# Thermoelastic Coupling Response of an Unbounded Solid with a Cylindrical Cavity Due to a Moving Heat Source

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Equations

- (i)
- Coupled dynamical thermoelasticity (CTE theory) [2]: ${\tau}_{\theta}={\tau}_{q}=0$ and $\varrho =1$

- (ii)
- Lord and Shulman generalized thermoelasticity theory (L–S theory) [3]: ${\tau}_{\theta}=0$, ${\tau}_{q}={\tau}_{0}$ and $\varrho =1$

- (iii)

- (iv)

- (v)

## 3. Problem Construction

## 4. Closed-Form Solution

- The surface of the cylindrical cavity is exposed to a harmonically varying heat

- The mechanical boundary condition is considered as the surface of the cylindrical cavity is traction free

## 5. Validation of Results

#### 5.1. First Validation Example

- The G–N theory gives the smallest absolute field of all field quantities.
- The other theories CTE and L–S give suitable results for the field quantities.
- Three values $N=3$, 4, and 5 have been used for the RDPL theory while the simple dual-phase-lag (SDPL) theory is described with $N=1$.
- The most accurate results are given by using the RDPL theory.
- For the RDPL theory the temperature, displacement, and hoop stress are slightly increasing with the increase in many terms $N$, while the dilatation, radial stress, and axial stress are slightly decreasing. The increasing and decreasing amounts may be un-sensitive when $N\ge 5$.

#### 5.2. Second Validation Example

#### 5.3. Additional Applications

#### 5.3.1. Effect of Angular Frequency of Thermal Vibration

#### 5.3.2. Effect of Velocity of Heat Source

#### 5.3.3. Effect of Dimensionless Time

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\alpha}_{t}$ | thermal expansion coefficient $\left({\mathrm{K}}^{-1}\right)$ |

${C}_{e}$ | specific heat at uniform strain $\left(\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${\delta}_{ij}$ | Kronecker delta function |

${e}_{\phi \phi}$ | hoop strain |

${e}_{rr}$ | radial strain |

$e$ | dilatation |

${e}_{ij}$ | linear strain tensor |

$\vartheta $ | the velocity of heat source $\left({\mathrm{m}\mathrm{s}}^{-1}\right)$ |

$\gamma \equiv \left(3\lambda +2\mu \right){\alpha}_{t}$ | thermal modulus $\left({\mathrm{N}\mathrm{m}}^{-2}{\mathrm{K}}^{-1}\right)$ |

$H\left(t\right)$ | Heaviside unit step function |

$k$ | coefficient of heat conductivity $\left({\mathrm{W}\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${k}^{*}$ | rate of thermal conductivity of an isotropic material $\left({\mathrm{W}\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

$\lambda ,\mu $ | Lame’s constants $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

$\rho $ | material density $\left({\mathrm{kg}\mathrm{m}}^{-3}\right)$ |

$R$ | The radius of the cylindrical cavity $\left(\mathrm{m}\right)$ |

$\left(r,\phi ,z\right)$ | cylindrical coordinates system |

${\sigma}_{ij}$ | stress tensor components $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

${\sigma}_{\phi z},{\sigma}_{zr},{\sigma}_{r\phi}$ | shear stresses $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

${\sigma}_{\phi \phi}$ | hoop stress $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

${\sigma}_{rr}$ | radial stress $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

${\sigma}_{zz}$ | axial stress $\left({\mathrm{N}\mathrm{m}}^{-2}\right)$ |

$s$ | Laplace parameter |

$\theta =T-{T}_{0}$ | temperature change $\left(\mathrm{K}\right)$ |

${\theta}_{0}$ | thermal constant $\left(\mathrm{K}\right)$ |

${T}_{0}$ | environment temperature $\left(\mathrm{K}\right)$ |

${\tau}_{q}$ | phase-lag of heat flux $\left(\mathrm{s}\right)$ |

${\tau}_{\theta}$ | phase-lag of temperature gradient $\left(\mathrm{s}\right)$ |

${\tau}_{0}$ | first relaxation time $\left(\mathrm{s}\right)$ |

$\omega $ | angular frequency of thermal vibration $\left({\mathrm{rad}\mathrm{s}}^{-1}\right)$ |

