
==== Front
J Evol Equ
J Evol Equ
Journal of Evolution Equations
1424-3199
1424-3202
Springer International Publishing Cham

38989137
989
10.1007/s00028-024-00989-0
Article
The viscoelastic paradox in a nonlinear Kelvin–Voigt type model of dynamic fracture
http://orcid.org/0000-0002-0932-0022
Caponi Maicol maicol.caponi@unina.it

1
Carbotti Alessandro 2
Sapio Francesco 3
1 https://ror.org/05290cv24 grid.4691.a 0000 0001 0790 385X Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, 80126 Naples, Italy
2 https://ror.org/03fc1k060 grid.9906.6 0000 0001 2289 7785 Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Via Per Arnesano, 73100 Lecce, Italy
3 https://ror.org/048qkq105 grid.452056.0 Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
8 7 2024
8 7 2024
2024
24 3 6328 5 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox.

Keywords

Dynamic fracture
Cracking domains
Elastodynamics
Nonlinear viscoelasticity
Monotone operators
Energy-dissipation balance
Viscoelastic paradox
Mathematics Subject Classification

35L53
35A01
35Q74
47H05
74D10
74R10
http://dx.doi.org/10.13039/100007195 Università degli Studi di Napoli Federico II 21-UNINA-EPIG-172 Caponi Maicol http://dx.doi.org/10.13039/100007388 Compagnia di San Paolo 21-UNINA-EPIG-172 Caponi Maicol http://dx.doi.org/10.13039/100012740 Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni CUP_E53C22001930001 Caponi Maicol http://dx.doi.org/10.13039/501100021856 Ministero dell’Università e della Ricerca 20223L2NWK CUP_E53C22001930001 Carbotti Alessandro http://dx.doi.org/10.13039/501100002428 Austrian Science Fund project TAI 293 Sapio Francesco Università degli Studi di Napoli Federico IIOpen access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.

issue-copyright-statement© Springer Nature Switzerland AG 2024
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pmcIntroduction

In the derivation of a mathematical model for dynamic crack propagation, the two fundamental facts that must be taken into account are the laws of elastodynamics and the (dynamic) Griffith criterion. The first one states that the displacement of the deformation must solve the elastodynamics system away from the crack, while the second one dictates how the crack grows in time. More precisely, the Griffith criterion (see [16, 18]), originally formulated in the quasi-static setting, explains that there is a balance between the mechanical energy dissipated during the evolution and the energy used to increase the crack, which is supposed to be proportional to the area increment of the crack itself.

The first step to address the study of a model of dynamic fracture is to find the solution to the elastodynamics system when the evolution of the crack is prescribed. From the mathematical point of view, this leads to the study of the following system in a time-dependent domain:1.1 u¨(t)-div(σ(t))=f(t)inΩ\Γt,t∈[0,T],

with some prescribed boundary and initial conditions. In the above formulation, Ω⊂Rd is an open bounded set with Lipschitz boundary which represents the reference configuration of the material, Γt⊂Ω¯ is a (d-1)-dimensional closed set that models the crack at time t, u(t):Ω\Γt→Rd is the displacement of the deformation, σ(t) is the Cauchy stress tensor, and f(t) is a forcing term. Once found the displacement u that solves (1.1) with a prescribed crack evolution t↦Γt, we determine the pairs displacement-crack which satisfy the Griffith energy-dissipation balance. Finally, we select the “right” crack evolution according to some maximal dissipation principle.

In the easiest case of a pure elastic material, the system  (1.1) is coupled with the following constitutive law involving the Cauchy stress and the strain tensors:1.2 σ(t)=Ceu(t)inΩ\Γt,t∈[0,T],

where C is the elasticity tensor, which is fourth-order positive definite on the space of symmetric matrices Rsymd×d, and eu=12(∇u+∇uT) is the strain tensor. In this setting, the Griffith criterion reads as1.3 12‖u˙(t)‖22+12‖eu(t)‖22+Hd-1(Γt\Γ0)=12‖u˙(0)‖22+12‖eu(0)‖22+workofexternalforces

for all t∈[0,T]. We point out that the first two terms in the left-hand side of the above identity correspond to the mechanical energy (the sum of kinetic and elastic energy), while the term Hd-1(Γt\Γ0) models the energy used to increase the crack from Γ0 to Γt.

In the literature, we can find several mathematical results for the model associated with (1.1) and (1.2). As for the existence of a solution when the evolution t↦Γt is prescribed, we refer to [10, 13] for the antiplane case, that is when u(t):Ω\Γt→R is a scalar function and eu is replaced by ∇u, and [4, 26] for the general case. Regarding the determination of the crack evolution t↦Γt, we have only partial results. For example, we cite [5], where the authors characterize in the antiplane case and for d=2 the pairs displacement-crack which satisfy the energy-dissipation balance, and [11, 12] in which for d=2 the authors study the coupled problem under a suitable notion of maximal dissipation.

Viscoelastic materials, which exhibit both viscous and elastic behaviors when undergoing deformations, are another class widely studied in the literature. One of the simplest mathematical model is the Kelvin–Voigt one, where the constitutive law between the Cauchy stress and the strain tensors reads as1.4 12‖u˙(t)‖22+12‖eu(t)‖22+Hd-1(Γt\Γ0)=12‖u˙(0)‖22+12‖eu(0)‖22+workofexternalforces

where C and B are the elasticity and the viscosity tensors, respectively. For the Kelvin–Voigt model, the Griffith criterion leads to the following energy-dissipation balance1.5 12‖u˙(t)‖22+12‖eu(t)‖22+Hd-1(Γt\Γ0)+∫0t‖eu˙(s)‖22ds=12‖u˙(0)‖22+12‖eu(0)‖22+work of external forces

for all t∈[0,T]. Notice that, with respect to formula (1.3), in (1.5) we need to take into account also the energy dissipated by the viscous term, which is given by ∫0t‖eu˙(s)‖22ds.

In [10, 26], we can find existing results for the linear viscoelastic problem (1.1) and (1.4), when the evolution of the crack is prescribed. Unfortunately, in those papers, it is also shown that the Griffith energy-dissipation balance (1.5) holds without the term Hd-1(Γt\Γ0). As a consequence, there is no pair displacement-crack which satisfies (1.5), unless the crack does not grow in time, i.e., Γt=Γ0 for all t∈[0,T]. This phenomenon, which says that in the linear Kelvin–Voigt model the crack can not propagate, is well-known in mechanics as the viscoelastic paradox, see for instance [25, Chapter 7]. We point out that, if the viscosity tensor B is allowed to degenerate in a neighborhood of the moving crack, the viscoelastic paradox does not occur, as shown in [6]. For other versions of linear constitutive laws in the framework of viscoelastic materials, we refer for example to [7–9, 23].

More recently, viscoelastic materials in which the constitutive relation is nonlinear and given in an implicit form have been also considered. For example, in [3], the authors study the following elastodynamic system in a domain without cracks:1.6 u¨(t)-div(σ(t))=f(t)inΩ,t∈[0,T],

with the implicit constitutive law1.7 G(σ(t))=eu(t)+eu˙(t)inΩ,t∈[0,T],

where G:Rsymd×d→Rsymd×d is a nonlinear monotone operator which satisfies suitable p-growth assumptions. In particular, the prototypical models studied are1.8 G1(ξ):=|ξ|p-2ξforp>1,G2(ξ)=ξ(1+|ξ|a)1aforp=1,witha>0.

As explained by Bulíček, Patel, Süli, and Şengül in their paper [3] (see also [21]), linear models may be inaccurate to describe real phenomena, while implicit constitutive theories allow for a more general structure in modeling than explicit ones. Moreover, as shown by Rajagopal in [22], the nonlinear relationship between the stress and the strain can be obtained after linearizing the strain, and so it make sense to consider implicit constitutive relations in the contest of small deformations. Under suitable assumptions on the initial data and on the nonlinear term G, Bulíček, Patel, Süli, and Şengül in [3] prove existence and uniqueness of solutions to the problem (1.6) and (1.7) via the Galerkin approximation.

The aim of our paper is to study the model of viscoelastic materials with implicit constitutive law of [3], in the framework of dynamic crack propagation. More precisely, we consider the elastodynamics system (1.1) with the constitutive relation1.9 G(σ(t))=eu(t)+eu˙(t)inΩ\Γt,t∈[0,T],

where G:Rsymd×d→Rsymd×d is a nonlinear monotone operator which satisfies suitable p-growth assumptions (more precisely (G1)–(G3) in Sect. 2). Since the linear growth p=1 is hard to handle even in the case with no cracks, we restrict ourselves to the range p∈(1,2∗), where 2∗:=2dd-2 is the Sobolev critical exponent. The condition p<2∗, which also appears in [3], is needed to ensure that the displacement u(t) is an element of L2(Ω\Γt;Rd). Indeed, from (1.9), we easily deduce that u(t) lives in the Sobolev space W1,p′(Ω\Γt;Rd), being p′=pp-1 the Hölder conjugate exponent of p, which is compactly embedded in L2(Ω\Γt;Rd) whenever p<2∗. This simplifies the mathematical formulation of the problem. An interesting question, which is out of the scope of this paper, is whether this condition can be removed.

Our first result is Theorem 2.8, where we prove the existence of a solution to the problem (1.1) and (1.9) when the crack evolution t↦Γt is prescribed, under suitable conditions on the data and on the nonlinear term G. The proof of Theorem 2.8 follows the main ideas of [3], adapted to our setting. First, since the Galerkin approximation does not fit well with the framework of time-dependent domains, we use the discretization-in-time scheme exploited in [10]. Moreover, since we want to consider nonlinear operators which are not strictly monotone, we regularize G in order to invert the relation (1.9). This allows us to write the Cauchy tensor in terms of the displacement and to switch from the formulation (1.1) and (1.9) to a simpler system. More precisely, we fix n∈N and we search a discrete-in-time approximate solution to (1.1) and (1.9) with G replaced by its regularization. Then, we perform a discrete energy estimate (see Lemma 3.3), which allows us to pass to the limit as n→∞ to obtain a pair (u,σ) which solves (1.1). We prove that the displacement u is more regular in time, and by using a standard technique in the monotone operator theory, we show that (u,σ) satisfies also the implicit constitutive relation (1.9). We conclude this part of the paper with Theorem 3.10, where we prove that there is at most one pair (u,σ) with the same regularity of the solution of Theorem 2.8 that solves (1.1) and (1.9) for a prescribed crack evolution t↦Γt.

