ORIGINAL PAPER Open Access

Memory-dependent derivative approach onmagneto-thermoelastic transverselyisotropic medium with two temperaturesIqbal Kaur1* , Parveen Lata2 and Kulvinder Singh3

Abstract

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

Keywords: Thermoelastic, Transversely isotropic, Magneto-thermoelastic, Memory-dependent derivative, Time delay,Kernel function, Lord-Shulman model

IntroductionMagneto-thermoelasticity deals with the relations of themagnetic field, strain and temperature. It has wide appli-cations such as geophysics, examining the effects of theearth's magnetic field on seismic waves, emission of elec-tromagnetic radiations from nuclear devices and damp-ing of acoustic waves in a magnetic field. In recent years,inspired by the successful applications of fractional cal-culus in diverse areas of engineering and physics, gener-alized thermoelasticity (GTE) models have been furthercomprehensive into temporal fractional ones to expressmemory dependence in heat conductive sense.The MDD is defined in an integral form of a common

derivative with a kernel function. The kernels in physicallaws are important in many models that describe phys-ical phenomena including the memory effect. Wang andLi (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 theLord-Shulman (LS) theory of generalized thermoelasti-city to represent memory dependence and recognized asa memory-dependent LS model. This innovative modelmight be useful to the fractional models owing to thefollowing arguments. First, the new model is unique inits form, while the fractional-order models have differentmodifications (Riemann-Liouville, Caputo and othermodels). Second, the physical meaning of the new modelis clearer due to the essence of the MDD definition.Third, the new model is depicted by integer-order differ-entials and integrals, which is more convenient in nu-merical calculation as compared to the fractionalmodels. Finally, the kernel function and time delay ofthe MDD can be arbitrarily chosen; thus, the model ismore flexible in applications than the fractional models,in which the significant variable is the fractional-orderparameter (2016). Ezzat et al. (2014, 2015, 2016) dis-cussed some solutions of one-dimensional problems ob-tained with the use of the memory-dependent LS modelof generalized thermoelasticity.

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* Correspondence: bawahanda@gmail.com1Department of Mathematics, Government College for Girls, Palwal,Kurukshetra, Haryana, IndiaFull list of author information is available at the end of the article

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 https://doi.org/10.1186/s40712-020-00122-2

Ezzat et al. (2016) discussed a generalized model oftwo-temperature thermoelasticity theory with time delayand Kernel function and Taylor theorem with memory-dependent derivatives involving two temperatures. Ezzatand El-Bary (2017) applied the magneto-thermoelasticmodel to a one-dimensional thermal shock problem offunctionally graded half space based on the MDD. Ezzatet al. (2017) proposed a mathematical model of electro-thermoelasticity for heat conduction with a MDD.Aldawody et al. (2018) proposed a new mathematicalmodel of generalized magneto-thermo-viscoelasticitytheories with MDD of dual-phase-lag heat conductionlaw. Despite this, several researchers worked on differenttheory of thermoelasticity such as Marin (1995, 2010),Mahmoud (2012), Riaz et al. (2019), Marin et al. (2013,2016), Kumar and Chawla (2013), Sharma and Marin(2014), Kumar and Devi (2016), Bijarnia and Singh(2016), Othman et al. (2017), Ezzat and El-Barrry (2017),Bhatti et al. (2019a, b, 2020), Youssef (2013, 2016), Lataet 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 trans-versely isotropic magneto-thermoelastic medium withtwo temperatures. In this article, the memory-dependentderivatives (MDD) theory is revisited and it is adoptedto analyse the effect of MDD in a homogeneous trans-versely isotropic magneto-thermoelastic solid. The prob-lem is solved using Laplace transforms and Fouriertransform technique. The components of displacement,conductive temperature and stress components in thehomogeneous medium in the context of generalizedthermoelasticity using LS (Lord-Shulman) theory is dis-cussed and obtained in analytical form. The effect ofmemory-dependent derivatives is representedgraphically.

Basic equationsFollowing Kumar et al. (2016), the simplified Maxwell’slinear equation of electrodynamics for a slowly movingand perfectly conducting elastic solid are

curl h!¼ j

!þ ε0∂ E!∂t

; ð1Þ

curl E!¼ − μ0

∂ h!∂t

; ð2Þ

E!¼ − μ0

∂ u!∂t

þ H!

0

� �; ð3Þ

div h!¼ 0: ð4Þ

Maxwell stress components following Kumar et al.(2016) are given by

Tij ¼ μ0 Hihj þ H jhi −Hkhkδij� �

: ð5ÞEquation of motion for a transversely isotropic ther-

moelastic medium and taking into account Lorentz force

tij; j þ Fi ¼ ρ€ui; ð6Þwhere Fi ¼ μ0ð j

!� H!

