Nonsimple material problems addressed by the Lagrange s identity
|
|
- Clarence Price
- 5 years ago
- Views:
Transcription
1 Marin et al.oundary Value Problems 23, 23:35 R E S E A R C H Open Access Nonsimple material problems addressed by the Lagrange s identity Marin I Marin *, Ravi P Agarwal 2 and SR Mahmoud 3,4 * Correspondence: m.marin@unitbv.ro Department of Mathematics, University of rasov, rasov, Romania Full list of author information is available at the end of the article Abstract Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. y using the Lagrange identity, we prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients. Keywords: Lagrange identity; nonsimple materials; uniqueness; continuous dependence Introduction Even classical elasticity does not consider the inner structure, the material response of materials to stimuli depends in a relevant way on its internal structure. Thus, it has been needed to develop some new mathematical models for continuum materials where this kind of effects was taken into account. Some of them are nonsimple elastic solids. It is known that from a mathematical point of view, these materials are characterized by the inclusion of higher-order gradients of displacement in the basic postulates. The theory of nonsimple elastic materials was first proposed by Toupin in his famous article ]. Also, among the first studies devoted to this material, we must mention those belonging to Green and Rivlin 2] and Mindlin 3]. The interest to introduce high-order derivatives consists in the fact that the possible configurations of the materials are clarified more and more finely by the values of the successive higher gradients. As it is known, the constitutive equations of nonsimple elastic solids are known to contain first- and second-order gradients, both contributing to dissipation. It is then interesting to understand the relevance of the two different dissipation mechanisms which can appear in the theory. In fact, the simultaneous presence of both mechanisms can be analyzed as well, with inessential changes in the proofs. In that situation, the behavior turns out to be the same as if only the higher-order dissipation appears in the equations. In the last decade many studies have been devoted to nonsimple materials. We remember only three of them, differing in issues addressed, though in essence they are dedicated to nonsimple materials. So, in the paper of Pata and Quintanilla 4], the theory is linearized, and a uniqueness result is presented. Also,the study 5] of Martinez and Quintanilla is devoted to study the incremental problem in the thermoelastic theory of nonsimple elastic materials. 23 Marin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2 Marinetal.oundary Value Problems 23, 23:35 Page 2 of 4 A linearized theory of thermoelasticity for nonsimple materials is derived within the framework of extended thermodynamics in the paper of Ciarletta 6]. The theory is linearized, and a uniqueness result is presented. A Galerkin-type solution of the field equations and fundamental solutions for steady vibrations are also studied. Previous papers on the uniqueness and continuous dependence in elasticity or thermoelasticity were based almost exclusively on the assumptions that the elasticity tensor or thermoelastic coefficients are positive definite see, for instance, the paper 7]. In other papers, authors recourse the energy conservation law in order to derive the uniqueness or continuous dependence of solutions. For instance, a uniqueness result was indicated in paper 8] of Green and Lindsay by supplementing the restrictions arising from thermodynamics with certain definiteness assumptions. We want to outline that there are many papers which employ the various refinements of the Lagrange identity, of which we remember only a few, namely papers 9, ] and]. Also, a lot of papers are dedicated to Cesaro means, as 2 4]and9] for instance. The objective of our study is to examine by a new approach the mixed initial-boundary value problem in the context of thermoelasticity of nonsimple materials. The approach is developed on the basis of Lagrange identity and its consequences. Therefore, we establish the uniqueness and continuous dependence of solutions with respect to body forces, body couple, generalized external body load and heat supply. We also deduce the continuous dependence of solutions of our problem with respect to initial data and, finally, with respect to thermoelastic coefficients. The results are obtained for bounded regions of the Euclidian three-dimensional space. We point out, again, that the results are obtained without recourse to the energy conservation law or to any boundedness assumptions on the thermoelastic coefficients. Also, we avoid the use of definiteness assumptions on the thermoelastic coefficients. 2 asic equations We assume that a bounded region of the three-dimensional Euclidian space R 3 is occupied by a nonsimple elastic body, referred to the reference configuration and a fixed system of rectangular Cartesian axes. Let denote the closure of and call the boundary of the domain. We consider to be a piecewise smooth surface and designate by n i the components of the outward unit normal to the surface. Letters in boldface stand for vector fields. We use the notation v i to designate the components of the vector v in the underlying rectangular Cartesian coordinates frame. Superposed dots stand for the material time derivative. We employ the usual summation and differentiation conventions: the subscripts are understood to range over integer, 2, 3. Summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate. The spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. We refer the motion of the body to a fixed system of rectangular Cartesian axes Ox i, i =,2,3. Let us denote by u i the components of the displacement vector and by θ the temperature measured from the constant absolute temperature θ of the body in its reference state. As usual, we denote by σ ij the components of the stress tensor and by m ijk the components of the hyperstress tensor over. We here will use the theory and the notation in the way developed by Iesan in his book, which tackles also nonsimple materials 5].
3 Marinetal.oundary Value Problems 23, 23:35 Page 3 of 4 The equations of motion from thermoelasticity of nonsimple materials are as follows see also 6]: σ ji,j + m sji,sj + ϱf i = ϱü i. The equation of energy is given by θ Ṡ = q i,i + ϱr. 2 For an anisotropic and homogeneous nonsimple thermoelastic material, the constitutive equations have the form σ ij = A ijrs ε rs + ijpqr χ pqr + E ij θ, m ijk = rsijk ε rs + C ijkmnr χ mnr + D ijk θ, S = E ij ε ij D ijk χ ijk + c θ b i θ,i, T q i = θ b i θ k ij θ,j. 3 The kinematic characteristics of the body components of the strain tensors ε ij and χ ijk are defined by means of the geometric equations ε ij = 2 u i,j + u j,i, χ ijk = u k,ij. 4 In the above equations, we have used the following notations: - f i the components of body force; - ϱ is the reference constant mass density; - σ ij and m ijk are the components of the stress; - S is the entropy per unit mass; - r is the heat supply per unit mass; - q i are the components of heat flux vector. Also, the coefficients A ijrs, ijpqr, C ijkmnr, E ij, D ijk, b i, a and k ij are the characteristic constitutive constants of the material and they satisfy the following symmetry relations: A ijrs = A rsij = A jirs, ijpqr = jipqr = ijqpr, C ijkpqr = C pqrijk = C jipqr, E ij = E ji, D ijk = D jik, k ij = k ji. 5 Also, the second law of thermodynamics implies that k ij ξ i ξ j, ξ i. We denote by t i the components of surface traction and by q the heat flux. These quantities are defined by t i = σ ji n j, q = q i n i at regular points of the surface. Here, n i are the components of the outward unit normal of the surface.