In the second part of the paper, we aim to study the validity of the Griffith energy-dissipation balance for the implicit nonlinear model (1.1) and (1.9). At first, in Theorem 4.1 we prove that the mechanical energy of every regular solution to problem (1.1) and (1.9) (in particular, of the one found in Theorem 2.8) satisfies the implicit energy balance12‖u˙(t)‖22+∫0t∫Ωσ(s,x)·eu˙(s,x)dxds=12‖u˙(0)‖22+work of external forces

for every t∈[0,T]. Then, we consider the strictly monotone operator G(ξ)=|ξ|p-2ξ, so that our problem reduces to the nonlinear Kelvin–Voigt system1.10 u¨(t)-div(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t)))=f(t)inΩ\Γt,t∈[0,T].

In this setting, the Griffith energy-dissipation balance takes the form1.11 12‖u˙(t)‖22+1p′‖eu(t)‖p′′+Hd-1(Γt\Γ0)+∫0t∫Ω|eu(s,x)+eu˙(s,x)|p′-2(eu(s,x)+eu˙(s,x))·eu˙(s,x)dxds-∫0t∫Ω|eu(s,x)|p′-2eu(s,x)·eu˙(s,x)dxds=12‖u˙(0)‖22+1p′‖eu(0)‖p′′+workofexternalforces

for every t∈[0,T]. In particular, the energy dissipated by the viscous term is given by∫0t∫Ω|eu(s,x)+eu˙(s,x)|p′-2(eu(s,x)+eu˙(s,x))·eu˙(s,x)dxds-∫0t∫Ω|eu(s,x)|p′-2eu(s,x)·eu˙(s,x)dxds≥0,

which reduces to the corresponding term in (1.5) for p=2 (notice that this term is nonnegative due to the monotonicity of G-1(η):=|η|p′-2η). For this particular choice of G, in Corollary 4.3 we derive that the energy dissipation balance proved in Theorem 4.1 can be rewritten just in terms of the displacement u as (4.7). Therefore, the pair displacement-crack given by Theorem 2.8 satisfies (1.11) if and only if Γt=Γ0 for every t∈[0,T], i.e., when the crack does not grow in time. This shows that also the nonlinear Kelvin–Voigt model of dynamic fracture exhibits the viscoelastic paradox, as it happens in [10, 26] for the corresponding linear model.

We conclude the introduction by observing that the corresponding phase-field model of dynamic crack propagation has been analyzed in [20] (see also [21]). This is the one in which, roughly speaking, for a fixed ϵ>0 the crack set is replaced by a function vϵ which is 0 in a ϵ-neighborhood of the crack and 1 far from it. More precisely, in [20] the author proved that there exists a pair (uϵ,vϵ) which satisfies the elastodynamics system with the implicit constitutive law and the Griffith energy-dissipation balance for both the nonlinearities in (1.8). Therefore, it could be interesting to understand in a future paper if there is a connection between these two models and, in particular, if the viscoelastic paradox can also occur in the phase-field setting.

The rest of the paper goes as follows: in Sect. 2 we introduce the mathematical framework of our model of dynamic fracture for viscoelastic material, and we fix the main assumptions on the reference set, the crack evolution, and the nonlinearity G in the constitutive law. Moreover, in Definition 2.3 we give the notion of (weak) solution to problem (1.1) and (1.9), and we state our main existence result, which is Theorem 2.8. In Sect. 3 we prove Theorem 2.8 by performing a discretization-in-time scheme together with a regularization of the nonlinearity G. At first, we find an approximate solution in each node of the discretization of the regularized model. Then, in Lemma 3.3 we prove a discrete energy estimate, which allows us to pass to the limit when the parameter of the discretization and regularization goes to 0. Finally, we show that under suitable regularity assumptions the solution is unique. We conclude the paper with Sect. 4, where we prove that every regular solution to (1.1) and (1.9) satisfies the energy-dissipation identity of Theorem 4.1. Afterwards, we consider the nonlinear Kelvin–Voigt system (1.10), and we use the energy-dissipation identity to show that this model exhibits the viscoelastic paradox.

Notation and formulation of the model

Notation

The space of m×d matrices with real entries is denoted by Rm×d; in case m=d, the subspace of symmetric matrices is denoted by Rsymd×d. For any A,B∈Rd×d we denote with A·B the Frobenius scalar product, namely A·B:=Tr(ATB). Given a function u:Rd→Rm, we denote its Jacobian matrix by ∇u, whose components are (∇u)ij:=∂jui for i∈{1,⋯,m} and j∈{1,⋯,d}; when u:Rd→Rd, we use eu to denote the symmetric part of the gradient, namely eu:=12(∇u+∇uT). Given a tensor field A:Rd→Rm×d, by divA we mean its divergence with respect to rows, namely (divA)i:=∑j=1d∂jAij for i∈{1,⋯,m}.

We denote the d-dimensional Lebesgue measure by Ld and the (d-1)-dimensional Hausdorff measure by Hd-1; given a bounded open set Ω with Lipschitz boundary, by ν we mean the outer unit normal vector to ∂Ω, which is defined Hd-1-a.e. on the boundary. The Lebesgue and Sobolev spaces on Ω are defined as usual; the boundary values of a Sobolev function are always intended in the sense of traces. When there is no ambiguity, we simply write ‖·‖p to denote the norm in Lp(Ω;Rk) for all p∈[1,∞] and k∈N.

The norm of a generic Banach space X is denoted by ‖·‖X; when X is a Hilbert space, we use (·,·)X to denote its scalar product. We denote by X′ the dual of X and by ⟨·,·⟩X′ the duality product between X′ and X. Given two Banach spaces X1 and X2, the space of linear and continuous maps from X1 to X2 is denoted by L(X1;X2); given A∈L(X1;X2) and u∈X1, we write Au∈X2 to denote the image of u under A.

Given an open interval (a,b)⊂R and q∈[1,∞], we denote by Lq(a,b;X) the space of Lq functions from (a, b) to X; we use Wk,q(a,b;X) to denote the Sobolev space of functions from (a, b) to X with derivatives up to order k in Lq(a,b;X). Given u∈W1,q(a,b;X), we denote by u˙∈Lq(a,b;X) its derivative in the sense of distributions. When dealing with an element u∈W1,q(a,b;X) we always assume u to be the continuous representative of its class, and therefore, the pointwise value u(t) of u is well defined for all t∈[a,b]. We use Cw0([a,b];X) to denote the set of weakly continuous functions from [a, b] to X, namely the collection of maps u:[a,b]→X such that t↦⟨x′,u(t)⟩X′ is continuous from [a, b] to R, for all x′∈X′.

Mathematical framework

Let T>0 and d∈N with d≥2. Let Ω⊂Rd be a bounded open set (which represents the reference configuration of the body) with Lipschitz boundary. Let ∂DΩ be a Borel subset of ∂Ω, on which we prescribe the Dirichlet condition, ∂NΩ its complement in ∂Ω, and Γ⊂Ω¯ the prescribed crack path. As in [6, 7], we assume the following hypotheses on the geometry of the crack and the Dirichlet part of the boundary: Γ is a closed set with Ld(Γ)=0 and Hd-1(Γ∩∂Ω)=0;

Ω\Γ is the union of two disjoint bounded open sets Ω1 and Ω2 with Lipschitz boundary;

∂DΩ∩∂Ωi contains the graph of a Lipschitz function θi over a non-empty open subset of Rd-1 for all i∈{1,2};

{Γt}t∈[0,T] is a family of closed subsets of Γ satisfying Γs⊆Γt for all 0≤s≤t≤T.

We recall that the set Γt represents the prescribed crack at time t∈[0,T] inside Ω.

Thanks to (E1)–(E4) for all q∈[1,∞] the space Lq(Ω\Γt;Rd) coincides with Lq(Ω;Rd) for all t∈[0,T]. In particular, we can extend a function u∈Lq(Ω\Γt;Rd) to a function in Lq(Ω;Rd) by setting u=0 on Γt. Moreover, for all q∈[1,∞) the trace of u∈W1,q(Ω\Γ;Rd) is well defined on ∂Ω and there exists a constant Ctr>0, depending on Ω, Γ, and q, such that2.1 ‖u‖Lq(∂Ω;Rd)≤Ctr‖u‖W1,q(Ω\Γ;Rd)for allu∈W1,q(Ω\Γ;Rd).

Hence, we can define the spaceWD1,q(Ω\Γ;Rd):={u∈W1,q(Ω\Γ;Rd):u=0on∂DΩ}.

Furthermore, by using the second Korn inequality in Ω1 and Ω2 (see, e.g., [19, Theorem 2.4]) and taking the sum we can find a positive constant CK, depending on Ω, Γ, and q, such that2.2 ‖∇u‖Lq(Ω;Rd×d)≤CK(‖u‖Lq(Ω;Rd)q+‖eu‖Lq(Ω;Rsymd×d)q)1qforallu∈W1,q(Ω\Γ;Rd).

Similarly, thanks to the Korn-Poincaré inequality (see, e.g., [19, Theorem 2.7]) we obtain also the existence of a constant CKP, depending on Ω, Γ, q, and ∂DΩ, such that2.3 ‖u‖W1,q(Ω\Γ;Rd)≤CKP‖eu‖Lq(Ω;Rsymd×d)for allu∈WD1,q(Ω\Γ;Rd).

