Numerical investigation of dynamic brittle fracture via gradient damage models

Variational framework

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.

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

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.

Homogeneous fracture toughness case

In the first case a homogeneous plate will be considered. This antiplane tearing example is physically similar to the 1-d film peeling problem which can be studied using the classical Griffith’s theory of dynamic fracture. According to [35] and [2], the crack speed, with respect to the loading displacement \(U=kt\) or to the physical time t, as a function of the loading velocity k is given by

$$\begin{aligned} \frac{\mathrm {d} l}{\mathrm {d} U}(k)=\sqrt{\frac{\mu H}{G_\mathrm {c}+\rho Hk^2}}\quad \text {or}\quad \frac{\mathrm {d} l}{\mathrm {d} t}(k)=\sqrt{\frac{\mu Hk^2}{G_\mathrm {c}+\rho Hk^2}} \end{aligned}$$

(27)

from which we retrieve the quasi-static limit \(\mathrm {d} l/\mathrm {d} U(0)=\sqrt{\mu H/G_\mathrm {c}}\) predicted in [2] and the dynamic limit as the shear wave speed \(\mathrm {d} l/\mathrm {d} t(\infty )=\sqrt{\mu /\rho }\), which is a classical result in the Griffith’s theory of dynamic fracture [36]. We also observe that for low loading speeds \(k\approx 0\), the dynamic crack speed \(\mathrm {d} l/\mathrm {d} t\approx k\sqrt{\mu H/G_\mathrm {c}}\) scales linearly in k, which agrees with the remarks given in [25]. Comparisons between the numerical results using the dynamic gradient model and this theoretic result (27) with \(G_\mathrm {c}\) replaced by \((G_\mathrm {c})_\mathrm {eff}\) are illustrated in Fig. 5.

Despite the transverse wave reflection present in the two-dimensional numerical model, a very good quantitative agreement is found between them. In particular, as it is also observed in [25], the numerically obtained crack speed indeed approaches the limiting shear wave speed when the loading speed increases. The explicit time-stepping Algorithm 2 as well as its implementation work fine even at supersonic loading speeds.

The conventional dynamic energy release rate (21) is numerically computed and the validity of the asymptotic Griffith’s law is analyzed by varying the inner radius r of virtual perturbations defined in Fig. 2. During the propagation phase \(\dot{l}_t>0\), three arbitrary time instants are taken when the crack length attains respectively \(l_t\approx 1.6\), \(l_t\approx 2\) and \(l_t\approx 2.4\). An evident r-dependence of \(G^\alpha _t\) is illustrated in Fig. 6, where the ratio \(R/r=\frac{5}{2}\) is fixed.

In LEFM, it is known that the J-integral is directly related to the dynamic stress intensity factors at the crack tip, see [36]. In gradient damage models however, there is no more stress singularities. When r is small, we go directly into the process zone dominated by damage-induced strain softening and \(G^\alpha _t\rightarrow 0\) is expected as \(r\rightarrow 0\). However, as r increases, \(G^\alpha _t\) captures well the outer mechanical fields. An equivalent energy release rate can thus be defined, and according to the asymptotic behaviors of \(\gamma _t\), we have the desired result \(G^\alpha _t=\gamma _t\rightarrow (G_\mathrm {c})_\mathrm {eff}\).

We will then turn to the evolution of the conventional dynamic energy release rate when the existing crack initiates and further propagates. From the above r-dependence analysis, a fixed inner radius \(r=2\ell \) is used which should already correctly capture the far mechanical fields. The crack length \(l_t\) given by (26) as well as the calculated \(G^\alpha _t\) are given as a function of the loading displacement in Fig. 7, where three separate calculations corresponding to three loading speeds k are reported.

Recall that an initial crack of length 1 is present in the body and we observe \(G^\alpha _t=0\) before the waves arrive at the initial crack tip. When the energy release rate \(G^\alpha _t\) at the initial crack tip attains the fracture toughness \((G_\mathrm {c})_\mathrm {eff}\), the existing crack initiates and then propagates with the equality \(G^\alpha _t=(G_\mathrm {c})_\mathrm {eff}\) if the spatial and temporal numerical discretization errors are ignored. Indeed this equality is not enforced algorithmically during the solving of the \((\mathbf {u},\alpha )\) evolution which is instead determined by Algorithm 2. We may conclude that the crack-tip evolution (initiation and propagation) is well governed by the asymptotic Griffith’s law (23) in the dynamic gradient damage model, when outer fields are considered.

