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1 This article was downloaded by: [University of Salerno] On: 4 November 8 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 7954 Registered office: Mortimer House, 37-4 Mortimer Street, London WT 3JH, UK Journal of Thermal Stresses Publication details, including instructions for authors and subscription information: On Some Growth-Decay Results in Thermoelasticity of Porous Media M. Ciarletta a ; S. Chiri b a Department of Information Engineering and Applied Mathematics DIIMA), University of Salerno, Fisciano Sa), Italy b Faculty of Mathematics, Al. I. Cuza University of Iai, Iai, Romania Online Publication Date: October 6 To cite this Article Ciarletta, M. and Chiri, S.6)'On Some Growth-Decay Results in Thermoelasticity of Porous Media',Journal of Thermal Stresses,9:,95 94 To link to this Article: DOI:.8/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Thermal Stresses, 9: 95 94, 6 Copyright Taylor & Francis Group, LLC ISSN: print/5-74x online DOI:.8/ ON SOME GROWTH-DECAY RESULTS IN THERMOELASTICITY OF POROUS MEDIA M. Ciarletta Department of Information Engineering and Applied Mathematics DIIMA), University of Salerno, Fisciano Sa), Italy S. Chiriţă Faculty of Mathematics, Al. I. Cuza University of Iaşi, Iaşi, Romania Downloaded By: [University of Salerno] At: 4:9 4 November 8 Keywords: In this paper we study the spatial behaviour for a large class of isotropic and homogeneous porous thermoelastic materials for which the constitutive coefficients are supposed to satisfy some relaxed positive definiteness conditions. By using some appropriate measures, we are able to establish results describing the spatial behaviour of transient and steady-state solutions in these enlarged classes of thermoelastic porous materials. Spatial behaviour; Steady-state; Thermoelastic porous materials; Transient solutions INTRODUCTION The theory of elastic porous materials has been developed by Goodman and Cowin [], Nunziato and Cowin [] and Cowin and Nunziato [3] for describing the deformation of continuum with voids in which the matrix material is elastic and the interstices are void of material. The basic concept underlying this theory is that of a material for which the bulk density is written as the product of two fields, the density field of the matrix material and the volume fraction field. This representation was employed by Nunziato and Cowin [] to develop a nonlinear theory of elastic materials with voids. The linear theory of elastic materials with voids has been developed by Cowin and Nunziato [3]. The intended applications of the theory of elastic materials with voids are to geological materials such as rock and soils and to manufactured porous materials. The first investigations in the theory of thermoelastic materials with voids are due to Nunziato and Cowin [] and Ieşan [4]. The linear theory of thermoelastic materials with voids was presented in [4] see also [5]). The spatial behaviour of solutions in elastostatics of cylinders made of a porous elastic material was studied by Chiriţă [6] and Ieşan and Quintanilla [7]. The spatial behaviour of the transient and steady-state solutions in linear theory of thermoelastic materials with voids has been studied by Chiriţă and Scalia [8] and Received 7 November 5; accepted 8 January 6. Address correspondence to Stan Chiriţă, Faculty of Mathematics, Al. I. Cuza University of Iaşi, Blvd. Carol I, No., Iaşi 756, Romania. schirita@uaic.ro 95

