Variational framework
The variational ingredients of the dynamic gradient damage model as well as the induced governing equations are firstly recalled in this section. Let us consider a two-dimensional isotropic body \(\Omega \). We place ourselves under the small displacement condition. This is a plausible hypothesis for brittle materials when large rotations are also not expected. Damage and subsequent fracture occur more easily in tension than in compression, thus tension-compression asymmetry formulations in the sense of [10, 11] for instance are in general needed. Nevertheless the numerical experiments considered here do not require the use of such formulations. Hence we adhere to the original approach of [1, 19] where damage acts on the sound elastic energy density \(\psi _0\) symmetrically under tension and compression. Assuming these hypotheses, the elastic energy of the domain \(\Omega \) is given by
$$\begin{aligned} \mathcal {E}(\mathbf {u}_t,\alpha _t)= & {} \int _\Omega \psi \bigl (\varvec{\varepsilon }(\mathbf {u}_t),\alpha _t\bigr ) \,\mathrm {d}\mathbf {x}=\int _\Omega a(\alpha _t)\psi _0\bigl (\varvec{\varepsilon }(\mathbf {u}_t)\bigr ) \,\mathrm {d}\mathbf {x}\nonumber \\= & {} \int _\Omega \frac{1}{2}a(\alpha _t)\mathsf {A}\varvec{\varepsilon }(\mathbf {u}_t)\cdot \varvec{\varepsilon }(\mathbf {u}_t)\,\mathrm {d}\mathbf {x}\end{aligned}$$
(1)
where \(\mathsf {A}\) is the standard isotropic Hooke’s elasticity tensor and \(\varvec{\varepsilon }(\mathbf {u}_t)=\frac{1}{2}(\nabla \mathbf {u}_t+\nabla ^\mathsf {T}\mathbf {u}_t)\) the linearized strain. By definition, the stress tensor conjugate to the strain measure reads \(\varvec{\sigma }_t=a(\alpha _t)\mathsf {A}\varvec{\varepsilon }(\mathbf {u}_t)\), with \(\alpha \mapsto a(\alpha )\) a non-dimensional damage constitutive function describing stiffness degradation in the bulk. Concerning the kinetic energy, we admit total mass conservation and no damage dependence of the material density, which leads to its classical definition
$$\begin{aligned} \mathcal {K}(\dot{\mathbf {u}}_t)=\int _\Omega \kappa (\dot{\mathbf {u}}_t)\,\mathrm {d}\mathbf {x}=\int _\Omega \frac{1}{2}\rho \dot{\mathbf {u}}_t\cdot \dot{\mathbf {u}}_t\,\mathrm {d}\mathbf {x}. \end{aligned}$$
(2)
We now turn to the definition of the dissipated energy which quantifies the amount of energy consumed in the damage process. In the phase-field terminology, it is also called the regularized crack functional. It is due to the fact that this energy is closely related to the Griffith-like surface energy of the phase-field representation of cracks according to the \(\Gamma \)-convergence theory [2]. Dynamics should not influence the definition of such functional. We will hence use the same definition used in [7] for quasi-static calculations
$$\begin{aligned} \mathcal {S}(\alpha _t)=\int _\Omega \varsigma (\alpha _t,\nabla \alpha _t)\,\mathrm {d}\mathbf {x}=\int _\Omega \frac{G_\mathrm {c}}{c_w}\left( \frac{w(\alpha _t)}{\ell }+\ell \nabla \alpha _t\cdot \nabla \alpha _t\right) \,\mathrm {d}\mathbf {x}. \end{aligned}$$
(3)
Contrary to local strain-softening constitutive models, here the damage dissipation mechanism becomes non-local and localization is systematically accompanied by finite energy consumption, due to the presence of the gradient term. In (3), the function \(\alpha \mapsto w(\alpha )\) is another non-dimensional damage constitutive law characterizing local damage dissipation. This function along with the former stiffness degradation function \(a(\alpha )\) contribute to the damage constitutive behavior of the material and should also satisfy certain physical properties [6] which we do not reproduce here. In this paper we adopt the following constitutive functions
$$\begin{aligned} a(\alpha )=(1-\alpha )^2\qquad \text {and}\qquad w(\alpha )=\alpha . \end{aligned}$$
(4)
Concerning local damage dissipation, a quadratic function \(w(\alpha )=\alpha ^2\) originally proposed in [19] is widely used among the phase-field community [8, 13–15]. It can be regarded as the Ambrosio and Tortorelli elliptic regularization of the Griffith functional based on their work on image segmentation. In our brittle fracture modeling context however, the use of \(w(\alpha )=\alpha \) should be preferred since it guarantees the existence of a purely elastic domain and provides a non-null threshold for damage evolution, see [6]. The link between the gradient damage description of cracks and the Griffith’s one lies in the definition of the normalization factor \(c_w\). The fracture toughness \(G_\mathrm {c}\), i.e. the energy required to create a unit Griffith-like crack surface, can be identified in the gradient damage terminology as the energy dissipated during the optimal damage band creation. Using this identification along with a direct calculation in a 1-d setting, we obtain
$$\begin{aligned} c_w=4\int _0^1\sqrt{w(\beta )}\,\mathrm {d}\beta \end{aligned}$$
(5)
which gives \(c_w=\frac{8}{3}\) for \(w(\alpha )=\alpha \). We refer the readers again to [6, 20] for a more detailed discussion on the relationship between these gradient damage and fracture parameters.