0Þi are the components of Lorentzforce. The constitutive relations for a transversely iso-tropic thermoelastic medium are given by

tij ¼ Cijklekl − βijT : ð7Þand

βij ¼ Cijklαij; ð8Þ

eij ¼ 12

ui; j þ uj;i� �

; i; j ¼ 1; 2; 3: ð9Þ

T ¼ φ − aijφ;ij; ð10Þαij = αiδij, βij = βiδij, Kij = Kiδij, i is not summed.Here, Cijkl(Cijkl =Cklij =Cjikl =Cijlk) are elastic parame-

ters and having symmetry (Cijkl =Cklij =Cjikl = Cijlk). Thebasis of these symmetries of Cijkl is due to the following:

i. The stress tensor is symmetric, which is onlypossible if (Cijkl = Cjikl)

ii. If a strain energy density exists for the material, theelastic stiffness tensor must satisfy Cijkl = Cklij

iii. From stress tensor and elastic stiffness tensor,symmetries infer (Cijkl = Cijlk) and Cijkl = Cklij =Cjikl = Cijlk

Following Bachher (2019), heat conduction equationfor an anisotropic media is given by

Kijφ;ij ¼ 1þ χDχ� �

ρCET þ βijT 0eijÞ;�

ð11Þ

For the differentiable function f(t), Wang and Li(2011) introduced the first-order MDD with respect tothe time delay χ > 0 for a fixed time t:

Dχ f tð Þ ¼ 1χ

Z t

t − χK t − ξð Þ f 0

ξð Þdξ; ð12Þ

The choice of the kernel function K(t − ξ) and thetime delay parameter χ is determined by the materialproperties. The kernel function K(t − ξ) is differentiablewith respect to the variables t and ξ. The motivation forsuch a new definition is that it provides more insightinto the memory effect (the instantaneous change ratedepends on the past state) and also better physicalmeaning, which might be superior to the fractionalmodels. This kind of the definition can reflect the mem-ory effect on the delay interval [t − χ, t], which varieswith time. They also suggested that the kernel form K(t

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 2 of 13

− 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 canbe understood as the degree of the past effect on thepresent. Therefore, the forms [(ξ − t)/χ + 1]2 and [(ξ −t)/χ + 1]1/4 may be more practical because they aremonotonic functions: K(t − ξ) = 0 for the past time t − χand K(t − ξ) = 1 for the present time t, i.e. it is easilyconcluded that the kernel function K(t − ξ) is a mono-tonic function increasing from zero to unity with time.The right side of the MDD definition given above can beunderstood as a mean value of f (ξ) on the past interval[t − χ, t] with different weights. Generally, from theviewpoint of applications, the function K(t − ξ) shouldsatisfy the inequality 0 ≤ K(t − ξ) < 1 for ξ ∈ [t − χ, t].Therefore, the magnitude of the MDD Dχ f(t) is usuallysmaller than that of the common derivative f (t). It canalso be noted that the common derivative d/dt is thelimit of Dχ as κ → 0. Following Ezzat et al. (2014, 2015,2016), the kernel function K(t − ξ) is taken here in theform

K t − ξð Þ ¼ 1 −2bχ

t − ξð Þ a2

χ2t − ξð Þ2

¼1

1þ ξ − tð Þ=χξ − t þ 1

1þ ξ − tð Þ=χ½ �2

8>><>>:a ¼ 0; b ¼ 0

a ¼ 0; b ¼ 1=2a ¼ 0; b ¼ 1=χa ¼ 1; b ¼ 1

ð13Þ

where a and b are constants. It should be also men-tioned that the kernel in the fractional sense is singular,while that in the MDD model is non-singular. The ker-nel can be now simply considered a memory manager.The comma is further used to indicate the derivativewith respect to the space variable, and the superimposeddot represents the time derivative.

Method and solution of the problemWe consider a homogeneous transversely isotropicmagneto-thermoelastic medium initially at a uniformtemperature T0, permeated by an initial magnetic field

H!

0 ¼ ð0;H0; 0Þ acting along the y-axis. The rectangularCartesian co-ordinate system (x, y, z) having origin onthe surface (z = 0) with z-axis pointing vertically into themedium is introduced. From the generalized Ohm’s law

E!¼ − μ0H0 − €w; 0; €uð Þ ð14Þ

j!¼ −

∂Hy∂z

− ε0Ex; 0; −∂Hy∂x

− ε0EzÞ:�

ð15Þ

F!¼ μ0H

20

∂e∂x

− ε0μ0€u

� �; 0; μ0H

20

∂e∂z

− ε0μ0€w

� �� �ð16Þ

In addition, we consider the plane such that all parti-cles on a line parallel to y-axis are equally displaced, sothat the equations of displacement vector (u, v,w) andconductive temperature φ for transversely isotropic ther-moelastic solid are given by

u ¼ u x; z; tð Þ; v ¼ 0;w ¼ w x; z; tð Þ and φ ¼ φ x; z; tð Þ:ð17Þ

Now, using the transformation on Eqs. (5) and (12)following Slaughter (2002) with the aid of (16) and (17)yields

C11∂2u∂x2

þ C13∂2w∂x∂z

þ C44∂2u∂z2

þ ∂2w∂x∂z

� �− β1

∂∂x

φ − a1∂2φ∂x2

þ a3∂2φ∂z2

� �� �

þμ0H20

∂e∂x

− ε0μ0€u

� �¼ ρ

∂2u∂t2

;

ð18Þ

C13 þ C44ð Þ ∂2u∂x∂z

þ C44∂2w∂x2

þ C33∂2w∂z2

− β3∂∂z

φ − a1∂2φ∂x2

þ a3∂2φ∂z2

� �� �

þμ0H20

∂e∂z

− ε0μ0€w

� �¼ ρ

∂2w∂t2

;

