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Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far-field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen's extension of Noll's theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips. [ABSTRACT FROM AUTHOR]

The present paper provides an analytical solution for a periodic array of two collinear and symmetric cracks (P-TCSC) under remote tension. This is achieved by representing the multiple collinear cracks problem as the contact problem with discrete ligament regions, and the governing equations are obtained as integral equations with Cauchy-type kernel. Closed-form expressions are derived for the crack opening profile, normal stress distribution and mode I stress intensity factors (SIFs), which can reduce to the classical solutions of two collinear and symmetric cracks (TCSC) or a periodic row of collinear cracks with equal length and equal spacing (PCEE) under special conditions. Finite element analysis is also performed to validate the analytical solutions obtained. Different from the TCSC case, results show that crack initiation for P-TCSC seems more complicated depending on a combination of two nondimensional parameters, and a SIFs map for P-TCSC is further constructed to give a more precise evaluation. The proposed method relies solely on solving the integral equations with Cauchy-type kernel combined with the corresponding boundary conditions without a prior knowledge of the complex potential function in traditional complex variable method of plane elasticity, and it may find application in plastic zone evaluation and fracture criteria of collinear cracks. [ABSTRACT FROM AUTHOR]

This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy, where the clamped boundary is partially debonded along the thickness direction. Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates, incorporating the Gurtin-Murdoch surface elasticity theory. For two typical cases of constant bending moment and uniform shear force in the debonded segment, the associated problems are reduced to two mixed boundary value problems. By solving the resulting mixed boundary value problems using the Fourier integral transform, a new type of singular integral equation with two Cauchy kernels is obtained for each case, and the exact solutions in terms of the fundamental functions are determined using the Poincare-Bertrand formula. Asymptotic elastic fields near the debonded tips including the bending moment, effective shear force, and bulk stress components exhibit the oscillatory singularity. The dependence relations among the singular fields, the material constants, and the plate's thickness are analyzed for partially debonded cantilever micro-plates. If surface energy is neglected, these results reduce the bending fracture of a macroscale partially debonded cantilever plate, which has not been previously reported. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox. [ABSTRACT FROM AUTHOR]

We show that the linear brittle Griffith energy on a thin rectangle Γ -converges after rescaling to the linear one-dimensional brittle Euler–Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy. [ABSTRACT FROM AUTHOR]

We study the nucleation and development of crack patterns in thin composite fibers under tension in this work. A fiber comprises an elastic core and an outer layer of a weaker brittle material. In recent tensile experiments on such composites, multiple cracks were observed to develop simultaneously on the outer layer. We propose here a simple one-dimensional model to predict such phenomena using the inverse-deformation approach to fracture. We idealize the problem as two axially loaded rods coupled by a linear interfacial condition. The latter can be regarded as an adhesive that resists slip between the two materials. One rod is modeled as a brittle material, and the other a linearly elastic material, both undergoing finite deformations. [ABSTRACT FROM AUTHOR]

We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions. [ABSTRACT FROM AUTHOR]

In this work, space-time formulations and Galerkin discretizations for phase-field fracture optimal control problems are considered. The fracture irreversibility constraint is formulated on the time-continuous level and is regularized by means of penalization. The optimization scheme is formulated in terms of the reduced approach and then solved with a Newton method. To this end, the state, adjoint, tangent, and adjoint Hessian equations are derived. The key focus is on the design of appropriate function spaces and the rigorous justification of all Fréchet derivatives that require fourth-order regularizations. Therein, a second-order time derivative on the phase-field variable appears, which is reformulated as a mixed first-order-in-time system. These derivations are carefully established for all four equations. Finally, the corresponding time-stepping schemes are derived by employing a dG(r ) discretization in time. [ABSTRACT FROM AUTHOR]

Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. For this reason, discrete terms and energy contributions from the ultrathin rod's lateral surface appear in the limiting functional. Fracture energy in the Γ -limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds, and torsional cracks. In special cases, the cell formula can be significantly simplified—we illustrate this by the example of a full crack and also show that the energy of a mere fold is strictly lower for a class of models. Our approach applies e.g. to atomistic systems with Lennard–Jones-type potentials and is motivated by the research of ceramic nanowires. [ABSTRACT FROM AUTHOR]

We prove a compactness result in GBD which also provides a new proof of the compactness theorem in GSBD , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities. [ABSTRACT FROM AUTHOR]

