Abstract
In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions, we also show a uniqueness result. Finally, we exhibit an example where the energy-dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the theory of dynamic fracture, the deformation of an elastic material evolves according to the elastodynamics system, while the evolution of the crack follows Griffith’s dynamic criterion, see [13]. This principle, originally formulated in [11] for the quasi-static setting, states that there is an exact balance between the energy released during the evolution and the energy used to increase the crack, which is postulated to be proportional to the area increment of the crack itself.
For an antiplane displacement, the elastodynamics system leads to the following wave equation
with some prescribed boundary and initial conditions. Here, \(\Omega \subset {\mathbb {R}}^d\) is an open bounded set with Lipschitz boundary, which represents the cross section of the material, the closed set \(\Gamma _t\subset {{\overline{\Omega }}}\) models the crack at time t in the reference configuration, \(u(t):\Omega {\setminus }\Gamma _{t}\rightarrow {\mathbb {R}}\) is the antiplane displacement, and f is a forcing term. In this case, Griffith’s dynamic criterion reads
where \({\mathcal {E}}(t)\) is the total energy at time t, given by the sum of kinetic and elastic energy, and \({\mathcal {H}}^{d-1}\) is the \((d-1)\)-dimensional Hausdorff measure.
From the mathematical point of view, the first step to study the evolution of the fracture is to solve the wave equation (1.1) when the evolution of the crack is assigned, see, for example, [2, 3, 7, 14, 18] (we refer also to [6, 10, 16] for the case of a one-dimensional model). When we want to take into account the viscoelastic properties of the material, Kelvin–Voigt’s model is the most common one. If no crack is present, this leads to the damped wave equation
As it is well known, the solutions to (1.2) satisfy the energy-dissipation balance
When we consider a crack in a viscoelastic material, Griffith’s dynamic criterion becomes
For a prescribed crack evolution, this model was already considered by Dal Maso and Larsen [3] in the antiplane case and more in general by Tasso [18] for the vector-valued case. As proved in the quoted papers, the solutions to (1.2) on a domain with a prescribed time-dependent crack, i.e., with \(\Omega \) replaced by \(\Omega {\setminus }\Gamma _t\), satisfy (1.3) for every time. This equality implies that (1.4) cannot be satisfied unless \(\Gamma _t=\Gamma _0\) for every t. This phenomenon was already well known in mechanics as the viscoelastic paradox, see, for instance, [17, Chapter 7].
To overcome this problem, we modify Kelvin–Voigt’s model by considering a possibly degenerate viscosity term depending on t and x. More precisely, we study the following equation
On the function \(\Psi :(0,T)\times \Omega \rightarrow {\mathbb {R}}\), we only require some regularity assumptions (see (2.7)); a particularly interesting case is when \(\Psi \) assumes the value zero on some points of \(\Omega \), which means that the material has no longer viscoelastic properties in such a zone.
The main result of this paper is Theorem 3.1, in which we show the existence of a weak solution to (1.5). This is done in the more general case of linear elasticity, that is, when the displacement is vector-valued and the elastic energy depends only on the symmetric part of its gradient. To this aim, we first perform a time discretization in the same spirit of [3], and then we pass to the limit as the time step goes to zero by relying on energy estimates; as a byproduct, we obtain the energy-dissipation inequality (4.4). By using the change of variables method implemented in [7, 14], we also prove a uniqueness result, but only in dimension \(d=2\) and when \(\Psi (t)\) vanishes on a neighborhood of the tip of \(\Gamma _t\).
We complete our work by providing an example in \(d=2\) of a weak solution to (1.5) for which the fracture can grow while balancing the energy. More precisely, when the cracks \(\Gamma _t\) move with constant speed along the \(x_1\)-axis and \(\Psi (t)\) is zero in a neighborhood of the crack tip, we construct a function u which solves (1.5) and satisfies
Notice that this is the natural extension of Griffith’s dynamic criterion (1.4) to this setting.
The paper is organized as follows. In Sect. 2, we fix the notation adopted throughout the paper, we list the standard assumptions on the family of cracks \(\{\Gamma _t\}_{t\in [0,T]}\) and on the function \(\Psi \), and we specify the notion of weak solution to problem (1.5). In Sect. 3, we state our main existence result (Theorem 3.1) and we implement the time discretization method. We conclude the proof of Theorem 3.1 in Sect. 4, where we show the validity of the initial conditions and the energy-dissipation inequality (4.4). Section 5 deals with uniqueness: under stronger regularity assumptions on the cracks sets, in Theorem 5.5 we prove the uniqueness of a weak solution, but only when the space dimension is \(d=2\). To this aim, we assume also that the function \(\Psi \) is zero in a neighborhood of the crack tip. We conclude with Sect. 6, where in dimension \(d=2\), we show an example of a moving crack that satisfies Griffith’s dynamic energy-dissipation balance (1.6).
2 Notation and preliminary results
The space of \(m\times d\) matrices with real entries is denoted by \({\mathbb {R}}^{m\times d}\); in case \(m=d\), the subspace of symmetric matrices is denoted by \({\mathbb {R}}^{d\times d}_\mathrm{{sym}}\). Given two vectors \(v_1,v_2\in {\mathbb {R}}^d\), their Euclidean scalar product is denoted by \(v_1\cdot v_2\in {\mathbb {R}}\) and their tensor product is denoted by \(v_1\otimes v_2\in {\mathbb {R}}^{d\times d}\); we use \(v_1\odot v_2\in {\mathbb {R}}^{d\times d}_\mathrm{{sym}}\) to denote the symmetric part of \(v_1\otimes v_2\), namely \(v_1\odot v_2:=\frac{1}{2}(v_1\otimes v_2+v_2\otimes v_1)\). Given \(A\in {\mathbb {R}}^{m\times d}\), we use \(A^T\) to denote its transpose; we use \(A_1\cdot A_2\in {\mathbb {R}}\) to denote the Euclidean scalar product of two matrices \(A_1,A_2\in {\mathbb {R}}^{d\times d}\).
The partial derivatives with respect to the variable \(x_i\) are denoted by \(\partial _i\). Given a function \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^m\), we denote its Jacobian matrix by \(\nabla f\), whose components are \((\nabla f)_{ij}:=\partial _j f_i\), \(i=1,\dots ,m\), \(j=1,\dots ,d\). For a tensor field \(F:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^{m\times d}\), by \(\mathrm{div}F\), we mean the divergence of F with respect to rows, namely \((\mathrm{div}F)_i:=\sum _{j=1}^d\partial _jF_{ij}\), for \(i=1,\dots ,m\).
The d-dimensional Lebesgue measure is denoted by \({\mathcal {L}}^d\) and the \((d-1)\)-dimensional Hausdorff measure by \(\mathcal H^{d-1}\). We adopted standard notations for Lebesgue and Sobolev spaces on open subsets of \({\mathbb {R}}^d\); given an open set \(\Omega \subseteq {\mathbb {R}}^d\), we use \(||\cdot ||_\infty \) to denote the norm of \(L^\infty (\Omega ;{\mathbb {R}}^m)\). The boundary values of a Sobolev function are always intended in the sense of traces. Given an open bounded set \(\Omega \) with Lipschitz boundary, we denote by \(\nu \) the outer unit normal vector to \(\partial \Omega \), which is defined \({\mathcal {H}}^{d-1}\)-a.e. on the boundary.
Given a Banach space X, its norm is denoted by \(\Vert \cdot \Vert _X\); if X is an Hilbert space, we use \((\cdot ,\cdot )_X\) to denote its scalar product. The dual space of X is denoted by \(X'\), and we use \(\langle \cdot , \cdot \rangle _{X'}\) to denote the duality product between \(X'\) and X. Given two Banach spaces \(X_1\) and \(X_2\), the space of linear and continuous maps from \(X_1\) to \(X_2\) is denoted by \({\mathscr {L}}(X_1;X_2)\); given \(\mathbb A\in {\mathscr {L}}(X_1;X_2)\) and \(u\in X_1\), we write \({\mathbb {A}} u\in X_2\) to denote the image of u under \({\mathbb {A}}\).
Given an open interval \((a,b)\subseteq {\mathbb {R}}\), \(L^p(a,b;X)\) is the space of \(L^p\) functions from (a, b) to X. Given \(u\in L^p(a,b;X)\), we denote by \({\dot{u}}\in {\mathcal {D}}'(a,b;X)\) its distributional derivative. The set of continuous functions from [a, b] to X is denoted by \(C^0([a,b];X)\). Given a reflexive Banach space X, \(C_w^0([a,b];X)\) is the set of weakly continuous functions from [a, b] to X, namely, it is the collection of maps \(u:[a,b]\rightarrow X\) such that \(t\mapsto \langle x',u(t)\rangle _{X'}\) is continuous from [a, b] to \(\mathbb {R}\) for every \(x'\in X'\).
