Fractional Ultrafast Laser–Induced Thermo-Elastic Behavior In Metal Films

In this work, a new mathematical model of modification heat conduction for an isotropic generalized thermoelasticity is derived using the methodology of fractional calculus. Some theorems of generalized thermoelasticity follow as limit cases. An ultrafast fractional thermoelasticity model utilizing the modified hyperbolic heat conduction model and the generalized fractional thermoelastic theory was formulated to describe the thermoelastic behavior of a thin metal film irradiated by a femtosecond laser pulse. The temporal profile of the ultrafast laser was regarded as being non-Gaussian. An analytical–numerical technique based on the Laplace transform was used to solve the governing equations and the time histories of the temperature, displacement and stress in a gold film were analyzed. Some comparisons have been shown in figures to estimate the effects of the relaxation time and fractional order parameter α on all the studied fields.     

Waves in nonlocal thermoelastic solids of type II

The present article deals with the investigation of the propagation of thermoelastic plane harmonic waves in a nonlocal thermoelastic medium. The Green and Naghdi theory II (without energy dissipation) of generalized thermoelasticity with elastic nonlocal effect is adopted to address this problem. The problem of reflection of thermoelastic waves due to an incident coupled longitudinal elastic wave from the rigid and thermally insulated boundary of a homogeneous, isotropic nonlocal thermoelastic half-space is also studied. The amplitude ratios of the reflected waves and their respective energy ratios are determined analytically. For a particular model, the effect of elastic nonlocality parameter on the variations of phase speeds, attenuation coefficients, amplitude ratios and corresponding energy ratios of the reflected waves is presented graphically and analysis of these results is given.     

Investigation on the dynamic responses of a generalized thermoelastic problem with variable properties and nonlocal effect

To explore the dynamic responses of generalized thermoelastic problems with nonlocal effect in microtemporal scale, a finite length thermoelastic rod subjected to a moving heat source is modeled and investigated in generalized thermoelasticity. The rod is fixed at both ends and its material properties are temperature-dependent. The corresponding governing equations are first given and then reduced into one-dimensional ones with temperature-dependent properties assumed to be functions of reference temperature. Subsequently, the equations after normalization are solved together with the initial conditions and the boundary conditions by means of Laplace transform and its numerical inversion. The distributions of the nondimensional temperature, displacement, and stress are obtained and illustrated graphically. In calculation, the effects of the velocity of the heat source, the temperature-dependent properties and the nonlocal parameter on the distributions of the considered variables are emphatically examined and discussed in detail. The results show that the velocity of the heat source and the variable properties markedly influence the variations of the considered variables, while the nonlocal parameter barely influences the variations of the nondimensional temperature, slightly influences the variations of the peak value of the nondimensional displacement and significantly influences the variations of the peak value of the nondimensional stress.     

Two-dimensional electromagneto-thermoelastic coupled problem under fractional order theory of thermoelasticity

The dynamic response of a two-dimensional generalized thermal shock problem is investigated in the context of the fractional order theory of thermoelasticity proposed by Sherief et al. To demonstrate the solution process, a thermoelastic half-space subjected to a thermal shock on its bounding surface is considered in detail. The governing equations for the problem are formulated and then solved by normal mode analysis. The distributions of the considered nondimensional temperature, displacement, and stress are obtained and illustrated graphically. The effect of fractional order parameter on the variations of the considered variables is investigated, and the results show that the fractional order parameter has significant influence on the variations of the considered variables.     

Fundamental solution of steady oscillations equations in nonlocal thermoelastic medium with voids

Abstract

A new nonlocal theory of generalized thermoelasticity with voids based on Eringen’s nonlocal elasticity is established. The propagation of plane harmonic waves in nonlocal thermoelastic medium with voids is investigated in the context of dual-phase-lag model of generalized thermoelasticity. There exist three longitudinal waves, namely elastic (E-mode), thermal (T-mode) and volume fraction (V-mode) in addition to transverse waves which get decoupled from the rest of motion and not affected by thermal and volume fraction fields. The fundamental solution of the system of differential equations in case of steady oscillations in terms of the elementary functions has been constructed. The effect of nonlocal parameter and the effect of voids on phase velocities, attenuation coefficients and penetration depths are presented graphically.     

