# The Fractional Strain Influence on a Solid Sphere under Hyperbolic Two-Temperature Generalized Thermoelasticity Theory by Using Diagonalization Method

Authors: Hamdy M. Youssef; Alaa A. El-Bary; Eman A. N. Al-Lehaibi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This article constructs a mathematical model based on fractional-order deformations for a one-dimensional, thermoelastic, homogenous, and isotropic solid sphere. In the context of the hyperbolic two-temperature generalized thermoelasticity theory, the governing equations have been established. Thermally and without deformation, the sphere’s bounding surface is shocked. The singularities of the functions examined at the center of the world were decreased by using L’Hopital’s rule. Numerical results with different parameter fractional-order values, the double temperature function, radial distance, and time have been graphically illustrated. The two-temperature parameter, radial distance, and time have significant effects on all the studied functions, and the fractional-order parameter influences only mechanical functions. In the hyperbolic two-temperature theory as well as in one-temperature theory (the Lord-Shulman model), thermal and mechanical waves spread at low speeds in the thermoelastic organization.


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## Body

## 1. Introduction

The subject matter’s primary concern is how to detect an exact model that simulates thermoelastic material’s behavior. Researchers and authors have developed several models in which the waves in solids and thermoelastic materials have been transmitted. But not all these models succeed as one of the metrics of a good model is to give the experimental results with a finite speed of progress of mechanical and thermal waves. To talk about thermomechanical transition models with elastic materials, a wide area cannot be limited to study. Therefore, we will discuss some recent models that need to be addressed.The thermoelasticity model has been developed by Chen and Gurtin, based on two different conductive and dynamic temperatures. The temperature differential is proportionate to the heat source [1]. In the two-temperature thermoelastic theory, Warren and Chen investigated wave propagation [2]. However, before Youssef updates this theory and creates a two-temperature generalized thermostat model, there will be no study of that theory [3]. In many applications and inquiries, Youssef and several other writers used this model [4–13]. Youssef and El-Bary have proven that the classical two-temperature generalized thermoelasticity model does not provide heat wave transmission at a finite speed [14]. Therefore, Youssef and El-Bary have changed this model and implemented a new two-temperature model based on different thermal conductivity rules, called hyperbolic two-temperature generalized thermoelasticity [14]. In that model, Youssef suggested that it is proportional to the heat supply to differentiate between conductive temperature and dynamic temperature acceleration. The speed of thermal wave propagation is limited in this model. The two-temperature theory is considered as a type of generalization of the heat conduction equation. It helps us to separate the conductive thermal wave and dynamical thermal wave and study each wave and its effects on the materials.The fact that fractional systems possess memory justifies this generalization, as the time evolution of romantic relationships is naturally impacted by memory [15, 16]. The fractional-order dynamic model could model various real materials more adequately than integer-order ones and provide a more adequate description of many actual dynamical processes [17, 18]. Thus, some new thermoelastic models based on the concept of fractional calculus were introduced. Magin and Royston developed the first model using the fractional deformation derivative characterizing the material’s behavior [19]. A Hookean solid is the derivative’s zero-order, while a Newtonian fluid is the one-order. The intermediate spectrum of the heat gives different orders of thermoviscoelastic substances [19].Depending on the factional-order strain, Youssef launched another new theory of general thermoelasticity. The relationship between stress and strain is seen as a new and distinct addition to Duhamel-Neumann [20]. Youssef has solved thermoelasticity with fractional sequence strain in one dimension, half of the space of Biot, Lord-Shulman, Green-Lindsay, and Green-Naghdi type II [20]. Youssef solved many applications of thermoelasticity of infinite thermoelastic spherical medium [21–24]. Mukhopadhyay and Kumar studied the general thermoelastic interactions with the cavity in the unlimited medium [25]. Many authors solved problems of thermoelastic medium with spherical cavity [26–35].Most of the authors who regarded their applications as the spherical medium faced the singularity problem in its center. Few authors, for example, Thibault and others, were able to solve this situation in the thermoelastic solid sphere using L’Hopital’s rule to resolve this problem [36].In the present investigation, a solid sphere in the context of hyperbolic two-temperature generalized thermoelasticity theory based on the fractional-order strain definition will be studied. The advantages of the current article are that it is considered more generalized model than the models that precede it and contains the results of a new theory with comparing it to the results of previous theories.


