On Kelvin-Voigt model and its generalizations

Transcription

1 Nečas Center for Mathematical Moeling On Kelvin-Voigt moel an its generalizations M. Bulíček, J. Málek an K. R. Rajagopal Preprint no Research Team 1 Mathematical Institute of the Charles University Sokolovská 83, Praha 8

2 ON KELVIN-VOIGT MODEL AND ITS GENERALIZATIONS MIROSLAV BULÍČEK, JOSEF MÁLEK, AND K.R. RAJAGOPAL Abstract. We consier a generalization of the Kelvin-Voigt moel where the elastic part of the Cauchy stress epens non-linearly on the linearize strain an the issipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity graient. The assumption that the Cauchy stress epens non-linearly on the linearize strain can be justifie if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, an linearizes uner the assumption that the isplacement graient is small. Long-time an large ata existence, uniqueness an regularity properties of weak solution to such a generalize Kelvin-Voigt moel are establishe for the non-homogeneous mixe bounary value problem. The main novelty consists in incluing nonlinear issipation into the analysis. 1. Introuction The classical Kelvin-Voigt viscoelastic soli (see Kelvin [6], Voigt [8]) can be viewe as a mixture of a linearize elastic soli an a linearly viscous flui that co-exist. The one-imensional moel is represente as a linear spring in parallel with a linearly viscous ashpot. A generalization of the mechanical analog is to consier a non-linear spring in parallel with a non-linearly viscous ashpot. Such a one-imensional moel can be appropriately generalize to obtain a three imensional moel. Recently, Rajagopal [4] (see also [5]) has consiere generalizations of the classical Kelvin- Voigt moel wherein he allows for both the elastic soli an viscous flui to be escribe through implicit constitutive relations. That is, the elastic soli is escribe by a constitutive relation of the form 1 (1.1) H(T e, B) =, where T e is the stress in the elastic soli, an B the right Cauchy-Green stretch, while the flui is escribe by a constitutive relation of the form (1.) G(T v, D) =, where T v is the stress in the flui, an D is the symmetric part of the velocity graient. The Cauchy stress T in the viscoelastic soli is given by (1.3) T = T e + T v. Key wors an phrases. Kelvin-Voigt moel, viscoelastic soli, non-linear wave equation, weak solution, large ata existence, uniqueness, regularity. Miroslav Bulíček thanks the Jinřich Nečas Center for Mathematical Moeling (the project LC65 finance by MŠMT) for its support. Josef Málek s contribution is a part of the research project MSM finance by MŠMT; the support of GAČR 1/9/917 is also acknowlege. K. R. Rajagopal thanks the National Science Founation an the Jinřich Nečas Center for Mathematical Moeling for their support. 1 The kinematical quantities in the introuction are efine in the next section. 1

3 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL The aitive ecomposition reflects the fact that the boy uner consieration is a mixture of coexisting components. The above moel is one for a compressible viscoelastic boy. If one is intereste in a moel for an incompressible material, we replace T e an T v in (1.1) an (1.) by S e an S v, where S e an S v are the eviatoric part of the stress in the elastic an viscous boies an (1.3) takes the form (1.6) T = pi + S e + S v or T = pi + S with S = S e + S v. In the case of the incompressible flui we woul replace (1.1) an (1.) by (1.7) ĤH(S e, B) =, an (1.8) ĜG(S v, D) =. In general, one woul have to solve (1.1) (1.3) an the balance of mass an linear momentum simultaneously, or, in the case of an incompressible boy, the moifie equations (1.7), (1.8) an (1.6), an the balance of linear momentum an the constraint of incompressibility, simultaneously. Solving such systems, for compressible an incompressible viscoelastic boies, woul seem much more aunting than what one oes usually, that is, to substitute the expression for the stress in the balance of linear momentum to obtain a partial ifferential equation for the isplacement fiel. However, such a proceure increases the orer of the partial ifferential equation for the isplacement fiel one eals with an leas to issues concerning the orer of the ifferentiability an hence the space of functions in which one seeks a solution. It is not merely the mathematical issue concerning the ifferentiability of solutions that warrants one to look at the type of generalizations consiere by Rajagopal [4]. In fact, the much more important an interesting reason for consiering such moels is the ability to consier general elastic response that cannot be capture within the context of either Cauchy elasticity or Green elasticity an the issipative response that allows one to consier fluis that have a threshol, such as Bingham flui 3, an/or fluis whose viscosity epens on the mean normal stress in the flui an the symmetric part of the velocity graient. Such elastic an issipative response is escribe by the implicit moels (1.1) an (1.), respectively. (In the case of an incompressible boy, such response is given by (1.7) an (1.8).) The Cauchy stress in a classical incompressible three-imensional Kelvin-Voigt soli is given by (1.4) T = pi + µb + ηd, where pi is the reaction stress ue to the constraint of incompressibility, B is the right Cauchy-Green stretch tensor an D is the symmetric part of the velocity graient. Also µ enotes the shear moulus of the elastic component an η is twice the shear viscosity of the flui component. The compressible counterpart to (1.4) takes the form (1.5) T = µb + ηd. While the moel (1.4) is use often to escribe a variety of polymeric solis an geological materials, the moel (1.5) is not use that often. The moel (1.5) might be appropriate for a class of compressible polymeric foams. 3 If one accepts the efinition that a flui is a boy that cannot support a shear stress, then the notion of a Bingham flui is an untenable concept. However, the moel seems to capture the behavior of certain flui-like boies within the context of observable length scales, time scales an force scales.