${Q}_{0}$ | strength of heat source $\left({\mathrm{W}\mathrm{m}}^{-3}\right)$ |

$\delta $ | delta function |

$\stackrel{\rightharpoonup}{q}$ | heat flux vector $\left({\mathrm{W}\mathrm{m}}^{-2}\right)$ |

${u}_{r}$ | radial displacement $\left(\mathrm{m}\right)$ |

${u}_{\varphi}$ | hoop displacement $\left(\mathrm{m}\right)$ |

${u}_{z}$ | axial displacement $\left(\mathrm{m}\right)$ |

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**Figure 1.**The temperature $\overline{\theta}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 2.**The dilatation $\overline{e}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 3.**The radial displacement $\overline{u}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 4.**The radial stress ${\overline{\sigma}}_{1}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 5.**The hoop stress ${\overline{\sigma}}_{2}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 6.**The axial stress ${\overline{\sigma}}_{3}$ across the radial direction of the cylindrical cavity conferring to all theories.

**Figure 7.**Effect of $\omega $ on temperature $\overline{\theta}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 8.**Effect of $\omega $ on dilatation $\overline{e}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 9.**Effect of $\omega $ on radial displacement $\overline{u}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 10.**Effect of $\omega $ on radial stress ${\overline{\sigma}}_{1}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 11.**Effect of $\omega $ on radial stress ${\overline{\sigma}}_{2}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 12.**Effect of $\omega $ on radial stress ${\overline{\sigma}}_{3}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 13.**Effect of $\vartheta $ on temperature $\overline{\theta}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 14.**Effect of $\vartheta $ on dilatation $\overline{e}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 15.**Effect of $\vartheta $ on radial displacement $\overline{u}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 16.**Effect of $\vartheta $ on radial stress ${\overline{\sigma}}_{1}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 17.**Effect of $\vartheta $ on radial stress ${\overline{\sigma}}_{2}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 18.**Effect of $\vartheta $ on radial stress ${\overline{\sigma}}_{3}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 19.**Effect of $t$ on temperature $\overline{\theta}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 20.**Effect of $t$ on dilatation $\overline{e}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 21.**Effect of $t$ on radial displacement $\overline{u}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 22.**Effect of $t$ on radial stress ${\overline{\sigma}}_{1}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 23.**Effect of $t$ on radial stress ${\overline{\sigma}}_{2}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Figure 24.**This is a figure. Schemes follow the same formatting Effect of $t$ on radial stress ${\overline{\sigma}}_{3}$ across the radial direction of the cylindrical cavity utilizing the RDPL model.

**Table 1.**Effect of the velocity of heat source $\vartheta $ on the field variables of different thermoelasticity theories with $t=0.03$, $r=1.2$, $\omega =0$.

$\mathit{\vartheta}$ | Variable | CTE | G–N | L–S | SDPL | RDPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

17 | $\overline{\theta}$ | 3.5206798 | 0.0074277 | 4.5558651 | 3.4642684 | 3.6460648 | 3.9083885 | 4.2446350 |

${e}^{*}$ | −6.4643864 | −0.7652114 | −10.4814114 | −6.5686154 | −6.8240864 | −7.0147188 | −7.2539067 | |

${u}^{*}$ | −50.6571091 | 0.0540479 | 0.0234225 | −37.4998105 | −4.5146261 | 5.7141745 | 11.3103719 | |

${\overline{\sigma}}_{1}$ | 16.5744843 | −0.0362606 | −4.6655407 | 11.3944541 | −1.9149801 | −6.2492888 | −8.8146180 | |

${\overline{\sigma}}_{2}$ | −23.7087643 | 0.0104058 | −4.5968524 | −18.4174099 | −5.4752409 | −1.6683198 | 0.2210782 | |

${\overline{\sigma}}_{3}$ | −3.5517066 | −0.0111005 | −4.6061725 | −3.4957955 | −3.6788182 | −3.9420569 | −4.2794514 | |

20 | $\overline{\theta}$ | 4.1072506 | −0.0029614 | 4.4306514 | 4.0800238 | 4.3206529 | 4.6090319 | 4.9665501 |

${e}^{*}$ | −6.5105278 | 0.3051538 | −10.1923716 | −6.6189071 | −6.8807134 | −7.0727326 | −7.3110451 | |

${u}^{*}$ | −0.2306539 | 0.0382336 | 0.0231236 | 2.4286536 | 2.5890897 | 1.9194700 | 1.3751839 | |

