setting
Thus, (3.3.4) can be interpreted as
We now expand in a Maclaurin series of powers
Now we are going to study the effect of fractional order theory of thermoelasticity proposed by H.H. Sherief et al. in the problem encountered in Chapter 3. We now have a new heat equation which consists of new fractional derivatives of order obtained from the Caputo definition. Our aim is to see if as we get back the Lord and Shulman (L-S) solutions.
Once again, we transform our governing equations into spherically symmetric form before non-dimensionalising them. Only the radial component is considered. All initial and boundary conditions are kept the same as in Chapter 3 for comparison purposes. The Laplace Transform is used again but since our characteristic equation is too complicated, we are unable to invert it directly to obtain the solutions. Short time approximations are not valid too.
Hence, we use Mathematica to find numerical approximations of the displacement, temperature and stresses. We observe their behaviour under different values of and plot them on a graph.
Finally we compare the graphs with the L-S Model (where 1).
4.2 Formulation of the problem
We use the same assumptions as in Chapter 3 here, with certain changes. We have a new heat equation of the form
The remaining equations remain the same as before.
Upon non-dimensionalising the governing equations, we obtain the following 1-D equations :
Note: We use the same dimensional variables like in Chapter 3 for comparison purposes later.
We assume the same boundary and initial conditions as in Chapter 3 to study the behaviours of Eqn (4.2.2)-(4.2.5).
Setting in the governing equations (4.2.2) and (4.2.3), we obtain:
On eliminating we get
4.3 Transform Solution
Taking Laplace transform in the time domain, Eqn (4.3.1)
and Eqn (4.2.8) becomes
The above equation can be rewritten in the form
where and are the roots of the biquadratic equation
Now using Helmholtz equation and using regularity condition that as , we obtain the following solution for Eqn (4.3.4):
Using the Eqn (4.3.6), we have the general solution for displacement and temperature in the Laplace transform version in the form
On solving Eqn (4.3.5), we have
4.4 Inversion of Laplace Transform
On substituting Eqn (4.3.9) and (4.3.10) in our equations for and we find that we obtain complicated expressions because of the presence of . Thus we are unable to invert our equations directly using Laplace transform or even short time approximations as we have for division by zero and Maclaurin expansion is not valid under such conditions.
Hence, we use the Talbot algorithm to find numerical approximations to determine the inverse transforms.
4.5 Numerical Results and Discussion
A copper material is considered for the graph as in the previous chapters. We solve the problem by taking
while varying the values of r.
We then compare the graphs we obtain for each component for the three different values of obtained.
We can see from Fig 4.6.1-4.6.4 that there are only minor differences with the different values of .
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