3 96 M. CIARLETTA AND S. CHIRIŢĂ Downloaded By: [University of Salerno] At: 4:9 4 November 8 Scalia, Pompei and Chiriţă [9], respectively, under the assumption concerning the positive definiteness of the constitutive coefficients. Recently, there have been prepared novel foam structures with negative Poisson s ratios and their mechanical behaviour and structure evaluated see, e.g., Lakes [] and Caddock and Evans []). Such auxetic or anti-rubber materials expand laterally when stretched, in contrast to ordinary materials. Some anisotropic polymer foams have been prepared which exhibit a Poisson s ratio exceeding see Lee and Lakes []). Materials of the above sorts are expected to have interesting mechanical properties, such as high energy absorption and fracture resistance, which may be useful in applications. Possible applications of such materials in prevention of pressure sores or ulcers are outlined by Wang and Lakes [3]. Saint-Venant end effects for materials with negative Poisson s ratio are studied by Lakes [4]. In this paper, we describe some methods, relevant to the study of spatial behaviour for the steady-state and transient solutions in a linear isotropic homogeneous thermoelastic body with voids, under mild conditions concerning the positive definiteness of the constitutive coefficients, including the class of auxetic materials. Concerning the transient solutions we establish some spatial estimates describing the exponential decay of appropriate measures with respect to the distance to the support of the given data. For the case of an unbounded body we consider a measure for which we are able to prove that the decay rate is quadratic in terms of the distance to the support of the given data. For the study of the time-harmonic vibrations, we consider a right cylinder made of an isotropic and homogeneous thermoelastic material with voids and introduce two appropriate cross-sectional measures. The treatment of this problem leads to a second-order differential inequality furnishing the information concerning the spatial behaviour of the amplitude of the vibration, provided the frequency of the harmonic vibration is lower than a certain critical frequency. Such results are established for the same enlarged classes of materials with voids as for the transient solutions. FORMULATION OF THE PROBLEM Let B be a bounded or unbounded regular region of the physical space 3 with the piecewise smooth boundary surface B. We designate by n the outward unit normal vector to the boundary. The region B is filled with an isotropic and homogeneous thermoelastic porous material. We select a rectangular system of coordinates and note that vectors and tensors will have components denoted by Latin subscripts ranging over 3. Summation over repeated subscripts and other typical conventions for differential operations are implied such as a superposed dot or a comma followed by a subscript to denote partial derivative with respect to time or the corresponding cartesian coordinate, respectively. Further, we suppress the dependence upon the spatial variable when no confusion may occur. In the context of linear theory of thermoelasticity theory for a porous solid developed by Ieşan [4] see also [5]), the equations of motion are t jij + f i = ü i h jj + g + l = κ ) T = q jj + S in B

4 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 97 In the above equations we have used the following notation: u i are the components of the displacement vector field, is the change in volume fraction from the reference volume fraction, T is the constant reference temperature, t ij are the components of the stress tensor, h i are the components of the equilibrated stress vector, g is the intrinsic equilibrated force, is the entropy, q i are the components of the heat flux vector, f i are the components of the body force vector, l is the extrinsic equilibrated body force and S is the heat supply. Moreover, and κ are the bulk mass density and the equilibrated inertia in the reference state. The constitutive equations for the linear theory of isotropic and homogeneous thermoelastic porous media are [4, 5] Downloaded By: [University of Salerno] At: 4:9 4 November 8 t ij = e rr ij + e ij + b ij ij h j = j g = be rr + m ) = e rr + a + m q j = k j in B is the temperature variation from the reference temperature T,,, a, b, k, m,,, and are constant constitutive coefficients, ij is the Kronecker s delta and e ij = u ij + u ji 3) The set B represents the closure of B. The internal energy density E associated with the kinematic fields u i, is defined by E = e rre ss + e ij e ij + be rr + + i i 4) and it is positive definite if and only if see, e.g., Cowin and Nunziato [3]) > > > 3 + > 3 + > 3b 5) As we will see later, the hypotheses of positive definiteness on the constitutive coefficients defined by 5) will be relaxed. To this end, we consider an isotropic, homogeneous thermoelastic porous solid so that the basic Eqs. ) to 3) of the linear dynamic theory reduce to [4, 5] u jrr + + u rrj + b j j + f j = ü j rr bu rr + m + l = κ 6) k rr T u jj at mt = S The initial conditions are u j = u j u j = u j = = = on B 7) u j, u j,, and are prescribed continuous functions.

