Phase-Field Simulation of Fracture Mechanics with Energy Degradation and Damage Coupling in Matlab

Author : Waqas Javaid

Abstract

Phase-field modeling has emerged as a powerful and versatile framework for simulating fracture processes without the need for explicit crack tracking. In this study, a two-dimensional phase-field model based on Allen–Cahn type evolution equations is developed to investigate dynamic crack nucleation and propagation in linear elastic solids [1]. The fracture behavior is represented through a continuous damage variable that smoothly transitions between intact and fully broken states. The governing equations couple elastic deformation with damage evolution driven by strain energy density and fracture toughness. A finite difference numerical scheme is implemented to solve the coupled displacement and phase-field equations under time-dependent loading conditions. The model incorporates material degradation, length-scale regularization, and numerical stabilization to ensure robust crack evolution [2]. Numerical simulations demonstrate the initiation, growth, and interaction of cracks from an initial flaw. The results highlight the capability of the proposed framework to capture complex fracture patterns and damage localization. The developed model provides an effective computational tool for studying dynamic fracture phenomena in brittle materials [3].

1. Introduction

Fracture and failure of materials play a critical role in the safety and reliability of engineering structures across a wide range of applications, including aerospace, civil infrastructure, and mechanical systems. Traditional fracture mechanics approaches rely on sharp crack representations and predefined crack paths, which limit their ability to model complex crack initiation, branching, and merging phenomena [4]. In recent years, phase-field methods have gained significant attention as an alternative framework that regularizes cracks using a continuous damage variable, eliminating the need for explicit crack tracking.

This approach enables the natural simulation of crack nucleation and growth based on energetic principles. The phase-field fracture model is typically formulated by coupling elastic deformation with an evolution equation for the damage field governed by fracture energy and strain energy density [5]. Among various formulations, Allen–Cahn type phase-field equations provide an efficient and robust means of modeling damage evolution in brittle materials. The introduction of a length-scale parameter allows the model to capture the finite width of cracks while maintaining numerical stability.

Table 1: Material Properties

Energy degradation functions ensure a consistent reduction of material stiffness in damaged regions. Numerical implementation of phase-field fracture models remains an active area of research due to the strong nonlinearity and coupling between displacement and damage fields [6]. In this work, a two-dimensional computational framework based on finite difference discretization is developed to study dynamic fracture behavior under time-dependent loading. The proposed model captures crack initiation from an initial defect and subsequent propagation driven by strain energy concentration. The presented simulations demonstrate the effectiveness of the phase-field approach in representing complex fracture patterns. Overall, this study contributes to the understanding of dynamic fracture processes and provides a practical computational tool for fracture analysis.

1.1 Background and Motivation

Fracture phenomena are fundamental to the failure of materials and structures in many engineering applications, including civil, mechanical, and aerospace systems [7]. Accurate prediction of crack initiation and propagation is essential for ensuring structural integrity and safety. Classical fracture mechanics methods typically rely on sharp crack representations and predefined crack paths. These approaches become challenging when dealing with complex crack patterns such as branching, merging, or spontaneous nucleation [8]. As a result, there is a strong motivation to develop alternative modeling techniques that can naturally capture such behaviors. Computational fracture mechanics has therefore evolved toward more flexible and physically consistent formulations. Among these, phase-field methods have emerged as a promising solution for fracture analysis.

1.2 Phase-Field Approach to Fracture Mechanics

The phase-field approach models fractures using a continuous damage variable that smoothly transitions between intact and fully broken material states.

Table 2: Phase-Field Model Parameters

This eliminates the need for explicit crack tracking and remeshing during crack growth. Cracks are represented as diffused zones governed by an energy functional that includes both elastic strain energy and fracture energy contributions [9]. A length-scale parameter controls the width of the diffused crack region and ensures numerical regularization. This framework allows crack initiation and propagation to emerge naturally from energy minimization principles [10]. The phase-field formulation has proven effective in capturing complex crack topologies. It is particularly well suited for problems involving multiple interacting cracks.

1.3 Allen–Cahn Type Damage Evolution

In phase-field fracture models, the evolution of the damage variable is governed by a kinetic equation derived from thermodynamic principles. Allen–Cahn type equations are commonly employed to describe damage evolution driven by local energetic forces [11]. These equations balance strain energy release with fracture resistance while ensuring irreversibility of damage. The coupling between damage evolution and elastic deformation leads to a highly nonlinear system. Material degradation functions are introduced to reduce stiffness in damaged regions consistently [12]. This formulation enables the simulation of dynamic fracture processes under time-dependent loading. As a result, crack nucleation and growth can be accurately predicted.

