In this undertaking, we will analyze look into the thermoelastic interactions caused by a uninterrupted point heat beginning in a homogenous and isotropic boundless thermoelastic organic structure, by utilizing the additive theory of thermoelasticity without energy dissipation. We are traveling to work out the job utilizing the same boundary conditions utilizing different thermoelastic theoretical accounts, viz. Green and Naghdi II, Lord and Shulman and Fractional Order theory of thermoelasticity and compare the consequences for supplanting, temperature, radial emphasis and transverse emphasis.
First, allow us specify the keywords: Generalised thermoelasticity, point heat beginning.
The generalization of thermoelasticity arises from a modifying the Fourier ‘s jurisprudence of heat conductivity.
What is thermoelasticity?
Thermoelasticity is the alteration in size and form of a solid stuff as its temperature fluctuates. Objects which are more elastic will spread out more and those which are less elastic will spread out less. Scientists use their cognition of thermoelasticity to invent new stuffs that withstand heat alterations better without interrupting.
Following, we have a point [ heat ] beginning. A point beginning [ of heat ] is a individual identifiable localized beginning [ of heat ] .
In mathematical modeling, these beginnings can be approximated as a mathematical point to simplify analysis. It is good to observe that the size of the existent beginning need non needfully be little physically. It is comparative to the graduated table where it is being used. For illustration, in uranology, stars are treated as point beginnings.
Now allow us see the assorted thermoelastic theoretical accounts we have.
1.2 COUPLED/CLASSICAL THERMOELASTICITY ( CTE ) -BIOT [ 1956 ]
The classical theory of thermoelasticity ( CTE- Biot [ 1956 ] ) is based on the heat conductivity equation of Fourier which assumes that the thermic perturbations propagate at infinite velocity. This anticipation seems unrealistic physically, particularly in state of affairss affecting really short transient continuances, sudden high heat flux exposures, and really low temperatures near the absolute nothing.
These facets have caused much disturbance in the field of heat extension.
The classical theoretical account for extension of heat is:
( 1.2.1 )
( Fourier Law )
( 1.2.2 )
which yields the parabolic heat equation:
The government equations are:
( 1.2.3 )
( 1.2.4 )
( 1.2.5 )
The over-dot represents partial derived functions with regard to clip, T.
1.3 EXTENDED THERMOELASTICITY-Lord & A ; Shulman [ 1967 ]
Lord and Shulman proposed generalized thermoelasticity equations by modifying the Fourier ‘s heat conductivity equation taking into history the clip needed for acceleration of the heat flow. That is our energy equation which consists of two conjugate partial differential equations, includes a relaxation clip and is of the signifier:
( 1.3.1 )
The stress- temperature dealingss is:
. ( 1.3.2 )
The equation of gesture is:
( 1.3.3 )
We note that the lone manner the above equations are different from the regulating equations of CTE is the presence of and on taking it we get back our original equations. This invariable is the relaxation clip which will do certain that the heat conductivity equation will foretell finite velocities of heat extension, V,
1.4 TEMPERATURE-RATE-DEPENDENT THERMOELASTICITY
Green & A ; Lindsay [ 1972 ]
Green and Lindsay on the other manus, did alternations with the constitutive and the energy equation, without nevertheless altering the Fourier heat conductivity equation. The thermoelastic equations involve two new relaxation parametric quantities, and this clip in the stress- temperature relation and the heat conductivity.
The regulating equations about speed, temperature and emphasiss are given severally as follows:
We note that by doing and, the theory reduces to CTE and L-S severally.
1.5 Green & A ; Naghdi Models
( a ) THERMOELASTICITY WITHOUT ENERGY DISSIPATION- GN II
Green and Naghdi ( 1992 ) developed a to the full non-linear theory which permits heat transmittal at finite velocity without any heat loss. In this theoretical account thermal-displacement gradient is considered as a constituent variable and is different from the old theoretical accounts since it does non suit for energy loss.
The GN II theoretical account considers undamped thermoelastic moving ridges and is known as the theory of thermoelasticity without energy disspation.
The government equations are:
The velocity of the moving ridge is given by:
Thermoelasticity With Energy Dissipation, G-N III
Based on their old theoretical account, Green and Naghdi developed another thermoelasticity theory. This clip, they included enery dissipation. There is an excess to the heat equation.
Another manner GN-III differs from GN-II is that it admits damped moving ridges.
The regulating equations in vector signifier are
The finite velocity is given by:
1.6. Fractional Order Model
Fractional concretion is a subdivision of mathematical analysis that investigate the possibility of widening derived functions and integrals to non-integer orders. It is by and large known than integer-order derived functions have clear physical and geographical readings. However in the fractional order theoretical account, which is merely a new born in the universe of mathematics, is still on the manner to germinate. For more than 300 old ages, there was no clear reading of this theoretical account.
