Abstract
In this work, a novel mathematical model of thermoelastic, homogenous, isotropic, and infinite medium with a spherical cavity has been constructed. Under the hyperbolic twotemperature GreenNaghdi theory of thermoelasticity typeI and typeIII with fractionalorder strain, the governing equations have been established. The bounding surface of the cavity has been thermally loaded by a ramptype heat and is connected to a rigid foundation which prevents volumetric strain. Different values of the fractionalorder and twotemperature parameters have shown numerical results for the dynamical and conductive temperature increment, strain, displacement, and average of principal stresses, which are graphically applicable to all the functions studied. The fractionalorder parameter has significant effects on stress and strain distributions, while it has a limited effect on the dynamical and conductive temperatures increment. The hyperbolic twotemperature parameter has significant effects on all studied functions based on GreenNaghdi models of type1 and typeII. Moreover, the ramptime heat parameter has a significant impact on all the studied functions under all the studied models of thermoelasticity.
Highlights
 This research examined how skin tissue responds to a steady heat flux caused by a thermoelectrical shock on the bounding plane in terms of temperature reaction and response.
 The fractionalorder strain parameter significantly affects strain and stress.
 The voltage, resistance, and duallag time values significantly affect the distribution of temperature increments and the propagation of thermal waves through skin tissue.
 The Tzou model is the most effective model that accurately predicts the physical behaviour of skin tissue.
 GreenNaghdi typeI thermal wave propagation profiles are smaller than typeIII.
 Strain reduces mechanical wave propagation profile with a fractionalorder and hyperbolic twotemperature heat conduction model.
1. Introduction
How to find a precise model that accurately represents the behaviour of thermoelastic materials is the main issue of the topic. Researchers and writers have created several models that depict how waves are transferred through solids and thermoelastic materials. However, not all of these models are successful because one of the criteria for a good model is to reproduce experimental findings with waves of mechanical and thermal propagation moving at a finite speed. No one field of study can be used to discuss thermomechanical transition models that apply to elastic materials [1].
Based on the idea of fractional calculus, some fresh thermoelastic models were presented. The first model describing the behaviour of the material was created by Magin and Royston using the fractional deformation derivative [2]. The derivative's zero order is a Hookean solid, while it's one order is a Newtonian fluid. The split order of thermoviscoelastic materials and the intermediate heat exchange spectrum [2].
Youssef introduced a different new theory of global thermoelasticity based on the fractionalorder strain. It is believed that the connection between stress and strain is a novel and distinctive contribution to DuhamelNeumann's theory [3]. Youssef has solved the typeII problems of Biot, LordShulman, GreenLindsay, and thermoelasticity in one dimension with fractional sequence strain [3].
Based on two distinct conductive and dynamic temperatures, Chen and Gurtin created the thermoelasticity model. The relationship between the temperature difference and the heat source [4]. Warren and Chen looked at the propagation of waves in the twotemperature thermoelastic theory [5]. However, there won't be any research on that theory before Youssef updates it and develops a twotemperature generalized thermostat model [6]. Youssef and many other authors applied this concept in several applications and enquiries [713]. Youssef and ElBary validated the twotemperature generalized thermoelasticity model, although it does not give a fixed speed for thermal wave propagation [14]. To replace this model, Youssef and ElBary created the hyperbolic twotemperature generalized thermoelasticity model, which is based on new thermal conductivity principles [14]. Youssef proposed that the difference between conductive temperature and dynamic temperature acceleration in that model be proportional to the heat supply. In this model, the rate of thermal wave propagation is constrained. Youssef found solutions to several uses of the infinite thermoelastic spherical media [1518]. In the limitless media, Mukhopadhyay and Kumar investigated the universal thermoelastic interactions with the cavity [19]. Many authors used spherical cavities to solve problems involving thermoelastic mediums [2029].
In the current work, with a spherical cavity, a novel mathematical model of a thermoelastic, homogenous, isotropic, and infinite medium will be developed. The governing equations will be constructed for the hyperbolic twotemperature GreenNaghdi theory of thermoelasticity type1 and typeIII based on fractionalorder strain consideration. A ramptype heat will thermally load the cavity's boundary, and it is attached to a rigid foundation to avoid volumetric strain. Different fractionalorder and twotemperature parameter values will be produced numerical findings for the strain, displacement, average principle stresses, and dynamical and conductive temperature increment, which are visually relevant to all the functions under study.