ð19Þ

K1∂2φ∂x2

þ K 3∂2φ

∂z2¼ 1þ χDχ

� �T0 β1

∂u∂x

þ β3∂w∂z

Þ þ ρCE φ − a1∂2φ

∂x2− a3

∂2φ

∂z2g�

��ð20Þ

where from (7)

β1 ¼ C11 þ C12ð Þα1 þ C13α3;β3 ¼ 2C13α1 þ C33α3;e ¼ ∂u∂x

þ ∂w∂z

And stress strain relations given by (5) after using (9)and (17) becomes

t11 x; z; tð Þ ¼ C11∂u∂x

þ C13∂w∂z

− β1 φ − a1∂2φ

∂x2þ a3

∂2φ

∂z2

� �� �;

ð21Þ

t22 x; z; tð Þ ¼ c12∂u∂x

þ c13∂w∂z

− β1 φ − a1∂2φ

∂x2þ a3

∂2φ

∂z2

� �� �;

ð22Þ

t33 x; z; tð Þ ¼ C13∂u∂x

þ C33∂w∂z

− β3 φ − a1∂2φ

∂x2þ a3

∂2φ

∂z2

� �� �;

ð23Þ

t13 x; z; tð Þ ¼ C44∂u∂z

þ ∂w∂x

� �; ð24Þ

t12 x; z; tð Þ ¼ t23 x; z; tð Þ ¼ 0: ð25Þ

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 3 of 13

We assume that medium is initially at rest. The undis-turbed state is maintained at reference temperature T0.Then, we have the initial and regularity conditions givenby

u x; z; 0ð Þ ¼ 0 ¼ u x; z; 0ð Þ;w x; z; 0ð Þ ¼ 0 ¼ w x; z; 0ð Þ;φ x; z; 0ð Þ¼ 0 ¼ φ x; z; 0ð Þ for z≥0; − ∞ < x < ∞; u x; z; tð Þ¼ w x; z; tð Þ ¼ φ x; z; tð Þ ¼ 0 for t > 0 when z→∞:

To simplify the solution, the following dimensionlessquantities are used:

ðx′; z′Þ ¼ 1Lðx; zÞ; ðu′;w′Þ ¼ ρc21

Lβ1T0ðu;wÞ; ða′1; a′3Þ

¼ 1

L2ða1; a3Þ; ρC2

1 ¼ C11φ′ ¼ φ

T 0; ðt′11; t′13; t′33Þ

¼ 1β1T0

ðt11; t13; t33Þ; t′ ¼ C1

Lt:

ð26ÞMaking use of dimensionless quantities defined by

(26) in Eqs. (18)–(20), after suppressing the primes,yields

1þ δ5ð Þ ∂2u∂x2

þ δ4 þ δ5ð Þ ∂2w

∂x∂zþ δ2

∂2u∂z2

þ ∂2w∂x∂z

� �−

∂∂x

φ − a1∂2φ∂x2

þ a3∂2φ∂z2

� �� �

¼ ε0μ20H20

ρþ 1

� �∂2u∂t2

;

ð27Þ

δ1 þ δ5ð Þ ∂2u∂x∂z

þ δ2∂2w∂x2

þ δ3 þ δ5ð Þ ∂2w∂z2

−β3β1

∂∂z

φ − a1∂2φ∂x2

þ a3∂2φ∂z2

� �� �

¼ ε0μ20H20

ρþ 1

� �∂2w∂t2

;

ð28Þ

K1∂2φ∂x2

þ K3∂2φ

∂z2¼ 1þ χDχ

� �∂∂t

δ6∂u∂x

þ δ8∂w∂z

� �þ δ7

∂∂t

φ − a1∂2φ∂x2

− a3∂2φ

∂z2

� � ;

ð29Þwhere

δ1 ¼ C13 þ C44

C11; δ2 ¼ C44

C11; δ3 ¼ C33

C11; δ4 ¼ C13

C11; δ5

¼ β1T 0μ0H20

Lρ2C41

; δ6 ¼ Lβ21T 0

ρC1; δ8 ¼ Lβ1β3T 0

ρC1; δ7

¼ ρCEC1L:

Laplace transforms is defined by

L f x; z; tð Þ½ � ¼Z∞0

e − st f x; z; tð Þdt ¼ ~f x; z; sð Þ: ð30Þ

with basic properties

L∂ f∂t

� �¼ s f x; z; sð Þ − f x; z; 0ð Þ; ð31Þ

L∂2 f∂t2

� �¼ s2 f x; z; sð Þ − sf x; z; 0ð Þ − ∂ f

∂t

� �t¼0

: ð32Þ

Fourier transforms of a function f with respect to vari-able x with ξ as a Fourier transform variable is definedby

F ~f x; z; sð Þn o

¼ f ξ; z; sð Þ ¼Z∞− ∞

~f x; z; sð Þeiξxdx: ð33Þ

With basic properties: if f(x) and first (n − 1) deriva-tives of f(x) vanish identically as x→ ± ∞ , then