Let T be a positive real number and let \(\Omega \subset {\mathbb {R}}^d\) be an open bounded set with Lipschitz boundary. Let \(\partial _D\Omega \) be a (possibly empty) Borel subset of \(\partial \Omega \) and let \(\partial _N\Omega \) be its complement. We assume the following hypotheses on the geometry of the cracks:
-
(E1)
\(\Gamma \subset {{\overline{\Omega }}}\) is a closed set with \({\mathcal {L}}^d(\Gamma )=0\) and \(\mathcal H^{d-1}(\Gamma \cap \partial \Omega )=0\);
-
(E2)
for every \(x\in \Gamma \), there exists an open neighborhood U of x in \({\mathbb {R}}^d\) such that \((U\cap \Omega ){\setminus }\Gamma \) is the union of two disjoint open sets \(U^+\) and \(U^-\) with Lipschitz boundary;
-
(E3)
\(\{\Gamma _t\}_{t\in [0,T]}\) is a family of closed subsets of \(\Gamma \) satisfying \(\Gamma _s\subset \Gamma _t\) for \(0\le s\le t\le T\).
Thanks to (E1)–(E3), the space \(L^2(\Omega {\setminus }\Gamma _t;{\mathbb {R}}^m)\) coincides with \(L^2(\Omega ;{\mathbb {R}}^m)\) for every \(t\in [0,T]\) and \(m\in {\mathbb {N}}\). In particular, we can extend a function \(u\in L^2(\Omega {\setminus }\Gamma _t;{\mathbb {R}}^m)\) to a function in \(L^2(\Omega ;{\mathbb {R}}^m)\) by setting \(u=0\) on \(\Gamma _t\). Moreover, the trace of \(u\in H^1(\Omega {\setminus }\Gamma )\) is well defined on \(\partial \Omega \). Indeed, we may find a finite number of open sets with Lipschitz boundary \(U_j\subset \Omega {\setminus }\Gamma \), \(j=1,\dots m\), such that \(\partial \Omega {\setminus }(\Gamma \cap \partial \Omega )\subset \cup _{j=1}^m\partial U_j\). Since \({\mathcal {H}}^{d-1}(\Gamma \cap \partial \Omega )=0\), there exists a constant \(C>0\), depending only on \(\Omega \) and \(\Gamma \), such that
Similarly, we can find a finite number of open sets \(U_j\subset \Omega {\setminus }\Gamma \), \(j=1,\dots m\), with Lipschitz boundary, such that \(\Omega {\setminus }\Gamma =\cup _{j=1}^m U_j\). By using second Korn’s inequality in each \(U_j\) (see, e.g., [15, Theorem 2.4]) and taking the sum over j, we can find a constant \(C_K\), depending only on \(\Omega \) and \(\Gamma \), such that
for every \(u\in H^1(\Omega {\setminus }\Gamma ;{\mathbb {R}}^d),\) where \(Eu\) is the symmetric part of \(\nabla u\), i.e., \(Eu:=\frac{1}{2}( \nabla u+ \nabla u^T)\).
For every \(t\in [0,T]\), we define
Notice that in the definition of \(V_t\), we are considering only the distributional gradient of u in \(\Omega {\setminus }\Gamma _t\) and not the one in \(\Omega \). The set \(V_t\) is a Hilbert space with respect to the following norm:
To simplify our exposition, we set \(H:=L^2(\Omega ;{\mathbb {R}}^m)\) and \(H_N:=L^2(\partial _N\Omega ;{\mathbb {R}}^m)\); for every \(m\in {\mathbb {N}}\), we always identify the dual of H by H itself and \(L^2(0,T;L^2(\Omega ;{\mathbb {R}}^m))\) by \(L^2((0,T)\times \Omega ;{\mathbb {R}}^m)\).
Thanks to (2.2), the space \(V_t\) coincides with the usual Sobolev space \(H^1(\Omega {\setminus }\Gamma _t;{\mathbb {R}}^d)\). Therefore, by (2.1), it makes sense to consider for every \(t\in [0,T]\) the set
which is a Hilbert space with respect to \(||\cdot ||_{V_t}\). Moreover, by combining (2.2) with (2.1), we derive also the existence of a constant \(C_{tr}>0\) such that
Let \({\mathbb {C}},{\mathbb {B}}:\Omega \rightarrow {\mathscr {L}}({\mathbb {R}}^{d\times d}_\mathrm{{sym}};{\mathbb {R}}^{d\times d}_\mathrm{{sym}})\) be two fourth-order tensors satisfying
for two positive constants \(\lambda _1,\lambda _2\) independent of x. Consider a function \(\Psi :(0,T)\times \Omega \rightarrow {\mathbb {R}}\) satisfying
Given \(f\in L^2(0,T;H)\), \(w\in H^2(0,T;H)\cap H^1(0,T;V_0)\), \(g\in H^1(0,T;H_N)\), \(u^0\in V_0\) with \(u^0-w(0)\in V_0^D\), and \(u^1\in H\), we want to find a solution to the viscoelastic dynamic system
satisfying the following boundary and initial conditions
As usual, the Neumann boundary conditions are only formal, and their meaning will be specified in Definition 2.4.
Throughout the paper, we always assume that the family \(\{\Gamma _t\}_{t\in [0,T]}\) satisfies (E1)–(E3), as well as \({\mathbb {C}}\), \({\mathbb {B}}\), \(\Psi \), f, w, g, \(u^0\), and \(u^1\) the previous hypotheses. Let us define the following functional spaces:
Remark 2.1
In the classical viscoelastic case, namely when \(\Psi \) is identically equal to 1, the solution u to system (2.8) has derivative \({\dot{u}}(t)\in V_t\) for a.e. \(t\in (0,T)\) with \(E{\dot{u}}\in L^2(0,T;H)\). For a generic \(\Psi \), we expect to have \(\Psi E{\dot{u}}\in L^2(0,T;H)\). Therefore, \({\mathcal {W}}\) is the natural setting when looking for a solution to (2.8). Indeed, from a distributional point of view, we have
and \(E(\Psi \dot{u}),\nabla \Psi \odot \dot{u}\in L^2(0,T;H)\) if \(u\in {\mathcal {W}}\), thanks to (2.7).
Remark 2.2
The set \({\mathcal {W}}\) coincides with the space of functions \(u\in H^1(0,T;H)\) such that \(u(t)\in V_t\) and \(\Psi (t){\dot{u}}(t)\in V_t\) for a.e. \(t\in (0,T)\), and satisfying
This is a consequence of the strong measurability of the maps \(t\mapsto u(t)\) and \(t\mapsto \Psi (t){\dot{u}}(t)\) from (0, T) into \(V_T\), which gives that (2.13) is well defined and \(u,\Psi \dot{u}\in L^2(0,T;V_T)\). To prove the strong measurability of these two maps, it is enough to observe that \(V_T\) is a separable Hilbert space and that the maps \(t\mapsto {\dot{u}}(t)\) and \(t\mapsto \Psi (t){\dot{u}}(t)\) from (0, T) into \(V_T\) are weakly measurable. Indeed, for every \(\varphi \in C^\infty _c(\Omega {\setminus }\Gamma _T)\), the maps
are measurable from (0, T) into \({\mathbb {R}}\), and \(C^\infty _c(\Omega {\setminus }\Gamma _T)\) is dense in \(L^2(\Omega )\).
Lemma 2.3
The spaces \({\mathcal {V}}\) and \({\mathcal {W}}\) are Hilbert spaces with respect to the following norms:
Moreover, \({\mathcal {V}}^D\) is a closed subspace of \({\mathcal {V}}\).
Proof
It is clear that \(||\cdot ||_{{\mathcal {V}}}\) and \(||\cdot ||_{{\mathcal {W}}}\) are norms on \({\mathcal {V}}\) and \(\mathcal W\) induced by scalar products. We just have to check the completeness of such spaces with respect to these norms.
