Generalized thermo-viscoelasticity with memory-dependent derivatives

https://doi.org/10.1016/j.ijmecsci.2014.10.006Get rights and content

Highlights

  • We derive the heat conduction equation with time-delay in thermoelasticity theory.

  • The new model of thermo-viscoelasticity is applied to one-dimensional problem of a half-space.

  • The memory-dependent derivatives are better than the fractional one for reflecting the memory effect.

Abstract

A new generalized thermo-viscoelasticity theory with memory-dependent derivatives is constructed. The governing coupled equations with time-delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to one-dimensional problem of a half-space. The bounding surface is taken traction free and subjected to a time dependent thermal shock. The Laplace transforms technique is used to obtain the general solution in a closed form. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusions about the new theory are given. The predictions of the theory are discussed and compared with dynamic classical coupled theory.

Graphical abstract

The variation of heat flux for different forms of kernal function K(t, ξ)

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Introduction

The linear theory of elasticity is of paramount importance in the stress analysis of steel, which is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e.g., concrete, wood, and coal. However, the theory does not apply to the behavior of many of the new synthetic materials of the elastomer and polymer type, e.g., polymethyl-methacrylate (Perspex), polyethylene, and polyvinyl chloride.

Linear viscoelastic materials are rheological materials that exhibit time temperature rate-of-loading dependence. When their response is not only a function of the current input, but also of the current and past input history, the characterization of the viscoelastic response can be expressed using the convolution (hereditary) integral. Tschoegl [1] has presented a general overview of time-dependent material properties. Gross investigated the mechanical-model representation of linear viscoelastic behavior results [2]. One can refer to Atkinson and Craster [3] for a review of fracture mechanics and generalizations to the viscoelastic materials, and Rajagopal and Saccomandi [4] for non-linear theory.

Physical observations and results of the conventional coupled dynamic thermoelasticity theories involving infinite speed of thermal signals, which were based on the mixed parabolic–hyperbolic governing equations of Biot [5] are mismatched. To remove this paradox, the conventional theories of thermoelasticity have been generalized, where the generalization is in the sense that these theories involve a hyperbolic-type heat transport equation supported by experiments which exhibit the actual occurrence of wave type heat transport in solids, called the second sound effect. The first is due to Cattaneo [6] who obtained a wave-type heat equation by postulating a new law of heat conduction to replace the classical Fourier law.

Several generalizations to the coupled theory are introduced. One can refer to Ignaczak [7] and Chandrasekharaiah [8] for a review. Hetnarski and Ignaczak [9] described the modern approaches to the analytical treatment of dynamical thermoelasticity.

Within the theoretical contributions to thermo-viscoelasticity theory are the proofs of uniqueness theorems under different conditions by Ezzat and El Karamany [10], [11] and the boundary element formulation was done by El-Karamany and Ezzat [12], [13]. The fundamental solutions for the cylindrical region were obtained by Ezzat [14]. Ezzat at el. [15] solved some problems in thermo-viscoelasticity with thermal relaxation by using the state space approach [16]. Ezzat [17] investigated the relaxation effects on the volume properties of an electrically conducting viscoelastic material.

In the last decade, considerable interest in fractional calculus has been stimulated by the applications in different areas of physics and engineering. Recently, some efforts have been done to modify the classical Fourier law of heat conduction by using the fractional calculus [18], [19], [20], [21], [22], [23].

Diethelm [24] has developed Caputo [25] derivative to beDaαf(t)=atKα(tξ)f(m)(ξ)dξwithKα(tξ)=(tξ)mα1Γ(mα)where Kα(tξ) is the kernel function and f(m) denotes the common m-order derivative, which has specific physical meaning.

The memory-dependent derivative is defined in an integral form of a common derivative with a kernel function on a slipping interval. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory dependent differential equation has more expressive force.

Wang and Li [26] introduced a memory-dependent derivative (MDD), the first order memory-dependent derivative of function f is simply defined in an integral form of a common derivative with a kernel function on a slipping interval, in the formDωf(x,t)=1ωtωtK(tξ)f(x,ξ)dξwhere ω is the time delay and K(tω) is the kernel function in which they can be chosen freely. Wang and Li indicated that the memory effect requires weight 0K(tξ)1 for ξ[tω,t) so that the magnitude of memory-dependent derivative, Dωf(x,t) is usually smaller than that of the common partial derivative f(x,t)t. The kernel form K(tξ) can also be chosen freely, such as 1, ξt+1, and [((ξt)/ω)+1]p, where p=0.25,1,2, etc. which may be more practical. They are a monotone function with K=0 for the past time tξ and K=1 for the present time t. In case, K(tξ)1, we haveDωf(x,t)=1ωtωtfξ(x,ξ)dξ=f(x,t)f(x,tω)ωf(x,t)t.This means that the common partial derivative /t is the limit of Dω as ω0.