F∂n~f x; z; sð Þ

∂xn

( )¼ iξð Þn f ξ; z; sð Þ ð34Þ

Applying Laplace and Fourier transforms defined by(31)–(34) on Eqs. (24)-(29) yields

− 1þ δ5ð Þξ2uþ δ4 þ δ5ð ÞiξDwþ δ2 D2uþ iξDw

� �− iξ φ − − a1ξ

2φþ a3D2φ

� �� �¼ ε0μ20H

20

ρþ 1

� �s2u;

ð35Þ

δ1 þ δ5ð ÞiξDu − δ2ξ2w

þ δ3 þ δ5ð ÞD2w −β3β1

D φ − − a1ξ2φþ a3D

2φ� �� �

¼ ε0μ20H20

ρþ 1

� �s2w;

ð36Þ

− K1ξ2φþ K 3D

2φ ¼ 1þ Gð Þs δ6iξuþ δ8Dwð Þ þ δ7s φ − − a1ξ

2φþ a3D2φ

� �� � �;

ð37Þwhere

D ¼ ddz

;G ¼ τ0χ

1 − e − sχð Þ 1 −2bχs

þ 2a2

χ2s2

� �- a2 − 2bþ 2a2

χs

� �e − sχ

:

These equations on further simplifications become

− 1þ δ5ð Þξ2 − ε0μ20H20

ρþ 1

� �s2 þ δ2D2

u ξ; z; sð Þ

þ δ4 þ δ5 þ δ2ð ÞiξD½ �w ξ; z; sð Þ

þ − iξð Þ 1þ a1ξ2 − a3D

2 �

φ ξ; z; sð Þ ¼ 0;

ð38Þ

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 4 of 13

δ1 þ δ5ð ÞiξD½ �u ξ; z; sð Þ þ − δ2ξ2 þ δ3 þ δ5ð ÞD2 −

ε0μ20H20

ρþ 1

� �s2

w ξ; z; sð Þ − β3

β1D 1þ a1ξ

2 − a3D2

�φ ξ; z; sð Þ ¼ 0;

ð39Þ1þ Gð Þsδ6iξ½ �u ξ; z; sð Þ þ 1þ Gð Þsδ8D½ �w ξ; z; sð Þ

þ − K1ξ2 þ K 3D

2 þ 1þ Gð Þδ7s 1þ a1ξ2 − a3D

2� � �

φ ξ; z; sð Þ ¼ 0;

ð40ÞThe above equations can be written as

ζ1 þ δ2D2 �

u ξ; z; sð Þ þ ζ2D½ �w ξ; z; sð Þ þ ζ3 þ ζ4D2

�φ ξ; z; sð Þ ¼ 0;

ð41Þζ5D½ �u ξ; z; sð Þ þ ζ6 þ ζ7D

2 �

w ξ; z; sð Þ þ ζ8Dþ ζ9D3

�φ ξ; z; sð Þ ¼ 0;

ð42Þζ10½ �u ξ; z; sð Þ þ ζ11½ �w ξ; z; sð Þ þ ζ12 þ ζ13D

2 �

φ ξ; z; sð Þ ¼ 0;

ð43ÞEquations (41)–(43) will have a non-trivial solution if

determinant of coefficient matrix of ðu; w; φÞ vanishes,and thus we obtain the characteristic equation as

AD6 þ BD4 þ CD2 þ E� �

u; w; φð Þ ¼ 0; ð44Þwhere

D ¼ ddz

;A ¼ − δ2ζ11ζ9 þ ζ13δ2ζ7;

B ¼ δ2ζ6ζ13 þ ζ7ζ1ζ13 þ δ2ζ7ζ12 − δ2ζ11ζ8 − ζ1ζ9ζ11 − ζ5δ2ζ13 þ δ2ζ10ζ9 þ ζ11ζ4ζ5 − ζ10ζ7ζ4;

C ¼ ζ13ζ1ζ6 þ δ2ζ6ζ12 þ ζ1ζ7ζ12 − ζ1ζ11ζ8 þ ζ4ζ8ζ9 − ζ5δ2ζ12 − ζ10δ2ζ8 þ ζ11ζ3ζ5 − ζ3ζ7ζ10;

E ¼ ζ12ζ1ζ6 − ζ10ζ6ζ3;

ζ1 ¼ − 1þ δ5ð Þξ2 − ε0μ20H20

ρþ 1

� �s2; ζ2 ¼ δ4 þ δ5 þ δ2ð Þiξ; ζ3 ¼ − iξ 1þ a1ξ

2� �;

ζ4 ¼ a3iξ; ζ5 ¼ δ1 þ δ5ð Þiξ; ζ6 ¼ − δ2ξ2 −

ε0μ20H20

ρþ 1

� �s2; ζ7 ¼ δ3 þ δ5ð Þ;

