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ORIGINAL PAPER Open Access Axisymmetric deformation in transversely isotropic magneto-thermoelastic solid with GreenNaghdi III due to inclined load Iqbal Kaur * and Parveen Lata Abstract The axisymmetric problem in two-dimensional transversely isotropic magneto-thermoelastic (TIMT) solid due to inclined load with GreenNaghdi (GN)-III theory and two temperature (2T) has been studied. The Laplace and Hankel transform has been used to get the expressions of temperature distribution, displacement, and stress components with the horizontal distance in the physical domain. The effect of GreenNaghdi theories of type I, II, and III theories of thermoelasticity has been studied graphically on the resulting quantities. A special case for the magneto-thermoelastic isotropic medium has also been studied. Keywords: Transversely isotropic, Magneto-thermoelastic, Mechanical and thermal stresses, Axisymmetric deformation Introduction The study of deformation in a thermoelastic medium is one of the wide and dynamic domains of con- tinuum dynamics. It is well known that all the rotating large bodies have angular velocity, as well as magnetism; therefore, the thermoelastic interactions in a rotating medium under magnetic field are of importance. The study of thermoelasticity is beneficial to analyze the deformation field such as geothermal engineering, advanced aircraft structure design, ther- mal power plants, composite engineering, geology, high-energy particle accelerators, and many developing technologies. Eubanks and Sternberg (1954) discussed the axisymmet- ric issue of elasticity concept for a transversely isotropy medium. Vendhan and Archer (1978) electrostatically an- alyzed transversely isotropic (TI) finite stress-free cylin- ders with lateral surfaces using displacement potential. Green and Naghdi (1992, 1993) dealt with the linear and the nonlinear theories of the thermoelastic body with and without energy dissipation. Three new thermoelastic theories were proposed by them, based on entropy equa- lity. Their theories are known as GN-I, GN-II, and GN-III theories of thermoelasticity. On linearization, type I becomes the classical heat equation, whereas on linearization, type II as well as type III theories predicts the finite speed of thermal wave. Savruk (1994) discussed the axisymmetric deformation of a TI body containing cracks. Tarn et al. (2009) analyzed the axisymmetric and stress dispersal in a TI roundabout barrel-shaped body utilizing Hamiltonian variational definition through Legendres change. Liang and Wu (2012) discussed the axisymmetric deformation of one TI cylinder with the Lure method. Mahmoud (2012) considered the impact of relaxation times, the rotation, and the initial stress on Rayleigh waves. Shi et al. (2016) presented the thermomagnetoelectroelastic field in a heterogeneous annular multi-ferric composite plate with thermal loadings which is uniformly distributed on the boundaries. Kumar et al. (2016a, 2016b) studied the conflicts of thermomechanical sources in a TI homoge- neous thermoelastic rotating medium with magnetic effect as well as two temperature applied to the thermoelasticity GN-III theories. Li et al. (2016) presented a set of axisym- metric solutions of the thermoelastic field in a heteroge- neous circular plate which could either simply reinforced or fastened persuaded by the external thermal load. Including these, various researchers dealt with various theory of thermoelasticity such as Marin (Marin 1997a, © The Author(s). 2020 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. * Correspondence: [email protected] Department of Basic and Applied Sciences, Punjabi University, Patiala, Punjab, India Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 https://doi.org/10.1186/s40712-019-0111-8
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  • ORIGINAL PAPER Open Access

    Axisymmetric deformation in transverselyisotropic magneto-thermoelastic solid withGreen–Naghdi III due to inclined loadIqbal Kaur* and Parveen Lata

    Abstract

    The axisymmetric problem in two-dimensional transversely isotropic magneto-thermoelastic (TIMT) solid due toinclined load with Green–Naghdi (GN)-III theory and two temperature (2T) has been studied. The Laplace andHankel transform has been used to get the expressions of temperature distribution, displacement, and stresscomponents with the horizontal distance in the physical domain. The effect of Green–Naghdi theories of type I, II,and III theories of thermoelasticity has been studied graphically on the resulting quantities. A special case for themagneto-thermoelastic isotropic medium has also been studied.

