Effect of internal state variables in thermoelasticity of microstretch bodies

Transcription

1 DOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 43,016, Effect of internal state variables in thermoelasticity of microstretch bodies Marin Marin and Sorin Vlase Abstract First, we formulate the mixed initial boundary value problem in the theory of thermoelastic microstretch bodies having certain internal state variables. Then by using some approachable computing techniques and the known Gronwall s inequality we will prove that the presence of internal state variables do not influence the uniqueness of solution of the mixed problem. The purpose of the theory of thermo-microstretch elastic solids is to eliminate discrepancies between classical elasticity and experiments. This is a theory of thermoelasticity with microstructure that include intrinsic rotations and microstructural expansion and contractions. The classical elasticity failed to present acceptable results when the effects of material microstructure were known to contribute significantly to the body s overall deformations, for example, in the case of granular bodies with large molecules e.g. polymers, graphite or human bones. These cases are becoming increasingly important in the design and manufacture of modern day advanced materials, as small-scale effects become paramount in the prediction of the overall mechanical behaviour of these materials. Key Words: thermoelasticity, microstretch body, internal state variables, uniqueness of solution, Gronwall s inequality. 010 Mathematics Subject Classification: Primary 35A5, 35G46, 74A60; Secondary 74H5, 80A0. Received: Revised: Accepted:

2 MICROSTRETCH ODIES 4 Other intended applications of this theory are to composite materials reinforced with chopped fibers and various porous materials. We must outline that the theory of thermo-microstretch elastic solids was first elaborated by Eringen, [4]. This theory can be useful in the applications which deal with porous materials as geological materials, solid packed granular materials and many others. On the other hand, materials which operate at elevated temperatures will invarianbly be subjected to heat flow at some time during normal use. Such heat flow will involve a non-linear temperature distribution which will inevitable give rise to thermal stresses. For these reasons, the development, design and selection of materials for high temperature applications requires a great deal of care. The role of the pertinent material properties and other variables which can affect the magnitude of thermal stress must be considered. The main difficulty of the thermo-microstretch materials is the large number of the thermoelastic coefficients and, as such, the problem of their determination in the laboratory. Yet many authors consider that this problem will be solved in future. Already, in the isotropic case, when strongly decreases the number of coefficients, they are calculated as can be seen in many works due to Eringen [4] or Iesan and Quintanilla [8]. The present paper must be considered as a first step to a better understanding of microstretch and thermal stress in the study of above ennumerated materials. Some considerations on the propagation of plane waves in a microstretch thermoelastic diffusion solid of infinite extent can be find in the paper [9]. The Newton interpolating series proposed in [6] can be used successfully to approximate the solution of boundary value problems. The impact of initial mechanical deformation on the propagation of plane waves in a linear elastic isotropic solid, is shown in [7], while the algorithm proposed in [5] may be useful in addressing the solutions of mixed initial boundary value problem in the context of microstretch bodies. Different considerations are given to the solution of the mixed problem in thermoelasticity of microstretch bodies as in papers [10]-[15]. Interest to consider the internal state variables as a means to estimate mechanical properties has grown rapidly in recent years. The theories of internal state variables in different kind of materials represent a material length scale and are quite sufficient for a large number of the solid mechanics applications. The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, n, is

