Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual nonpenetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.
This article is part of the theme issue 'Non-smooth variational problems and applications'.
(i) classical linear elasticity is exploited without viscosity; (ii) not only existence but also uniqueness of a solution is ensured; (iii) contact law is formulated in terms of displacement, which is considered to be more realistic.
In this paper, we propose to impose a contact condition to linear combination of normal displacement and normal velocity on the interface with some constant coefficient δ > 0; see (2.2a) below. Since δ = 0 and δ = ∞ correspond to the contact conditions in displacement and in velocity, respectively, it can be regarded as an intermediate between them. We call (2.2a) the Signorini contact condition of dynamic type (hereinafter, referred to as SCD condition). With the SCD and Tresca friction conditions, we prove unique existence of a strong solution for the linearized elastodyanmic equations, thus having properties (i) and (ii). Moreover, property (iii) is also approached by our model because δ > 0 can be fixed to an arbitrarily small value (however it is not possible to make exactly δ = 0). An expository interpretation of our result may be that making the Signorini contact condition in displacement 'dynamic a bit' (recall that boundary conditions having quantities with time derivative are called dynamic) leads to some stabilization effect to the system. We expect that this fact has some connection with Baumgarte-like stabilization techniques known in numerical simulations of non-smooth mechanics (see [17]), which is to be investigated in the future. The present result will also be of basic interest when we make an attempt to more involved crack problems, e.g. propagation and singular behaviour of crack tips. This paper is organized as follows. In §2, we introduce notation and the precise mathematical setting to be studied. In §3, a variational inequality formulation as well as the definition of a strong solution is introduced, and we present the main theorem. Section 4 is devoted to its proof based on regularization of a variational inequality and Galerkin's method. The strategy basically follows our previous study [15]; nevertheless, the analysis, in particular a priori estimates and a uniqueness proof, becomes more intricate to deal with the contact condition.

Preliminaries (a) Notation
Let Ω ⊂ R 3 be a bounded domain with a smooth boundary ∂Ω consisting of two parts Γ D = ∅ and Γ N that are mutually disjoint. Let Γ be a two-dimensional closed smooth interface that separates Ω into two subdomains Ω ± , that is, We assume that ∂Ω ± satisfy the Lipschitz condition and that ∂Ω ± ∩ Γ D = ∅. A crack is supposed to be represented by an open subset Γ c of Γ such that Γ c ⊂ Γ \ ∂Γ (namely, Γ c Γ ); we refer to Ω c := Ω \ Γ c as the domain with a crack. The unit normal vector associated with ∂Ω is denoted by ν ∂Ω , and the unit normal vector on Γ pointing from Ω − to Ω + is denoted by ν. The geometric situation explained so far is schematically summarized in figure 1.
We mainly deal with functions defined in Ω c in this paper. For such a function u, we let u ± := u| Ω ± be its restrictions to subdomains Ω ± . If u ± are smooth enough, we define the jump discontinuity of u across Γ by For function spaces, we employ the usual Lebesgue spaces L p (Ω c ) (1 ≤ p ≤ ∞) and the Sobolev space H 1 (Ω c ), which have the characterization and Accordingly, their norms are given by , which is the Lions-Magenes space (see [18]).
Functions and function spaces that are vector-or tensor-valued are written with bold fonts, e.g. u ∈ H 1 (Ω c ) = H 1 (Ω c ) 3 , whereas fine fonts mean scalar quantities. We denote the inner products of L 2 (Ω c ) by (·, ·), and those of L 2 (Γ N ), L 2 (Γ c ) by (·, ·) Γ N , (·, ·) Γ c (the same notation will also be used for vectors and tensors). We also exploit the notation of Bochner spaces L p (0, T; X) and W k,p (0, T; X) for a positive constant T and a Banach space X, where k > 0 is an integer and 1 ≤ p ≤ ∞. Finally, the dual space of X is denoted by X * .

