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This investigation is devoted to exhibit the thermal damage of skin tissue exposed to a moving heat source. Modelling of the problem is performed by adopting dual phase lag theory of bio heat transfer in context of memory dependent derivative. Laplace transform technique has been adopted to represent the analytical solutions of temperature and thermal damage of skin tissue. Thermal damages to the tissues are calculated by the extent of the denatured protein employing with the Arrhenius equation. In order to predict the significance of memory dependent derivative approach, computational results of temperature and thermal damage are evaluated in the frame of different kernel functions as well as time-delay. For the purpose of exhibiting the attractiveness of the present model, obtained results are compared with the results corresponding to the absence of memory dependent derivative. Also, the impact of the velocity of moving heat source has been precisely investigated on temperature variation and thermal damage of skin tissue using quantitative results. Authors' believe that this study will be helpful to study the thermal treatment of several diseases such as hyperthermia. [ABSTRACT FROM AUTHOR]

The primary focus of the present contribution is to develop a new system of differential equations describing the nonlocal thermoelasticity theory. The problem deals with the thermoelastic interaction in a nonhomogeneous thermoelastic layer induced by absorbing penetrating laser radiation throughout its volume. The heat conduction equation has been formulated in the context of memory dependent three-phase lag model. The surfaces of the layer are thermally insulated and free of traction. Employing the integral Laplace transform technique, the solutions of the field quantities have been achieved in the transformed domain. The corresponding solutions in the space–time domain have been derived using the numerical inversion of the Laplace transform by Riemann-sum approximation technique. The numerical estimates of the field quantities have been depicted graphically to address the effects of nonlocality and memory effects on the temperature profile, displacement and stress components. The effect of the delay time, nonhomogeneity and various choices of linear and nonlinear kernel functions in the heat transport law have also been reported. [ABSTRACT FROM AUTHOR]

This paper is concerned with the influence of memory-dependent heat transport law on rotating thermoelastic medium with voids via three-phase-lag. The entire pervious medium is rotating with a invariant angular haste, where the bounding airplane is subordinated to a thermal shock and is free of tractions. By employing the normal mode analysis, the exact expressions for the displacement components, stresses, temperature distribution and change in volume fraction field have been depicted graphically in the presence and the absence of gyration and memory-dependent outgrowth. The effect of the voiding is also bandied in the literature review. [ABSTRACT FROM AUTHOR]

Enlightened by the memory dependent derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of the magnetic field and moving heat source in a rod in the context of three-phase lag theory of thermoelasticity based on Eringen's nonlocal elasticity. Both ends of the rod were fixed, and heat is insulated. Employing Laplace transform as a tool, the problem was transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain was obtained by applying the inverse Laplace transform. Numerical calculation for temperature, displacement and stress within the rod was carried out and displayed graphically. The effect of the moving heat source speed, time instance, memory dependent derivative, magnetic field and nonlocality on temperature, displacement, and stress was studied. Finally, three animated were added for easy understanding of the effect of moving the heat source on the studied fields. As per authors concern, the concept of the animated has been presented for the first time in this kinds of literature till now. [ABSTRACT FROM AUTHOR]

The main concern of this article is to deal with the thermoelastic interaction in a functionally graded thermoelastic rod being enlightened by the memory-dependent derivative. This article investigates the transient phenomena due to the influence of an induced magnetic field of constant intensity and due to the presence of a moving heat source of constant velocity in the context of three-phase lag model of generalized thermoelasticity. Employing the Laplace and the Fourier transforms as tool, the problem has been constructed in the transformed domain. The inversions of the Fourier transform have been carried out using residual calculus whereas the numerical inversions of the Laplace transform have been performed employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod are carried out and have been demonstrated graphically. The results also demonstrate how the nonhomogeneity parameter and the speed of the moving heat source influence the thermophysical quantities. It is observed that the temperature, thermally induced displacement, and stress of the rod are found to decrease at large source speed. Also, significant differences on the thermophysical quantities are revealed due to the influence of magnetic field, nonhomogeneity, and memory effect also. [ABSTRACT FROM AUTHOR]