Finally, for all q∈(2dd+2,∞] the embedding W1,q(Ω\Γ;Rd)↪L2(Ω;Rd) is continuous and compact.

We fix p∈(1,2∗), where 2∗ is the Sobolev conjugate of 2, defined as2∗:=∞ford=2,2dd-2ford>2.

Notice that p∈(1,2∗) if and only if p′∈(2dd+2,∞), where p′:=pp-1 is the Hölder conjugate exponent of p. We set H:=L2(Ω;Rd) and we define the following spacesV:=W1,p′(Ω\Γ;Rd)andVt:=W1,p′(Ω\Γt;Rd)for allt∈[0,T].

We point out that in the definition of V and Vt, we are considering only the distributional gradient of u in Ω\Γ and in Ω\Γt, respectively, and not the one in Ω. Taking into account (2.2), we shall use on the set Vt (and also on the set V) the equivalent norm‖u‖Vt:=‖u‖p′′+‖eu‖p′′1p′for allu∈Vt.

Furthermore, by (2.1), we can consider the setsVD:={u∈V:u=0on∂DΩ},VtD:={u∈Vt:u=0on∂DΩ}for allt∈[0,T],

which are closed subspaces of V and Vt, respectively.

Remark 2.1

Since p∈(1,2∗), by exploiting (E1)–(E4) we derive that for all t∈[0,T] the space VtD is a separable reflexive Banach space with embeddingVtD↪Hcontinuous, compact, and dense.

In particular, the aforementioned condition on p is used to deduce the compactness of VtD in H. Therefore, the embedding H↪(VtD)′, which is defined by2.4 ⟨h,u⟩(VtD)′:=(h,u)Hforh∈Handu∈VtD,

is continuous, and the same holds true also for Vt, V, and VD.

Let us consider a nonlinear operator G:Rsymd×d→Rsymd×d satisfying the following assumptions: there exists a convex function ϕ:Rsymd×d→R of class C1 such that G(ξ)=∇ϕ(ξ) for all ξ∈Rsymd×d;

there exist constants b1>0 and b2≥0 such that G(ξ)·ξ≥b1|ξ|p-b2 for all ξ∈Rsymd×d;

there exists a constant b3>0 such that |G(ξ)|≤b3(1+|ξ|p-1) for all ξ∈Rsymd×d.

Remark 2.2

The assumption (G1) implies that G is continuous and monotone, i.e.,2.5 (G(ξ1)-G(ξ2))·(ξ1-ξ2)≥0for allξ1,ξ2∈Rsymd×d.

Moreover, up to add a constant, we always assume that ϕ(0)=0.

Given f∈L2(0,T;H);

z∈W2,p′(0,T;V0)∩W2,2(0,T;H);

u0,u1∈V0 such that u0-z(0)∈V0D and u1-z˙(0)∈V0D;

we study the following dynamic viscoelastic system with implicit nonlinear constitutive law:2.6 u¨(t)-div(σ(t))=f(t)inΩ\Γt,t∈[0,T],G(σ(t))=eu(t)+eu˙(t)inΩ\Γt,t∈[0,T],

equipped with the boundary conditions2.7 u(t)=z(t)on∂DΩ,t∈[0,T],

2.8 σ(t)ν=0on∂NΩ∪Γt,t∈[0,T],

where ν denotes the outward unit normal to ∂Ω, and the initial conditions2.9 u(0)=u0,u˙(0)=u1inΩ\Γ0.

Notice that in (2.6)–(2.9) the explicit dependence on x is omitted to enlighten notation. As usual, the Neumann boundary conditions are only formal, and their meaning will be explained in Remark 2.4.

From now on we always assume that p∈(1,2∗) and that (E1)–(E4), (G1)–(G3), and (D1)–(D3) are satisfied. Let us define the following functional spaces:V:={φ∈W1,p′(0,T;V)∩W1,∞(0,T;H):φ(t)∈Vtfor allt∈[0,T]},D:={φ∈Cc1(0,T;V):φ(t)∈VtDfor allt∈[0,T]}.

Similarly to [14], we introduce the following notion of weak solution.

Definition 2.3

(Weak solution) A pair (u,σ)∈V×Lp(0,T;Lp(Ω,Rsymd×d)) is a weak solution to the nonlinear viscoelastic system (2.6)–(2.8) if (i) u(t)-z(t)∈VtD for all t∈[0,T];

(ii) the following identity holds 2.10 -∫0T(u˙(t),φ˙(t))Hdt+∫0T(σ(t),eφ(t))p,p′dt=∫0T(f(t),φ(t))Hdtforallφ∈D

where (g,h)p,p′:=∫Ωg(x)·h(x)dx for all g∈Lp(Ω;Rsymd×d) and h∈L′(Ω;Rsymd×d);

(iii) the constitutive law 2.11 G(σ(t))=eu(t)+eu˙(t)inΩ\Γtfor a.e.t∈[0,T]

is satisfied.

Remark 2.4

The Neumann boundary conditions (2.8) are formally used to pass from the strong formulation (2.6)–(2.8) to the weak formulation (2.10). Notice that, if u(t), σ(t), and Γt are sufficiently regular, then (2.8) can be deduced from (2.10) by using integration by parts in space.

We want to give a meaning to the initial conditions (2.9) for a weak solution (u,σ) to (2.6)–(2.8). To this aim, we first recall the following result (see, for instance [15, Chapitre XVIII, §5, Lemme 6]).

Lemma 2.5

Let X, Y be reflexive Banach spaces such that X↪Y continuously. Then,L∞(0,T;X)∩Cw0([0,T];Y)=Cw0([0,T];X).

Moreover, we need the following regularity result for the weak solutions to (2.6)–(2.8).

Lemma 2.6

Let (u,σ)∈V×Lp(0,T;Lp(Ω,Rsymd×d)) be a weak solution to the nonlinear viscoelastic system (2.6)–(2.8). Then, u∈W2,q(0,T;(V0D)′), where q:=min{2,p}. In particular u∈C0([0,T];V) and u˙∈Cw0([0,T];H).

Proof

Let q be given as in the statement. We define Λ∈Lq(0,T;(V0D)′) as follows:⟨Λ(t),v⟩(V0D)′:=-(σ(t),ev)p,p′+(f(t),v)Hfor allv∈V0Dand for a.e.t∈[0,T].

Let us consider a test function ψ∈Cc1(0,T), then for all v∈V0D the function φ(t):=ψ(t)v satisfies2.12 φ∈Cc1(0,T;V),φ(t)∈V0D⊂VtDfor allt∈[0,T].

Thanks to (2.10), since φ∈D from (2.12), we can write-∫0T(u˙(t),v)Hψ˙(t)dt=-∫0T(σ(t),ev)p,p′ψ(t)dt+∫0T(f(t),v)Hψ(t)dt=∫0T⟨Λ(t),v⟩(V0D)′ψ(t)dt,

which implies by (2.4)〈-∫0Tu˙(t)ψ˙(t)dt,v〉(V0D)′=〈∫0TΛ(t)ψ(t)dt,v〉(V0D)′for allv∈V0D.

Hence, we get2.13 -∫0Tu˙(t)ψ˙(t)dt=∫0TΛ(t)ψ(t)dtin(V0D)′for allψ∈Cc1(0,T).

Since u˙∈L∞(0,T;H)↪L∞(0,T;(V0D)′) then identity (2.13) impliesu∈W2,q(0,T;(V0D)′).

Therefore u˙∈W1,q(0,T;(V0D)′)↪C0([0,T];(V0D)′). Since u˙∈L∞(0,T;H), by Lemma 2.5 we deduce that u˙∈Cw0([0,T];H). Finally, we have W1,p′(0,T;V)↪C0([0,T];V) hence u∈C0([0,T];V). □

If (u,σ)∈V×Lp(0,T;Lp(Ω;Rsymd×d)) is a weak solution to (2.6)–(2.8), then u(t) and u˙(t) are well defined as functions of V and H, respectively, for all t∈[0,T]. Therefore, it makes sense to evaluate them at time t=0 in order to make consistent the following definition.

Definition 2.7

(Initial conditions) We say that a weak solution (u,σ)∈V×Lp(0,T;Lp(Ω;Rsymd×d)) to the nonlinear viscoelastic system (2.6)–(2.8) satisfies the initial conditions (2.9) ifu(0)=u0inV,u˙(0)=u1inH

The main existence result of this paper is the following theorem.

Theorem 2.8

There exists a weak solution (u,σ)∈V×Lp(0,T;Lp(Ω;Rsymd×d)) to the nonlinear viscoelastic system (2.6)–(2.8) satisfying the initial conditions (2.9). Moreover, u∈W2,2(0,T;H).

The proof of Theorem 2.8 is postponed to the next section. We point out that the displacement u of the solution found in Theorem 2.8 is more regular in time, more precisely u¨∈L2(0,T;H). This regularity is used at the end of Sect. 3 to prove a uniqueness result for the nonlinear viscoelastic system  (2.6)–(2.9). Moreover, we exploit such a regularity in Sect. 4 to show the energy-dissipation balance of Theorem 4.1. This identity implies the viscoelastic paradox, which is discussed at the end of the paper.

Existence of solutions

This section is devoted to the proof of Theorem 2.8. As explained in the introduction, the main idea is to combine the discretization-in-time scheme of [10] with the regularization of the nonlinear operator G introduced in [3]. Therefore, we rephrase the system (2.6) in a simpler way, and we use Browder-Minty Theorem to find a sequence of approximate solutions in each node of the discretization scheme. Then in Lemma 3.3 we prove a discrete energy estimate and we use a compactness argument to obtain a pair (u,σ) which solves (2.10) (see Lemma 3.8). Finally, in Lemma 3.9, by performing a standard argument in the theory of nonlinear monotone operators we show the validity of the constitutive law (2.11).