(b) Problem formulation
We assume that Ω c is regarded as a reference configuration (or non-deformed state) of an elastic body. The deformation of the body may be described by a displacement field u : royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 380:  If the constitutive law of the material is based on isotropic linear elasticity, the stress tensor is given by where λ, μ are Lamé constants such that μ > 0 and 3λ + 2μ > 0, I is the unit tensor, and E(u) = (∇u + (∇u) )/2 means the linearized strain tensor. The dynamic deformation of the body is governed by the hyperbolic system where ρ is the density which is a positive constant, the prime stands for the time derivative (i.e. u = ∂ 2 t u), f is the external body force and T > 0 stands for a fixed time length. As for the boundary conditions, we consider where F is a prescribed traction on Γ N . At t = 0, the initial displacement and velocity fields are given as Before stating the interface conditions on the crack, we introduce the normal and tangential components of the displacement, velocity and traction on Γ , restricted from Ω ± , by together with their jumps In this paper, we consider the Signorini contact condition of dynamic type (SCD condition) and Tresca friction condition on the crack Γ c as follows: and where δ ∈ (0, ∞] is a constant, g = g(t, x) ≥ 0 is a given function. Several remarks are in order. First, σ ν := σ + ν = σ − ν and σ τ := σ + τ = σ − τ are well-defined as singlevalued functions on Γ c because they have no jump by (2.2). Second, if δ = 0 in (2.2a) then we formally recover the usual non-penetration condition introduced in [ δ = ∞ then we arrive at the contact condition in terms of velocity given by Eck et al. [11]. To see this we equivalently rewrite (2.2a), with γ := δ −1 , as and set γ = 0. For simplicity of presentation, we mainly deal with the SCD condition in the form (2.3) with γ ∈ [0, ∞) rather than (2.2a) in the subsequent analysis.
is mainly due to the mathematical reason as explained in the Introduction. From a modelling viewpoint, it can be regarded as a first-order approximation to the case δ = 0, i.e. the usual non-penetration condition [[u ν ]] ≥ 0. We see that the SCD condition allows for interpenetration of the crack, which is not physically feasible and may be a restriction in applications. However, it remains realistic for a short time interval in the case of no initial slip velocity on the crack (e.g. for the first-and usually strongest-wave of an earthquake as mentioned in [11], Chapter 5).
(ii) If g in (2.2b) is replaced by F |σ ν | (F ≥ 0 is a coefficient), then the resulting condition is known as the Coulomb friction law, which is mentioned in the Introduction.

Variational formulations (a) Variational inequality
As discussed in the previous section, the strong form of the initial boundary value problem considered in this paper is represented as follows: Let us derive a weak formulation to this problem assuming that u is smooth enough in [0, T] × (Ω \ Γ c ). To this end we introduce the following function spaces and convex cone: = 0 on Γ c and the fact that the outer unit normal w.r.t. Ω ± on Γ is ∓ν. By (2.1) we see that It follows from (3.1d) and (3.1e) that .
This is a variational inequality of hyperbolic type that is equivalent to the strong form (3.1), provided that there is a classical solution, as seen below.
and only if the following hold: Then the last equality of (3.1e) also follows. This proves that u solves (3.1).

(b) Main result
In view of proposition 3.1, let us define a solution of (3.1) based on its variational form.

Remark 3.3.
For second-order hyperbolic problems, one usually considers a weak solution in W 1,∞ (0, T; L 2 (Ω c )) ∩ L ∞ (0, T; H 1 (Ω c )). However, this class would not be appropriate for dynamic elasticity problems with friction where the trace of velocity explicitly appears on an interface. We also note that in the Kelvin-Voigt viscoelastic case, a natural class of a weak solution becomes W 1,∞ (0, T; L 2 (Ω c )) ∩ H 1 (0, T; H 1 (Ω c )), avoiding this issue. Now we are ready to state our main result in this paper. H 1 (0, T; H), F ∈ H 2 (0, T; L 2 (Γ N )), and let g ∈ H 2 (0, T; L 2 (Γ c )) be non-negative. We assume that u 0 ∈ V,u 0 ∈ V and that they satisfy the following compatibility conditions: Then there exists a unique strong solution of (3.1).

(c) Regularized problem
It is not easy to directly construct a solution of the time-dependent variational inequality (3.4) because it contains non-differentiable relations. To see this, we introduce two convex functions whose subdifferentials β := ∂ψ and α := ∂ϕ are maximal monotone graphs given by We then observe that the SCD and Tresca conditions in (3.1) are concisely expressed as To address the difficulty that β and α are multi-valued functions and non-differentiable, we approximate ψ and ϕ by the following functions which are convex and W 3,∞ ∩ C 2 : where > 0 is a constant and [x] − := max{−x, 0} for x ∈ R. Their derivatives β := dψ /dx and α := ∇ϕ are given by which are monotone and W 2,∞ ∩ C 1 .
With this preparation we consider the following regularized problem denoted by (VI) : find In the proposition below, we find that ( VI) is equivalent to the following variational equality problem denoted by (VE) : find u (t) ∈ V such that u (0) = u 0 , u (0) =u 0 and