3 MICROSTRETCH ODIES 43 usually equal to the order of the system s defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function s denominator after it has been reduced to a proper fraction. It is important to understand that converting a state space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. The theory of bodies with internal state variables has been first formulated for the thermo-viscoelastic materials see, for instance Chirita [3]. Then the internal state variables has been considered for different kind of materials. The study [16] of Nachlinger and Nunziato is dedicated to the internal state variables approach of finite deformations without heat conduction in the one-dimensional case. In the paper [17] the authors describe how the so-called ammann internal state variable constitutive approach, which has proven highly successful in modelling deformation processes in metals, can be applied with great benefit to silicate rocks and other geological materials in modelling their deformation dynamics. In its essence, the internal state variables theory provides a constitutive framework to account for changing history states that arise from inelastic dissipative microstructural evolution of a polycrystalline solid. A thermodynamically consistent framework is proposed for modeling the hysteresis of capillarity in partially saturated porous media in the paper [19]. The paper [] presents the formulation of a constitutive model for amorphous thermoplastics using a thermodynamic approach with physically motivated internal state variables. The formulation follows current internal state variable methodologies used for metals and departs from the spring-dashpot representation generally used to characterize the mechanical behavior of polymers. Anand and Gurtin develop in the paper [1] a continuum theory for the elastic-viscoplastic deformation of amorphous solids such as polymeric and metallic glasses. Introducing an internal-state variable that represents the local free-volume associated with certain metastable states, the authors are able to capture the highly non-linear stress-strain behavior that precedes the yield-peak and gives rise to post-yield strain softening. In the study [18], is presented a formulation of state variable based gradient theory to model damage evolution and alleviate numerical instability associated within the post-bifurcation regime. This proposed theory is developed using basic microforce balance laws and appropriate state variables within a consistent thermodynamic framework. The theory provides a strong coupling and consistent framework to prescribe energy storage and dissipation

4 MICROSTRETCH ODIES 44 associated with internal damage.. asic equations We consider thermoelastic microstretch body with internal state variables which occupy an open region of three-dimensional Euclidean space R 3, at time t = 0, in it s reference configuration. The boundary of the domain, denoted by, is a closed, bounded and piece-wise smooth surface which allows us the application of the divergence theorem. A fixed system of rectangular Cartesian axes is used and we adopt the Cartesian tensor notations. The points in are denoted by x i or x. The variable t is the time and t [0, t 0. We shall employ the usual summation over repeated subscripts while subsripts preceded by a comma denote the partial differentiation with respect to the spatial argument. Also, we use a superposed dot to denote the partial differentiation with respect to t. The Latin indices are understood to range over the integers 1,, 3, while the Greek subsripts have the range 1,,..., n. In the following we designate by n i the components of the outward unit normal to the surface. The closure of the domain, denoted by, means =. Also, the spatial argument and the time argument of a function will be ommited when there is no likelihood of confusion. The behaviour of a thermoelastic microstretch body is characterized by the following kinematic variables: u i = u i x, t, ϕ i = ϕ i x, t, x, t [0, t 0 where u i are the components of the displacement field and ϕ i - the components of the microrotation vector. The fundamental system of field equations, in the theory of microstretch thermoelastic bodies with internal state variables, consists of: - the equations of motion: - the energy equation: t ij, j + ϱf i = ϱü i, m ji, j + ε ijk t jk + ϱg i = I ij ϕ j ; 1 T 0 η = q i, i + ϱr;

5 MICROSTRETCH ODIES 45 - the constitutive equations: t ij = A ijmn ε mn + mnij γ mn A ij θ + a ijα ω α, m ij = ijmn ε mn + C ijmn γ mn ij θ + b ijα ω α, η = A ij ε ij + ij γ ij a θ g α ω α, 3 q i = C ijk ε jk + D ijk γ jk + d i θ + f iα ω α + K ij θ, j ; - the geometric equations: ε ij = u j, i + ε ijk ϕ k, γ ij = ϕ j, i. 4 The internal state variables are denoted by ξ α, α = 1,,..., n, but in the linear theory, we denote by ω α the internal state variables measured from the internal state variables ξ 0 α of the initial state. Also, the temperature θ represents the difference between the absolute temperature T and the temperature T 0, T 0 > 0, of the initial state. Thus we have: ξ α = ξ 0 α + ω α, T = T 0 + θ. 5 Within the linear approximation, from the entropy production inequality, it follows see, for instance, [1]: where ω α = f α, 6 f α = g ijα ε ij + h ijα γ ij + p α θ + q αβ ω β + r iα θ, i. 7 In the above relations we have used the following notations: - ϱ - the constant mass density; - t ij, ma ij - the components of the stress tensors; - I ij - the coefficients of inertia; - F i - the components of body force per unit mass; - G i - the components of body couple per unit mass; - r - the heat supply per unit mass and unit time; - η - the entropy per unit mass; - q i - the components of the heat flux; - ε ij, γ ij - the kinematic characteristics of the strain tensors. The above coefficients A ijmn, ijmn,..., A ij,..., C ijk,..., g ijα,..., r iα are functions of x and characterize the thermoelastic properties of the material with internal state variable the constitutive coefficients. K ij is the conductivity tensor.