## 2. The Governing Equations

Consider a perfect, thermoelastic, conducting, and isotropic spherical body that occupies the regionξ=r,ψ,ϕ:0≤r≤a,0≤ψ≤2π,0≤ϕ<2π. We use a spherical coordinative system r,ψ,ϕ that displays the radial coordinate, colatitude, and longitude of a spherical system, without any forces on the body and initially calming where r is the sphere radius, as in Figure 1. When there is no latitude or longitudinal variance, the symmetry condition is fulfilled. Both state functions depend on the distance and time of the radius.Figure 1
The isotropic homogeneous thermoelastic solid sphere.We note that, due to spherical symmetry, the displacement components have the form(1)ur,uψ,uϕ=ur,t,0,0.The equation of motion [10, 20] is(2)ρu¨=λ+2μ1+τ1αDtα∂e∂r−γ∂TD∂r.The constitutive equations with damage mechanics variable [10, 20] are(3)σrr=1+τ1αDtα2μerr+λe−γTD−T0,(4)σψψ=1+τ1αDtα2μeψψ+λe−γTD−T0,(5)σϕϕ=1+τ1αDtα2μeϕϕ+λe−γTD−T0,(6)σrϕ=σϕψ=σrψ=0.The strain components are(7)err=∂u∂r,eψψ=eϕϕ=ur,erϕ=eϕψ=erψ=0,where e is the cubical dilatation and it is given by(8)e=err+2eϕϕ=∂u∂r+2ur=1r2∂r2u∂r.The hyperbolic two-temperature heat conduction equations take the forms [10, 14, 20](9)K∇2TC=ρCE∂∂t+τ0∂2∂t2TD+γT01+τ1αDtα∂∂t+τ0∂2∂t2e,(10)T¨D=T¨C−c2∇2TC,where cm/s is the hyperbolic two-temperature parameter [14] and ∇2=1/r2∂/∂rr2∂/∂r.The Riemann–Liouville fractional integralIαft description is used in the above equations written in a convolution type form [20, 37](11)Iαft=1Γα∫0tt−υα−1fυdυ,t>0,α>0,which provides Caputo with the form of fractional derivatives:(12)Dtαft=I−αft=1Γ1−α∫0tfξt−ξαdξ,0<α<1,ft,α=0,f′t,α=1.We consider thatφ=TC−T0 and θ=TD−T0 are the conductive and dynamical temperature increments, respectively. Then equations (2)–(5), (9), and (10) take the forms(13)ρu¨=λ+2μ1+τ1αDtα∂e∂r−γ∂θ∂r,(14)σrr=1+τ1αDtα2μerr+λe−γθ,(15)σψψ=1+τ1αDtα2μeψψ+λe−γθ,(16)σϕϕ=1+τ1αDtα2μeϕϕ+λe−γθ.Equation (13) can be rewritten to be in