    Keywords: Transversely isotropic, Magneto-thermoelastic, Mechanical and thermal stresses, Axisymmetricdeformation

    IntroductionThe study of deformation in a thermoelastic mediumis one of the wide and dynamic domains of con-tinuum dynamics. It is well known that all therotating large bodies have angular velocity, as well asmagnetism; therefore, the thermoelastic interactionsin a rotating medium under magnetic field are ofimportance. The study of thermoelasticity is beneficialto analyze the deformation field such as geothermalengineering, advanced aircraft structure design, ther-mal power plants, composite engineering, geology,high-energy particle accelerators, and many developingtechnologies.Eubanks and Sternberg (1954) discussed the axisymmet-

    ric issue of elasticity concept for a transversely isotropymedium. Vendhan and Archer (1978) electrostatically an-alyzed transversely isotropic (TI) finite stress-free cylin-ders with lateral surfaces using displacement potential.Green and Naghdi (1992, 1993) dealt with the linear andthe nonlinear theories of the thermoelastic body with andwithout energy dissipation. Three new thermoelastictheories were proposed by them, based on entropy equa-lity. Their theories are known as GN-I, GN-II, and GN-III

    theories of thermoelasticity. On linearization, type Ibecomes the classical heat equation, whereas onlinearization, type II as well as type III theories predictsthe finite speed of thermal wave. Savruk (1994) discussedthe axisymmetric deformation of a TI body containingcracks.Tarn et al. (2009) analyzed the axisymmetric and stress

    dispersal in a TI roundabout barrel-shaped body utilizingHamiltonian variational definition through Legendre’schange. Liang and Wu (2012) discussed the axisymmetricdeformation of one TI cylinder with the Lure method.Mahmoud (2012) considered the impact of relaxationtimes, the rotation, and the initial stress on Rayleigh waves.Shi et al. (2016) presented the thermomagnetoelectroelasticfield in a heterogeneous annular multi-ferric compositeplate with thermal loadings which is uniformly distributedon the boundaries. Kumar et al. (2016a, 2016b) studied theconflicts of thermomechanical sources in a TI homoge-neous thermoelastic rotating medium with magnetic effectas well as two temperature applied to the thermoelasticityGN-III theories. Li et al. (2016) presented a set of axisym-metric solutions of the thermoelastic field in a heteroge-neous circular plate which could either simply reinforcedor fastened persuaded by the external thermal load.Including these, various researchers dealt with various

    theory of thermoelasticity such as Marin (Marin 1997a,

    © The Author(s). 2020 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

    * Correspondence: [email protected] of Basic and Applied Sciences, Punjabi University, Patiala, Punjab,India

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 https://doi.org/10.1186/s40712-019-0111-8

    http://crossmark.crossref.org/dialog/?doi=10.1186/s40712-019-0111-8&domain=pdfhttp://orcid.org/0000-0002-2210-7701http://creativecommons.org/licenses/by/4.0/mailto:[email protected]

  • 1997b; Marin 1998; Marin 1999), Marin (Marin 2008;Marin 1997a, 1997b), Ezzat et al. (2012), Atwa (2014),Marin et al. (2013), Marin (2016), Marin and Baleanu(2016), Bijarnia and Singh (2016), Sharma et al. (Sharmaet al. 2015a, 2015b; 2016; 2017), Ezzat et al. (2017), Lata(2018), Lata et al. (2016), Marin and Öchsner (2017),Othman and Marin (2017), Ezzat et al. (2017), Chauthaleand Khobragade (2017), Kumar et al. (2017), Lata andKaur (2018), Shahani and Torki (2018), Lata and Kaur(2019a, 2019b, 2019c, 2019d), Kaur and Lata (2019a,2019b), Bhatti and Lu (2019a, 2019b), and Marin et al.(2019).Despite these, very less work has been done in ther-

    momechanical interactions in TIMT rotating solid withGN-III theory, with two temperature in the axisym-metric medium. Remembering these contemplations,analytic expressions for the displacement components,stress components, and temperature distribution in two-dimensional homogeneous, TIMT rotating solids withGN-III theories, with two temperature have beenderived.