ζ8 ¼ − 1þ a1ξ2� � β3

β1; ζ9 ¼

β3β1

a3; ζ10 ¼ 1þ Gð Þsδ6iξ½ �; ζ11 ¼ 1þ Gð Þsδ8;ζ12 ¼ − K1ξ

2 þ 1þ Gð Þδ7s 1þ a1ξ2� � �

; ζ13 ¼ K3 − 1þ Gð Þδ7sa3

The roots of Eq. (44) are ± λi, (i = 1, 2, 3). The solu-tion of Eqs. (41)–(43) satisfying the radiation conditionthat ðu; w; φÞ→0 as z→∞ can be written as

u ξ; z; sð Þ ¼X3

j¼1Aje

− λ jz; ð45Þ

w ξ; z; sð Þ ¼X3

j¼1d jAje

− λ jz; ð46Þ

φ ξ; z; sð Þ ¼X3

j¼1l jA je

− λ jz; ð47Þ

where Aj, j = 1, 2, 3 being undetermined constants and djand lj are given by

d j ¼δ2ζ13λ

4j þ ζ13ζ1 − ζ10ζ4 þ δ2ζ12ð Þλ2j þ ζ1ζ12 − ζ10ζ3

ζ13ζ7 − ζ1ζ9ð Þλ4j þ ζ13ζ6 þ ζ12ζ7 − ζ1ζ8ð Þλ2j þ ζ12ζ6;

ð48Þ

l j ¼δ2ζ7λ

4j þ δ2ζ6 þ ζ1ζ7 − δ2ζ5ð Þλ2j þ ζ6ζ1

ζ13ζ7 − ζ1ζ9ð Þλ4j þ ζ13ζ6 þ ζ12ζ7 − ζ1ζ8ð Þλ2j þ ζ12ζ6:

ð49Þ

t11 ξ; z; sð Þ ¼ iξu ξ; z; sð Þ þ δ4Dw ξ; z; sð Þ − β3β1

1þ a1ξ2 − a3D

2 �

φ ξ; z; sð Þ;

ð50Þ

t33 ξ; z; sð Þ ¼ δ4iξu ξ; z; sð Þ þ δ3Dw ξ; z; sð Þ − β3β1

1þ a1ξ2 − a3D

2 �

φ ξ; z; sð Þ;

ð51Þ

t13 ξ; z; sð Þ ¼ δ2 Du ξ; z; sð Þ þ iξw ξ; z; sð Þð Þ: ð52Þ

Thus, using (45)–(47) in Eqs. (50)–(52) gives

t11 ξ; z; sð Þ ¼ iξX3

j¼1Aje

− λ jz þ δ4DX3

j¼1d jAje

− λ jz

− 1þ a1ξ2 − a3D

2 �X3

j¼1l jAje

− λ jz;

ð53Þ

t33 ξ; z; sð Þ ¼ δ4iξX3

j¼1Aje

− λ jz þ δ3DX3

j¼1d jAje

− λ jz

−β3β1

1þ a1ξ2 − a3D

2 �X3

j¼1l jAje

− λ jz;

ð54Þ

t13 ξ; z; sð Þ ¼ δ2 DX3

j¼1Aje

− λ jz þ iξX3

j¼1d jAje

− λ jz� �

:

ð55Þ

which on further simplification gives

t11 ξ; z; sð Þ ¼ ð iξ − δ4λ1d1 − 1þ a1ξ2 − a3λ1

2 �l1

� �� �e − λ1zA1

þ iξ − δ4λ2d2 − 1þ a1ξ2 − a3λ2

2 �l2

� �� �e − λ2zA2

þ iξ − δ4λ3d3 − 1þ a1ξ2 − a3λ3

2 �l3

� �� �e − λ3zA3Þ;

ð56Þ

t33 ξ; z; sð Þ ¼ ðδ4iξ − δ3λ1d1 −β3β1

1þ a1ξ2 − a3λ1

2 �l1

� �� �e − λ1zA1

þ δ4iξ − δ3λ2d2 −β3β1

1þ a1ξ2 − a3λ2

2 �l2

� �� �e − λ2zA2

þ δ4iξ − δ3λ3d3 −β3β1

1þ a1ξ2 − a3λ3

2 �l3

� �� �e − λ3zA3Þ;

ð57Þ

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 5 of 13

t13 ξ; z; sð Þ ¼ − λ1 þ iξd1ð Þδ2A1e− λ1z þ − λ2 þ iξd2ð Þδ2A2e

− λ2z

þ − λ3 þ iξd3ð Þδ2A3e− λ3z:

ð58Þwhich can be further simplified as

t11 ξ; z; sð Þ ¼ S1A1 þ S2A2 þ S3A3ð Þ ð59Þt33 ξ; z; sð Þ ¼ M1A1 þM2A2 þM3A3ð Þ ð60Þt13 ξ; z; sð Þ ¼ N1A1 þ N2A2 þ N3A3ð Þ ð61Þ

where

S j ¼ iξ − δ4d jλ j − l j 1þ a1ξ2� �

− a3λ2j

h in oe − λ jz

ð62Þ

N j ¼ δ2 − λ j þ iξd j� �

e − λ jz; ð63Þ

Mj ¼ δ4iξ − δ3d jλ j −β3β1

l j 1þ a1ξ2� �

− a3λ2j

h i� �e − λ jz;