the form(17)ρe¨=λ+2μ1+τ1αDtα∇2e−γ∇2θ.The following nondimensional variables are used for convenience [5, 9, 10]:(18)r′,u′,a′=coηr,u,a,t′,τ′,τo′,τ1′=co2ηt,τ,τo,τ1,θ′,φ′=1T0θ,φ,σ′=σμ.Then, we obtain(19)e¨=1+τ1αDtα∇2e−b∇2θ,(20)∇2φ=∂∂t+τo∂2∂t2θ+ε∂∂t+τo∂2∂t2e,(21)θ¨=φ¨−c˜2∇2φ,(22)σrr=1+τ1αDtαβ2e−2ur−ε1θ,(23)σψψ=1+τ1αDtαβ2e−2∂u∂r−ε1θ,(24)σϕϕ=1+τ1αDtαβ2e−2∂u∂r−ε1θ,where γ=3λ+2μαT, co2=λ+2μ/ρ, η=ρCE/K, ε=γ/ρCE, ε1=γTo/μ, β=λ+2μ/μ1/2, b=ε1/β2, and c˜2=c2/co2.The primes are suppressed for simplicity.The operator∇2=1/r2∂/∂rr2∂/∂r is singular at r=0; however, if symmetry prevails, the singularity will be reduced by the following L’Hopital’s rule [36]:(25)∇2e,θ,φ=limr⟶01r2∂∂rr2∂e,θ,φ∂r=limr⟶0∂2e,θ,φ∂r2+2r∂e,θ,φ∂r=∂2e,θ,φ∂r2+2∂2e,θ,φ∂r2.Then, we get(26)∇2e,θ,φ=3∂2∂r2e,θ,φ,satisfying the boundary condition(27)∂∂re,θ,φ|r=0=0.Hence, we have(28)∇2er,t=3∂2er,t∂r2,∇2θr,t=3∂2θr,t∂r2,∇2φr,t=3∂2φr,t∂r2.By applying the form in (26) in equations (19)–(21), we obtain(29)e¨=31+τ1αDtα∂2e∂r2−3b∂2θ∂r2,3∂2φ∂r2=∂∂t+τo∂2∂t2θ+ε∂∂t+τo∂2∂t2e,θ¨=φ¨−3c˜2∂2φ∂r2.We apply the Laplace transform defined as(30)ℓft=f¯s=∫0∞fte−stdt.The Laplace transform of the fractional derivative is defined as [37](31)ℓDtαft=sαf¯s−DtαI1−αf0+,0<α<1.The following initial conditions have been assumed:(32)DtαI1−αθr,0+,φr,0+,er,0+=0,0<α<1,∂θr,t∂t|t=0=∂φr,t∂t|t=0=∂er,t∂t|t=0=0.Thus, equations (19)–(24) have the forms(33)s2e¯=31+τ1αsα∂2e¯∂r2−3b∂2θ¯∂r2,(34)3∂2φ¯∂r2=s+τos2θ¯+εs+τos2e¯,(35)θ¯=φ¯−δ2∂2φ¯∂r2,(36)σ¯rr=1+τ1αsαβ2e¯−2u¯r−ε1θ¯,(37)σ¯ψψ=1+τ1αsαβ2e¯−2∂u¯∂r−ε1θ¯,(38)σ¯ϕϕ=1+τ1αsαβ2e¯−2∂u¯∂r−ε1θ¯,(39)e¯=1r2∂r2u¯∂r,where δ2=3c˜2/s2.Substituting equation (35) into equations (33) and (34), we get(40)∂2∂r2−α1e¯=β1∂2φ¯∂r2−δ2β1∂4φ¯∂r4,(41)∂2φ¯∂r2=α2φ¯+α3e¯,where α1=s2/31+τ1αsα, β1=b/1+τ1αsα, α2=s+τos2/3+δ2s+τos2, and α3=εs+τos2/3+δ2s+τos2.Substituting equation (41) into equation (40), we obtain(42)∂2e¯∂r2=α4φ¯+α5e¯,where α4=β1α21−δ2α2/1+δ2β1α3 and α5=α1+β1α31−δ2α2/1+δ2β1α3.