2. The problem formulation
Consider a perfect, thermoelastic, conducting, and isotropic body with a spherical cavity that occupies the region $\xi =\left\{\left(r,\psi ,\varphi \right):a\le r<\infty ,0\le \psi \le 2\pi ,0\le \varphi <2\pi \right\}$. We use a spherical coordinative system $\left(r,\psi ,\varphi \right)$ 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 Fig. 1. When there are no latitude and longitudinal variance is the symmetry condition fulfilled. Both state functions depend on the distance and time of the radius.
Fig. 1The isotropic homogeneous thermoelastic solid sphere with a spherical cavity
We note that due to spherical symmetry, the displacement components have the form:
The equations of motion [3, 13]:
The constitutive equations with damage mechanics variable [3, 13]:
The strain components are:
and
where $e$ is the cubical dilatation and is given by:
The hyperbolic twotemperature heat conduction equations take the forms [3, 13, 14].
The heat conduction equations which have been proposed by GreenNaghdi take the following form [29]:
The unified Eq. (6) could be used for the two types of GreenNaghdi theories as follows:
The setting $\stackrel{~}{K}=0$ represents the GreenNaghdi typeI model.
The setting $\stackrel{~}{K}={K}^{\mathrm{*}}/K$ represents the GreenNaghdi typeIII model, where ${K}^{\mathrm{*}}=\left(\lambda +2\mu \right){C}_{v}/4$ is the characteristic of GreenNaghdi theory, $K$ is the usual thermal conductivity, and the unit of the quantity $\left(\stackrel{~}{K}\right)$ is ${\mathrm{s}}^{1}$ and:
where $c$ (m/s) is the hyperbolic twotemperature parameter [14], and ${\nabla}^{2}=\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left({r}^{2}\frac{\partial}{\partial r}\right)$.
The Riemann – Liouville fractional integral ${I}^{\alpha}f\left(t\right)$ description is used in the above equations written in a convolutiontype form [3, 30]:
that provides Caputo with the form of fractional derivatives:
We consider that $\phi =\left({T}_{C}{T}_{0}\right)$ and $\theta =\left({T}_{D}{T}_{0}\right)$are the conductive and dynamical temperature increments, respectively. Then the Eqs. (2)(5), (10) and (11) take the forms:
The Eq. (14) can be rewritten to be in the form:
The following nondimensional variables are used for convenience [8, 12, 13]:
$\left\{\theta \text{'},\phi \text{'}\right\}=\frac{1}{{T}_{0}}\left\{\theta ,\phi \right\},{\sigma}^{\text{'}}=\frac{\sigma}{\lambda +2\mu}.$
Then, we obtain:
The prims have been deleted for simplicity.
3. Problem formulation in the Laplace transform domain
The Laplace transform will be applied which is defined as follows:
where the inversion of the Laplace transform may be calculated numerically by the following iteration:
where $Re$ denotes the real part, while $i$ defines the unit imaginary number. Numerous numerical tests have been conducted to determine if the value of $\kappa $ may meet the relation $\kappa t\approx 4.7\mathrm{}$[31, 32].