Transient responses of nanosandwich structure based on size-dependent generalized thermoelastic diffusion theory

Nowadays, the nanosandwich structure stands as one of the most promising candidates for nanodevices and nanocomposites (e.g., nanogenerator, nano-energy harvester) due to its superior mechanical properties. The size-dependent response analysis of nanosandwich structure under extreme in-service environment is of great importance for its fabrication and exploitation. This motivates us to study the transient responses of nanosandwich structure subjected to the symmetrical transient thermal and chemical potential shock loads, which are both imposed on its upper and lower bounding surfaces. It is further assumed that thermal contact resistance and diffusional contact impedance at the interface of each two adjacent layers are zero with ideal adhesion. For each isotropic homogeneous elastic layer, the governing equations are formulated in the context of size-dependent generalized thermoelastic diffusion theory and then are solved by using Laplace transformation techniques. The effects of elastic nonlocal parameter, fractional order parameters as well as material characteristic parameters on structural transient responses are also analyzed and discussed. The present study not only provides a comprehensive understanding of the size-dependent thermoelastic diffusion coupling of nanosandwich structure, but also offers basic guidelines for optimal design.     

Waves in dual-phase-lag thermoelastic materials with voids based on Eringen’s nonlocal elasticity

The main idea of the present work is to extend Eringen’s theory of nonlocal elasticity to generalized thermoelasticity with dual-phase-lag and voids. Then we study the propagation of time harmonic plane waves in an infinite nonlocal dual-phase-lag thermoelastic medium with voids. Three sets of coupled dilatational waves and an independent transverse wave may travel with distinct speeds through the medium. All these waves are found to be dispersive in nature. The coupled dilatational waves are damping, while the transverse wave is undamped in a certain range of the angular frequency. Coupled dilatational waves are found to be influenced by the presence of voids, thermal field and elastic nonlocal parameter, while the transverse wave is found to be influenced by the nonlocal parameter, but independent of void and thermal parameters. For a particular model, the effects of angular frequency, elastic nonlocality parameter, and some voids and thermal parameters on the wave speeds and damping coefficients of all the propagating waves have been studied numerically. Some comparisons are made between the results obtained for local and nonlocal cases. All the computed results have been depicted graphically and explained in details.     

A size-dependent generalized thermoelastic diffusion theory and its application

To capture the transient responses for the thermally and chemically shocked structure at micro or nanoscale, the present work is devoted to establish a size-dependent generalized thermoelastic diffusion theory within the thermodynamic framework. The uniqueness theorem and reciprocity theorem are, respectively, obtained. The corresponding generalized variational principle is developed using the semi-inverse method. In numerical implementation, a semi-infinite medium subjected to thermal and chemical shock at one end is considered and solved by the Laplace transformation. Numerical results are obtained and illustrated graphically. It can be concluded that the nonlocal scale parameter has a significant affect on the displacement and stress, which is excessively important in determining the material’s failure in complex environment. In addition, the numerical results show that the temperature, chemical potential, stress, and concentration are greatly influenced by the fractional order parameter.     

A Problem on Elastic Half Space Under Fractional Order Theory of Thermoelasticity

The present work is concerned with the solution of a problem on fractional order theory of thermoelasticity for an elastic medium. We investigate the thermoelastic interactions inside the medium by employing the fractional order theory of thermoelasticity, recently advocated by Sherief et al. (Int. J. Solids Struct., 47, 269–275, 2010). State space approach together with the Laplace transform technique is used to obtain the general solution of the problem. The general solution is then applied to three specific problems on an elastic half space, whose boundary is subjected to (i) a thermal shock (i.e., a step input in temperature and zero stress), (ii) a normal load (i.e., a step input in stress and zero temperature change) and (iii) a ramp type increase in temperature and zero stress. To observe the variations of displacement, temperature and stress inside the half-space we compute the numerical values of the field variables for a particular material by utilizing a numerical method of Laplace inversion. The effects of fractional order parameter on the variations of different fields inside the medium are analyzed graphically.     

