Dynamic Crack Growth in Viscoelastic Materials with Memory

In this paper we introduce a model of dynamic crack growth in viscoelastic materials, where the damping term depends on the history of the deformation. The model is based on a dynamic energy dissipation balance and on a maximal dissipation condition. Our main result is an existence theorem in dimension two under some a priori regularity constraints on the cracks.


Introduction
We consider the problem of crack growth in a viscoelastic cracked material with memory governed by the system ü(t) − div (C + V)Eu(t) + div ˆt −∞ e τ −t VEu(τ ) dτ = f (t), (1.1) where u, Eu, and ü, are the displacement, the symmetric part of its gradient, and its second derivative with respect to time, C and V are the elasticity and viscosity tensors, while f is the external load.For this model the stress at time t is given by Moreover, as in [6,13] we assume that we know the displacement u on (−∞, 0] and we want to solve (1.1) on [0, T ], for given T > 0. Under this assumption, it is convenient to write (1.1) in the form ü(t) − div(σ 0 (t)) = 0 (t) t ∈ [0, T ], (1.3) where σ 0 (t) := CEu(t) + VEu(t) − ˆt 0 e τ −t VEu(τ ) dτ, ( and u 0 is a function that represents the displacement on (−∞, 0], namely u(s) = u 0 (s) for every s ∈ (−∞, 0].For physical details regarding the right-hand side of Eq. (1.4) see, e.g., [20].
In this paper we study the problem on a bounded open set Ω ⊂ R 2 .The crack at time t ∈ [0, T ] is a 1-dimensional closed subset Γ t of Ω and the irreversibility of crack growth means that Γ t ⊆ Γ τ if t ≤ τ.For technical reasons we assume that the shape of the cracks and their dependence on time is sufficiently regular, with precise a priori estimates.
In the case of smooth functions, equation (1.3) is satisfied on Ω\Γ t (see also Fig. 1) with suitable boundary conditions (on the Dirichlet part ∂ D Ω, on the Neumann part ∂ N Ω, and on Γ t ) and with prescribed initial conditions.Namely, u and {Γ t } t∈[0,T ] satisfy ü(t) − div(σ 0 (t)) = 0 (t) in Ω\Γ t , (1.7) ) σ ± 0 (t)ν = F ± 0 (t)ν on Γ t , (1.10) u(0) = u 0 and u(0) = u 1 (1.11) for every t ∈ [0, T ], where u D is the Dirichlet condition, u 0 is the initial condition for the displacement, u 1 is the initial condition for the velocity, ν is the unit normal, and the symbol ± in (1.10) denotes suitable limits on each side of Γ t .We note that, as consequence of (1.3)- (1.6), the Neumann conditions on Γ t are not zero.In the paper we consider a weak formulation (see Definition 2.10) which coincides with the one in (1.7)- (1.11) under suitable regularity assumptions.
When {Γ t } t∈[0,T ] is prescribed, problem (1.7)- (1.11) has been studied in [5,18].More precisely, in [18] an existence theorem is proved, while in [5] one can find results regarding uniqueness and continuous dependence of u on the data (in particular on the cracks).
In the model considered in our paper the unknown of the problem is the family of cracks {Γ t } t∈[0,T ] which, in the spirit of [7,8], must satisfies the following conditions: Condition (a) is a dynamic version of Griffith's criterion (see [14] for the quasistatic case and [16] for the dynamic problem).
The main result of this paper is that, given initial and boundary conditions satisfying suitable hypotheses, there exists a family {Γ t } t∈[0,T ] satisfying (a) and (b) (see Theorem 4.3).
The proof follows the lines of [8], where a similar problem is studied for the case of pure elastodynamics.To deal with the memory term appearing in (1.4), we use the results of [5,18].In particular the continuous dependence on the data obtained in [5] is a fundamental tool for a compactness argument that plays a key role in the proof of Theorem 4.3.
The structure of the paper is the following: • in Sect. 2 we give a precise formulation of the problem and we give all the preliminary results; • in Sect. 3 we define the class of cracks {Γ t } t∈[0,T ] such that the energy balance described in a) is satisfied and we prove a compactness result; • in Sect. 4 we define the maximal dissipation condition and we prove the main result of the paper (Theorem 4.3).