In this work, external loads are applied to the body \(\Omega \) only through a prescribed displacement \(t\mapsto \mathbf {U}_t\) on a subset \(\partial \Omega _U\) of the boundary. It will be defined in the admissible displacement space \(\mathcal {C}_t\) which reads
$$\begin{aligned} \mathcal {C}_t=\{\mathbf {u}_t:\Omega \rightarrow \mathbb {R}^\mathrm {dim}|\mathbf {u}_t =\mathbf {U}_t\text { on }\partial \Omega _U \}. \end{aligned}$$
Damage is modeled as an irreversible defect evolution. Its admissible space will be built from the current damage state \(0\le \alpha _t\le 1\) and it is defined by
$$\begin{aligned} \mathrm {D}(\alpha _t)=\{\beta _t:\Omega \rightarrow [0,1]|0\le \alpha _t\le \beta _t\le 1\}. \end{aligned}$$
(6)
It can be seen that a damage field \(\beta _t\) is admissible, if and only if it is accessible from the current damage state \(\alpha _t\) verifying the irreversibility condition, i.e. the damage only grows. In order to formulate the temporal displacement-damage evolution as a boundary value problem (Hamilton’s principle), we consider an arbitrary interval of time \(I=[0,T]\) and fix the values of \((\mathbf {u},\alpha )\) at both time ends denoted by \(\mathbf {u}_{\partial I}=(\mathbf {u}_0,\mathbf {u}_T)\) and \(\alpha _{\partial I}=(\alpha _0,\alpha _T)\). This leads to the following admissible evolution spaces
$$\begin{aligned} \mathcal {C}(\mathbf {u})=\Bigl \{\,\mathbf {v}:I\times \Omega \rightarrow \mathbb {R}^\mathrm {dim}\Bigm |\mathbf {v}_t\in \mathcal {C}_t\text { for all}\,t \in I \text { and }\mathbf {v}_{\partial I}=\mathbf {u}_{\partial I}\,\Bigr \} \end{aligned}$$
and
$$\begin{aligned} \mathcal {D}(\alpha )=\Bigl \{\,\beta :I\times \Omega \rightarrow [0,1]\Bigm |\beta _t\in \mathrm {D}(\alpha _t)\text { for all} \,t\in I \text { and }\beta _{\partial I}=\alpha _{\partial I}\,\Bigr \}. \end{aligned}$$
With all the variational ingredients set, we are now in a position to form the space-time action integral given by
$$\begin{aligned} \mathcal {A}(\mathbf {u},\alpha )=\int _I\mathcal {L}(\mathbf {u}_t,\dot{\mathbf {u}}_t,\alpha _t) \,\mathrm {d}t=\int _I\bigl (\mathcal {E}(\mathbf {u}_t,\alpha _t)+\mathcal {S} (\alpha _t)-\mathcal {K}(\dot{\mathbf {u}}_t)\bigr )\,\mathrm {d}t \end{aligned}$$
(7)
and announce the following three physical principles governing the coupled two-field \((\mathbf {u},\alpha )\) time-continuous dynamic gradient damage problem:
-
1
Irreversibility the damage \(t\mapsto \alpha _t\) is a non-decreasing function of time.
