# Thermodynamic Relations among Isotropic Material Properties in Conditions of Plane Shear Stress

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## Abstract

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## 1. Introduction

## 2. The Gibbs Equation and Generalized Free Energy

## 3. Relations among the Material Properties

#### 3.1. Entropy Representation

#### 3.1.1. $ds(T,p,\tau )=ds(T,p,\gamma )$

#### 3.1.2. $ds(T,p,\gamma )=ds(T,\u03f5,\tau )$

#### 3.1.3. $ds(T,\u03f5,\tau )=ds(T,\u03f5,\gamma )$

#### 3.1.4. $ds(T,\u03f5,\gamma )=ds(T,p,\tau )$

#### 3.2. Volume Representation

#### 3.2.1. $d\u03f5(T,p,\tau )=d\u03f5(s,p,\tau )$

#### 3.2.2. $d\u03f5(s,p,\tau )=d\u03f5(T,p,\gamma )$

#### 3.2.3. $d\u03f5(T,p,\gamma )=d\u03f5(s,p,\gamma )$

#### 3.2.4. $d\u03f5(T,p,\tau )=d\u03f5(T,p,\gamma )$

#### 3.3. Shear-Angle Representation

#### 3.3.1. $d\gamma (T,p,\tau )=d\gamma (s,p,\tau )$

#### 3.3.2. $d\gamma (s,p,\tau )=d\gamma (T,\u03f5,\tau )$

#### 3.3.3. $d\gamma (T,\u03f5,\tau )=d\gamma (s,\u03f5,\tau )$

#### 3.3.4. $d\gamma (T,p,\tau )=d\gamma (s,\u03f5,\tau )$

## 4. Application

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Results of experiments on a Boom clay conducted at constant volume (i.e., constant $\u03f5$). Data available from [33]. (

**a**) Temperature path with respect to the shear angle during the heating experiment at constant $\u03f5$ and $\tau $. Interpolating function: $370-150.17{e}^{-0.38\gamma}$. (

**b**) Shear stress-angle relationship at constant $\u03f5$ and T (294 K). (

**c**) Thermal shear deformation at constant $\u03f5$ as a function of $\gamma $, computed as $d\gamma /dT$. (

**d**) Isothermal shear compliance at constant $\u03f5$ computed as ${S}_{T,\u03f5}=d\gamma /d\tau $.

**Figure 2.**Difference between iso-$\tau $ and iso-$\gamma $ heat capacity at constant volume, ${c}_{v,\tau}-{c}_{v,\gamma}$, computed from Equation (76).

**Table 1.**Material properties as derived from the Gibbs free energy. The first row contains the extensive variable to differentiate, while the first column contains the operators.

s | $\mathit{\u03f5}$ | $\mathit{\gamma}$ | |
---|---|---|---|

$\frac{\partial}{\partial T}$ | $\frac{{c}_{p,\tau}}{T}$ | ${\alpha}_{\tau}$ | ${\beta}_{p}$ |

$\frac{\partial}{\partial p}$ | $-{\alpha}_{\tau}$ | $-{k}_{T,\tau}$ | $-{\eta}_{T}$ |

$\frac{\partial}{\partial \tau}$ | ${\beta}_{p}$ | ${\eta}_{T}$ | ${S}_{T,p}$ |

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**MDPI and ACS Style**

Porporato, A.; Calabrese, S.; Hueckel, T. Thermodynamic Relations among Isotropic Material Properties in Conditions of Plane Shear Stress. *Entropy* **2019**, *21*, 295.
https://doi.org/10.3390/e21030295

**AMA Style**

Porporato A, Calabrese S, Hueckel T. Thermodynamic Relations among Isotropic Material Properties in Conditions of Plane Shear Stress. *Entropy*. 2019; 21(3):295.
https://doi.org/10.3390/e21030295

**Chicago/Turabian Style**

Porporato, Amilcare, Salvatore Calabrese, and Tomasz Hueckel. 2019. "Thermodynamic Relations among Isotropic Material Properties in Conditions of Plane Shear Stress" *Entropy* 21, no. 3: 295.
https://doi.org/10.3390/e21030295