5 98 M. CIARLETTA AND S. CHIRIŢĂ In what follows, for convenience, we will consider the following boundary conditions u j = = = on 8) t jk n j = t k h j n j = h q j n j = q on 9) Downloaded By: [University of Salerno] At: 4:9 4 November 8 n j are the components of the outward unit vector normal to the boundary surface B, and are subsets of B so that = B, = and t j, h and q are prescribed functions on. Under the positive definiteness conditions 5) for the potential energy density E and by assuming >, κ >, the spatial behaviour of the transient solutions of the initial-boundary value problem defined by the relations 6) 9), has been established by Chiriţă and Scalia [8]. Throughout this paper we will consider classical solutions of the initial-boundary value problem, that is we will consider the triplet of functions u j twice continuous differentiable with respect to the spatial and time variables, satisfying the relations 6) 9). We have to outline that our method of proof can be used also for appropriate classes of weak solutions. SPATIAL BEHAVIOUR OF THE TRANSIENT SOLUTIONS Throughout this section we will establish the spatial behaviour of the transient solutions of the initial-boundary value problem under hypotheses on the constitutive coefficients different from those used in Chiriţă and Scalia [8] and Pompei and Scalia [5]. In fact, we will establish our results for appropriate complementary classes of porous thermoelastic materials by using the method developed by Chiriţă and Ciarletta [6]. Clearly, the union of these classes is essentially larger than the class of porous thermoelastic materials described by the relation 5). This task is possible because we introduce some new measures associated with the solutions, essentially different from those used by Chiriţă and Scalia [8] and Pompei and Scalia [5]. Our analysis is motivated by the existence of the novel foam structures for which the internal energy density E is not always a positive definite quadratic form. For fixed T>, we consider the support D T of the initial and boundary data and the body supplies on the time interval T and further, for convenience, we assume that it is a bounded regular region. Of course, D T is the union of the set of all x B for which u i x or u i x or x or x or x or b i x or lx or Sx for some T and the set of all x for which t i x or hx or qx for some T Further, we introduce the following sets D r = { x B Sxr D T B r = B\D r r Br r = B r \B r r <r

6 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 99 Sxr is the closure of the ball with radius r and center at x. Finally, we denote by S r the subsurface of B r contained into inside of B and whose outward unit normal vector is forwarded to the exterior of D r. We can observe that the initial and boundary data and the body forces are null on B r, S r. First Measure and Related Estimates Throughout this subsection we will assume that > κ > > > a> k> > > > 3b ) Downloaded By: [University of Salerno] At: 4:9 4 November 8 Corresponding to a solution U = u j of the initial-boundary value problem we introduce the following function t [ r t = e s u i ss ji s + sh j s + s ] q S r T j s n j da ds ) is a positive parameter at hand and s ji = S ji ij S ji = u ij + + u rr ij + b ij ) This function is defined on I T, I is the interval if B is an unbounded body, ) I is the interval L, L = max x B miny D T y x,ifb is a bounded body. In this last case, from the relation ) and the definition of S r and D r,wecan conclude that L t = for all t T. By using Eq. 6), we get ] [ u i s ji + h j + T q j = j t K + W + a + k T j j f i u i l S T 3) K = u i u i + κ 4) Wu = W u + W u + W 3 5) W u = + u + u + 33) u + + u u + u u 33 + u 33 u + + bu + u + u 33 W u = u + u + u 3 + u 3 + u 3 + ) u 3 6) W 3 = j j

7 9 M. CIARLETTA AND S. CHIRIŢĂ The definition for S r, B r, the divergence theorem and the relations ), 3) lead to r t r t t [ = e s u i ss ji s + sh j s + s ] q Br r T j s n j da ds t = e s{ Br r s Ks + Ws + a s + k } T j s j s dv ds 7) for r <r, t T. Taking into account 7) and the definition for S r and B r,it is a simple matter to obtain Downloaded By: [University of Salerno] At: 4:9 4 November 8 r r t = e t Kt + Wt + a t da S r t { [ e s Ks + Ws + a s ] + k } S r T j s j s da ds 8) Obviously, if > and κ > then the quadratic form K in the variables u u u 3 κ is positive definite and if > then the form W 3 in the variables 3 is positive definite. Moreover, W is a positive definite quadratic form in the variables u u u 3 u 3 u 3 u 3 if and only if >. On the other hand, the quadratic form W in the variables u u u 33 is positive definite if and only if >, >, >, > 3b. The eigenvalues of the matrix of the quadratic form W are = = 3 = { } b 4 = { } b Consequently, the relations ) and 8) imply that r t is a non-increasing function with respect to r, for all t T. Since L t = for all t T,it follows that for a bounded body we have r t for r t I T. On the other hand, by means of the Schwarz inequality and the arithmeticgeometric mean inequalities, we have u i s ji n j + h j n j [ ui u i + κ ) + s ij s ij + κ )] h jh j 9) for any arbitrary positive constant. Furthermore, from the relation ) we get s ji s ji + S ji S ji ) for all > )