-
2
First-order stability the first-order action variation is non-negative with respect to arbitrary admissible displacement and damage evolutions
$$\begin{aligned} \mathcal {A}'(\mathbf {u},\alpha )(\mathbf {v}-\mathbf {u},\beta -\alpha )\ge 0\quad \text { for all} \;\mathbf {v}\in \mathcal {C}(\mathbf {u}) \text { and all }\;\beta \in \mathcal {D}(\alpha ). \end{aligned}$$
(8)
-
3
Energy balance the only energy dissipation is due to damage
$$\begin{aligned} \mathcal {H}_t=\mathcal {H}_0+\int _0^t\Biggl (\int _\Omega \bigl (\varvec{\sigma }_s\cdot \varvec{\varepsilon }(\dot{\mathbf {U}}_s)+\rho \ddot{\mathbf {u}}_s\cdot \dot{\mathbf {U}}_s\bigr )\,\mathrm {d}\mathbf {x}\Biggr )\,\mathrm {d}s \end{aligned}$$
(9)
where the total energy is defined by
$$\begin{aligned} \mathcal {H}_t=\mathcal {E}(\mathbf {u}_t,\alpha _t)+\mathcal {S}(\alpha _t)+\mathcal {K} (\dot{\mathbf {u}}_t). \end{aligned}$$
Remark
In the first-order stability condition (8), we evaluate the directional (Gâteaux) derivative of the action integral (7) at \((\mathbf {u},\alpha )\), a possible solution to the dynamic evolution problem, in the direction \((\mathbf {v}-\mathbf {u},\beta -\alpha )\) which corresponds to a perturbation. Formally, using the Lagrangian \(\mathcal {L}\), we have
$$\begin{aligned} \mathcal {A}'(\mathbf {u},\alpha )(\mathbf {w},\beta -\alpha )=\int _I\left( \frac{\partial \mathcal {L}}{\partial \mathbf {u}_t}(\mathbf {s}_t)(\mathbf {w}_t)+\frac{\partial \mathcal {L}}{\partial \dot{\mathbf {u}}_t}(\mathbf {s}_t)(\dot{\mathbf {w}}_t)+\frac{\partial \mathcal {L}}{\partial \alpha _t}(\mathbf {s}_t)(\beta _t-\alpha _t)\right) \,\mathrm {d}t\,, \end{aligned}$$
where \(\mathbf {w}=\mathbf {v}-\mathbf {u}\) denotes a displacement evolution variation and \(\mathbf {s}_t=(\mathbf {u}_t,\dot{\mathbf {u}}_t,\alpha _t)\) corresponds to a state of the dynamical system.
Remark
Remark that (8) is written as a variational inequality to take into account the unilateral effects introduced by the irreversibility condition in the definition of the damage admissible space (6). It can be regarded as an extension of Hamilton’s principle applied to systems with irreversible dissipation. The energy balance condition (9) complements the first-order stability condition (8) which ensures that energy could only be dissipated through damage (or phase-field like fracture).
By developing the directional derivative of the action integral (7), further physical insights into the first-order stability condition (8) can be obtained if sufficient spatial and temporal regularities of the involved fields are assumed. Testing (8) with \(\beta =\alpha \), we obtain after an integration by parts in the time domain
$$\begin{aligned} \mathcal {A}'(\mathbf {u},\alpha )(\mathbf {w},0)=\int _I\left( \int _\Omega \bigl (\varvec{\sigma }_t\cdot \varvec{\varepsilon }(\mathbf {w}_t)+\rho \ddot{\mathbf {u}}_t\cdot \mathbf {w}_t\bigr )\,\mathrm {d}\mathbf {x}\right) \,\mathrm {d}t=0\quad \text { for all} \,\mathbf {w}_t\in \mathcal {C}_0\,, \end{aligned}$$
where \(\mathcal {C}_0\) is the associated linear space of \(\mathcal {C}_t\), i.e. defined by
$$\begin{aligned} \mathcal {C}_0=\{\mathbf {u}_t:\Omega \rightarrow \mathbb {R}^\mathrm {dim}|\mathbf {u}_t= \mathbf {0}\text { on }\partial \Omega _U\}. \end{aligned}$$
This leads thus to the weak elastic-damage dynamic wave equation
$$\begin{aligned} \int _\Omega \bigl (\varvec{\sigma }_t\cdot \varvec{\varepsilon }(\mathbf {w}_t)+\rho \ddot{\mathbf {u}}_t\cdot \mathbf {w}_t\bigr )\,\mathrm {d}\mathbf {x}=0\quad \text { for all} \, \mathbf {w}_t\in \mathcal {C}_0. \end{aligned}$$
(10)
Compared to the classical elastodynamic equation, we note that here the stress tensor is modulated by the stiffness degradation function \(\varvec{\sigma }_t=a(\alpha _t)\mathsf {A}\varvec{\varepsilon }(\mathbf {u}_t)\).
We now turn to the governing equation for damage evolution induced from the first-order stability condition (8). We observe that the admissible damage space \(\mathrm {D}(\alpha _t)\) defined in (6) is convex. Due to the arbitrariness of the temporal variation of \(\beta \), testing (8) now with \(\mathbf {v}=\mathbf {u}\) gives the Euler’s inequality condition stating the partial minimality of the total energy with respect to the damage variable under the irreversible constraint for every \(t\in I\)
$$\begin{aligned} \mathcal {E}(\mathbf {u}_t,\alpha _t)+\mathcal {S}(\alpha _t)\le \mathcal {E} (\mathbf {u}_t,\beta _t)+\mathcal {S}(\beta _t)\quad \text { for all} \,\beta _t\in \mathrm {D}(\alpha _t). \end{aligned}$$
(11)
Although the same energy minimization principle (11) holds also for quasi-static gradient damage models [6], here the displacement field \(\mathbf {u}_t\) is governed by the elastic-damage dynamic wave equation (10). As will be shown through subsequent numerical experiments, this has a direct impact on the apparent crack evolution when damage is propagating along a specific curve. In this work the energy minimization principle (11) will be numerically solved directly at the structural scale by a specific bound-constrained convex optimization algorithm to guarantee the irreversibility condition. The equivalent pointwise conditions of (11) and the energy balance condition (9) can be readily derived by evaluating the inequality \(\mathcal {A}'(\mathbf {u},\alpha )(\mathbf {0},\beta -\alpha )\ge 0\) and performing a temporal derivative of the total energy in (9). This yields a strong formulation in the form of the Kuhn–Tucker conditions which govern local damage evolution at a particular material point. Due to the presence of the damage gradient, the criterion is described by an elliptic type equation in space involving the Laplacian of the damage. Damage growth is not possible until a certain non-local threshold is reached and all energy dissipated in the body during such process until time t is recorded by the crack functional \(\mathcal {S}(\alpha _t)\). For a detailed derivation of the pointwise conditions, interested readers are referred to [6].
