Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 49-58

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1 Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, THERMOELASTICITY WITH FRACTIONAL ORDER STRAIN FOR DIPOLAR MATERIALS WITH VOIDS Lavinia CODARCEA - MUNTEANU 1 and Marin MARIN 2 Abstract The goal of this article is to combine the theory based on fractional order of strain with the theory of thermoelasticity of dipolar bodies with voids, in order to obtain the basic thermoelasticity equations with fractional order strain for dipolar materials with voids Mathematics Subject Classification: 73B30, 73C35, 73C60. Key words: dipolar, voids, fractional order strain, thermoelasticity. 1 Introduction A theory of linear elastic materials with voids or vacuous pores was presented by S.C.Cowin and J.W Nunziato in [4], where they applied the theory to some problems of technological interest. In fact, this theory is a linearization of a theory described by them earlier in [14], where they presented a nonlinear theory for the behaviour of porous solids. They developed the first study on porous materials made by M.A.Goodman and S.C.Cowin in [5]. The presence of small pores (or voids) in the conventional continuum model is introduced by assigning an additional degree of freedom to each particle, namely the fraction of elementary volume that is possibly found void of mater. This additional degree of freedom is useful in order to develop the mechanical behaviour of porous solids in which the matrix material is elastic and the interstices are voids of material [11]. D. Ieşan has studied a linear theory of thermoelastic materials with voids in [9], treating, on the one hand, some general theorems (uniqueness, reciprocal and variational theorems) and, on the other hand, some problems of equilibrium. The first results on dipolar bodies theory, which is a part of a multipolar structures theory, were published by R.D.Mindlin in [13], A.E.Green and R.S.Rivlin in [6]. 1 Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania, codarcealavinia@unitbv.ro 2 Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania, m.marin@unitbv.ro

2 50 Lavinia Codarcea-Munteanu and Marin Marin In the theory of dipolar continua, each material point is constrained to deform homogeneously. In that case, the degrees of freedom for each particle are three translations and nine micro-deformations [12]. In [10], M.Marin applied the general results from the theory of elliptic equations, in order to obtain the existence and uniqueness of the generalized solutions for the boundary value problems in elasticity of dipolar materials with voids. H.M.Youssef has derived in [17] a new theory based on fractional order of strain (fraction order Duhamel-Neumann stress-strain relation). We combine this theory with the theory of thermoelasticity of dipolar bodies with voids, in order to obtain the basic equations of the thermoelasticity with fractional order strain for dipolar materials with voids. 2 Notations and basic equations Considering a thermoelastic, anisotropic dipolar material with voids, which occupies a properly regular region of the three - dimensional Euclidean space R 3, we suppose that is bounded by a C 1,1 boundary, and the closure of is noted by. The body s movement is reported to a fixed system of rectangular Cartesian axes Ox i (i = 1, 2, 3), and throughout this article we adopt the Cartesian tensor notation. The convention, according to which the material time derivative is represented by a superposed dot stand, and a comma followed by a subscript representing partial derivatives, is used in this paper. At the same time, on repeated indices we use the Einstein summation, and the time argument or the spatial argument of a function will be omitted when the confusion is ruled out. In the points are denoted by x i or x. The variable t is the time and t [0, t 0 ). The thermoelastic dipolar body with voids has a behaviour characterized by the following kinematic variables: u i = u i (x, t), Φ ij = Φ ij (x, t), ν = ν(x, t), (x, t) [0, t 0 ) where u = (u i ) i=1,3 is the displacement vector field, Φ = (Φ ij ) 1,j 3 is the dipolar tensor field and ν is the volume fraction field corresponding to voids. We have the following fundamental equations: the equations of motion: (t ji + η ji ),j + ρ 0 F i = ρ 0 ü i in (0, ) µ ijk,i + η jk + ρ 0 M jk = I ks Φjs in (0, ) (1) the equation of energy: ρ 0 T η = ρ 0 Q + q i,i (2)