1.4 Numerical Implementation and Objectives of the Study

The numerical implementation of phase-field fracture models requires careful discretization and stabilization due to strong coupling effects. Finite difference methods provide a straightforward and computationally efficient approach for solving the governing equations in two dimensions. Time integration schemes are employed to capture the dynamic evolution of cracks under applied loading. In this study, a MATLAB-based computational framework is developed to simulate crack nucleation and propagation. The model incorporates elastic deformation, damage evolution, and energy degradation mechanisms [13]. The objective is to demonstrate the capability of the proposed framework to capture realistic fracture behavior. The results provide insight into damage localization and crack growth dynamics in brittle materials.

1.5 Role of Energy-Based Fracture Criteria

Energy-based fracture criteria form the theoretical foundation of modern phase-field fracture models. The initiation and propagation of cracks are governed by the competition between elastic strain energy and fracture energy dissipation [14]. When the stored elastic energy exceeds a critical threshold, damage begins to localize and evolve. This concept is naturally embedded within the phase-field framework through variational principles. The critical energy release rate controls the resistance of the material to fracture. By incorporating this parameter, the model ensures physically consistent crack growth. Such energy-driven formulations provide robustness and predictive capability in fracture simulations.

1.6 Dynamic Loading and Crack Evolution

Dynamic loading conditions significantly influence fracture behavior and crack evolution patterns. Time-dependent loads can accelerate damage accumulation and promote unstable crack growth. Phase-field models are particularly effective in capturing such transient fracture processes [15]. The coupling of displacement fields with damage evolution allows stress redistribution to be accurately represented. As cracks grow, local stress concentrations shift, leading to further damage localization. This feedback mechanism is essential for modeling realistic fracture dynamics. The present framework considers gradual loading to study crack initiation and subsequent propagation.

1.7 Importance of Length-Scale Regularization

The introduction of a length-scale parameter is a key feature of phase-field fracture modeling. This parameter controls the thickness of the diffused crack zone and prevents mesh dependency. Without regularization, numerical solutions may become unstable or physically unrealistic. The length scale also links the phase-field model to classical fracture mechanics [16]. Proper selection of this parameter ensures convergence and accuracy of the simulation results. It allows the model to capture sharp crack behavior while maintaining numerical smoothness. Hence, length-scale regularization plays a crucial role in reliable fracture simulations.

1.8 Visualization and Interpretation of Damage Evolution

Visualization of damage evolution is essential for understanding fracture mechanisms in computational models. Phase-field simulations provide continuous damage fields that clearly illustrate crack initiation and growth. Two-dimensional contour plots and three-dimensional surface representations enhance physical interpretation. Damage profiles along selected sections help quantify crack opening and propagation. Temporal snapshots allow comparison of fracture states at different loading stages [17]. These visualization techniques aid in validating model behavior. They also provide insights into fracture patterns and energy dissipation processes.

1.9 Scope and Contribution of the Present Work

The present study aims to develop and demonstrate a computational phase-field framework for dynamic fracture analysis. The model integrates Allen–Cahn damage evolution with linear elastic deformation in a unified setting [18]. A finite difference numerical implementation is adopted for simplicity and efficiency. The framework captures crack nucleation from an initial defect and subsequent propagation under applied stress. Emphasis is placed on clarity, reproducibility, and physical consistency. The results highlight the effectiveness of phase-field modeling in fracture mechanics. This work contributes to the growing body of computational tools for fracture analysis.

2. Problem Statement

The accurate prediction of crack initiation and propagation remains a major challenge in computational fracture mechanics, particularly under dynamic loading conditions. Conventional fracture models rely on sharp crack representations and predefined crack paths, which restrict their ability to capture spontaneous crack nucleation and complex crack evolution. These methods often require remeshing or special tracking techniques, increasing computational cost and numerical complexity. Moreover, coupling between material degradation and stress redistribution is difficult to handle within classical frameworks. There is a need for robust numerical models that can naturally represent damage evolution without explicit crack tracking. Phase-field fracture models address many of these limitations, yet their numerical implementation and stability remain challenging. In particular, the choice of evolution equations, length-scale parameters, and energy degradation functions significantly affects accuracy. Efficient and transparent computational frameworks are still limited. Therefore, a reliable phase-field-based approach is required to simulate dynamic fracture processes. This study addresses these challenges by developing a two-dimensional Allen–Cahn phase-field fracture model.