Merely late, have we found more about its importance and maps. It is used in the survey of viscoelastic stuffs, every bit good as many Fieldss in scientific discipline and technology including fluid flow, in electrical web, electromagnetic theory and chance.
The most popular definitions of the fractional derived functions are those of Riemann-Liouville and Grunwald-Letnikov [ 6 ] .
The Riemann-Liouville definition is given as:
where is the Gamma map.
The Grunwald-Letnikov definition is
where, is the integer portion of ten and H is the clip measure.
The derivation of with regard to is.
Now we want to cognize the derived function of utilizing a non-integer value. Let us see the stairss of the computation.
The general instance is:
where, is the non whole number value such that and k is the power of the map
The derived function of with is. Upon distinguishing a 2nd clip utilizing the derived function 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 ) theoretical account is used to analyze the thermoelastic interactions caused by a uninterrupted point heat beginning in a homogenous and isotropic boundless thermoelastic organic structure, pretermiting any organic structure forces by utilizing the additive theory of thermoelasticity without energy dissipation.
We have to happen looks for the supplanting, temperature and emphasis Fieldss. First, we express our regulating equations in a one dimensional spherical system, so we non- dimensionalise them. The Laplace Transform is used to work out the job and we get an exact solution. We will besides be able to detect the behaviors of each constituent by their discontinuities.
Finally we represent some numerical informations for the supplanting, temperature and stresses diagrammatically for a Cu – like stuff.
2.2 Formulation of the Problem
In our state of affairs, we are sing a unidimensional job of the different constituents. The thermoelastic interactions caused by the beginning are spherically symmetric in nature, that is they depend merely on and. There are besides no organic structure forces present here.
Therefore, the displacement constituent consisting of merely the radial constituent is of the signifier
where R is the distance from the beginning.
For convenience, we apply a non-dimensionalising strategy to the regulating equations ( 1.2.1 ) , ( 1.2.2 ) and ( 1.2.3 ) utilizing these given transmutations:
( 2.2.1 )
where is a standard length and is a standard velocity.
Applying ( 2.2.1 ) to ( 1.2.1 ) – ( 1.2.3 ) and dropping the primes for convenience, we get the undermentioned set of non-dimensional equations:
( 2.2.2 )
( 2.2.3 )
( 2.2.4 )
( 2.2.5 )
( 2.2.6 )
Note that and the non-dimensional velocities of strictly elastic dilatational and shear moving ridges severally, is the non-dimensional velocity of strictly thermic moving ridge and that is the matching parametric quantity.
Suppose ab initio, the point beginning is at remainder in its undeformed province. The following homogenous initial conditions hold for:
( 2.2.7 )
( 2.2.8 )
( 2.2.9 )
Now if we substitute
( 2.2.10 )
into ( 2.2.2 ) and ( 2.2.3 ) , we get the undermentioned equations:
( 2.2.11 )
( 2.2.12 )
Extinguishing from the above two equations we get
( 2.2.13 )
This equation serves as the regulating equation for. Once we solve it under suited conditions and happen 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 organic structure is ab initio stationary in its original province with its temperature alteration and temperature rate equal to zero. We besides assume that the point heat beginning doing thermoelastic interactions, at clip T & gt ; 0, is specified by
( 2.3.1 )
Here, is a changeless, is the Dirac Delta and H ( T ) is the Heaviside unit measure map.
Now we will work out our job utilizing the Laplace Transform defined as:
Taking Laplace Transform of equation ( 2.2.13 ) under homogenous initial conditions, with given by ( 2.3.1 ) , we get
( 2.3.2 )
Now, Eqn 2.3.2 can be rewritten in the signifier
, ( 2.3.3 )
Where and are the roots of the biquadratic equation
( 2.3.4 )
Then utilizing the Helmholtz equation
and enforcing the regularity status that as we get the undermentioned solution for Eqn ( 2.3.3 ) :
( 2.3.5 )
We consider merely the positive existent roots and of Eqn ( 2.3.4 ) .
Using Laplace Transform to ( 2.2.7 ) and ( 2.2.8 ) we obtain the undermentioned solutions for and:
( 2.3.6 )
( 2.3.7 )
2.4 Exact Solutions
Given that we have obtained the solution for the supplanting and temperature, we are now traveling to happen the field variables in clip.