4 Marinetal.oundary Value Problems 23, 23:35 Page 4 of 4 Along with the system of field equations -4, we consider the following initial conditions: u i x,=c i x, u i x,=d i x, θx,=σ x, x 6 and the following prescribed boundary conditions: u i = ū i on, t, t i = σ ji n j = t i on c, t, θ = θ on 2, t, q = q i n i = q on c 2, t, 7 where t is some instant that may be infinite. Also, and 2 with respective complements c and c 2 are subsets of the surface such that c = 2 c 2 =, c = 2 c 2 =. Assume that c i, d i, σ, ū i, t i, θ and q are prescribed smooth functions in their domains. To avoid repeating the regularity assumptions, we assume from the beginning that: i all constitutive coefficients are continuously differentiable functions on ; ii ϱ is continuous on ; iii f i and r are continuous functions on, t ; iv c i, d i and σ are continuous functions on ; v ū i and θ are continuous functions on, t and 2, t,respectively; vi t i and q are piecewise regular functions on c, t and c 2, t, respectively, and continuous in time. Taking into account the constitutive equations 3, from and2 we obtain the following system of equations: ϱü i = A ijrs u r,sj + ijpqr u r,pqj + E ij θ,j + mnsji u m,nsj + C sjimnr u r,mnsj + D sji θ,sj + ϱf i, 8 c θ + θ E ij u i,j + D ijk u k,ij =k ij θ,j,i + ϱr. y a solution of the mixed initial boundary value problem of the theory of thermoelasticity of nonsimple bodies in the cylinder =, t, we mean an ordered array u i, θ which satisfies the system of equations 8 for all x, t, the boundary conditions 7 and the initial conditions 6. 3 Main result Let us consider f t, x andgt, x, two functions assumed to be twice continuously differentiable with respect to the time variable t. y direct calculations, it is easy to deduce that d dt f ġ ḟg=ḟ ġ + f g fg ḟ ġ = f g fg. For the sake of simplicity, the spatial argument and the time argument of the functions f t, x and gt, x are omitted because there is no likelihood of confusion.
5 Marinetal.oundary Value Problems 23, 23:35 Page 5 of 4 In the above equality, we substitute the functions f t, x andgt, x by the functions U i x, t andv i x, t, respectively, which also are assumed to be twice continuously differentiable with respect to the time variable, and then we obtain the following well known Lagrange identity: ϱx U i x, t V i x, t U i x, tv i x, t ] dv t = ϱx U i x, s V i x, s Ü i x, sv i x, s ] dv ds + ϱx U i x, V i x, U i x,v i x, ] dv. 9 Let us denote by u α i, θ α α =, 2 two solutions of the mixed initial boundary value problem defined by 8, 6and7 which correspondto thesame boundary data and same initial data, but to different body forces and heat supplies, f α i, r α α =,2, respectively. We introduce the following notations: v i = u 2 i u i, χ = θ 2 θ, t ij = σ 2 ij σ ij, m ijk = μ 2 ijk μ ijk, η = S2 S, p i = q 2 i q i, F i = f 2 i f i, P = r 2 r. The constitutive equations become t ij = A ijrs v r,s + ijpqr v p,qr + E ij χ, μ ijk = rsijk v r,s + C ijkmnr v r,mn + D ijk χ, η = E ij v i,j D ijk v k,ij + c χ b i χ,i, T p i = θ b i χ k ij χ,j. So, we deduce that the differences v i, χ satisfy the following equations and conditions: - the equations of motion ϱ v i = A ijrs v r,sj + ijpqr v r,pqj + E ij χ,j + mnsji v m,nsj + C sjimnr v r,mnsj + D sji χ,sj + ϱf i ; 2 - the equations of energy c χ + θ E ij v i,j + D ijk v k,ij =k ij χ,j,i + ϱp; 3 - the initial conditions v i x,=, v i x,=, χx,=, x ; 4
6 Marinetal.oundary Value Problems 23, 23:35 Page 6 of 4 - the boundary conditions v i x, t= on, t, t ji x, tn j = on c, t, χx, t= on 2, t, p i x, tn i = on c 2, t. 5 We are now in a position to prove the first basic result. Theorem For the differences v i, χ of two solutions of the mixed initial boundary value problem 8, 6 and 7,the Lagrangeidentitybecomes 2 ϱv i t v i t dv + t = ds t + t t k ij χ,i ξ dξ χ,j ξ dξ dv θ ϱ v i 2t sf i s v i sf i 2t s ] dv ϱ χs θ s χ2t s 2t s Pξ dξ Pξ dξ ] dv ds, t, t. 6 2 Proof ecause of the linearity of the problem defined by 8, 6 and7, we deduce that the differences v i, χ represent the solution of a mixed initial boundary value problem analogous to 8, 6 and7, namely the problem consisting of equations 2 and3 with loads F i,respectivelyp, the initial conditions 4 and boundary conditions 5. y setting U i x, s=v i x, s, V i x, s=v i x,2t s, s, 2t], t, t, 2 then the identity 9, after some straightforwardcalculation, becomes t 2 ϱv i tv i t dv = ds ϱ v i 2t s v i s v i 2t sv i s ] dv, 7 where we have used the fact that the initial and boundary data are null. We shall eliminate the inertial terms on the right-hand side of the relation 7bymeans of the equations of motion for the differences v i, χ. So, in view of equation 2, we have ϱ v i 2t s v i s v i 2t sv i s ] = { v i 2t s A ijrs v r,s s+ ijpqr v r,pq s+e ij χs + rskji v r,sk s+c kjipqr v r,pqk s+d kji χ,k s ]},j { v i s A ijrs v r,s 2t s+ ijpqr v r,pq 2t s+e ij χ2t s + rskji v r,sk 2t s+c kjipqr v r,pqk 2t s+d kji χ,k 2t s ]},j A ijrs v r,s sv i,j 2t s ijpqr v r,pq sv i,j 2t s E ij χsv i,j 2t s