So,|Dωf(x,t)||f(x,t)t|=|limω0f(x,t+ω)f(x,t)ω|

Sherief et al. [27] introduced the fractional order theory of thermoelasticity, in which the heat conduction equation was assumed to be the form qi+ταqitα=kθ,i

Recently, an interesting application of memory-dependent derivative is given by Yu et al. [28]. They introduced the memory-dependent derivative (MDD) instead of fractional calculus, into the rate of heat flux in Lord–Shulman generalized thermoelasticity theory [29], to denote memory-dependence, as,qi+ωq̇i=kθ,iEq. (5) has more clear physical meaning.

In the current work, a modified law of heat conduction including both the heat flux and its memory-dependent derivative replaces the conventional Fourier׳s law in thermo-viscoelasticity. The resulting non-dimensional coupled equations of generalized thermo-viscoelasticity with memory-dependent derivative together with the Laplace transforms techniques are applied to a specific problem of a half-space subjected to thermal shock and traction free surface. A direct approach is introduced to obtain the solutions in the Laplace transform domain for different forms of kernel functions. The inversion of Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier expansion techniques proposed by Honig and Hirdes [30]. In this theory, the coupled thermo-viscoelastic model is used. This implies infinite speeds of propagation of thermo-viscoelastic waves. The solutions are represented graphically for different values of time-delay and different forms of kernel function.

Section snippets

Differentiation of memory dependent derivative

Let nN, I:[a,b] and f:IR be such that f and all its derivatives up to f(n) are continuous on I, and f(n+1) exists on (a,b). Then,Dωnf(x,t)=1ωtωtK(tξ)fξ(n)(x,ξ)dξ

Derivation of heat conduction equation with time-delay in thermoelasticity

The classical Fourier׳s law, in which relates the heat flux vector q to the temperature gradientq(x,t)=κT(x,t)

The energy equation in terms of the heat conduction vector q in the context of thermoelasticity theory is given by [5]t(ρCET(x,t)+γToe(x,t))=q(x,t)+Q(x,t).

Using relation (5), we get the generalized heat conduction law for the considered new generalized theory with time-delayq(x,t+ω)=q(x,t)+ωDωq(x,t)

From a mathematical viewpoint, the Fourier law (7) in the theory of generalized

The formulation of the physical problem

The governing equations for generalized thermo-viscoelasticity consist of

  • 1.

    The equation of motion in the absence of body forcesρ2uit2=σji,j.

  • 2.

    The constitutive equation (17)Sij=0tR(tτ)eij(x,τ)τdτ=R˘(eij),whereSij=σijσkk3δij,and R(t) is relaxation function given byR˘(t)=2μ[1A0teβttα1dt],where α, β and A are non-dimensional empirical constants and Γ(α) is the Gamma function, 0<α<1,β>0,0A<βΓ(α),R˘(t)>0,ddtR˘(t)<0.

  • 3.

    The kinematic relationsεij=12(ui,j+uj,i),eij=εije3δij,e=εkk.

  • 4.

    The

Solution in the Laplace transform domain

Appling the Laplace transform with parameter s defined by the formulasL{g(x,t)}=g¯(x,s)=0estg(x,t)dt,on both sides of Eqs. (30), (31), (32), we get(D2αs2)u¯=αDθ,D2θ¯=s(1+G)(θ¯+εDu¯),σ¯=1αu¯xθ¯,and the boundary conditions (28) becomeθ¯(0,s)=f¯(s),σ¯(0,s)=0,whereD=x,L{R˘2ux2}=sR¯(s)2u¯x2,α=1/(1+sR¯),R¯(s)=4μ3sKo[1AΓ(α)(s+β)α],G(s)=(1esω)(12bωs+2a2ω2s2)(a22b+2a2ωs)esω,L{ωDωf(t)}=F(s){[(1esω)],m=n=0[11ωs(1esω)],m=0,n=12[(1esω)1s(1esω)+ωesω],m=0,n=ω2[(12ωs)+2ω2s2(1

Inversion of the Laplace transforms

We shall now outline the method used to invert the Laplace transforms in the above equations. Let f¯(s) be the Laplace transform of a function f(t). The inversion formula for Laplace transforms can be written as Honig and Hirdes [30]f(t)=edt2πeityf¯(d+iy)dy,where d is an arbitrary real number greater than all the real parts of the singularities of f¯(s).

Expanding the function h(t)=exp(dt)f(t) in a Fourier series in the interval [0, 2L], we obtain the approximate formulaf(t)fN(t)=12c0+k=1N

Numerical results and discussion

In this section, we aim to illustrate numerical results of the analytical expressions obtained in the previous section and elucidate the influence of time-delay ω on the behavior of the field quantities. In order to interpret the numerical computations, we consider material properties of a Polymethyl Methacrylate (Plexiglas) material. Following the values of physical constants are shown in Table 1 [17].

The calculations were carried out for the function f (t), which represents a time dependent

Conclusion

  • The main goal of this work is to introduced a generalized model for the Fourier law of heat conduction with time-delay and the kernel function by using the definition for reflecting the memory effect (instantaneous change rate depends on the past state).

  • According to this new theory, we have to construct a new classification for materials according to their, time-delay ω where this parameter becomes a new indicator of its ability to conduct heat in conducting medium.

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