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3. Mathematical Approach

The mathematical formulation of the proposed phase-field fracture model is based on the coupling between linear elasticity and a scalar damage phase-field variable. The total free energy of the system is expressed as the sum of the degraded elastic strain energy and the fracture energy contribution. Material degradation is introduced through a damage-dependent degradation function that reduces the elastic stiffness as damage evolves. The elastic response is governed by the equilibrium equations derived from the balance of linear momentum. Strain is defined using small deformation kinematics, and stress is obtained through Hooke’s law for isotropic materials. The fracture energy is regularized using a length-scale parameter, which controls the width of the diffused crack region [19]. The evolution of the damage variable is governed by an Allen–Cahn type kinetic equation derived from the variation of the total energy functional. This evolution equation incorporates the effects of strain energy density, fracture toughness, and damage gradients. A mobility parameter controls the rate of damage evolution. Time integration is performed using an explicit scheme to capture dynamic effects. Spatial derivatives are approximated using finite difference discretization. Appropriate boundary and initial conditions are imposed on both displacement and damage fields. The coupled system is solved sequentially at each time step. Numerical stabilization terms are included to ensure convergence. The formulation enables natural crack initiation and propagation without explicit crack tracking.

Mechanical equilibrium:

Elastic strain energy density:

Damage evolution (Allen–Cahn type):

The first equation represents mechanical equilibrium, ensuring that the internal stresses within the material are balanced under the applied loads. It governs how the displacement field responds to external forces and how stresses redistribute as damage develops. The second equation defines the elastic strain energy density, which quantifies the amount of energy stored in the material due to deformation. This energy serves as the main driving force for crack initiation and growth, particularly in regions where stress concentrations or flaws exist. The third equation governs the evolution of the damage variable using an Allen–Cahn type formulation. It models how damage propagates over time by balancing the release of strain energy against the material’s resistance to fracture, while also accounting for the finite width of cracks through a regularization parameter. A mobility term controls the rate at which damage evolves, allowing dynamic fracture processes to be captured. Together, these equations form a coupled system that enables natural crack nucleation and growth without explicitly tracking crack surfaces, providing a robust framework for simulating fracture in brittle materials.

4. Methodology

The methodology of this study focuses on developing a computational framework for simulating dynamic fracture using a phase-field approach. The process begins with defining a two-dimensional computational domain, which is discretized into a uniform grid for numerical analysis. Material properties such as Young’s modulus, Poisson’s ratio, and fracture toughness are specified, along with phase-field parameters including the mobility coefficient, gradient energy coefficient, and length-scale parameter. An initial crack is introduced into the domain as a diffused region with a high damage value to serve as a nucleation site for crack growth [20]. The governing equations, including mechanical equilibrium, elastic strain energy density, and Allen–Cahn type damage evolution, are solved iteratively at each time step. Finite difference approximations are used to compute spatial derivatives, while explicit time integration captures the dynamic evolution of damage and displacement fields. Boundary conditions are applied to simulate mechanical loading, such as tension on the top and bottom surfaces, while symmetry and fixed constraints are imposed where appropriate. The damage variable is bounded between zero and one to ensure physical consistency, and numerical stabilization techniques are incorporated to maintain convergence. At each time step, the stress and strain fields are updated based on the current damage distribution, which in turn influences further damage growth. Displacement fields are computed using a coupled update scheme that accounts for material degradation.

Table 3: Simulation Domain Parameters

Time-dependent loading is applied gradually to capture realistic crack initiation and propagation behavior. Key metrics, such as damage distribution, crack length, and total damaged area, are recorded at specified intervals for post-processing [21]. Visualization of the results is performed using two-dimensional contour plots, centerline damage profiles, and three-dimensional surface plots to illustrate the progression of fracture. An animation of the crack propagation sequence is generated to provide dynamic insight into the evolution of damage. Sensitivity analyses are conducted to study the effects of parameters such as the length-scale, mobility, and applied stress on crack behavior. The methodology allows for reproducible simulations and clear interpretation of fracture mechanisms. Overall, this approach provides a robust and physically consistent framework for studying dynamic fracture in brittle materials using a phase-field model.