Let \(\{\varphi _k\}_k\subset {\mathcal {V}}\) be a Cauchy sequence. Then, \(\{\varphi _k\}_k\) and \(\{{\dot{\varphi }}_k\}_k\) are Cauchy sequences, respectively, in \(L^2(0,T;V_T)\) and \(L^2(0,T;H)\), which are complete Hilbert spaces. Thus, there exists \(\varphi \in L^2(0,T;V_T)\) with \({\dot{\varphi }}\in L^2(0,T;H)\) such that \(\varphi _k\rightarrow \varphi \) in \(L^2(0,T;V_T)\) and \({\dot{\varphi }}_k\rightarrow {\dot{\varphi }}\) in \(L^2(0,T;H)\). In particular, there exists a subsequence \(\{\varphi _{k_j}\}_j\) such that \(\varphi _{k_j}(t)\rightarrow \varphi (t)\) in \(V_T\) for a.e. \(t\in (0,T)\). Since \(\varphi _{k_j}(t)\in V_t\) for a.e. \(t\in (0,T)\), we deduce that \(\varphi (t)\in V_t\) for a.e. \(t\in (0,T)\). Hence, \(\varphi \in \mathcal V\) and \(\varphi _k\rightarrow \varphi \) in \({\mathcal {V}}\). With a similar argument, we can prove that \({\mathcal {V}}^D\subset {\mathcal {V}}\) is a closed subspace.
Let us now consider a Cauchy sequence \(\{u_k\}_k\subset {\mathcal {W}}\). We have that \(\{u_k\}_k\) and \(\{\Psi \dot{u}_k\}_k\) are Cauchy sequences, respectively, in \({\mathcal {V}}\) and \(L^2(0,T;V_T)\), which are complete Hilbert spaces. Thus, there exist two functions \(u\in {\mathcal {V}}\) and \(z\in L^2(0,T;V_T)\) such that \(u_k\rightarrow u\) in \({\mathcal {V}}\) and \(\Psi \dot{u}_k\rightarrow z\) in \(L^2(0,T;V_T)\). Since \(\dot{u}_k\rightarrow \dot{u}\) in \(L^2(0,T;H)\) and \(\Psi \in L^\infty ((0,T)\times \Omega )\), we also have that \(\Psi \dot{u}_k\rightarrow \Psi \dot{u}\) in \(L^2(0,T;H)\), which gives that \(z=\Psi \dot{u}\). Finally, let us prove that \(\Psi (t){\dot{u}}(t)\in V_t\) for a.e. \(t\in (0,T)\). By the fact that \(\Psi \dot{u}_k\rightarrow \Psi \dot{u}\) in \(L^2(0,T;V_T)\), there exists a subsequence \(\{\Psi \dot{u}_{k_j}\}_j\) such that \(\Psi (t)\dot{u}_{k_j}(t)\rightarrow \Psi (t){\dot{u}}(t)\) in \(V_T\) for a.e. \(t\in (0,T)\). Since \(\Psi (t)\dot{u}_{k_j}(t)\in V_t\) for a.e. \(t\in (0,T)\), we deduce that \(\Psi (t){\dot{u}}(t)\in V_t\) for a.e. \(t\in (0,T)\). Hence, \(u\in {\mathcal {W}}\) and \(u_k\rightarrow u\) in \({\mathcal {W}}\). \(\square \)
We are now in position to define a weak solution to (2.8)–(2.11).
Definition 2.4
(Weak solution) We say that \(u\in {\mathcal {W}}\) is a weak solution to system (2.8) with boundary conditions (2.9)–(2.11) if \(u-w\in \mathcal V^D\) and
for every \(\varphi \in {\mathcal {V}}^D\) such that \(\varphi (0)=\varphi (T)=0\).
Notice that the Neumann boundary conditions (2.10) and (2.11) can be obtained from (2.14), by using integration by parts in space, only when u(t) and \(\Gamma _t\) are sufficiently regular.
Remark 2.5
If \(\dot{u}\) is regular enough (for example, \(\dot{u}\in L^2(0,T;V_T)\) with \({\dot{u}}(t)\in V_t\) for a.e. \(t\in (0,T)\)), then we have \(\Psi E\dot{u}=E(\Psi \dot{u})-\nabla \Psi \odot \dot{u}\). Therefore, (2.14) is coherent with the strong formulation (2.8). In particular, for a function \(u\in {\mathcal {W}}\), we can define
so that Eq. (2.14) can be rephrased as
for every \(\varphi \in {\mathcal {V}}^D\) such that \(\varphi (0)=\varphi (T)=0\).
Definition 2.6
(Initial conditions) We say that \(u\in {\mathcal {W}}\) satisfies the initial conditions (2.12) if
3 Existence
We now state our main existence result, whose proof will be given at the end of Sect. 4.
Theorem 3.1
There exists a weak solution \(u\in {\mathcal {W}}\) to (2.8)–(2.11) satisfying the initial conditions \(u(0)=u^0\) and \(\dot{u}(0)=u^1\) in the sense of (2.16). Moreover \(u\in C_w([0,T];V_T)\), \(\dot{u}\in C_w([0,T];H)\cap H^1(0,T;(V_0^D)')\), and
To prove the existence of a weak solution to (2.8)–(2.11), we use a time discretization scheme in the same spirit of [3]. Let us fix \(n\in {\mathbb {N}}\) and set
We define
For every \(k=1,\dots ,n\), let \(u_n^k\in V_T\), with \(u_n^k-w_n^k\in V_n^k\), be the solution to
for every \(v\in V_n^k,\) where
The existence of a unique solution \(u_n^k\) to (3.1) is an easy application of Lax–Milgram’s theorem.
Remark 3.2
Since \(\delta u_n^k\in V_{(k-1)\tau _n}\), then \(\Psi _n^kE\delta u_n^k=E(\Psi _n^ku_n^k)-\nabla \Psi _n^k\odot u_n^k\), so that the discrete equation (3.1) is coherent with the weak formulation given in (2.14).
In the next lemma, we show a uniform estimate for the family \(\{u_n^k\}_{k=1}^n\) with respect to \(n\in {\mathbb {N}}\) that will be used later to pass to the limit in the discrete equation (3.1).
Lemma 3.3
There exists a constant \(C>0\), independent of \(n\in {\mathbb {N}}\), such that
Proof
We fix \(n\in {\mathbb {N}}\). To simplify the notation, we set
By taking as test function \(v=\tau _n(\delta u_n^k-\delta w_n^k)\in V_n^k\) in (3.1), for \(k=1,\dots ,n\), we obtain
where
Thanks to the following identities
and by omitting the terms with \(\tau _n^2\), which are non-negative, we derive
We fix \(i\in \{1,\dots ,n\}\) and sum over \(k=1,\dots , i\) to obtain the following discrete energy inequality
where \({\mathcal {E}}_0:=\frac{1}{2}||u^1 ||_H^2+\frac{1}{2}(\mathbb CEu^0,Eu^0)_{H}\). Let us now estimate the right-hand side in (3.3) from above. By (2.3) and (2.4), we have
For the other term involving \(g_n^k\), we perform the following discrete integration by parts
Hence, for every \(\epsilon \in (0,1)\), by using (2.3) and Young’s inequality, we get
where \(C_\epsilon \) is a positive constant depending on \(\epsilon \). Thanks to Jensen’s inequality, we can write
so that (3.8) can be further estimated as
for some positive constants \(\tilde{C}_\epsilon \) and \(\tilde{C}\), with \(\tilde{C}_\epsilon \) depending on \(\epsilon \). Similarly to (3.7), we can say
from which we deduce that for every \(\epsilon >0\)
where \({\bar{C}}_\epsilon \) is a positive constant depending on \(\epsilon \). We estimate from above the last term in the right-hand side of (3.3) in the following way
By considering (3.3)–(3.12) and using (2.6), we obtain
for two positive constants \({\hat{C}}_\epsilon \) and \({\hat{C}}\), with \({\hat{C}}_\epsilon \) depending on \(\epsilon \). We choose \(\epsilon <\frac{1}{2}\min \left\{ 1,\frac{\lambda _1}{C_{tr}^2}\right\} \) to derive the following estimate
where \(C_1\) and \(C_2\) are two positive constants depending only on \(u^0\), \(u^1\), f, g, and w. Thanks to a discrete version of Gronwall’s lemma (see, e.g., [1, Lemma 3.2.4]), we deduce the existence of a constant \(C_3>0\), independent of i and n, such that