    Basic equationsThe field equations with and without energy dissipation,without body forces and heat sources for an anisotropicthermoelastic medium following Lata and Kaur (2019d),are:

    ti j ¼ Ci jklekl − βi jT ; ð1Þ

    Ki jφ;i j þ K �i jφ:

    ;i j ¼ βi jT 0€ei j þ ρCE €T : ð2Þ

    and the equation of motion for a medium rotatinguniformly and Lorentz force is

    ti j; j þ Fi ¼ ρf€ui þ ðΩ� ðΩ� uÞi þ ð2Ω� u: Þig;

    ð3ÞwhereΩ =Ωn, n is a unit vector signifying the direction of

    the rotating axis.

    Fi ¼ μ0�j!� H!0

    �;

    T ¼ φ−aijφ;ij; ð4Þ

    βij ¼ Cijklαij; ð5Þ

    eij ¼ 12 ui; j þ uj;i� �

    :i ¼ 1; 2; 3 ð6Þ

    βij ¼ βiδij;Kij ¼ Kiδij;K �ij ¼ K�i δij; i is not summed

    HereCijkl having symmetry Cijkl ¼ Cklij ¼ Cjikl ¼ Cijlk� �

    :

    Method and formulation of the problemConsider a TIMT homogeneous medium with an initialtemperature T0. We consider a cylindrical polar coord-inate system (r, θ, z) with symmetry about the z-axis.For a plane axisymmetric problem, v = 0, and u, w, and φare independent of θ. Additionally, we take

    Ω ¼ 0;Ω; 0ð Þ:

    and

    J2 ¼ 0:

    The density components J1and J3 are given as

    J1 ¼ −ε0μ0H0∂2w∂t2

    ; ð7Þ

    J3 ¼ ε0μ0H0∂2u∂t2

    : ð8Þ

    Using the appropriate transformation followingSlaughter (2002) on Eqs. (1)–(3) to determine the condi-tions for TI thermoelastic solid with 2T and with andwithout energy dissipation, we get

    C11

    ∂2u∂r2

    þ 1r∂u∂r

    −1r2u

    !þ C13

    ∂2w∂r∂z

    !

    þ C44 ∂2u∂z2

    þ C44

    ∂2w∂r∂z

    !−β1

    ∂∂r

    (φ−a1

    ∂2φ∂r2

    þ 1r∂φ∂r

    !−a3

    ∂2φ∂z2

    )−μ0 J3H0

    ¼ ρ ∂2u∂t2

    −Ω2uþ 2Ω ∂w∂t

    !; ð9Þ

    ðC11 þ C44Þ

    ∂2u∂r∂z

    þ 1r∂u∂z

    !þ C44

    ∂2w∂r2

    þ 1r∂w∂r

    !þ C33 ∂

    2w∂z2

    −β3∂∂z

    (φ−a1ð∂

    ∂r2

    þ 1r∂φ∂r

    !−a3

    ∂2φ∂z2

    )þ μ0 J1H0

    ¼ ρ ∂2w∂t2

    −Ω2w−2Ω∂u∂t

    !; ð10Þ

    K1 þ K�1∂∂t

    � �∂2φ∂r2

    þ 1r∂φ∂r

    � �þ K3 þ K �3

    ∂∂t

    � �∂2φ∂z2

    ¼ T 0 ∂2

    ∂t2β1

    ∂u∂r

    þ β3∂w∂z

    � �þ ρCE ∂

    2

    ∂t2φ−a1

    ∂2φ∂r2

    þ 1r∂φ∂r

    � �−a3

    ∂2φ∂z2

    � :