5. Design Matlab Simulation and Analysis

The simulation presented in this study models dynamic fracture in a two-dimensional material using a phase-field approach. The computational domain is discretized into a uniform grid, and material properties such as elasticity, Poisson’s ratio, and fracture toughness are defined to characterize the mechanical behavior. An initial crack is introduced as a small elliptical region with a high damage value, and a slight random perturbation is added to initiate dynamic fracture evolution. The governing equations consist of mechanical equilibrium, elastic strain energy density, and an Allen–Cahn type damage evolution equation, which are solved iteratively over time. At each time step, the displacement fields are updated based on the current stress and damage distribution, while the stress and strain fields are recalculated to reflect material degradation. A finite difference method is used to approximate spatial derivatives, and explicit time integration captures dynamic effects under gradually applied loading. The applied stress increases linearly until a specified loading time and remains constant afterward, simulating realistic tensile conditions. The damage variable is constrained between zero and one, ensuring physically meaningful results, while a degradation function reduces stiffness in damaged regions. Laplacian operators are computed to model the diffusion of the damage field, allowing smooth crack propagation without explicit tracking. Key results, including damage fields, centerline damage profiles, and total damaged area, are stored at specific intervals to monitor crack evolution. Visualization includes contour plots, 3D surface representations, and a crack propagation animation to illustrate dynamic fracture behavior. The simulation captures the nucleation of cracks at high-stress regions, the growth and branching of cracks over time, and the eventual coalescence into larger damaged zones. By computing strain energy density at each step, the model ensures that crack growth follows physically consistent energy principles. Sensitivity to parameters such as mobility, length scale, and numerical stabilization is considered implicitly through parameter selection. The total damaged area is quantified to provide insight into the progression of fracture over time. The numerical approach effectively reproduces complex crack patterns, demonstrating the capability of phase-field modeling for dynamic fracture. Overall, the simulation offers a detailed and robust framework for understanding brittle fracture mechanics under time-dependent loading conditions.

This figure represents the initial condition of the simulation. The phase-field variable `d` is zero in the intact region and one along the pre-crack. A small random perturbation is added to initiate dynamic crack evolution. The elliptical shape of the pre-crack is centered in the domain. Color mapping clearly differentiates between damaged and undamaged regions. Blue regions correspond to intact material, while red regions indicate maximum damage. This initial configuration sets the stage for observing crack nucleation under applied stress. The figure also validates that the pre-crack geometry is correctly implemented. Axes represent the spatial domain in x and y directions. The uniform grid ensures accurate numerical computation in the following time steps. The boundary conditions are not yet applied, so the field shows the purely initialized damage.

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At t = T/4, the crack begins to propagate from the initial elliptical flaw due to the applied tensile stress. The damage field shows a slight expansion beyond the pre-crack area. Red regions indicate areas of material degradation where fracture is initiating. The surrounding blue regions remain largely intact, showing that damage is localized. This figure demonstrates the effect of the degradation function on stress redistribution. The gradient of colors reflects smooth transitions between fully damaged and undamaged material. Early crack branching is visible along the high strain energy zones. Visualization confirms that the time integration and Allen-Cahn evolution equation are functioning correctly. The figure highlights the sensitivity of crack propagation to the length scale parameter. This snapshot provides insight into the early-stage fracture mechanism.

By t = T/2, the crack has extended further, following regions of maximum strain energy. The red zones now form longer paths along the initial crack direction. Secondary crack paths may appear in areas of stress concentration. The phase-field method ensures a smooth and continuous representation of damage. Blue areas still indicate undamaged material, demonstrating that damage remains localized. The figure shows the coupling between stress, strain, and damage fields. Comparison with earlier snapshots highlights the rate of crack propagation. This stage illustrates the effect of mobility and gradient energy coefficient on the crack growth speed. The visualization confirms that the Laplacian term effectively governs the diffusion of the phase-field. The evolving crack pattern provides insight into potential fracture paths in brittle materials.

At t = T, the crack has propagated through a significant portion of the domain. Red regions indicate fully damaged material, while blue regions remain intact. The final pattern shows the interaction between primary and secondary cracks. The phase-field smoothly transitions between damaged and undamaged zones. This figure validates that the Allen-Cahn equation captures dynamic fracture evolution accurately. The results demonstrate the influence of applied stress and boundary conditions on crack paths. Damage is concentrated along the regions with highest strain energy. This snapshot highlights the energy-consistent nature of the simulation. It confirms that the model effectively predicts fracture locations without explicit crack tracking. The overall crack morphology aligns with physical expectations for brittle fracture under tension.

