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A basic law of motion for a dynamic crack propagating in a brittle material is derived in the case of two space dimensions. The only basic assumptions for this purpose are the energy conservation law and a variational inequality following the well known Hamilton's principle. Our study is developed within Griffith's framework, that is under the assumption that crack surface energy is proportional to crack length. It is shown that the speed and direction of the crack can be found without further assumptions. Moreover, the corresponding law is local, and if expressed in terms of the stress intensity factors yields the principle of local symmetry previously proposed for quasi-static evolution., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We introduce a three-dimensional dynamic crack propagation law, which is derived from Hamilton's principle. The result is an extension of a previous one obtained, corresponding to the two-dimensional case. As a matter of fact, in an orthogonal plane to the crack front, the geometric condition to be satisfied over the path is the same as in two dimensions. The third mode enters only through the energy release rate. The fact that the physics of the problem is locally two dimensional is a consequence of the virtual motions allowed in the set of admissible crack configurations., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Double network (DN) gels, consisting of two contrasting interpenetrated polymer networks, exhibit large resistances to crack initiation and propagation. For practical applications, crack resistances of materials in highly stretched states are important. In this study, we investigated the crack growth behaviors of DN gels in stretched states by inducing crack seeds into these gels under various degrees of tension. This examination enables the analysis of crack propagation for a wide range of bulk energy release rate G. The power-law elationship between G and crack growth velocity v was investigated for DN gels with different tensile behaviors. We found that for brittle and unnecking DN gels, the velocities changed from slow mode (quasi-stationary fracture) to fast mode (dynamic fracture) with an increase in G, similar to that of a single network gel; in contrast, for necking DN gels having higher crack resistances than those of the unnecking DN gels, only fast modes were observed once the G is above a threshold. Real-time birefringence observation reveals a large damage zone around the crack tip at G slightly lower than the threshold, while the damage zone is hardly observed at G higher than the threshold. The results indicate that, for the necking DN gels, crack initiation has a large energy barrier owing to the formation of the damage zone; once this barrier is overcome, the excess energy release accelerates the crack propagation and therefore the gels exhibit dynamic fracture.

Double network (DN) gels, consisting of two contrasting interpenetrated polymer networks, exhibit large resistances to crack initiation and propagation. For practical applications, crack resistances of materials in highly stretched states are important. In this study, we investigated the crack growth behaviors of DN gels in stretched states by inducing crack seeds into these gels under various degrees of tension. This examination enables the analysis of crack propagation for a wide range of bulk energy release rate G. The power-law elationship between G and crack growth velocity v was investigated for DN gels with different tensile behaviors. We found that for brittle and unnecking DN gels, the velocities changed from slow mode (quasi-stationary fracture) to fast mode (dynamic fracture) with an increase in G, similar to that of a single network gel; in contrast, for necking DN gels having higher crack resistances than those of the unnecking DN gels, only fast modes were observed once the G is above a threshold. Real-time birefringence observation reveals a large damage zone around the crack tip at G slightly lower than the threshold, while the damage zone is hardly observed at G higher than the threshold. The results indicate that, for the necking DN gels, crack initiation has a large energy barrier owing to the formation of the damage zone; once this barrier is overcome, the excess energy release accelerates the crack propagation and therefore the gels exhibit dynamic fracture.

Double network (DN) gels, consisting of two contrasting interpenetrated polymer networks, exhibit large resistances to crack initiation and propagation. For practical applications, crack resistances of materials in highly stretched states are important. In this study, we investigated the crack growth behaviors of DN gels in stretched states by inducing crack seeds into these gels under various degrees of tension. This examination enables the analysis of crack propagation for a wide range of bulk energy release rate G. The power-law elationship between G and crack growth velocity v was investigated for DN gels with different tensile behaviors. We found that for brittle and unnecking DN gels, the velocities changed from slow mode (quasi-stationary fracture) to fast mode (dynamic fracture) with an increase in G, similar to that of a single network gel; in contrast, for necking DN gels having higher crack resistances than those of the unnecking DN gels, only fast modes were observed once the G is above a threshold. Real-time birefringence observation reveals a large damage zone around the crack tip at G slightly lower than the threshold, while the damage zone is hardly observed at G higher than the threshold. The results indicate that, for the necking DN gels, crack initiation has a large energy barrier owing to the formation of the damage zone; once this barrier is overcome, the excess energy release accelerates the crack propagation and therefore the gels exhibit dynamic fracture.