Solving the biquadratic Eqn ( 2.3.4 ) , we find that
( 2.4.2 )
( 2.4.3 )
( 2.4.4 )
Substituting from Eqn ( 2.4.1 ) into the transform solutions ( 2.3.6 ) and ( 2.3.7 ) and using the opposite Laplace transforms of the looks obtained, we get the exact looks for and:
( 2.4.5 )
( 2.4.6 )
( 2.4.7 )
The emphasiss are now obtained from Eqn ( 2.2.4 ) and ( 2.2.5 ) :
( 2.4.8 )
( 2.4.9 )
To corroborate our computations, we straight verify that ( 2.4.5 ) , ( 2.4.6 ) , ( 2.4.8 ) and ( 2.4.9 ) satisfy the regulating Eqn ( 2.2.5 ) and ( 2.2.6 ) with given by ( 2.3.1 ) , the constituent dealingss ( 2.2.7 ) and ( 2.2.8 ) with the homogenous initial conditions and the regularity conditions. We find that ( 2.4.5 ) , ( 2.4.6 ) , ( 2.4.8 ) and ( 2.4.9 ) are so exact solutions, in closed signifier, for, , and We note that these are made up of two parts, each one corresponding to a moving ridge traveling with finite velocity, and
Using ( 2.4.2 ) and ( 2.4.3 ) , we find that. and if, and if. The perturbations that we are sing consist of two distinct coupled moving ridges, 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 about zero for. This implies that there exists a certain blink of an eye of clip where the points of the organic structure which are past the faster wave front do non see any perturbation. Therefore, the effects of heat beginning are restricted to a time-dependent delimited part environing the beginning.
2.5 Analysing Discontinuities
After analyzing solutions ( 2.4.5 ) , ( 2.4.6 ) , ( 2.4.8 ) and ( 2.4.9 ) , we can detect the discontinuities experienced by, , and at the wave fronts, . These are:
( 2.5.1 )
( 2.5.2 )
( 2.5.3 )
( 2.5.4 )
Note represents the discontinuity of the map across the wave front, .
Expression ( 2.5.1 ) shows that supplanting is uninterrupted at both wave fronts.
Expressions ( 2.5.2 ) – ( 2.5.4 ) show us that there are positive leaps in temperature and emphasis at the slower wave front, while at the faster wavefront the leap in temperature is positive and in emphasis is negative. We can besides detect that these leaps are reciprocally relative to
2.6 Numeric Consequences
For intents of numerical rating, we consider a Cu like stuff with invariables and.
Using looks ( 2.4.2 ) and ( 2.4.3 ) the values of and are obtained as:
Graphs are so plotted for.
The graph shows that supplanting is uninterrupted for all positive values of including the location wave front. additions bit by bit up to of the faster wave front and becomes zero beyond this point.
Here we see that decreases bit by bit in two intervals: and undergoing finite leaps at and and it disappears identically for Calculation shows that is about equal to at the point found merely behind the e-wavefront and 0.031765 at the point merely beyond this wave front. so decreases and comes to the value 0.01003 at the point merely before the wave front and leaps to zero beyond that wave front.
Fig 2.6.3 shows that is compressive in the interval, sing leaps at and and disappears identically for. It increases in the interval and attacks -0.025551 about at the point merely before the e-wavefront ( the slower forepart ) . Immediately beyond that point, it jumps down to the value. Then increases once more, making a maximal value at the point merely behind the faster wave front ( -wavefront ) . Across this wave front, jumps to go zero.
Using 2.5.2-2.5.4, we calculate the discontinuities in and and hive away them in the undermentioned tabular array:
It can be noted that Fig 2.6.1-2.6.4 show that and leap boundlessly at which is the place of the heat beginning.
While is uninterrupted and experience no leap, and leap at the place of the -wavefront, that is, at and at the place of the -wavefront, at
In this Chapter, we used the GN-II theoretical account to analyze the one dimensional thermoelastic interctions without energy dissipation due to a point heat beginning. We used Laplace Transform to work out the job and got exact solutions for supplanting, temperature and emphasis Fieldss by taking the opposite Laplace transform. Then we plotted the graph of each constituent to detect their behavior by maintaining changeless ( and changing. Thus it was found that was uninterrupted and experience no leaps while and were discontinuous.
In this subdivision, we use the Lord and Shulman theoretical account to look into the thermoelastic interctions without energy dissipation due to a point heat beginning. The same processs as earlier are followed.
Here excessively, we determine the supplanting, temperature and emphasis distributions. Laplace Transform is once more used to work out the job. However, since the characteristic roots are excessively complicated, analytical inversion is non possible. Hence, we use little clip estimates and utilizing the package Mathematica, we obtain the opposite transforms.
Finally, we illustrate the consequences diagrammatically for a Cu like stuff.