Proof.
Although the proof is standard, we present it for completeness. Let u be a solution of (VI) . Taking v = ±hw + γ u (t) + u (t) with arbitrary h > 0 and w ∈ V, dividing by h, and letting h → 0, we deduce (VE) from the relations Conversely, let u be a solution of (VE) . Note that, since ψ and ϕ are convex, for all w ∈ V. Setting this w in such a way that w + γ u (t) + u (t) = v and using (3.7), we arrive at (3.6).
As a result of proposition 3.6, it suffices to solve an equation problem for obtaining u . Furthermore, since it follows from (3.7) that we expect that u should converge to a solution of the original problem (3.1) as → 0. Justification of this fact, which is actually the idea to prove theorem 3.4, is the task of the next section.

Proof of main result
In this section, we establish existence in §4a-d and uniqueness in §4e. Coercivity of a(·, ·) in V, that is, a(v, v) ≥ C v 2 H 1 (Ω c ) ∀v ∈ V, which is justified by Korn's inequality (e.g. [14]), will be frequently used in the proof. Here and in what follows, C represents a generic constant depending only on the domain Ω c , Lamé constants λ, μ and density ρ. We will also write C(f , g), etc. in order to indicate dependency on other quantities.
The inequality above allows us to define the norm of V as v V := a(v, v) 1/2 , whereas we use v H := v L 2 (Ω c ) .

(a) Galerkin approximation
We apply Galerkin's method to solve (3.7). Since V ⊂ H 1 (Ω c ) is separable, there exist countable members w 1 , w 2 , . . . , ∈ V, which are linearly independent, such that ∞ We may assume that u 0 ,u 0 ∈ V m for m ≥ 2 (otherwise one can add u 0 andu 0 to the members {w k } m k=1 ). For m = 2, 3, . . ., the Galerkin approximation problem consists in determining c k (t) Applying Hölder's and Young's inequalities to terms involving γ on the right-hand side yields where we have used |α (·)| ≤ 1 and the trace inequality Integration of both sides with respect to t gives In particular,  of presentation; if we keep this term, we obtain (4.5) below), we rephrase this estimate as If γ = 0, we find from Gronwall's inequality that Otherwise we further integrate (4.3) with respect to t, with B 1 (t) := t 0 A(s) ds, to get so that, by Gronwall's inequality, which concludes where we have used the fact that β and ∇α are non-negative. Applying Hölder's and Young's inequalities to the first three and the sixth terms on the righthand side, together with |α (·)| ≤ 1 and the trace inequality where the eighth term on the right-hand side equals Hölder's and Young's inequalities, combined with the relations where the last contribution owes to γ ( It remains to estimate u (0) H . For this purpose we make t = 0 and take v = u (0) ∈ V m in (4.1) to see Noting that a(u 0 , u m (0)) and using the compatibility conditions, we deduce Substituting this into (4.6), we proceed as in the previous subsection assuming ≤ 1. If γ = 0, Gronwall's inequality gives us If γ > 0, we further integrate (4.6) to have Applying Gronwall's inequality above and substituting the resulting estimate into (4.6), in which 2ργ |(u m (t), u m (t))| is bounded by The argument of the passage to the limits m → ∞ and → 0 is basically similar to ( [15], Section 3.7), the essential difference lying in the verification of the constraint γ u(t) + u (t) ∈ K. However, for the sake of completeness we present the whole proof. First let us consider the limit m → ∞ for fixed ∈ (0, 1]. As a consequence of the a priori estimates as m → ∞. Here, we note the compact embedding W 1,∞ (0, T; L 2 (Ω ± )) ∩ L ∞ (0, T; H 1 (Ω ± )) → C([0, T]; L 2 (Ω ± )) (see [19]) and the compactness of the trace operator H 1 (Ω ± ) → L 3 (Γ c ) (e.g. [20]). It then follows that For arbitrary η ∈ C ∞ 0 (0, T) and v ∈ V m (m = 2, 3, . . . ), we find from (4.1) that Letting m → ∞, using (4.8), and applying the dominated convergence theorem, we have Since ∞ m=1 V m = V and η is arbitrary, we conclude (3.7), that is, u is a solution of (VE) and also of (VI) by virtue of proposition 3. ≤ C(f , F, g, u 0 ,u 0 , T, γ ). (4.9) Next we consider the limit → 0. By (4.9), there exist a subsequence of {u }, denoted by the same symbol, and some u ∈ In fact, the former inequality above results from the following weak convergence:  (u, u ) + γ a(u, u) dt,