8 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 9 As was shown by Ciarletta, Chiriţă and Passarella [7], from the relations ), ) and 6) we obtain S ji S ji M W + W ) { M = max [ ] } b ) By using the relations ) and ) in 9), we obtain the following estimate u i s ji n j + h j n j K + [ M + + ]W ) 3) κ Downloaded By: [University of Salerno] At: 4:9 4 November 8 On the other hand, from the relation ) we get q T j n j T a + ) a q jq j a + k k T a T j j for all > 4) Concluding, the relations 3) and 4) give u is ji n j + h j n j + q T j n j K + [ M + + ] κ W [ ) + ] a T a + k a for all strictly positive constants,,. At this stage we request the equalities = [ ] M + + = 3 + ) + κ a T = k a That is, we set k T j j 5) 6) = c = M + + κ = k [ ] M + + 7) a κ is the positive root of the following algebraic equation + + k ) 3 3 = 8) κ M at M a M a M Therefore, with the above choices, relations ), 8), and 5) furnish the following first-order differential inequality r t+ c r t r for all r t I T 9)

9 9 M. CIARLETTA AND S. CHIRIŢĂ As in [6], we can exploit the differential inequality 9) in order to prove for a bounded body the following estimate r t texp ) r for all r t I T 3) c Let us now study the case of an unbounded body. We can conclude that either r t for all r t I T and then a spatial decay estimate of the type described by 3) holds true, or there exists a value rt such that rt t< and then r t < for all r>rt and t T and, moreover, [ ] r t r t texp r r t c r>r t t T 3) Downloaded By: [University of Salerno] At: 4:9 4 November 8 Let us consider further that we have r t for all r t I T.In view of the spatial decay 3) we can introduce the new measure r t = = r t td { r [ d e s S u i ss ji s + sh j s + s ] } q T j s n j da ds By means of the relations ) and 3) and the definition of S r, we have r s s d e q S T j sn j da = = r r = r = S r e s d e s k s s S T n da ) s k d e s da S n T e s k sda d S T 3) k sda 33) T and hence the measure r t becomes t { } r t = d e s u i ss ji s + sh j sn j da ds r S t + e s k sda ds 34) S r T Obviously, from 8), 3), and 34), we obtain t r t = r d e t u i ts ji t + th j tn j da + S S r e t k tda 35) T

10 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 93 and r r t = e t B r Kt + Wt + a tdv t { e s B r Ks + Ws + a s + } k T j s j s dv ds 36) r r t = e t S r Kt + Wt + a tda t { + e s S r Ks + Ws + a s + } k T j s j s da ds 37) Downloaded By: [University of Salerno] At: 4:9 4 November 8 Furthermore, by using the estimate 3) with [ = c = M + + ] κ + ) 3 κ M a 3 M a M = 38) in 35), we get t r t c r d e t Kt + Wt + a tda + k e t S at S r a tda Concluding, from the relations 36), 37), and 39) we obtain the following differential inequality t r t c r t + k r at r By using the change of function r t = exp at c 4k t ) 39) r t r t T 4) ) at c exp k r L r t 4) we can write the differential inequality 4) in the following form k L at r r t L r t 4) t We further note that L r = for r> L t= e at c 4k t t for t L r t uniformly in t as r 43) By means of the maximum principle for parabolic differential inequalities, we have L r t r t r t 44)

