d
Laboratoire de M´ecanique des Solides, Ecole Polytechnique, 91128 Palaiseau, France
Abstract
In its numerical implementation, the variational approach to **brittle** **fracture** approximates the crack evolution in an elastic solid through the use of gradient damage models. In the present paper, we first formulate the quasi-static evolution problem for a general class of such dam- age models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress-displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant mate- rial parameters, such as limit stress and internal length. Finally, the one-dimensional analytical results are compared with the numerical solution of the evolution problem in a two dimensional setting.

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interfacial damage variable strain Lamé coefficients Pa adhesion history function J m2 brittle fracture history function J m3 porosity stress Pa damage variable grain size m crack energy[r]

has already been opened. Meshless methods for **brittle** **fracture** have also been introduced by [PKA + 05].
2.3 Real-Time **Fracture** Simulation
The first physically-based real-time system for frac- ture has been proposed in [MMDJ01], where bodies composed of a few number of elements breaking in real-time. Later, M ¨uller et. al. presented another frame- work for real time **fracture** that leverages the com- putational gains of corotational formulation of FEM techniques [MG04]. The authors demonstrated their techniques for ductile **fracture**. However, using their method for **brittle** **fracture** would imply to use smaller time steps, that would compromise the performances. Recently, [PO09] proposed a complete solution target- ing game industry, in which the graphical and physi- cal meshes are dissociated through the use of so-called splinters. When **fracture** surfaces are defined along the mesh boundaries or when remeshing techniques are used, there is no need for a fragment generation step (the fragments are already defined by subsets of ele- ments of the initial mesh, or cut elements). However, it is also possible to define fragments as subsets of the nodes of the initial body [MBF05], and to generate the geometric surface of **fracture** separately.

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1. Introduction
Despite its many successes, Griffith’s theory of **brittle** **fracture** [ 51 ] and its heir, Linear Elastic **Fracture** Mechanics (LEFM), still faces many challenges. In order to identify crack path, additional branching criteria whose choice are still unsettled have to be considered. Accounting for scale effects in LEFM is also challenging, as illustrated by the following example: Consider a reference structure of unit size rescaled by a factor L. The critical loading at the onset of **fracture** scales then as 1/ √ L, leading to a infinite nucleation load as the structure size approaches 0, which is inconsistent with experimental observation for small structures [ 10 , 55 , 30 ].

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The fundamental link between gradient damage models and Griffith model of **fracture** me- chanics relies strongly on the variational structure of both models, hence justifying the necessity to work with standard damage models. Indeed, gradient damage models appear as an ellip- tic approximation of the variational **fracture** mechanics problem. The variational approach of **brittle** **fracture** recast the evolution problem for the cracked state of a body as a minimality principle for an energy functional sum of the elastic energy and the energy dissipated to create the crack [ 19 ]. On the basis of the results of the mathematical theory of image segmentation and free-discontinuity problems [ 34 , 3 ], [ 12 ] approximate the minima of this energy functional through the minimization of a regularized elliptic functional that may be mechanically inter- preted as the energy of a gradient damage model with an internal length. Mathematical results based on Gamma-Convergence theory show that when the internal length of a large class of gradient damage models tends to zero, the global minima of the damage energy functional tend towards the global minima of the energy functional of Griffith **brittle** **fracture** [ 14 ]. The same is true for the corresponding quasi-static evolutions ruled by a global minimality principle [ 22 ]. Similar variational approximations of **brittle** **fracture** have emerged in the community of physi- cists adopting Ginzburg-Landau theories to study phase transitions [ 25 , 23 ], producing relevant results [ 43 ]. Nowadays the so-called phase-field models of **fracture** are extensively adopted in computational mechanics to study **fracture** phenomena [ 27 , 20 , 44 , 26 , 33 , 10 , 28 ].

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Mots clés — Dynamic **fracture**, Gradient damage models, Variational principles, Finite element imple- mentation
1 Introduction
Gradient damage models as formulated in a pure variational setting [1] provide, through strain and dam- age localization in narrow bands representing a regularized description of cracks, a complete and unified framework of **brittle** **fracture** including the onset and the space-time quasi-static propagation of cracks with possible complex topologies, see [2, 3] and references therein. The presence of the damage gradi- ent confirms the non-local nature of the model and induces naturally by dimensional analysis a material internal length. From the damage mechanics point of view, local damage models are mathematically ill-posed where damage localization is possible without any additional energy dissipation resulting in a spurious mesh dependence of the FEM results [4, 5]. The introduction of the damage gradient can thus be seen as a regularization of the classical continuum damage models to overcome this difficulty although other techniques are also available [6]. The link between damage and **fracture** can be estab- lished on one hand through Γ-convergence theories in terms of global minima of the total energy as long as this internal length is small before the size of the structure [7]. On the other hand, it is shown in [8] using matched asymptotic analysis, that the damage evolution ruled a priori by three physical principles of irreversibility, local directional stability and energy balance satisfies apparently the classical Griffith criterion through the definition of a fictitious energy release rate G of the outer problem and a material toughness G c proportional to the local damage dissipation and the internal length. This gradient damage

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Received: 29 September 2008 / Accepted: 19 February 2010 © Springer Science+Business Media, LLC 2010
Abstract We revisit in a 2d setting the notion of energy release rate, which plays a
pivotal role in **brittle** **fracture**. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an applica- tion, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result poten- tially renders obsolete in our opinion a longstanding debate in **fracture** mechanics on the correct criterion for kinking.