By combining this last estimate with (3.13) and (2.6), we finally get (3.2) and we conclude. \(\square \)
We now want to pass to the limit into the discrete equation (3.1) to obtain a weak solution to (2.8)–(2.11). We start by defining the following approximating sequences of our limit solution
Notice that \(u_n\in H^1(0,T;H)\) with \(\dot{u}_n(t)=\delta u_n^k=\tilde{u}^+_n(t)\) for \(t\in ((k-1)\tau _n,k\tau _n)\) and \(k=1,\dots ,n\). Let us approximate \(\Psi \) and w by
Lemma 3.4
There exists a function \(u\in {\mathcal {W}}\), with \(u-w\in {\mathcal {V}}^D\), such that, up to a not relabeled subsequence
Proof
Thanks to Lemma 3.3, the sequences \(\{u_n\}_n\subset H^1(0,T;H)\cap L^\infty (0,T;V_T)\), \(\{u_n^\pm \}_n\subset L^\infty (0,T;V_T)\), and \(\{{\tilde{u}}_n^\pm \}_n\subset L^\infty (0,T;H)\) are uniformly bounded. By Banach–Alaoglu’s theorem, there exist \(u\in H^1(0,T;H)\) and \(v\in L^2(0,T;V_T)\) such that, up to a not relabeled subsequence
Since there exists a constant \(C>0\) such that
we can conclude that \(u=v\). Moreover, given that \( u^-_n(t)=u^+_n(t-\tau _n)\) for \(t\in (\tau _n,T)\), \(\tilde{u}^+_n(t)={\dot{u}}_n(t)\) for a.e. \(t\in (0,T)\), and \(\tilde{u}^-_n(t)={\tilde{u}}^+_n(t-\tau _n)\) for \(t\in (\tau _n,T)\), we deduce
By (3.2), we derive that the sequences \(\{E(\Psi ^+_n\tilde{u}^+_n)\}_n\subset L^2(0,T;H)\) and \(\{\nabla \Psi ^+_n\odot \tilde{u}^+_n\}_n\subset L^2(0,T;H)\) are uniformly bounded. Indeed, there exists a constant \(C>0\) independent of n such that
Therefore, there exist \(w_1,w_2\in L^2(0,T;H)\) such that, up to a further not relabeled subsequence
We want to identify the limit functions \(w_1\) and \(w_2\). Consider \(\varphi \in L^2(0,T;H)\), then
where \(\varphi ^\mathrm{{sym}}:=\frac{\varphi +\varphi ^T}{2}\). Since and \(\varphi ^\mathrm{{sym}}\nabla \Psi ^+_n \xrightarrow [n\rightarrow \infty ]{L^2(0,T;H)}\varphi ^\mathrm{{sym}}\nabla \Psi \) by dominated convergence theorem, we obtain
and so \(w_1=\nabla \Psi \odot {\dot{u}}\). Moreover, for \(\phi \in L^2(0,T;H)\), we have
thanks to and \(\Psi ^+_n\phi \xrightarrow [n\rightarrow \infty ]{L^2(0,T;H)}\Psi \phi \), again implied by dominated convergence theorem. Therefore, , from which \(E(\Psi ^+_n\tilde{u}^+_n)\xrightarrow [n\rightarrow \infty ]{{\mathcal {D}}'(0,T;H)}E(\Psi {\dot{u}})\), that gives \(w_2=E(\Psi {\dot{u}})\). In particular, we have \(\Psi \dot{u}\in L^2(0,T;V_T)\). By arguing in a similar way, we also obtain
Let us check that \(u\in {\mathcal {W}}\). To this aim, let us consider the following set
We have that F is a (strong) closed convex subset of \(L^2(0,T;V_T)\), and so by Hahn–Banach’s theorem, the set F is weakly closed. Notice that \(\{u^-_n\}_n,\{\Psi ^-_n\tilde{u}^-_n\}_n\subset F\), indeed
Since and , we conclude that \(u,\Psi \dot{u}\in F\). Finally, to show that \(u-w\in {\mathcal {V}}^D\), we observe
Therefore, \(\{u_n^-w_n^-\}_n\subset \{v\in L^2(0,T;V_T): v(t)\in V_t^D\hbox { for a.e.}\ t\in (0,T)\}\), which is a (strong) closed convex subset of \(L^2(0,T;V_T)\), and so it is weakly closed. Since and \(w_n^-\xrightarrow [n\rightarrow \infty ]{L^2(0,T;V_0)}w\), we get that \(u(t)-w(t)\in V_t^D\) for a.e. \(t\in (0,T)\), which implies \(u-w\in {\mathcal {V}}^D\). \(\square \)
We now use Lemma 3.4 to pass to the limit in the discrete equation (3.1).
Lemma 3.5
The limit function \(u\in {\mathcal {W}}\) of Lemma 3.4 is a weak solution to (2.8)–(2.11).
Proof
We only need to prove that \(u\in {\mathcal {W}}\) satisfies (2.14). We fix \(n\in {\mathbb {N}}\), \(\varphi \in C_c^1(0,T;V_T)\) such that \(\varphi (t)\in V_t^D\) for every \(t\in (0,T)\), and we consider
and the approximating sequences
If we use \(\tau _n\varphi _n^k\in V_n^k\) as test function in (3.1), after summing over \(k=1,\ldots ,n\), we get
By these identities
from (3.16), we deduce
Thanks to (3.14), (3.15), and the following convergences
we can pass to the limit in (3.17), and we get that the function \(u\in {\mathcal {W}}\) satisfies (2.14) for every \(\varphi \in C_c^1(0,T;V_T)\) such that \(\varphi (t)\in V_t^D\) for every \(t\in (0,T)\). Finally, by using a density argument (see [8, Remark 2.9]), we conclude that \(u\in {\mathcal {W}}\) is a weak solution to (2.8)–(2.11). \(\square \)
4 Initial conditions and energy-dissipation inequality
To complete our existence result, it remains to prove that the function \(u\in {\mathcal {W}}\) given by Lemma 3.5 satisfies the initial conditions (2.12) in the sense of (2.16). Let us start by showing that the second distributional derivative \(\ddot{u}\) belongs to \(L^2(0,T;(V_0^D)')\). If we consider the discrete equation (3.1), for every \(v\in V_0^D\subseteq V_n^k\), with \(\Vert v\Vert _{V_0}\le 1\), we have
Therefore, taking the supremum over \(v\in V_0^D\) with \(||v ||_{V_0}\le 1\), we obtain the existence of a positive constant C such that
If we multiply this inequality by \(\tau _n\) and we sum over \(k=1,\dots ,n\), we get
Thanks to (4.1) and Lemma 3.3, we conclude that \(\sum _{k=1}^n\tau _n\Vert \delta ^2 u^k_n\Vert ^2_{(V_0^D)'}\le {\tilde{C}}\) for every \(n\in {\mathbb {N}}\) for a positive constant \({\tilde{C}}\) independent on \(n\in {\mathbb {N}}\). In particular, the sequence \(\{\tilde{u}_n\}_n\subset H^1(0,T;(V_0^D)')\) is uniformly bounded (notice that \(\dot{{\tilde{u}}}_n(t)=\delta ^2 u_n^k\) for \(t\in ((k-1)\tau _n,k\tau _n)\) and \(k=1,\dots ,n\)). Hence, up to extract a further (not relabeled) subsequence from the one of Lemma 3.4, we get
and by using the following estimate
we conclude that \(w_3={\dot{u}}\).
Let us recall the following result, whose proof can be found, for example, in [9].
Lemma 4.1
Let X, Y be two reflexive Banach spaces such that \(X\hookrightarrow Y\) continuously. Then
Since \(H^1(0,T;(V^D_0)')\hookrightarrow C^0([0,T],(V^D_0)')\), by using Lemmas 3.4 and 4.1, we get that our weak solution \(u\in {\mathcal {W}}\) satisfies
By (3.14) and (4.2), we hence obtain
so that \(u(0)=u^0\) and \(\dot{u}(0)=u^1\), since \(u_n(0)=u^0\) and \(\tilde{u}_n(0)=u^1\).
To prove that
we will actually show
This is a consequence of the following energy-dissipation inequality which holds for the weak solution \(u\in {\mathcal {W}}\) of Lemma 3.5. Let us define the total energy as
Notice that the total energy \({\mathcal {E}}(t)\) is well defined for every \(t\in [0,T]\) since \(u\in C_w^0([0,T];V_T)\) and \(\dot{u}\in C_w^0([0,T];H)\), and that \({\mathcal {E}}(0)=\frac{1}{2}|| u^1 ||_H^2+\frac{1}{2}(\mathbb CEu^0,Eu^0)_H\).