11 94 M. CIARLETTA AND S. CHIRIŢĂ k r t r t = at r t r> t > r = L r = r t= L t t r t L r t uniformly in) t as r 45) The solution of the above problem is given by r t = at k t r t 3 e at 4kt r e at c 4k d 46) Downloaded By: [University of Salerno] At: 4:9 4 November 8 Therefore, the relations 44) and 45) give L r t e at c k r L tr t 47) at r t = k t rs 3 at e 4ks r + at c 4k ) s ds 48) The function r t can be estimated as in Horgan, Payne, and Wheeler [8] to obtain kt r at c r t at r c e 4k t e at 4kt r for r>c t 49) t Thus, from the relations 4), 47), and 49) we get the following estimate kt r t t at r r c e t at c 4k t e at c k r at 4kt r for r>c t t T 5) Second Measure and Related Estimates Throughout this subsection we will assume the following hypotheses > κ > > > > << + > 3b a> k> 5) We now introduce the following time-weighted function on I T: t [ r t = e s u i sr ji s + sh j s + s ] q S r T j s n j da ds 5)

12 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 95 r ji = T ji ij T ji = u ij + + u ji + b ij 53) The relations 6) and 53) give ] [ u i r ji + h j + T q j = j t K + W + a + k T j j f i u i l S T 54) in which Downloaded By: [University of Salerno] At: 4:9 4 November 8 W u = W u + W u + W 3 55) W u = + u + u + ) ) u 33 + b u + u + u 33 + W u = u + u + u 3 + u 3 + u 3 + ) u 3 56) + + ) u u + u 3 u 3 + u 3 u 3 Clearly, K is positive definite if and only if >, κ >, while W 3 is positive definite if and only if >. Moreover, the quadratic form W in the variables u u u 3 u 3 u 3 u 3 is positive definite if and only if <, + >. The eigenvalues of the matrix B of this quadratic form are = = 3 = 4 = 5 = 6 = + 57) Further, the quadratic form W in the variables u u u 33 is positive definite if and only if >, + >, + > 3b. The matrix of the quadratic form W has the following eigenvalues ˆ = ˆ = + ˆ 34 = { + + ± } + + b 58) In view of the results established in [7], we can conclude that T ij T ij ˇ M W + ˆ MW 59) ˇ M = max + { ˆ M = max + [ ]} 6) + + b Taking into account the definition for S r B r and the divergence theorem, then the relations 5) and 53) imply that r t r t t = e s{ Br r s Ks + W s + a s + k } T j s j s dv ds 6)

13 96 M. CIARLETTA AND S. CHIRIŢĂ and therefore, we get r r t = e t Kt + W t + a tda S r t { e s S r Ks + W s + a s + k } T j s j s da ds 6) Moreover, from 5) we have t r t = [ e t S r ] u i tr ji t + th j t + t q T j t n j da 63) Downloaded By: [University of Salerno] At: 4:9 4 November 8 In view of a procedure similar to that in the above subsection, we can obtain the following differential inequality c = r t+ c M + + κ { M = max κ + r t r for all r t I T 64) b ) } 65) and is the positive root of the algebraic equation + + k ) 3 3 κ M at M a M a M = 66) Thus, from 64) we obtain the following estimate r t texp c ) r for all r t I T 67) for a bounded body. While for an unbounded body we can conclude that either r t for all r t I T and then a spatial decay estimate of the type described by 67) holds true, or there exists a value r t such that r t t< and then we have [ r t r t c texp ) ] r r t for all r t r t T 68) Furthermore, by assuming that r t for all r t I T, we can introduce the new measure r t = r td 69)