6 MICROSTRETCH ODIES 46 In the case of a homogeneous medium these quantities are constants. The constitutive coefficients obey to the following symmetry relations A ijmn = A mnij, ijmn = mnij, A ij = A ji, C ijmn = C mnij, a ijα = a jiα, K ij = K ji. 8 We supplement the above equations with the following initial conditions u i x s, 0 = u 0 i x s, u i x s, 0 = u 1 i x s, ϕ i x s, 0 = ϕ 0 i x s, ϕ i x s, 0 = ϕ 1 i x s, 9 θ x s, 0 = θ 0 x s, ω α x s, 0 = ω 0 α x s, x s and the prescribed boundary conditions u i = ũ i, on 1 [0, t 0 ], t i t ij n j = t i, on c 1 [0, t 0 ], ϕ i = ϕ i, on [0, t 0 ], m i m ij n j = m i, on c [0, t 0 ], 10 θ = θ, on 3 [0, t 0 ], q q i n i = q, on c 3 [0, t 0 ]. In 10 the surfaces 1,, 3 and respective complements c 1, c, c 3 are subsets of the boundary which satisfay the relations 1 c 1 = c = 3 c 3 = 1 c = c = 3 c 3 = The functions u 0 i, u1 i, ϕ0 i, ϕ1 i, θ0 ω 0 α, ũ i, t i, ϕ i, m i, θ and q, used in the above conditions 9 and 10, are assumed be prescribed in their domain of definition. In conclusion, the mixed initial boundary value problem of the thermoelasticity of microstretch bodies with internal variables consists of the equations 1, and 6, the initial conditions 9 and the boundary conditions 10. y a solution of this problem we mean an ordered array u i, ϕ i, θ, ω α satisfying the Eqns. 1, and 6 and the conditions 9 and Main results In this section we shall deduce some estimations and then, as a consequence, we obtain in simple manner the uniqueness theorem of the solution of the above problem. In order to prove these results, we shall need the following assumptions - i the mass density ϱ is strictly positive, i.e. ϱ x s ϱ 0 > 0, on ;

7 MICROSTRETCH ODIES 47 - ii there exists a positive constant λ 1 such that I ij ξ i ξ j λ 1 ξ i ξ i, ξ i ; - iii the specific heat a from 3 4 is strictly positive, i.e. a x s a 0 > 0, on ; - iv the constitutive tensors A ijmn, ijmn and C ijmn are positive definite, in the following sense: A ijmn ξ ij ξ mn dv λ ξ ij ξ ij dv, ξ ij ijmn ξ ij ξ mn dv λ 3 ξ ij ξ ij dv, ξ ij C ijmn ξ ij ξ mn dv λ 4 ξ ij ξ ij dv, ξ ij where λ, λ 3 and λ 4 are positive constants; - v the symmetric part K ij of the thermal conductivity tensor K ij is positive definite, in the sense that there exists a positive constant µ such that K ij ξ i ξ j dv µ ξ i ξ i dv, for all vectors ξ i. Consider that our mixed problem has two solutions u ν i, ϕ ν i, θ ν, ω α ν, ν = 1, two solutions of our initial boundary value problem. ecause of the linearity of the problem, their difference is also solution of the problem. We denote by v i, ψ i, ϑ, w α the differences, v i = u i u 1 i, ψ i = ϕ i ϕ 1 i, ϑ = θ θ 1, w α = ω α ω α 1 In order to prove the desired uniquness theorem, it suffice to prove that the above considered problem, consists of the equations 1, and 6 and the conditions 9 and 10, in which F i = G i = r = 0 u 0 i = u 1 i = ϕ 0 i = ϕ 1 i = θ 0 = ω 0 α = 0 and ũ i = t i = ϕ i = m ij = θ = q = 0