4 KELVIN-VOIGT VISCOELASTIC SOLID 3 If one is intereste in the strains being small 4, then (1.1) woul take the form (1.9) H(T e, ε) =, where ε is the linearize strain. While the special subclass of moels (1.1) ε = H (T e ) (H is in general a non-linear function) can be properly justifie (see Rajagopal [4]), the moel (1.11) T e = H(ε), where H is non-linear is inconsistent, if one starts with the notion of a Cauchy elastic boy, as one has alreay presume the nonlinear part of the strain is to be ignore. However, if (1.1) is invertible, then one can justify the use of (1.11), but only as approximation that stems from a totally ifferent stanpoint (see Rajagopal [4] for a etaile iscussion of the issues). In this paper, we shall consier both compressible an incompressible viscoelastic solis that are mixtures of an elastic soli given, in the case of a compressible soli, by (1.11), with the constitutive relation for the flui given by (1.1) T v = G(D). At this juncture it woul be appropriate to point out that there have been several papers that consier partial ifferential equations that are motivate as stemming from a consieration of the equations of motion for a viscoelastic material; one such an example is the recent paper by Tvet [7]. Unfortunately, the motivation is not very accurate. If the boy uner consieration is an incompressible viscoelastic soli, then the equations (1.1) in [7] shoul contain a pressure graient an if it is a compressible material, the ensity shoul appear in the governing equations (1.1) of the paper an one woul also have to solve the balance of mass that gets couple to Eq.(1.1) by the ensity. The papers by Tvet [7], Frieman an Nečas [] an others are primarily concerne with mathematical issues, their relevance to physics take a back seat. For instance in [7] one is not mae aware of what strain measure an stress measure are being use, or for that matter whether they are ealing with with a non-linear measure of strain or the linearize strain. Of course, in one imension these issues o not arise (as also pointe out in Sect. 3 below), however the author clearly mentions that he is ealing with vector-value functions. In this paper we stuy the existence of a weak solution (an its uniqueness an regularity) to equations (1.3), (1.11) an (1.1) an the balance of linear momentum (an the balance of mass that is in our setting however uncouple from the balance of linear momentum), an we also investigate the same to the governing equations for an incompressible viscoelastic boy. When we cast a compressible viscoelastic generalization of a Kelvin-Voigt soli as (1.13) B = H 1 (T e ), an (1.14) D = G 1 (T v ), 4 The linearization that leas to classical linearize elasticity is not that the strains be small, rather the assumption is that the isplacement graient be small which implies that max u = o(δ), < δ << 1. x B,t R The above conition implies that both the strains an rotations are small an not merely that the strains are small.

5 4 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL with T being given by (1.3), we cannot substitute for the stress T given by (1.3) into the balance of linear momentum, but we nee to solve the system of partial ifferential equations given by (1.13), (1.14), (1.3) an the balance of mass an linear momentum simultaneously. We now have equations for the unknowns ϱ, u, T e, T v an T. In the case of an incompressible material, the constitutive relations take the form (1.15) B = H (S e ), an (1.16) D = G (S v ). We will have to solve (1.15), (1.16), (1.6), the constraint of incompressibility an the balance of linear momentum simultaneously for the unknowns p, u, S e, S v an T. We shall make a few brief remarks concerning the linearize version for the compressible moel. In this case, we consier the constitutive relations to be given by (1.17) ε = H 3 (T e ), an (1.18) ε = G 3 (T v ). We woul nee to solve (1.17), (1.18) in conjunction with the balance of mass an linear momentum. If we are to restrict ourselves to solving Quasi-Static problems wherein the inertial term in the balance of linear momentum is neglecte, the problem simplifies consierably as we have to solve (1.17), (1.18), (1.3) in conjunction with (1.19) iv T =. The balance of mass gets uncouple from the other equations an can be use to etermine the ensity. Thus, the unknowns are u, T e, T v, T an the appropriate equations are (1.17), (1.18), (1.3) an (1.19).. Kinematics an Balance laws Let B enote a boy an let κ t : B E, where E is a three imensional Eucliean space, be a family of placers parameterize by time t [, ) an κ t (B) the configuration of the boy at time t. We shall assume that for any t the placers are one to one. Such a family of placers is calle a motion an one can ientify the motion with a mapping χ κ : κ (B) R κ t (B) such that 5 (.1) x = χ κ (X, t). The isplacement u is efine as the ifference between x an X that are relate through (.1), i.e., (.) u(t, X) = χ κ (t, X) X. (.3) The velocity v an the eformation graient F are efine through v = χ κ t = u t, (.4) F = χ κ X. 5 In what follows we suppress notation of the epenence of the quantities on κt.