Theorem 4.2
The weak solution \(u\in {\mathcal {W}}\) to (2.8)–(2.11), given by Lemma 3.5, satisfies for every \(t\in [0,T]\) the following energy-dissipation inequality
where \(\Psi E\dot{u}\) is the function defined in (2.15) and \({\mathcal {W}}_\mathrm{{tot}}(t)\) is the total work on the solution u at time \(t\in [0,T]\), which is given by
Remark 4.3
From the classical point of view, the total work on the solution u at time \(t\in [0,T]\) is given by
where \({\mathcal {W}}_\mathrm{{load}}(t)\) is the work on the solution u at time \(t\in [0,T]\) due to the loading term, which is defined as
and \({\mathcal {W}}_\mathrm{{bdry}}(t)\) is the work on the solution u at time \(t\in [0,T]\) due to the varying boundary conditions, which one expects to be equal to
being \(H_D:=L^2(\partial _D\Omega ;{\mathbb {R}}^d)\). Unfortunately, \(\mathcal W_\mathrm{{bdry}}(t)\) is not well defined under our assumptions on u. Notice that when \(\Psi \equiv 1\) on a neighborhood U of the closure of \(\partial _N\Omega \), then every weak solution u to (2.8)–(2.11) satisfies \(u\in H^1(0,T;H^1((\Omega \cap U){\setminus }\Gamma ;{\mathbb {R}}^d))\), which gives that \(u\in H^1(0,T;H_N)\) by our assumptions on \(\Gamma \). Hence, the first term of \({\mathcal {W}}_\mathrm{{bdry}}(t)\) makes sense and satisfies
The term involving the Dirichlet datum w is more difficult to handle since the trace of \(({\mathbb {C}}Eu+\Psi ^2{\mathbb {B}}E\dot{u})\nu \) on \(\partial _D\Omega \) is not well defined even when \(\Psi \equiv 1\) on a neighborhood of the closure of \(\partial _D\Omega \). If we assume that \(u\in H^1(0,T;H^2(\Omega {\setminus }\Gamma ;{\mathbb {R}}^d))\cap H^2(0,T;L^2(\Omega ;{\mathbb {R}}^d))\) and that \(\Gamma \) is a smooth manifold, then we can integrate by part Eq. (2.14) to deduce that u satisfies (2.8). In this case, \(({\mathbb {C}}Eu+\Psi ^2{\mathbb {B}}E\dot{u})\nu \in L^2(0,T;H_D)\) and by using (2.8), together with the divergence theorem and the integration by parts formula, we deduce
Hence, the definition of total work given in (4.5) is coherent with the classical one (4.6). Notice that if u is the solution to (2.8)–(2.11) given by Lemma 3.5, then (4.5) is well defined for every \(t\in [0,T]\), since \(g\in C^0([0,T];H_N)\), \(\dot{w}\in C^0([0,T];H)\), \(u\in C_w^0([0,T];V_T)\), and \(\dot{u}\in C_w^0([0,T];H)\). In particular, the function \(t\mapsto \mathcal W_\mathrm{{tot}}(t)\) from [0, T] to \({\mathbb {R}}\) is continuous.
Proof
Fixed \(t\in (0,T]\), for every \(n\in {\mathbb {N}}\), there exists a unique \(j\in \{1,\dots ,n\}\) such that \(t\in ((j-1)\tau _n,j\tau _n]\). After setting \(t_n:=j\tau _n\), we can rewrite (3.3) as
where
Thanks to (3.2), we have
The last convergences and (4.3) imply
and since \(\Vert u_n^+(t)\Vert _{V_T}+\Vert \tilde{u}_n^+(t)\Vert _H\le C\) for every \(n\in {\mathbb {N}}\), we get
By the lower semicontinuity properties of \(v\mapsto ||v ||_H^2\) and \(v\mapsto ({\mathbb {C}}Ev,Ev)_H\), we conclude
Thanks to Lemma 3.4 and (2.15), we obtain
so that
since \(t\le t_n\) and \(v\mapsto \int _0^t(\mathbb {B}v,v)_H\,\mathrm {d}s\) is a non-negative quadratic form on \(L^2(0,T;H)\). Let us study the right-hand side of (4.7). Given that we have
we can deduce
In a similar way, we can prove
since the following convergences hold
It remains to study the behavior as \(n\rightarrow \infty \) of the terms
Thanks to formula (3.10), we have
By arguing as before, we hence deduce
thanks to (4.8) and by these convergences
Notice that in the last convergence, we used the continuity of w from [0, T] in H. Similarly, we have
so that we get
thanks to (4.8), the continuity of \(s\mapsto g(s)\) in \(H_N\), and the fact that
By combining (4.9)–(4.16), we deduce the validity of the energy-dissipation inequality (4.4) for every \(t\in (0,T]\). Finally, for \(t=0\), the inequality trivially holds since \(u(0)=u^0\) and \(\dot{u}(0)=u^1\). \(\square \)
We now are in position to prove the validity of the initial conditions.
Lemma 4.4
The weak solution \(u\in {\mathcal {W}}\) to (2.8)–(2.11) of Lemma 3.5 satisfies
In particular, u satisfies the initial conditions (2.12) in the sense of (2.16).
Proof
By sending \(t\rightarrow 0^+\) into the energy-dissipation inequality (4.4) and using that the functions \(u\in C_w^0([0,T];V_T)\) and \(\dot{u}\in C_w^0([0,T];H)\), we deduce
since the right-hand side of (4.4) is continuous in t, \(u(0)=u^0\), and \(\dot{u}(0)=u^1\). Therefore, there exists \(\lim _{t\rightarrow 0^+}{\mathcal {E}}(t)={\mathcal {E}}(0)\). By using the lower semicontinuity of \(t\mapsto ||{\dot{u}}(t) ||^2_H\) and \(t\mapsto ({\mathbb {C}}Eu(t),Eu(t))_H\), we derive
Finally, since we have
we deduce (4.17). In particular, the functions \(u:[0,T]\rightarrow V_T\) and \(\dot{u}:[0,T]\rightarrow H\) are continuous at \(t=0\), which implies (2.16). \(\square \)
We can finally prove Theorem 3.1.
Proof of Theorem 3.1
It is enough to combine Lemmas 3.5 and 4.4. \(\square \)
Remark 4.5
We have proved Theorem 3.1 for the d-dimensional linear elastic case, namely when the displacement u is a vector-valued function. The same result is true with identical proofs in the antiplane case, that is, when the displacement u is a scalar function and satisfies (1.5).
5 Uniqueness
In this section, we investigate the uniqueness properties of system (2.8) with boundary and initial conditions (2.9)–(2.12). To this aim, we need to assume stronger regularity assumptions on the crack sets \(\{\Gamma _t\}_{t\in [0,T]}\) and on the function \(\Psi \). Moreover, we have to restrict our problem to the dimensional case \(d=2\), since in our proof, we need to construct a suitable family of diffeomorphisms which maps the time-dependent crack \(\Gamma _t\) into a fixed set, and this can be explicitly done only for \(d=2\) (see [7, Example 2.14]).
We proceed in two steps; first, in Lemma 5.2, we prove a uniqueness result in every dimension d, but when the cracks are not increasing, that is, \(\Gamma _T=\Gamma _0\). Next, in Theorem 5.5, we combine Lemma 5.2 with the finite speed of propagation theorem of [5] and the uniqueness result of [8] to derive the uniqueness of a weak solution to (2.8)–(2.12) in the case \(d=2\).
Let us start with the following lemma, whose proof is similar to that one of [8, Proposition 2.10].
Lemma 5.1
Let \(u\in {\mathcal {W}}\) be a weak solution to (2.8)–(2.11) satisfying the initial condition \(\dot{u}(0)=0\) in the following sense
Then u satisfies
for every \(\varphi \in {\mathcal {V}}^D\) such that \(\varphi (T)=0\), where \(\Psi E\dot{u}\) is the function defined in (2.15).
Proof
We fix \(\varphi \in {\mathcal {V}}^D\) with \(\varphi (T)=0\) and for every \(\epsilon >0\), we define the following function
We have that \(\varphi _{\epsilon }\in {\mathcal {V}}^D\) and \(\varphi _{\epsilon }(0)=\varphi _{\epsilon }(T)=0\), so we can use \(\varphi _{\epsilon }\) as test function in (2.14). By proceeding as in [8, Proposition 2.10], we obtain
It remains to consider the terms involving \({\mathbb {B}}\) and g.
We have
hence, by the dominated convergence theorem, we get
By combining together all the previous convergences, we get the thesis. \(\square \)
We now state the uniqueness result in the case of a fixed domain, that is, \(\Gamma _T=\Gamma _0\). We follow the same ideas of [12], and we need to assume
while on \(\Gamma _0\), we do not require any further hypotheses.
Lemma 5.2
(Uniqueness in a fixed domain) Assume (5.1) and \(\Gamma _T=\Gamma _0\). Then the viscoelastic dynamic system (2.8) with boundary and initial conditions (2.9)–(2.12) (the latter in the sense of (2.16)) has a unique weak solution.