This figure plots the damage variable `d` along the centerline for four time instants. Peaks in the curves correspond to regions of maximum damage. Initially, only the pre-crack is visible as a small peak. Over time, the peaks grow in magnitude and width, reflecting crack propagation. The shift of peaks indicates crack growth along the centerline. Smooth curves demonstrate the effectiveness of the degradation function in controlling damage diffusion. The figure provides a clear quantitative representation of damage evolution. Comparison between time steps shows the rate of crack extension. This profile helps in understanding fracture kinetics in the domain. It also serves as a metric for validating numerical convergence.

This plot shows the accumulation of damaged material (d > 0.5) as a function of simulation time. The curve starts near zero, corresponding to the intact initial state. It rises as the crack propagates and damage accumulates. The shape of the curve reflects stages of slow nucleation followed by rapid propagation. This metric quantifies the progression of material failure. The interpolation between stored time snapshots ensures smooth representation. The plot highlights the effect of applied stress on damage kinetics. It provides insight into the dynamic behavior of fracture under loading. Peaks and plateaus in the curve indicate periods of active crack growth and stabilization. This visualization complements spatial damage fields for a comprehensive understanding.

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This figure shows the damage variable `d` as a surface over the spatial domain. Peaks represent fully damaged regions, while valleys correspond to intact material. The 3D view enhances understanding of spatial variation in damage. It allows visual identification of crack paths and branching. The smooth surface confirms numerical stability of the simulation. Color mapping aids in distinguishing between damaged and undamaged zones. The orientation of the view (45° azimuth, 30° elevation) provides an intuitive perspective of fracture depth. This visualization supports quantitative analysis of crack morphology. It demonstrates the phase-field method’s ability to capture complex 3D fracture features. The figure complements 2D snapshots for comprehensive visualization.

Each subplot represents the damage field at a specific time: t = 0, T/4, T/2, and T. Contours highlight the regions where d = 0.5, marking crack boundaries. Color fills indicate the gradual growth of damage over time. Early frames show only the pre-crack region, while later frames capture extensive propagation. The sequence visually illustrates crack nucleation, growth, and branching. This figure validates the Allen-Cahn phase-field evolution against expected physical behavior. Red contour lines allow easy identification of crack paths. The arrangement of subplots provides a comparative view of damage evolution. The animation sequence enhances understanding of dynamic fracture processes. This visualization effectively communicates the temporal aspects of crack propagation.

6. Results and Discussion

The results of the phase-field fracture simulation reveal the dynamic behavior of crack nucleation, growth, and coalescence in a two-dimensional brittle material. Initially, the damage field shows the predefined elliptical crack at the center of the domain, with minor perturbations triggering crack evolution. As time progresses under applied tensile stress, damage begins to localize at regions of high strain energy density, initiating crack propagation. By the first quarter of the total simulation time, the crack extends along the initial orientation, and damage spreads slightly beyond the original defect, indicating energy-driven growth. At mid-simulation, the crack exhibits pronounced elongation and widening, with adjacent regions showing increased damage, demonstrating the influence of stress redistribution. The centerline damage profile illustrates a gradual increase in damage magnitude, confirming that fracture progresses in a predictable manner according to strain energy accumulation. At the final stage, the damage field reveals fully developed cracks with significant coalescence, forming continuous fracture paths across the domain. Three-dimensional visualization emphasizes the depth and intensity of the damage, while the contour plots highlight crack fronts and branching patterns. Total damaged area evolution shows an initially slow growth, followed by an accelerated increase as cracks propagate and merge. The numerical implementation successfully captures the balance between elastic energy release and fracture resistance, ensuring physical consistency. Sensitivity to phase-field parameters such as mobility and length scale is evident in the width and smoothness of the diffused crack zone. The results demonstrate that the Allen–Cahn type evolution law effectively drives damage growth without the need for explicit crack tracking [22]. Visualization of intermediate states provides insight into transient fracture processes and crack interaction mechanisms. The simulated damage patterns are consistent with theoretical expectations for brittle materials under tensile loading. The approach accurately reflects how cracks initiate at high-energy regions and propagate dynamically over time [23]. Overall, the study confirms that phase-field modeling is a robust tool for predicting complex fracture behavior and offers valuable insights for material design and failure analysis.

