## MEMORY DEPENDENT RESPONSE IN AN INFINITELY LONG THERMOELASTIC SOLID CIRCULAR CYLINDER

**Authors:**Lamba N.K.^{1}, Deshmukh K.C.^{2}**Affiliations:**- Department of Mathematics, Shri Lemdeo Patil Mahavidyalaya
- Department of Mathematics, R.T.M. Nagpur University

**Issue:**No 1 (2024)**Pages:**5-12**Section:**ARTICLES**URL:**https://ered.pstu.ru/index.php/mechanics/article/view/4162**DOI:**https://doi.org/10.15593/perm.mech/2024.1.01- Cite item

# Abstract

Memory-dependent derivatives (MDD) have physical meaning, and compared to fractional derivatives, they are more suitable and convenient for temporal remodeling. In this study, the temperature and stress distributions in an infinitely extended generalized thermally elastic solid circular cylinder have been investigated by utilizing the concept of a memory-dependent heat conduction model. The homogeneous, isotropic, infinitely long solid circular cylinder is considered to have a lateral surface that is free of traction and is subjected to a known surrounding temperature. In the domain of the integral Laplace transform, the problem is worked out, and its complex inversion is performed numerically using the Fourier series expansion method. The material properties of copper metal are chosen for the purpose of numerical computation, and the thermoelastic impact of time delay on temperature distribution, displacement distribution, and thermal stresses are represented graphically. Also, time delay's effect on temperature history, displacement history, and thermal heat transfer stress history are shown, respectively. This study could also benefit mathematicians and researchers involved in the development of thermoelasticity, as it accounts for the memory-related derivatives that are useful in explaining the behaviour of a variety of physical processes. The thermal fluctuations captured by various factors with memory-dependent responses are used in engineering applications to realistically design machines or structures.

# Full Text

In this current decade, many researchers have started a new contribution on memory-based derivatives. In 2011, Wang and Li [1] gave the concept of MDD (memory-dependent derivatives) by surpassing the fractional derivative. Currently, MDD has developed as an emerging field of fractional calculus that is continually expanding due to its ability to reflect memory-dependent responses in various physical processes. Despite being specified on an interval, the fractional derivative (FD) mainly captures local change. The physical meaning of MDD is significantly more noticeable than that of FD. The MDD-dependent weight is reflected by the kernel function, while the time delay demonstrates the duration of the memory effect. Available research shows that the memory-related derivative is more appropriate for temporal modelling which is useful in describing the thermal effect of solid bodies. Memory-dependent derivatives are capable of replacing fractional derivatives and having useful applications in the field of thermoelasticity, thermoelectricity, particle physics and vibration mechanics etc. For more details on the fractional-related theory of thermal response, refer to the contents developed by many renowned researchers, as shown in references [2–13]. El-Karamany and Ezzat [14] constructed generalized thermoelastic theory with time delay and investigated MDD problem for a half space Cartesian body. Sun and Wang [15] reconstructed the memory dependent heat model based on the assumption that temperature increases slowly after heat transfer. Xue et al. [16] investigated thermal stresses in hollow cylinder containing internal and surface crack by using integral transform technique by considering heat conduction model with memory dependent derivatives. Ma and Gao [17] studied dynamic response of infinite hollow cylinder under thermal shock based on generalized nonlocal thermoelastic theory and MDD theory. Recently, Verma et al. [18] determined the impact of memory in hygrothermoelastic hollow cylinder by theory of uncoupled-coupled heat and moisture. Marin [19, 20] contributed the thermoelastodynamics study on bodies with void. Abouelregal et al. [21 to 26] elaborated the study of thermoelasticity based on higher-order memorydependent derivative with time delay and discussed vibrational behaviour in nanobeam. Atta [27] analyzed fractional operator response on thermal diffusion problem of infinite medium. Yu et al. [28] developed a novel thermoelastic model using memory-related derivatives and described its applications. Hussein [29] determined the solution by considering the circular cylinder problem under fractional theory. Chepurnenko and Turina [30] determined the stressstrain relation for multilayer beams using the finite element method. Ogorodnikov et al. [31] constructed the mathematical model for nonlinear viscoelasticity and discussed the impact of fractional integro-differentiation. Also, various other authors contributed to the field and evaluated the impact of fractional responses on different bodies, as highlighted in [32–38]. According to the aforementioned literature review, the Memory dependent derivative surpasses the fractional derivative on the basis of significant improvements in memoryrelated response. Thus, the authors have the opportunity to investigate the impact of MDD in an infinitely long, solid circular cylinder whose lateral face is free from traction and exposed to a specified ambient temperature. To obtain the solution of temperature and stress distribution, Laplace transform is utilized and its inversion is carried out numerically by Fourier series expansion method as given by Honig and Hirdes [39]. Obtained thermoelastic responses are represented graphically and the effects of MDD are discussed.# About the authors

### N. K. Lamba

Department of Mathematics, Shri Lemdeo Patil Mahavidyalaya

### K. C. Deshmukh

Department of Mathematics, R.T.M. Nagpur University

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