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An important aspect for phase field models of **brittle** **fracture** is their ability to correctly reproduce the impact of load- ing conditions. Specifically, to describe the asymmetric behavior of **brittle** materials (ie, tension/compression asymmetry), different strategies have been employed. The phase field model of Reference 3 uses the decomposition of the strain tensor into positive and negative parts to include tension/compression asymmetry. An alternative option, which has been pro- posed by Amor et al, 18 is based on the separation of the strain tensor into spherical and deviatoric contributions. Closure effects are then handled by considering the impact of damage on stiffness properties differently depending on the sign of the spherical strain tensor. These strategies have been compared to each other by Ambati et al, 19 who also proposed a hybrid formulation to reduce the computational cost of finite element simulations of **brittle** **fracture** with the PFM. Recently, another alternative, which considers the crack's orientation to define the driving stress and the corresponding driving strain potential, has been introduced by Steinke et al. 20 They postulate a directional split analysis of the crack closure behaviour of isotropic linear elastic material, which overcomes some issues of the previous volumetric-deviatoric and spectral splits.

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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 is then interpreted as a mate- rial 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 oth- ers. Secondly, the tension-compression asymmetry phenomenon as observed for **brittle** materials can be easily formulated directly in the gradient damage model. The resulting sharp interface **fracture** model as → 0 remains unclear and inversely the elliptic reg- ularization 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 tension-compression asymme- try 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 non-interpenetration condition.

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Keywords: In-situ tests, **brittle** **fracture**, stress analysis,
cleavage criterion, polycrystalline modelling. 1 INTRODUCTION
Neutron irradiation ageing causes a temper embrittlement of low alloy steels [1] which shifts their ductile to **brittle** transition range to higher temperatures. As a result, to assess pressure vessel steels integrity during a pressurized thermal shock (in case of an accidental loss of coolant for instance), it becomes necessary to consider their **brittle** behavior. It is thus very important to characterize the mechanical properties of such un-irradiated materials (especially the toughness as well as the mechanisms responsible for **brittle** **fracture** (nucleation and propagation of cracks)) at very low temperatures and to define relevant **brittle** **fracture** criteria [2], in order to predict their service life.

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3.1. **Brittle** **fracture** model
Modelling **fracture** mechanics is not an easy task. Several methods are available in the literature, e.g. XFEM [ 32 –
34 ], remeshing [ 35 – 37 ], Cohesive Zone Models [ 38 – 40 ] or Variational Approach [ 41 , 42 ]. Concerning applications to ice, all these methods have been applied: both XFEM [ 43 ] and remeshing [ 44 , 45 ] are used for modeling ice-floe /sea- ice fractures; Cohesive Zone Model have been successfully applied for sea-ice and structure interactions [ 46 – 48 ] but also for aircraft (or similar) applications [ 49 , 50 ]; cohesive and adhesive failures of ice on an aluminium airfoil have been investigated thanks to a variational approach in the context of electrothermal de-icing systems [ 29 – 31 ]. Regarding electromechanical ice protection systems, Budinger et al. compared a computed energy release rate to a given **fracture** toughness, according to Gri ffith’s theory [ 27 ], to determine if a **fracture** is unstable or not [ 22 ]. To the authors’ knowledge, it is the only attempt to numerically study **fracture** propagation in this context. As an energy balance approach, the work of Budinger et al. took the same flavor as in [ 31 ] but assumed **fracture** mechanisms to be known.

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Based on this picture, the emergence of shear localization in “well-aged” glasses, accompanying the macroscopic stress drop, was interpreted in the framework of the nonequilibrium phase coexistence and first-order discontinuous transition. These findings were independently validated in the context of the elastoplastic models and mean-field estimations in Ref. [ 29 ] by arguing that the statistical distribution of local instability thresholds will depend on the extent of struc- tural heterogeneities in the quenched glass and, therefore, is accountable for the **brittle** **fracture**-like transition. In the present study, however, the ductile-to-**brittle** transition is not controlled by the initial annealing of the system (initial het- erogeneity is statistically the same for different samples) but by the interparticle friction μ.