    ð11Þ

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 2 of 9

  • Constitutive relations are

    trr ¼ c11err þ c12eθθ þ c13ezz−β1T ;tzr ¼ 2c44erz;

    tzz ¼ c13err þ c13eθθ þ c33ezz−β3T ;tθθ ¼ c12err þ c11eθθ þ c13ezz−β3T ;

    ð12Þ

    where

    erz ¼ 12∂u∂z

    þ ∂w∂r

    � �;

    err ¼ ∂u∂r ;eθθ ¼ ur ;

    ezz ¼ ∂w∂z ;

    T ¼ φ−a1 ∂2φ∂r2

    þ 1r∂φ∂r

    � �−a3

    ∂2φ∂z2

    ;

    β1 ¼ c11 þ c12ð Þα1 þ c13α3;β3 ¼ 2c13α1 þ c33α3:

    We consider that a primary medium is at rest. Therefore,the preliminary and symmetry conditions are assumed as

    u r; z; 0ð Þ ¼ 0 ¼ u: r; z; 0ð Þ;w r; z; 0ð Þ ¼ 0 ¼ w: r; z; 0ð Þ;

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

    To simplify the solution, the following dimensionlessquantities are introduced

    r0 ¼ r

    L; z

    0 ¼ zL; t

    0 ¼ c1Lt;u

    0 ¼ ρc21

    Lβ1T 0u;w

    0 ¼ ρc21

    Lβ1T 0w;

    T0 ¼ T

    T 0; t

    0zr ¼

    tzrβ1T 0

    ; t0zz ¼

    tzzβ1T0

    ;φ0 ¼ φ

    T 0; a

    01 ¼

    a1L2

    ;

    a03 ¼

    a3L2

    ; h0 ¼ h

    H0;Ω

    0 ¼ LC1

    Ω:

    ð13ÞApplying the dimensionless quantities introduced in

    (13) on Eqs. (9)–(11) and subsequently suppressing theprimes and using the following Laplace and Hankeltransforms

    f r; z; sð Þ ¼Z∞0

    f r; z; tð Þe−stdt; ð14Þ

    ~f ξ; z; sð Þ ¼Z ∞0

    f r; z; sð Þr Jn rξð Þdr: ð15Þ

    on the resulting quantities, we obtain

    −ξ2 þ δ2D2−s2δ7 þΩ2� �

    ~uþ δ1Dξ−2Ωsð Þ~wþ −ξ 1þ a1ξ2

    � �þ a3ξD2� �~φ ¼ 0;ð16Þ

    δ1Dξ þ 2Ωsð Þ~uþ δ3D2−δ2ξ2−s2δ7 þΩ2� �

    ~w

    −β3β1

    D 1þ ξ2a1� �

    −a3D2

    �� �~φ ¼ 0;

    ð17Þ

    −δ6s2ξ~u−β3β1

    δ6s2D~wþ�−δ8s2 1þ ξ2a1−a3D2

    � �−ξ2 K1 þ δ4sð Þ

    þD2 K 3 þ δ5sð ÞÞ~φ ¼ 0:ð18Þ

    where

    δ1 ¼ c13 þ c44c11 ; δ2 ¼c44c11

    ; δ3 ¼ c33c11 ; δ4 ¼K�1C1L

    ; δ5

    ¼ K�3C1L

    ; δ6 ¼ −T 0β21

    ρ; δ7 ¼ ε0μ

    20H

    20

    ρþ 1; δ8

    ¼ −ρCEC21:The non-trivial solution of (16)–(18) by eliminating ~u,

    ~w, and ~φ yields

    AD6 þ BD4 þ CD2 þ E ¼ 0 ð19Þwhere

    A ¼ δ2δ3ζ11−ζ9δ2ζ7;B ¼ δ2ζ3ζ11 þ δ3ζ1ζ11 þ δ2δ3ζ10−δ2ζ7ζ10−ζ7ζ1ζ9−ζ22ζ11þζ2ζ4ζ9 þ ζ8ζ7ζ2−δ3ζ4ζ8;