Let us fix n∈N and set

We define Gn:Rsymd×d→Rsymd×d asGn(ξ):=G(ξ)+1n|ξ|p-2ξfor allξ∈Rsymd×d.

Notice that Gn still satisfies (G1)–(G3) with ϕ replaced byϕn(ξ):=ϕ(ξ)+1np|ξ|pfor allξ∈Rsymd×d,

and with b3 replaced by b3+1. Since Gn is strictly monotone, by the standard theory of monotone operators there exists the inverse operator Gn-1:Rsymd×d→Rsymd×d, which is still strictly monotone. Moreover, if we introduce the Legendre transform ϕn∗ of ϕn, defined asϕn∗(η):=supξ∈Rsymd×d{η·ξ-ϕn(ξ)}for allη∈Rsymd×d,

by (G1)–(G3) we have that ϕn∗:Rsymd×d→R is still a convex function of class C1 and Gn-1 satisfies3.1 Gn-1(η)=∇ϕn∗(η)for allη∈Rsymd×d,

3.2 Gn-1(η)·η≥c1|η|′-c2for allη∈Rsymd×d,

3.3 |Gn-1(η)|≤c3(1+|η|p′-1)for allη∈Rsymd×d,

for suitable constants c1,c3>0 and c2≥0 independent of n∈N. Furthermore, if we define η0:=G(0)=Gn(0), by the assumption ϕ(0)=0 (see Remark 2.2) we haveϕn∗(η0)=-ϕn(0)=0.

Therefore, thanks to the convexity of ϕn∗ we derive3.4 ϕn∗(η)≥ϕn∗(η0)+Gn-1(η0)·(η-η0)=0for allη∈Rsymd×d,

3.5 ϕn∗(η)≤ϕn∗(η0)+Gn-1(η)·(η-η0)≤c4(1+|η|′)for allη∈Rsymd×d,

for a suitable constant c4>0 independent of n∈N.

For all k∈{1,⋯,n} we search for a function unk∈V with unk-znk∈Vnk satisfying the following identity3.6 (δ2unk,φ)H+(Gn-1(eunk+eδunk),eφ)p,p′=(fnk,φ)Hfor allφ∈Vnk,

where3.7 δunk:=unk-unk-1τn,δ2unk:=δunk-δunk-1τnfork∈{1,⋯,n}.

To this aim, we find a function vnk∈Vnk which solves3.8 (δ2vnk+δ2znk,φ)H+(Gn-1(evnk+eδvnk+eznk+eδznk),eφ)p,p′=(fnk,φ)Hfor allφ∈Vnk,

where δznk and δ2znk are defined similarly to (3.7) starting from znk. Indeed, the function vnk∈Vnk solves (3.8) if and only if unk:=vnk+znk∈V satisfies unk-znk=vnk∈Vnk and (3.6).

To solve (3.8), we consider the family of nonlinear operators Fnk:Vnk→(Vnk)′ defined by⟨Fnk(v),φ⟩(Vnk)′:=1τn2(v+vnk-2-2vnk-1+τn2δ2znk-τn2fnk,φ)H+(Gn-1((1+1τn)ev-1τnvnk-1+eznk+eδznk),eφ)p,p′

for v,φ∈Vnk. It is clear that vnk∈Vnk solves (3.8) if and only if3.9 Fnk(vnk)=0in(Vnk)′.

To find a solution to (3.9) we need the following result, whose proof can be found in [2, 17].

Theorem 3.1

(Browder-Minty) Let X be a reflexive Banach space and let F:X→X′ be a monotone, hemicontinuous, and coercive operator. Then, F is surjective. Moreover, if F is strictly monotone, then F is also injective.

Let us show that Fnk satisfies the hypotheses of Theorem 3.1.

Proposition 3.2

For every n∈N and k∈{1,⋯,n} the nonlinear operator Fnk:Vnk→(Vnk)′ is strictly monotone, coercive, and hemicontinuous.

Proof

Let us fix n∈N and k∈{1,⋯,n}. We start by proving that Fnk is a strictly monotone operator, i.e.,⟨Fnk(v)-Fnk(w),v-w⟩(Vnk)′>0for allv,w∈Vnkwithv≠w.

By the definition of Fnk, for all v,w∈Vnk with v≠w we have3.10 ⟨Fnk(v)-Fnk(w),v-w⟩(Vnk)′=1τn2‖v-w‖H2+(Gn-1(cnev+hnk)-Gn-1(cnew+hnk),ev-ew)p,p′,

wherecn:=1+1τn>0,hnk:=-1τnevnk-1+eznk+eδznk∈Vnk.

By using in (3.10) the monotonicity of Gn-1 with η1=cnev+hnk and η2=cnew+hnk, we can write⟨Fnk(v)-Fnk(w),v-w⟩(Vnk)′≥1τn2‖v-w‖H2>0,

which shows the strictly monotonicity of Fnk.

To prove the coerciveness of Fnk, we have to show that3.11 ⟨Fnk(v),v⟩(Vnk)′‖v‖Vnk→∞as‖v‖Vnk→∞.

Notice that⟨Fnk(v),v⟩(Vnk)′=dn‖v‖H2+dn(ℓnk,v)H+1cn(Gn-1(cnev+hnk),cnev+hnk)p,p′-1cn(Gn-1(cnev+hnk),hnk)p,p′,

wheredn:=1τn2>0,ℓnk:=vnk-2-2vnk-1+τn2δ2znk+τn2fnk∈H.

Thanks to (3.2), (3.3), and Young inequality, for all ε>0 we have3.12 1cn(Gn-1(cnev+hnk),cnev+hnk)p,p′-1cn(Gn-1(cnev+hnk),hnk)p,p′≥c1cn‖cnev+hnk‖′′-c2cnLd(Ω)-1cn‖Gn-1(cnev+hnk)‖p‖hnk‖p′≥c1cn‖cnev+hnk‖′′-c2cnLd(Ω)-εppcn‖Gn-1(cnev+hnk)‖pp-1p′cnε′‖hnk‖′′≥c1cn‖cnev+hnk‖′′-c2cnLd(Ω)-1p′cnε′‖hnk‖′′-εppcn(2p-1c3p‖cnev+hnk‖′′+2p-1c3pLd(Ω))=1cn(c1-2p-1c3pεpp)‖cnev+hnk‖′′-1p′cnε′‖hnk‖′′-1cn(c2+2p-1c3pεpp)Ld(Ω).

In particular, the Korn-Poincaré inequality (2.3) yieldscn′CKP′‖v‖Vnk′≤‖cnev‖′′≤2p′-1‖cnev+hnk‖′′+2p′-1‖hnk‖′′.

Hence, from (3.12) we deduce3.13 1cn(Gn-1(cnev+hnk),cnev+hnk)p,p′-1cn(Gn-1(cnev+hnk),hnk)p,p′≥cnp′-12p′-1CKP′(c1-2p-1c3pεpp)‖v‖Vnk′-1cn(c1-2p-1c3pεpp+1p′ε′)‖hnk‖′′-1cn(c2+2p-1c3pεpp)Ld(Ω).

By applying again Young inequality, we can write3.14 dn‖v‖H2+dn(ℓnk,v)H≥dn2‖v‖H2-dn2‖ℓnk‖H2.

If we choose0<ε<c1p2p-1c3p1p,

thanks to (3.13) and (3.14) we obtain the existence a positive constant K1 such that3.15 ⟨Fnk(v),v⟩(Vnk)′≥K1‖v‖H2+‖v‖Vnk′-‖hnk‖′′-‖ℓnk‖H2-1.

Clearly, we have3.16 ‖hnk‖′′+‖ℓnk‖H2+1‖v‖Vnk→0as‖v‖Vnk→∞.

Moreover, we can write3.17 ‖v‖H2+‖v‖Vnk′‖v‖Vnk≥‖v‖Vnkp′-1→∞as‖v‖Vnk→∞.

Thanks to (3.15)–(3.17) we get (3.11).

To prove the hemicontinuity of Fnk, we need to show that for all u,v,w∈Vnk there exists t0=t0(u,v,w) such that the function [-t0,t0]∋t↦⟨Fnk(v+tu),w⟩(Vnk)′ is continuous in t=0. We fix u,v,w∈Vnk and we notice that⟨Fnk(v+tu),w⟩(Vnk)′=dn(v+ℓnk,w)H+dnt(u,w)H+(Gn-1(cn(ev+teu)+hnk),ew)p,p′.

Moreover, we can write3.18 Gn-1(cn(ev+teu)+hnk)·ew→t→0a.e.Gn-1(cnev+hnk)·ew,

and thanks to (3.3) we get3.19 |(Gn-1(cn(ev+teu)+hnk),ew)p,p′|≤1p‖Gn-1(cn(ev+teu)+hnk)‖pp+1p′‖ew‖′′≤2p-1c3pp‖cn(ev+teu)+hnk‖p′′+2p-1c3ppLd(Ω)+1p′‖ew‖′′≤K2(‖cnev+hnk‖p′′+‖eu‖p′′+‖ew‖′′+1),

for a positive constant K2. By using (3.18), (3.19), and dominate convergence theorem we obtain3.20 (Gn-1(cn(ev+teu)+hnk),ew)p,p′→t→0(Gn-1(cnev+hnk),ew)p,p′.

Since dnt(u,w)H→0 as t→0, by (3.20) we have⟨Fnk(v+tu),w⟩(Vnk)′→t→0dn(v+ℓnk,w)H+(Gn-1(cnev+hnk),ew)p,p′=⟨Fnk(v),w⟩(Vnk)′,

□

Thanks to Theorem 3.1 and Proposition 3.2, we obtain that for all n∈N and k∈{1,⋯,n} the nonlinear operator Fnk:Vnk→(Vnk)′ is bijective, and hence there exists a unique vnk∈Vnk which solves (3.8). As a consequence, the function unk=vnk+znk∈V is the unique solution to (3.6).