14 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 97 so that we are lead to the counterpart of 5) in the form kt r at c r t t at r c e 4k t e at c k r at 4kt r for r> c t t T 7) t c = [ M + + ] κ + + ) 3 κ M a 3 = 7) M a M Downloaded By: [University of Salerno] At: 4:9 4 November 8 SPATIAL BEHAVIOUR OF THE STEADY-STATE SOLUTIONS Here we discuss the problem of spatial behaviour of the steady vibrations. For this purpose we stipulate that the region B from here on refers to the interior of a prismatic cylinder with parallel plane ends. The rectangular Cartesian coordinate frame is supposed to be chosen in such a way that one end of the cylinder lies in the x Ox plane and contains the origin. We suppose that the length of the cylinder is L and that D x3 represents the bounded cross-section at distance x 3 from the x Ox plane. The boundary D of each cross-section is assumed to be a piecewise smooth simple closed curve. The Greek subscripts range over,. In what follows we shall assume that the external body supplies are absent. Moreover, we assume the following boundary conditions for all t T u j = = = on D L D L t T 7) u 3j + u j3 + u rr + b j3 = t j e it 73) 3 = he it k 3 = qe it for all x x x 3 D t T t i, h, and q are prescribed functions on x x D and is a positive prescribed parameter. Other boundary conditions can be viewed but essentially for our considerations is to conserve the boundary condition 7). In this section we discuss the problem of steady-state vibrations assuming that u j =Rev j x e it =Rex e it =Rex e it 74) Ref represents the real part of f. Then the equations of motion 6) reduce to v jrr + + v rrj + b j j + v j = 75) rr bv rr + m + κ = 76) k rr it v rr iat imt = 77) while the boundary conditions 7) and 73) reduce to v j = = = on D L D L 78) v 3j + v j3 + v rr + b j3 = t j 3 = h k 3 = q for all x x x 3 D 79)

15 98 M. CIARLETTA AND S. CHIRIŢĂ In what follows, we will study the spatial behaviour of the amplitude v j of the harmonic vibration described by the relation 74). To this end, we note that v j is the solution of the boundary value problem defined by the Eqs. 75) 77) and by the boundary conditions 78) and 79). Downloaded By: [University of Salerno] At: 4:9 4 November 8 First Measure Throughout this subsection we will assume that the hypotheses described by the relation ) hold true. Then we associate with the solution v j of the problem the function on L defined by [ x 3 = v r v r + + v 3 v ] k da D x3 T [ + v v 3 + v v 3 + bv 3 + v 3 v 3 + v 3 ] dv B x3 x 3 L 8) a superposed bar denotes the complex conjugate and is a prescribed parameter at our hand and B x3 = D x 3 L. We have to note that the boundary condition 78) implies that L = 8) In what follows we will prove that this function is convex on L, for appropriate values of the parameters and. To this end, we note that x 3 = D x3 [ v r v r3 + v r v r3 + + v 3 v rr + v 3 v rr + bv 3 + v 3 v 3 + v ] k T da x 3 L 8) and, by the relation 78), we have L = 83) Furthermore, on the basis of Eqs. 75) to 77) and by using the boundary conditions 78), we get x 3 = D x3 { w + r r + k T r r v r v r + κ [ iv rr + + iv rr ] m [ i + + i ]} da 84)

16 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 99 w = w + w 85) w = + v v + v v + v 33 v v v + v v + v v 33 + v v 33 + v 33 v + v 33 v + + b [ v + v + v 33 + v + v + v 33 ] 86) w = v v + v v + v 3 v 3 + v 3 v 3 + v 3 v 3 + v 3 v 3 87) On the basis of the assumption ) we deduce that w m v v + v v + v 33 v ) Downloaded By: [University of Salerno] At: 4:9 4 November 8 { m = min } b Moreover, we deduce that 89) w m v jk v jk + 9) In view of the boundary conditions 78), we have v r v r da v r v r da D x3 D x3 da da da da D x3 D Dx3 x3 D x3 is the first eigenvalue corresponding to the membrane problem for the section D x3. Then, the relations 9) and 9) imply { x 3 m v rk v rk + m + + k D x3 T [ iv rr + + iv rr ] m [ i + + i ] )} v rk v rk + κ da 9) Furthermore, from 9) we get ] x 3 { m [v rk m + i rk ][v rk m i rk D x3 [ + m + m + i m + + M v rk v rk + M + M 3 9) ][ ] m i m + } da 93)