    C ¼ ζ1ζ3ζ11 þ δ2ζ10ζ5 þ δ3ζ3ζ10−ζ9ζ1ζ6−ζ22ζ10 þ ζ2ζ6ζ8þ ζ3ζ2ζ9−ζ3ζ8δ3−ζ4ζ8ζ5 þ 4Ω2s2ζ11;

    E ¼ ζ5ζ1ζ10−ζ8ζ5ζ3 þ 4Ω2s2ζ10:ζ1 ¼ −ξ2−s2δ7 þΩ2;ζ2 ¼ δ1ξ;ζ3 ¼ −ξ a1ξ2 þ 1

    � �;

    ζ4 ¼ a3ξ;ζ5 ¼ −δ2ξ2−s2δ7 þΩ2;ζ6 ¼ −

    β3β1

    1þ a1ξ2� �

    ;

    ζ7 ¼ a3β3β1

    ;

    ζ8 ¼ −ξδ6s2;ζ9 ¼ −

    β3β1

    δ6s2;

    ζ10 ¼ −δ8s2 1þ a1ξ2� �

    −ξ2 K1 þ δ4sð Þ;ζ11 ¼ K 3 þ δ5sð Þ þ δ8s2a3:

    The solutions of Eq. (19) can be written as

    ûðξ; z; sÞ ¼X3

    j¼1Aje−λ jz; ð20Þ

    ŵ ξ; z; sð Þ ¼X3

    j¼1d jAje−λ jz; ð21Þ

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 3 of 9

  • φ̂ ξ; z; sð Þ ¼X3j¼1

    l jA je−λ jz; ð22Þ

    where Aj being arbitrary constants, ±λj represents theroots of Eq. (19) and dj and lj are given by

    d j ¼δ2ζ11λ

    4j þ ðζ11ζ1−ζ4ζ8 þ δ2ζ10Þλ2j þ ζ1ζ10−ζ8ζ3

    ðδ3ζ11−ζ7ζ9Þλ4j þ ðδ3ζ10 þ ζ5ζ11−ζ9ζ6Þλ2j þ ζ5ζ10;

    l j ¼δ2δ3λ

    4j þ ðδ2ζ5 þ ζ1δ3−ζ22Þλ2j þ ζ1ζ5 þ 4Ω2s2

    ðδ3ζ11−ζ7ζ9Þλ4j þ ðδ3ζ10 þ ζ5ζ11−ζ9ζ6Þλ2j þ ζ5ζ10;

    ftzz ¼XAjðξ; sÞη je−λ jz; ð23Þetrz ¼XAj ξ; sð Þμ je−λ jz; ð24Þetrr ¼XAj ξ; sð ÞQje−λ jz; ð25Þ

    where

    η j ¼ δ11ξ−δ3λ jd j−β3β1

    1þ a1ξ2� �

    l j þ β3β1a3l jλ

    2j ; ð26Þ

    μ j ¼ δ2 −λ j þ ξd j� �

    ;

    Qj ¼ δ10 þ 1ð Þξ−δ11λ jd j−l j 1þ a1ξ2� �þ a3l jλ2j ;

    i; j ¼ 1; 2; 3:

    Boundary conditionsThe boundary conditions when normal force and tan-gential load are applied to the half-space (z = 0) are

    tzz r; z; tð Þ ¼ −F1ψ1 rð ÞH tð Þ; ð27Þtrz r; z; tð Þ ¼ −F2ψ2 rð ÞH tð Þ; ð28Þ∂φ r; z; tð Þ

    ∂zþ hφ r; z; tð Þ ¼ 0: ð29Þ

    where ψ1(r) and ψ2(r) are the vertical and the tangen-tial load applied on along the r-axis and

    HðtÞ ¼(

    1 t > 0

    0 t < 0XAj ξ; sð Þη j ¼ −F1ψ1 ξð Þ;XAj ξ; sð Þμ j ¼ −F2ψ2 ξð Þ;X

    Aj ξ; sð ÞP j ¼ 0: where;P j ¼ l j −λ j þ h� �

    :