Let us define3.21 σnk:=Gn-1(eunk+eδunk)for allk∈{1,⋯,n}.

In the next lemma, we show a uniform energy estimate with respect to n for the family {(unk,σnk)}k=1n, which will be used to pass to the limit as n→∞ in the discrete equation (3.6).

Lemma 3.3

There exists a positive constant C1, independent of n∈N, such that3.22 maxi∈{1,⋯,n}‖uni‖V+maxi∈{1,⋯,n}‖δuni‖H+∑i=1nτn‖δuni‖V′+∑i=1nτn‖σni‖pp≤C1.

Proof

We take φ=τn(unk-znk)∈Vnk as a test function in (3.6). Therefore, we obtain3.23 τn(Gn-1(eunk+eδunk),eunk-eznk)p,p′=τn(fnk,unk-znk)H-τn(δ2unk,unk-znk)H.

We fix i∈{1,⋯,n} and by summing in (3.23) over k∈{1,⋯,i} we obtain3.24 ∑k=1iτn(Gn-1(eunk+eδunk),eunk)p,p′=∑k=1iτn(Gn-1(eunk+eδunk),eznk)p,p′+∑k=1iτn(fnk,unk-znk)H-∑k=1iτn(δ2unk,unk-znk)H.

Now we use φ=τn(δunk-δznk)∈Vnk as a test function in (3.6) and we get3.25 ‖δunk‖H2-(δunk-1,δunk)H+τn(Gn-1(eunk+eδunk),eδunk)p,p′=τn(fnk,δunk-δznk)H+τn(Gn-1(eunk+eδunk),eδznk)p,p′+τn(δ2unk,δznk)H.

By means of the following identity‖δunk‖H2-(δunk-1,δunk)H=12‖δunk‖H2-12‖δunk-1‖H2+τn22‖δ2unk‖H2,

from (3.25) we infer12‖δunk‖H2-12‖δunk-1‖H2+τn22‖δ2unk‖H2+τn(Gn-1(eunk+eδunk),eδunk)p,p′=τn(fnk,δunk-δznk)H+τn(Gn-1(eunk+eδunk),eδznk)p,p′+τn(δ2unk,δznk)H,

and, by summing again over k∈{1,⋯,i} we get3.26 12‖δuni‖H2-12‖u1‖H2+∑k=1iτn(Gn-1(eunk+eδunk),eδunk)p,p′≤∑k=1iτn(fnk,δunk-δznk)H+∑k=1iτn(Gn-1(eunk+eδunk),eδznk)p,p′+∑k=1iτn(δ2unk,δznk)H.

By considering together (3.24) and (3.26) we get12‖δuni‖H2+∑k=1iτn(Gn-1(eunk+eδunk),eunk+eδunk)p,p′≤12‖u1‖H2+∑k=1iτn(Gn-1(eunk+eδunk),eznk+eδznk)p,p′+∑k=1iτn(fnk,unk+δunk-znk-δznk)H+∑k=1iτn(δ2unk,znk+δznk)H-∑k=1iτn(δ2unk,unk)H.

Thanks to (3.1)–(3.3) and the Korn-Poincaré inequality (2.3) we deduce from the previous estimate3.27 12‖δuni‖H2+c1CKP′∑k=1iτn‖unk+δunk‖V′≤c2TLd(Ω)+12‖u1‖H2+∑k=1iτn(Gn-1(eunk+eδunk),eznk+eδznk)p,p′+∑k=1iτn(fnk,unk+δunk-znk-δznk)H+∑k=1iτn(δ2unk,znk+δznk)H-∑k=1iτn(δ2unk,unk)H.

Let us now estimate the right-hand side of (3.27) from above. We can write3.28 ∑k=1iτn(fnk,unk+δunk)H≤‖f‖L2(0,T;H)2+12∑k=1iτn‖unk‖H2+12∑k=1iτn‖δunk‖H2,

3.29 ∑k=1iτn(fnk,znk+δznk)H≤‖f‖L2(0,T;H)2+T2‖z‖L∞(0,T;H)2+12‖z˙‖L2(0,T;H)2.

Moreover3.30 ∑k=1iτn(Gn-1(eunk+eδunk),eznk+eδznk)p,p′≤εpp∑k=1iτn‖Gn-1(eunk+eδunk)‖pp+1p′ε′∑k=1iτn‖eznk+eδznk‖p′′≤2p-1c3pεpp∑k=1iτn‖unk+δunk‖V′+2p-1c3pTεppLd(Ω)+2p′-1Tp′ε′‖z‖L∞(0,T;V)′+2p′-1p′ε′‖z˙‖L′(0,T;V)′.

Notice that the following discrete integration by parts formulas hold3.31 ∑k=1iτn(δ2unk,znk+δznk)H=(δuni,zni+δzni)H-(δun0,zn0+δzn0)H-∑k=1iτn(δunk-1,δznk+δ2znk)H,

3.32 ∑k=1iτn(δ2unk,unk)H=(δuni,uni)H-(δun0,un0)H-∑k=1iτn(δunk-1,δunk)H.

Since3.33 ∑k=1iτn‖δunk-1‖H2=∑k=0i-1τn‖δunk‖H2≤T‖u1‖H2+∑k=1iτn‖δunk‖H2,

thanks to (3.31) we can write for all ε>03.34 |∑k=1iτn(δ2unk,znk+δznk)H|≤ε2‖δuni‖H2+12ε‖zni+δzni‖H2+‖u1‖H‖z(0)+z˙(0)‖H+12∑k=1iτn‖δunk-1‖H2+12∑k=1iτn‖δznk+δ2znk‖H2≤K1ε+ε2‖δuni‖H2+12∑k=1iτn‖δunk‖H2,

where K1ε is a positive constant depending on ε. Moreover, since uni=∑k=1iτnδunk+u0 for all i∈{1,⋯,n}, the discrete Hölder inequality gives us3.35 ‖uni‖H≤∑k=1iτn‖δunk‖H+‖u0‖H≤T12∑k=1iτn‖δunk‖H212+‖u0‖H.

Hence from (3.32), (3.33), and (3.35) we deduce3.36 ∑k=1iτn(δ2unk,unk)H≤ε2‖δuni‖H2+12ε‖uni‖H2+‖u1‖H‖u0‖H+12∑k=1iτn‖δunk-1‖H2+12∑k=1iτn‖δunk‖H2≤K2ε+ε2‖δuni‖H2+1+Tε∑k=1iτn‖δunk‖H2,

where K2ε is a positive constant depending on ε. Furthermore3.37 12∑k=1iτn‖unk‖H2≤T‖u0‖H2+T2∑k=1iτn‖δuni‖H2.

If we consider together (3.27)–(3.37), we get(12-ε)‖δuni‖H2+c1CKP′-2p-1c3pεpp∑k=1iτn‖unk+δunk‖V′≤K3ε1+∑k=1iτn‖δunk‖H2,

where K3ε is a positive constant depending on ε. By choosing0<ε<min12,(c1pCKP′2p-1c3p)1p

we get the existence of a positive constant K4 independent of n and i such that3.38 ‖δuni‖H2+∑k=1iτn‖unk+δunk‖V′≤K41+∑k=1iτn‖δunk‖H2.

By defining ani:=‖δuni‖H2 for all i∈{1,⋯,n}, from (3.38) we deriveani≤K41+∑k=1iτnankfor alli∈{1,⋯,n},

and taking into account a discrete version of Gronwall lemma (see, e.g., [1, Lemma 3.2.4]) we deduce that the family {ani}i=1n is bounded by a positive constant K5 independent of i and n, i.e.,3.39 ‖δuni‖H2≤K5foralli∈{1,⋯,n}andn∈N.

By using (3.38) and (3.39) we get the existence of a positive constant K6 independent of n such that3.40 maxi∈{1,⋯,n}‖δuni‖H2+∑i=1nτn‖uni+δuni‖V′≤K6.

In particular, by (3.3) and (3.21) it holds3.41 ∑i=1nτn‖σni‖pp≤2p-1c3p∑i=1nτn‖euni+eδuni‖p′′+2p-1c3pTLd(Ω)≤2p-1c3pK6+2p-1c3pTLd(Ω).

To get the last estimate in (3.22) we set bnk:=(1+τn)k for k∈{0,⋯,n} and we notice that3.42 bnk-bnk-1τn=bnk-1for allk∈{1,⋯,n}.

From (3.42) we can write3.43 bnkunk-bn0un0=∑j=1k(bnjunj-bnj-1unj-1)=∑j=1kτnbnj-bnj-1τnunj+∑j=1kτnbnj-1unj-unj-1τn=∑j=1kτnbnj-1(unj+δunj).

Since1≤(1+τn)k≤(1+τn)n=1+TnnTT≤eT,

from (3.43) we deduce the existence of a positive constant K7 such that3.44 ‖unk‖V′≤K71+∑j=1kτn‖unj+δunj‖V′≤K71+K6.

As a consequence of this, we obtain3.45 ∑j=1kτn‖δunj‖V′≤2p′-1∑j=1kτn(‖unj+δunj‖V′+‖unj‖V′)≤2p′-1K6+TK7+TK7K6.

Hence by considering together (3.40), (3.41), (3.44), and (3.45) we get (3.22). □

As a consequence of (3.22) and of the particular form of equation (3.6), we derive also a uniform bound on the discrete second time derivative of unk in the space H. This allows us to find in the limit as n→∞ a weak solution to (2.6)–(2.8) with displacement u∈W2,2(0,T;H).

Corollary 3.4

There exists a constant C2, independent of n∈N, such that3.46 ∑k=1nτn‖δ2unk‖H2≤C2.