Formulation of the Problem
The We give a precise definition of the admissible cracks of our model using a suitable class of curves.The following definitions and results are based on [7,8].The curves are always parameterized using the arc-length parameter s and for a given curve γ : When it is clear from the context we omit the dependence on γ and we write Γ and Γ s instead of Γ γ and Γ γ s .In order to describe the initial crack, we fix a curve γ 0 : [a 0 , 0] → Ω such that γ 0 (a 0 ) ∈ ∂Ω, γ 0 (s) ∈ Ω for every s ∈ (a 0 , 0] and we define the initial crack as We suppose that γ 0 is of class C 3,1 and that it is transversal to ∂Ω at γ 0 (a 0 ) (there exists an isosceles triangle contained in Ω with vertex in γ 0 (a 0 ) and axis parallel to γ 0 (a 0 )).We fix two constants r > 0 and L > 0 and we now define the space of admissible crack paths.
where γ (i) denotes the i-th derivative of γ.
We fix γ 0 , r, and L such that G r,L = ∅.We have to describe the dependence of the crack length on the time.We fix two constants μ > 0 and M > 0 which bound the speed of the crack tip and some higher order derivatives of the crack length with respect to time, respectively.Definition 2.4.Let T 0 < T 1 .The class S reg μ,M (T 0 , T 1 ) is composed of all nonnegative functions satisfying the following conditions: , where dots denote derivatives with respect to time.We denote by S piec μ,M (T 0 , T 1 ) the set of all functions s ∈ C 0 ([T 0 , T 1 ]) such that there exists a finite subdivision The minimal set {τ 0 , τ 1 , . . ., τ k } for which this property holds is denoted by sing(s).
μ,M (T 0 , T 1 ), with s(T 1 ) ≤ b γ , the time dependent cracks corresponding to these functions are given by Γ γ s(t) := γ([a 0 , s(t)]) for all t ∈ [T 0 , T 1 ], and the corresponding cracked domains are For simplicity of notation we sometimes denote Γ γ s(t) by Γ s(t) , when γ is clear from the context.See Fig. 1.
Remark 2.5.Under our assumptions, the cracks are described by curves starting from ∂Ω.This assumption is used in several points.For example, it is necessary to apply the results of [5,7,8].
We now define the functional spaces that will be used in order to give the definition of weak solution of the viscoelastic problem (1.7)- (1.11).
We define R 2×2 as the space of real 2 × 2 matrix and R 2×2 sym as the space of real 2 × 2 symmetric matrices.The Euclidean scalar product between the matrices A and B is denoted by , where A T denotes the transpose matrix of A. For any pair of vector spaces we define L(X; Y ) as the space of linear and continuous maps form X into Y.Let 0 < λ < Λ be two fixed constants.We now define the space of tensors that satisfy suitable conditions regarding regularity and symmetry.See also [8,Definition 3.1].

Definition 2.7. We define E(λ, Λ) as the set of all maps
) We now fix the following maps where C(x) and V(x) respectively represent the elasticity and viscosity tensor at the point , with s(T 1 ) ≤ b γ , we now introduce the function spaces that will be used in the precise formulation of problem (1.7)- (1.11).

Lemma 2.8. Let γ ∈ G r,L and let
Then there exists a constant K, depending only on Ω and Γ, such that for every u ∈ H 1 (Ω\Γ; R 2 ), where • denotes the L 2 norm.
We note that in Lemma 2.8 it is possible to find a smooth extension of Γ up to the boundary of Ω. From this observation, we obtain that Lemma 2.8 is an immediate consequence of [5,Lemma 2.2].
Remark 2.9.Let γ ∈ G r,L and let Γ := γ([a 0 , b γ ]; R 2 ).Then, using a localization argument (see, e.g., [5]), we can prove that the trace operator is well defined and continuous from We set (2.9) Since L 2 (Γ) = 0, we have the embedding V γ → H × H given by v → (v, Dv) and we can see the distributional gradient Dv on Ω\Γ as a function defined a.e. on Ω, which belongs to H.For every finite dimensional Hilbert space Y the symbols (• , •) and • denote the scalar product and the norm in the L 2 (Ω; Y ), according to the context.The space V γ is endowed with the norm (2.10) For every s ∈ [a 0 , b γ ] we define where Γ s = γ([a 0 , s]) and u| ∂ D Ω denotes the trace of u on ∂ D Ω.We note that V γ s and V γ,D s are closed linear subspaces of V γ .For every s ∈ S piec μ,M (T 0 , T 1 ) and for every t ∈ [T 0 , T 1 ] the spaces V γ s(t) and V γ,D s(t) are defined as in (2.11) with s = s(t).We define which is a Hilbert space with the norm where the dot denotes the distributional derivative with respect to t. Moreover we set which is a closed linear subspace of V γ,s (T 0 , T 1 ) and we define which is a Banach space with the norm Moreover, it is convenient to introduce the space of weakly continuous functions with values in a Banach space X with topological dual X * , defined by When it is clear from the context we will omit the dependence on γ or s in the functional spaces, writing V, V s(t) , V D s(t) , V(T 0 , T 1 ), V D (T 0 , T 1 ), and In particular v(T 0 ) and v(T 1 ) are well defined elements of H, for every v ∈ V(T 0 , T 1 ).
We set (2.17) On the forcing term (t) of (1.7) we assume that where are prescribed functions and the divergence of a matrix valued function is the vector valued function whose components are obtained taking the divergence of the rows.The Dirichlet boundary condition on ∂ D Ω is obtained by prescribing a function where V 0 is V s for s = 0.It is not restrictive to assume that for every t ∈ [0, T ] u D (t) = 0 a.e. on {x ∈ Ω | dist(x, ∂Ω) ≥ r}. (2.21) We are now in a position to give the definition of weak solution for the viscoelastic problem.