    Solving Eqs. (27)–(29) with the aid of (20)–(25), weobtain

    u ¼ F1fψ1 ξð ÞΛ

    X3j¼1

    Λ1 jθ j

    " #þ F2fψ2 ξð Þ

    Λ

    X3j¼1

    Λ2 jθ j

    " #; ð30Þ

    ~w ¼ F1fψ1 ξð ÞΛ

    X3j¼1

    d jΛ1 jθ j

    " #þ F2fψ2 ξð Þ

    Λ

    X3j¼1

    d jΛ2 jθ j

    " #;

    ð31Þ

    ~φ ¼ F1fψ1 ξð ÞΛ

    X3j¼1

    l jΛ1 jθ j

    " #þ F2fψ2 ξð Þ

    Λ

    X3i¼1

    l jΛ2 jθ j

    " #;

    ð32Þ

    etrr ¼ F1fψ1 ξð ÞΛ X3j¼1 QjΛ1 jθ j" #

    þ F2fψ2 ξð ÞΛ

    X3j¼1

    QjΛ2 jθ j

    " #;

    ð33Þ

    etzr ¼ F1fψ1 ξð ÞΛ X3j¼1 μ jΛ1 jθ j" #

    þ F2fψ2 ξð ÞΛ

    X3j¼1

    μ jΛ2 jθ j

    " #;

    ð34Þ

    ftzz ¼ F1fψ1 ξð ÞΛ X3j¼1

    η jΛ1 jθ j

    " #

    þ F2fψ2 ξð ÞΛ

    X3j¼1

    η jΛ2 jθ j

    " #;

    ð35Þ

    Fig. 1 Geometry of the problem

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 4 of 9

  • where

    Λ11 ¼ −μ2P3 þ P2μ3;Λ12 ¼ μ1P3−P1μ3;Λ13 ¼ −μ1P2 þ P1μ2;Λ21 ¼ η2P3−P2η3Λ22 ¼ −η1P3 þ P1η3Λ23 ¼ η1P2−P1η2

    Λ ¼ −η1Λ11−η2Λ12−η3Λ13; j ¼ 1; 2; 3:

    Special casesConcentrated normal force (CNF)The CNF applied on the half-space is taken as

    ψ1ðrÞ ¼δðrÞ2πr

    ;ψ2ðrÞ ¼δðrÞ2πr

    : ð36Þ

    Applying Hankel transform, we get

    ψ̂1ðξÞ ¼1

    2πξ; ψ̂2ðξÞ ¼

    12πξ

    : ð37Þ

    The solution of Eqs. (30)–(35) with CNF is obtained using (37).

    Uniformly distributed force (UDF)Let a uniform force F1/constant temperature F2 beapplied over a uniform circular region of radius a.We obtained the solution with UDF applied on thehalf-space by taking

    ψ1ðrÞ ¼ ψ2ðrÞ ¼Hða−rÞπa2

    ; ð38Þ

    where H(a − r) is a Heaviside function. The Hankeltransforms of ψ1(r) and ψ2(r)are given by

    ψ̂1 ξð Þ ¼ ψ̂2 ξð Þ ¼J1 ξað Þ2πaξ

    � ; ξ≠0: ð39Þ

    The solution of Eqs. (30)–(35) with UDF is obtainedusing (39).

    Fig. 5 Deviations of trr with x (with CNF)

    Fig. 4 Deviations of φ with x (with CNF)

    Fig. 3 Deviations of w with x (with CNF)

    Fig. 2 Deviations of u with x (with CNF)

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 5 of 9

  • ApplicationsWe considered an inclined load (F0/unit length) ap-plied on a uniform circular region and its inclinationwith the z-axis is θ (see Fig. 1), we have

    F1 ¼ F0 cosθ and F2 ¼ F0 sinθ ð40Þ

    Using Eq. (40) in Eqs. (30)–(35) and with the aid of Eqs.(37) and (39), we obtain displacement components, stresscomponents, and conductive temperature with uniformlydistributed force and concentrated force on the surface ofTIMT body with and without energy dissipation.

