Memory-dependent derivative approach on magneto-thermoelastic transversely isotropic medium with two temperatures

The aim of the present investigation is to examine the memory-dependent derivatives (MDD) in 2D transversely isotropic homogeneous magneto thermoelastic medium with two temperatures. The problem is solved using Laplace transforms and Fourier transform technique. In order to estimate the nature of the displacements, stresses and temperature distributions in the physical domain, an efficient approximate numerical inverse Fourier and Laplace transform technique is adopted. The distribution of displacements, temperature and stresses in the homogeneous medium in the context of generalized thermoelasticity using LS (Lord-Shulman) theory is discussed and obtained in analytical form. The effect of memory-dependent derivatives is represented graphically.


Introduction
Magneto-thermoelasticity deals with the relations of the magnetic field, strain and temperature. It has wide applications such as geophysics, examining the effects of the earth's magnetic field on seismic waves, emission of electromagnetic radiations from nuclear devices and damping of acoustic waves in a magnetic field. In recent years, inspired by the successful applications of fractional calculus in diverse areas of engineering and physics, generalized thermoelasticity (GTE) models have been further comprehensive into temporal fractional ones to express memory dependence in heat conductive sense.
The MDD is defined in an integral form of a common derivative with a kernel function. The kernels in physical laws are important in many models that describe physical phenomena including the memory effect. Wang and Li (2011) introduced the concept of a MDD. Yu et al. (2014) introduced the MDD as an alternative of fractional calculus into the rate of the heat flux in the Lord-Shulman (LS) theory of generalized thermoelasticity to represent memory dependence and recognized as a memory-dependent LS model. This innovative model might be useful to the fractional models owing to the following arguments. First, the new model is unique in its form, while the fractional-order models have different modifications (Riemann-Liouville, Caputo and other models). Second, the physical meaning of the new model is clearer due to the essence of the MDD definition. Third, the new model is depicted by integer-order differentials and integrals, which is more convenient in numerical calculation as compared to the fractional models. Finally, the kernel function and time delay of the MDD can be arbitrarily chosen; thus, the model is more flexible in applications than the fractional models, in which the significant variable is the fractional-order parameter (2016). Ezzat et al. (2014Ezzat et al. ( , 2015Ezzat et al. ( , 2016 discussed some solutions of one-dimensional problems obtained with the use of the memory-dependent LS model of generalized thermoelasticity. Ezzat et al. (2016) discussed a generalized model of two-temperature thermoelasticity theory with time delay and Kernel function and Taylor theorem with memorydependent derivatives involving two temperatures.  applied the magneto-thermoelastic model to a one-dimensional thermal shock problem of functionally graded half space based on the MDD.  proposed a mathematical model of electrothermoelasticity for heat conduction with a MDD. Aldawody et al. (2018) proposed a new mathematical model of generalized magneto-thermo-viscoelasticity theories with MDD of dual-phase-lag heat conduction law. Despite this, several researchers worked on different theory of thermoelasticity such as Marin (1995Marin ( , 2010, Mahmoud (2012), Riaz et al. (2019), Marin et al. (2013Marin et al. ( , 2016, Kumar and Chawla (2013), Sharma and Marin (2014), Kumar and Devi (2016), Bijarnia and Singh (2016), , Ezzat and El-Barrry (2017), Bhatti et al. (2019aBhatti et al. ( , b, 2020, Youssef (2013Youssef ( , 2016, Lata et al. (2016), Othman and Marin (2017), Lata and Kaur (2019c, d, e) and Zhang et al. (2020).
In spite of these, not much work has been carried out in memory-dependent derivative approach for transversely isotropic magneto-thermoelastic medium with two temperatures. In this article, the memory-dependent derivatives (MDD) theory is revisited and it is adopted to analyse the effect of MDD in a homogeneous transversely isotropic magneto-thermoelastic solid. The problem is solved using Laplace transforms and Fourier transform technique. The components of displacement, conductive temperature and stress components in the homogeneous medium in the context of generalized thermoelasticity using LS (Lord-Shulman) theory is discussed and obtained in analytical form. The effect of memory-dependent derivatives is represented graphically.