Definition 2.10. (Solution for visco-elastodynamics with cracks)
and let u 1 ∈ H.We say that u is a weak solution of the problem of visco-elastodynamics on the cracked domains Ω\Γ s(t) , t ∈ [T 0 , T 1 ], with initial conditions u 0 and u 1 , if where (V D s(T 0 ) ) * denotes the topological dual of V D s(T 0 ) .
Existence of the solution for the viscoelastic problem (2.22)-(2.24) is given by [18] for Ω ⊂ R d with d ≥ 1 and under more general assumptions on the regularity of the cracks.Uniqueness and continuous dependence on the data are proved in [5] under the assumption that the constant μ, which controls the speed of the crack tip in Definition 2.4, satisfies 0 < μ < μ 0 , (2.30) where the constant μ 0 is not explicitly defined in terms of the data of the problem.
Using the fact that d = 2 in our work, we will prove that uniqueness and continuous dependence can be obtained under the explicit assumption where λ are the constants that appears in Definition 2.7 respectively.In order to prove this results, we have to define an auxiliary problem, which can be interpreted as the elastodynamics problem with elasticity tensor replaced by A.

Definition 2.13. (Solution for elastodynamics with cracks)
and let u 1 ∈ H.We say that v is a weak solution of the problem of elastodynamics on the cracked domains Ω\Γ s(t) , t ∈ [T 0 , T 1 ], with initial conditions u 0 and u The existence of the solution can be proved without any constraint on the speed of the crack tip and can be found, for instance, in [18], where the author considered a more general problem.To prove the uniqueness, it is enough to consider the case F = 0.Under this assumption, the proof of uniqueness is given by [8].
With the following result we obtain a better regularity with respect to time.

Proposition 2.16. Under the same assumption of Theorem 2.15, let v be the unique solution of problem (2.32)-(2.34). Then
Proof.In the case F = 0, a solution for the elastodynamics with cracks in the sense of [8] is also a solution in the sense of Definition 2.13.By uniqueness, the two solutions coincide.In particular, we get that, if F = 0, the solution is in If the forcing term F is not zero, we can use same approximation argument used in [5,Lemma 4.7].Then for every ε > 0 there exists We define v ε as the solution of the elastodynamic problem in Definition 2.13 with F replaced by F ε .Since F ε is regular in space we have that for all t ∈ [0, T ] and for all ψ ∈ V.It follows that v ε is a solution in the sense of Definition 2.13 with f and F respectively replaced by f − divF ε and 0. By the results of [8] we have that Using the continuous dependence on the forcing terms given by [5, Proposition 3.5] and (2.40), we obtain that In particular, we get that v We now fix the notation that will be useful in order to give the main results concerning continuous dependence on the data.
Let 0 ≤ T 0 < T 1 ≤ T, let γ k ∈ G r,L be a sequence of cracks paths, and let s k ∈ S piec μ,M (T 0 , T 1 ), with s k (T 1 ) ≤ b γ k , be a sequence of crack lengths.We define

.42)
We define u k as the weak solution of k-th viscoelastic problem on the cracked domains Ω\Γ (2.45)Moreover, we define v k as the weak solution of k-th problem of elastodynamics on the cracked domains Ω\Γ