7 Marinetal.oundary Value Problems 23, 23:35 Page 7 of 4 rskji v r,sk sv i,j 2t s C kjipqr v r,pqk sv i,j 2t s D kji χ,k sv i,j 2t s + A ijrs v r,s 2t sv i,j s+ ijpqr v r,pq 2t sv i,j s+e ij χ2t sv i,j s + rskji v r,sk 2t sv i,j s+c kjipqr v r,pqk 2t sv i,j s+d kji χ,k 2t sv i,j s + ϱ F i sv i 2t s F i 2t sv i s ]. After we use the symmetry relations 5, this equality takes on the form ϱ v i 2t s v i s v i 2t sv i s ] = { v i 2t s A ijrs v r,s s+ ijpqr v r,pq s+e ij χs + rskji v r,sk s+c kjipqr v r,pqk s+d kji χ,k s ]},j { v i s A ijrs v r,s 2t s+ ijpqr v r,pq 2t s+e ij χ2t s + rskji v r,sk 2t s+c kjipqr v r,pqk 2t s+d kji χ,k 2t s ]},j + E ij χ2t svi,j s χsv i,j 2t s ] + D kji χ,k 2t sv i,j s D kji χ,k sv i,j 2t s + ϱ F i sv i 2t s F i 2t sv i s ]. 8 We integrate by parts equality 8, and after using boundary conditions 5, we get the equality ϱ v i 2t s v i s v i 2t sv i s ] dv t = Eij v i,j s+d kji v i,kj s ] χ2t s dv ds t Eij v i,j 2t s+d kji v i,kj 2t s ] χs dv ds t + ϱ F i sv i 2t s F i 2t sv i s ] dv ds. 9 Now we integrate equation 3ontheinterval,s] and take into account the zero initial data in 4 so that we obtain the relation E ij v i,j s+d kji v i,kj s = c χs s χ,j z dz ϱ s Pz dz, θ θ,i θ s, t. 2 After we multiply in equality 2 byχ2t s and use a similar result obtained for E ij v i,j 2t s+d kji v i,kj 2t s, multiplied by χs, we find t Eij v i,j s+d kji v i,kj s ] χ2t s dv ds t Eij v i,j 2t s+d kji v i,kj 2t s ] χs dv ds
8 Marinetal.oundary Value Problems 23, 23:35 Page 8 of 4 = θ ϱ + k ij χ,i 2t s θ χs 2t s s χ,j z dz k ij χ,i s Pz dz χ2t s If we introduce 2 into9, we obtain ϱ v i 2t s v i s v i 2t sv i s ] dv = θ + + θ t k ij χ,i 2t s ϱ χs 2t s s χ,j z dz k ij χ,i s Pz dz χ2t s 2t s s ] χ,j z dz dv ] Pz dz dv. 2 2t s s ] χ,j z dz dv ] Pz dz dv ϱ F i sv i 2t s F i 2t sv i s ] dv ds. 22 ased on the symmetry of the tensor k ij,weget t = = d k ij θ ds θ k ij θ k ij s 2t s ] χ,i ξ dξ χ,j ξ dξ dv ds s 2t s ] χ,i ξ dξ χ,j ξ dξ ds dv t d ds t t χ,i ξ dξ χ,j ξ dξ On the other hand, integrating by parts, we obtain t ds k ij θ t = ds χ,i s d k ij θ ds 2t s s From 23and24wededuce t s ds k ij χ,i 2t s θ s = k ij χ,i ξ dξ θ dv. 23 s ] χ,j ξ dξ χ,j 2t s χ,i ξ dξ dv 2t s ] χ,i ξ dξ χ,j ξ dξ dv. 24 2t s ] χ,j ξ dξ dv ] χ,j ξ dξ dv. 25 χ,j ξ dξ χ,i 2t s 2t s We now substitute 25 in22, and so we are led to equality 6. With this, the proof of Theorem is completed. Remark It is important to note that the identity 6 is just like in the classical thermoelasticity see 7]. The identity 6 constitutes the basis on which we shall prove the uniqueness and the continuous dependence results.
9 Marinetal.oundary Value Problems 23, 23:35 Page 9 of 4 We proceed first to obtain the uniqueness of the solution of the mixed initial boundary value problem defined by 8, 6and7. Theorem 2 Assume that the conductivity tensor k ij is positive definite in the sense there exists a positive constant k such that k ij ξ i ξ j k ξ i ξ i, ξ i. Also, we suppose that the symmetry relations 5 are satisfied. If 2 is not empty or cx on, then the mixed initial boundary value problem in thermoelastodynamics of nonsimple materials has at most one solution. Proof Suppose, by contrary, that our mixed problem defined by 8, 6 and7 hastwo solutions u α i, θ α α =, 2 that correspond to the same initial and boundary data, to the same body force and the same heat supply. If we denote by v i = u 2 i u i, χ = θ 2 i θ i, 26 then we shall prove that v i x, t=, ψ i x, t=, δx, t=, χx, t=, x, t, t. 27 It is clear that the differences v i, ψ i, δ, χfrom26alsorepresentasolutionofourproblem but with null body force and null heat supply. If we write the identity 6 for this particular case, we have t t 2 ϱv i t v i t dv + k ij χ,i ξ dξ χ,j ξ dξ dv =. θ Now, we integrate this equality on the interval, s], s, t /2 and obtain s τ τ ϱv i sv i s dv + k ij χ,i ξ dξ χ,j ξ dξ dv dτ =. θ Taking into account the properties of ϱ and k ij, the above identity proves that v i x, t=, χ,i x, t=, x, t, t /2. 28 If 2 is not empty, considering the boundary conditions 7, then from 28wededuce that 27 holds. Ifax, from the equation of energy written for the differences, we get χ =.However,χ vanishes initially, such that 27againholdstrue. If t is infinite, then the proof of Theorem 2 is complete. If t is finite, then we set v i x, t = v i x, t =, χ x, t = 2 2 2
10 Marin et al.oundary Value Problems 23, 23:35 Page of 4 and repeat the above procedure on the interval t /2, t /2 + t /4] such that we extend the conclusion 27on, 3t /4, and so on. Finally, we obtain 27on, t and this concludes the proof of Theorem 2. We are ready to state and prove the continuous dependence theorem with regard to body force and heat supply on the compact subintervals of the interval, t for the solution of the mixed initial boundary value problem defined by the system of equations 8, the initial conditions 6 and the boundaryconditions 7. Theorem 3 Suppose the same conditions as in Theorem 2. Let u α i, θ α α =,2be two solutions of our mixed problem which correspond to the same initial and boundary data but to different body force and different heat supply,f α i, P α α =,2,where F i = f 2 i f i, P = r2 r. Moreover, we suppose that there exists t, t such that t t ϱf i tf i t dv dt M 2, ϱv i tv i t dv dt K 2, t t ϱ t θ 2 Pξ dξ dv dt M2 2, ϱ θ χ 2 t dv dt Q Then we have the following estimate: s t t ϱv i sv i s dv + k ij χ,i ξ dξ χ,j ξ dξ dv dt θ t ] /2 t K ϱf i tf i t dv dt + t Q t ϱ t 2 /2 Pξ dξ dv dt], 3 θ where v i s and χs are the differences defined in 26 and s, t /2. Proof We will use the identity 6. On the right-hand side of this identity, we employ the Schwarz inequality for each integral. For instance, we have t ϱv i 2t sf i s dv ds t ] /2 t ] /2 ϱf i sf i s dv ds ϱv i 2t sv i 2t s dv ds t ] /2 2t ] /2 = ϱf i sf i s dv ds ϱv i sv i s dv ds K t ϱf i sf i s dv ds] /2, where, at last, we use the substitution 2t s s. t