6 KELVIN-VOIGT VISCOELASTIC SOLID 5 We assume that et F > in κ (B). The right an left Cauchy-Green stretch tensors are efine through (.5) C = F T F, B = FF T, (A T enotes transpose to A) respectively, an the Green-St. Venant an the Almansi-Hamel strains are efine through E = 1 (C I), e = 1 ( I B 1), respectively. The linearize strain is efine through (.6) ε = 1 ( X u + ( X u) T ). The velocity graient L an its symmetric part D are efine through (.7) L = x v, D = 1 (L + LT ). Any property ϕ associate with a boy B can be expresse as (.8) ϕ = ˆϕ(t, X) = ϕ(t, x), which results to the introuction of the following Lagrangean an Eulerian temporal an spatial erivatives: ϕ ˆϕ (.9) := t t, ϕ,t := ϕ t, Xϕ = ˆϕ X, xϕ := ϕ x. The Lagrangean an Eulerian ivergence operators will be expresse as iv X an iv x, respectively. It follows from the notations an efinitions given above, in particular from (.9) an (.3) that ϕ (.1) t = ϕ,t + x ϕ v. Since we consier, in this stuy, processes that take place at uniform temperature we provie merely the formulation for the balance of mass an balance of linear momentum. Their Eulerian formulation is escribe by (.11) ϱ,t + iv x (ϱv) = an (ϱv),t + iv x (ϱv v) = iv x T in κ t (B), while their Lagrangean formulation is (.1) t (ϱ et F) = an ϱ v t = iv ( X (et F)TF T ) in κ (B), where ϱ : R enotes the initial istribution of the ensity, i.e., ϱ(, ) = ϱ. Note that in the Lagrangean formulation the balance of linear momentum inclues the initial (given) ensity an if T is inepenent of ϱ the equation for v can be solve separately from the equation for ϱ. If the boy is incompressible, i.e., iv v = in the Eulerian escription an et F = 1 in the Lagrangean framework, the Eulerian formulation of the balance of equations is given by (.13) iv v =, ϱ,t + x ϱ v = an ϱ (v,t + iv x (v v)) = x p + iv x S in κ t (B), while the Lagrangean form reuces to (.14) et F = 1, t ϱ = an ϱ v t = ( Xp + iv X TF T ) in κ (B).

7 6 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL Finally, in the linearize theory, when the isplacement graient is small an the Cauchy stress T an the Piola-Kirchoff stress (et F)TF T are of the same orer O(1), the governing equations for a compressible boy then rea (.15) ϱ(t, )(1 + tr ε(t, )) = ϱ an ϱ v t = iv X T in κ (B). It follows from (.15) 1 that ϱ an ϱ are of the same orer as tr ε is of the orer O(δ). Thus, in the linearize theory, it oes not matter whether one uses ϱ or ϱ. For the (inhomogeneous) incompressible soli, the governing equations take the form (.16) ϱ(t, ) = ϱ an ϱ v t = Xp + iv X S in κ (B). Since we focus our analysis on the moel for a compressible boy, we eal with the equation (.15) where T is given through the constitutive relations (1.3), (1.11) an (1.1). Thus, we are intereste in unerstaning the properties of the weak solution to relevant initial an bounary value problems (for given initial ensity, isplacement an velocity, an for given isplacement or traction on the bounary) associate with the system of partial ifferential equations (.17) ϱ u t = iv[ H(ε) + G(D)], where iv stans for iv X. Note that the graient of the pressure appears at the right-han sie of (.) if the boy is incompressible. In Sections 4 6, where the mathematical analysis of the initial an bounary value problem associate to (.17) is presente, we shall however use a slightly ifferent notation to express (1.11) an (1.1), namely (.18) T e = H(D(u)), an (.19) T v = G(D(v)), where D(u) an D(v) enote the symmetric part of the graients of the isplacement an the velocity respectively. Then the equation (.17) takes the form u (.) ϱ = iv[h(d(u)) + G(D(v))]. t While (.) is obtaine for small graient of the isplacement, we will show next that one can en-up with the scalar equation of the form (.) if one restricts ourself to special oneimensional eformations an start with the fully non-linear moel (it means that the graient of the isplacement is not necessarily small). 3. A special motion Let us consier the special shearing motion of an incompressible Kelvin-Voigt soli (1.4), with a view towars a generalization which will follow, given by (3.1) x = X + u(y, t), y = Y, z = Z with the pressure fiel being given by (3.) p = p(t, x).

8 KELVIN-VOIGT VISCOELASTIC SOLID 7 The above motion (3.1) automatically satisfies the constraint of incompressibility. It immeiately follows that the balance of linear momentum reuces to ϱ u,tt = p,x + µu,yy + ηu,yyt, (3.3) = p,y, = p,z. This implies that the pressure can almost be a function of time an linear in x. Let us seek a solution wherein the pressure is a constant; in this case the problem reuces to (3.4) ϱ u,tt = µu,yy + ηu,yyt. In the above equation the ensity ϱ is given. We coul also stuy the above motion within the context of a compressible Kelvin-Voigt soli (1.5). Even in this case we obtain only one non-trivial component for the balance of linear momentum, namely (3.4). If one consiers generalizations of both the incompressible an compressible Kelvin-Voigt moels given by T e = H(B), an T v = G(D), with T given by (1.3), then in the case of simple shear we obtain that (3.5) ϱ u,tt = [H 1 (u,y )],y + [G 1 (u,yt )],y, where ϱ is given. To conclue, we aime to show that the governing system of equations to a class of fully nonlinear Kelvin-Voigt moels reuces, at a simple shearing motion, to the form (.) without necessity to restrict ourselves to small eformation graients. 4. Existence, uniqueness an regularity of weak solution 4.1. Equations an assumptions. Let := κ (B) R be an open boune omain with Lipschitz bounary that contains two smooth parts Γ D an Γ N such that = Γ D Γ N. For such an for T >, an for given ϱ : R, u : R, v : R, u D : (, T ) Γ D R an g : (, T ) Γ N R, we consier the following problem: to fin u : (, T ) R fulfilling (4.1) ϱ u,tt iv T = in (, T ), u(, ) = u in, u,t (, ) = v in, u = u D on (, T ) Γ D, Tn = g on (, T ) Γ N, where the stress tensor T is suppose to be of the form { G(D(u,t (t, x))) + H(D(u(t, x))) for compressible boy, T(t, x) := G(D(u,t (t, x))) + H(D(u(t, x))) p(t, x)i for incompressible boy, where pi enotes the unetermine part of the stress. In the latter case, we have to a the constraint of incompressibility expresse in the form iv u = iv v =. Regaring the tensorial functions G an H we assume that they are continuous on R an that there are r, q (1, ) an C 1, C > such that for arbitrary D 1, D R the following