17 9 M. CIARLETTA AND S. CHIRIŢĂ M = m M = m + κ M 3 = T k 3 m m 3 + m + m m m + ) 94) Concluding from 93), we can see that x 3 for all x 3 L 95) if we choose the parameter so that T ) 3 m + k m m + 96) Downloaded By: [University of Salerno] At: 4:9 4 November 8 and assume that the frequency of the given vibration satisfies = min { m m + κ < 97) } k m m + T m 3 + m + 3 Throughout this subsection we will assume the relations 96) and 97) hold true. Then the relation 95) proves that x 3 is a non-decreasing function with respect to x 3 on L so that, by using 83), we obtain 98) x 3 L = for all x 3 L 99) Thus, x 3 is a non-increasing function on L and hence, by using 8), we get x 3 for all x 3 L ) On the other hand, by using the relations 83) and 93) we get [ x 3 M v rk v rk + M + M 3 ] dv ) B x3 Now, by using the arithmetic geometric inequality and the Schwarz inequality, from 8) we obtain [ x 3 max + v r v r + + ] k da T D x3 + B x3 [ + v v + v 3 v 3 + b v 3 v T v 3 v 3 + ] dv ) T

18 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 9 Further, we shall use the inequalities described in 9) to obtain max + x 3 v D x3 rk v rk + + ) k da T + m v rk v rk + m + m 3 ) dv 3) B x3 m = + +b+t m = b m 3 = T 4) Downloaded By: [University of Salerno] At: 4:9 4 November 8 By combining the relations 93), ), and 3) we get the following second order differential inequality x 3 + c 3 x 3 c 4 x 3 for all x 3 L 5) with { m c 3 = max m m } 3 6) M M M 3 { max + c 4 = max } k 7) M M T M 3 It follows from the inequality 5) and a well-known Comparison Principle a generalisation of the curve under chord property for convex functions [9]) that x 3 is bounded above by x 3, the solution of the associated differential equation with the same boundary conditions, that is On this basis we obtain x 3 c 3 x c 3 x 4 c 3 = 4 = L = 8) x 3 er r L x 3 e r x 3 for all x e r r L 3 L 9) r = ) c c 3 + c3 + 4c 4 4 r = ) c c 3 c3 + 4c 4 4 ) This last result gives x 3 e r x 3 for all x 3 L ) provided the relations 96) and 97) hold true.

19 9 M. CIARLETTA AND S. CHIRIŢĂ Second Measure In the remainder of this paper we shall assume that the hypotheses described by the relation 5) hold true. Then we introduce the following measure [ x 3 = v r v r + + v 3 v ] k da D x3 T [ + v v 3 + v v 3 + bv 3 + v 3 v 3 + v 3 ] dv B x3 x 3 L ) Downloaded By: [University of Salerno] At: 4:9 4 November 8 is a positive parameter at our hand. Furthermore, we have { x 3 = w + r r + k T r r v r v r + κ D x3 iv rr + + i v rr } m i + + i da 3) w = w + w 4) w = + v v + v v + v 33 v 33 + b [ v + v + v 33 + v + v + v 33 ] + 5) w = v v + v v + v 3 v 3 + v 3 v 3 + v 3 v 3 + v 3 v v v + v v + v 3 v 3 + v 3 v 3 + v 3 v 3 + v 3 v 3 6) On the basis of the assumption 5) we deduce that w ˇ mv v + v v + v 33 v 33 + w ˆ mv v + v v + v 3 v 3 + v 3 v 3 + v 3 v 3 + v 3 v 3 7) ˇ m = min + { ˆ m = min + [ b ]} 8) Thus, we deduce that w m v jk v jk + m = minˇ m ˆ m 9) Furthermore, from 3) we get x 3 { M v jk v jk + M + M 3 } da D x3 )

20 SPATIAL BEHAVIOUR IN LINEAR-POROUS THERMOELASTICITY 93 M = m M = m + κ M 3 = T k 3 m m 3 + m + m m m + ) ) Thus, we have x 3 for all x 3 L ) Downloaded By: [University of Salerno] At: 4:9 4 November 8 provided that and with { = min m T 3 + k m m + κ m m + ) 3) < 4) k m m + T m 3 + m + 3 } 5) Therefore, as in the above subsection we can obtain the following spatial estimate r = c c 3 4 c 3 = max x 3 e r x 3 for all x 3 L 6) c c 4 { m M m M m 3 M 3 } { max + c 4 = max M M k T M 3 } 7) CONCLUDING REMARKS In the present paper we introduced the new measures ), 3), 5), 69), and 8), ) in order to study the spatial behaviour of the transient and steadystate solutions, respectively. The measures defined by the relations ), 3), and 8) are suitable for the class of porous elastic materials described by the relation ); while the measures 5), 69), and ) are useful in the class described by the relation 5). These classes of porous elastic materials are not coincident with the class defined by 5) and treated in [8, 9]. The method presented here is believed to be used successfully for many novel foam structures with negative Poisson s ratio, because the relations ) and 5) can include such materials, not included in the class defined by 5).

