Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures
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ORIGINAL PAPER
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Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures Iqbal Kaur1* , Parveen Lata2 and Kulvinder Singh3
Abstract The aim of the present investigation is to examine the memory-dependent derivatives (MDD) in 2D transversely isotropic homogeneous magneto thermoelastic medium with two temperatures. The problem is solved using Laplace transforms and Fourier transform technique. In order to estimate the nature of the displacements, stresses and temperature distributions in the physical domain, an efficient approximate numerical inverse Fourier and Laplace transform technique is adopted. The distribution of displacements, temperature and stresses in the homogeneous medium in the context of generalized thermoelasticity using LS (Lord-Shulman) theory is discussed and obtained in analytical form. The effect of memory-dependent derivatives is represented graphically. Keywords: Thermoelastic, Transversely isotropic, Magneto-thermoelastic, Memory-dependent derivative, Time delay, Kernel function, Lord-Shulman model
Introduction Magneto-thermoelasticity deals with the relations of the magnetic field, strain and temperature. It has wide applications such as geophysics, examining the effects of the earth's magnetic field on seismic waves, emission of electromagnetic radiations from nuclear devices and damping of acoustic waves in a magnetic field. In recent years, inspired by the successful applications of fractional calculus in diverse areas of engineering and physics, generalized thermoelasticity (GTE) models have been further comprehensive into temporal fractional ones to express memory dependence in heat conductive sense. The MDD is defined in an integral form of a common derivative with a kernel function. The kernels in physical laws are important in many models that describe physical phenomena including the memory effect. Wang and Li (2011) introduced the concept of a MDD. Yu et al. (2014) introduced the MDD as an alternative of * Correspondence: [email protected] 1 Department of Mathematics, Government College for Girls, Palwal, Kurukshetra, Haryana, India Full list of author information is available at the end of the article
fractional calculus into the rate of the heat flux in the Lord-Shulman (LS) theory of generalized thermoelasticity to represent memory dependence and recognized as a memory-dependent LS model. This innovative model might be useful to the fractional models owing to the following arguments. First, the new model is unique in its form, while the fractional-order models have different modifications (Riemann-Liouville, Caputo and other models). Second, the physical meaning of the new model is clearer due to the essence of the MDD definition. Third, the new model is depicted by integer-order differentials and integrals, which is more convenient in numerical calculation as compared to the fractional models. Finally, the kernel function and time delay of the MDD can be arbitrarily chosen; thus, the model is more flexib
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