9 8 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL conitions hol: (A1) (A) 1 (G(D 1 ) G(D )) (D 1 D ) C 1 D 1 D (1 + D 1 + s(d D 1 ) ) r s, 1 (H(D 1 ) H(D )) (D 1 D ) C 1 D 1 D (1 + D 1 + s(d D 1 ) ) q s, G(D 1 ) G(D ) C D 1 D H(D 1 ) H(D ) C D 1 D Moreover, we assume that 1 1 (1 + D 1 + s(d D 1 ) ) r s, (1 + D 1 + s(d D 1 ) ) q s. (A3) G() = an H() =. Note that (A1) (A3) lea to the stanar coercivity an growth conitions (see for example [3] for etails an note that the precise value of the constants C 1, C may iffer from C 1 an C use in (A1) an (A)) (4.) (4.3) G(D) D C 1 ( D r 1), H(D) D C 1 ( D q 1), G(D) C (1 + D ) r 1, H(D) C (1 + D ) q 1. Since, the methos we use later are inepenent of solving the compressible or the incompressible case, we will focus only on the compressible one in what follows. However, all results that we shall prove can be formulate also for incompressible boies. Before we efine the notion of a weak solution we first introuce the function spaces neee later. The Lebesgue spaces L p () an the Sobolev spaces W 1,p () are efine, for 1 p, in a stanar way. Moreover, we efine W 1,p Γ D () := {u W 1,p (); u ΓD = }. In aition, if u L p () an v L p () we use the notation (u, v) := uv x. Similarly, we also abbreviate bounary integrals. The same comment concerns vector- an tensor-value functions as well. Finally, for any Banach space X, we enote X := } X. {{.. X }. times Next, we efine the notion of a weak solution to (4.1). Definition 4.1. Let C,1, T > an G, H satisfy (4.) (4.3). In aition, assume that g L r (, T ; (W 1 1 r,r (Γ N ) ) ), u W 1,q (), v L () an u D W 1,r (, T ; W 1 1 r,r (Γ D ) ). Moreover, let m, M be such that < m ϱ (x) M < for a.a. x. We say that u is a weak solution (4.1) if (4.4) u L (, T ; W 1,q () ) W 1, (, T ; L () ) W 1,r (, T ; W 1,r () ), ϱ u,tt L min(r,q ) (, T ; (W 1,max(r,q) Γ D () ) ), u (,T ) ΓD = u D, u,t (,T ) ΓD = v D := (u D ),t, the following weak formulation of (4.1) hols (4.5) T ϱ u,tt, ϕ + ( G(D(u,t )) + H(D(u)), D(ϕ) ) T t = g, ϕ ΓN t for all ϕ L max(r,q) (, T ; W 1,max(r,q) Γ D () ),

10 KELVIN-VOIGT VISCOELASTIC SOLID 9 an the initial conitions are met in the following sense ( (4.6) lim u,t (t) v + u(t) u q ) t 1,q =. + Theorem 4.1 (Existence an Uniqueness). Assume that all the assumptions of Definition 4.1 are satisfie. Let G, H satisfy (A1) 1, (A) an (A3) with 1 < q r <. Moreover, let u W 1, () an ũ D W,r (, T ; W 1,r () ) be such that u ũ D (, ) W 1,r Γ D () an ũ D (,T ) Γ = u D. Then there exists unique weak solution to (4.1). Before formulating the results concerning the regularity properties of the weak solution to (4.1) we slightly strengthen the assumptions (A1) an (A) concerning G an H. We shall suppose that G an H are continuously ifferentiable on R an that there are C 1, C > such that for arbitrary A, D R the following conitions hol: (A1*) (A*) C 1 (1 + A ) r D G D (A) (D D) C (1 + A ) r D H D (A) (D D) C (1 + A ) q D Theorem 4. (Regularity). Let all the assumptions of Theorem 4.1 be satisfie an H an G fulfil (A1*) (A*). i) If in aition ϱ C,1 () an v W 1, loc () then (4.7) (4.8) (4.9) (1 + D(v) ) r D( v) L (, T ; L loc () ), v L (, T ; W 1, loc () ), v,t L r (, T ; L r loc () ). ii) Moreover, if ϱ C,1 (), v W 1, (), g W 1,r (, T ; W 1+ 1 r,r (Γ D )) an either Dirichlet or Neumann bounary conitions hol on connecte components of the bounary, then (4.1) (4.11) (4.1) v,t L (, T ; L () ), (1 + D(v) ) r D(v,t ) L (, T ; L () ), (1 + D(v) ) r D( v) L (, T ; L () ). The main novelty of Theorem 4.1 an Theorem 4. consists in allowing a nonlinear epenence of G on the symmetric part of the velocity graient D(v) (here we are able to analyze the case r > ) an investigating the problem with non-constant (given) ensity. Moreover, we o not nee to assume the existence of potentials to H an G in orer to prove the results state in both theorems. Such generalizations were not consiere in the earlier stuies by Tvet [7], an were solve only in some special cases (G growing at most linearly with D(v), an = or C C 1 being small) by Frieman an Nečas []. We also remark that we coul also prove that v L (, T ; W 1,r () ) if we assume the existence of a potential to G. Since one of our aims was to avoi the assumptions of this type, we o not inclue corresponing regularity result in the statement of Theorem 4..