${\overline{\sigma}}_{1}$ | −4.0778064 | −0.0093305 | −4.5374406 | −5.1097870 | −5.4167623 | −5.4405303 | −5.5837524 | |

${\overline{\sigma}}_{2}$ | −4.2302789 | 0.0196396 | −4.4703704 | −3.1454027 | −3.3234490 | −3.8791992 | −4.4544390 | |

${\overline{\sigma}}_{3}$ | −4.1384990 | 0.0044260 | −4.4795715 | −4.1117924 | −4.3536781 | −4.6429788 | −5.0016408 |

**Table 2.**Effect of the velocity of heat source $\vartheta $ on the field variables of different thermoelasticity theories with $t=0.03$, $r=1.2$, $\omega =20$.

$\mathit{\vartheta}$ | Variable | CTE | G–N | L–S | SDPL | RDPL | ||
---|---|---|---|---|---|---|---|---|

$\mathit{N}=1$ | $\mathit{N}=3$ | $\mathit{N}=4$ | $\mathit{N}=5$ | |||||

17 | $\overline{\theta}$ | 3.2835398 | 0.0074277 | 4.5528182 | 3.2447447 | 3.4508927 | 3.7188299 | 4.0555735 |

${e}^{*}$ | −6.2860288 | −0.7652114 | −10.4740672 | −6.3932523 | −6.6542973 | −6.8464693 | −7.0855337 | |

${u}^{*}$ | −50.6572509 | 0.0540479 | 0.0234210 | −37.4999432 | −4.5147434 | 5.7140628 | 11.3102642 | |

${\overline{\sigma}}_{1}$ | 16.8133885 | −0.0362606 | −4.6624229 | 11.6157097 | −1.7181357 | −6.0580749 | −8.6239015 | |

${\overline{\sigma}}_{2}$ | −23.4708247 | 0.0104058 | −4.5937709 | −18.1970973 | −5.2793005 | −1.4779982 | 0.4109050 | |

${\overline{\sigma}}_{3}$ | −3.3137105 | −0.0111005 | −4.6030904 | −3.2754301 | −3.4828312 | −3.7516908 | −4.0895818 | |

20 | $\overline{\theta}$ | 3.8701106 | −0.0029614 | 4.4276045 | 3.8605001 | 4.1254808 | 4.4194734 | 4.7774886 |

${e}^{*}$ | −6.3321702 | 0.3051538 | −10.1850275 | −6.4435440 | −6.7109243 | −6.9044831 | −7.1426721 | |

${u}^{*}$ | −0.2307957 | 0.0382336 | 0.0231221 | 2.4285209 | 2.5889724 | 1.9193583 | 1.3750762 | |

${\overline{\sigma}}_{1}$ | −3.8389023 | −0.0093305 | −4.5343228 | −4.8885315 | −5.2199179 | −5.2493164 | −5.3930359 | |

${\overline{\sigma}}_{2}$ | −3.9923393 | 0.0196396 | −4.4672888 | −2.9250901 | −3.1275086 | −3.6888775 | −4.2646122 | |

${\overline{\sigma}}_{3}$ | −3.9005030 | 0.0044260 | −4.4764894 | −3.8914270 | −4.1576911 | −4.4526127 | −4.8117711 |

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**MDPI and ACS Style**

Zenkour, A.M.; Mashat, D.S.; Allehaibi, A.M.
Thermoelastic Coupling Response of an Unbounded Solid with a Cylindrical Cavity Due to a Moving Heat Source. *Mathematics* **2022**, *10*, 9.
https://doi.org/10.3390/math10010009

**AMA Style**

Zenkour AM, Mashat DS, Allehaibi AM.
Thermoelastic Coupling Response of an Unbounded Solid with a Cylindrical Cavity Due to a Moving Heat Source. *Mathematics*. 2022; 10(1):9.
https://doi.org/10.3390/math10010009

**Chicago/Turabian Style**

Zenkour, Ashraf M., Daoud S. Mashat, and Ashraf M. Allehaibi.
2022. "Thermoelastic Coupling Response of an Unbounded Solid with a Cylindrical Cavity Due to a Moving Heat Source" *Mathematics* 10, no. 1: 9.
https://doi.org/10.3390/math10010009