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Variational principle and continuous dependence results on the generalized poro-thermoelasticity theory with one relaxation parameter

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Abstract

The present work is devoted to a study of the generalized poro-thermoelasticity theory and aims to derive the variational principle and continuous dependence results in the context of this theory. The basic field equations are considered for an isotropic and homogeneous fluid-saturated poro-thermoelastic medium. With the concept of incorporating initial conditions into the field equations, an alternative characterization of the present mixed initial-boundary value problem is presented. By taking into account of this alternative characterization, a convolution type variational principle is derived in the context of this generalized poro-thermoelasticity theory. Further, a continuous dependence result of solution on initial data and external supply terms (heat source and body force) is established. Uniqueness of solution of the present problem is also shown to be followed from this result.

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Abbreviations

\(\beta \) :

Porosity of the material

\(\sigma _{ij}\) :

Stress components for the solid phase

\(\sigma \) :

Stress for the fluid phase

\(b_{i}^{\mathrm{s}}, b_{i}^{\mathrm{f}}\) :

Body force of the solid and the fluid phases

\(u_{i}\) :

\(i{\mathrm{th}}\) displacement component of the solid phase

\(U_{i}\) :

\(i{\mathrm{th}}\) displacement component of the fluid phase

\(\rho ^{s*}, \rho ^{f*}\) :

Density of the solid and fluid phases

\(\rho ^{\mathrm{s}}\) :

\(\rho ^{\mathrm{s}}=\left( 1-\beta \right) \rho ^{s*},\) density of the solid phase per unit volume of bulk

\(\rho ^{\mathrm{f}}\) :

\(\rho ^{\mathrm{f}}=\beta \rho ^{f*}\), density of the fluid phase per unit volume of bulk

\(\rho \) :

\(\rho =\rho ^{\mathrm{s}}+\rho ^{\mathrm{f}}\), density of the aggregate

\(\rho _{11}\) :

\(\rho _{11}=\rho ^{\mathrm{s}}-\rho _{12}\), mass coefficient of the solid phase

\(\rho _{22}\) :

\(\rho _{22}=\rho ^{\mathrm{f}}-\rho _{12}\), mass coefficient of the fluid phase

\(\rho _{12}\) :

Coefficient of dynamics coupling

\(h^{\mathrm{s}}, h^{\mathrm{f}}\) :

Heat source in the solid and the fluid phases

\(q_{i}^{\mathrm{s}}, q_{i}^{\mathrm{f}}\) :

Heat flux for the solid and the fluid phases

\(\epsilon _{ij}\) :

Strain component of the solid phase

\(\epsilon \) :

Strain component of the fluid phase

\(C_{E}^{\mathrm{s}}, C_{E}^{\mathrm{f}}\) :

Specific heat at constant strain of the solid and the fluid phases

\(C_{E}^{\mathrm{sf}}\) :

Couplings of specific heat between the phases

\(\eta ^{\mathrm{s}}\) :

Entropy for the solid phase per unit mass of aggregate

\(\eta ^{\mathrm{f}}\) :

Entropy for the fluid phase per unit mass of aggregate

\(k^{\mathrm{s}}, k^{\mathrm{f}}\) :

Thermal conductivity of the solid and the the fluid phases

\(\alpha ^{\mathrm{s}}\) :

Coefficient of thermal expansion of the solid phase

\(\alpha ^{\mathrm{f}}\) :

Coefficient of thermal expansion of the fluid phase

\(\alpha ^{fs}, \alpha ^{\mathrm{sf}}\) :

Thermoelastic couplings between the solid and fluid phases

\(\tau _{0}^{\mathrm{s}}\) :

Relaxation time of the solid phase

\(\tau _{0}^{\mathrm{f}}\) :

Relaxation time of the fluid phase

\(\theta ^{\mathrm{s}}\) :

\(\theta ^{\mathrm{s}}=T^{\mathrm{s}}-T_{0}\), temperature increment of the solid phase

\(\theta ^{\mathrm{f}}\) :

\(\theta ^{\mathrm{f}}=T^{\mathrm{f}}-T_{0}\), temperature increment of the fluid phase

\(T_{0}\) :

Reference temperature

\(\lambda ,\mu ,R, Q\) :

Poroelastic coefficients

P:

\(P=3\lambda +2\mu \)

\(F_{11}\) :

\(F_{11}=\rho C_{E}^{\mathrm{s}}\)

\(F_{22}\) :

\(F_{22}=\rho C_{E}^{\mathrm{f}}\)

\(R_{11}\) :

\(R_{11}=\alpha ^{\mathrm{s}}P+\alpha ^{fs}Q\)

\(R_{22}\) :

\(R_{22}=\alpha ^{\mathrm{f}}R+3\alpha ^{\mathrm{sf}}Q\)

\(R_{12}\) :

\(R_{12}=\alpha ^{\mathrm{f}}Q+\alpha ^{\mathrm{sf}}P\)

\(R_{21}\) :

\(R_{21}=3\alpha ^{\mathrm{s}}Q+\alpha ^{fs}R\)

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Acknowledgements

The authors thankfully acknowledge the constructive suggestions by reviewers and editor to improve the quality of the present paper. One of the authors (Komal Jangid) thankfully acknowledges the full financial assistance from the Council of Scientific and Industrial Research (CSIR), India, as the JRF fellowship (File. No. 09/1217(0057)/2019-EMR-I) to carry out this research work.

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  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Komal Jangid & Santwana Mukhopadhyay

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  1. Komal Jangid
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  2. Santwana Mukhopadhyay
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Correspondence to Komal Jangid.

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Jangid, K., Mukhopadhyay, S. Variational principle and continuous dependence results on the generalized poro-thermoelasticity theory with one relaxation parameter. Continuum Mech. Thermodyn. 34, 867–881 (2022). https://doi.org/10.1007/s00161-022-01101-y

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