Proof
Let \(u_1,u_2\in {\mathcal {W}}\) be two weak solutions to (2.8)–(2.11) with initial conditions (2.12). The function \(u:=u_1-u_2\) satisfies
hence, by Lemma 5.1, it solves
for every \(\varphi \in {\mathcal {V}}^D\) such that \(\varphi (T)=0\). We fix \(s\in (0,T]\) and consider the function
Since \(\varphi _s\in {\mathcal {V}}^D\) and \(\varphi _s(T)=0\), we can use it as test function in (5.3) to obtain
In particular, we deduce
which implies
since \(u(0)=0=\varphi _s(s)\). From the distributional point of view, the following equality holds
indeed, for all \(v\in C^{\infty }_c(0,T;H)\), we have
In particular, \(\Psi Eu\in H^1(0,T;H)\subset C^0([0,T],H)\), so that by (5.2)
which yields \(\Psi (0)Eu(0)=0\). Thanks to (5.5) and to property \(\Psi u\in H^1(0,T;H)\), we deduce
and by integrating on [0, s], we get
since \(E\varphi _s(s)=0=\Psi (0)Eu(0)\) and \(E{\dot{\varphi }}_s=Eu\) in (0, s). By combining the previous inequality with (5.4) and using the coercivity of the tensor \({\mathbb {C}}\), we derive
Let us set \(\xi (t):=\int _0^t u(\tau )\mathrm {d}\tau \), then
from which we deduce
where \(C:=2||{\mathbb {B}} ||_\infty ||{\dot{\Psi }} ||_\infty ^2\). Therefore, if we set \(s_0:=\frac{\lambda _1}{4C}\), for all \(s\le s_0\), we obtain
By Gronwall’s lemma, the last inequality implies \(E\xi (s)=0\) for all \(s\le s_0\). Hence, thanks to (5.6), we get \(\Vert u(s)\Vert ^2_H\le 0\) for all \(s\le s_0\), which yields \(u(s)=0\) for all \(s\le s_0\). Since \(s_0\) depends only on \({\mathbb {C}}\), \({\mathbb {B}}\), and \(\Psi \), we can repeat this argument starting from \(s_0\), and with a finite number of steps, we obtain \(u\equiv 0\) on [0, T]. \(\square \)
In order to prove our uniqueness result in the case of a moving crack, we need two auxiliary results, which are [4, Theorem 6.1] and [8, Theorem 4.3]. For the sake of the readers, we rewrite below the statements without proof.
The first one ([4, Theorem 6.1]) is a generalization of the well-known result of finite speed of propagation for the wave equation. Given an open bounded set \(U\subset {\mathbb {R}}^d\), we define by \(\partial _LU\) the Lipschitz part of the boundary \(\partial U\), which is the collection of points \(x\in \partial U\) for which there exist an orthogonal coordinate system \(y_1,\dots ,y_d\), a neighborhood V of x of the form \(A\times I\), with A open in \({\mathbb {R}}^{d-1}\) and I open interval in \({\mathbb {R}}\), and a Lipschitz function \(g:A\rightarrow I\), such that \(V\cap U:=\{(y_1,\dots ,y_d)\in V:y_d<g(y_1,\dots ,y_{d-1})\}\). Moreover, given a Borel set \(S\subseteq \partial _LU\), we define
Notice that \(H_S(U;{\mathbb {R}}^d)\) is a Hilbert space, and we denote its dual by \(H^{-1}_S(U;{\mathbb {R}}^d).\)
Theorem 5.3
(Finite speed of propagation) Let \(U\subset {\mathbb {R}}^d\) be an open bounded set and let \(\partial _LU\) be the Lipschitz part of \(\partial U\). Let \(S_0\) and \(S_1\) be two Borel sets with \(S_0\subseteq S_1\subseteq \partial _LU\), and let \(\mathbb C:U\rightarrow {\mathscr {L}}({\mathbb {R}}^{d\times d}_\mathrm{{sym}};{\mathbb {R}}^{d\times d}_\mathrm{{sym}})\) be a fourth-order tensor satisfying (2.4)–(2.6). Let
be a solution to
with initial conditions \(u(0)=0\) and \(\dot{u}(0)=0\) in the sense of \(L^2(U;{\mathbb {R}}^d)\) and \(H^{-1}_{S_1}(U;{\mathbb {R}}^d)\), respectively. Then
for every \(t\in [0,T]\).
Proof
See [4, Theorem 6.1]. \(\square \)
The second one ([8, Theorem 4.3]) is a uniqueness result for the weak solutions of the wave equation in a moving domain. Let \({\hat{H}}\) be a separable Hilbert space, and let \(\{\hat{V}_t\}_{t\in [0,T]}\) be a family of separable Hilbert spaces with the following properties:
- (i):
-
for every \(t\in [0,T]\), the space \({\hat{V}}_t\) is contained and dense in \({\hat{H}}\) with continuous embedding;
- (ii):
-
for every \(s,t\in [0,T]\), with \(s<t\), \({\hat{V}}_s\subset {\hat{V}}_t\) and the Hilbert space structure on \({\hat{V}}_s\) is the one induced by \({\hat{V}}_t\).
Let \(a:{\hat{V}}_T\times {\hat{V}}_T\rightarrow {\mathbb {R}}\) be a bilinear symmetric form satisfying the following conditions:
- (iii):
-
there exists \(M_0\) such that
$$\begin{aligned} |a(u,v)|\le M_0||u ||_{{\hat{V}}_T}||v ||_{{\hat{V}}_T}\quad \text {for every }u,v\in {\hat{V}}_T; \end{aligned}$$ - (iv):
-
there exists \(\lambda _0>0\) and \(\nu _0\in {\mathbb {R}}\) such that
$$\begin{aligned} a(u,u)\ge \lambda _0||u ||_{{\hat{V}}_T}^2-\nu _0||u ||_{\hat{H}}^2\quad \text {for every }u\in {\hat{V}}_T. \end{aligned}$$
Assume that
-
(U1)
for every \(t\in [0,T]\), there is a continuous and linear bijective operator \(Q_t:{\hat{V}}_t\rightarrow {\hat{V}}_0\), with continuous inverse \(R_t:{\hat{V}}_0\rightarrow {\hat{V}}_t\);
-
(U2)
\(Q_0\) and \(R_0\) are the identity maps on \({\hat{V}}_0\);
-
(U3)
there exists a constant \(M_1\) independent of t such that
$$\begin{aligned}&||Q_tu ||_{{\hat{H}}}\le M_1 ||u ||_{{\hat{H}}}\quad \text {for every }u\in {\hat{V}}_t,&||R_tu ||_{{\hat{H}}}\le M_1 ||u ||_{{\hat{H}}}\quad \text {for every }u\in {\hat{V}}_0,\\&||Q_tu ||_{{\hat{V}}_0}\le M_1 ||u ||_{{\hat{V}}_t}\quad \text {for every }u\in {\hat{V}}_t,&||R_tu ||_{{\hat{V}}_t}\le M_1 ||u ||_{{\hat{V}}_0}\quad \text {for every }u\in {\hat{V}}_0. \end{aligned}$$
Since \({\hat{V}}_t\) is dense in \({\hat{H}}\), (U3) implies that \(Q_t\) and \(R_t\) can be extended to continuous linear operators from \({\hat{H}}\) into itself, still denoted by \(Q_t\) and \(R_t\). We also require
-
(U4)
for every \(v\in {\hat{V}}_0\), the function \(t\mapsto R_tv\) from [0, T] into \({\hat{H}}\) has a derivative, denoted by \(\dot{R}_t v\);