Numerical implementation
This section describes a numerical implementation of the above continuous two-field evolution problem. In practice it consists of solving numerically the elastic-damage dynamic wave equation (10) coupled with the total energy minimization (11). The irreversibility condition will be automatically enforced during the bound-constrained minimization process. The time-discrete model which we describe below should converge to the continuous one when the time increment becomes small, see [21]. In particular, the energy balance condition (9) will be hence automatically satisfied.
A spatial discretization is performed based on a mesh \(\Omega _h\) of the original domain \(\Omega \). The ideal properties of this mesh are indicated in [6] in order to achieve a better modeling of fracture via phase-field approaches. The displacement \(\mathbf {u}_t\) and the damage field \(\alpha _t\) will be discretized by linear isoparametric finite elements. For two-dimensional plane problems, an arbitrary element possesses at every node 3 nodal degrees of freedom corresponding to 2 components of the displacement and 1 scalar value of the damage. The symbols \(\underline{\mathbf {u}}\) and \(\underline{\varvec{\alpha }}\) are used to denote the current global displacement and damage nodal vectors. Inside a given element \(\Omega _\mathrm {e}\in \Omega _h\), their local nodal vectors \(\underline{\mathbf {u}}^\mathrm {e}\) and \(\underline{\varvec{\alpha }}^\mathrm {e}\) are used to perform an interpolation of the displacement and damage fields as well as their derivatives
$$\begin{aligned}&\mathbf {u}_t(\mathbf {x})=\mathbf {N}(\mathbf {x})\underline{\mathbf {u}}^\mathrm {e}\quad \text {and}\quad \varvec{\varepsilon }(\mathbf {u}_t)(\mathbf {x})=\mathbf {B}(\mathbf {x})\underline{\mathbf {u}}^\mathrm {e}\, , \\&\alpha _t(\mathbf {x})=\mathbf {N}_\alpha (\mathbf {x})\underline{\varvec{\alpha }}^\mathrm {e}\quad \text {and} \quad \nabla \alpha _t(\mathbf {x})=\mathbf {B}_\alpha (\mathbf {x})\underline{\varvec{\alpha }}^\mathrm {e}\end{aligned}$$
where \(\mathbf {N}\)’s and \(\mathbf {B}\)’s are respectively the interpolation and differentiation matrices.
After spatial discretization the elastic-damage dynamic wave equation (10) becomes
$$\begin{aligned} \mathbf {M}\ddot{\underline{\mathbf {u}}}=-\mathbf {F}_\mathrm {int}(\underline{\mathbf {u}},\underline{\varvec{\alpha }}) \end{aligned}$$
(12)
where \(\mathbf {M}\) refers to the classical consistent mass matrix which will be lumped using the traditional row-sum technique described for example in [22]. The internal force vector \(\mathbf {F}_\mathrm {int}\) is assembled from the elementary vectors given by
$$\begin{aligned} \mathbf {F}_\mathrm {int}^\mathrm {e}=\int _{\Omega _\mathrm {e}}\mathbf {B}^\mathsf {T}\varvec{\sigma }\bigl (\mathbf {B} \underline{\mathbf {u}}^\mathrm {e},\mathbf {N}_\alpha \underline{\varvec{\alpha }}^\mathrm {e}\bigr )\,\mathrm {d}\mathbf {x}=\int _{\Omega _\mathrm {e}}\mathbf {B}^\mathsf {T}\bigl (a(\mathbf {N}_\alpha \underline{\varvec{\alpha }}^\mathrm {e})\mathbf {A}\mathbf {B}\underline{\mathbf {u}}^\mathrm {e}\bigr )\,\mathrm {d}\mathbf {x}\end{aligned}$$
(13)
where \(\mathbf {A}\) is the Voigt representation of the elasticity tensor \(\mathsf {A}\).