## 3. The Diagonalization Method

We can rewrite equations (41) and (42) in a matrix form as follows [38]:(43)ddrφ¯e¯φ′¯e′¯=00100001α2α300α4α500φ¯eφ′¯e′¯.For simplicity, we write the system in (43) as a homogenous system of linear first-order differential equation as [38](44)dZrdr=AZr,where Zr=φ¯re¯rφ′¯re′¯r and A=00100001α2α300α4α500.MatrixA has four linearly independent eigenvectors; hence, we can construct a matrix V from the eigenvectors of matrix A such that V−1AV=W, where W is a diagonal matrix [38].If we make the substitutionZ=VY in system (44), then(45)VY′=AVY,Y′=V−1AVY=WY,which gives(46)y1′y2′y3′y4′=k10000k20000k30000k4y1y2y3y4,where ±k1and±k2 are the eigenvalues of matrix A or the roots of the characteristic equation(47)k4−Lk2+M=0,where(48)L=k12+k22=α2+α5,M=k12k22=α2α5−α3α4,k2=−k1,k4=−k3.SinceW is a diagonal matrix, then system (46) is uncoupled, making each differential equation in the system have the form yi′=kiyi,i=1,2,3,4. The solution to each of these linear equations is yi=ciekix,i=1,2,3,4. Hence, the general solution of system (46) can be written as the column vector [38]:(49)Y=c1ek1rc2e−k1rc3ek2rc4e−k2r.Then, the final solution of system (44) is(50)Zr=VYr.MatrixV from the eigenvectors of matrix A takes the form(51)V=α3k1k12−α2−α3k1k12−α2α3k2k2−α2−α3k2k2−α21k1−1k11k2−1k2α3k12−α2α3k12−α2α3k2−α2α3k2−α21111.Substituting equations (49) and (51) into equation (50), we get(52)φ¯re¯rφ′¯re′¯r=α3k1k12−α2−α3k1k12−α2α3k2k2−α2−α3k2k2−α21k1−1k11k2−1k2α3k12−α2α3k12−α2α3k2−α2α3k2−α21111c1ek1rc2e−k1rc3ek2rc4e−k2r.The boundary conditions in (27) and equation (52) give that(53)c1=−c2,c3=−c4.Hence, we obtain(54)φ¯r,s=α3∑i=12Aicoshkirkiki2−α2,(55)e¯r,s=∑i=12Aicoshkirki.To get the constantsA1andA2, we must apply the boundary conditions at r=a; we consider the sphere when r=a is thermally shocked as follows:(56)φa,t=φoHt,where Ht is the Heaviside unit step function and φo is constant.For the mechanical boundary condition, we consider the surface of the sphere whenr=a is connected to a rigid foundation, which can stop any displacement u|r=a=0, offering zero volumetric deformation from equation (39); hence, we have(57)ea,t=0.Applying Laplace transform on equations (56) and (57), we obtain(58)φ¯a,s=φos,e¯a,s=0.Applying the boundary conditions in equations (54) and (55), we get the following system:(59)∑i=12Aicoshkiakiki2−α2=φ0sα3,(60)∑i=12Aicoshkiaki=0.By solving system (59) and (60) by using the relations between the roots (48), we get(61)A1=φ0α4k1sk12−k22coshk1a,A2=−φ0α4k2sk12−k22coshk2a.Hence, we have(62)φ¯r,s=φ0α3α4sk12−k22coshk1rk12−α2coshk1a−coshk2rk22−α2coshk2a,(63)e¯r,s=φ0α4sk22−k12coshk1rcoshk1a−coshk2rcoshk2a.To obtain the displacement function, we will use equations (39) and (63) as(64)u¯r,s=1r2∫r2e¯r,s∂r.The singularity in (64) can be removed by using L’Hopital’s rule twice in a row as follows [36]:(65)u¯r,s=limr⟶0∫r2e¯r,s∂rr2=limr⟶0r2e¯r,s2r=re¯r,s2.Hence, we have(66)u¯r,s=φ0α4r2sk22−k12coshk1rcoshk1a−coshk2rcoshk2a.We use the average of the three principal stress components for equations (36)–(38) to achieve a straightforward way of distribution to be as follows:(67)σ¯r,s=σrr¯+σψψ¯+σϕϕ¯3=β2−431−De¯r,s−ε11−Dθ¯r,s.The techniques of Riemann-sum approximation are used to obtain numerical solutions of studied functions in the time domain. In this method, any function can be transformed into the Laplace by using the following formula [39]:(68)ft=eκtt12f¯κ+Re∑n=1N−1nf¯κ+inπt,where “i” is the well-known unit of imaginary number and “Re” gives the real part. Many computational studies have shown that value κ meets the relationship for convergence with faster procedures. κt≈4.7 [39].After calculating the Laplace transformations, the stress-strain energy can be obtained in the following formula [40]:(69)ϖr,t=12σijr,teijr,t.For the current model, the stress-strain energy is in the form(70)ϖr,t=12σrrerr+σψψeψψ+σϕϕeϕϕ.After eliminating the term with a small value, we get(71)ϖr,t≈121+τ1αDtαβ2e2r,t−ε1er,tθr,t.