Proof

Let us define vnk:=unk+δunk∈V for all k∈{1,⋯,n} and n∈N. By equation (3.6) we deduce that vnk solves the following equation(δvnk,φ)H+(Gn-1(evnk),eφ)p,p′=(fnk+δunk,φ)Hfor allφ∈Vnk.

We take φ:=τn(δvnk-δznk-δ2znk)∈Vnk as a test function in (3.6). We fix i∈{1,⋯,n} and by summing over k∈1,⋯i we get3.47 ∑k=1iτn‖δvnk‖H2+∑k=1i(Gn-1(evnk),evnk-evnk-1)p,p′=∑k=1iτn(fnk+δunk,δvnk)H-∑k=1iτn(fnk+δunk,δznk+δ2znk)H+∑k=1iτn(δvnk,δznk+δ2znk)H+∑k=1iτn(Gn-1(evnk),eδznk+eδ2znk)p,p′.

Let us now estimate the right-hand side of (3.47) from above. Thanks to (3.22) we can write3.48 ∑k=1iτn(fnk+δunk,δvnk)H≤12ε‖f‖L2(0,T;H)2+TC122ε+ε∑k=1iτn‖δvnk‖H2,

3.49 ∑k=1iτn(fnk+δunk,δznk+δ2znk)H≤‖f‖L2(0,T;H)2+TC12+‖z˙‖W1,2(0,T;H)2,

3.50 ∑k=1iτn(δvnk,δznk+δ2znk)H≤ε∑k=1iτn‖δvnk‖H2+12ε‖z˙‖W1,2(0,T;H)2.

Moreover3.51 ∑k=1iτn(Gn-1(evnk),eδznk+eδ2znk)p,p′≤1p∑k=1iτn‖Gn-1(evnk)‖pp+1p′∑k=1iτn‖eδznk+eδ2znk‖p′′≤2p-1c3pp∑k=1iτn‖unk+δunk‖V′+2p-1c3pTpLd(Ω)+2p-1p′‖z˙‖W1,p′(0;T;V)′≤4p-1c3pp(TC1p+C1)+2p-1c3pTpLd(Ω)+2p-1p′‖z˙‖W1,p′(0;T;V)′.

Finally, by (3.1) and the convexity of ϕn∗ we have3.52 ∑k=1i(Gn-1(evnk),evnk-evnk-1)p,p′≥∑k=1i∫Ωϕn∗(evnk(x))dx-∫Ωϕn∗(evnk-1(x))dx=∫Ωϕn∗(evni(x))dx-∫Ωϕn∗(evn0(x))dx.

By combining (3.47)–(3.52) with the bound (3.5) for ϕn∗, we deduce the existence of a positive constant Kε, which depends on ε, but it is independent of n and i, such that3.53 (1-2ε)∑k=1iτn‖δvnk‖H2+∫Ωϕn∗(evni(x))dx≤Kεfor alli∈{1,⋯,n}.

By choosing ε=14 and using (3.4), from (3.22) and (3.53) we deduce (3.46). □

We now want to pass to the limit as n→∞ in the discrete equation (3.6) to obtain a weak solution (u,σ) to the nonlinear viscoelastic system (2.6)–(2.8), according to Definition 2.3. We start by defining the following interpolation sequences of {(unk,σnk)}k=1n:un(t):=unk+(t-kτn)δunk,u~n(t):=δunk+(t-kτn)δ2unk,t∈[(k-1)τn,kτn],k∈{1,⋯,n},un+(t):=unk,u~n+(t):=δunk,t∈((k-1)τn,kτn],k∈{1,⋯,n},un+(0):=un0=u0,u~n+(t):=δun0=u1,un-(t):=unk-1,u~n-(t):=δunk-1,t∈[(k-1)τn,kτn),k∈{1,⋯,n},un-(T):=unn,u~n-(T):=δunn,σn+(t):=σnk,t∈((k-1)τn,kτn],k∈{1,⋯,n}.

By means of this notation, we can state the following convergence lemma.

Lemma 3.5

There exists a pair (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) such that, up to a not relabeled subsequence3.54

3.55

Moreover3.56

3.57

Proof

Thanks to Lemma 3.3 and the estimate (3.46), the sequences{un}n⊂W1,p′(0,T;V)∩W1,∞(0,T;H),{u~n}n⊂L′(0,T;V)∩W1,2(0,T;H),{σn+}n⊂Lp(0,T;Lp(Ω;Rsymd×d)),

are uniformly bounded with respect to n∈N. Indeed, we have‖un‖W1,p′(0,T;V)′≤Tmaxk∈{0,⋯,n}‖unk‖V′+∑k=1nτn‖δunk‖V′,‖un‖W1,∞(0,T;H)≤maxk∈{0,⋯,n}‖unk‖H+maxk∈{1,⋯,n}‖δunk‖H,‖u~n‖L′(0,T;V)′≤2∑k=1nτn‖δunk‖V′+‖u1‖V′,‖u~n‖W1,2(0,T;H)2≤Tmaxk∈{0,⋯,n}‖δunk‖H2+∑k=1nτn‖δ2unk‖H2,‖σn+‖Lp(0,T;Lp(Ω;Rsymd×d))p=∑k=1nτn‖σnk‖pp.

By Banach-Alaoglu theorem and Lemma 2.5 there exist three functions u∈W1,p′(0,T;V)∩W1,∞(0,T;H), v∈L′(0,T;V)∩W1,2(0,T;H), and σ∈Lp(0,T;Lp(Ω;Rsymd×d)) such that, up to a not relabeled subsequence3.58

and

Thanks to (3.46) we get‖u˙n-u~n‖L2(0,T;H)2≤τn2∑k=1nτn‖δ2unk‖H2≤C2τn2→n→∞0,

from which we deduce that v=u˙.

By (3.22) also the sequences3.59 {un±}n⊂L∞(0,T;V),{u~n±}n⊂L′(0,T;V)∩L∞(0,T;H),

are uniformly bounded. Moreover, by using again (3.22) and (3.46) we have‖un-un+‖L∞(0,T;H)≤τnmaxk∈{1,⋯,n}‖δunk‖H≤C1τn→n→∞0,‖un+-un-‖L∞(0,T;H)≤τnmaxk∈{1,⋯,n}‖δunk‖H≤C1τn→n→∞0,‖u~n-u~n+‖L2(0,T;H)2≤τn2∑k=1nτn‖δ2unk‖H2≤C2τn2→n→∞0,‖u~n+-u~n-‖L2(0,T;H)2≤τn2∑k=1nτn‖δ2unk‖H2≤C2τn2→n→∞0.

We combine (3.58) and (3.59) with the previous convergences to derive

Finally, by (3.58) for all t∈[0,T] we have

Thanks to (3.22) and (3.46), for all t∈[0,T] we get‖un±(t)‖V≤C1,‖un+(t)-un(t)‖H≤C1τn→n→∞0,‖un+(t)-un-(t)‖H≤C1τn→n→∞0,‖u~n±(t)‖H≤C1,‖u~n+(t)-u~n(t)‖H2≤τn∑k=1nτn‖δ2unk‖H2≤C2τn→n→∞0,‖u~n+(t)-u~n-(t)‖H2=τn∑k=1nτn‖δ2unk‖H2≤C2τn→n→∞0,

which imply (3.56) and (3.57). □

In view of the compactness of the embedding V↪H (see Remark 2.1), we deduce also the following strong convergences.

Corollary 3.6

Let (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) be the pair of functions given by Lemma 3.5. Then, we have3.60 un+→n→∞L2(0,T;H)u,u~n+→n→∞L2(0,T;H)u˙.

Proof

By Lemma 3.5 we know that the following sequences{un}n⊂W1,p′(0,T;V)∩W1,∞(0,T;H),{u~n}n⊂L′(0,T;V)∩W1,2(0,T;H),

are uniformly bounded with respect to n. Since the embedding V↪H is compact, by Aubin-Lions lemma (see for example [24, Theorem 3]), we deriveun→n→∞L2(0,T;H)u,u~n→n→∞L2(0,T;H)u˙.

Moreover, we have‖un-un+‖L2(0,T;H)2≤τn2∑k=1nτn‖δunk‖H2≤TC1τn2→n→∞0,‖u~n-u~n+‖L2(0,T;H)2≤τn2∑k=1nτn‖δ2unk‖H2≤C2τn2→n→∞0,

which imply (3.60). □

We want to prove that the pair (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) of Lemma 3.5 is a weak solution to the nonlinear viscoelastic system (2.6)–(2.8) with initial conditions (2.9). To this aim, we need to check (i)–(iii) of Definition 2.3 and that u(0)=u0 in V and u˙(0)=u1 in H. We start by showing that the function u∈V∩W2,2(0,T;H) satisfies the Dirichlet boundary conditions and the initial conditions.

Lemma 3.7

The function u∈V∩W2,2(0,T;H) of Lemma 3.5 satisfies (i) of Definition 2.3 and the initial conditions u(0)=u0 in V and u˙(0)=u1 in H.

Proof

By (3.56) we have

Hence, u(0)=u0 in V and u˙(0)=u1 in H. Moreover, since z∈C0([0,T];V0) and thanks to (3.57), we have for all t∈[0,T]

Thus, u(t)-z(t)∈VtD for all t∈[0,T], being VtD a closed subspace of V. □

With the next lemma, we show that the pair (u,σ) solves the weak formulation (2.10) of the elastodynamics system.

Lemma 3.8

The pair (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) of Lemma 3.5 satisfies (ii) of Definition 2.3.

Proof

We fix n∈N and a function φ∈D. We consider the following functionsφnk:=φ(kτn)fork∈{0,⋯,n},δφnk:=φnk-φnk-1τnfork∈{1,⋯,n},

and the piecewise-constant approximating sequencesφn+(t):=φnk,φ~n+(t):=δφnk,fn+(t):=fnk,fort∈((k-1)τn,kτn],k∈{1,⋯,n}.