11 Marin et al.oundary Value Problems 23, 23:35 Page of 4 We proceed analogously with other integrals in the identity 6. Finally, we integrate the resulting inequality over, s], s, t /2] and we obtain the inequality 3 andthe proof of Theorem 3 is complete. In the following theorem, we use the estimate 3 in order to deduce a continuous dependence result upon initial data. Theorem 4 Assume that the symmetry relations 5 are satisfied. Consider u i, θ, u i + v i, θ + χ two solutions of the mixed initial boundary value problem defined by 8, 6 and 7 which correspond to the same body force and heat supply and to the same boundary data, but to different initial data u i, u i, θ, u 2 i, u 2 i, θ 2, where u 2 i = u i + a i, u2 i = u i + a i, θ 2 = θ + d. Here the perturbations a i, a i, d obey the following restrictions: ϱ a i a i + a i a i dv M 2 3, whereweusedthenotation T ϱ η2 dv M2 4, η x= cx θ d x E ij xa i,j x D ijkxa k,ij x. Using perturbation v i and χ, we define the functions U i x, t and x, t by U i x, t= t s v i x, τ dτ ds, x, t= t s χx, τ dτ ds. 3 If the functions U i, satisfy the conditions 29, then we have the following estimate: t s s ϱu i tu i t dv + k ij,i ξ dξ,j ξ dξ dv ds θ t t K + t 2 /2 ϱa i a i dv + ] /2 t 2 /2 + t3 /3 ϱa i a i dv + t 7/2 Q T /2 2 ϱ η2 dv, t, t ] Proof Integrating by parts in 3, we deduce U i x, t= t t sv i x, s ds, x, t= t t sχx, s ds.
12 Marin et al.oundary Value Problems 23, 23:35 Page 2 of 4 It is easy to prove that the difference functions v i, χ satisfy the equations of motion and the equation of energy as in 8, but with null loads f i = r =. Also, by direct calculations, we deduce that the difference functions satisfy the initial conditions in the form v i x,=a i x, v ix,=a i x, χx,=d x, x. Then, a straightforward calculation proves that the functions U i, definedin3satisfy the equations of motion and the equation of energy as in 8, but with the following body force and heat supply: f i x, t=a i x+ta i x, rx, t= θ ] cx ϱ t d x E ij xa i,j θ x D ijkxa k,ij x. y using these specifications, the estimate 32 follows from 3 andtheorem4 is concluded. Finally, we obtain a continuous dependence result of the solution to problems 8, 6 and 7 upon the thermoelastic coefficients, again as a consequence of Theorem 3. Theorem 5 Assume that the symmetry relations 5 are satisfied and consider u i, θ, u i + v i, θ + χ two solutions of the mixed initial boundary value problem defined by 8, 6 and 7 which correspond to the same body force and heat supply and to the same boundary and initial data, but to different thermoelastic coefficients A ijrs, ijpqr, C ijkmnr, E ij, D ijk, b i, k ij, c, A ijrs + A ijrs, ijpqr + ijpqr, C ijkmnr + C ijkmnr, E ij + E ij, D ijk + D ijk, b i + i, k ij + K ij, c + C. Suppose that the perturbations v i, χ satisfy the conditions 29. Then any solution u i, θ of the initial boundary value problem defined by 8, 6 and 7 that satisfies the condition t ui,j u i,j + u i,jk u i,jk + u i,j u i,j + θ,j θ,j + θ,jk θ,jk + θ 2 dv ds M 2 5 depends continuously on the thermoelastic coefficients on the interval, t /2] in t s s ϱv i tv i t+ k ij χ,i ξ dξ χ,j ξ dξ dv ds. θ
13 Marin et al.oundary Value Problems 23, 23:35 Page 3 of 4 Proof A straightforward calculation proves that the perturbations v i, χoftwosolutions verify the equations of motion and the equation of energy with the following body force load and heat supply: ϱf i = A ijrs u 2 r,sj + ijpqru 2 r,pqj + E ijθ 2,j + rskji u 2 r,skj + C kjipqru 2 r,pqkj + D kjiθ 2,kj, ϱp = K ij θ 2,j,i θ C θ θ 2 E ij u 2 i,j D ijk u 2 k,ij Thus the problem is analogous to the problem from Theorem 4. Therefore, according to the estimates 32and3, we obtain the desired result.. 4 Concluding remarks The uniqueness theorem and the continuous dependence theorems were proved without recourse to any conservation laws or to any boundedness assumptions on the thermoelastic coefficients. In various papers, the existence of the solution to the mixed initial boundary value problem defined by 8, 6 and7 is obtained by assuming some strong restrictions. For instance, in the paper 8]the existence of the solution isobtained under assumption that the internal energy density is positive definite. Competing interests The authors declare that they have no competing interests. Authors contributions MIM proposed main results of the paper and verified all calculations and demonstrations. RPA proposed the method of demonstration of results, without using a sophisticated mathematical apparatus. Also, he controlled the final shape of the paper. SRM performed all calculations and demonstrations and took into account the suggestions given by MIM. Author details Department of Mathematics, University of rasov, rasov, Romania. 