11 1 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL For our analysis it is suitable to rewrite (4.1) in the form (4.13) ϱ v,t iv G(D(v)) iv H(D(u)) = in (, T ), u,t = v in (, T ), u(, x) = u, v(, x) = v in, u = u D, v = v D on (, T ) Γ D, (G(D(v)) + H(D(u)))n = g on (, T ) Γ N. 5. Proof of Theorem Galerkin approximation. Since W 1,r Γ () is separable there is a basis {w n } n=1 that is in aition orthonormal in L (). Let P n enote the projection W 1,r Γ D () to the span generate by {w i } n i=1 an un := P n (u ũ D ()) + ũ D () an v n := P n (v ṽ D ()) + ṽ D (). We look for the approximate solution u n of the form n u n (t, x) := c i (t)w i (x) + ũ D (t, x) satisfying (note that v n enotes u n,t) (5.1) i=1 (ϱ v n,t, w i ) + (G(D(v n )), D(w i )) +(H(D(u n )), D(w i )) = g, w i ΓN for all i = 1,,..., n, u n (, ) = u n an v n (, ) : = u n,t(, ) = v n. It follows from the Carathéoory theory (see [1, Chapter ]) that there exists a local-in-time solution to (5.1). Moreover, using the uniform estimates erive in the next subsection, we can exten such a solution to the whole time interval [, T ]. 5.. Uniform estimates. Multiplying the i-th equation in (5.1) by c i (t) an taking the sum over i = 1,..., n, we obtain (recall that ṽ D = (ũ D ),t ) the following ientity (5.) (ϱ v n,t, v n ṽ D ) + (G(D(v n )), D(v n ṽ D )) + (H(D(u n )), D(v n ṽ D )) = g, v n ṽ D ΓN, which we rewrite in the form (ϱ (v n ṽ D ),t, v n ṽ D ) + (G(D(v n )), D(v n )) = g, v n ṽ D ΓN (5.3) + (ϱ (ṽ D ),t, ṽ D v n ) + (G(D(v n )), D(ṽ D )) (H(D(u n )), D(v n ṽ D )). Next, using the assumptions (A1) 1, (A) an (A3) (an their consequences (4.) an (4.3)), we get (note that we assume q ) (5.4) t ϱ (v n ṽ D ) + C 1 D(v n ) r r C 1 + g, v n ṽ D ΓN + (ϱ (ṽ D ),t, ṽ D v n ) + C 1 + D(v n ) r 1 r D(ṽ D ) r + C (1 + D(u n ) ) ( D(v n ) + D(ṽ D ) ) x. Then, aing the inequality t D(un ) = (D(u n ), D(v n )) D(v n ) D(u n ) x

12 KELVIN-VOIGT VISCOELASTIC SOLID 11 to (5.4) an using the continuous embeing of W 1,r () into W 1 1 r,r (Γ N ), the assumptions concerning ṽ D, the Höler an the Young inequalities we obtain (5.5) ( ϱ (v n ṽ D ) + D(u n ) ) + C1 D(v n ) r r t ( ) C 1 + g r (W 1 1 r,r (Γ N )) + ṽ D r 1,r + ϱ (ṽ D ),t r (W 1,r Γ ()) + D(un ). D Finally, using the Gronwall lemma an the Korn inequality, an combining them with the assumptions concerning g, ϱ an ũ D, we euce from (5.5) the following uniform estimate (5.6) ( T sup v n + u n 1,) + v n r 1,r t C(u, v, g, ϱ, ũ D, ) C. t (,T ) Consequently, using (4.3) an (5.1) we euce that (5.7) sup H(D(u n )) + t (,T ) T G(D(v n )) r r t + T ϱ v n,t r ( ) W 1,r t C. Γ () D 5.3. Limit n. Having the estimates (5.6) (5.7), using (4.3) an the continuity of the projection P n, we can fin (not relabele) subsequences such that (5.8) (5.9) (5.1) (5.11) (5.1) (5.13) (5.14) (5.15) u n u strongly in W 1,r (), v n v strongly in L (), u n u weakly in L (, T ; W 1, () ), v n v weakly in L (, T ; L () ), v n v weakly in L r (, T ; W 1,r () ), ϱ v n,t ϱ v,t weakly in L r (, T ; (W 1,r Γ () ) ), G(D(v n )) G weakly in L r (, T ; L r () ), H(D(u n )) H weakly in L (, T ; L () ). Having on han (5.1) (5.15), it is then easy to take the limit in (5.1) to obtain (5.16) T ϱ v,t, w + (G, D(w)) + (H, D(w)) t = T g, w t, for all w L r (, T ; W 1,r Γ () ), T (, T ], where in aition v = u,t. Thus, it remains to show that H = H(D(u)) an G = G(D(v)). For this purpose it is enough to show that there is a subsequence that we enote again as the original sequence such that (5.17) D(v n ) D(v) an D(u n ) D(u) a.e. in (, T ). To o this, we integrate (5.) over the time interval (, T ), where T (, T ) is arbitrary, take the limit n, use weak lower semicontinuity of the norm an the convergence properties