3 Thermoelasticity with fractional order strain for dipolar materials with voids 51 the geometric equations: the balance of equilibrated forces: the initial conditions: 2ε ij = u i,j + u j,i γ ij = u j,i Φ ij χ ijk = Φ jk,i (3) σ i,i + ξ + ρ 0 L = ρ 0 k ν in (0, ) (4) u i (x, 0) = u 0 i (x) u i (x, 0) = u 1 i (x) Φ ij (x, 0) = Φ 0 ij (x) Φ ij (x, 0) = Φ 1 ij (x) ν(x, 0) = ν 0 (x) ν(x, 0) = ν 1 (x) θ(x, 0) = θ 0 (x) x (5) the boundary conditions: u i (x, t) = ũ i Φ ij (x, t) = Φ ij ν(x, t) = ν θ(x, t) = θ (x, t) [0, ) (6) where u 0 i, u1 i, Φ0 ij, Φ1 ij, ν0, ν 1, θ 0, ũ i, Φ ij, ν and θ are prescribed functions. We postulate the conservation of energy in the form: (ρ 0 u i ü i + I ks Φjk Φjs + ρ 0 k ν ν)dv + ρ 0 ėdv = = ρ 0 (F i u i + M jk Φjk + L ν + Q)dV + (t i u i + µ jk Φjk + σ ν + q)da. (7) The following notations were used in the above equations: ρ 0 is the constant reference density; t ij, η ij, µ ijk are the components of stress tensors; F i are the components of the body force per unit mass; M jk are the components of the dipolar body force per unit mass; k is the coefficient of equilibrated inertia; L is the extrinsic equilibrated body force per unit mass associated to voids; Q is the heat supply per unit mass; t i are the components of the stress vector; µ jk is the dipolar stress tensor; q is the heat flux vector; I ks are the coefficients of microinertia; σ is the equilibrated stress corresponding to ν.

4 52 Lavinia Codarcea-Munteanu and Marin Marin θ is the temperature measured from the constant absolute temperature T 0 of the body considered in the reference state. Using the Green and Rivlin method, we consider a second motion, different from the given motion only by a superposed constant rigid translation. All characteristics of the body remain unchanged for the new motion. The principle of conservation of energy and the other characteristics of the body are still valid if we replace u by u i + α i respectively Φ ij by Φ ij + β ij, where α i and β ij are arbitrary constants. Using the divergence theorem and considering the fact that constants α i and β ij are arbitrary, we obtain the relations: t i := (t ji + η ji )n j µ jk := µ ijk n i q := q i n i on (8) σ := σ i n i where n i are the components of the outward unit normal to the boundary surface. In this article we use the heat conduction equations introduced by Cattaneo, see [2],[15],[16] equations that take the next form: q i + τ 0 q i = K ij T,j, i = 1, 2, 3; (9) where K ij is the thermal conductvity tensor, and τ 0 is the relaxation time [7],[8]. The fractional derivative with respect to time, introduced by Caputo [1] and used in [15],[17], is defined as: D β t f(t) = 1 t Γ(1 β) 0 f (τ) dτ, 0 β 1 (10) (t τ) β where Γ is the Gamma function. Multiplying, following [3], the (1) 1 relation by u i, we obtain: then, we have: (t ji + η ji ),j u i + ρ 0 F i u i = ρ 0 ü i u i, (11) ρ 0 u i ü i dv = (t ji + η ji ),j u i dv + Multiplying, the (1) 2 relation by Φ jk, we get the relation: ρ 0 F i u i dv. (12) µ ijk,i Φjk + η jk Φjk + ρ 0 M jk Φjk = I ks Φjs Φjk. (13) Naturally, we deduce: I ks Φjs Φjk dv = (µ ijk,i Φjk + η jk Φjk + ρ 0 M jk Φjk )dv. (14)

5 Thermoelasticity with fractional order strain for dipolar materials with voids 53 Multiplying the (4) relation by ν we obtain: so, we can write: σ i,i ν + ξ ν + ρ 0 L ν = ρ 0 k ν ν (15) ρ 0 k ν νdv = (σ i,i ν + ξ ν + ρ 0 L ν)dv. (16) Introducing the relations: (8),(12),(14),(16) into (7) we have the following relations: + (t ji + η ji ),j u i dv + + σ i,i νdv + µ ijk Φjk n i da + ξ νdv = ρ 0 ėdv + µ ijk,i Φjk dv + η jk Φjk dv + ρ 0 QdV + (t ji + η ji )n j u i da+ σ i νn i da + q i n i da. Using the theorem of divergence, the previous relation becomes: ηjk Φjk dv + ξ νdv + ρ 0 ėdv = ρ 0 QdV + (t ji + η ji ) u i,j dv + µ ijk Φjk,i dv + σ i ν,i dv + q i,i dv. Rewriting the above relation, we get: (17) (18) [ρ 0 ė + η jk Φjk + ξ ν ρ 0 Q (t ji + η ji ) u i,j µ ijk Φjk,i σ i ν,i q i,i ]dv = 0. (19) Considering that is an arbitrary domain, we deduce from (19) the following equality: ρ 0 ė = (t ji + η ji ) u i,j + µ ijk Φjk,i η jk φjk + q i,i + ρ 0 Q ξ ν + σ i ν,i (20) The Helmholtz free energy Φ = e T η Φ = Φ( ε ij, γ ij, χ ijk, T, T,i, ν, ν,i ) e = e( ε ij, γ ij, χ ijk, T, T,i, ν, ν,i ) η = η( ε ij, γ ij, χ ijk, T, T,i, ν, ν,i ) q = q( ε ij, γ ij, χ ijk, T, T,i, ν, ν,i ). ( ) were ε ij = 1 + τ β D β t ε ij and τ is the mechanical relaxation parameter. (21)