2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA. 3 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia. 4 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt. Acknowledgements We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions. Received: 4 March 23 Accepted: 4 May 23 Published: 2 May 23 References. Toupin, RA: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 7, Green, AE, Rivlin, RS: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 7, Mindlin, RD: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 6, Pata, V, Quintanilla, R: On the decay of solutions in nonsimple elastic solids with memory. J. Math. Anal. Appl. 363, Martinez, F, Quintanilla, R: On the incremental problem in thermoelasticity of nonsimple materials. Z. Angew. Math. Mech. 78, Ciarletta, M: Thermoelasticity of nonsimple materials with thermal relaxation. J. Therm. Stresses 98, Wilkes, NS: Continuous dependence and instability in linear thermoelasticity. SIAM J. Appl. Math., Green, AE, Lindsay, KA: Thermoelasticity. J. Elast. 2, Levine, HA: An equipartition of energy theorem for weak solutions of evolutionary equations in Hilbert space. J. Differ. Equ. 24, Gurtin, ME: The dynamics of solid-solid phase transitions. Arch. Ration. Mech. Anal. 4, Marin, M: Lagrange identity method for microstretch thermoelastic materials. J. Math. Anal. Appl. 363, Goldstein, JA, Sandefur, JT: Asymptotic equipartition of energy for differential equations in Hilbert space. Trans. Am. Math. Soc. 29, Marin, M: A partition of energy in thermoelasticity of microstretch bodies. Nonlinear Anal., Real World Appl. 4, Day, WA: Means and autocorrections in elastodynamics. Arch. Ration. Mech. Anal. 73,
14 Marin et al.oundary Value Problems 23, 23:35 Page 4 of 4 5. Iesan, D: Thermoelastic Models of Continua. Kluwer Academic, Dordrecht Iesan, D: Thermoelasticity of nonsimple materials. J. Therm. Stresses 62-4, Knops, RJ, Payne, LE: On uniqueness and continuous dependence in dynamical problems of linear thermoelasticity. Int. J. Solids Struct. 6, Iesan, D, Quintanilla, R: Thermal stresses in microstretch bodies. Int. J. Eng. Sci. 43, doi:.86/ Cite this article as: Marin et al.: Nonsimple material problems addressed by the Lagrange s identity. oundary Value Problems 23 23:35.
Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 83-90
Bulletin of the Transilvania University of Braşov Vol 10(59), No. 1-2017 Series III: Mathematics, Informatics, Physics, 83-90 GENERALIZED MICROPOLAR THERMOELASTICITY WITH FRACTIONAL ORDER STRAIN Adina
More informationBulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 49-58
Bulletin of the Transilvania University of Braşov Vol 10(59), No. 2-2017 Series III: Mathematics, Informatics, Physics, 49-58 THERMOELASTICITY WITH FRACTIONAL ORDER STRAIN FOR DIPOLAR MATERIALS WITH VOIDS
More informationOn some exponential decay estimates for porous elastic cylinders
Arch. Mech., 56, 3, pp. 33 46, Warszawa 004 On some exponential decay estimates for porous elastic cylinders S. CHIRIȚĂ Faculty of Mathematics, University of Iaşi, 6600 Iaşi, Romania. In this paper we
More informationEffect of internal state variables in thermoelasticity of microstretch bodies
DOI: 10.1515/auom-016-0057 An. Şt. Univ. Ovidius Constanţa Vol. 43,016, 41 57 Effect of internal state variables in thermoelasticity of microstretch bodies Marin Marin and Sorin Vlase Abstract First, we
More informationUniqueness in thermoelasticity of porous media with microtemperatures
Arch. Mech., 61, 5, pp. 371 382, Warszawa 29 Uniqueness in thermoelasticity of porous media with microtemperatures R. QUINTANILLA Department of Applied Mathematics II UPC Terrassa, Colom 11, 8222 Terrassa,
More informationOn the Three-Phase-Lag Heat Equation with Spatial Dependent Lags
Nonlinear Analysis and Differential Equations, Vol. 5, 07, no., 53-66 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/nade.07.694 On the Three-Phase-Lag Heat Equation with Spatial Dependent Lags Yang
More informationINTERNAL STATE VARIABLES IN DIPOLAR THERMOELASTIC BODIES
Hacettepe Journal of Mathematics and Statistics Volume 43 1 14, 15 6 INTERNAL STATE VARIALES IN DIPOLAR THERMOELASTIC ODIES M. Marin a, S. R. Mahmoud b c, and G. Stan a Received 7 : 1 : 1 : Accepted 3
More informationOn temporal behaviour of solutions in Thermoelasticity of porous micropolar bodies
DOI: 1.2478/auom-214-14 An. Şt. Univ. Ovidius Constanţa Vol. 22(1),214, 169 188 On temporal behaviour of solutions in Thermoelasticity of porous micropolar bodies Marin Marin and Olivia Florea Abstract
More informationMathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY ( 65-7 (6 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity H. M.