    Particular cases

    a) If we take K�ij≠0, Eq. (2) is GN-III theory, andthus we obtain the solution of (30)–(35) forTIMT solid with rotation and GN-III theory.

    b) Equation (2) becomes GN-II theory if we takeK�ij ¼ 0, and we obtain the solution of (30)–(35) forTIMT solid with rotation and GN-II theory.

    c) If we take Kij = 0, the equation of GN-III theoryreduces to the GN-I theory, which is identical with

    the classical theory of thermoelasticity, and thus weobtain the solution of (30)–(35) for TIMT solidwith rotation and GN-I theory.

    d) If C11 ¼ C33 ¼ λþ 2μ;C12 ¼ C13 ¼ λ;C44 ¼ μ;α1 ¼ α3 ¼ α0 ; a1 ¼ a3 ¼ a; b1 ¼ b3 ¼ b;K1 ¼ K3 ¼ K ; K�1 ¼ K�3 ¼ K � , we obtain thesolution of (30)–(35) for TIMT materials withrotation and with GN-III theory.

    Inversion of the transformationTo obtain the solution to the problem in the physical domainfollowing Sharma et al. (2015a, 2015b), invert the transformsin Eqs. (30)–(35) by inverting the Hankel transform using

    f �ðr; z; sÞ ¼Z∞0

    ξ~f ðξ; z; sÞ JnðξrÞdξ: ð41Þ

    The integral in Eq. (41) is calculated using Romberg’sintegration through adaptive step size as defined in Presset al. (1986).

    Fig. 9 Deviations of w with x (with UDF)

    Fig. 8 Deviations of u with x (with UDF)

    Fig. 7 Deviations of tzz with x (with CNF)

    Fig. 6 Deviations of trz with x (with CNF)

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 6 of 9

  • Numerical results and discussionFor determining the theoretical results and influenceof GN-I, GN-II, and GN-III theories of thermoelasti-city, the physical data for cobalt material has beenconsidered from Dhaliwal and Singh (1980) as

    c11 ¼ 3:07� 1011 N m−2; c33 ¼ 3:581� 1011N m−2; c13 ¼ 1:027�1010 N m−2; c44 ¼ 1:510� 1011 N m−2; β1 ¼ 7:04�106N m−2 deg−1; β3 ¼ 6:90� 106N m−2 deg−1; ρ ¼ 8:836�103kg m−3; CE ¼ 4:27� 102jkg−1 deg−1;K1 ¼ 0:690�102W m−1 deg−1;K3 ¼ 0:690� 102W m−1 K−1;T0¼ 298 K; H0 ¼ 1 J m−1nb−1; ε0 ¼ 8:838� 10−12 F m−1; L¼ 1; and Ω ¼ 0:5:

    Using these values, the graphical illustrations of dis-placement components (u and w), conductivetemperature φ, normal force stress tzz, tangential stresstzr, and radial stress trr,for a TIMT solid with GN-IIItheory and with 2T due to inclined load, have been il-lustrated. The numerical calculations have been obtainedby developing a FORTRAN program using the abovevalues for cobalt material.

    i. The black line with a square symbol relates toK�ij≠0 for TIMT solid with rotation and GN-IIItheory.

    ii. The red line with a circle symbol relates to K �ij ¼ 0for TIMT solid with rotation and GN-II theory.

    iii. The blue line with a triangle symbol relates toKij = 0 the GN theory of type I.

    Case 1: Concentrated normal forceFigures 2, 3, 4, 5, 6, and 7 illustrate the deviations of u andw, conductive temperature φ, and trr, trz, and tzz for a TIMTmedium with concentrated normal force, with rotation, anddue to inclined load. From the graph, we find that that the dis-placement component (u) and conductive temperature φ de-creases while stress components ( trr, trz, and tzz) anddisplacement component (w) show a sharp increase and thenan oscillatory pattern with an amplitude difference.

