State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory [Youssef, H., 2005a. The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, J. Appl. Math. Mech., 26(4), 4827; Youssef, H., 2005b. Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math., 1–8]. The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to thermal shock and traction free. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature parameter.     

Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity

The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.     

Some theorems on two-temperature generalized thermoelasticity

The aim of the present work is to establish a reciprocal principle of Betti type in the context of linear theory of two-temperature generalized thermoelasticity (Youssef in IMA J Appl Math 71:383–390, 2006; Arch Appl Mech 75:553–565, 2006) for homogeneous and isotropic body. Generalizations of the theorems of Somigliana and Green to two-temperature generalized thermoelasticity are also established on the basis of our reciprocal principle.     

State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source

In this work,a model of two-temperature generalized thermoelasticity without energy dissipation for an elastic half-space with constant elastic parameters is constructed.The Laplace transform and state-space techniques are used to obtain the general solution for any set of boundary conditions.The general solutions are applied to a specific problem of a half-space subjected to a moving heat source with a constant velocity.The inverse Laplace transforms are computed numerically,and the comparisons are shown in figures to estimate the effects of the heat source velocity and the two-temperature parameter.     

Two-temperature theory in generalized magneto-thermoelasticity with two relaxation times

The model of one-dimensional equations of the two-temperature generalized magneto-thermoelasticity theory with two relaxation times in a perfect electric conducting medium is established. The state space approach developed in Ezzat (Can J. Phys. Rev. 86(11):1241–1250, 2008) is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform techniques are applied to a specific problem of a half-space subjected to thermal shock and traction-free surface. The inversion of the Laplace transforms is carried out using a numerical approach. Some comparisons have been shown in figures to estimate the effects of the temperature discrepancy and the applied magnetic field.     

Two-temperature generalized thermoelasticity under ramp-type heating by finite element method

In this work, a general finite element model is proposed to analyze transient phenomena in thermoelastic half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in IMA J. Appl. Math. 71(3):383–390, 2006). A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. A finite element scheme is presented for the high accuracy numerical purpose. The numerical solutions of the non-dimensional governing partial differential equations of the problem have been shown graphically and some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the parameter of two-temperature.     

Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source

In this work, a problem of thermoelastic interactions in an elastic infinite medium with cylindrical cavity thermally shocked at its bounding surface and subjected to moving heat source with constant velocity has been solved. The governing equations are taken in the context of two-temperature generalized thermoelasticity theory (Youssef model). The analytical solution with direct approach in the Laplace transforms domain has been obtained. The derived analytical expressions have been computed for specific situations. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are represented graphically with comparisons by one-temperature generalized thermoelasticity (Lord–Shulman model).     

An Optimal Condition of Compactness for Elasticity Problems Involving one Directional Reinforcement

This paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007) and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Γ-convergence approach of Briane and Camar-Eddine (J. Math. Pures Appl. 88:483–505, 2007).     

Reliable treatment of a new analytical method for solving MHD boundary-layer equations

The purpose of this study is to implement a new analytical method which is a combination of the homotopy analysis method (HAM) and the Padé approximant for solving magnetohydrodynamic boundary-layer flow. The solution is compared with the numerical solution. Comparisons between the HAM–Padé and the numerical solution reveal that the new technique is a promising tool for solving MHD boundary-layer equations. The effects of the various parameters on the velocity and temperature profiles are presented graphically form. Favorable comparisons with previously published works (Crane, J. Appl. Math. Phys. 21:645–647, 1970, and Vajravelu and Hadjinicolaou, Int. J. Eng. Sci. 35:1237–1244, 1997) are obtained. It is predicted that HAM–Padé can have wide application in engineering problems (especially for boundary-layer and natural convection problems).     