8 MICROSTRETCH ODIES 48 imply that v i = ψ i = ϑ = w α = 0, in [0, t 0 ], provided that the hypotheses i - v hold. So, we have a new problem, let say P 0, defined by the following equations t ij, j = ϱü i, m ij, j + ε ijk t jk = I ij ϕ j 11 T 0 η = q i, i 1 with the initial conditions u i x s, 0 = 0, u i x s, 0 = 0, ϕ i x s, 0 = 0, ω α = f α, 13 ϕ i x s, 0 = 0, θ x s, 0 = 0, ω α x s, 0 = 0, x s 14 and the boundary conditions u i = 0, on 1 [0, t 0 ], t i t ij n j = 0, on c 1 [0, t 0 ], ϕ i = 0, on [0, t 0 ], m i m ij n j = 0, on c [0, t 0 ], 15 θ = 0, on 3 [0, t 0 ], q q i n i = 0, on c 3 [0, t 0 ]. Together with these equations and conditions we take into account the constitutive relations 3 and 7. In order to prove that the problem P 0 admits the null solution, we will show that the function yt defined by yt = ui u i + ϕ i ϕ i + ε ij ε ij + γ ij γ ij + θ + ω α ω α dv vanishes on [0, t 0 ]. To this aim, we first prove some useful estimations. Theorem 1. If the ordered array u i, ϕ i, θ, ω α is a solution of the problem P 0, then the following relation holds 1 A ijmn ε ij ε mn + ijmn ε ij γ mn + ijmn γ ij γ mn + +a ijα ε ij ω α + b ijα γ ij ω α + aθ + ϱ u i u i + I ij ϕ i ϕ j dv = 16 t ] = a ijα ε ij ω α + b ijα γ ij ω α + g α θ ω α 1T0 q i θ, i dv ds. 0

9 MICROSTRETCH ODIES 49 Proof. In view of constitutive equations 3 and the symmetry relations 8, we obtain t ij ε ij + m ij γ ij + ηθ = +A ijmn ε ij ε mn + mnij ε mn ε ij A ij ε ij θ + a ijα ε ij ω α + + ijmn ε mn γ ij + C mnij γ mn γ ij ij γ ij θ + b ijα γ ij ω α + +A ij ε ij θ + ij γ ij θ + a θθ g α ω α θ = 17 = 1 A ijmn ε ij ε mn + mnij ε ij γ mn + C ijmn γ ij γ mn + aθ + t +a ijα ε ij ω α + b ijα γ ij ω α a ijα ε ij ω α b ijα γ ij ω α g α θ ω α. On the other hand, in view of equations 11, 1 and the geometric relations 4, we are lead to the relation t ij ε ij + m ij γ ij + ηθ = t ij u j, i + ε ijk ϕ k + m ij ϕ j, i + 1 T 0 q i, iθ = = t ij u j, i t ij, i u j +ε ijk ϕ k t ij +m ij ϕ j, i m ij, i ϕ j + = t ij u j + m ij ϕ j + 1 q iθ 1 T 0, i t From the equalities 17 and 18 we obtain 1 T 0 q iθ 1 q iθ, i 18, i T 0 ϱ u i u i + I ij ϕ i ϕ j 1 T 0 q iθ, i 1 t A ijmnε ij ε mn + mnij ε ij γ mn + C ijmn γ ij γ mn + +a ijα ε ij ω α + b ijα γ ij ω α + aθ + ϱ u i u i + I ij ϕ i ϕ j = 19 = t ij u j + m ij ϕ j + 1 q i θ 1 q i θ, i + T 0 T 0, i +a ijα ε ij ω α + b ijα γ ij ω α + g α θ ω α Now, we integrate relation 19 over the domain. y using the divergence theorem and the boundary conditions 15, we conclude that 1 A ijmn ε ij ε mn + mnij ε ij γ mn + C ijmn γ ij γ mn + t +a ijα ε ij ω α + b ijα γ ij ω α + aθ + ϱ u i u i + I ij ϕ i ϕ j dv = 0 = a ijα ε ij ω α + b ijα γ ij ω α + g α θ ω α 1T0 q i θ, i dv. Finally, we integrate the equality 0 on [0, t] and, by using the null initial condition 14, we arrive at the desired result 16.