More informationStructural Continuous Dependence in Micropolar Porous Bodies
Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 Structural Continuous Dependence in Micropolar Porous odies M. Marin 1,, A.M. Abd-Alla 3,4, D. Raducanu 1 and S.M. Abo-Dahab 3,5 Abstract:
More informationResearch Article Weak Solutions in Elasticity of Dipolar Porous Materials
Mathematical Problems in Engineering Volume 2008, Article ID 158908, 8 pages doi:10.1155/2008/158908 Research Article Weak Solutions in Elasticity of Dipolar Porous Materials Marin Marin Department of
More informationOn vibrations in thermoelasticity without energy dissipation for micropolar bodies
Marin and Baleanu Boundary Value Problems 206 206: DOI 0.86/s366-06-0620-9 R E S E A R C H Open Access On vibrations in thermoelasticity without energy dissipation for micropolar bodies Marin Marin * and
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More informationChapter 3. Forces, Momentum & Stress. 3.1 Newtonian mechanics: a very brief résumé
Chapter 3 Forces, Momentum & Stress 3.1 Newtonian mechanics: a very brief résumé In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration when acted on by external
More informationOn Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers
Continuum Mech. Thermodyn. (1996) 8: 247 256 On Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers I-Shih Liu Instituto de Matemática Universidade do rio de Janeiro, Caixa Postal
More informationQualitative results on mixed problem of micropolar bodies with microtemperatures
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 2 December 2017, pp. 776 789 Applications and Applied Mathematics: An International Journal AAM Qualitative results on
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [University of Salerno] On: 4 November 8 Access details: Access Details: [subscription number 73997733] Publisher Taylor & Francis Informa Ltd Registered in England and
More informationBasic results for porous dipolar elastic materials
asic results for porous dipolar elastic materials MARIN MARIN Department of Mathematics, University of rasov Romania, e-mail: m.marin@unitbv.ro Abstract. This paper is concerned with the nonlinear theory
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
More informationThermodynamics for fluid flow in porous structures
Communications to SIMAI Congress, ISSN 1827-9015, Vol. 1 (2006) DOI: 10.1685/CSC06105 Thermodynamics for fluid flow in porous structures M.E. Malaspina University of Messina, Department of Mathematics
More informationThermodynamics of non-simple elastic materials
Journal of Elasticity, Vol. 6, No.4, October 1976 NoordhotT International Publishing - Leyden Printed in The Netherlands Thermodynamics of non-simple elastic materials R. C. BATRA ME Department, The University
More informationA hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams
A hierarchy of higher order and higher grade continua Application to the plasticity and fracture of metallic foams Samuel Forest Centre des Matériaux/UMR 7633 Mines Paris ParisTech /CNRS BP 87, 91003 Evry,
More informationThe Effect of Heat Laser Pulse on Generalized Thermoelasticity for Micropolar Medium
Mechanics and Mechanical Engineering Vol. 21, No. 4 (2017) 797 811 c Lodz University of Technology The Effect of Heat Laser Pulse on Generalized Thermoelasticity for Micropolar Medium Mohamed I. A. Othman
More informationContinuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms
Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive
More informationModule-3: Kinematics
Module-3: Kinematics Lecture-20: Material and Spatial Time Derivatives The velocity acceleration and velocity gradient are important quantities of kinematics. Here we discuss the description of these kinematic
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by:[youssef, Hamdy M.] On: 22 February 2008 Access Details: [subscription number 790771681] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered
More information2 Basic Equations in Generalized Plane Strain
Boundary integral equations for plane orthotropic bodies and exterior regions G. Szeidl and J. Dudra University of Miskolc, Department of Mechanics 3515 Miskolc-Egyetemváros, Hungary Abstract Assuming
More informationDr. Parveen Lata Department of Basic and Applied Sciences, Punjabi University, Patiala, Punjab, India.
International Journal of Theoretical and Applied Mechanics. ISSN 973-685 Volume 12, Number 3 (217) pp. 435-443 Research India Publications http://www.ripublication.com Linearly Distributed Time Harmonic
More informationResearch Article Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 7, Article ID 5636, 1 pages doi:1.1155/7/5636 Research Article Finite Element Formulation of Forced Vibration Problem of a Prestretched
More informationAn Irreducible Function Basis of Isotropic Invariants of A Third Order Three-Dimensional Symmetric Tensor
An Irreducible Function Basis of Isotropic Invariants of A Third Order Three-Dimensional Symmetric Tensor Zhongming Chen Jinjie Liu Liqun Qi Quanshui Zheng Wennan Zou June 22, 2018 arxiv:1803.01269v2 [math-ph]
More informationOn the linear problem of swelling porous elastic soils
J. Math. Anal. Appl. 269 22 5 72 www.academicpress.com On the linear problem of swelling porous elastic soils R. Quintanilla Matematica Aplicada 2, ETSEIT Universitat Politècnica de Catalunya, Colom 11,
More informationIranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16
Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16 THE EFFECT OF PURE SHEAR ON THE REFLECTION OF PLANE WAVES AT THE BOUNDARY OF AN ELASTIC HALF-SPACE W. HUSSAIN DEPARTMENT
More informationRelaxation methods and finite element schemes for the equations of visco-elastodynamics. Chiara Simeoni
Relaxation methods and finite element schemes for the equations of visco-elastodynamics Chiara Simeoni Department of Information Engineering, Computer Science and Mathematics University of L Aquila (Italy)
More informationAnalytical formulation of Modified Upper Bound theorem
CHAPTER 3 Analytical formulation of Modified Upper Bound theorem 3.1 Introduction In the mathematical theory of elasticity, the principles of minimum potential energy and minimum complimentary energy are
More informationA CONTINUUM MECHANICS PRIMER
A CONTINUUM MECHANICS PRIMER On Constitutive Theories of Materials I-SHIH LIU Rio de Janeiro Preface In this note, we concern only fundamental concepts of continuum mechanics for the formulation of basic
More informationStress of a spatially uniform dislocation density field
Stress of a spatially uniform dislocation density field Amit Acharya November 19, 2018 Abstract It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising
More informationChapter 0. Preliminaries. 0.1 Things you should already know
Chapter 0 Preliminaries These notes cover the course MATH45061 (Continuum Mechanics) and are intended to supplement the lectures. The course does not follow any particular text, so you do not need to buy
More informationExistence and uniqueness of the weak solution for a contact problem
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (216), 186 199 Research Article Existence and uniqueness of the weak solution for a contact problem Amar Megrous a, Ammar Derbazi b, Mohamed
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Volume 19, 2013 http://acousticalsociety.org/ ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Engineering Acoustics Session 1aEA: Thermoacoustics I 1aEA7. On discontinuity
More informationIntroduction to Continuum Mechanics
Introduction to Continuum Mechanics I-Shih Liu Instituto de Matemática Universidade Federal do Rio de Janeiro 2018 Contents 1 Notations and tensor algebra 1 1.1 Vector space, inner product........................