-
(U5)
there exists \(\eta \in (0,1)\) such that
$$\begin{aligned} ||\dot{R}_t Q_tv ||^2_{{\hat{H}}}\le \lambda _0(1-\eta )||v ||_{\hat{V}_t}^2\quad \text {for every }v\in {\hat{V}}_t; \end{aligned}$$ -
(U6)
there exists a constant \(M_2\) such that
$$\begin{aligned} ||Q_tv-Q_sv ||_{{\hat{H}}}\le M_2||v ||_{\hat{V}_s}(t-s)\quad \text {for every } 0\le s<t\le T \text { and every } v\in \hat{V}_s; \end{aligned}$$ -
(U7)
for very \(t\in [0,T)\) and for every \(v\in {\hat{V}}_t\), there exists an element of \({\hat{H}}\), denoted by \(\dot{Q}_tv\), such that
$$\begin{aligned} \lim _{h\rightarrow 0^+}\frac{Q_{t+h}v-Q_tv}{h}=\dot{Q}_t v\text { in }{\hat{H}}. \end{aligned}$$
For every \(t\in [0,T]\), define
We assume that there exists a constant \(M_3\) such that
-
(U8)
the maps \(t\mapsto \alpha (t)(u,v)\), \(t\mapsto \beta (t)(u,v)\), \(t\mapsto \gamma (t)(u,v)\), and \(t\mapsto \delta (t)(u,v)\) are Lipschitz continuous and for a.e. \(t\in (0,T)\), their derivatives satisfy
$$\begin{aligned}&|{\dot{\alpha }}(t)(u,v)|\le M_3||u ||_{{\hat{V}}_0}||v ||_{\hat{V}_0}\quad \text {for }u,v\in {\hat{V}}_0,\\&|{\dot{\beta }}(t)(u,v)|\le M_3||u ||_{{\hat{V}}_0}||v ||_{\hat{V}_0}\quad \text {for }u,v\in {\hat{V}}_0,\\&|{\dot{\gamma }}(t)(u,v)|\le M_3||u ||_{{\hat{V}}_0}||v ||_{\hat{H}}\quad \text {for }u\in {\hat{V}}_0\text { and }v\in {\hat{H}},\\&|{\dot{\delta }}(t)(u,v)|\le M_3||u ||_{{\hat{H}}}||v ||_{\hat{H}}\quad \text {for }u,v\in {\hat{H}}. \end{aligned}$$
Theorem 5.4
(Uniqueness for the wave equation) Assume that \({\hat{H}}\), \(\{{\hat{V}}_t\}_{t\in [0,T]}\), and a satisfy (i)–(iv) and that (U1)–(U8) hold. Given \(u^0\in {\hat{V}}_0\), \(u^1\in {\hat{H}}\), and \(f\in L^2(0,T;{\hat{H}})\), there exists a unique solution
to the wave equation
satisfying the initial conditions \(u(0)=u^0\) and \(\dot{u}(0)=u^1\) in the sense that
Proof
See [8, Theorem 4.3]. \(\square \)
We now are in position to prove the uniqueness theorem in the case of a moving domain. We consider the dimensional case \(d=2\), and we require the following assumptions:
-
(H1)
there is a \(C^{2,1}\) simple curve \(\Gamma \subset {{\overline{\Omega }}}\subset {\mathbb {R}}^2\), parametrized by arc-length \(\gamma :[0,\ell ]\rightarrow {{\overline{\Omega }}}\), such that \(\Gamma \cap \partial \Omega =\gamma (0)\cup \gamma (\ell )\) and \(\Omega {\setminus }\Gamma \) is the union of two disjoint open sets with Lipschitz boundary;
-
(H2)
there exists a non-decreasing function \(s:[0,T]\rightarrow (0,\ell )\) of class \(C^{1,1}\) such that \(\Gamma _t=\gamma ([0,s(t)])\);
-
(H3)
\(|\dot{s}(t)|^2< \frac{\lambda _1}{C_K}\), where \(\lambda _1\) is the ellipticity constant of \({\mathbb {C}}\) and \(C_K\) is the constant that appears in Korn’s inequality in (2.2).
Notice that hypotheses (H1) and (H2) imply (E1)–(E3). We also assume that \(\Psi \) satisfies (5.1) and there exists a constant \(\epsilon >0\) such that for every \(t\in [0,T]\)
Theorem 5.5
Assume \(d=2\) and (H1)–(H3), (5.1), and (5.7). Then the system (2.8) with boundary conditions (2.9)–(2.11) has a unique weak solution \(u\in {\mathcal {W}}\) which satisfies the initial conditions \(u(0)=u^0\) and \(\dot{u}(0)=u^1\) in the sense of (2.16).
Proof
As before, let \(u_1,u_2\in {\mathcal {W}}\) be two weak solutions to (2.8)–(2.11) with initial conditions (2.12). Then \(u:=u_1-u_2\) satisfies (5.2) and (5.3) for every \(\varphi \in {\mathcal {V}}^D\) such that \(\varphi (T)=0\). Let us define
and assume by contradiction that \(t_0<T\). Consider first the case in which \(t_0>0\). By (H1), (H2), (5.1), and (5.7), we can find two open sets \(A_1\) and \(A_2\), with \(A_1\subset \subset A_2\subset \subset \Omega \), and a number \(\delta >0\) such that for every \(t\in [t_0-\delta ,t_0+\delta ]\), we have \(\gamma (s(t))\in A_1\), \(\Psi (t,x)=0\) for every \(x\in {\overline{A}}_2\), and \((A_2{\setminus } A_1)\setminus \Gamma \) is the union of two disjoint open sets with Lipschitz boundary. Let us define
Since every function in \({\hat{V}}^1\) can be extended to a function in \(V_{t_0-\delta }^D\), by classical results for linear hyperbolic equations (see, e.g., [9]), we deduce \({\ddot{u}}\in L^2(t_0-\delta ,t_0+\delta ;({\hat{V}}^1)')\) and that u satisfies for a.e. \(t\in (t_0-\delta ,t_0+\delta )\)
Moreover, we have \(u(t_0)=0\) as element of \( {\hat{H}}^1\) and \(\dot{u}(t_0)=0\) as element of \(({\hat{V}}^1)'\), since \(u(t)\equiv 0\) in \([t_0-\delta ,t_0)\), \(u\in C^0([t_0-\delta ,t_0];{\hat{H}}^1)\), and \(\dot{u}\in C^0([t_0-\delta ,t_0];({\hat{V}}^1)')\). We are now in position to apply the result of finite speed of propagation of Theorem 5.3. This theorem ensures the existence of the third open set \(A_3\), with \(A_1\subset \subset A_3\subset \subset A_2\), such that, up to choose a smaller \(\delta \), we have \(u(t)=0\) on \(\partial A_3\) for every \(t\in [t_0,t_0+\delta ]\), and both \((\Omega {\setminus } A_3)\setminus \Gamma \) and \(A_3{\setminus } \Gamma \) are union of two disjoint open sets with Lipschitz boundary.
In \(\Omega {\setminus } A_3\), the function u solves
for every \(\varphi \in L^2(t_0-\delta ,t_0+\delta ;{\hat{V}}^2)\cap H^1(t_0-\delta ,t_0+\delta ;{\hat{H}}^2)\) such that \(\varphi (t_0-\delta )=\varphi (t_0+\delta )=0\), where
Since \(u(t)=0\) on \(\partial _D\Omega \cup \partial A_3\) for every \(t\in [t_0-\delta ,t_0+\delta ]\) and \(u(t_0-\delta )=\dot{u}(t_0-\delta )=0\) in the sense of (2.16) (recall that \(u\equiv 0\) in \([t_0-\delta ,t_0)\)), we can apply Lemma 5.2 to deduce \(u(t)=0\) in \(\Omega {\setminus } A_3\) for every \(t\in [t_0-\delta ,t_0+\delta ]\).