We now turn to the spatially-discretized damage problem. It can be observed that the use of the damage constitutive law (4) leads to a total damageable energy \(\mathcal {E}+\mathcal {S}\) which is a second-order quadratic polynomial with respect to the damage vector \(\underline{\varvec{\alpha }}\). The energy minimization principle (11) involves thus the following functional
$$\begin{aligned} q_{\underline{\mathbf {u}}}(\underline{\varvec{\alpha }})=\frac{1}{2}\underline{\varvec{\alpha }}^\mathsf {T}\mathbf {H}(\underline{\mathbf {u}})\underline{\varvec{\alpha }}-\mathbf {b}(\underline{\mathbf {u}})^\mathsf {T}\underline{\varvec{\alpha }}. \end{aligned}$$
(14)
where the Hessian matrix \(\mathbf {H}\) and the second member vector \(\mathbf {b}\) can be assembled from the elementary matrix and vector given by
$$\begin{aligned} \mathbf {H}^\mathrm {e}= & {} \int _{\Omega _\mathrm {e}}\left( 2\psi _0(\mathbf {B}\underline{\mathbf {u}}^\mathrm {e}) \mathbf {N}_\alpha ^\mathsf {T}\mathbf {N}_\alpha +2w_1\ell ^2 \mathbf {B}_\alpha ^\mathsf {T}\mathbf {B}_\alpha \right) \,\mathrm {d}\mathbf {x}\,, \\ \mathbf {b}^\mathrm {e}= & {} \int _{\Omega _\mathrm {e}}\left( 2\psi _0\bigl (\mathbf {B} \underline{\mathbf {u}}^\mathrm {e}\bigr )-w_1\right) \mathbf {N}_\alpha \,\mathrm {d}\mathbf {x}\end{aligned}$$
with \(w_1=G_\mathrm {c}/(c_w\ell )\). They remain constant during the solving process of the damage problem, since they depend solely on the current displacement state \(\underline{\mathbf {u}}\).
We now consider an arbitrary discretization \((t^n)\) of the time interval of interest I where the superscript n denotes a quantity evaluated at the n-th time step. We will mainly focus on the time stepping procedures bringing the current known states \((\underline{\mathbf {u}}^n,\dot{\underline{\mathbf {u}}}^n,\ddot{\underline{\mathbf {u}}}^n,\underline{\varvec{\alpha }}^n)\) to the next time step \((\underline{\mathbf {u}}^{n+1},\dot{\underline{\mathbf {u}}}^{n+1},\ddot{\underline{\mathbf {u}}}^{n+1},\underline{\varvec{\alpha }}^{n+1})\). In the time-continuous model the elastic-damage dynamic wave equation (10) and the damage minimality condition (11) are coupled in the first-order stability principle (8). After temporal discretization \(\underline{\mathbf {u}}\) and \(\underline{\varvec{\alpha }}\) evaluated at the last time step \(t=t^n\) and the current time step \(t=t^{n+1}\) are in general involved in an implicit fashion. However, we observe that the energy minimization principle (11) for damage is not a genuine time evolution problem since time dependence is only introduced via the irreversibility condition. In the space-time discrete model at time \(t=t^{n+1}\), it reads
$$\begin{aligned} q_{\,\underline{\mathbf {u}}^{n+1}}(\underline{\varvec{\alpha }}^{n+1})\le q_{\,\underline{\mathbf {u}}^{n+1}}(\underline{\mathbf {\beta }}) \quad \text { for all} \, \underline{\mathbf {\beta }} \text { that}\quad 0\le \underline{\varvec{\alpha }}^n\le \underline{\mathbf {\beta }}\le 1 \end{aligned}$$
(15)
where the Hessian matrix and the second member vector in (14) are evaluated at \(\underline{\mathbf {u}}^{n+1}\). The Eq. (15) can be interpreted as a numerical minimization problem of the quadratic functional q under the irreversible constraint that the current sought damage state \(\underline{\varvec{\alpha }}^{n+1}\) is pointwise within the bound \([\underline{\varvec{\alpha }}^n,1]\)
$$\begin{aligned} \underline{\varvec{\alpha }}^{n+1}={\text {arg min}}q_{\,\underline{\mathbf {u}}^{n+1}}(\cdot )\text { subjected to the constraints }0\le \underline{\varvec{\alpha }}^n\le \underline{\varvec{\alpha }}^{n+1}\le 1. \end{aligned}$$
(16)
The next damage state \(\underline{\varvec{\alpha }}^{n+1}\) can thus be accurately calculated as long as the next deformation state \(\underline{\mathbf {u}}^{n+1}\) is known. The gradient projection conjugate gradient algorithm initially proposed in [23] is used to solve (16) in an iterative fashion. It is designed for quadratic bound-constrained minimization problems. Due to the bound constraint, approximate solutions \(\underline{\varvec{\beta }}\) to (16) can be defined using the projected gradient \([\mathbf {g}]\) of which the i-th component is given by
$$\begin{aligned} 0\overset{?}{\approx }[\mathbf {g}]_i= {\left\{ \begin{array}{ll} \partial _i q &{} \text {if}\quad \underline{\varvec{\beta }}_i\in (\underline{\varvec{\alpha }}_i^n,1)\,, \\ \min (\partial _i q, 0) &{} \text {if} \quad \underline{\varvec{\beta }}_i=\underline{\varvec{\alpha }}_i^n\,, \\ \max (\partial _i q, 0) &{} \text {if} \quad \underline{\varvec{\beta }}_i=1. \end{array}\right. } \end{aligned}$$
At each solving iteration, the method consists of several gradient projections to appoximately identify the active nodes, i.e. those either \(\underline{\varvec{\alpha }}_i^{n+1}=\underline{\varvec{\alpha }}_i^n\) or \(\underline{\varvec{\alpha }}_i^{n+1}=1\). Then it applies the preconditioned conjugate gradient method to minimize an unconstrained reduced problem of the free variables, i.e. those satisfying \(\underline{\varvec{\alpha }}_i^n<\underline{\varvec{\alpha }}_i^{n+1}<1\). The method proceeds to the next iteration until convergence. Interested readers are referred to [23] for a more detailed explanation of the algorithm. The GPCG method is implemented in the parallel linear algebra library PETSc [24]. We also use this library for manipulation of sparse matrices and vectors, similarly to the previous work of [5, 6].