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• Quality control. Each step in the algorithm is a source of error: (i) choice of the set of degrees of freedom for which the solution is searched in a reduced space (ii) choice of the reduced basis (iii) choice of the accuracy for the local solution (iv) choice of the system reduction (v) choice of the time stepping parameters. These choices should be made within an encompassing framework to ensure that the error associated with each of them can be controlled in the same manner. Therefore, one needs to find a relevant error criterion and build a control process, capable of ensuring that the above choices introduce the same order of error in the solution so that the global error be not dominated by any of them in particular. In terms of further applications, this method should be validated using more advanced models for **brittle** **fracture**, in particular nonlocal continuum damage or softening plasticity models, for which the distinction between local and global effects is not trivial. The extension to ductile failure and/or geometric nonlinearities should also be possible, the global corrections being expected to efficiently handle the “smooth” part of the nonlinearities, while the local/global scheme should treat separately the localisation of defects.

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Case 1 (pure cohesive). In this simplified test, only the ice layer is simulated. To mimick the presence of the alu- minium substrate which undergoes the flexural mode on its neutral line, displacements are imposed on the adhesive interface—bottom boundary of the ice layer equivalently. These displacements are extracted after a purely elastic computation: initialization of Algorithm 1 . Domain and boundary condition are shown in Figure 7 . The crack behav- ior is displayed in Figure 9 (left). The crack nucleates on the first antinode where the tensile stress is maximum. Note that the elastic energy spliting ( 7 ) forbids the degradation of its compressive part and, consequently, crack nucleation in compression (second antinode). It then propagates through the entire ice thickness. Once reaching the ice bottom boundary, the **fracture** branches and the crack continues its propagation along the bottom boundary.

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for different time intervals.. log size for different time intervals.. log size for different time intervals.. log size for different time intervals.. The mathematical mod[r]

Algorithm 1. Adhesive/**brittle** failure computation
where b and d are set to 10 3 .
The algorithm can be viewed in the following manner. For each debonded configuration, a bulk damage computation is performed to check whether no corresponding compatible bulk failure state exists. Bulk damage and interfacial damage interact through the boundary conditions. Indeed, a increase in interfacial damage b will change the boundary conditions. In turn this change will impact the deformed state of the material, from which is deduced bulk damage. The convergence loops are performed in this order as, in the case studied here at least, adhesive debonding is the initiating process and bulk failure is the terminating one (as shall be shown in the following sections). Therefore, the loops are performed in this order to be physically consistent with the application at hand. Combining both models in a more general coupled framework is the focus of current work.

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Received: 27 November 2016; Accepted: 19 December 2016; Published: 21 December 2016
Abstract: The **fracture** behaviors of quasi-**brittle** materials are commonly specimen size (size effect) and crack size (boundary effect) dependent. In this study, a new failure model is developed for characterizing the size and boundary effects. The derivative of the energy release rate is firstly introduced to predict the nominal strength dominated by the strength mechanism. Combined with the energy criterion for the energy mechanism, an asymptotic model is developed to capture the effect of any crack size on the nominal strength, and its expression for geometrically similar specimens is also established, which is able to characterize the size effect. Detailed comparisons of the proposed model with the size effect law and the boundary effect model are performed, respectively. The nominal strength predictions based on the proposed model are validated with the experimental results of cracked three-point bending beam specimens made of concrete, of limestone and of hardened cement paste and compared with the model predictions given by the size effect law and the boundary effect model.

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In this paper, studies are conducted on the silica glass material, which can be considered to be a homogeneous, isotropic and perfectly **brittle** elastic material. This re- search is related to the subsurface damage of silica glass due to surface polishing. In a preliminary study (5), discrete element models were used to obtain qualita- tively good agreement with experiments. The current challenge is to develop a 3D DEM spherical model to quantitatively simulate silica glass as a continuous me- dia. The first step of simulating the elastic behavior of silica (as represented by the Young’s modulus and Pois- son’s ratio), was achieved using the cohesive beam bond model. This method was detailed in a previous paper (6).

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2 Center for Biomedical Computing, Simula Research Laboratory, Oslo, Norway
3 Institut Jean Le Rond d’Alembert, Sorbonne Universit´es, UPMC, Univ Paris 06, CNRS, UMR 7190, France
SUMMARY
The variational approach to **fracture** is effective for simulating the nucleation and propagation of complex crack patterns, but is computationally demanding. The model is a strongly nonlinear non-convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill-conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss-Seidel iteration and employ over-relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton’s method, to further reduce the time to solution; and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton’s method. We investigate the improvements in efficiency on several examples from the literature; the new solver is 5–6 × faster on a majority of the test cases considered.

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[10] B. Bourdin, G. A. Francfort, J.-J. Marigo, The variational approach to **fracture**, J. Elasticity 91 (2008), 5–148. [11] G. Buttazzo, “Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation”, Pitman
Research Notes in Mathematics Series 203, Longman Scientific & Technical, Harlow, 1989.
[12] A. Chambolle, An approximation result for special functions with bounded deformation, J. Math. Pures Appl. (9) 83 (2004), 929–954.