ð64Þ

Boundary conditionsThe appropriate boundary conditions for the thermallyinsulated boundaries z = 0 are

t33 x; z; tð Þ ¼ − F1ψ1 xð ÞH tð Þ; ð65Þt31 x; z; tð Þ ¼ − F2ψ2 xð ÞH tð Þ; ð66Þ∂φ∂z

x; z; tð Þ þ hφ ¼ 0: ð67Þ

where h→ 0 corresponds to thermally insulated surfaceand h→∞ corresponds to isothermal surface, F1 and F2are the magnitude of the forces applied and ψ1(x) andψ2(x) specify the vertical and horizontal load distributionfunction along x-axis.Applying dimensionless conditions (26) and suppress-

ing primes and then applying the Laplace and Fouriertransform defined by (30) and (33) on the boundary con-ditions (65)–(67) and using (47), (60) and (61), the threeequations in three variables A1, A2, A3 are

t33 ξ; 0; sð Þ ¼ M1A1 þM2A2 þM3A3 ¼ − F1ψ1 ξð Þs

;

ð68Þ

t13 ξ; 0; sð Þ ¼ N1A1 þ N2A2 þ N3A3ð Þ ¼ − F2ψ2 ξð Þs

;

ð69ÞDφ ξ; 0; sð Þ þ hφ ξ; 0; sð Þ ¼ R1A1 þ R2A2 þ R3A3ð Þ ¼ 0;

ð70Þ

where Rj = − λjlj + hlj, note at z = 0, e − λ jz ¼ 1; in (62–64)for values of S j;Mj;N j and D ¼ ∂

∂z :

Case I: Thermally insulated boundariesWhen h = 0 solving (68)–(70), the values of A1, A2, A3

are obtained as

A1 ¼ − F1ψ1 ξð ÞΛ11 þ F2ψ2 ξð ÞΛ21

Λs; ð71Þ

A2 ¼ F1ψ1 ξð ÞΛ12 − F2ψ2 ξð ÞΛ22

Λs; ð72Þ

A3 ¼ F1ψ1 ξð ÞΛ13 − F2ψ2 ξð ÞΛ23

Λs; ð73Þ

where

Λ11 ¼ −N2R3 þ R2N3;Λ12 ¼ N1R3 − R1N3;Λ13 ¼ −N1R2 þ R1N2;Λ21 ¼ M2R3 − R2M3;Λ22 ¼ −M1R3 þ R1M3;Λ23 ¼ M1R2 − R1M2;Λ ¼ −M1Λ11 −M2Λ12 −M3Λ13;

Rj ¼ − λ jl j:

The components of displacement, conductivetemperature, normal stress and tangential stress are ob-tained from (45) to (47) and (59) to (61) by putting thevalues of A1, A2, A3 from (71) to (73) as

u ¼ F1ψ1 ξð ÞsΛ

X3

i¼1Λ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1Λ2ie

− λizh i

;

ð74Þ

w ¼ F1ψ1 ξð ÞsΛ

X3

i¼1diΛ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1diΛ2ie

− λizh i

;

ð75Þ

φ ¼ F1ψ1 ξð ÞsΛ

X3

i¼1liΛ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1liΛ2ie

− λizh i

;

ð76Þ

ct11 ¼ F1ψ1 ξð ÞsΛ

X3

i¼1SiΛ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1SiΛ2ie

− λizh i

;

ð77Þ

ct13 ¼ F1ψ1 ξð ÞsΛ

X3

i¼1NiΛ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1NiΛ2ie

− λizh i

;

ð78Þ

ct33 ¼ F1ψ1 ξð ÞsΛ

X3

i¼1MiΛ1ie

− λizh i

þ F2ψ2 ξð ÞsΛ

X3

i¼1MiΛ2ie

− λizh i

;

ð79Þ

Case II: Isothermal boundariesWhen h→∞ ,solving (68)–(70), using Cramer’s rule, thevalues of A1, A2, A3 are obtained as

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 6 of 13

A1 ¼ − F1ψ1 ξð ÞΛ�11 þ F2ψ2 ξð ÞΛ�

21

Λ�s; ð80Þ

A2 ¼ F1ψ1 ξð ÞΛ�12 − F2ψ2 ξð ÞΛ�

22

Λ�s; ð81Þ

A3 ¼ F1ψ1 ξð ÞΛ�13 − F2ψ2 ξð ÞΛ�

23

Λ�s; ð82Þ

where

Λ�11 ¼ −N2R

�3 þ R�

2N3;Λ�

12 ¼ N1R�3 − R�

1N3;Λ�

13 ¼ −N1R�2 þ R�

1N2;Λ�

21 ¼ M2R�3 − R�

2M3;Λ�

22 ¼ −M1R�3 þ R�

1M3;Λ�

23 ¼ M1R�2 − R�

1M2;Λ� ¼ −M1Λ�

11 −M2Λ�12 −M3Λ�

13;

where R�j ¼ l j.