It remains then to specify the temporal discretization method used for the \(\mathbf {u}\)-problem. In this work we adopt the classical Newmark-\(\beta \) integrator, which assumes the following time-stepping procedure
$$\begin{aligned} \dot{\underline{\mathbf {u}}}^{n+1}= & {} \dot{\underline{\mathbf {u}}}^n+\frac{\Delta t}{2}\left( \ddot{\underline{\mathbf {u}}}^n+\ddot{\underline{\mathbf {u}}}^{n+1}\right) \,, \end{aligned}$$
(17)
$$\begin{aligned} \underline{\mathbf {u}}^{n+1}= & {} \underline{\mathbf {u}}^n+\Delta t\dot{\underline{\mathbf {u}}}^n+\frac{1-2\beta }{2} \Delta t^2\ddot{\underline{\mathbf {u}}}^n+\beta \Delta t^2\ddot{\underline{\mathbf {u}}}^{n+1}. \end{aligned}$$
(18)
The explicit method \(\beta =0\) should be preferred mainly in terms of computational efficiency for applications where the loading speed or the crack propagation speed is comparable to the material speed of sound. The implicit method \(0<\beta \le \frac{1}{2}\) may be suitable for intermediate situations between a quasi-static and an explicit dynamic calculation. From (12), the determination of the the new acceleration \(\ddot{\underline{\mathbf {u}}}^{n+1}\) requires the knowledge of the new deformation state \(\underline{\mathbf {u}}^{n+1}\) which itself determines the new damage field at time \(t=t^{n+1}\) via (16). For the implicit Newmark method \(\beta \ne 0\), (18) can thus be regarded as a nonlinear equation in \(\underline{\mathbf {u}}^{n+1}\), where nonlinearity results from the irreversibility condition when minimizing the total energy (16). To decouple the \((\underline{\mathbf {u}}^{n+1},\underline{\varvec{\alpha }}^{n+1})\) problem, we use a staggered time-stepping procedure as used in [13, 14, 25] among others. The idea is to update the acceleration \(\ddot{\underline{\mathbf {u}}}^{n+1}\) while fixing the damage state at its previous known value \(\underline{\varvec{\alpha }}^n\). When a relatively small time-step is used, it is expected that the damage increment \(\underline{\varvec{\alpha }}^{n+1}-\underline{\varvec{\alpha }}^n\) is bounded and the staggered time-discrete problem will converge to the continuous one, cf. [21]. Introducing the displacement prediction at time \(t=t^{n+1}\)
$$\begin{aligned} \tilde{\underline{\mathbf {u}}}^{n+1}=\underline{\mathbf {u}}^n+\Delta t\dot{\underline{\mathbf {u}}}^n+\frac{1-2\beta }{2}\Delta t^2\ddot{\underline{\mathbf {u}}}^n, \end{aligned}$$
from (18) we obtain the linear system for \(\underline{\mathbf {u}}^{n+1}\)
$$\begin{aligned} \mathbf {M}\frac{\underline{\mathbf {u}}^{n+1}-\tilde{\underline{\mathbf {u}}}^{n+1}}{\Delta t^2}=-\beta \mathbf {F}_\mathrm {int}(\underline{\mathbf {u}}^{n+1},\underline{\varvec{\alpha }}^n)=-\beta \mathbf {K}(\underline{\varvec{\alpha }}^n)\underline{\mathbf {u}}^{n+1} \end{aligned}$$
(19)
where \(\mathbf {K}\) is the standard stiffness matrix corresponding to the previous damage state \(\underline{\varvec{\alpha }}^n\). The time-stepping procedure for the dynamic gradient damage model based on the implicit Newmark-\(\beta \) method in a prediction-correction form is summarized in Algorithm 1
Remark
After temporal discretization, the elastic-damage dynamic wave equation (10) and the damage minimality condition (11) can also be solved in a monolithic fashion as described for example in [8, 14]. Due to the irreversible condition contained in (16), the GPCG method for instance should be included in the monolithic solver to ensure that the damage variable is subject to a bound constraint during solving. Future work could be devoted to a detailed analysis of these schemes in terms of computational efficiency.