If we use τnφnk∈Vnk as a test function in (3.6), after summing over k∈{1,⋯,n}, we get3.61 ∑k=1nτn(δ2unk,φnk)H+∑k=1nτn(σnk,eφnk)p,p′=∑k=1nτn(fnk,φnk)H.

Since φn0=φnn=0 we obtain∑k=1nτn(δ2unk,φnk)H=∑k=1n(δunk,φnk)H-∑k=1n(δunk-1,φnk)H=∑k=0n-1(δunk,φnk)H-∑k=0n-1(δunk,φnk+1)H=-∑k=0n-1τn(δunk,δφnk+1)H=-∑k=1nτn(δunk-1,δφnk)H=-∫0T(u~n-(t),φ~n+(t))Hdt,

and from (3.61) we deduce3.62 -∫0T(u~n-(t),φ~n+(t))Hdt+∫0T(σn+(t),eφn+(t))p,p′dt=∫0T(fn+(t),φn+(t))Hdt.

Thanks to (3.55) and the convergencesφn+→n→∞L′(0,T;V)φ,φn+→n→∞L2(0,T;H)φ,φ~n+→n→∞L2(0,T;H)φ˙,fn+→n→∞L2(0,T;H)f

we can pass to the limit in (3.62), and we get that the pair (u,σ)∈V×Lp(0,T;Lp(Ω;Rsymd×d)) satisfies (ii) of Definition 2.3. □

Finally, we have that the pair (u,σ) satisfies the constitutive law (2.11).

Lemma 3.9

The pair (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) of Lemma 3.5 satisfies (iii) of Definition 2.3.

Proof

In order to verify the constitutive law, we use a modification of Minty method, as done in [3, 20]. Since u˙∈W1,2(0,T;H), by integrating by parts in (2.10) we deduce that (u,σ) solve3.63 ∫0T(u¨(t),φ(t))Hdt+∫0T(σ(t),eφ(t))p,p′dt=∫0T(f(t),φ(t))Hdtfor allφ∈D.

Let us now consider a function φ∈L′(0,T;V)∩L2(0,T;H) with φ(t)∈VtD for a.e. t∈[0,T]. Then, there exists a sequence of functions {φn}n⊂D such thatφn→n→∞L′(0,T;V)φ,φn→n→∞L2(0,T;H)φ.

This can be done, for example, by considering a sequence {ωn}n⊂Cc1((2n,T-2n)) with 0≤ωn≤1 in [0, T] for all n∈N and such that ωn(t)→1 as n→∞ for all t∈(0,T), and a sequence {ρn}n⊂Cc1((0,1n)) with ρn≥0 and ∫Rρndt=1 for all n∈N, and definingφn:=ρn∗(ωnφ)for alln∈N

(see also [14, Lemma 2.8]). By testing (3.63) with φn and passing to the limit as n→∞ can deduce that the pair (u,σ) satisfies3.64 ∫0T(u¨(t),φ(t))Hdt+∫0T(σ(t),eφ(t))p,p′dt=∫0T(f(t),φ(t))Hdt

for all φ∈L′(0,T;V)∩L2(0,T;H) with φ(t)∈VtD for a.e. t∈[0,T]. Notice thatu˙-z˙∈L′(0,T;V)∩L2(0,T;H),u˙(t)-z˙(t)∈VtDfora.e.t∈[0,T],

since u(t)-u(t-h)h-z(t)-z(t-h)h∈VtD for all t∈(0,T] and h∈(0,t), andu(t)-u(t-h)h-z(t)-z(t-h)h→u˙(t)-z˙(t)for a.e.t∈[0,T]ash→0.

Hence, by using φ:=u+u˙-z-z˙ in (3.64) we get3.65 ∫0T(σ(t),eu(t)+eu˙(t))p,p′dt=∫0T(f(t),u(t)+u˙(t))Hdt-∫0T(u¨(t),u(t)+u˙(t))Hdt-∫0T(f(t),z(t)+z˙(t))Hdt+∫0T(u¨(t),z(t)+z˙(t))Hdt+∫0T(σ(t),ez(t)+ez˙(t))p,p′dt.

We now consider equation (3.6) and we use φ=τn(unk+δunk-znk-δznk) as test function. By summing over k∈{1,⋯,n} we get∑k=1nτn(Gn-1(eunk+eδunk),eunk+eδunk)p,p′=∑k=1nτn(fnk,unk+δunk)H-∑k=1nτn(δ2unk,unk+δunk)H-∑k=1nτn(fnk,znk+δznk)H+∑k=1nτn(δ2unk,znk+δznk)H+∑k=1nτn(Gn-1(eunk+eδunk),eznk+eδznk)p,p′.

By using the notations introduced before, we can rewrite the previous identity as3.66 ∫0T(σn+(t),Gn(σn+(t)))p,p′dt=∫0T(Gn-1(eun+(t)+eu~n+(t)),eun+(t)+eu~n+(t))p,p′dt=∫0T(fn+(t),un+(t)+u~n+(t))Hdt-∫0T(u~˙n(t),un+(t)+u~n+(t))Hdt-∫0T(fn+(t),zn+(t)+z~n+(t))Hdt+∫0T(u~˙n(t),zn+(t)+z~n+(t))Hdt+∫0T(σn+(t),ezn+(t)+ez~n+(t))p,p′dt.

Now we pass to the limit in (3.66) as n→∞. Thanks to the strong convergencesfn+→n→∞L2(0,T;H)f,zn+→n→∞L′(0,T;V)z,zn+→n→∞L2(0,T;H)z,z~n+→n→∞L′(0,T;V)z˙,z~n+→n→∞L2(0,T;H)z˙

and the convergences in (3.54), (3.55), and (3.60) we deduce that there existslimn→∞∫0T(σn+(t),Gn(σn+(t)))p,p′dt=∫0T(f(t),u(t)+u˙(t))Hdt-∫0T(u¨(t),u(t)+u˙(t))Hdt-∫0T(f(t),z(t)+z˙(t))Hdt+∫0T(u¨(t),z(t)+z˙(t))Hdt+∫0T(σ(t),ez(t)+ez˙(t))p,p′dt=∫0T(σ(t),eu(t)+eu˙(t))p,p′dt,

in view of (3.65). Notice that by (3.22)3.67 ‖G(σn+)-eun+-eu~n+‖L′(0,T;L′(Ω;Rsymd×d))′=‖G(σn+)-Gn(σn+)‖L′(0,T;L′(Ω;Rsymd×d))′=1n′‖σn+‖Lp(0,T;Lp(Ω;Rsymd×d))p≤C1n′→n→∞0,,

which giveslimn→∞∫0T(σn+(t),G(σn+(t)))p,p′dt=limn→∞∫0T(σn+(t),Gn(σn+(t)))p,p′dt=∫0T(σ(t),eu(t)+eu˙(t))p,p′dt.

Moreover, thanks to (G3) and (3.22) the sequence {G(σn+)}n⊂L′(0,T;L′(Ω;Rsymd×d)) is uniformly bounded. Hence, by (3.54) and (3.67) we derive

We combine these two facts and we obtain that for all w∈Lp(0,T;Lp(Ω;Rsymd×d))0≤limn→∞∫0T(σn+(t)-w(t),G(σn+(t))-G(w(t)))p,p′dt=∫0T(σ(t)-w(t),eu(t)+eu˙(t)-G(w(t)))p,p′dt.

In particular, we take w:=σ-kb with b∈Lp(0,T;Lp(Ω;Rsymd×d)) and k>0, and by dividing by k we get0≤∫0T(b(t),eu(t)+eu˙(t)-G(σ(t)-kb(t)))p,p′dt.

Since G is continuous, by sending k→0+ we deduce0≤∫0T(b(t),eu(t)+eu˙(t)-G(σ(t)))p,p′dt

for all b∈Lp(0,T;Lp(Ω;Rsymd×d)). This implies the constitutive law (2.11). □

We can finally prove our main existence result Theorem 2.8.

Proof of Theorem 2.8

It is enough to combine Lemma 3.5 with Lemmas 3.7–3.9. □

We conclude this section with a uniqueness result in the space (V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) for the weak solutions (u,σ) to the system (2.6)–(2.8) which satisfy the initial conditions (2.9).

Theorem 3.10

Let (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) be a weak solution to the nonlinear viscoelastic system (2.6)–(2.8) satisfying the initial conditions (2.9). Then, the function u is unique. Moreover, if G is strictly monotone, also σ is unique.

Proof

Let (u1,σ1),(u2,σ2)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) be two weak solutions to the nonlinear viscoelastic system (2.6)–(2.8) satisfying the initial conditions (2.9).

We fix s∈(0,T]. If we set u:=u1-u2∈V∩W2,2(0,T;H), by arguing as in (3.64), we derive that u satisfies the following identity3.68 ∫0s(u¨(t),φ(t))Hdt+∫0s(σ1(t)-σ2(t),eφ(t))p,p′dt=0

for all φ∈L′(0,s;V)∩L2(0,s;H) with φ(t)∈VtD for a.e. t∈[0,s]. Moreover, we have3.69 u(0)=u˙(0)=0,u(t)+u˙(t)∈VtDfora.e.t∈[0,T].

Thanks to (3.69) we can use u+u˙ as test function in (3.68), and we get3.70 ∫0s(u¨(t),u(t)+u˙(t))Hdt=-∫0s(σ1(t)-σ2(t),eu(t)+eu˙(t))p,p′dt.

By taking into account (2.5) and (2.11), by (3.69) we have3.71 ∫0s(σ1(t)-σ2(t),eu(t)+eu˙(t))p,p′dt=∫0s(σ1(t)-σ2(t),G(σ1(t))-G(σ2(t)))p,p′dt≥0.