21 94 M. CIARLETTA AND S. CHIRIŢĂ On the other hand, we have to outline that we can combine our estimates 3), 5), 67), 7), ), and 6) with those predicted in [8, 9] in order to obtain complete information about the spatial behaviour of the transient and steady-state solutions. REFERENCES Downloaded By: [University of Salerno] At: 4:9 4 November 8. M. A. Goodman and S. C. Cowin, A Continuum Theory for Granular Materials, Arch. Rational Mech. Anal., vol. 44, pp , 97.. J. W. Nunziato and S. C. Cowin, A Non-linear Theory of Elastic Materials with Voids, Arch. Rational Mech. Anal., vol. 7, pp. 75, S. C. Cowin and J. W. Nunziato, Linear Elastic Materials with Voids, J. Elasticity, vol. 3, pp. 5 47, D. Ieşan, A Theory of Thermoelastic Materials with Voids, Acta Mechanica, vol. 6, pp , D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Boston/Dordrecht/London, S. Chiriţă, Some Growth and Decay Estimates for a Cylinder Made of an Elastic Material with Voids, Rev. Roum. Math. Pures et Appl., vol. 39, pp. 7 6, D. Ieşan and R. Quintanilla, Decay Estimates and Energy Bounds for Porous Elastic Cylinders, Z. Angew. Math. Phys. ZAMP), vol. 46, pp. 68 8, S. Chiriţă and A. Scalia, On the Spatial and Temporal Behaviour in Linear Thermoelasticity of Materials with Voids, J. Thermal Stresses, vol. 4, pp ,. 9. A. Scalia, A. Pompei, and S. Chiriţă, On the Behavior of Steady Time harmonic Oscillations in Thermoelastic Materials with Voids, J. Thermal Stresses, vol. 7, pp. 9 6, 4.. R. Lakes, Foam Structures with a Negative Poisson s Ratio, Science, vol. 35, pp. 38 4, B. D. Caddock and K. E. Evans, Microporous Materials with Negative Poisson s Ratios: I. Microstructure and Mechanical Properties; II. Mechanisms and Interpretation, J. Phys. D: Appl. Phys., vol., pp , , T. Lee and R. S. Lakes, Anisotropic Polyurethane Foam with Poisson s Ratio Greater Than, J. Mater. Sci., vol. 3, pp , Y. C. Wang and R. S. Lakes, Analytical Parametric Analysis of the Contact Problem of Human Buttocks and Negative Poisson s Ratio Foam Cushions, Inter. J. Solids Struct., vol. 39, pp ,. 4. R. S. Lakes, Saint Venant End Effects for Materials with Negative Poisson s Ratios, J. Appl. Mechanics, vol. 59, pp , A. Pompei and A. Scalia, On the Asymptotic Spatial Behavior in Linear Thermoelasticity of Materials with Voids, J. Thermal Stresses, vol. 5, pp ,. 6. S. Chiriţă and M. Ciarletta, Time Weighted Surface Power Function Method for the Study of Spatial Behaviour in Dynamics of Continua, Eur. J. Mech. A/Solids, vol. 8, pp , M. Ciarletta, S. Chiriţă, and F. Passarella, Some Results on the Spatial Behavior in Linear Porous Elasticity, Arch. Mech., vol. 57, pp , C. O. Horgan, L. E. Payne, and L. T. Wheeler, Spatial Decay Estimates in Transient Heat Conduction, Quar. Appl. Math., vol. 4, pp. 9 7, J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations: An Introduction, CRC Press, Boca Raton, FL, 995.