In the explicit case when \(\beta =0\), it turns out that the time evolution system in \((\underline{\mathbf {u}},\underline{\varvec{\alpha }})\) is automatically decoupled and the two subproblems separately in \(\underline{\mathbf {u}}^{n+1}\) and in \(\underline{\varvec{\alpha }}^{n+1}\) can be independently solved one from the other at every time step. Introducing the middle-step velocity
$$\begin{aligned} \dot{\underline{\mathbf {u}}}^{n+1/2}=\dot{\underline{\mathbf {u}}}^n+\frac{\Delta t}{2}\ddot{\underline{\mathbf {u}}}^n\,, \end{aligned}$$
the explicit time-stepping procedure for the discretized dynamic gradient damage model is summarized in Algorithm 2.
The initialization phase for the above implicit and explicit time-stepping procedure is described in Algorithm 3. We observe that the initial damage is recomputed \(\underline{\varvec{\alpha }}^{-1}\mapsto \underline{\varvec{\alpha }}^0\) in the step 2. The role of \(\underline{\varvec{\alpha }}^{-1}\) is to bring some a priori knowledge of the damage field resulting from a previous calculation or more frequently to represent an initial crack \(\underline{\varvec{\alpha }}^{-1}=1\) on \(\Gamma _0\). The initial step 2 thus renders it compatible with the initial displacement condition and the energy minimization structure.
Both the implicit and explicit time-stepping Algorithms 1 and 2 are implemented as a Python package named “FEniCS Dynamic Gradient Damage”. It is based on the FEniCS Project [26] for automated solution of partial differential equations.
Energy release rate in dynamic gradient damage models
The definition of an energy release rate in gradient damage models which competes with the fracture toughness \(G_\mathrm {c}\) can be found in [27] under quasi-static conditions. It is found that the damage evolution, when seen as a propagating crack band concentrated along a certain path, is governed by Griffith’s law in an asymptotic sense when the internal length \(\ell \) is small compared to any other structural length. A theoretic derivation of these similar concepts in dynamics is presented in [28]. Here we will summarize our findings and introduce some useful quantities that establish the link from gradient damage to fracture.
We assume that damage is contained in a thin band described by a parametrized curve \(\Gamma _t=\{\mathbf {x}\in \Omega |\alpha _t(\mathbf {x})=1\}\) representing the crack with its current tip \(\mathbf {P}_t\). Our definition of energy release rates in dynamics is based on shape derivative techniques, see for example [29] for an application of these methods in fracture mechanics. A virtual extension \(\varvec{\theta }_t\) of the crack tip in the current propagation direction is introduced. This function \(\varvec{\theta }_t\) should verify certain properties discussed in [29]. In particular, we have \(\varvec{\theta }_t(\mathbf {P}_t)=\varvec{\tau }_t\) where \(\varvec{\tau }_t\) refers to the current crack propagation direction. Moreover, It does not alter the crack lip shape, that is \(\varvec{\theta }_t\cdot \mathbf {n}=0\) on the crack lip \(\Gamma _t\) with \(\mathbf {n}\) the unit normal vector. A widely used definition of the virtual perturbation is recalled as follows. Suppose that the crack \(\Gamma _t\) lies on the x-axis and its current crack tip \(\mathbf {P}_t\) is propagating along the \(\mathbf {e}_1\) direction. The virtual perturbation \(\varvec{\theta }_t\) which introduces a fictive crack advance admits the form \(\varvec{\theta }_t=\theta _t\mathbf {e}_1\). The construction of the continuous scalar field \(0\le \theta _t\le 1\) parametrized by two radii \(r<R\) is given in Fig. 2.