    Case II: Uniformly distributed force (UDF)Figures 8, 9, 10, 11, 12, and 13 illustrate the deviations ofu and w, conductive temperature φ, and trr, trz, and tzz fora TIMT medium with UDF and with rotation, and due to

    Fig. 13 Deviations of tzzwith x (with UDF)

    Fig. 12 Deviations of the trzwith x (with UDF)

    Fig. 11 Deviations of trr with x (with UDF)

    Fig. 10 Deviations of φ with x (with UDF)

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 7 of 9

  • inclined load. The displacement components (u), stresscomponents ( trr, trz, and tzz), and temperature φ decreasesharply and then show the small oscillatory pattern, whilethe displacement component (w) first increases during theinitial range of distance near the loading surface and fol-lows the small oscillatory pattern for the rest of thevalues of distance.

    ConclusionsIn the above research, we conclude:

    � The components of displacement, stress, andtemperature distribution for TIMT solid withGN-III theory, with 2T with inclined load, arecalculated numerically.

    � The study motivates to consider magneto-thermoelastic materials as an inventive field of ther-moelastic solids. The shape of curves demonstratesthe effect of various GN theories and rotation on thebody and fulfills the purpose of the study.

    � The outcomes of this research are extremelyhelpful in the 2-D problem with dynamicresponse of inclined load in TIMT medium withrotation which is beneficial to detect thedeformation field such as geothermal engineering,advanced aircraft structure design, thermal powerplants, composite engineering, geology, high-energy particle accelerators, geophysics, auditoryrange, and geomagnetism. The proposed model issignificant to different problems in thermoelasti-city and thermodynamics.

    Nomenclature

    Symbol Name of Symbol SI Unit Symbol Name ofSymbol

    SI Unit

    δij Kronecker delta, ω Frequency Hz

    Cijkl Elasticparameters,

    Nm−2 βij Thermal elasticcouplingtensor,

    Nm−2K−1

    τ0 Relaxation Time s Ω AngularVelocity of theSolid

    s−1

    Fi Components ofLorentz force

    N T Absolutetemperature,

    K

    H!

    0Magnetic fieldintensity vector

    Jm−1nb−1 eij Strain tensors, Nm−2

    φ conductivetemperature,

    Wm−1K−1 j! Current Density

    VectorAm−2

    tij Stress tensors, Nm−2

    u! DisplacementVector

    m

    μ0 Magneticpermeability

    Hm−1 T0 Referencetemperature,

    K

    ui Components ofdisplacement,

    m ε0 Electricpermeability

    Fm−1

    (Continued)

    Symbol Name of Symbol SI Unit Symbol Name ofSymbol

    SI Unit

    ρ Medium density, Kgm−3 δ(t) Dirac’s deltafunction

    CE Specific heat, JKg−1K−1 Kij Materialistic

    constant,Wm−1K−1

    αij Linear thermalexpansioncoefficient,

    K−1 K�ij Thermalconductivity,

    Ns−2K−1

    H(t) Heaviside unitstep function

    F1, F2 magnitude ofapplied forces

    N

    AcknowledgementsNA

    Authors’ contributionsThe research work is done by IK under the supervision of PL. Both authorsread and approved the final manuscript.

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

    Availability of data and materialsFor the numerical outcomes, the physical data of cobalt material has beenconsidered from Dhaliwal and Singh (1980).

    Competing interestsThe authors declare that they have no competing interests.

    Received: 6 September 2019 Accepted: 31 October 2019

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    Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

    Kaur and Lata International Journal of Mechanical and Materials Engineering (2020) 15:3 Page 9 of 9

    AbstractIntroductionBasic equationsMethod and formulation of the problemBoundary conditionsSpecial casesConcentrated normal force (CNF)Uniformly distributed force (UDF)

    ApplicationsParticular casesInversion of the transformationNumerical results and discussionCase 1: Concentrated normal forceCase II: Uniformly distributed force (UDF)

    ConclusionsNomenclature

    AcknowledgementsAuthors’ contributionsFundingAvailability of data and materialsCompeting interestsReferencesPublisher’s Note


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