This paper develops a deep learning tool based on neural processes (NPs) called the Peri-Net-Pro, to predict the crack patterns in a moving disk and classifies them according to the classification modes with quantified uncertainties. In particular, image classification and regression studies are conducted by means of convolutional neural networks (CNNs) and NPs. First, the amount and quality of the data are enhanced by using peridynamics to theoretically compensate for the problems of the finite element method (FEM) in generating crack pattern images. Second, case studies are conducted with the prototype microelastic brittle (PMB), linear peridynamic solid (LPS), and viscoelastic solid (VES) models obtained by using the peridynamic theory. The case studies are performed to classify the images by using CNNs and determine the suitability of the PMB, LBS, and VES models. Finally, a regression analysis is performed on the crack pattern images with NPs to predict the crack patterns. The regression analysis results confirm that the variance decreases when the number of epochs increases by using the NPs. The training results gradually improve, and the variance ranges decrease to less than 0.035. The main finding of this study is that the NPs enable accurate predictions, even with missing or insufficient training data. The results demonstrate that if the context points are set to the 10th, 100th, 300th, and 784th, the training information is deliberately omitted for the context points of the 10th, 100th, and 300th, and the predictions are different when the context points are significantly lower. However, the comparison of the results of the 100th and 784th context points shows that the predicted results are similar because of the Gaussian processes in the NPs. Therefore, if the NPs are employed for training, the missing information of the training data can be supplemented to predict the results. [ABSTRACT FROM AUTHOR]

We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy. [ABSTRACT FROM AUTHOR]

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model, which cannot describe certain nonequilibrium phenomena by finite collisions of particles, decreasing the fidelity of the solution. Based on an alternative formulation of the targeted essentially non-oscillatory (TENO) scheme, a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems. The numerical flux is constructed by the TENO interpolation of the solution and its derivatives, instead of being reconstructed from the physical flux. The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations. The numerical methods are verified by several classical plasma problems. The results show that compared with the original TENO scheme, the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation. [ABSTRACT FROM AUTHOR]

The surface fracture toughness is an important mechanical parameter for studying the failure behavior of air plasma sprayed (APS) thermal barrier coatings (TBCs). As APS TBCs are typical multilayer porous ceramic materials, the direct applications of the traditional single edge notched beam (SENB) method that ignores those typical structural characters may cause errors. To measure the surface fracture toughness more accurately, the effects of multilayer and porous characters on the fracture toughness of APS TBCs should be considered. In this paper, a modified single edge V-notched beam (MSEVNB) method with typical structural characters is developed. According to the finite element analysis (FEA), the geometry factor of the multilayer structure is recalculated. Owing to the narrower V-notches, a more accurate critical fracture stress is obtained. Based on the Griffith energy balance, the reduction of the crack surface caused by micro-defects is corrected. The MSEVNB method can measure the surface fracture toughness more accurately than the SENB method. [ABSTRACT FROM AUTHOR]

A prestressed elastic medium containing a mode-III crack is studied by means of the couple stress theory (CST). Based on the CST under initial stresses, a governing differential equation along with a mixed boundary value problem is established. The singularities of the couple stress and force stress near the crack tips are analyzed through the asymptotic crack-tip fields resulting from the characteristic expansion method. To determine their intensity, a hypersingular integral equation is derived and numerically solved with the help of the Chebyshev polynomial. The obtained results show a strong size-dependence of the out-of-plane displacement on the crack and the couple stress intensity factor (CSIF) and the force stress intensity factor (FSIF) around the crack tips. The symmetric part of the shear stress has no singularity, and the skew-symmetric part related to the couple stress exhibits an r−3/2 singularity, in which r is the distance from the crack tip. The initial stresses also affect the crack tearing displacement and the CSIF and FSIF. [ABSTRACT FROM AUTHOR]

We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads. [ABSTRACT FROM AUTHOR]

We study the thermoelectric field for an electrically and thermally insulated coated hole of arbitrary shape embedded in an infinite nonlinearly coupled thermoelectric material subject to uniform remote electric current density and uniform remote energy flux. A conformal mapping function for the coating and matrix is introduced, which simultaneously maps the hole boundary and the coating-matrix interface onto two concentric circles in the image plane. Using analytic continuation, we derive a general solution in terms of two auxiliary functions. The general solution satisfies the insulating conditions along the hole boundary and all of the continuity conditions across the perfect coating-matrix interface. Once the two auxiliary functions have been obtained in the elementary-form, the four original analytic functions in the coating and matrix characterizing the thermoelectric fields are completely and explicitly determined. The design of a neutral coated circular hole that does not disturb the prescribed thermoelectric field in the thermoelectric matrix is achieved when the relative thickness parameter and the two mismatch parameters satisfy a simple condition. Finally, the neutrality of a coated circular thermoelectric inhomogeneity is also accomplished. [ABSTRACT FROM AUTHOR]