We then perform a reparametrization of the damage field \(\alpha _t\) by the arc length \(l_t\) of the crack \(\Gamma _t\). When \(\ell \) is small by comparison with the dimension of the body, the first-order stability condition (8), the energy balance (9) as well as an essential singularity analysis lead to a Griffith-like evolution law governing the phase-field crack length evolution
$$\begin{aligned} \dot{l}_t\ge 0\,,\quad G^\alpha _t-\gamma _t\le 0\quad \text {and}\quad (G^\alpha _t-\gamma _t)\dot{l}_t=0. \end{aligned}$$
(20)
In (20), the conventional dynamic energy release rate \(G^\alpha _t\), which plays the role of “G” in Griffith’s law, is defined by
$$\begin{aligned} G_t^\alpha= & {} \int _\Omega \Bigl (\bigl (\kappa (\dot{\mathbf {u}}_t)-\psi \bigl (\varvec{\varepsilon }(\mathbf {u}_t), \alpha _t\bigr )\bigr )\textit{ div } \varvec{\theta }_t+\varvec{\sigma }_t\cdot (\nabla \mathbf {u}_t\nabla \varvec{\theta }_t)+\rho \ddot{\mathbf {u}}_t\cdot \nabla \mathbf {u}_t\varvec{\theta }_t\nonumber \\&+\rho \dot{\mathbf {u}}_t \cdot \nabla \dot{\mathbf {u}}_t\varvec{\theta }_t\Bigr )\,\mathrm {d}\mathbf {x}. \end{aligned}$$
(21)
It is formally very similar to the classical dynamic energy release rate in linear elastic fracture mechanics (LEFM) written as a volume integral with the help of the virtual perturbation \(\varvec{\theta }_t\), see for example [30]. Note however that in the gradient damage model the elastic energy density \(\psi \) and the stress tensor \(\varvec{\sigma }_t\) are modulated by the stiffness degradation function \(a(\alpha )\) and consequently stress singularity automatically disappears at the crack tip. Written in the form of (21), the conventional dynamic energy release rate involves an integral in the elements and hence is more convenient and accurate compared to a line integral (e.g. the J-integral) in a finite element calculation. The use of the traditional J-integral in the sense of [31] to calculate an effective energy release rate in a gradient damage modeling of fracture can be found in [17, 20] for instance.
The quantity \(\gamma _t\) in (20) which plays the role of “\(G_\mathrm {c}\)” in the classical Griffith’s law is the damage dissipation rate
$$\begin{aligned} \gamma _t=\int _\Omega \left( \varsigma (\alpha _t,\nabla \alpha _t)\textit{ div }\varvec{\theta }_t-2w_1 \ell ^2\nabla \alpha _t\cdot \nabla \varvec{\theta }_t\nabla \alpha _t\right) \,\mathrm {d}\mathbf {x}\,, \end{aligned}$$
(22)
where \(w_1=G_\mathrm {c}/(c_w\ell )\). Formally it is defined as the derivative of the dissipated energy (3) with respect to the crack length \(l_t\), thus quantifying the energy dissipated due to damage per crack advance.
Remark
The conventional dynamic energy release rate (21) and the damage dissipation rate (22) admit also a J-like path integral representation involving a generalized Eshelby tensor used frequently in configurational force approaches such as [32, 33] and references therein. During crack propagation, we have
$$\begin{aligned} \widehat{J}_t=\lim _{\epsilon \rightarrow 0}\int _{C_\epsilon }\widehat{\mathbf {J}}_t \mathbf {n}\cdot \varvec{\tau }_t\,\mathrm {d}\mathbf {s}=G_t^\alpha -\gamma _t\,, \end{aligned}$$
where \(\mathbf {n}\) denotes the normal pointing out of the ball of radius \(\epsilon \) centered at the crack tip \(B_\epsilon (\mathbf {P}_t)\) with \(C_\epsilon =\partial B_\epsilon (\mathbf {P}_t)\) its boundary. The generalized dynamic \(\widehat{\mathbf {J}}_t\) tensor is defined by
$$\begin{aligned} \widehat{\mathbf {J}}_t=\Bigl (\psi \bigl (\varvec{\varepsilon }(\mathbf {u}_t),\alpha _t\bigr )+ \kappa (\dot{\mathbf {u}}_t)+\varsigma (\alpha _t,\nabla \alpha _t)\Bigr )\mathbb {I}- \nabla \mathbf {u}_t^\mathsf {T}\varvec{\sigma }_t-2w_1\ell ^2\nabla \alpha _t\otimes \nabla \alpha _t\,, \end{aligned}$$
where \(w_1=G_\mathrm {c}/(c_w\ell )\). Interested readers are referred to [28] for a detailed discussion on this point.
With the help of a two-scale approach, the inner damage problem near the crack tip and the outer LEFM problem far from the crack band can be separated [27]. It can be shown that if the inner radius r of the virtual perturbation defined in Fig. 2 is sufficiently big with respect to the internal length \(\ell \), \(G_t^\alpha \) defines an equivalent dynamic energy release rate \(G_t\) corresponding to the outer mechanical fields. Similarly, the damage dissipation rate \(\gamma _t\) will converge to the fracture toughness \(G_\mathrm {c}\) defined in the crack functional (3). The following asymptotic Griffith’s law is obtained when the outer fields are considered as r increases
$$\begin{aligned} \dot{l}_t\ge 0\,,\quad G_t-G_\mathrm {c}\le 0\quad \text {and}\quad (G_t-G_\mathrm {c})\dot{l}_t=0. \end{aligned}$$
(23)