## 4. Numerical Results and Discussion

For numerical analysis, copper is the thermoelastic substance for which the various physical constants’ values are taken as follows [24]:(72)K=386kg mk−1s−3,αT=1.7810−5k−1,ρ=8954kgm−3,To=293k,CE=383.1m2k−1s−2,μ=3.861010kgm−1s−2,λ=7.761010kgm−1s−2.Thus, we get the nondimensional values of the problem as(73)b=0.01047,ε1=0.0419,ε=1.6086β2=4,φ0=1.0,τo=0.02,τo=0.01.For a large range of the dimensionless radial distance,r0≤r≤8.0, the numerical results from conductive and dynamic temperature increments, pressure, changes, average stress, and stress-power distributions were shown at the instant value of dimensionless time t=1.0.Figures2–7 were obtained for various values of the two-temperature parameter c˜=0.0,0.5, which gives δ2=3c˜2/s2=0.0,30.52,30.5/s2, where the value δ=0.0 represents the L-S one-temperature model; it has been figured in solid curves. The value c˜=0.5 represents two cases; the first case is δ2=30.52, which represents the classical two-temperature model and has been figured with dash curves, while the second case is δ2=30.5/s2, which represents the hyperbolic two-temperature model and has been figured in dot curves. The numerical results of those figures have been calculated when the fractional order parameter α=0.5.Figure 2
The conductive temperature increment distribution for different models.Figure 3
The dynamical temperature increment distribution for different models.Figure 4
The volumetric deformation distribution for different models.Figure 5
The displacement distribution for different models.Figure 6
The average stress distribution for different models.Figure 7
The stress-strain energy distribution for different models.Figure2 shows the conductive temperature increment distributions, and it is seen that all the curves start from the position r=8.0 with the value φr=8.0=1.0 , which is the value of the thermal shock on the bounding surface of the sphere. All the curves have the same behavior but with different values. The two-temperature parameter has a significant effect on conductive temperature distribution. At the endpoint of the sphere's radius, the one-temperature and hyperbolic two-temperature model curves fall to zero values. It means that the conductive thermal wave has propagated with a finite speed in the context of one-temperature and hyperbolic two-temperature models, while the classical two-temperature model does not.Figure3 shows the dynamical temperature increment distributions, where it is noted that all the curves start from the position r=8.0 with different values θ|one−temp.=1.0, which is the value of the thermal shock on the bounding surface of the sphere θ|Class.two−temp.=0.98 and θ|hyp.two−temp.=0.75. All the curves have the same behavior with various values. The two-temperature parameter has a significant effect on the dynamical temperature distribution. At the endpoint of the sphere’s radius, the one-temperature and hyperbolic two-temperature model curves fall to zero values. It means that the dynamical thermal wave has propagated with a finite speed in the context of one-temperature and hyperbolic two-temperature models, while the classical two-temperature model does not.The volumetric distribution of the strain is illustrated in Figure4, and three curves begin at r=8.0, a position with zero value er=8.0=0.0 , which agrees with the boundary condition on the sphere’s bounding surface. All the curves have the same attitude with different values. The one-temperature and hyperbolic two-temperature models’ curves have a sharp point in the same position with different values. In contrast, the classical two-temperature model has a peak point in different situations. The absolute values of the maximum volumetric deformation take the following order:(74)emaxOne−temp.>emaxHyp. two−temp.>emaxClass. two−temp..Figure5 shows the displacement distributions, and it is seen that the three curves start from the position r=8.0 with zero value ur=8.0=0.0. All the curves have the same attitude with various values. The one-temperature and hyperbolic two-temperature models’ curves have sharp points in the same position with different values. In contrast, the classical two-temperature model has a peak point in different situations. The absolute values of the maximum displacement take the following order:(75)umaxOne−temp.>umaxHyp. two−temp.