The internal length \(\ell \) plays a rather subtle role during the propagation phase. The crack tip is governed by the asymptotic Griffith’s law (23) if and only if a separation of scales between the inner damage problem and the outer LEFM is possible, i.e. only when the internal length is sufficiently small compared to any other structural length [27]. Although \(\ell \) is indeed hidden in (23), the validity of the latter depends directly on it. Below we present the simulation results with a fixed loading speed \(k=0.2\) and three small enough internal lengths. As can be seen from Fig. 8, the crack evolution is globally conforming with Griffith’s law, as long as the involved quantities are calculated with a virtual perturbation \(\varvec{\theta }_t\) capturing correctly the far fields. Here according to Fig. 6, we use an inner radius adapted with the internal length \(r=2\ell \), which should produce an error less than \(3\,\%\).

The stress distribution along a vertical slice \(\{(x,y)\in \mathbb {R}^2|x=l_t\}\) passing by the current crack tip \(\mathbf {P}_t\) should illustrate and highlight the separation of scales when \(\ell \) is small. For the sake of simplicity, we consider a stationary crack \([0,2]\times \{0\}\) and solve the static problem with the gradient damage model and the LEFM model (linear elastic body with a sharp crack embedded in the domain). We can verify from Fig. 9 that the LEFM develops a well-known inverse square root singularity for the two stress components \(\sigma _{13}\) and \(\sigma _{23}\) and their near-tip fields are well approximated by the theoretic asymptotic solutions.

On the other hand, the gradient damage model provides a better modeling of the stress field near the crack tip as their values are bounded. A good matching can be observed far from the crack tip and the discrepancy with the outer LEFM model is concentrated within a process zone proportionally dependent on the internal length. When \(\ell \) is very large, the process zone could cover the whole structural domain and a separation of scales is no longer possible. In this case the asymptotic Griffith’s law (23) is not applicable since we are no longer dealing with a fracture mechanics problem.

Discontinuous fracture toughness cases

The homogeneous antiplane tearing problem is firstly solved by the dynamic gradient damage model and the above first-order quasi-static gradient damage model. In the dynamic calculation a small loading speed \(k=0.001\approx 0.2\,\% c\) is assumed and we use the unconditionally stable implicit Newmark scheme as described in Algorithm 1, with \(\beta =\frac{1}{4}\). The time step is set to \(\Delta t=10\Delta t_\mathrm {CFL}\), where \(t_\mathrm {CFL}\) is defined in (25). In Fig. 10 we plot the crack length evolution as well as the conventional energy release rate \(G^\alpha _t\) both for the dynamic model and the first-order quasi-static model. It is recalled that the static \(G^\alpha _t\) can be simply obtained by setting \(\dot{\mathbf {u}}_t\) and \(\ddot{\mathbf {u}}_t\) to zero in (21). We observe that these two solutions coincide, and both present a time-continuous crack evolution (initiation and propagation) conforming to the asymptotic Griffith’s law (23). The numerically computed quasi-static crack speed (with respect to \(U=kt\)) is compared in Table 2 to the analytical value \(\sqrt{\mu H/G_\mathrm {c}}\) announced in [2]. A very good agreement can be found if the numerically amplified fracture toughness \((G_\mathrm {c})_\mathrm {eff}\) is used in the formula.

	Numerical	Theoretic	Error
Quasi-static crack speed	4.326	4.391	1.5 %

We then turn to the case where the fracture toughness jumps suddenly from a lower value \(K_1=0.01\) to a higher one \(K_2=2K_1=0.02\) at \(x=2\). The unconditionally stable implicit Newmark scheme with \(\beta =\frac{1}{4}\) is used again with a time increment \(\Delta t=10\Delta t_\mathrm {CFL}\). As can be observed from Fig. 11 the convergence of the dynamic model toward the quasi-static one is verified and the crack initiates and propagates following Griffith’s law. A temporary arrest phase is present shortly after the crack reaches the interface at \(x=2\). Due to continuous loading the energy release rate increases and the crack then restarts and begins to propagate in the second material when the energy release rate \(G^\alpha _t\) attains the higher fracture toughness \(K_2\).

However, for the case where the fracture toughness \(K_1=2K_2=0.02\) suddenly drops to a smaller value \(K_2=0.01\) at \(x=1\) (exceptionally here the initial crack length is \(\frac{1}{4}\)), a relatively good matching can only be found before and after the jump phase produced at the discontinuity, both in terms of the crack length evolution and the energy release rate. It is exactly at the jump phase that these two models strongly disagree, cf. Fig. 12.

