Generalized thermo-viscoelasticity with memory-dependent derivative: uniqueness and reciprocity
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O R I G I NA L
Indranil Sarkar · Basudeb Mukhopadhyay
Generalized thermo-viscoelasticity with memory-dependent derivative: uniqueness and reciprocity
Received: 8 March 2020 / Accepted: 19 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This article theoretically demonstrates the reciprocity and uniqueness theorems for generalized theory of thermo-viscoelasticity involving memory-dependent derivative (MDD). To prove the theorems, a thermo-viscoelastic initial-boundary value problem under the domain of three-phase-lag (TPL) model is taken into consideration for an isotropic, homogeneous medium. The theorems are proved with the help of the Laplace transform of the thermophysical quantities. Finally, a few special cases in the generalized theory of thermo-elasticity and thermo-viscoelasticity with MDD and without MDD are derived from the present model. Keywords Thermo-viscoelasticity · Uniqueness · Reciprocity · Three-phase-lag · Mixed initial-boundary value problem · Memory-dependent derivative
1 Introduction The study of motion and forces in solids, liquids and gases and the deformation or flow of these materials is explained through numerous models which constitute the subject of continuum mechanics. Elasticity is the elementary model and is relevant whenever instantaneous response of the material in a purely mechanical framework is a good enough approximation. Very frequently, materials, when subjected to externally applied loads, exhibit both the features of elastic solids and viscous fluids through simultaneous dissipation and storage of mechanical energy. This type of phenomena is nicely elaborated in the light of viscoelasticity in which, whether or not thermal effects are taken into consideration, the mechanical response characteristic is taken to be influenced by the preceding behavior of the material itself. Linear viscoelasticity, in which stress and strain are related linearly at any given time, is a reasonable approximation to the time-dependent behavior of polymers, metals and ceramics at relatively low temperature and at relatively low stress [1]. Due to the overnight development of plastic industry and polymer science and also the broad utilization of composite materials under high temperature in modern technology, the literature has committed increasing attention to thermo-viscoelastic materials due to its numerous applications from industry to technology such as material science, engineering science, geology, solid-state physics, polymers and composites, concrete technology and plastic processing. [2–4]. Gross [5] formulated the mechanical model of linear viscoelasticity. Contributions in the mathematical theory of viscoelasticity include the works of several researchers such as Gurtin and Sternberg [6], Stratonova [7] I. Sarkar (B) · B. Mukhopadhyay Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Shibpur, Howrah, West Bengal 711103, India E-mail: [email protected] B. Mukhopadhyay E-mail: [email protected]
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