The components of displacement, conductivetemperature, normal stress and tangential stress are ob-tained from (45) to (47) and (59) to (61) by putting thevalues of A1, A2, A3 from (80) to (82) as

u ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1Λ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1Λ�

2ie− λiz

h i;

ð83Þ

w ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1liΛ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1diΛ2ie

− λizh i

;

ð84Þ

φ ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1liΛ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1liΛ2ie

− λizh i

;

ð85Þ

ct11 ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1SiΛ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1SiΛ�

2ie− λiz

h i;

ð86Þ

ct13 ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1NiΛ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1NiΛ�

2ie− λiz

h i;

ð87Þ

ct33 ¼ F1ψ1 ξð ÞsΛ�

X3

i¼1MiΛ�

1ie− λiz

h iþ F2ψ2 ξð Þ

s�X3

i¼1MiΛ�

2ie− λiz

h i;

ð88Þ

ApplicationsWe consider a normal line load F1 per unit length actingin the positive z-axis on the plane boundary z = 0 alongthe y-axis and a tangential load F2 per unit length, actingat the origin in the positive x-axis. Suppose an inclinedload, F0 per unit length is acting on the y-axis and its in-clination with z-axis is θ (see Fig. 1), we have

F1 ¼ F0cosθ and F2 ¼ F0sinθ: ð89Þ

Special casesConcentrated forceThe solution due to concentrated normal force on thehalf space is obtained by setting

ψ1 xð Þ ¼ δ xð Þ;ψ2 xð Þ ¼ δ xð Þ; ð90Þwhere δ(x) is Dirac delta function.Applying Fourier transform defined by (33) on (90)

yields

ψ1 ξð Þ ¼ 1; ψ2 ξð Þ ¼ 1: ð91ÞFor case I, using (91) in Eqs. (74)–(79) and for case II,

using (91) in Eqs. (83)–(88), the components of displace-ment, stress and conductive temperature are obtainedfor case I and case II, respectively.

Uniformly distributed forceThe solution due to uniformly distributed force appliedon the half space is obtained by setting

ψ1 xð Þ;ψ2 xð Þ ¼ 1 if j x j ≤m0 if j x j> m

�ð92Þ

The Fourier transforms of ψ1(x) and ψ2(x) with respectto the pair (x, ξ) for the case of a uniform strip load ofnon-dimensional width 2m applied at origin of co-ordinate system x = z = 0 in the dimensionless formafter suppressing the primes becomes

ψ1 ξð Þ ¼ ψ2 ξð Þ ¼ 2 sin ξmð Þξ

� �; ξ≠0: ð93Þ

Fig. 1 Inclined load over a transversely isotropicmagneto-thermoelastic solid

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 7 of 13

For case I, using (93) in Eqs. (74)–(79) and for case II,using (93) in Eqs. (83)–(88), the components of displace-ment, stress and conductive temperature are obtainedfor case I and case II.

Linearly distributed forceThe solution due to linearly distributed force applied onthe half space is obtained by setting

ψ1 xð Þ;ψ2 xð Þf g ¼ 1 −j x jm

if j x j ≤m0 if j x j> m

(ð94Þ

Here, 2 m is the width of the strip load, and applyingthe transform defined by (33) on (94), we get

ψ1 ξð Þ ¼ ψ2 ξð Þ ¼ 2 1 − cos ξmð Þð Þξ2m

� �; ξ≠0: ð95Þ

For case I, using (95) in Eqs. (74)–(79) and for case II,using (95) in Eqs. (83)–(88), the components of displace-ment, stress and conductive temperature are obtainedfor case I and case II, respectively.

Inversion of the transformationTo find the solution of the problem in physical domainand to invert the transforms in Eqs (74)–(79) and forcase II in Eqs. (83)–(88), here, the displacement compo-nents, normal and tangential stresses and conductivetemperature are functions of z, the parameters of La-place and Fourier transforms s and ξ respectively and

hence are of the form f ðξ; z; sÞ. To find the function ~f ðx; z; tÞ in the physical domain, we first invert the Fouriertransform using

~f x; z; sð Þ ¼ 12π

Z ∞

− ∞e − iξx f ξ; z; sð Þdξ

¼ 12π

Z ∞

− ∞cos ξxð Þ f e − isin ξxð Þ f oj jdξ; ð96Þ

where fe and fo are respectively the odd and even partsof f ðξ; z; sÞ: We obtain Fourier inverse transform by re-placing s by ω in (96). Following Honig and Hirdes

(1984), the Laplace transform function ~f ðx; z; sÞ can beinverted to f(x, z, t) for problem I by

f x; z; tð Þ ¼ 12πi

Zeþi∞

e − i∞

~f x; z; sð Þe − stds: ð97Þ

The last step is to calculate the integral in Eq. (97).The method for evaluating this integral is described inPress (1986).

Numerical results and discussionTo demonstrate the theoretical results and effect ofmemory-dependent derivatives, the physical data for co-balt material, which is transversely isotropic, is takenfrom Kumar et al. (2016) given as

C11 ¼ 3:07� 1011Nm − 2;C33

¼ 3:581� 1011Nm − 2;C13

¼ 1:027� 1010Nm − 2;C44

¼ 1:510� 1011Nm − 2; β1¼ 7:04� 106Nm − 2deg − 1; β3¼ 6:90� 106Nm − 2deg − 1; ρ¼ 8:836� 103Kgm − 3;CE

¼ 4:27� 102 jKg − 1deg − 1;K1

¼ 0:690� 102Wm − 1Kdeg − 1;K3

¼ 0:690� 102Wm − 1K − 1;K�1

¼ 1:313� 102Wsec;K �3 ¼ 1:54� 102Wsec;T0

¼ 298K;H0 ¼ 1Jm − 1nb − 1; ε0¼ 8:838� 10 − 12 Fm − 1; L ¼ 1:

The values of rotation Ω and magnetic effect H0

are taken as 0.5 and 10, respectively. The softwareFORTRAN has been used to determine the compo-nents of displacement, stress and conductivetemperature. A comparison has been made to showthe effect of kernel function of MDD on the variousquantities.