More informationA note on stability in three-phase-lag heat conduction
Universität Konstanz A note on stability in three-phase-lag heat conduction Ramón Quintanilla Reinhard Racke Konstanzer Schriften in Mathematik und Informatik Nr. 8, März 007 ISSN 1430-3558 Fachbereich
More informationEFFECT OF DISTINCT CONDUCTIVE AND THERMODYNAMIC TEMPERATURES ON THE REFLECTION OF PLANE WAVES IN MICROPOLAR ELASTIC HALF-SPACE
U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 2, 23 ISSN 223-727 EFFECT OF DISTINCT CONDUCTIVE AND THERMODYNAMIC TEMPERATURES ON THE REFLECTION OF PLANE WAVES IN MICROPOLAR ELASTIC HALF-SPACE Kunal SHARMA,
More informationTHE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM
Internat. J. Math. & Math. Sci. Vol., No. 8 () 59 56 S67 Hindawi Publishing Corp. THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM ABO-EL-NOUR N. ABD-ALLA and AMIRA A. S. AL-DAWY
More informationBLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF TYPE WITH ARBITRARY POSITIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 6 6, No. 33, pp. 8. ISSN: 7-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF
More informationMechanics of Materials and Structures
Journal of Mechanics of Materials and Structures FUNDAMENTAL SOLUTION IN THE THEORY OF VISCOELASTIC MIXTURES Simona De Cicco and Merab Svanadze Volume 4, Nº 1 January 2009 mathematical sciences publishers
More informationOn the Numerical Modelling of Orthotropic Large Strain Elastoplasticity
63 Advances in 63 On the Numerical Modelling of Orthotropic Large Strain Elastoplasticity I. Karsaj, C. Sansour and J. Soric Summary A constitutive model for orthotropic yield function at large strain
More informationModels for dynamic fracture based on Griffith s criterion
Models for dynamic fracture based on Griffith s criterion Christopher J. Larsen Abstract There has been much recent progress in extending Griffith s criterion for crack growth into mathematical models
More information4 Constitutive Theory
ME338A CONTINUUM MECHANICS lecture notes 13 Tuesday, May 13, 2008 4.1 Closure Problem In the preceding chapter, we derived the fundamental balance equations: Balance of Spatial Material Mass ρ t + ρ t
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationExistence and Uniqueness of the Weak Solution for a Contact Problem
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. x (215), 1 15 Research Article Existence and Uniqueness of the Weak Solution for a Contact Problem Amar Megrous a, Ammar Derbazi b, Mohamed Dalah
More informationChapter 3 Stress, Strain, Virtual Power and Conservation Principles
Chapter 3 Stress, Strain, irtual Power and Conservation Principles 1 Introduction Stress and strain are key concepts in the analytical characterization of the mechanical state of a solid body. While stress
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More informationThe Rotating Inhomogeneous Elastic Cylinders of. Variable-Thickness and Density
Applied Mathematics & Information Sciences 23 2008, 237 257 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. The Rotating Inhomogeneous Elastic Cylinders of Variable-Thickness and
More informationTime Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time
Tamkang Journal of Science and Engineering, Vol. 13, No. 2, pp. 117 126 (2010) 117 Time Harmonic Inclined Load in Micropolar Thermoelastic Medium Possesing Cubic Symmetry with One Relaxation Time Praveen
More informationPhysical Conservation and Balance Laws & Thermodynamics
Chapter 4 Physical Conservation and Balance Laws & Thermodynamics The laws of classical physics are, for the most part, expressions of conservation or balances of certain quantities: e.g. mass, momentum,
More informationNon-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3
Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Tommaso Ruggeri Department of Mathematics and Research Center of Applied Mathematics University of Bologna January 21, 2017 ommaso
More informationMomentum and Energy. Chapter Conservation Principles
Chapter 2 Momentum an Energy In this chapter we present some funamental results of continuum mechanics. The formulation is base on the principles of conservation of mass, momentum, angular momentum, an
More informationFixed point theorems for a class of maps in normed Boolean vector spaces
RESEARCH Fixed point theorems for a class of maps in normed Boolean vector spaces Swaminath Mishra 1, Rajendra Pant 1* and Venkat Murali 2 Open Access * Correspondence: pant. rajendra@gmail.com 1 Department
More informationMaximax rearrangement optimization related to a homogeneous Dirichlet problem
DOI 10.1007/s40065-013-0083-0 M. Zivari-Rezapour Maximax rearrangement optimization related to a homogeneous Dirichlet problem Received: 26 June 2012 / Accepted: 1 July 2013 The Author(s) 2013. This article
More informationMath Tune-Up Louisiana State University August, Lectures on Partial Differential Equations and Hilbert Space
Math Tune-Up Louisiana State University August, 2008 Lectures on Partial Differential Equations and Hilbert Space 1. A linear partial differential equation of physics We begin by considering the simplest
More information(2) 2. (3) 2 Using equation (3), the material time derivative of the Green-Lagrange strain tensor can be obtained as: 1 = + + +
LAGRANGIAN FORMULAION OF CONINUA Review of Continuum Kinematics he reader is referred to Belytscho et al. () for a concise review of the continuum mechanics concepts used here. he notation followed here
More informationNon-Conventional Thermodynamics and Models of Gradient Elasticity
entropy Article Non-Conventional Thermodynamics and Models of Gradient Elasticity Hans-Dieter Alber 1, Carsten Broese 2, Charalampos Tsakmakis 2, * and Dimitri E. Beskos 3 1 Faculty of Mathematics, Technische
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sci. Technol., 3() (0), pp. 7-39 International Journal of Pure and Applied Sciences and Technology ISSN 9-607 Available online at www.ijopaasat.in Research Paper Reflection of Quasi
More informationEXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM
1 EXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM V. BERTI and M. FABRIZIO Dipartimento di Matematica, Università degli Studi di Bologna, P.zza di Porta S. Donato 5, I-4126, Bologna,
More informationExponential decay for the solution of semilinear viscoelastic wave equations with localized damping
Electronic Journal of Differential Equations, Vol. 22(22), No. 44, pp. 1 14. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Exponential decay
More informationSome New Results on Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy
Some New Results on Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy Patrizia Pucci 1 & James Serrin Dedicated to Olga Ladyzhenskaya with admiration and esteem 1. Introduction.