Moreover, since u∈W2,2(0,T;H), we derive3.72 ∫0s(u¨(t),u(t)+u˙(t))Hdt=12‖u˙(s)‖H2+(u˙(s),u(s))H-∫0s‖u˙(t)‖H2dt,

and by Young inequality3.73 |(u˙(s),u(s))H|≤14‖u˙(s)‖H2+‖u(s)‖H2≤14‖u˙(s)‖H2+T∫0s‖u˙(t)‖H2dt.

Hence, by (3.70)–(3.73), for every s∈(0,T] we obtain3.74 14‖u˙(s)‖H2-(T+1)∫0s‖u˙(t)‖H2dt≤12‖u˙(s)‖H2+(u˙(s),u(s))H-∫0s‖u˙(t)‖H2dt≤0.

In particular, sinceddse-4(T+1)s∫0s‖u˙(t)‖H2dt=e-4(T+1)s‖u˙(s)‖H2-4(T+1)∫0s‖u˙(t)‖H2dtfora.e.s∈[0,T],

thanks to (3.74) we have that the function s↦e-4(T+1)s∫0s‖u˙(t)‖H2dt is decreasing on [0, T], from which we deduce∫0s‖u˙(t)‖H2dt=0for alls∈[0,T].

Therefore, u˙≡0 on [0, T], which implies u≡c for some constant c∈H. By (3.69), we have 0=u(0)=c, that is u1=u2.

Finally, if G is strictly monotone, by G(σ1)-G(σ2)=eu+eu˙=0, we conclude that σ1=σ2. □

Energy-dissipation balance and the viscoelastic paradox

In Theorem 2.8, we proved the existence of a solution (u,σ) to the nonlinear viscoelastic system (2.6)–(2.8). As observed in Lemma 3.9, the displacement u obtained via the discretization-in-time scheme is more regular in time, more precisely u∈W2,2(0,T;H). This regularity allows us to prove the following energy-dissipation balance.

Theorem 4.1

Every weak solution (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) to the nonlinear viscoelastic system (2.6)–(2.8) satisfies the energy-dissipation balance4.1 12‖u˙(s)‖H2+∫0s(σ(t),eu˙(t))p,p′dt=12‖u˙(0)‖H2+W(0,s;u,σ)for alls∈[0,T],

where W(0,s;u,σ) is the total work of (u,σ) on the time interval [0,s]⊆[0,T], defined asW(0,s;u,σ):=∫0s(f(t),u˙(t)-z˙(t))Hdt+∫0s(u¨(t),z˙(t))Hdt+∫0s(σ(t),ez˙(t))p,p′dtforalls∈[0,T].

Proof

We fix s∈(0,T]. By arguing as in (3.64), we derive that the pair (u,σ)∈(V∩W2,2(0,T;H))×Lp(0,T;Lp(Ω;Rsymd×d)) satisfies∫0s(u¨(t),φ(t))Hdt+∫0s(σ(t),eφ(t))p,p′dt=∫0s(f(t),φ(t))Hdt

for all φ∈L′(0,s;V)∩L2(0,s;H) with φ(t)∈VtD for a.e. t∈[0,s]. Hence, if we use φ:=u˙-z˙ we obtain∫0s(u¨(t),u˙(t))Hdt+∫0s(σ(t),eu˙(t))p,p′dt=W(0,s;u,σ)for alls∈[0,T].

Finally, since u∈W2,2(0,T;H), we can use the identity∫0s(u¨(t),u˙(t))Hdt=12‖u˙(s)‖H2-12‖u˙(0)‖H2for alls∈[0,T]

to derive (4.1). □

We conclude the paper by showing that in the nonlinear Kelvin–Voigt model, which is the one associated with the monotone operator4.2 G(ξ):=|ξ|p-2ξforξ∈Rsymd×d,

the solution to the system (2.6)–(2.8) found in Theorem 2.8 satisfies another energy-dissipation balance, which is (4.7). This implies that the crack can not propagate in time, i.e., also the nonlinear Kelvin–Voigt model of dynamic fracture exhibits the viscoelastic paradox, as discussed in the introduction.

We assume that G is defined by (4.2). Therefore, G satisfies the assumptions (G1)–(G3) and in addition it is strictly monotone. In particular, G is invertible and its inverse is given byG-1(η)=|η|p′-2ηforη∈Rsymd×d.

In this case, the system (2.6) reduces to4.3 u¨(t)-div(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t)))=f(t)inΩ\Γt,t∈[0,T],

with boundary conditions4.4 u(t)=z(t)on∂DΩ,t∈[0,T],

4.5 |eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t))ν=0on∂NΩ∪Γt,t∈[0,T],

and initial conditions4.6 u(0)=u0,u˙(0)=u1inΩ\Γ0.

According to Definition 2.3, we say that u∈V is a weak solution to the nonlinear Kelvin–Voigt system (4.3)–(4.5) if u(t)-z(t)∈VtD for all t∈[0,T] and the following identity holds:-∫0T(u˙(t),φ˙(t))Hdt+∫0T(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t)),eφ(t))p,p′dt=∫0T(f(t),φ(t))Hdt

for all φ∈D. By Theorems 2.8 and 3.10 we know that there exists a unique weak solution u∈V∩W2,2(0,T;H) to (4.3)–(4.5) which satisfies the initial conditions (4.6). Moreover, by Theorem 4.1 the function u satisfies the identity (4.1).

We want to show that the energy-dissipation balance (4.1) can be rephrased just in terms of u. Given u∈V∩W2,2(0,T;H), we define the mechanical energy E at time s∈[0,T] asE(s;u):=12‖u˙(s)‖H2+1p′‖eu(s)‖p′′,

the energy dissipated by the viscous term V on the time interval [0,s]⊆[0,T] asV(0,s;u):=∫0s(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t))-|eu(t)|p′-2eu(t),eu˙(t))p,p′dt,

and the total work W on the time interval [0,s]⊆[0,T] asW(0,s;u):=∫0s(f(t),u˙(t)-z˙(t))Hdt+∫0s(u¨(t),z˙(t))Hdt+∫0s(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t)),ez˙(t))p,p′dt.

Remark 4.2

For p=2, we haveV(0,s;u)=∫0s‖eu˙(t)‖H2dt,

which corresponds to the viscous dissipation term in the linear Kelvin–Voigt model. Moreover, since G-1 satisfies (G1), we deduce that(G-1(η1)-G-1(η2))·(η1-η2)≥0for allη1,η2∈Rsymd×d,

and by choosing η1=eu(t)+eu˙(t) and η2=eu(t) we deriveV(0,s;u)≥0for everys∈[0,T].

Therefore, V can be seen as the analogous of the viscous dissipation term in the nonlinear setting.

Thanks to Theorem 4.1 and (4.2), we derive the following result.

Corollary 4.3

Every weak solution u∈V∩W2,2(0,T;H) to the nonlinear Kelvin–Voigt system (4.3)–(4.5) satisfies the energy-dissipation balance4.7 E(s;u)+V(0,s;u)=E(0;u)+W(0,s;u)for alls∈[0,T].

Proof

By Theorem 4.1, we know that u satisfies the energy dissipation balance (4.1). Moreover, for the nonlinear operator G given by (4.2) we observe that∫0s(σ(t),eu˙(t))p,p′dt=∫0s(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t)),eu˙(t))p,p′dt=∫0s(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t))-|eu(t)|p′-2eu(t),eu˙(t))p,p′dt+∫0s(|eu(t)|p′-2eu(t),eu˙(t))p,p′dt=∫0s(|eu(t)+eu˙(t)|p′-2(eu(t)+eu˙(t))-|eu(t)|p′-2eu(t),eu˙(t))p,p′dt+1p′‖eu(s)‖p′′-1p′‖eu(0)‖p′′.

Indeed, u∈W1,p′(0,T;V), which implies that the map t↦‖eu(t)‖p′′ is absolutely continuous on [0, T] withddt‖eu(t)‖p′′=p′(|eu(t)|p′-2eu(t),eu˙(t))p,p′for a.e.t∈[0,T].

By combining the previous identity with (4.1) we derive (4.7). □

As a consequence of Corollary 4.3, we deduce that for every weak solution u∈V∩W2,2(0,T;V) to the nonlinear Kelvin–Voigt system (4.3)–(4.5) the crack can not grow in time. Indeed, as explained in the introduction, according to the Griffith criterion there is a balance between the mechanical energy dissipated and the energy used to increase the crack. In the nonlinear Kelvin–Voigt model (4.3)–(4.5), this reads asE(s;u)+Hd-1(Γt\Γ0)+V(0,s;u)=E(0;u)+W(0,s;u)for alls∈[0,T].

Since the energy dissipated by the crack growth, which is Hd-1(Γt\Γ0), does not appear in (4.7), we derive that for the weak solution u∈V∩W2,2(0,T;H) to (4.3)–(4.5) given by Theorem 2.8 we must have Hd-1(Γt\Γ0)=0 for every t∈[0,T]. Hence, the crack associated with u does not increase in time. We point out that this phenomenon, called viscoelastic paradox, is the same which arises in linear Kelvin–Voigt models, as shown in [10, 26].

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). M.C. acknowledges the support of the project STAR PLUS 2020 - Linea 1 (21-UNINA-EPIG-172) “New perspectives in the Variational modeling of Continuum Mechanics” from the University of Naples Federico II and Compagnia di San Paolo, and of the INdAM - GNAMPA Project “Equazioni differenziali alle derivate parziali di tipo misto o dipendenti da campi di vettori” (Project number CUP_E53C22001930001). A.C. acknowledges the support of the INdAM - GNAMPA Project “Problemi variazionali per funzionali e operatori non-locali” (Project number CUP_E53C22001930001) and of the MUR PRIN project “Elliptic and parabolic problems, heat kernel estimates and spectral theory” (Project number 20223L2NWK). F.S. acknowledges the financial support received from the Austrian Science Fund (FWF) through the project TAI 293.

Funding

Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.

Publisher's Note

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