1. The solid line with square symbol represents a =0.0, b = 0.0, K(t − ξ) = 1,

2. The dashed line with circle symbol represents a¼ 0:0; b ¼ 1

2 ;Kðt − ξÞ ¼ 1þ ðξ − tÞ=χ,3. The dotted line with triangle symbol represents a

¼ 0:0; b ¼ 1χ ;Kðt − ξÞ ¼ ξ − t þ 1,

4. The dash-dotted line with diamond symbol repre-sents a = 1, b = 1, K(t − ξ) = [1 + (ξ − t)/χ]2

Figures 2, 3, 4, 5, 6 and 7 shows the variations of thedisplacement components (u and w), conductivetemperature φ and stress components ( t11, t13 and t33)for a transversely isotropic magneto-thermoelasticmedium with concentrated force and with combined ef-fects of rotation, two temperatures with different kernelfunction K(t − ξ), respectively. The displacement compo-nents (u and w), conductive temperature φ and stresscomponents ( t11 and t13) illustrate the opposite behav-iour for the kernel function a ¼ 0:0; b ¼ 1

χ ;Kðt − ξÞ¼ ξ − t þ 1 but with other kernel functions shows thesame pattern. However, stress component t33 illustratethe opposite behaviour for the kernel function [1 + (ξ− t)/χ]2 but with other kernel functions shows thesame pattern. These components vary (increases ordecreases) during the initial range of distance near

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 8 of 13

the loading surface of the inclined load and follow asmall oscillatory pattern for the rest of the range ofdistance.

ConclusionFrom the above study, the following is observed:

� Displacement components (u and w), conductivetemperature φ and stress components ( t11, t13 and

t33) for a transversely isotropic magneto-thermoelastic medium with concentrated force andwith combined effects of two-temperature model ofthermoelasticity with different kernel functions ofMDD, respectively

� In order to estimate the nature of the displacements,stresses and temperature distributions in the physicaldomain, an efficient approximate numerical inverseFourier and Laplace transform technique is adopted.

Fig. 2 Variation of the displacement component u with distance x

Fig. 3 Variation of the displacement component w with distance x

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 9 of 13

� Moreover, the magnetic effect of two temperatures,rotation, and the angle of inclination of the appliedload plays a key part in the deformation of all thephysical quantities.

� K(t − ξ) = [1 + (ξ − t)/χ]2 shows the more oscillatorynature for the displacement components and stresscomponents.

� The result gives the inspiration to study magneto-thermoelastic materials with memory-dependentderivatives as an innovative domain of applicablethermoelastic solids.

� The shape of curves shows the impact of Kernelfunction on the body and fulfils the purpose of thestudy.

Fig. 4 Variation of the conductive temperature φ with distance x

Fig. 5 Variation of the stress component t11with distance x

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 10 of 13

� The outcomes of this research are extremely helpfulin the 2-D problem with dynamic response ofmemory-dependent derivatives in transverselyisotropic magneto-thermoelastic medium with twotemperatures which is advantageous to successfulapplications of memory dependence in heatconductive sense.

Nomenclatureδij Kronecker deltaCijkl Elastic parametersβij Thermal elastic coupling tensorT Absolute temperatureT0 Reference temperatureφ conductive temperature

Fig. 6 Variation of the stress component t13with distance x

Fig. 7 Variation of the stress component t33with distance x

Kaur et al. International Journal of Mechanical and Materials Engineering (2020) 15:10 Page 11 of 13

tij Stress tensorseij Strain tensorsui Components of displacementρ Medium densityCE Specific heataij Two-temperature parametersαij Linear thermal expansion coefficientKij Thermal conductivityTij Maxwell stress tensorχ Time delayμ0 Magnetic permeabilityK(t − ξ) Kernel functionu! Displacement vector

H!

0 Magnetic field intensity vector

j!

Current density vectorFi Components of Lorentz forceτ0 Relaxation timeε0 Electric permeabilityδ(t) Dirac’s delta function

h!

Induced magnetic field vector

E!

Induced electric field vector

AcknowledgementsNot applicable.

Authors’ contributionsThe work is carried by the corresponding author under the guidance andsupervision of Dr. Parveen Lata and Dr. Kulvinder Singh. The author(s) readand approved the final manuscript.

FundingNo fund/grant/scholarship has been taken for the research work.

Availability of data and materialsFor the numerical results, cobalt material has been taken for thermoelasticmaterial from Kumar et al. (2016).

Competing interestsThe authors declare that they have no competing interests.

Author details1Department of Mathematics, Government College for Girls, Palwal,Kurukshetra, Haryana, India. 2Department of Basic and Applied Sciences,Punjabi University, Patiala, Punjab, India. 3Kurukshetra University, Kurukshetra,Haryana, India.

Received: 22 April 2020 Accepted: 4 November 2020

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