The Laplace transform of the fractional derivative is defined as [30]:
We assume the following initial conditions:
Then, we obtain:
Eliminating $\stackrel{}{\theta}$ from the Eqs. (33)(35), we obtain:
where:
${\alpha}_{4}=\frac{{s}^{2}}{\left(s+\stackrel{~}{K}+{\stackrel{~}{c}}^{2}\right)},{\alpha}_{5}=\frac{{\epsilon}_{1}{s}^{2}\left(1+{\tau}^{\alpha}{s}^{\alpha}\right)}{\left(s+\stackrel{~}{K}+{\stackrel{~}{c}}^{2}\right)}.$
Eliminating $\stackrel{}{e}$ from the Eqs. (38) and (39), we get:
Eliminating $\stackrel{}{\phi}$ from the Eqs. (39) and (40), we obtain:
where $L=\frac{\left({\alpha}_{1}+{\alpha}_{4}+{\alpha}_{2}{\alpha}_{5}\right)}{\left(1+{\alpha}_{3}{\alpha}_{5}\right)}$, $M=\frac{{\alpha}_{1}{\alpha}_{4}}{\left(1+{\alpha}_{3}{\alpha}_{5}\right)}$, and $\pm {k}_{1},\pm {k}_{2}$ are the roots of the following characteristic equation:
The general solutions of the Eqs. (41) and (42) must be bounded at infinity, thus, they take the following forms:
From the Eq. (39), we obtain:
Thus, we have:
To get the parameters ${A}_{1}$ and ${A}_{2}$, we must apply the boundary conditions, so we will consider that the surface of the cavity is subjected to a thermal loading with a function of time only as follows:
Moreover, we will consider that the surface of the cavity is connected to a rigid foundation which can stop any cubical deformation, i.e., we have:
After applying Laplace transform, we obtain:
By using the conditions Eqs. (50) and (51) into the Eqs. (44) and (47), we obtain the following system of linear equations:
By solving the above system, we get:
Hence, we have the solutions in the Laplace transform domain as follows:
Now, we must determine the thermal loading function $f\left(t\right)$, so we will consider that the thermal loading function is a ramptype heating which takes the form [7, 33, 34]:
where ${t}_{0}$ is called the ramptime heat parameter.
After applying the Laplace transform to the above equation, we obtain:
That completes the solutions in the Laplace transform domain.
To obtain the stress distribution, we can sum the Eq. (35)(37), then we obtain the average value of the principal stresses components as follows:
4. Numerical results
For the numerical results, silicon (Si) has been taken as the thermoelastic semiconducting material, for which we take the following values of the different physical constants [7, 3335]: $\mu =$5.46×10^{10} kg m^{1} s^{2}, $\lambda =$3.64×10^{10} kg m^{1} s^{2}, $\rho =$2330 kg m^{3}, ${C}_{v}=$695 m^{2} K^{1} s^{2}, ${\alpha}_{T}=$3.0×10^{6} K^{1}, $K=$ 150 kg m k^{1} s^{3}, ${\phi}_{0}=$1.0.
The numerical results of the dynamic temperature increment, conductive temperature increment, volumetric deformation, displacement, average stress, and stressstrain energy distributions have been figured with a wide range of the dimensionless radial distance $r\left(1\le r\le 4\right)$ and at the instant value of dimensionless time $t=$ 1.0.
5. Discussions
Figs. 2 and 3 show the studied function distributions of GreenNaghdi typeI and typeIII, respectively, with various values of the fractionalorder parameter $\alpha =$(0.0, 0.4, 0.6, 0,9) at a time $t=2.0$ and ${t}_{0}=2.0$ to sand on the effect of the fractionalorder parameter.
Fig. 2The studied function distributions of GreenNaghdi typeI with various values of the fractionalorder parameter
a) The conductive temperature increment
b) The dynamic temperature increment
c) The volumetric strain
d) The average of principal stresses
Fig. 2(a), 3(a), 2(b), and 3(b) represent the conductive and dynamical temperature increments, respectively. It is noted that the value of the fractionalorder strain parameter has a very limited effect on the thermal wave.
Figs. 2(c) and 3(c) represent the volumetric strain, and it is noted that the fractionalorder of strain parameter has a significant effect where the absolute value of the peak point of the volumetric strain increases when the value of the fractionalorder strain parameter increases. Moreover, the maximum absolute value of the volumetric strain occurs when the consideration of the fractionalorder strain does not exist.
Figs. 2(d) and 3(d) represent the average value of the principal stresses, and it is noted that the fractionalorder of strain parameter has a small effect.
Also, we can see that the value of the dynamical temperature increment based on typeI is smaller than its value based on typeIII. While the absolute value of the principal stresses based on typeI is greater than its value based on typeIII.
Figs. 4 and 5 show the studied functions distributions of GreenNaghdi typeI and typeII, respectively, based on onetemperature and hyperbolic twotemperature models ($c=$0.0 and $c\ne $0.0) and various values of the fractionalorder parameter $\alpha =$ (0.0, 0.5) at a time $t=$ 2.0 and ${t}_{0}=$ 2.0 to sand on the effect of the fractionalorder parameter on the two studied models.
Figs 4(a), 4(b), 5(a), and 5(d) show that the hyperbolic twotemperature parameter has significant effects on the conductive and dynamical temperature increment or the thermal wave in general for the two studied types I and III.