	Quasi-static	Dynamic
Numerical	1.465	1.995
Theoretic	\(\sqrt{2}\)	2
Error	3.6 %	0.25 %

Here due to the unstable crack propagation during the jump, the explicit Newmark scheme is used for the dynamic calculation with \(\Delta t=\Delta t_\mathrm {CFL}\). When the crack arrives at the discontinuity, the first-order quasi-static numerical model underestimates the crack jump and predicts no further crack arrest, by relating directly the static energy release rate \(G^\alpha _t\) to the fracture toughness \(K_2\) just after the jump. For the dynamic model, the jump length is bigger and a subsequent temporary crack arrest is observed, as the dynamic energy release rate oscillates with a high frequency but remains smaller than the fracture toughness \(K_2\) after the jump. We observe that in both cases the jump takes place at \(x\approx 0.9\) somewhat prior to the fracture toughness discontinuity \(x=1\). We suspect that this is due to the damage regularization of cracks with a half-band \(D=2\ell =0.1\) using the constitutive laws of (4). If this effect is ignored, the crack length after the jump is recorded in Table 3 for each case. From the static energy release rate evolution, we see that the crack length \(l_\mathrm {m}\) after the jump predicted in the first-order quasi-static numerical model is governed by \(G(l_\mathrm {m})=G_\mathrm {c}(l_\mathrm {m})\) from which authors of [35] find \(l_\mathrm {m}=\sqrt{K_1/K_2}=\sqrt{2}\). However their dynamic analysis shows that the crack length after the jump \(l_\mathrm {c}\) should instead be given by the total (quasi-static) energy conservation principle \(\mathcal {P}(1)=\mathcal {P}(l_\mathrm {c})\), which results in \(l_\mathrm {c}=K_1/K_2=2\). We see from Table 3 that our dynamic gradient damage model indeed reproduces this correct value.

To better analyze the jump phase, energy evolutions are investigated against the crack length in Fig. 13.

In the quasi-static case we pick the total energy \(\mathcal {P}=\mathcal {E}+\mathcal {S}\) while in the dynamic case we plot separately the static energy \(\mathcal {P}=\mathcal {E}+\mathcal {S}\) and the kinetic one \(\mathcal {K}\). Before and sufficiently after the jump a good agreement between these two potential energies can be found. We observe that the (incorrect) quasi-static jump i.e., an unstable or brutal crack propagation) is accompanied by a slight loss of the total energy \(\Delta \mathcal {P}_\mathrm {stat.}\), contradicting the balance condition (29). This phenomenon has already been observed by several authors such as [2, 6, 10, 25]. On the one hand, it can be regarded as a numerical issue as the effective implementation of the quasi-static model is solely based on the first-order stability condition (28). For this particular problem based on quasi-static energy conservation we could predict a correct quasi-static crack evolution toward which the dynamic solution converges when the loading speed becomes small, see [35]. On the other hand, from a theoretic point of view, it is already known in [38] that there may not exist an energy-conserving evolution which also respects the stability criterion at every time. Moreover even equipped with the energy balance condition, the quasi-static model may still differ from the dynamic analysis [39]. A natural and physical remedy for all general unstable crack propagation cases is to introduce inertial effects. In Fig. 13 the dynamic jump process is continuous (the crack propagates at a finite speed bounded by the shear wave speed) compared to the quasi-static one where the jump occurs necessarily in a discontinuous fashion between two iterations. We verify the conclusions drawn in [35] that the kinetic energy \(\mathcal {K}\) plays only a transient role in this problem, as it attains a finite value during the jump and becomes again negligible after. The dynamic potential energy \(\mathcal {P}=\mathcal {E}+\mathcal {S}\) after the jump is slightly bigger that its value before the jump, due to the fact that the loading speed \(k=0.001\) is small but not zero.

During the jump, the crack propagates at a speed comparable to the material speed of sound which, according to [35], is given by

$$\begin{aligned} v_\mathrm {jump}=\frac{\left( \sqrt{\widehat{K}_1+\epsilon ^2}+\epsilon \right) ^2- \widehat{K}_2}{\left( \sqrt{\widehat{K}_1+\epsilon ^2}+\epsilon \right) ^2+ \widehat{K}_2}\cdot c \end{aligned}$$

(30)

with the non-dimensional fracture toughness \(\widehat{K}_i=K_i/(2\mu H)\) and the normalized loading speed \(\epsilon =k/c\). The crack length evolution during the jump is illustrated in Fig. 14.

Due to transverse wave reflection in this 2-d problem, the crack propagates during this interval with a small fluctuation of period T approximately corresponding to the first standing wave between the boundary and the crack \(T\approx 2H/c\approx 4.5\). That is why we calculate from Fig. 14 only the initial crack speed at jump for comparison in Table 4. A good agreement can be found between the numerical and the theoretic ones.