13 1 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL (5.8) (5.15) an finally obtain (5.18) lim sup n T T (G(D(v n )), D(v n ṽ D )) + (H(D(u n )), D(v n ṽ D )) t ϱ v,t, v ṽ D + g, v ṽ D ΓN t 1 ϱ (v(t ) ṽ D (T )) + 1 ϱ (v ṽ D ()). Next, setting w := v ṽ D in (5.16) an comparing the result with (5.18) we observe that (note that here, we use that fact that v() = v, which will be prove in the next subsection an the proof is base only on (5.16)) (5.19) lim sup n T (G(D(v n )), D(v n )) + (H(D(u n )), D(v n )) t T Finally, using (5.14), (5.15) an (5.19), it is easy to euce that for all T (, T ) (5.) lim sup n T (G + H, D(v)) t. (G(D(v n )) G(D(v)) + H(D(u n )) H(D(u)), D(v n v)) t. Next, for arbitrary but fixe T 1 (, T ) we efine Q 1 := (, T 1 ) an we set 1 (5.1) J n : = C 1 D(v n v) (1 + D(v) + sd(v n v) ) r s x t, Q 1 (5.) I n : = C Q 1 D(v n v) D(u n u) 1 an we observe that (5.), (A1) 1 an (A) leas to 6 (1 + D(u) + sd(u n u) ) q s x t, (5.3) J n g(n) + I n, where lim sup g(n) =. n This is our starting point to achieve (5.17). Using (5.4) u n (t) u(t) = t v n (τ) v(τ) τ + u n u, an the fact that q, we arrive at T1 T1 t I n C D(v n v) D(u n u ) t + C D(v n v) D(v n v) t (5.5) T1 ( T1 ) g(n) + C D(v n v) D(v n v) s t, where we use (5.8) an (5.1) to conclue that that the first integral on the right han sie of (5.5) tens to zero as n. Applying the Höler inequality to the secon integral on the right han sie of (5.5) we observe from (5.3) that T1 (5.6) J n g(n) + C T 1 D(v n v) t 6 We use the symbol g(n) to enote any quantity fulfilling lim supn g(n) = ; its exact efinition can change from line to line.

14 Since r it follows from the efinition of J n that (5.7) KELVIN-VOIGT VISCOELASTIC SOLID 13 T1 J n C 1 D(v n v) t. Setting T 1 := C 1 C, the inequalities (5.6) an (5.7) finally imply that T1 (5.8) D(v n v) x t = D(v n v) g(n). Q 1 To prove the strong convergence (5.8) on the whole time cyliner (, T ) we continue inuctively. We efine T k := kt 1 an assume that (5.8) hols on Q k := (, T k ). The aim is to show that (5.8) hols also on Q k+1. We procee analogously as in (5.)-(5.5), replacing T 1 by T k+1 (in particular in the efinitions of J n an I n ). Doing so, we are able to euce from the strong convergence (5.8) on Q k that Tk+1 t J n g(n) + C D(v n v) D(v n v) s t Tk+1 g(n) + C D(v n t v) D(v n v) s t T k Tk+1 g(n) + C D(v n t v) D(v n v) s t T k T k g(n) + (T k+1 T k ) C }{{} =T 1 Tk+1 T k D(v n v) t g(n) + 1 J n. Consequently, the convergence properties (5.8) on Q k+1 follow. (5.17), at least for a selecte subsequence. Thus, (4.5) hols. This completes the proof of 5.4. Initial conitions. In this subsection we prove that the initial ata meet (4.6). Note that this property has been alreay use in the proof of point-wise convergence of D(v n ). Thus, we cannot incorporate (4.5) into our consieration. We can however use (5.16). First, we integrate (5.1) with respect to time over (, t) an observe that (5.9) (ϱ v n (t),w i ) + = t t (G(D(v n )), D(w i )) + (H(D(u n )), D(w i )) τ g, w i ΓN τ + (ϱ v n, w i ) for all i = 1,,..., n. Next, we use (5.1) (5.15) (note that ϱ v C weak (, T ; L () ), which follows from (5.11), (5.13) an the embeing theorem) to obtain for all time t (, T ) (5.3) (ϱ v(t), w i ) + = t t (G, D(w i )) + (H, D(w i )) τ g, w i ΓN τ + (ϱ v, w i ) for all i N.

15 14 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL Finally, since {w n } n=1 that is a basis for W 1,r Γ D () that is ense in L () we conclue from (5.3) (5.31) lim t + (ϱ v(t), w) = (ϱ v, w) for all w L (). Next, integrating (5.5) over (, t), applying the Gronwall lemma, letting n an using weak lower semicontinuity of the L -norm, we observe that (5.3) lim sup t + ϱ (v(t) ṽ D (t)) ϱ (v ṽ D ()). Then, we combine (5.31), (5.3) an the assumptions concerning ṽ D to get lim ϱ (v(t) v ) ( = lim ϱ (v(t) ṽ D (t)) + ϱ (ṽ D (t) v ) ) t + t + + lim t + (ϱ (v(t) ṽ D (t)), ṽ D (t) v ) (5.3),(5.31) ϱ (v ṽ D ()) + ϱ (ṽ D () v ) =. + (ϱ (v ṽ D ()), ṽ D () v ) Finally, using the fact that ϱ is boune from below, we obtain (4.6) 1. It remains to establish (4.6). We shall prove it for q =. We first multiply (5.1) consiere at time τ by τ an integrate the result over (, t). We obtain (5.33) (ϱ v n (t), tw i ) (ϱ u n (t), w i ) + (ϱ u n, w i ) = t (τg(d(v n (τ))), D(w i )) + (τh(d(u n (τ))), D(w i )) τ g(τ), w i ΓN for all i = 1,,..., n. Next, letting n in (5.33) we come to the conclusion that (5.34) lim t + (ϱ u(t), w) = (ϱ u, w) for all w L (). Then, we observe that the following simple relation hols for any τ t T : t ) (5.35) ϱ u(t) = (ϱ v s, u(t) + (ϱ u(τ), u(t)). τ Thus, first letting τ + an using (5.34) an then letting t + an using (5.34) again we observe that (5.36) lim sup t + ϱ u(t) ϱ u. Using (5.34) again, we conclue from (5.36) that (5.37) lim t + u(t) u =, which is the first part of (4.6). It follows from (5.1) an (5.37) that (5.38) lim t + ( u(t), w) = ( u, w) for all w W 1, (). τ