On the other hand in \(A_3\), by setting
we get that the function u solves
for every \(\varphi \in L^2(t_0-\delta ,t_0+\delta ;\hat{V}^3_{t_0+\delta })\cap H^1(t_0-\delta ,t_0+\delta ;{\hat{H}}^3)\) such that \(\varphi (t)\in {\hat{V}}^3_t\) for a.e. \(t\in (t_0-\delta ,t_0+\delta )\) and \(\varphi (t_0-\delta )=\varphi (t_0+\delta )=0\). Here, we would like to apply the uniqueness result of Theorem 5.4 for the spaces \(\{{\hat{V}}_t^3\}_{t\in [t_0-\delta ,t_0+\delta ]}\) and \(\hat{H}^3\), endowed with the usual norms, and for the bilinear form
As shown in [7, Example 2.14], we can construct \(\Phi ,\Lambda \in C^{1,1}([t_0-\delta ,t_0+\delta ]\times \overline{A}_3;{\mathbb {R}}^2)\) such that for every \(t\in [0,T]\), the function \(\Phi (t,\cdot ):{\overline{A}}_3\rightarrow {\overline{A}}_3\) is a diffeomorphism of \(A_3\) in itself with inverse \(\Lambda (t,\cdot ):{\overline{A}}_3\rightarrow {\overline{A}}_3\). Moreover, \(\Phi (0,y)=y\) for every \(y\in {\overline{A}}_3\), \(\Phi (t,\Gamma \cap {\overline{A}}_3)=\Gamma \cap {\overline{A}}_3\) and \(\Phi (t,\Gamma _{t_0-\delta }\cap {\overline{A}}_3)=\Gamma _t\cap {\overline{A}}_3\) for every \(t\in [t_0-\delta ,t_0+\delta ]\). For every \(t\in [t_0-\delta ,t_0+\delta ]\), the maps \((Q_tu)(y):=u(\Phi (t,y))\), \(u\in {\hat{V}}_t^3\) and \(y\in A_3\), and \((R_tv)(x):=v(\Lambda (t,x))\), \(v\in {\hat{V}}_{t_0-\delta }^3\) and \(x\in A_3\), provide a family of linear and continuous operators which satisfy the assumptions (U1)–(U8) of Theorem 5.4 (see [8, Example 4.2]). The only condition to check is (U5). The bilinear form a satisfies the following ellipticity condition
where \({\hat{C}}_K\) is the constant in Korn’s inequality in \(\hat{V}_{t_0+\delta }^3\), namely
Notice that for \(t\in [t_0-\delta ,t_0+\delta ]\)
from which we obtain
Hence, have to show the property
This is ensured by (H3). Indeed, as explained in [7, Example 3.1], we can construct the maps \(\Phi \) and \(\Lambda \) in such a way that
since \(|\dot{s}(t)|^2<\frac{\lambda _1}{C_K}\). Moreover, every function in \(\hat{V}^3_{t_0+\delta }\) can be extended to a function in \(H^1(\Omega {\setminus }\Gamma ;{\mathbb {R}}^d)\). Hence, for Korn’s inequality in \({\hat{V}}_{t_0+\delta }^3\), we can use the same constant \(C_K\) of \(H^1(\Omega {\setminus }\Gamma ;{\mathbb {R}}^d)\). This allows us to apply Theorem 5.4, which implies \(u(t)=0\) in \(A_3\) for every \(t\in [t_0,t_0+\delta ]\). In the case \(t_0=0\), it is enough to argue as before in \([0,\delta ]\), by exploiting (5.2). Therefore, \(u(t)=0\) in \(\Omega \) for every \(t\in [t_0,t_0+\delta ]\), which contradicts the maximality of \(t_0\). Hence \(t_0=T\), that yields \(u(t)=0\) in \(\Omega \) for every \(t\in [0,T]\). \(\square \)
Remark 5.6
Also Theorem 5.5 is true in the antiplane case, with essentially the same proof. Notice that, when the displacement is scalar, we do not need to use Korn’s inequality in (5.8) to get the coercivity in \({\hat{V}}_{t_0+\delta }^3\) of the bilinear form a defined before. Therefore, in this case in (H3), it is enough to assume \(|\dot{s}(t)|^2<\lambda _1\).
6 A moving crack satisfying Griffith’s dynamic energy-dissipation balance
We conclude this paper with an example of a moving crack \(\{\Gamma _t\}_{t\in [0,T]}\) and weak solution to (2.8)–(2.12) which satisfy the energy-dissipation balance of Griffith’s dynamic criterion, as happens in [4] for the purely elastic case. In dimension \(d=2\), we consider an antiplane evolution, which means that the displacement u is scalar, and \(\Omega :=\{x\in {\mathbb {R}}^2:|x|<R\}\), with \(R>0\). We fix a constant \(0<c<1\) such that \(cT<R\), and we set
Let us define the following function
where \(\text {Im}\) denotes the imaginary part of a complex number. Notice that the function S satisfies \(S\in H^1(\Omega \setminus \Gamma _0)\setminus H^2(\Omega \setminus \Gamma _0)\), and it is a weak solution to
Let us consider the function
and let w(t) be its restriction to \(\partial \Omega \). Since u(t) has a singularity only at the crack tip (ct, 0), the function w(t) can be seen as the trace on \(\partial \Omega \) of a function belonging to \( H^2(0,T;L^2(\Omega ))\cap H^1(0,T;H^1(\Omega {\setminus }\Gamma _0))\), still denoted by w(t). It is easy to see that u solves the wave equation
with boundary conditions
and initial data
Let us consider a function \(\Psi \) which satisfies the regularity assumptions (5.1) and condition (5.7), namely
with \(0<\epsilon <R-cT\). In this case, u is a weak solution, in the sense of Definition 2.4, to the damped wave equation
with forcing term f given by
and boundary and initial conditions
Notice that to derive the homogeneous Neumann boundary conditions on \(\Gamma _t\), we used \(\frac{\partial \dot{u}}{\partial \nu }(t)=\nabla {\dot{u}}(t)\cdot \nu =\partial _2{\dot{u}}(t)=0\) on \(\Gamma _t\). By the uniqueness result proved in the previous section, the function u coincides with that one found in Theorem 3.1. Thanks to the computations done in [4, Section 4], we know that u satisfies for every \(t\in [0,T]\) the following energy-dissipation balance for the undamped equation, where ct coincides with the length of \(\Gamma _t{\setminus }\Gamma _0\)
Moreover, we have
For every \(t\in [0,T]\), we compute
If we denote by \(\dot{u}^\oplus (t)\) and \(\dot{w}^\oplus (t)\) the traces of \({\dot{u}}(t)\) and \(\dot{w}(t)\) on \(\Gamma _t\) from above and by \(\dot{u}^\ominus (t)\) and \(\dot{w}^\ominus (t)\) the trace from below, thanks to the divergence theorem, we have
since \(u(t)=w(t)\) on \(\partial \Omega \), \(\Psi (t)=0\) on \(\partial B_\epsilon (t)\), and \(\partial _2{\dot{u}}(t)=0\) on \(\Gamma _t\). Therefore, for every \(t\in [0,T]\), we get
By combining (6.1)–(6.3), we deduce that u satisfies for every \(t\in [0,T]\) the following Griffith’s energy-dissipation balance for the viscoelastic dynamic equation
where in this case, the total work takes the form
Notice that equality (6.4) gives (1.6). This shows that in this model Griffith’s dynamic energy-dissipation balance can be satisfied by a moving crack, in contrast with the case \(\Psi =1\), which always leads to (1.3).
References
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005)
Caponi, M.: Linear hyperbolic systems in domains with growing cracks. Milan J. Math. 85, 149–185 (2017)
Dal Maso, G., Larsen, C.J.: Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22, 387–408 (2011)
Dal Maso, G., Larsen, C.J., Toader, R.: Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition. J. Mech. Phys. Solids 95, 697–707 (2016)
Dal Maso, G., Larsen, C.J., Toader, R.: Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition. J. Math. Pures Appl. 127, 160–191 (2019)
Dal Maso, G., Lazzaroni, G., Nardini, L.: Existence and uniqueness of dynamic evolutions for a peeling test in dimension one. J. Differ. Equ. 261, 4897–4923 (2016)
Dal Maso, G., Lucardesi, I.: The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data. Appl. Math. Res. Express 2017, 184–241 (2017)
Dal Maso, G., Toader, R.: On the Cauchy problem for the wave equation on time-dependent domains. J. Differ. Equ. 266, 3209–3246 (2019)
Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique pour les sciences et les techniques. Évolution: semi-groupe, variationnel. Masson, Paris (1988)
Dumouchel, P.E., Marigo, J.J., Charlotte, M.: Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution. Contin. Mech. Thermodyn. 20, 1–19 (2008)
Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221–A, 163–198 (1920)
Ladyzenskaya, O.A.: On integral estimates, convergence, approximate methods, and solution in functionals for elliptic operators. Vestnik Leningr. Univ. 13, 60–69 (1958)
Mott, N.F.: Brittle fracture in mild steel plates. Engineering 165, 16–18 (1948)
Nicaise, S., Sändig, A.M.: Dynamic crack propagation in a 2D elastic body: the out-of-plane case. J. Math. Anal. Appl. 329, 1–30 (2007)
Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and Its Applications, vol. 26. North-Holland Publishing Co., Amsterdam (1992)
Riva, F., Nardini, L.: Existence and uniqueness of dynamic evolutions for a one dimensional debonding model with damping. Preprint SISSA 28/2018/MATE (2018) (submitted for publication)
Slepyan, L.I.: Models and Phenomena in Fracture Mechanics. Foundations of Engineering Mechanics. Springer, Berlin (2002)
Tasso, E.: Weak formulation of elastodynamics in domains with growing cracks. Preprint SISSA 51/2018/MATE (2018) (submitted for publication)
Acknowledgements
The authors wish to thank Professors Gianni Dal Maso and Rodica Toader for having proposed the problem and for many helpful discussions on the topic. 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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Caponi, M., Sapio, F. A dynamic model for viscoelastic materials with prescribed growing cracks. Annali di Matematica 199, 1263–1292 (2020). https://doi.org/10.1007/s10231-019-00921-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-019-00921-1
Keywords
- Linear second-order hyperbolic systems
- Dynamic fracture mechanics
- Elastodynamics
- Viscoelasticity
- Cracking domains