ON THE PROPAGATION OF THERMOELASTIC WAVES IN MEDIA WITH LONG-RANGE COHESIVE FORCES

Abstract

After establishing the fundamental equations of nonlocal coupled thermoelasticity in Fourier space, nonlocal longitudinal thermoelastic waves in an infinite space are analyzed. Identification of the elastic dispersion equation obtained with its counterpart derived in lattice dynamics leads to values of the nonlocal elastic moduli. Similar identification of the velocity of thermal waves with its counterpart in second sound theory yields the values of the nonlocal thermal moduli. A numerical example involving solid helium is given.     

Application of fractional order theory of thermoelasticity to a bilayered structure with interfacial conditions

In this work, fractional order theory of thermoelasticity is applied to a bilayered structure being in imperfect thermal and mechanical contact. The model is subjected to a sudden heating at the traction-free end, assumed to be undisturbed at infinity. The heat conduction in each medium is described by the time-fractional heat conduction equations with two fractional order parameters, respectively. An analytical technique based on Laplace transform is adopted. Numerical results are computed and represented graphically, from which the effects of fractional derivative parameters of both media, thermal contact resistance, elastic wave impedance ratio on the responses are discussed.     

Three-Dimensional Fractional Order Generalized Thermoelastic Problem Under the Effect of Rotation in a Half Space

The current article is concerned with three-dimensional problems for a homogeneous, isotropic thermoelastic half-space subjected to a time-dependent heat source on the surface, which is traction-free under the effect of rotation. The governing equations are taken in the context of fractional order generalized thermoelasticity (Green–Lindsay) theory. Normal mode analysis technique and eigenvalue approaches have been used to solve the non-dimensional equations. Numerical results for temperature, thermal stress and displacement distributions are presented graphically and discussed.     

Eigenvalue approach to fractional order generalized thermoelasticity with line heat source in an infinite medium

Applying the eigenvalue approach method along with Laplace transform, a general solution scheme for the thermoelastic deformation of an unbounded transversely isotropic medium has been developed on fractional order generalized thermoelasticity with an instantaneous heat source. Solution has been achieved in Laplace transform domain for the perturbed temperature field and other field variables. Several graphs have been presented, and analyses of the results have been made.     

Thermoelastic Damping in Nanomechanical Resonators Based on Two-Temperature Generalized Thermoelasticity Theory

This work is concerned with the study of the thermoelastic damping of nanobeam resonators in the context of the two-temperature generalized thermoelasticity theory. An explicit formula of thermoelastic damping has been derived. Influences of the beam height, the relaxation time parameter, the two-temperature parameter and the isothermal value of frequency have been studied with some comparisons between the Biot model and Lord–Shulman model (L–S). Numerical results show that the values of thermal relaxation parameter and the two-temperature parameter have a strong influence on thermoelastic damping in nanoscales.     

Dynamic analysis to the fractional order thermoelastic diffusion problem of an infinite body with a spherical cavity and variable material properties

Abstract

Based on the generalized thermoelastic diffusion theory with fractional order derivative, the dynamic response of an infinite thermoelastic medium with a spherical cavity is investigated. The thermoelastic and diffusive properties of the medium are assumed to be temperature-dependent, and the medium is subjected to a thermal shock and a chemical potential shock at the inner surface of the spherical cavity simultaneously. The governing equations of the problem are formulated and then solved by Laplace transform together with its numerical inversion. The distributions of the non-dimensional temperature, displacement, radial stress, concentration and chemical potential are obtained and illustrated graphically. In calculation, the effects of the fractional order parameter and the temperature-dependent properties on the variations of the considered variables are presented and discussed in detail. The results show that the fractional order parameter and the temperature-dependent properties significantly influence the variations of all the considered variables. The present investigation may be valuable in heat and mass transfer, waste disposal or petroleum engineering, etc.     

