Numerical investigation of dynamic brittle fracture via gradient damage models
 Tianyi Li^{1}Email authorView ORCID ID profile,
 JeanJacques Marigo^{2},
 Daniel Guilbaud^{1, 3} and
 Serguei Potapov^{1, 4}
DOI: 10.1186/s403230160080x
© The Author(s) 2016
Received: 11 May 2016
Accepted: 12 August 2016
Published: 30 August 2016
Abstract
Background:
Gradient damage models can be acknowledged as a unified framework of dynamic brittle fracture. As a phasefield approach to fracture, they are gaining popularity over the last few years in the computational mechanics community. This paper concentrates on a better understanding of these models. We will highlight their properties during the initiation and propagation phases of defect evolution.
Methods:
The variational ingredients of the dynamic gradient damage model are recalled. Temporal discretization based on the Newmark\(\beta \) scheme is performed. Several energy release rates in gradient damage models are introduced to bridge the link from damage to fracture.
Results and discussion:
An antiplane tearing numerical experiment is considered. It is found that the phasefield crack tip is governed by the asymptotic Griffith’s law. In the absence of unstable crack propagation, the dynamic gradient damage model converges to the quasistatic one. The defect evolution is in quantitative accordance with the linear elastic fracture mechanics predictions.
Conclusion:
These numerical experiments provide a justification of the dynamic gradient damage model along with its current implementation, when it is used as a phasefield model for complex realworld dynamic fracture problems.
Keywords
Dynamic brittle fracture Gradient damage models Griffith’s theory Quasistatic limitsBackground
The internal length \(\ell \) induced naturally through dimensional analysis within the gradient damage model [1] admits several interpretations. From a geometric point of view, it controls the width of the damage band such that a sharp description of cracks can be retrieved in the limit \(\ell \rightarrow 0\). Meanwhile, it turns out that in such process the total energy in the gradient damage model actually converges in a certain sense to the Griffith functional defined in the variational approach to fracture [2]. The gradient damage model can thus be regarded as an elliptic regularization of the previous sharpinterface variational fracture model and the internal length \(\ell \) serves as a purely numerical parameter which should be as small as possible. This is the interpretation undertaken among others by [3–5] where the gradient damage model is applied to drying, thin films debonding, and other fracture mechanics problems.
On the other hand, the gradient damage model can also be acknowledged as a genuine model per se of brittle fracture, where the internal length \(\ell \) is then interpreted as a material parameter which contributes to the fracture or damage behavior of materials. This interpretation presents several advantages from a physical point of view. First of all, this additional length parameter could be related to the maximal stress that the material can sustain and hence introduces additional experimentally validated size effects which are not present in the Griffith model of fracture mechanics, see the work of [6–9] among others. Secondly, the tensioncompression asymmetry phenomenon as observed for brittle materials can be easily formulated directly in the gradient damage model. The resulting sharp interface fracture model as \(\ell \rightarrow 0\) remains unclear and inversely the elliptic regularization of the variational approach to fracture that actually accounts for unilateral contact between crack lips is still considered as a difficult task both from the physical and mathematical point of view, see [10]. Nevertheless, these tensioncompression asymmetry formulations as summarized for instance in [11] constitute an improvement of the original gradient damage model [6] and can be regarded as an approximation of the actual noninterpenetration condition.
In this paper we will adhere to the second viewpoint and interpret the gradient damage model as a phasefield approach to dynamic brittle fracture. The formulation of dynamic gradient damage models that extends the original quasistatic ones [1] is sketched in [12]. The governing equations derived from the variational principles resemble those of other phasefield models originated from the computational mechanics community [8, 13–15], with a particular choice of damage constitutive law. These models settle down a unified and coherent numerical framework covering the onset and the spacetime propagation of cracks with possible complex topologies and have been successfully applied to study various realworld dynamic fracture problems. Meanwhile, in the quasistatic setting, further physical insights into the gradient damage model seen as a phasefield model of fracture are given among others in [16, 17] where a comparison with the cohesive zone model and the Griffith’s linear elastic theory is conducted. In dynamics, more welldesigned numerical experiments should be performed to carry out such verification, see [18] for instance for an investigation of the phasefield crack speeds for plane problems. The main objective of this contribution is thus to provide a better understanding of the dynamic gradient damage model. We will highlight the properties of gradient damage models as a phasefield approach to fracture and mainly focus on the initiation and propagation phases of defect evolution. For that, an antiplane tearing numerical experiment is considered and an existing crack may initiate and then propagate along a predefined path in a modeIII condition. A comparison with the classical Griffith’s theory will be conducted in the cases when the loading speed is of the same order, or small, with respect to the wave speed.
Specifically this paper is organized as follows. The variational formulation of the dynamic gradient damage model is firstly recalled. An opensource numerical implementation is proposed to solve the elasticdamage dynamic problem. Some theoretic concepts are introduced to bridge the link between gradient damage and fracture. We then present and discuss the simulation results corresponding to crack propagation in an antiplane tearing situation. In the last section we will summarize the findings and indicate possible further research directions.
General notation conventions adopted in this paper are summarized here. Scalarvalued quantities will be denoted by italic Roman or Greek letters like the damage field \(\alpha _t\). Vectors, secondorder tensors and their matrix representations will be represented by boldface letters such as the displacement field \(\mathbf {u}_t\) and the stress tensor \(\varvec{\sigma }_t\). Higher order tensors considered as linear operators will be indicated by sansserif letters: the elasticity tensor \(\mathsf {A}\) for instance. Intrinsic notation is adopted and contraction on lowerorder tensors will be written without dots \(\mathsf {A}\varvec{\varepsilon }_t=\mathsf {A}_{ijkl}\varvec{\varepsilon }_{kl}\) (the summation convention is assumed). Inner products between two vectors or tensors of the same order will be denoted with a dot, such as \(\mathsf {A}\varvec{\varepsilon }_t\cdot \varvec{\varepsilon }_t=\mathsf {A}_{ijkl}\varvec{\varepsilon }_{kl}\varvec{\varepsilon }_{ij}\) (the summation convention is assumed). Time dependence will be indicated at the subscripts of the involved quantities, like \(\mathbf {u}:(t,\mathbf {x})\mapsto \mathbf {u}_t(\mathbf {x})\).
Methods
Variational framework
 1
Irreversibility the damage \(t\mapsto \alpha _t\) is a nondecreasing function of time.
 2Firstorder stability the firstorder action variation is nonnegative 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)
 3Energy balance the only energy dissipation is due to damagewhere the total energy is defined by$$\begin{aligned} \mathcal {H}_t=\mathcal {H}_0+\int _0^t\Biggl (\int _\Omega \bigl (\varvec{\sigma }_s\cdot \varvec{\varepsilon }(\dot{\mathbf {U}}_s)+\rho \ddot{\mathbf {u}}_s\cdot \dot{\mathbf {U}}_s\bigr )\,\mathrm {d}\mathbf {x}\Biggr )\,\mathrm {d}s \end{aligned}$$(9)$$\begin{aligned} \mathcal {H}_t=\mathcal {E}(\mathbf {u}_t,\alpha _t)+\mathcal {S}(\alpha _t)+\mathcal {K} (\dot{\mathbf {u}}_t). \end{aligned}$$
Remark
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 firstorder stability condition (8) which ensures that energy could only be dissipated through damage (or phasefield like fracture).
Numerical implementation
This section describes a numerical implementation of the above continuous twofield evolution problem. In practice it consists of solving numerically the elasticdamage dynamic wave equation (10) coupled with the total energy minimization (11). The irreversibility condition will be automatically enforced during the boundconstrained minimization process. The timediscrete 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.
Remark
After temporal discretization, the elasticdamage 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.
Both the implicit and explicit timestepping 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 quasistatic 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.
Remark
Results and discussion
In this section we will present and discuss a particular numerical experiment tailored to highlight the properties of the dynamic gradient damage model while focusing on the initiation and propagation phases of defect evolution. Specifically, we will investigate the fracture mechanics criterion for an existing phasefield crack to initiate, and then to propagate along a certain path.
Remark
If the original model is used, i.e. when the degradation function also acts on \(\frac{\partial u_t}{\partial x_1}\), numerically it is observed that for low propagation speeds crack curving (including kinking and branching) does not take place and the modification (24) produces the same response as the original model. However for higher propagation speeds (for example due to a larger loading velocity k), crack curving is observed (see for example [25]) and these two models no longer predict the same crack evolution. Crack path prediction is exactly the raison d’être of phasefield models of fracture. A thorough investigation of crack kinking/branching phenomena (as a function of crack speed for instance) is a very important task to which future work will be devoted. Nevertheless, the current contribution focuses on the behavior of gradient damage models when these dynamic instabilities (kinking, branching) are somehow suppressed (see for example [34] for an experimental investigation on this point), which permits a direct comparison with the classical Griffith’s theory of dynamic fracture.
 1
In the first case, the fracture toughness \(G_\mathrm {c}\) is assumed to be homogeneous throughout the domain. The loading speed is of the same order of the material speed of sound \(c=\sqrt{\mu /\rho }\) and we will use the explicit Newmark timestepping method.
 2
In the second case, \(G_\mathrm {c}\) may admit a spatial discontinuity in the propagation direction. We also prescribe a relatively small loading speed in order to investigate the quasistatic limit of the dynamic model. Depending on whether the crack propagation speed itself is smaller with respect to the speed of sound or not (the term unstable propagation often refers to this case), the implicit or the explicit Newmark method will be used.
Geometric, material and numerical parameters for the antiplane tearing experiment
\(\varvec{L}\)  \(\varvec{H}\)  \(\varvec{l}_{\mathbf {0}}\)  \(\varvec{\mu }\)  \(\varvec{\rho }\)  \(\varvec{G}_{\mathbf {c}}\)  \(\varvec{\ell }\)  \(\varvec{h}\)  \(\varvec{\Delta t}\) 