16 Thus, using the relation ( t u(t) = together with (5.38) we finally obtain (4.6). KELVIN-VOIGT VISCOELASTIC SOLID 15 τ ) v s, u(t) + ( u(τ), u(t)) 5.5. Uniqueness. Assume that u 1 an u are two weak solutions of (4.13) that satisfy the same initial an bounary ata. Set v 1 := u 1,t an v := u,t an enote w := u 1 u an z := v 1 v. Then subtracting (4.5) for u 1 from (4.5) for u an setting ϕ := z (note that this is an amissible test function) we come to the ientity (5.39) 1 t ϱ z + (G(D(v 1 )) G(D(v )), D(z)) = (H(D(u )) H(D(u 1 )), D(z)). Next, aing the inequality t D(w) D(w) D(z), to (5.39), an using (A1) 1, (A), the fact that q, an the Höler an the Young inequality, we obtain ( ϱ z + D(w) ) (5.4) + C1 D(z) C D(w) t, which implies, after applying the Gronwall lemma, that z in (, T ) (an consequently also w ). The proof of uniqueness is complete. 6. Proof of Theorem 4. In this section we establish regularity properties of any weak solution to (4.1) provie that ata are smooth enough, i.e., we will prove Theorem 4.. Here, we only give a formal proof, but the whole proceure can be one rigorously by using the stanar ifference quotient technique an by using uniqueness of weak solution. We start by proving the interior regularity result as formulate in the part i) of Theorem 4.. For arbitrary h > we set h := {x ; ist(x, ) h} an we consier a nonnegative ϕ D() such that ϕ = 1 in h. Then we multiply (4.13) 1 by iv(ϕ v) an integrate over. Doing so, we come, after integration by parts, to the ientity (note that all bounary integrals vanish ue to our choice of ϕ) (6.1) 1 t ϱ ϕ v + ( (G(D(v)) + H(D(u)), ϕ D( v)) = ( ϱ ϕ v,t, v) ( (G(D(v)) + H(D(u)), ϕ ϕ v). Next, we estimate the secon term on the left han sie of (6.1) by means of (A1*) (A*) an obtain ( (G(D(v)) + H(D(u)), ϕ D( v)) C 1 (1 + D(v) ) r D( v) ϕ x (6.) C (1 + D(u) ) q D( u) D( v) ϕ x.

17 16 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL To estimate the secon term on right han sie of (6.1) we use (A1*) (A*) an the fact that q an we euce that (6.3) ( (G(D(v)) + H(D(u)), ϕ ϕ v) C((1 + D(v) ) r D( v) ϕ, (1 + D(v) ) r v ) + C( D( u) ϕ, v ) C 1 ((1 + D(v) ) r D( v) ϕ x + C(1 + v r r + v + D( u)ϕ ). Inserting (6.3) an (6.) into (6.1), applying the Höler inequality to the first term on the right han sie of (6.1), an again using the fact that q an the Young inequality to estimate the secon term on the right han sie of (6.), we finally observe that (6.4) t ϱ ϕ v + C 1 (1 + D(v) ) r D( v) ϕ x C( v r r + v + D( u)ϕ ) + ϱ ϕ v,t ϕ r v r. In orer to estimate the last term on the right han sie of (6.4), we use (4.13) an (A1*) (A*) an observe that (6.5) v,t ϕ r iv(g(d(v)) + H(D(u)))ϕ r C ( (1 + D(v) ) r D( v)ϕ r + (1 + D(u) ) q ) D( u)ϕ r ) C ( (1 + D(v) ) r D( v)ϕ (1 + D(v) ) r r + D( u)ϕ r ( ) C (1 + D(v) ) r D( v)ϕ 1 + v r r + D( u)ϕ Upon inserting (6.5) into (6.4), we obtain (using also the Young inequality) t ϱ ( v)ϕ + (1 + D(v) ) r D( v) ϕ x (6.6) C ( D( u)ϕ + (1 + ϱ ϕ ) 1 + v r r + ϱ ϕ ). Next, aing the inequality to (6.6) we get (6.7) t D( u)ϕ D( u)ϕ D( v)ϕ 1 D( v)ϕ + 4 D( u)ϕ ( ϱ vϕ + D( u)ϕ ) + (1 + D(v) ) r D( v) ϕ x t C ( D( u)ϕ + g(t) ), where g(t) := (1 + ϱ ϕ ) 1 + v r r + ϱ ϕ L 1 (, T ) accoring to our assumptions. Thus, applying the Gronwall lemma, we obtain (4.7) (4.8). The regularity property (4.9) is then a consequence of (6.5) an (6.7). In the next part, we prove the properties (4.1) an (4.11) that concern the regularity of v,t. Although we will procee formally, one can obtain the same type result rigorously by eriving uniform estimates for the Galerkin approximations an conclue (4.1) an (4.11) for their (uniquely etermine) weak limit. Here, we ifferentiate (4.13) 1 with respect to time an take