>umaxClass. two−temp..Figure6 represents the average stress distributions, and we can see that the three curves start from the position r=8.0 with different values. The one- and two-temperature curves have the same behavior with different values, and each has a sharp point. The curve, by comparison, is distinct and smooth in the classical two-temperature model.Figure7 shows the stress-strain energy distribution, and it is noted that the three curves start from the position r=8.0 with zero values. The three curves have the same behavior with different values, and each one has a peak point. The maximum values of the stress-strain energy take the following order:(76)ϖmaxOne−temp.>ϖmaxHyp. two−temp.>ϖmaxClass.  two−temp..Figures8–13 have been done for different values of the fractional-order parameter α=0.0τ1=0,0.2,0.6,1.0 in the context of the hyperbolic two-temperature model to discuss its effects on the state of the studied functions. The case α=τ1=0.0 represents the normal stress-strain relations as the usual Hooke’s law of the elastic body. In contrast, the cases α=0.2,0.6 define the new cases between the normal elasticity and the viscoelasticity and α=1.0 represents the viscoelasticity.Figure 8
The conductive temperature increment distribution with various values of the fractional-order parameter.Figure 9
The dynamical temperature increment distribution with various values of the fractional-order parameter.Figure 10
The volumetric deformation distribution with various values of the fractional-order parameter.Figure 11
The displacement distribution with various values of the fractional-order parameter.Figure 12
The average stress distribution with various values of the fractional-order parameter.Figure 13
The stress-strain energy distribution with various values of the fractional-order parameter.Figures8 and 9 show that the fractional-order parameter has limited impacts on the dynamical and conductive temperature increment. This result was expected because the fractional-order parameter’s effect exists firmly in the stress-strain relation, which significantly influences the mechanical waves more than the thermal waves.Figure10 represents that the effect of the fractional-order parameter is significant on the volumetric strain distributions. All the curves start from the position r=8.0 with zero value, and every curve contains a peak except for the curve, which has no fraction τ1=0.0, and it has a sharp point. The maximum values of the absolute value of the volumetric deformation take the following order:(77)emaxα=1.0>emaxα=0.6>emaxα=0.2>emaxα=τ1=0.0.Figure11 indicates a significant influence on displacement distributions of the fractional-order parameter. The four curves begin at the location r=8.0, and each curve contains a peak except the curve, which does not have a fraction τ1=0.0, where it has a sharp point. The maximum values of the absolute value of the displacement take the following order:(78)umaxα=1.0>umaxα=0.6>umaxα=0.2>umaxα=τ1=0.0.Figure12 represents that the fractional-order parameter has significant impacts on stress distributions. All the curves start from the position r=8.0 with the same value, and each curve is smooth except the curve of the case, which has no fraction τ1=0.0, and it has a sharp point.Figure13 represents that the fractional-order parameter has significant effects on stress-strain energy distributions. The four curves start from the position r=8.0 with zero values, and each curve has a peak point. The values of the maximum stress-strain energy take the following order:(79)ϖmaxα=1.0>ϖmaxα=0.6>ϖmaxα=0.2>ϖmaxα=τ1=0.0.For the validation of the results, we can notice that the current results that are based on traditional integer derivative agree with the results in the past publications [25, 28, 29, 32, 41–43].


## 5. Conclusions

The numerical results conclude that the one-temperature model and the hyperbolic two-temperature theory of thermoelasticity generate thermomechanical waves that can propagate with limited speeds. Therefore, the hyperbolic two-temperature thermoelasticity model is a successful model to describe thermoelastic materials’ thermodynamical behavior. Moreover, the two-temperature parameter has significant impacts on the states of all the studied functions. The fractional-order strain parameter has weak effects on the thermal waves, while its effects on the mechanical waves are significant.



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