6 54 Lavinia Codarcea-Munteanu and Marin Marin Replacing ė = Φ + T η + T η and u i,j by ε ij into relation (20) we have: ρ 0 Φ+ρ0 T η+ρ 0 T η = (t ji +η ji ) ε ij +µ ijk Φjk,i η jk Φjk +q i,i +ρ 0 Q ξ ν +σ i ν,i. (22) Using χ ijk = Φ jk,i ; γ ij = u j,i Φ ij the relation (22) becomes: ρ 0 Φ = tij εij + η ij γ ij + µ ijk χ ijk + q i,i + ρ 0 Q ξ ν + σ i ν,i ρ 0 T η ρ 0 T η. (23) From relation (21), we get: ρ 0 Φ =ρ0 εij + ρ 0 γ ij + ρ 0 χ ijk + ρ 0 T ε ij γ ij χ ijk T + + ρ 0 T,i + ρ 0 ν ν + ρ 0 ν,i. ν,i T,i (24) Comparing relations (23) and (24), we obtain t ij = ρ 0 ε ij η ij = ρ 0 γ ij µ ijk = ρ 0 χ ijk η = T (25) ξ = ρ 0 ν σ i = ρ 0 ν,i T,i = 0 and which is the energy equation in our case. q i,i + ρ 0 Q = ρ 0 T η (26)

7 Thermoelasticity with fractional order strain for dipolar materials with voids 55 The free energy is given by the relation below: ρ 0 Φ( ε ij, γ ij, χ ijk, θ, ν, ν,i ) = 1 2 C ijmn ε ij ε mn + G mnij ε mn γ ij B ijmnγ ij γ mn + F mnrij ε ij χ mnr + D mnijk γ mn χ ijk A ijkmnrχ ijk χ mnr + d ijm ε ij ν,m + e ijm γ ij ν,m + f ijkm χ ijk ν,m + (27) g imν,i ν,m + a ij ε ij ν + b ij γ ij ν + c ijk χ ijk ν + d i ν,i ν ων2 α ij ε ij θ β ij γ ij θ γ ijk χ ijk θ 1 2 aθ2 a i ν,i θ bνθ. The free energy expression has the constitutive coefficients as functions in C 1 () and they satisfy the following symmetry relations: C ijmn = C mnij = C jimn, B ijmn = B mnij, G ijmn = G ijnm, F ijkmn = F jikmn, A ijkmnr = A mnrijk, a ij = a ji, g im = g mi, α ij = α ji. Taking into account (25), we deduce the following constitutive equations of the linear theory of thermoelasticity of dipolar bodies with voids using fractional order strain: t ij = C ijmn ε mn + G ijmn γ mn + F mnrij χ mnr + d ijm ν,m + a ij ν α ij θ = = C ijmn (1 + τ β D β t )ε mn + G ijmn γ mn + F mnrij χ mnr + d ijm ν,m + a ij ν α ij θ η ij = B ijmn γ mn + D ijmnr χ mnr + G ijmn ε mn + e ijm ν,m + b ij ν β ij θ = = B ijmn γ mn + D ijmnr χ mnr + G ijmn (1 + τ β D β t )ε mn + e ijm ν,m + b ij ν β ij θ µ ijk = A ijkmnr χ mnr + D mnijk γ mn + F ijkmn ε mn + f ijkm ν,m + c ijk ν γ ijk θ = = A ijkmnr χ mnr + D mnijk γ mn + F ijkmn (1 + τ β D β t )ε mn + f ijkm ν,m + c ijk ν γ ijk θ ρ 0 η = α ij ε ij + β ij γ ij + γ ijk χ ijk + aθ + a i ν,i + bν = = α ij (1 + τ β D β t )ε ij + β ij γ ij + γ ijk χ ijk + aθ + a i ν,i + bν ξ = a ij ε ij b ij γ ij c ijk χ ijk d i ν,i ων + bθ = = a ij (1 + τ β D β t )ε ij b ij γ ij c ijk χ ijk d i ν,i ων + bθ σ i = d mni ε mn + e mni γ mn + f mnri χ mnr + g im ν,m + d i ν a i θ = = d mni (1 + τ β D β t )ε mn + e mni γ mn + f mnri χ mnr + g im ν,m + d i ν a i θ. Using relation (25) 4 into (26), we have: (28)