10 MICROSTRETCH ODIES 50 Theorem. Let u i, ϕ i, θ, ω α be a solution of the problem P 0. Then there exists the positive constants m 1 and m such that the following relation hold ] [a ijα ε ij + b ijα γ ij + g α θ ω α 1T0 q i θ, i dv m 1 θ, i θ, j dv + m εij ε ij + γ ij γ ij + θ + ω α ω α dv. 1 Proof. Taking into account the relations 6, 7 and 3 4, we can write: a ijα ε ij ω α + b ijα γ ij ω α + g α θ ω α 1 T 0 q i θ, i = =a ijα ε ij +b ijα γ ij +g α θ g mnα ε mn +h mnα γ mn +p α θ+q αβ ω β +r kα θ, k 1 T 0 C kij ε ij + D kij γ ij + d k θ + f kα ω α + K kj θ, j θ, k = = 1 a ijαg mnα +a mnα g ijα ε ij ε mn +a ijα h mnα +b mnα g ijα ε ij γ mn + 1 b ijαh mnα +b mnα h ijα γ ij γ mn +a ijα p α +g ijα g α ε ij θ+a ijα q αβ ε ij ω β + b ijα p α + h ijα g α γ ij θ + a ijα r kα 1T0 C kij ε ij θ, k + g β q βα θω α + +b ijα q αβ γ ij ω β + b ijα r kα 1T0 D kij γ ij θ, k + g α p α θ + + g α r iα 1T0 d i θθ, i 1 f iα ω α θ, i 1 K ij θ, i θ, j T 0 T 0 For the sake of simplicity we introduce the following notations A ijmn = 1 a ijαg mnα + a mnα g ijα, ijmn = a ijα h mnα + b mnα g ijα, C ijmn = 1 b ijαh mnα + b mnα h ijα, D ij = a ijα p α + g ijα g α, E ij = b ijα p α + h ijα g α, ijα = a ijβ q βα, D ijα = b ijβ q βα, 3 A ijk = a ijα r kα 1 T 0 C kij, ijk = b ijα r kα 1 T 0 D kij, M = g α p α, L α = G β q βα, D i = g α r iα 1 T 0 d i, F iα = 1 T 0 f iα Now, we introduce 3 into, then we integrate over so that we are lead