Basic equations
Following , the simplified Maxwell's linear equation of electrodynamics for a slowly moving and perfectly conducting elastic solid are Maxwell stress components following  are given by Equation of motion for a transversely isotropic thermoelastic medium and taking into account Lorentz force where F i ¼ μ 0 ð j ! Â H ! 0 Þ i are the components of Lorentz force. The constitutive relations for a transversely isotropic thermoelastic medium are given by and Here, C ijkl (C ijkl = C klij = C jikl = C ijlk ) are elastic parameters and having symmetry (C ijkl = C klij = C jikl = C ijlk ). The basis of these symmetries of C ijkl is due to the following: i. The stress tensor is symmetric, which is only possible if (C ijkl = C jikl ) ii. If a strain energy density exists for the material, the elastic stiffness tensor must satisfy C ijkl = C klij iii. From stress tensor and elastic stiffness tensor, symmetries infer (C ijkl = C ijlk ) and C ijkl = C klij = C jikl = C ijlk Following Bachher (2019), heat conduction equation for an anisotropic media is given by For the differentiable function f(t), Wang and Li (2011) introduced the first-order MDD with respect to the time delay χ > 0 for a fixed time t: The choice of the kernel function K(t − ξ) and the time delay parameter χ is determined by the material properties. The kernel function K(t − ξ) is differentiable with respect to the variables t and ξ. The motivation for such a new definition is that it provides more insight into the memory effect (the instantaneous change rate depends on the past state) and also better physical meaning, which might be superior to the fractional models. This kind of the definition can reflect the memory effect on the delay interval [t − χ, t], which varies with time. They also suggested that the kernel form K(t − ξ) can also be chosen freely, e.g. as 1, ξ − t + 1, [(ξ − t)/χ + 1] 2 and [(ξ − t)/χ + 1] 1/4 . The kernel function can be understood as the degree of the past effect on the present. Therefore, the forms [(ξ − t)/χ + 1] 2 and [(ξ − t)/χ + 1] 1/4 may be more practical because they are monotonic functions: K(t − ξ) = 0 for the past time t − χ and K(t − ξ) = 1 for the present time t, i.e. it is easily concluded that the kernel function K(t − ξ) is a monotonic function increasing from zero to unity with time. The right side of the MDD definition given above can be understood as a mean value of f (ξ) on the past interval [t − χ, t] with different weights. Generally, from the viewpoint of applications, the function Therefore, the magnitude of the MDD D χ f(t) is usually smaller than that of the common derivative f (t). It can also be noted that the common derivative d/dt is the limit of D χ as κ → 0. Following Ezzat et al. (2014Ezzat et al. ( , 2015Ezzat et al. ( , 2016, the kernel function K(t − ξ) is taken here in the form where a and b are constants. It should be also mentioned that the kernel in the fractional sense is singular, while that in the MDD model is non-singular. The kernel can be now simply considered a memory manager. The comma is further used to indicate the derivative with respect to the space variable, and the superimposed dot represents the time derivative.

Boundary conditions
The appropriate boundary conditions for the thermally insulated boundaries z = 0 are ∂φ ∂z x; z; t ð Þþhφ ¼ 0: where h → 0 corresponds to thermally insulated surface and h → ∞ corresponds to isothermal surface, F 1 and F 2 are the magnitude of the forces applied and ψ 1 (x) and ψ 2 (x) specify the vertical and horizontal load distribution function along x-axis.