5  1  1  0.2  1  0.01  0.05  0.01  \(\Delta t_\mathrm {CFL}\) 
Homogeneous fracture toughness case
Despite the transverse wave reflection present in the twodimensional 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 timestepping Algorithm 2 as well as its implementation work fine even at supersonic loading speeds.
In LEFM, it is known that the Jintegral 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 damageinduced 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 \(G^\alpha _t=\gamma _t\rightarrow (G_\mathrm {c})_\mathrm {eff}\).
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 cracktip 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 stress distribution along a vertical slice \(\{(x,y)\in \mathbb {R}^2x=l_t\}\) 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 wellknown inverse square root singularity for the two stress components \(\sigma _{13}\) and \(\sigma _{23}\) and their neartip 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
 1
Irreversibility the damage \(t\mapsto \alpha _t\) is a nondecreasing function of time.
 2Firstorder stability the firstorder variation of the potential energy is nonnegative with respect to arbitrary admissible displacement and damage fieldswhere in the absence of external forces the potential energy is given by$$\begin{aligned} \mathcal {P}'(\mathbf {u}_t,\alpha _t)(\mathbf {v}_t\mathbf {u}_t,\beta _t\alpha _t)\ge 0\quad \text { for all} \quad \mathbf {v}_t\in \mathcal {C}_t \text { and all}\quad \beta _t\in \mathrm {D}(\alpha _t). \end{aligned}$$(28)$$\begin{aligned} \mathcal {P}(\mathbf {u}_t,\alpha _t)=\mathcal {E}(\mathbf {u}_t,\alpha _t)+\mathcal {S}(\alpha _t) \end{aligned}$$
 3Energy balance the only energy dissipation is due to damage$$\begin{aligned} \mathcal {P}_t=\mathcal {P}_0+\int _0^t\left( \int _\Omega \varvec{\sigma }_s\cdot \varvec{\varepsilon }(\dot{\mathbf {U}}_s)\,\mathrm {d}\mathbf {x}\right) \,\mathrm {d}s. \end{aligned}$$(29)
Comparison of the numerically computed quasistatic crack speed in the homogeneous case with the theoretic one \(\sqrt{\mu H/G_\mathrm {c}}\) given in [2]
Numerical  Theoretic  Error  