11 MICROSTRETCH ODIES 51 to ] [a ijα ε ij + b ijα γ ij + g α θ ω α 1T0 q i θ, i dv = = A ijmn ε ij ε mn + ijmn ε ij γ mn + C ijmn γ ij γ mn + D ij ε ij θ+ 4 +E ij γ ij θ + ijα ε ij ω α + D ijα γ ij ω α + A ijk ε ij θ, k + ijk γ ij θ, k + +Mθ 1 +L α θω α +D i θθ, i +F iα ω α θ, i dv K ij θ,i θ,j dv. T 0 On the terms in the right hand side of 4 we will use the Schwarz s inequality and the arithmetic - geometric mean inequality ab 1 a p + b p 5 through a convenient choice of the parameter p. For instance, we can write A ijk ε ij θ, k dv p 1 θ, i θ, j dv + M 1 p ε ij ε ij dv 1 and so on. Thus we have, for arbitrary positive constants p 1, p, p 3 and p 4 ] [a ijα ε ij + b ijα γ ij + g α θ ω α 1T0 q i θ, i dv µ + p 1 + p + p 3 + p 4 M + 1 p + M5 + M7 + M8 + M10 1 M + + M6 + M9 + M p θ, i θ, i dv + ε ij ε ij dv + 6 γ ij γ ij dv + ω α ω α dv M + 3 M p + M1 + 3 θ dv p + M where p 1, p, p 3 and p 4 are arbitrary positive constants. Also, in the inequality 6 we have used the notations M 1 =max A ijk A ijk x s, M =max ijk ijk x s, M 3 =max D iα D iα x s, M 4 = max F iα F iα x s, M 5 = max [A ijmn A ijmn x s ] 1/, M 6 = max [C ijmn C ijmn x s ] 1/, M 7 = max ijmn ijmn x s, M 8 =max D ij D ij x s, M 9 =max E ij E ij x s, M 10 =max ijα ijα x s, M 11 =max D ijα D ijα x s, M 1 = max M x s, M 13 =max L α L α x s,

12 MICROSTRETCH ODIES 5 Since the parameters p 1, p, p 3 and p 4 are arbitrary, we can choose them such that the number m 1 defined by m 1 = µ 1 p 1 + p + p 3 + p 4 is strictly positive. Also, if we choose the constant m as follows m = 1 max { M 1 p 1 + M 5 + M 7 + M 8 + M 10, M p + M 6 + M 9 + M , M 4 π 4 M 3 p 3 + M 8 + M 18 + M 19 +, + M 1 + 3, M 4 p 4 } + M13 + then we arrive to the estimate 1 and this conclude the proof of Theorem. Theorem 3. Let u i, ϕ i, θ, ω α be a solution of the problem P 0 and suppose that the assumptions i - v are satisfied. Then there exists a positive constant m 3 such that we have the following inequality ui u i + ϕ i ϕ i + ε ij ε ij + γ i γ i + θ + ω α ω α dv t m 3 ui u i + ϕ i ϕ i + ε ij ε ij + γ ij γ ij + θ + ω α ω α dv ds 7 0 for any t [0, t 0 ]. Proof. First, taking into account the hypotheses i - v, we have m 0 ui u i + ϕ i ϕ i + ε ij ε ij + γ ij γ ij + θ dv A ijmn ε ij ε mn + ijmn ε ij γ mn + C ijmn γ ij γ mn + where we have used the notation +a ijα ε ij ω α + b ijα γ ij ω α aθ + ϱ u i u i + I ij ϕ i ϕ j dv, 8 m 0 = min {ϱ, a, λ 1, λ, λ 3, λ 4 } Next, we use the Schwarz s inequality and the arithmetic - geometric mean inequality 5 to the right hand side of the relation 8. For arbitrary positive

13 MICROSTRETCH ODIES 53 constants p 5, p 6 and p 7, we have ijmn ε ij γ mn dv p 5 a ijα ε ij ω α dv p 6 b ijα γ ij ω α dv p 7 where we have used the notations ε ij ε ij dv + N 1 p 5 ε ij ε ij dv + N p 6 γ ij γ ij dv + N 3 p 7 γ ij γ ij dv, ω α ω α dv, 9 ω α ω α dv, N 1 = max ijmn ijmn x s, N = max a ijα a ijα x s, N 3 = max b ijα b ijα x s So, from 16, 1, 8 and 9 we are lead to the inequality m 0 ui u i + ϕ i ϕ i + ε ij ε ij + γ ij γ ij + θ dv p 5 + p N 6 ε ij ε ij dv + 1 p + p 7 γ ij γ ij dv + 5 N + p + N t 3 6 p ω α ω α dv m 1 θ, i θ, i dv ds t +m εij ε ij + γ ij γ ij + θ + ω α ω α dv ds 0 where t [0, t 0 ]. Now, by using the null initial conditions 15, the consitutive relation 7 and the equation 13, we arrive to the conclusion that: ω α ω α dv = = t 0 t 0 d ds ω α ω α dv ds = t 0 ω α ω α dv ds = g ijα ε ij + h ijα γ ij + p α θ + q αβ ω β + r i θ, i ω α dv ds 31 Now, by using, again, the Schwarz s inequality and the arithmetic - geometric mean inequality 5 to the right hand side of the relation 31. So, we deduce that for an arbitrary positive constant p 8 the following inequality hold: g ijα ε ij ω α + h ijα γ ij ω α + p α θω α + q αβ ω β ω α + r i ω α θ, i dv Q p 8 θ, i θ, i dv + 5 p + Q 3 + Q 4 + ω α ω α dv Q 1 ε ij ε ij dv + Q γ ij γ ij dv + θ dv,