The vibration of elastic micro-beam is of major significance due to wide applications of ultra-fast lasers, where size effect on heat conduction and elastic deformation increase and classical theory of thermoelastic coupling does not hold any more. Accordingly, the size-dependent thermoelastic model has been developed to deal with higher-order simple material to adopt both size effect of heat conduction and elasticity with the aids of extended irreversible thermodynamics and generalized free-energy. This survey demonstrates the effect of ultra-short-pulsed laser heating in the problem of coupled thermoelastic vibration of a microscale resonator, in which, the non-local heat transport equation is drafted in an integral form of a common derivative on a slipping interval by incorporating the three-phase-lag memory-dependent heat transfer in the context of two-temperature theory. Laplace transform and finite sinusoidal Fourier transform techniques have been employed to determine the lateral vibration of the beam and the scaled thermal moment within the beam. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Specific attention is paid to study the effect of thermal non-local parameter, energy absorption depth, memory-dependent derivative, the two-temperature parameter and delay time also. [ABSTRACT FROM AUTHOR]

The present article is aimed at studying the effect of an induced magnetic field in an initially stressed homogeneous, isotropic, thermoelastic half-space. The corresponding mathematical modeling has been formulated in the context the memory-dependent heat transport equation in the context of three-phase lag model of generalized thermoelasticity involving two-temperature theory. The bounding plane of the medium is subjected to different types of prescribed temperature and variable mechanical loading. Employing the Laplace transform, the analytical results for the distributions of the thermophysical quantities have been derived and the corresponding vector-matrix differential equation is solved with the help of state space approach. The numerical inversion of the Laplace transform is carried out with the help of an efficient and pragmatic algorithm based on the Riemann-sum approximation technique. Numerical computations for a copper material is performed and have been demonstrated graphically. The result provides a motivation to investigate the problem and draw conducting remarks due to the influence of the magnetic field, memory effect and time-delay parameter also. [ABSTRACT FROM AUTHOR]

The objective of the present study is to formulate the bio-heat model to study the variations of temperature profile and thermal damages within a spherical living tissue subjected to a thermal therapy, whose outer surface is thermally insulated. The heat conduction equation is formulated in the context of convolution-type Dual-phase (DP) lag memory-dependent derivative, having kernels to be power functions, from which, the Lord–Shulman model can be obtained as a particular case. The Laplace transform technique is implemented to solve the governing equations. The influences of the memory-dependent derivative and effect of time-delay parameters on the temperature of skin tissues and the thermal injuries are precisely demonstrated. The thermal injuries of the tissue are assessed by the denatured protein range. The numerical inversion of the Laplace transform is performed with the help of Zakian methods. The numerical outcomes of thermal injuries and temperatures have been represented graphically. Excellent predictive capability is demonstrated to identify an appropriate therapy to select different kernel functions for effective heating in hyperthermia treatment. Significant effect of the thermal therapy is reported due to the effect of delay time also. [ABSTRACT FROM AUTHOR]

The analysis of fracture is very important along with the miniaturization of the device and wide application of ultra-fast lasers, where size effect on heat conduction and elastic deformation increase and classical theory of thermoelastic coupling does not hold any more. Due to these, size-dependent thermoelastic model have been introduced for higher order simple material to adopt both the size effect of heat conduction and elasticity with the aids of extended irreversible thermodynamics and generalized free energy. With this motivation, the present study is now formulated to provide the nonlocal phenomena of a homogeneous, isotropic infinite space weakened by a finite linear mode I crack. The boundary of the crack is subjected to prescribed temperature and stress. The governing equations have been solved on employing the Laplace and the Fourier transforms, which reduces to four dual integral equations, the solution of which is equivalent to solving the Fredholm's integral equation of the first kind. The Bellman method has been used to compute numerical inversion of the Laplace transform. The result provides the direction how the nonlocality and size-dependence can control the fracture and what is the significant effect of various kernel functions and the effect of memory is also reported. [ABSTRACT FROM AUTHOR]