Case I: Thermally insulated boundaries
When h = 0 solving (68)- (70), the values of A 1 , A 2 , A 3 are obtained as where The components of displacement, conductive temperature, normal stress and tangential stress are obtained from (45) to (47) and (59) to (61) by putting the values of A 1 , A 2 , A 3 from (71) to (73) aŝ Case II: Isothermal boundaries When h→ ∞ ,solving (68)-(70), using Cramer's rule, the values of A 1 , A 2 , A 3 are obtained as Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 6 of 13 where The components of displacement, conductive temperature, normal stress and tangential stress are obtained from (45) to (47) and (59) to (61) by putting the values of A 1 , A 2 , A 3 from (80) to (82) aŝ

Applications
We consider a normal line load F 1 per unit length acting in the positive z-axis on the plane boundary z = 0 along the y-axis and a tangential load F 2 per unit length, acting at the origin in the positive x-axis. Suppose an inclined load, F 0 per unit length is acting on the y-axis and its inclination with z-axis is θ (see Fig. 1), we have Special cases

Concentrated force
The solution due to concentrated normal force on the half space is obtained by setting where δ(x) is Dirac delta function. Applying Fourier transform defined by (33) on (90) yieldŝ For case I, using (91) in Eqs. (74)-(79) and for case II, using (91) in Eqs. (83)-(88), the components of displacement, stress and conductive temperature are obtained for case I and case II, respectively.

Uniformly distributed force
The solution due to uniformly distributed force applied on the half space is obtained by setting The Fourier transforms of ψ 1 (x) and ψ 2 (x) with respect to the pair (x, ξ) for the case of a uniform strip load of non-dimensional width 2 m applied at origin of coordinate system x = z = 0 in the dimensionless form after suppressing the primes becomeŝ For case I, using (93) in Eqs. (74)-(79) and for case II, using (93) in Eqs. (83)-(88), the components of displacement, stress and conductive temperature are obtained for case I and case II.

Linearly distributed force
The solution due to linearly distributed force applied on the half space is obtained by setting Here, 2 m is the width of the strip load, and applying the transform defined by (33) on (94), we get For case I, using (95) in Eqs. (74)-(79) and for case II, using (95) in Eqs. (83)-(88), the components of displacement, stress and conductive temperature are obtained for case I and case II, respectively.

Inversion of the transformation
To find the solution of the problem in physical domain and to invert the transforms in Eqs (74)-(79) and for case II in Eqs. (83)-(88), here, the displacement components, normal and tangential stresses and conductive temperature are functions of z, the parameters of Laplace and Fourier transforms s and ξ respectively and hence are of the formf ðξ; z; sÞ. To find the functionf ð x; z; tÞ in the physical domain, we first invert the Fourier transform using where f e and f o are respectively the odd and even parts off ðξ; z; sÞ: We obtain Fourier inverse transform by replacing s by ω in (96). Following Honig and Hirdes (1984), the Laplace transform functionf ðx; z; sÞ can be inverted to f(x, z, t) for problem I by The last step is to calculate the integral in Eq. (97). The method for evaluating this integral is described in Press (1986). the loading surface of the inclined load and follow a small oscillatory pattern for the rest of the range of distance.

Conclusion
From the above study, the following is observed: Displacement components (u and w), conductive temperature φ and stress components ( t 11 , t 13 and t 33 ) for a transversely isotropic magnetothermoelastic medium with concentrated force and with combined effects of two-temperature model of thermoelasticity with different kernel functions of MDD, respectively In order to estimate the nature of the displacements, stresses and temperature distributions in the physical domain, an efficient approximate numerical inverse Fourier and Laplace transform technique is adopted. Moreover, the magnetic effect of two temperatures, rotation, and the angle of inclination of the applied load plays a key part in the deformation of all the physical quantities. K(t − ξ) = [1 + (ξ − t)/χ] 2 shows the more oscillatory nature for the displacement components and stress components.
The result gives the inspiration to study magnetothermoelastic materials with memory-dependent derivatives as an innovative domain of applicable thermoelastic solids. The shape of curves shows the impact of Kernel function on the body and fulfils the purpose of the study.