Decay Structure for Symmetric Hyperbolic Systems with Non-Symmetric Relaxation and its Application

This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima–Shizuta stability condition formulated in Umeda et al. (Jpn J Appl Math 1:435–457, 1984) and Shizuta and Kawashima (Hokkaido Math J 14:249–275, 1985) and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (for example, the Timoshenko system and the Euler–Maxwell system). Moreover, it had been already known that the dissipative structure of such systems is weaker than the standard type and is of the regularity-loss type (see Duan in J Hyperbolic Differ Equ 8:375–413, 2011; Ide et al. in Math Models Meth Appl Sci 18:647–667, 2008; Ide and Kawashima in Math Models Meth Appl Sci 18:1001–1025, 2008; Ueda et al. in SIAM J Math Anal 2012; Ueda and Kawashima in Methods Appl Anal 2012). Therefore our purpose in this paper is to formulate a new structural condition which includes the Kawashima–Shizuta condition, and to analyze the weak dissipative structure for general systems with non-symmetric relaxation.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Formulation of Multidimensional Scalar Conservation Laws

We show that Kruzhkov’s theory of entropy solutions to multidimensional scalar conservation laws (Kruzhkov in Mat Sb (N.S.), 81(123), 228–255, 1970) can be entirely recast in L ² and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Brenier (in C R Acad Sci Paris Ser I Math, 292, 563–566, 1981; in J Diff Equ, 50, 375–390, 1983; in SIAM J Numer Anal, 21, 1013–1037; in Methods Appl Anal, 11, 515–532, 2004), Giga and Miyakawa (in Duke Math J, 50, 505–515, 1983), and Tsai et al. (in Math Comp, 72, 159–181, 2003).     

On the Analytical Approach to Galin’s Stick-Slip Problem. A Survey

Galin’s classical work (PMM J Appl Math Mech 9:413–424, 1945) on the contact of a rigid flat-ended indenter with an elastic half-plane with partial slip was the first successful attempt to take into account friction in the problem of normal contact. As Galin was unable to find an exact solution of the formulated problem, the problem of contact with partial slip of a rigid punch with an elastic half-plane was challenged by many researchers. At the same time Galin’s seminal work stimulated development of solutions for other contact problems with friction that feature different punch geometries and different material responses. This paper presents an overview of the developments in the area of elastic contact with partial slip. In the spirit of Galin’s work the focus is placed on contributions with substantial analytical merit.      

State space approach to thermoelectric fluid with fractional order heat transfer

A new mathematical model for electromagnetic thermofluid equation heat transfer with thermoelectric properties using the methodology of fractional calculus is constructed. The governing coupled equations in the frame 11 of the boundary layer model are applied to variety problems. Laplace transforms and state space techniques (Ezzat Can J Phys Rev 86:1241–1250 in 2008) are used to get the solution of a thermal shock problem, a layer problem and a problem for the semi-infinite space in the presence of heat sources. According to the numerical results and its graphs, a parametric study of time-fractional order 0 < α ≤ 1, on temperature and the thermoelectric figure of merit are conducted.     

Saint-Venant’s Problem for Compound Piezoelectric Beams

The solution of the Saint-Venant’s Problem for a slender compound piezoelectric beam presented in this paper generalizes the recent solution by the authors and E. Harash (J. Appl. Mech. 11:1–10, 2007) for a homogeneous piezoelectric beam and the solution for a compound elastic beam developed by O. Rand and the first author (Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Boston, 2005). Justification for this approximation emerges from the St. Venant’s Principle. The stress, strain and (electrical) displacement components (“solution hypothesis”) are presented as a set of initially assumed expressions involving twelve tip loading parameters, six unknown weight coefficients, and three pairs of torsion/bending functions of two variables. Each pair of functions satisfies the so-called coupled non-homogeneous Neumann problem (CNNP) in the cross-sectional domain. The work develops concepts of the torsion/bending functions, the torsional rigidity and piezoelectric shear center, the tip coupling matrix, for a compound piezoelectric beam. Examples of exact and approximate solutions for rectangular laminated beams made of transtropic materials are presented.      

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in a unified system from which the field equations for coupled thermoelasticity as well as for generalized thermoelasticity can be easily obtained as particular cases. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating with different theories of thermoelasticity.     

Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].     

On the Impossibility, in Some Cases, of the Leray�CHopf Condition for Energy Estimates

Current proofs of time independent energy bounds for solutions of the time dependent Navier–Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of κ less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1–82, 1993), Hopf (Math Ann 117:764–775, 1941) and Finn (Acta Math 105:197–244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of κ, has been shown by Takeshita (Pac J Math 157:151–158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains.     

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.