Fractional order theory of thermo-viscoelasticity and application
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Fractional order theory of thermo-viscoelasticity and application Hany H. Sherief1 · Mohammed A. El-Hagary2
Received: 30 May 2018 / Accepted: 1 April 2019 © Springer Nature B.V. 2019
Abstract In this work, we derive a new fractional order theory for thermo-viscoelasticity. A uniqueness theorem for these equations is proved. A reciprocity theorem is also proved. A 1D problem for a viscoelastic half space is solved by using the Laplace transform technique. The solution in the transformed domain is obtained by a direct approach. The inverse transforms are obtained by using a numerical method. The temperature, displacement and stress distributions are computed and represented graphically. Keywords Fractional calculus · Half Space · Reciprocity theorem · Thermo-viscoelasticity · Uniqueness theorem
1 Introduction In recent years, viscoelastic materials became a very important study field. This is due to the massive use of polymers and composite materials in industry. The applications of these materials are various and numerous. For instance, they are used in the fabrication of medical diagnostic tools and also used in the NASA space programs. Also the investigation of seismic viscoelastic waves plays an important role for geophysical prospecting technology. The mechanical model of linear viscoelasticity was represented by Gross (1953). Many authors worked to develop this model and discussed the behavior of viscoelastic materials like Gurtin and Sternberg (1962), Stratonova (1971), Malyi (1976), and Li (1978). Pobedrya (1969), Il’yushin (1968), Kovalenko and Karnaukhov (1972) and Medri (1988) discussed the coupled theory of thermo-viscoelasticity and solved some problems in the context of this theory. The heat equation of this theory is a parabolic partial differential equation which predicts limited velocity of spread for heat waves contrary to physical observations. Due to this flaw, Sherief et al. (2011) introduced the generalized theory of thermoviscoelasticity with one relaxation time. This theory is an extension to the generalized theory
B M.A. El-Hagary
[email protected]
1
Department of Mathematics, University of Alexandria, Alexandria, Egypt
2
Department of Mathematics, Damiatta University, New Damiatta, Egypt
Mech Time-Depend Mater
of thermoelasticity which was introduced by Lord and Shulman (1967) and Dhaliwal and Sherief (1980). Elhagary (2013) has solved a thermo-mechanical shock problem for generalized theory of thermo-viscoelasticity. Sherief et al. (2015) solved a 2D problem for a half space in the generalized theory of thermo-viscoelasticity. Recently, fractional calculus has been used to describe an increasing numbers of physical processes, like, electromagnetism, astrophysics, quantum mechanics, and nuclear physics etc. Caputo and Mainardi (1971a, 1971b) and Caputo (1974) had got good experimental results when using fractional derivatives for description of viscoelastic materials and instituted the relationship between the theory of linear viscoelasticity and fractional order derivatives. Ad
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