14 MICROSTRETCH ODIES 54 where t [0, t 0 ] and we have used the notations Q 1 = max g ijα g ijα x s, Q = max h ijα h ijα x s, Q 3 = max p α p α x s, Q 4 = max [q αβ q αβ x s ] 1/, Q 5 = max r iα r iα x s, x s If we denote by m 4 the quantity { } Q m 4 = max 5 + Q 3 + Q 4 +, Q 1, Q, 1, p 8 then, from 31 and 3 we obtain the following inequality ω α ω α dv p 8p 9 t +m 4 p 9 0 t From 30 and 33 we obtain m 0 0 θ, i θ, i dv ds + ui u i + ϕ i ϕ i + θ dv + m 0 p 5 p 6 + m 0 N 1 p p 7 5 γ ij γ ij dv + t εij ε ij + γ ij γ ij + θ + ω α ω α dv ds 33 p 9 N p N 3 6 p 7 ε ij ε ij dv + ω α ω α dv dv 34 m + m 4 p 9 εij ε ij + γ ij γ ij + θ + ω α ω α dv ds 0 m 1 p 8p t 9 θ, i θ, i dv ds ecause the constants p 5, p 6, p 7, p 8 and p 9 are arbitrary, we can choose them so that m 5 m 0 p 5 p 6 > 0 m 6 m 0 N 1 p 5 m 7 p 9 N π 6 N 3 p 7 0 p 7 > 0, > 0, m 8 m 1 p 8p 9 > 0,

15 MICROSTRETCH ODIES 55 and thus from 34 we are lead to m + m 4 p 9 t εij ε ij + γ ij γ ij + θ + ω α ω α dv ds 0 m 0 ui u i + ϕ i ϕ i + θ dv + m 5 ε ij ε ij dv + +m 6 γ ij γ ij dv + m 7 ω α ω α dv + m 8 θ, i θ, i dv ds 35 m 9 ui u i + ϕ i ϕ i + ε ij ε ij + γ ij γ ij + θ + ω α ω α dv, where the semnification of the constant m 9 is { } m 10 = min m 0, m 5, m 6, m 7. Let us observe that t ui u i + ϕ ij ϕ ij + ε ij ε ij + γ ij γ ij + θ + ω α ω α dv ds 0 Finally, if we choose t 0 εij ε ij + γ ij γ ij + θ + ω α ω α dv ds 36 m + m 4 p 9 m 3 = m 9 then from 35 and 36 we arrive at the desired result 7 and Theorem 3 is proved. Theorem 1, Theorem and Theorem 3 form the basis of the main result of this study: the uniqueness of solution of mixed initial-boundary value problem for thermoelastic microstretch body with internal state variables. Theorem 4. Assume that the hypotheses i - v hold. Then there exists at most one solution of the problem defined by the equations 1, and 6 with the initial conditions 9 and the boundary conditions 10. Proof. Suppose that the mixed problem has two solutions. Then the difference of these solutions is solution for the above mentioned problem P 0. For our aim it is suffice to show that the function yt defined by yt = ui u i + ϕ ij ϕ ij + ε ij ε ij + γ ij γ ij + θ + ω α ω α dv