49Q20, 70G75, Integral representation, global method for relaxation, generalized special functions of bounded deformation, Korn-type inequalities, free discontinuity problems, 26A45, 49J45, 74R10 Keywords: Integral representation; global method for relaxation; free discontinuity problems; generalized special functions of bounded deformation; Korn-type inequalities; 26A45; 49J45; 49Q20; 70G75; 74R10 EN Integral representation global method for relaxation free discontinuity problems generalized special functions of bounded deformation Korn-type inequalities 26A45 49J45 49Q20 70G75 74R10 705 733 29 10/03/22 20221001 NES 221001 1 Introduction Integral representation results are a fundamental tool in the abstract theory of variational limits by -convergence or in relaxation problems (see [[32]]). In this paper we contribute to this topic by proving an integral representation result for a general class of energies arising in the modeling of linear elastic solids with surface discontinuities. functions: Integral representation and -convergence, Arch. Ration. [Extracted from the article]

We obtain linear elasticity as Γ -limit of finite elasticity under incompressibility assumption and Dirichlet boundary conditions. The result is shown for a large class of energy densities for rubber-like materials. [ABSTRACT FROM AUTHOR]

Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering. [ABSTRACT FROM AUTHOR]

Two efficient methods are presented to detect multiple cracks in 2D elastic bodies, based on the insights from Extended Finite Element. Adetection mesh is assigned to the cracked body and the responses are measured at the nodes. A finite element model with the same mesh is used to represent the uncracked state of the physical body. In the first method which is called Crack Detection based on Residual Error (CDRE), the residual error norm is calculated based on the uncracked body stiffness matrix and the cracked body responses. The contour of the error norm would show the crack pattern; the method is computationally efficient. In the second method that is coined as Crack Detection based on Stiffness Residual (CDSR), the crack locations are found based on the difference between the stiffness matrix of the cracked body and the uncracked body. The stiffness matrix of the cracked body is found by solving a dynamic inverse problem based on a modified Tikhonov regularization. The efficiency and accuracy of the identification method are enhanced by predicting the crack pattern by the CDRE method. Several examples are presented to show the accuracy and robustness of the methods in the presence of high noise levels. [ABSTRACT FROM AUTHOR]

Within the framework of Kirchhoff–Love plate theory, we analyze a variational model for elastic plates with rigid inclusions and interfacial cracks. The main feature of the model is a fully coupled nonpenetration condition that involves both the normal component of the longitudinal displacements and the normal derivative of the transverse deflection of the crack faces. Without making any artificial assumptions on the crack geometry and shape variation, we prove that the first-order shape derivative of the potential deformation energy is well defined and provide an explicit representation for it. The result is applied to derive the Griffith formula for the energy release rate associated with crack extension. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracking domains. [ABSTRACT FROM AUTHOR]

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly. [ABSTRACT FROM AUTHOR]

We prove an existence result for the fractional Kelvin–Voigt's model involving Caputo's derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt's model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique. [ABSTRACT FROM AUTHOR]

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case. [ABSTRACT FROM AUTHOR]

In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates. [ABSTRACT FROM AUTHOR]

In this paper, the counter – current imbibition phenomena in a heterogeneous porous media is studied with the consideration of two types of porous materials like volcanic and fine sand and Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. A simulation result is developed here to study the effect of heterogeneity, capillarity and relative permeability on saturation rate and to obtain an optimum recovery rate of the reservoir with the choices of some interesting parametric value.