8 56 Lavinia Codarcea-Munteanu and Marin Marin ( q i,i = ρ 0 Q + ρ 0 T 2 Φ ε ij T ε ij 2 Φ γ ij T γ ij 2 Φ χ ijk T χ ijk 2 Φ T 2 T 2 Φ ν T ν 2 Φ ν,i T ν,i The relation (29) can be rewritten as: [ ( q i,i = ρ 0 Q T ρ 0 T ε ij + ( ) ρ 0 χ ijk ρ 0 T χ ijk T ( ) ρ 0 ν + T ν T ). ) ε ij + T ( ρ 0 ν,i ( T ( ρ 0 ) ν,i ] γ ij ) T. ) γ ij + (29) (30) Replacing relations (25) into (30), the last one takes the following form: ( tij q i,i = ρ 0 Q T T ε ij + η ij T γ ij + µ ijk T η ρ 0 T T ξ T ν + σ ) i T ν,i. χ ijk (31) Using the constitutive equations (28), the relation (31) becomes: q i,i = ρ 0 Q + α ij T ε ij + β ij T γ ij + γ ijk T χ ijk + at T + bt ν + a i T ν,i. (32) Let T T 0 for linearity, so: q i,i = ρ 0 Q + α ij T 0 (1 + τ β D β t ) ε ij + β ij T 0 γ ij + γ ijk T 0 χ ijk + at 0 T + bt 0 ν + a i T 0 ν,i. (33) Using (6), the relations of Cattaneo, we can deduce: so: q i,i + τ 0 q i,i = ( K ij T,j ),i i, j = 1, 2, 3 (34) ( K ij T,j ),i = ρ 0 Q + α ij T 0 (1 + τ β D β t ) ε ij + β ij T 0 γ ij + γ ijk T 0 χ ijk + + at 0 T + bt 0 ν + a i T 0 ν,i τ 0 [ρ 0 Q α ij T 0 (1 + τ β D β t ) ε ij ] β ij T 0 γ ij γ ijk T 0 χ ijk at 0 T bt0 ν a i T 0 ν,i. (35)

9 Thermoelasticity with fractional order strain for dipolar materials with voids 57 References [1] Caputo, M., Linear model of dissipation whose Q is almost frequency independent - II, Geophys. J. R. Astron. Soc. 13 (5) (1967), , tb02303.x. [2] Cattaneo, C., Sulla conduzione del calore, Atti. Sem. Mat. Fis. Univ. Modena 3 (1948), [3] Chirilă, A., Generalized micropolar thermoelasticty with fractional order strain, Bulletin of the Transilvania University of Braşov, Series III: Mathematics, Informatics, Physics 10 (59)(2017), no. 1, [4] Cowin, S. C. and Nunziato, J. W., Linear elastic materials with voids, Journal of Elasticity 13 (1983), [5] Goodman, M. A. and Cowin, S. C., A continuum theory for granular materials, Archive for Rational Mechanics and Analysis, 44 (1972), [6] Green, A. E. and Rivlin, R. S., Multipolar continuum mechanics, Archive for Rational Mechanics and Analysis, 17 (1964), [7] Hetnarski, R. B., Thermal Stresses IV, Elsevier, Amsterdam, [8] Hetnarski, R.B. and Ignaczak, J., Generalized thermoelasticity, Journal of Thermal Stresses, 22 (1999), [9] Iesan, D., A theory of thermoelastic materials with voids, Acta Mechanica 60 (1 2) (1986), [10] Marin, M., On weak solutions in elasticity of dipolar bodies with voids, Journal of Computational and Applied Mathematics 82 (1997), [11] Marin, M., Agarwal, R. P., Codarcea, L., A mathematical model for three-phaselag dipolar thermoelastic bodies, Journal of Inequalities and Applications (2017) 2017:109,1 16, [12] Marin, M., Nicaise, S., Existence and stability results for thermoelastic dipolar bodies with double porosity, Continuum Mechanics and Thermodynamics, 28 (6)(2016), , https: //doi.org/ /s [13] Mindlin, R. D., Microstructure in linear elasticity, Archive for Rational Mechanics and Analysis, 16 (1964), [14] Nunziato, J. W. and Cowin, S. C., A nonlinear theory of elastic materials with voids, Archive for Rational Mechanics and Analysis, 72 (1979), [15] Podlubny, I., Fractional Differential Equations : An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press,New York, [16] Povstenko, Y. Z., Thermoelasticity that uses fractional heat conduction equation, Journal of Mathematical Sciences, 162 (2)(2009), [17] Youssef, H. M., Theory of generalized thermoelasticity with fractional order strain, Journal of Vibration and Control 22 (18)(2015), ,

10 58 Lavinia Codarcea-Munteanu and Marin Marin