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationOn the torsion of functionally graded anisotropic linearly elastic bars
IMA Journal of Applied Mathematics (2007) 72, 556 562 doi:10.1093/imamat/hxm027 Advance Access publication on September 25, 2007 edicated with admiration to Robin Knops On the torsion of functionally graded
More informationON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES
ON THE FOLIATION OF SPACE-TIME BY CONSTANT MEAN CURVATURE HYPERSURFACES CLAUS GERHARDT Abstract. We prove that the mean curvature τ of the slices given by a constant mean curvature foliation can be used
More information16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #8 Principle of Virtual Displacements
6. Techniques of Structural Analysis and Design Spring 005 Unit #8 rinciple of irtual Displacements Raúl Radovitzky March 3, 005 rinciple of irtual Displacements Consider a body in equilibrium. We know
More informationDAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE
Materials Physics and Mechanics 4 () 64-73 Received: April 9 DAMPING OF GENERALIZED THERMO ELASTIC WAVES IN A HOMOGENEOUS ISOTROPIC PLATE R. Selvamani * P. Ponnusamy Department of Mathematics Karunya University
More informationOn one system of the Burgers equations arising in the two-velocity hydrodynamics
Journal of Physics: Conference Series PAPER OPEN ACCESS On one system of the Burgers equations arising in the two-velocity hydrodynamics To cite this article: Kholmatzhon Imomnazarov et al 216 J. Phys.:
More information. D CR Nomenclature D 1
. D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the
More informationThe viscosity technique for the implicit midpoint rule of nonexpansive mappings in
Xu et al. Fixed Point Theory and Applications 05) 05:4 DOI 0.86/s3663-05-08-9 R E S E A R C H Open Access The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationPropagation of Rayleigh Wave in Two Temperature Dual Phase Lag Thermoelasticity
Mechanics and Mechanical Engineering Vol. 21, No. 1 (2017) 105 116 c Lodz University of Technology Propagation of Rayleigh Wave in Two Temperature Dual Phase Lag Thermoelasticity Baljeet Singh Department
More informationNewtonian Mechanics. Chapter Classical space-time
Chapter 1 Newtonian Mechanics In these notes classical mechanics will be viewed as a mathematical model for the description of physical systems consisting of a certain (generally finite) number of particles
More informationStability of Thick Spherical Shells
Continuum Mech. Thermodyn. (1995) 7: 249-258 Stability of Thick Spherical Shells I-Shih Liu 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro Caixa Postal 68530, Rio de Janeiro 21945-970,
More informationOther state variables include the temperature, θ, and the entropy, S, which are defined below.
Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive
More informationCH.1. THERMODYNAMIC FOUNDATIONS OF CONSTITUTIVE MODELLING. Computational Solid Mechanics- Xavier Oliver-UPC
CH.1. THERMODYNAMIC FOUNDATIONS OF CONSTITUTIVE MODELLING Computational Solid Mechanics- Xavier Oliver-UPC 1.1 Dissipation approach for constitutive modelling Ch.1. Thermodynamical foundations of constitutive
More informationResearch Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space
Applied Mathematics Volume 011, Article ID 71349, 9 pages doi:10.1155/011/71349 Research Article Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space Sukumar Saha BAS Division,
More informationNiiranen, Jarkko; Khakalo, Sergei; Balobanov, Viacheslav Strain gradient elasticity theories in lattice structure modelling
Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Niiranen, Jarkko; Khakalo, Sergei;
More informationDRIVING FORCE IN SIMULATION OF PHASE TRANSITION FRONT PROPAGATION
Chapter 1 DRIVING FORCE IN SIMULATION OF PHASE TRANSITION FRONT PROPAGATION A. Berezovski Institute of Cybernetics at Tallinn Technical University, Centre for Nonlinear Studies, Akadeemia tee 21, 12618
More informationINTRODUCTION TO CONTINUUM MECHANICS ME 36003
INTRODUCTION TO CONTINUUM MECHANICS ME 36003 Prof. M. B. Rubin Faculty of Mechanical Engineering Technion - Israel Institute of Technology Winter 1991 Latest revision Spring 2015 These lecture notes are
More informationResearch Article Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument
Journal of Applied Mathematics Volume 2012, Article ID 498073, 18 pages doi:10.1155/2012/498073 Research Article Oscillation Criteria of Certain hird-order Differential Equation with Piecewise Constant
More informationAvailable online Journal of Scientific and Engineering Research, 2016, 3(6): Research Article
vailable online www.jsaer.com 6 (6):88-99 Research rticle IN: 9-6 CODEN(U): JERBR -D Problem of Generalized Thermoelastic Medium with Voids under the Effect of Gravity: Comparison of Different Theories
More informationA maximum principle for optimal control system with endpoint constraints
Wang and Liu Journal of Inequalities and Applications 212, 212: http://www.journalofinequalitiesandapplications.com/content/212/231/ R E S E A R C H Open Access A maimum principle for optimal control system
More informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationLinearized theory of elasticity
Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark
More informationResearch Article Propagation of Plane Waves in a Thermally Conducting Mixture
International Scholarly Research Network ISRN Applied Mathematics Volume 211, Article ID 31816, 12 pages doi:1.542/211/31816 Research Article Propagation of Plane Waves in a Thermally Conducting Mixture
More informationResearch Article Global Attractivity of a Higher-Order Difference Equation
Discrete Dynamics in Nature and Society Volume 2012, Article ID 930410, 11 pages doi:10.1155/2012/930410 Research Article Global Attractivity of a Higher-Order Difference Equation R. Abo-Zeid Department
More informationFormulation of the problem
TOPICAL PROBLEMS OF FLUID MECHANICS DOI: https://doi.org/.43/tpfm.27. NOTE ON THE PROBLEM OF DISSIPATIVE MEASURE-VALUED SOLUTIONS TO THE COMPRESSIBLE NON-NEWTONIAN SYSTEM H. Al Baba, 2, M. Caggio, B. Ducomet
More informationInternal Variables and Generalized Continuum Theories
Internal Variables and Generalized Continuum Theories Arkadi Berezovski, Jüri Engelbrecht and Gérard A. Maugin Abstract The canonical thermomechanics on the material manifold is enriched by the introduction
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationDissipation Function in Hyperbolic Thermoelasticity
This article was downloaded by: [University of Illinois at Urbana-Champaign] On: 18 April 2013, At: 12:23 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954
More informationNEW APPROACH TO THE NON-CLASSICAL HEAT CONDUCTION
Journal of Theoretical and Applied Mechanics Sofia 008 vol. 38 No. 3 pp 61-70 NEW APPROACH TO THE NON-CLASSICAL HEAT CONDUCTION N. Petrov Institute of Mechanics Bulgarian Academy of Sciences Acad. G. Bonchev
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationOptimal design and hyperbolic problems
Mathematical Communications 4(999), 2-29 2 Optimal design and hyperbolic problems Nenad Antonić and Marko Vrdoljak Abstract. Quite often practical problems of optimal design have no solution. This situation
More informationA RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS
A RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS KATARINA JEGDIĆ, BARBARA LEE KEYFITZ, AND SUN CICA ČANIĆ We study a Riemann problem for the two-dimensional isentropic gas dynamics equations
More information