Dynamic problem of fractional order thermoelasticity for a thick circular plate with finite wave speeds

This article is aimed at determining the thermoelastic displacement, stress, and temperature in a thick circular plate of finite thickness and infinite extent whose lower and upper surfaces are traction free, subjected to a given axisymmetric temperature distribution. The problem is formulated in the context of fractional order thermoelasticity theory with finite wave speeds. Integral transform technique is used to obtain the general solution in Laplace transform domain. Inversion of the Laplace transforms is done using a numerical scheme. A mathematical model is prepared for a copper material plate. Thermoelastic stresses, temperature and displacement are shown graphically and the effects of fractional-order parameters are discussed.     

A NOTE ON THE THEORY OF THERMOELASTIC DAMPING

This paper consists of three parts. In the first part, the problem of the conversion of mechanical energy into heat is treated, while in the second a general theory of thermoelastic friction in three-dimensional finite bodies is developed. In the third part, explicit results are obtained for thermoelastic damping in vibrating elastic beams. All investigations are based on the linear theory of thermoelasticity.     

Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams

In this work, the thermal effect on the buckling response of the axially functionally graded (AFG) nanobeams is studied based on the nonlocal thermoelasticity theory. Size effects of elastic deformation and heat conduction are considered simultaneously. Non-uniform distribution of temperature along the longitudinal direction of the AFG nanobeams is taken into account and determined by the nonlocal heat conductive law. Equations of motion and the corresponding boundary conditions are derived with the aid of the variational principle within the sinusoidal shear deformation theory and the nonlocal thermoelasticity theory. Ritz method is used to obtain the solutions for the thermal buckling response of the AFG nanobeams with various boundary conditions. Numerical results addressing the significance of the AFG index, the nonlocal parameters of elasticity and heat conduction, and the transverse shear deformation on the buckling behavior are displayed. It is found that, in addition to the nonlocal effect of elasticity, the nonlocal heat conduction plays an important role in analyzing the thermal–mechanical behaviors of the FG nanostructures.     

MECHANICAL AND THERMAL SOURCES IN A GENERALIZED THERMOELASTIC HALF-SPACE

The disturbance due to mechanical point loads and thermal sources acting on the boundary of a homogeneous isotropic thermoelastic half-space has been investigated upon applying the Laplace and Hankel transforms in the context of generalized theories of thermoelasticity. The integral transforms have been inverted using a numerical technique to obtain the displacements, temperature, and stresses in the physical domain. The numerical technique expresses the integrand as a Fourier series representation with respect to the Laplace transform parameter and evaluates the inverse Hankel transform integral via Romberg integration with an adaptive stepsize after using the results from successive refinements of the extended trapezoidal rule followed by extrapolating the results to the limit when the stepsize tends to zero. The results for various physical quantities are computed and presented graphically. A comparison of the results for different generalized theories of thermoelasticity are also presented.     

Fractional Order Generalized Thermo-Piezoelectric Semi-Infinite Medium with Temperature-Dependent Properties Subjected to a Ramp-Type Heating

The generalized thermoelasticity theory that, based on a fractional order model, is used to solve a one-dimensional boundary value problem of a semi-infinite piezoelectric medium. The resulting formulation is applied to a half-space subjected to ramp-type heating and traction free. The generalized thermo-piezoelectricity model in an isotropic elastic medium with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of the reference temperature. The Laplace transform technique is used to obtain the general solution for any set of boundary conditions. The inverse Laplace transforms are numerically computed using the Fourier expansion techniques. The effects of fractional order and the ramping of heating parameters are studied and comparison with different theories of thermoelasticity are considered. The results are also compared to results obtained in the case of a temperature-independent modulus of elasticity.