We consider the problem of a semi-infinite mode III crack partially penetrating a three-phase elliptical inhomogeneity. The two elliptical interfaces are confocal, and the semi-infinite crack passes through the vertices of the two interfaces with its crack tip located at one of the two common foci of the two elliptical interfaces. Full-field closed-form solutions are derived by means of conformal mapping. In particular, elementary expressions of the local mode III stress intensity factor at the crack tip are obtained and discussed in detail. [ABSTRACT FROM AUTHOR]

An approximation, in the sense of Γ -convergence and in any dimension d ≥ 1 , of Griffith-type functionals, with p-growth ( p > 1 ) in the symmetrized gradient, is provided by means of a sequence of non-local integral functionals depending on the average of the symmetrized gradients on small balls. [ABSTRACT FROM AUTHOR]

In this work, an elastic-damage evolution analysis is carried out for a cylinder under torsion made of a material obeying a gradient damage model with softening. Both semi-analytical and asymptotic approaches are developed to analyze the elastic, axisymmetric and bifurcation stages. We show the existence of a fundamental branch where the damage field is asymmetric and localized within a finite thickness from the boundary. By minimizing a generalized Rayleigh quotient, the bifurcation time and modes are obtained as a function of the length scale ϵ = ℓ / R involving a material internal length and the cylinder radius. We will then focus on these size effects by assuming that ϵ is a small parameter in an asymptotic setting. After justification, specific spatial and temporal rescaled variables are introduced for the boundary layer problem. It is shown that the axisymmetric damage evolution and the bifurcation are governed by two universal functions independent of the length scale. The simulation results obtained by the semi-analytical approach are formally justified by the asymptotic methods. [ABSTRACT FROM AUTHOR]

We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step δ is smaller than the ellipticity parameter ε , we show the Γ -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no L p fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort. [ABSTRACT FROM AUTHOR]

Employing primarily analytic continuation, we derive analytical solutions to the anti-plane problem associated with a screw dislocation located anywhere inside an anisotropic tri-material composed of an intermediate anisotropic elastic layer of finite thickness sandwiched between two semi-infinite anisotropic elastic media. All three phases of the tri-material are monoclinic with the symmetry plane at x 3 = 0 in a Cartesian coordinate system. We obtain explicit expressions for each one of the three analytic functions defined in its respective anisotropic elastic phase. This allows for the complete determination of the stresses and displacement in the tri-material. In addition, we present simple yet concise expressions for the image force acting on the screw dislocation. An interfacial Zener-Stroh crack in the tri-material is discussed using the analytical solution developed for an interfacial screw dislocation. [ABSTRACT FROM AUTHOR]

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time. [ABSTRACT FROM AUTHOR]

An equivalent indentation method is developed for the external crack problem with a Dugdale cohesive zone in the both axisymmetric and two-dimensional (2D) cases. This is achieved based on the principle of superposition by decomposing the original problem into two simple boundary value problems, with one considering action of a constant traction within the cohesive zone, and the other corresponding to indentation by a rigid concave punch. Closed-form expressions are derived for the distributions of displacement and traction on the crack interface, which are consistent with the classical results in fracture mechanics. Results show that the interfacial traction distributions in the axisymmetric and 2D cases share the similar mathematical forms except for different coordinate parameters. Finite element analysis is performed to validate the obtained analytical solutions. The proposed method relies solely on a few contact solutions on the surface irrespective of a general elasticity solution in the whole body, and it may find applications in the external crack analysis and adhesive contact model involving functionally graded elastic solids or piezoelectric materials. [ABSTRACT FROM AUTHOR]

This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an L 2 -penalization in the phase field variable, driven by a viscous parameter δ > 0 , and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t. time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for δ > 0 and then consider the vanishing viscosity limit δ → 0 . [ABSTRACT FROM AUTHOR]

In this paper, we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet–Neumann conditions on the boundary. The only assumptions on the crack sets are to be (n - 1) -rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular, they might be dense; hence, the weak formulation must fall outside the usual context of Sobolev spaces and Korn's inequality. We prove existence of a solution for both the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions, we also show a uniqueness result. Finally, we exhibit an example where the energy-dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth. [ABSTRACT FROM AUTHOR]

We propose a phase–field model of dynamic fracture based on the Ambrosio–Tortorelli's approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in Larsen et al. (Math Models Methods Appl Sci 20:1021–1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy–dissipation balance, according to Griffith's criterion. Finally, we analyze the dynamic phase–field model of Bourdin et al. (Int J Fract 168:133–143, 2011) and Larsen (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, IUTAM Bookseries, vol 21. Springer, Dordrecht, 2010, pp 131–140) with no dissipative terms. [ABSTRACT FROM AUTHOR]

We prove an existence result for a model of dynamic fracture based on Griffith's criterion in the case of a prescribed crack path with a kink. [ABSTRACT FROM AUTHOR]