In the present analysis, the bio-heat equation has been studied in the context of memory responses which is defined in the form of convolution having kernels as power functions. The heat transport equation for this problem involving the memory-dependent derivative on a slipping interval in the context of Dual-phase (DP) lag model, which is used to study the thermal damage within the graded spherical skin tissue during the thermal therapy, whose outer surface is thermally insulated. Laplace transform technique is implemented to solve the governing equations. The influences of the memory-dependent derivative and time-dependent moving heat source velocity on the temperature of skin tissues and the thermal injuries are precisely investigated. The thermal injuries to the tissue are assessed by the denatured protein range using the current formulation in which the numerical inversion of the Laplace transform is carried out using Zakian method. The numerical outcomes of thermal injuries and temperatures have been represented graphically. Excellent predictive capability is demonstrated for identification of appropriate procedure to select different kernel functions to reach effective heating in hyperthermia treatment. The significant effect of the thermal therapy is reported due to the effect of nonhomogeneity and delay time also. [ABSTRACT FROM AUTHOR]

This present study deals with a novel mathematical model of generalized thermoelasticity in a fiber-reinforced, isotropic unbounded thermoelastic solid due to the presence of a continuous line heat source under the influence of magnetic field. The derivative is defined in an integral form of a common derivative on a slipping interval by incorporating memory-dependent heat transfer. Employing both Laplace and Hankel transforms as tools, analytical results for distributions of different fields have been derived with the help of potential function approach. The inversion of Hankel transform is performed analytically, whereas the numerical inversion of Laplace transform is carried out employing Zakian method and the method based on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory to exhibit the significance of the kernel functions and the time-delay parameter is constructed. The significant differences of the two different numerical schemes have also explained. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative and presence of reinforcement and magnetic field. Also, significant differences due to the choice of linear and nonlinear kernels in the heat equation is elaborated. [ABSTRACT FROM AUTHOR]

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of dual-phase lag (DPL) theory of thermoelasticity based on Eringen's nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for temperature, displacement and stress within the rod is carried out and displayed graphically. The effect of moving heat source speed, time instance, memory-dependent derivative, magnetic-field and nonlocality on temperature, displacement and stress are studied. [ABSTRACT FROM AUTHOR]

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of three-phase lag (TPL) theory of thermoelasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space-domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effect of moving heat source speed on temperature, stress, and temperature is studied. It is found from the distributions that the temperature, thermally induced displacement and stress of the rod are found to decrease at large source speed. For the better understanding of the effect of moving heat source on all the distributions, three animations are added. [ABSTRACT FROM AUTHOR]

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of Lord–Shulman (LS) theory of thermoelasticity based on Eringen's nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effects of moving heat source speed, time instance, memory-dependent derivative, magnetic field and nonlocality on temperature, stress, and temperature are studied. [ABSTRACT FROM AUTHOR]

Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hyperthermia cancer treatment. In the present analysis, the bioheat equation is studied in the context of memory responses. The heat transport equation for this problem involving the memory-dependent derivative (MDD) on a slipping interval in the context of three-phase (3P) lag model under two-temperature theory is formulated and is then used to study the thermal damage within the skin tissue during the thermal therapy. Laplace transform technique is implemented to solve the governing equations. The influences of the MDD and moving heat source velocity on the temperature of skin tissues are precisely investigated. The numerical inversion of the Laplace transform is carried out using Zakian method. The numerical outcomes of temperatures are represented graphically. Excellent predictive capability is demonstrated for identification of an appropriate procedure to select different kernel functions to reach effective heating in hyperthermia treatment. Significant effect of thermal therapy is reported due to the effect of delay time and the velocity of moving heat source as well. [ABSTRACT FROM AUTHOR]

This work deals with the study of transient responses in a two‐temperature generalized thermoelastic infinite medium having a cylindrical cavity due to a time‐dependent moving heat source where the conventional Fourier's law of heat conduction is modified by introducing a new Taylor's series expansion using memory‐dependent derivative (MDD). The resulting non‐dimensional equations are applied to a specific problem. A direct approach is introduced to obtain the analytical expressions of the physical quantities in the Laplace transform domain. The inversion of the Laplace transforms are carried out using the methods based on the Fourier series expansion technique. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are presented graphically. The effects of time‐delay and kernel function have been studied on the thermo‐physical quantities. [ABSTRACT FROM AUTHOR]