Quasistatic 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 quasistatic 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 \(K_2\).
Comparison of the numerical crack lengths after the jump with the theoretic predictions
Quasistatic  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 firstorder quasistatic 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 halfband \(D=2\ell =0.1\) 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 firstorder quasistatic 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 (quasistatic) 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.
In the quasistatic 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) quasistatic 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 quasistatic model is solely based on the firstorder stability condition (28). For this particular problem based on quasistatic energy conservation we could predict a correct quasistatic 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 energyconserving evolution which also respects the stability criterion at every time. Moreover even equipped with the energy balance condition, the quasistatic 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 quasistatic 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.
Due to transverse wave reflection in this 2d 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.
Conclusion

In the dynamic tearing example of a homogeneous plate, it is verified that the crack evolution is governed by the asymptotic Griffith’s law (23), as long as the material internal length is sufficiently small to establish a separation of scales between the inner damage problem and the outer LEFM problem. The conventional dynamic energy release rate is numerically computed and verified as a tool to translate gradient damage mechanics results in fracture mechanics terminology. We conduct a comparison with the 1d peeling problem [35] analytically studied with the classical Griffith’s theory of dynamic fracture. A good agreement between them can be found in terms of the crack speeds prediction as a function of the loading speed.

We then investigate the quasistatic limits of the dynamic gradient damage model. In the absence of brutal or unstable crack propagation where the classical static Griffith’s theory fails, the dynamic model converges to the firstorder quasistatic gradient damage model, when the loading speed decreases. However, when the crack may propagate at a speed comparable to the material speed of sound, the dynamic model should be preferred to correctly account for inertial effects. The crack evolution in the dynamic gradient damage model is in quantitative accordance with the LEFM predictions on the 1d peeling problem.
Abbreviations
 LEFM:

linear elastic fracture mechanics
 GPCG:

gradient projection conjugate gradient algorithm
Declarations
Author's contributions
TL and JJM have developed the theory part. TL has carried out the numerical implementation and has drafted the manuscript. JJM, DG and SP have supervised the different studies and the corrections of the draft. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets supporting the conclusions of this article are available in the FEniCS Dynamic Gradient Damage repository, https://bitbucket.org/litianyi/dynamicgradientdamage.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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