16 MICROSTRETCH ODIES 56 vanishes on the interval [0, t 0 ]. If we assume the contrary, i.e. yt 0, this is absurdum because the inequality 7 and Gronwall s inequality imply that yt 0 on [0, t 0 ] and Theorem 4 is concluded. Conclusion. The presence of internal state variables do not affect the uniqueness of solution of the mixed problem for microstretch thermoelastic materials. References [1] L. Anand, M.E. Gurtin, A theory of amorphous solids undergoing large deformations I nt. J. Solids Struct., Vol , [] J.L. ouvard, D.K. Ward, D. Hossain, E.. Marin, D.J. ammann, M.F. Horstemeyer, A general inelastic internal state variable model for amorphous glassy polymers, Acta Mechanica, Vol. 13, 1-010, [3] S. Chirita, On the linear theory of thermo-viscoelastic materials with internal state variables, Arch. Mech., Vol , [4] A.C. Eringen, Theory of thermo-microstretch elastic solids, Int. J. Engng. Sci., Vol , [5] C. Flaut, D. Savin, Some examples of division symbol algebras of degree 3 and 5, Carpathian Journal of Mathematics, 31: , 015. [6] G. Groza, M. Jianu, N. Pop, Infinitely differentiable functions represented into Newton interpolating series, Carpathian Journal of Mathematics, 303: , 014. [7] L. Harabagiu, O. Simionescu-Panait, Propagation of inhomogeneous plane waves in isotropic solid crystals, Ann. Sci. Univ. Ovidius Constanta, 3 3: 55-64, 015 [8] D. Iesan, R. Quintanilla, Some theorems in the theory of microstretch thermopiezo-electricity, Int. J. Engng. Sci., Vol. 45, 1007, 1-16 [9] R. Kumar, Wave propagation in a microstretch thermoelastic diffusion solid, Ann. Sci. Univ. Ovidius Constanta, 3 1: , 015. [10] M. Marin, On existence and uniqueness in thermoelasticity of micropolar bodies, Comptes Rendus, Acad. Sci. Paris, Serie II, vol. 31 1, , 1995

17 MICROSTRETCH ODIES 57 [11] M. Marin, Some basic theorems in elastostatics of micropolar materials with voids, J. Comp. Appl. Math., Vol. 701, , 1996 [1] M. Marin, C. Marinescu, Thermoelasticity of initially stressed bodies. Asymptotic equipartition of energies, Int. J. Eng. Sci., vol. 36 1, 73-86, 1998 [13] M. Marin, A domain of influence theorem for microstretch elastic materials, Nonlinear Analysis: R.W.A., vol. 115, , 010 [14] M. Marin, Lagrange identity method for microstretch thermoelastic materials, J. Math. Analysis and Applications, vol , pp , 010 [15] K. Sharma, M. Marin, Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids, Ann. Sci. Univ. Ovidius Constanta, vol. 014, [16] R.R. Nachlinger, J.W. Nunziato, Wave propagation and uniqueness theorem for elastic materials with ISV, Int. J. Engng. Sci., Vol , [17] J.A. Sherburn, M.F. Horstemeyer, D.J. ammann, J.R. aumgardner, Application of the ammann inelasticity internal state variable constitutive model to geological materials, Geophysical J. Int., Vol. 184, 3011, [18] K.N. Solanki, D.J. ammann, A thermodynamic framework for a gradient theory of continuum damage, American Acad. Mech. Conf., New Orleans, 008 [19] C. Wei, M.M. Dewoolkar, Formulation of capillary hysteresis with internal state variables, Water resources research, Vol. 4006, Marin MARIN Department of Mathematics and Computer Science, Transilvania University of rasov, rasov, Romania. m.marin@unitbv.ro Sorin VLASE Department of Mechanical Engineering, Transilvania University of rasov, rasov, Romania svlase@unitbv.ro

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