Fig. 3The studied function distributions of GreenNaghdi typeIII with various values of the fractionalorder parameter
a) The conductive temperature increment
b) The dynamic temperature increment.
c) The volumetric strain
d) The average of principal stresses
Fig. 4The studied functions distributions of GreenNaghdi typeI based on onetemperature and hyperbolic twotemperature and various values of the fractionalorder parameter
a) The conductive temperature increment
b) The dynamic temperature increment.
c) The volumetric strain
d) The average of principal stresses
Figs. 4(c) and 5(c) represent the volumetric strain distributions, and it is noted that the hyperbolic twotemperature parameter has a significant effect beside the fractionalorder strain parameter where the absolute value of the peak point of the volumetric strain for the two studied types take the following order:
The above equation indicates that considering strain with fractionalorder and hyperbolic twotemperature heat model leads to a decrease in the profile of the mechanical wave propagation.
Figs. 4(d) and 5(d) represent the average of principal stresses distributions, and it is noted that the twotemperature parameter has a significant effect besides the fractionalorder strain parameter.
Fig. 5The studied functions distributions of GreenNaghdi typeIII based on onetemperature and hyperbolic twotemperature and various values of the fractionalorder parameter
a) The conductive temperature increment
b) The dynamic temperature increment
c) The volumetric strain
d) The average of principal stresses
Figs. 6 and 7 show the studied functions distributions of GreenNaghdi typeI and typeII, respectively, based on onetemperature and hyperbolic twotemperature models ($c=$0.0 and $c\ne $0.0) and the fractionalorder parameter $\alpha =0.5$ when $t<{t}_{0}$ and $t>{t}_{0}$, respectively to stand on the effect of the ramptime heat parameter on all the studied functions.
Figs. 6(a), 6(b), 7(a), and 7(b) show that the ramptime heat parameter has significant effects on the conductive and dynamical temperature increments. Based on the case $t<{t}_{0}$, the profile of the thermal wave’s propagation is smaller than its propagation based on the case $t>{t}_{0}$ in the context of onetemperature and hyperbolic twotemperature models of GreenNaghdi typeI and typeIII.
Figs. 6(c), 6(d), 7(c), and 7(d) show that the ramptime heat parameter has significant effects on the volumetric strain and average of principal stresses. Based on the case $t<{t}_{0}$, the absolute value of the volumetric strain and average of principal stresses (the profile of the mechanical wave’s propagation) are smaller based on the case $t>{t}_{0}$ in the context of onetemperature and hyperbolic twotemperature models of GreenNaghdi typeI and typeIII.
Fig. 6The studied functions distributions of GreenNaghdi TypeI and typeIII based on one/twotemperature when α=0.5 and t<t0
a) The conductive temperature increment
b) The dynamic temperature increment.
c) The volumetric strain
d) The average of principal stresses
Fig. 7The studied functions distributions of GreenNaghdi TypeI and typeIII based on one/twotemperature when α=0.5 and t>t0
a) The conductive temperature increment
b) The dynamic temperature increment.
c) The volumetric strain
d) The average of principal stresses
6. Conclusions
1) The value of the fractionalorder strain parameter has limited effects on the conductive and dynamical temperature increments.
2) The value of the fractionalorder strain parameter has significant effects on strain and stress.
3) The hyperbolic twotemperature parameter has significant effects on the conductive and dynamical temperature increments, strain, and stress.
4) The ramptime heat parameter has significant effects on the conductive and dynamical temperature increments, strain, and stress.
5) The profile of the thermal wave’s propagation based on GreenNaghdi typeI is smaller based on GreenNaghdi typeIII.
6) The profile of the mechanical wave’s propagation based on GreenNaghdi typeI is greater based on GreenNaghdi typeIII.
7) Considering strain with a fractionalorder and hyperbolic twotemperature heat conduction model causes the profile of mechanical wave propagation to be reduced.
8) GreenNaghdi typeI and typeII based on hyperbolic twotemperature heat conduction models offer propagation of thermal and mechanical waves with finite speeds.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Youssef construct the model, consider the conditions, obtain the numerical solutions, review all the work, and submit for publishing. Alghamdi figure the numerical results, discuss the results, set the conclusion, and write all the work.
The authors declare that they have no conflict of interest.