.48)
We now state the result concerning continuous dependence on the data for the problem of elastodynamics.It will be used to prove the same result for the viscoelastic problem.
Theorem 2.17. ) ) Proof.In the case . In the general case, the result follows from the same approximation argument used in [5, Lemma 4.7, Proposition 4.9].Now we are in a position to obtain the same results for the viscoelastic system.
Theorem 2.18.Let γ ∈ G r,L , 0 ≤ T 0 < T 1 ≤ T, s ∈ S piec μ,M (T 0 , T 1 ), with s(T 1 ) ≤ b γ , and assume (2.7), (2.19)-(2.21)and (2.31).Let u 0 ∈ V s(T 0 ) , such that u 0 − u D (T 0 ) ∈ V D s(T 0 ) and let u 1 ∈ H. Then there exists a unique solution u of problem We can not apply directly [5, Theorem 2.7] because in general (2.30) is not satisfied.However, assuming (2.31) instead of (2.30) we can repeat all arguments of the proof of that theorem, which is based on existence and uniqueness for elastodynamics with cracks (in our case given by Theorem 2.15) and on a fixed point argument.
Proposition 2.19.Under the same assumptions of Theorem 2.18, let u be the unique solution of problem Proof.It is enough to apply Proposition 2.16 with F (t) replaced by The following theorem provides the continuous dependence on the data for the solution of the viscoelastic problem.Proof.As in the proof of Theorem 2.18, we cannot apply directly [5, Theorem 4.1], because in general (2.30) is not satisfied.However, assuming (2.31) instead of (2.30) we can repeat all arguments of the proof of that theorem, which is based on the continuous dependence on the data for elastodynamics with cracks (in our case given by Theorem 2.17) and on a results concerning the convergence of fixed points of a sequence of functions (see [5,Lemma 4.2]).

Energy Balance
In this section we study the problem of the dynamic energy-dissipation balance on a given cracked domain Ω\Γ γ s(t) for the solution of the viscoelastic problem.
for all u ∈ V(T 0 , T 1 ), for all t ∈ [T 0 , T 1 ].Since it is easy to check that L T 0 is bounded.Indeed, using the Hölder inequality it is possible to prove that For every t ∈ [T 0 , T 1 ] the sum of kinetic and elastic energy is given by For an interval [t 1 , t 2 ] ⊂ [T 0 , T 1 ] the dissipation due to viscosity between time t 1 and t 2 is given by Moreover, taking into account the dynamic Griffith's criterion (see [14,16]), we assume that the energy dissipated in the process of crack production on the interval [t 1 , t 2 ] is proportional to s(t 2 ) − s(t 1 ), which represents the length of the crack increment.For simplicity we take the proportionality constant equal to one.Finally, the work done between time t 1 and t 2 by the boundary and volume forces is ), s(T 0 ) = s 0 , and s(T 1 ) ≤ b γ , such that the unique weak solution u of the viscoelastic problem (2.22)-(2.24)satisfies the energy-dissipation balance is defined in the same way replacing s ∈ S reg μ,M ([T 0 , T 1 ]) by s ∈ S piec μ,M ([T 0 , T 1 ]).The class B reg (T 0 , T 1 ) is nonempty, as clarified by the following result, whose proof follows the lines of [11, Lemma 1] and [10, Proposition 2.7].Proposition 3.4.Under the assumption of Definition 3.3, the pair (γ, s), with s(t) = s 0 for every t ∈ [T 0 , T 1 ], belongs to B reg (T 0 , T 1 ).
(a) an energy dissipation balance (consistent with dynamic Griffith's theory) for the solution u of (1.7)-(1.11)(see Definition 3.3): the sum of the kinetic and elastic energies and of the energies dissipated by viscosity and crack growth balances the work done by the forces acting on the system; (b) a maximal dissipation condition, depending on a parameter η > 0 (see Definition 4.1), which forces the crack to run as fast as possible.
reference configuration of our problem is a bounded open set Ω ⊂ R 2 , with Lipschitz boundary ∂Ω and we assume that ∂Ω = ∂ D Ω ∪ ∂ N Ω, where ∂ D Ω and ∂ N Ω are disjoint (possibly empty) Borel sets, on which we prescribe Dirichlet and Neumann boundary conditions respectively.Moreover, we fix a time interval [0, T ], with T > 0.