# A Study of Deformations in a Thermoelastic Dipolar Body with Voids

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (a)
- $\rho \left(x\right)\ge {a}_{1},\phantom{\rule{0.277778em}{0ex}}{J}_{m}\left(x\right)\ge {a}_{2},\phantom{\rule{0.277778em}{0ex}}c\left(x\right)\ge {a}_{3}$, where the real constants ${a}_{1},{a}_{2},{a}_{3}$ are positive;
- (b)
- ${K}_{ij}$ is a positive definite tensor;
- (c)
- the quadratic form $\Psi $ is positive definite.

- -
- -
- the equation of energy:$$\begin{array}{c}\hfill {K}_{ij}{\left(\right)}_{{\theta}_{,j}},i={T}_{0}\left(\right)open="["\; close="]">{a}_{ij}{\dot{\mathtt{u}}}_{i,j}+{b}_{ij}\left(\right)open="("\; close=")">{\dot{\mathtt{u}}}_{j,i}-{\dot{\varphi}}_{ij}& +{c}_{ijk}{\dot{\varphi}}_{ij,k}+c\dot{\theta}\\ ,\phantom{\rule{0.277778em}{0ex}}\mathrm{in}\phantom{\rule{0.277778em}{0ex}}[0,\infty )\times D;\end{array}$$
- -
- the kinematic equations (2), which take place in cylinder $[0,\infty )\times D$;
- -
- the constitutive conditions (4), which take place in cylinder $[0,\infty )\times D$;
- -
- the initial restrictions (11), which take place in $\overline{D}$;
- -
- the conditions to the limit:$$\begin{array}{c}\hfill {\mathtt{u}}_{i}(x,t)=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{u},\phantom{\rule{0.277778em}{0ex}}{t}_{i}=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{u}^{c},\phantom{\rule{17.07182pt}{0ex}}\\ \hfill {\varphi}_{ij}(x,t)=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times ,\partial {D}_{\varphi}\phantom{\rule{0.277778em}{0ex}}{m}_{jk}=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{\varphi}^{c},\\ \hfill \theta (x,t)=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{\theta},\phantom{\rule{0.277778em}{0ex}}q=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{\theta}^{c},\phantom{\rule{22.76228pt}{0ex}}\\ \hfill \phi (x,t)=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{\phi},\phantom{\rule{0.277778em}{0ex}}h=0,\phantom{\rule{0.277778em}{0ex}}(x,t)\in [0,\infty )\times \partial {D}_{\phi}^{c}.\phantom{\rule{19.91684pt}{0ex}}\end{array}$$

## 3. Auxiliary Results

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 4. On Localization in Time of Solutions

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Marin, M.; Abbas, I.; Vlase, S.; Craciun, E.M.
A Study of Deformations in a Thermoelastic Dipolar Body with Voids. *Symmetry* **2020**, *12*, 267.
https://doi.org/10.3390/sym12020267

**AMA Style**

Marin M, Abbas I, Vlase S, Craciun EM.
A Study of Deformations in a Thermoelastic Dipolar Body with Voids. *Symmetry*. 2020; 12(2):267.
https://doi.org/10.3390/sym12020267

**Chicago/Turabian Style**

Marin, Marin, Ibrahim Abbas, Sorin Vlase, and Eduard M. Craciun.
2020. "A Study of Deformations in a Thermoelastic Dipolar Body with Voids" *Symmetry* 12, no. 2: 267.
https://doi.org/10.3390/sym12020267