18 KELVIN-VOIGT VISCOELASTIC SOLID 17 the scalar prouct of the result an v,t. Integrating the output over we obtain (for simplicity we restrict ourselves to homogeneous Dirichlet bounary conitions on (, T ) ) ( ) ( ) 1 G H (6.8) t ϱ v,t + D (D(v)), D(v,t ) D(v,t ) + D (D(u)), D(v) D(v,t ) =. It then follows from (A1*), (A*) an the fact that q that t ϱ v,t + C 1 (1 + D(v) ) r D(v,t ) x (1 + D(u) ) q D(v) D(v,t ) x (6.9) (1 + D(v) ) (r )/ D(v,t ) (1 + D(v) ) (4 r)/ x C 1 (1 + D(v) ) r D(v,t ) x D(v) 4 r Since 4 r r we easily euce (4.1) an (4.11) from (6.9) an (4.4) 1. The final part of this subsection is evote to the regularity properties near the bounary. For simplicity, we prove the result only for a part of the bounary where we prescribe homogeneous Dirichlet conition 7, i.e. ṽ D =. In aition we simplify the proof by assuming that this part of the bounary is locate at ( 1, 1) 1 {} an within the support of a smooth function ϕ D(( 1, 1) ) such that ϕ = 1 in the neighborhoo of a part of Γ D we are intereste in. Next, we efine i h f(x) := f(x + hei ) f(x). an we take, for any fixe i = 1,..., 1 an sufficiently small h >, ϕ i := i h (ϕ i h v) as a test function in (4.5). Note that this choice is possible since ϕ i = on [, T ]. Also note that the bounary integral in (4.5) vanishes. Hence, we have (6.1) (ϱ v,t, ϕ i ) + (G(D(v)) + H(D(u)), D(ϕ i )) =. We start by estimating the first term. One easily observes that 4 r. (6.11) Next, enoting (ϱ v,t, ϕ) = ( i h (ϱ v,t ), ϕ i h v) 1 t ϱ ϕ i h v Ch v,t ϕ i h v ϱ. 1 Ih i := D(v(x + he i ) v(x)) (1 + D(v(x + he i )) s(d(v(x + he i ) v(x))) ) r s, 7 The technique of the proof for inhomogeneous Dirichlet ata an for ΓN is the same.

19 18 M. BULÍČEK, J. MÁLEK, AND K.R. RAJAGOPAL we observe by using (A1*), (A*) an the Höler an the Young inequalities that (G(D(v)), D(ϕ i )) C 1 Ih i ϕ x (6.1) ( ( 1 ) ) 1 C (Ih i ) 1 ϕ, i h v (1 + D(v(x + he i )) s(d(v(x + he i ) v(x))) ) r s C 1 I i h ϕ x Ch (1 + v r). Similarly, using (A) an the fact that q, we get (6.13) (H(D(u)), D(ϕ i )) C ϕ D( i h u) D( i h v) x C 1 4 Thus, inserting (6.11) (6.13) into (6.1) an aing the inequality (r ) ϕ I i h x + C ϕd( i h u). t D( i h u)ϕ D( i h u)ϕ D( i h v)ϕ C 1 D( i h v)ϕ + C D( i h u)ϕ C 1 Ih i ϕ x + C D( i h u)ϕ we obtain (iviing the result by h ) ( h ϱ i h t v + h D( i ) h u)ϕ + h ϕ Ih i (6.14) x C(h D( i h u)ϕ + v,t ). At this juncture, we apply the Gronwall lemma to (6.14), using (4.1), an conclue that (6.15) sup h> which implies that T (6.16) T ϕ h Ih i x t C, (1 + D(v) ) r D( xi v) ϕ x t C for all i = 1,..., 1. Finally, we rewrite the ith equation in (4.1) 1 in the form (6.17) x G i (D(v)) = 1 xj H ij (D(u)) xj G ij (D(v)) ϱ (v i ),t. j=1 Multiplying (6.17) by x D(v), taking the sum over i = 1,...,, integrating the result over, an then using the assumptions (A1*) an (A*) we obtain (6.18) (1 + D(v) ) r D( x v) C D( x u) D( x v) + g(t), whereby g enotes a function satisfying, by (4.1) an (6.16), that g L 1 (, T ). Aing the inequality t D( x u) D( x v) D( x u) i=1

20 KELVIN-VOIGT VISCOELASTIC SOLID 19 to (6.18), using the Young inequality an applying the Gronwall lemma, we finally get (4.1) (incorporating (6.16) again). The proof of Theorem 4. is complete. References [1] E. A. Coington an N. Levinson. Theory of orinary ifferential equations. McGraw-Hill Book Company, Inc., New York-Toronto-Lonon, [] A. Frieman an J. Nečas. Systems of nonlinear wave equations with nonlinear viscosity. Pacific J. Math., 135(1):9 55, [3] J. Málek, J. Nečas, M. Rokyta, an M. Růžička. Weak an measure-value solutions to evolutionary PDEs. Chapman & Hall, Lonon, [4] K. R. Rajagopal. A note on a reappraisal an generalization of the Kelvin-Voigt Moel. Mechanics Research Communications, 36:3 35, 9. [5] K. R. Rajagopal. Rethinking constitutive relations. 9. [6] W. Thompson. On the elasticity an viscosity of metals. Proc. Roy. Soc. Lonon, A14:89 97, [7] B. Tvet. Quasilinear equations for viscoelasticity of strain-rate type. Arch. Ration. Mech. Anal., 189():37 81, 8. [8] W. Voigt. Ueber innere Reibung fester Körper, insbesonere er Metalle. Annalen er Physik, 83: , 189. Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech Republic aress: mbul86@karlin.mff.cuni.cz Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech Republic aress: malek@karlin.mff.cuni.cz Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845, USA aress: krajagopal@tamu.eu