3.2 Formulation of the Problem
We consider the undermentioned transmutations to change over the regulating equations ( 1.3.1 ) – ( 1.3.3 ) given in Chapter 1 into non-dimensional signifier:
( 3.2.1 )
Traveling through the same processs as earlier and not dimensionalising our regulating equations we obtain the followers:
( 3.2.2 )
( 3.2.3 )
( 3.2.4 )
( 3.2.5 )
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:
( 3.2.5 )
( 3.2.6 )
Extinguishing from the above two equations outputs:
( 3.2.7 )
3.3 Transform Solution
Taking Laplace Transform of ( 3.2.7 ) under homogenous conditions, with given by ( 2.3.1 ) , we obtain the equation
. ( 3.3.1 )
The above equation may be re-written in the signifier
, ( 3.3.2 )
and and are the positive roots of the characteristic equation
, ( 3.3.3 )
Using the Helmholtz equation and enforcing similar regularity status that every bit, we get the undermentioned solution for Eqn ( 3.3.1 ) :
. ( 3.3.4 )
Solution ( 3.3.2 ) is used in the Laplace Transform versions of ( 2.2.10 )
and ( 3.2.5 ) , we get the undermentioned solutions for and:
( 3.3.5 )
( 3.3.6 )
3.4 Inversion of Laplace Transform
Given that the solution of the biquadratic equation ( 3.3.3 ) contains square roots, it is hard to invert straight. So, since thermoelastic interactions are short lived, we can therefore use the short clip estimate technique to happen the square roots.
Hence we use the Initial Value Theorem
( 3.4.1 )
We expand the consequence utilizing Maclaurin series after puting
( 3.4.2 )
Therefore, ( 3.3.4 ) can be interpreted as
We now expand in a Maclaurin series of powers
Now we are traveling to analyze the consequence of fractional order theory of thermoelasticity proposed by H.H. Sherief et Al. in the job encountered in Chapter 3. We now have a new heat equation which consists of new fractional derived functions of order obtained from the Caputo definition. Our purpose is to see if as we get back the Lord and Shulman ( L-S ) solutions.
Once once more, we transform our regulating equations into spherically symmetric signifier before non-dimensionalising them. Merely the radial constituent is considered. All initial and boundary conditions are kept the same as in Chapter 3 for comparing intents. The Laplace Transform is used once more but since our characteristic equation is excessively complicated, we are unable to invert it straight to obtain the solutions. Short clip estimates are non valid excessively.
Therefore, we use Mathematica to happen numerical estimates of the supplanting, temperature and emphasiss. We observe their behavior 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 job
We use the same premises as in Chapter 3 here, with certain alterations. We have a new heat equation of the signifier
( 4.2.1 )
The staying equations remain the same as before.
Upon non-dimensionalising the regulating equations, we obtain the undermentioned 1-D equations:
( 4.2.2 )
( 4.2.3 )
( 4.2.4 )
( 4.2.5 )
Note: We use the same dimensional variables like in Chapter 3 for comparing intents subsequently.
We assume the same boundary and initial conditions as in Chapter 3 to analyze the behaviors of Eqn ( 4.2.2 ) – ( 4.2.5 ) .
Puting in the government equations ( 4.2.2 ) and ( 4.2.3 ) , we obtain:
( 4.2.6 )
( 4.2.7 )
On extinguishing we get
( 4.2.8 )
4.3 Transform Solution
( 4.3.1 )
Taking Laplace transform in the clip sphere, Eqn ( 4.3.1 )
( 4.3.2 )
and Eqn ( 4.2.8 ) becomes
( 4.3.3 )
The above equation can be rewritten in the signifier
( 4.3.4 )
where and are the roots of the biquadratic equation
( 4.3.5 )
Now utilizing Helmholtz equation and utilizing regularity status that every bit, we obtain the undermentioned solution for Eqn ( 4.3.4 ) :
( 4.3.6 )
Using the Eqn ( 4.3.6 ) , we have the general solution for supplanting and temperature in the Laplace transform version in the signifier
( 4.3.7 )
( 4.3.8 )
On work outing Eqn ( 4.3.5 ) , we have
( 4.3.9 )
( 4.3.10 )
4.4 Inversion of Laplace Transform
On replacing Eqn ( 4.3.9 ) and ( 4.3.10 ) in our equations for and we find that we obtain complicated looks because of the presence of. Thus we are unable to invert our equations straight utilizing Laplace transform or even short clip estimates as we have for division by nothing and Maclaurin enlargement is non valid under such conditions.
Therefore, we use the Talbot algorithm to happen numerical estimates to find the opposite transforms.
4.5 Numeric Consequences and Discussion
A Cu stuff is considered for the graph as in the old chapters. We solve the job by taking
while changing the values of R.
We so compare the graphs we obtain for each constituent for the three different values of obtained.
We can see from Fig 4.6.1-4.6.4 that there are merely minor differences with the different values of.
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[ 2 ] SPEZIALE, C.G. ( 2001 ) . On the coupled heat equation of additive thermoelasticity. Acta Mechanica 150, 121-126.
[ 3 ] SHERIEF, H.H. , EL-SAYED, A.M.A and EL-LATIEF, A.M. ABD. ( 2010 ) . Fractional order theory of thermoelasticity. International diary of Solids and Structures 47, 269-275.
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