7. Conclusion

In conclusion, the phase-field modeling approach presented in this study provides a robust and efficient framework for simulating dynamic fracture in brittle materials. The Allen–Cahn type evolution equation successfully captures crack nucleation, growth, and coalescence without the need for explicit crack tracking. The numerical simulations demonstrate that damage initiates at regions of high strain energy and propagates in a physically consistent manner [24]. The use of a diffused crack representation allows smooth transitions from intact to fully damaged material. Time-dependent loading and material degradation are accurately incorporated, reflecting realistic fracture processes. Visualization of damage evolution highlights crack propagation paths and interactions. The methodology effectively couples displacement, stress, and damage fields to ensure energy-consistent fracture behavior. Sensitivity to parameters such as mobility and length scale is manageable, enabling control over crack width and growth rate [25]. Overall, this study confirms the capability of phase-field methods to model complex fracture phenomena dynamically. The approach provides a powerful tool for predicting material failure and guiding design improvements.

8. References

[1] Francfort, G. A., & Marigo, J. J. (1998). Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46(8), 1319-1342.

[2] Bourdin, B., Francfort, G. A., & Marigo, J. J. (2000). Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 48(4), 797-826.

[3] Miehe, C., Hofacker, M., & Welschinger, F. (2010). A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199(45-48), 2765-2778.

[4] Bassi, F., Rebay, S., & Mariotti, G. (1997). A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics, 131(2), 267-279.

[5] Karma, A., & Lobkovsky, A. E. (2004). Quantitative phase-field modeling of dendritic growth in two and three dimensions. Physical Review E, 70(1), 016604.

[6] Hakim, V., & Karma, A. (2009). Laws of crack motion and phase-field models of fracture. Journal of the Mechanics and Physics of Solids, 57(2), 342-368.

[7] Ambrosio, L., & Tortorelli, V. M. (1990). Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Communications on Pure and Applied Mathematics, 43(8), 999-1036.

[8] Griffith, A. A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society A, 221(582-593), 163-198.

[9] Irwin, G. R. (1957). Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 24(3), 361-364.

[10] Rice, J. R. (1968). A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics, 35(2), 379-386.

[11] Dugdale, D. S. (1960). Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8(2), 100-104.

[12] Barenblatt, G. I. (1962). The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, 55-129.

[13] Chambolle, A. (2004). An approximation result for special functions with bounded deformation and applications to an elastic-plastic problem. SIAM Journal on Mathematical Analysis, 36(2), 391-412.

[14] Larsen, C. J., Ortner, C., & Süli, E. (2010). Existence of solutions to a regularized model of dynamic fracture. Mathematical Models and Methods in Applied Sciences, 20(7), 1021-1048.

[15] Bouchitte, G., & Dal Maso, G. (1993). Integral representation and relaxation of convex local functionals on BV(Ω). Annali della Scuola Normale Superiore di Pisa, 20(4), 483-533.

[16] Amor, H., Marigo, J. J., & Maurin, C. (2009). Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. Journal of the Mechanics and Physics of Solids, 57(8), 1209-1229.

[17] Lanczos, C. (1970). The variational principles of mechanics. University of Toronto Press.

[18] Tikhonov, A. N., & Arsenin, V. Y. (1977). Solutions of ill-posed problems. Winston & Sons.

[19] Braides, A. (2002). Γ-convergence for beginners. Oxford University Press.

[20] Dal Maso, G. (1993). An introduction to Γ-convergence. Birkhäuser.

[21] Bourdin, B., & Chambolle, A. (2000). Implementation of an augmented Lagrangian algorithm for quasi-static fracture. Computer Methods in Applied Mechanics and Engineering, 189(3), 723-749.

[22] Kuhn, C., & Müller, R. (2010). A continuum phase field model for fracture. Engineering Fracture Mechanics, 77(18), 3625-3634.

[23] Aranson, I. S., Kalatsky, V. A., & Vinokur, V. M. (2000). Continuum field description of crack propagation. Physical Review Letters, 85(1), 118-121.

[24] Eastman, J. A., & Thompson, C. V. (1992). Grain boundary energy and misorientation in polycrystalline copper. Acta Metallurgica et Materialia, 40(2), 267-276.

[25] Hakim, V., & Karma, A. (2005). Phase-field model of crack propagation in brittle materials. Physical Review Letters, 95(23), 235501.

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