The phase field approach is widely used to model fracture behaviors due to the absence of the need to track the crack topology and the ability to predict crack nucleation and branching. In this work, the asynchronous variational integrator (AVI) is adapted for the phase field approach of dynamic brittle fracture. The AVI is derived from Hamilton's principle and allows each element in the mesh to have its own local time step that may be different from others'. While the displacement field is explicitly updated, the phase field is implicitly solved, with upper and lower bounds strictly and conveniently enforced. In particular, two important variants of the phase field approach, the AT1 and AT2 models, are equally easily implemented. Several benchmark problems are used to study the performances of both the AT1 and AT2 models, and the results show that the AVI for the phase field approach significantly speeds up the computational efficiency and successfully captures the complicated dynamic fracture behavior.

As a simplified model, wood can be considered as consisting of hollow cells held together by a weaker middle lamella. We use a high-resolution finite element model, endowed with a phase field model for brittle fracture to model such a structure. The micro-structured discretised materials, with and without middle lamella, differ significantly from a homogenous continuum reference in terms of damage patterns. There is also significant difference between the models with and without middle lamella, since damage localises in the weaker middle lamella, which underlines the importance to include the middle lamella in mechanical analyses. At increased loading velocity, the damage becomes scattered over a larger region ahead of the crack tip, indicating that dynamic effects affect the crack behaviour and that dynamics need to be taken into consideration when loading rates are high or when crack propagation is unstable.

Polymeric materials, in particular composite materials, have seen an increase in use in naval applications. Under these conditions, structural materials need to be able to withstand dynamic loads such as those from underwater blast and wave slamming, as well as environmental factors like moisture absorption and ultraviolet radiation. Both conditions add challenges in the design process due to complex loading conditions and change in material behavior. This dissertation presents an experimental study to understand the dynamic fracture behavior of poly(methyl methacrylate) (PMMA) and carbon fiber/epoxy. Dynamic loading was obtained by launching projectiles from a gas gun onto the specimen. A combination of ultra high-speed photography and digital image correlation was used to gather full-field experimental data. First, a methodology was established using PMMA to obtain pure mode-II fracture loading conditions. This study generated valuable experimental results, which updated current data available to the fracture community. Sharp transitions from mode-II to mode-I fracture loading were observed in addition to the effect of crack-tip sharpness and loading conditions on the crack path.Next, carbon fiber/epoxy specimens were studied after developing an experimental setup capable of obtaining dynamic mode-II loading conditions. The effect of moisture absorption on the fracture behavior of unidirectional carbon fiber/epoxy samples was studied. Two different methods were used to obtain significant moisture contents, first, samples were hygrothermally aged at 65°C for 28 days and second, submersion in room temperature distilled water for 400 days. Additionally, a control group of samples with no moisture content was used to compare results. This study determined that moisture absorption decreased the fracture toughness of the material, however no differences were observed between the different aging methods.Last, unidirectional and woven carbon fiber samples were subject to

Polymeric materials, in particular composite materials, have seen an increase in use in naval applications. Under these conditions, structural materials need to be able to withstand dynamic loads such as those from underwater blast and wave slamming, as well as environmental factors like moisture absorption and ultraviolet radiation. Both conditions add challenges in the design process due to complex loading conditions and change in material behavior. This dissertation presents an experimental study to understand the dynamic fracture behavior of poly(methyl methacrylate) (PMMA) and carbon fiber/epoxy. Dynamic loading was obtained by launching projectiles from a gas gun onto the specimen. A combination of ultra high-speed photography and digital image correlation was used to gather full-field experimental data. First, a methodology was established using PMMA to obtain pure mode-II fracture loading conditions. This study generated valuable experimental results, which updated current data available to the fracture community. Sharp transitions from mode-II to mode-I fracture loading were observed in addition to the effect of crack-tip sharpness and loading conditions on the crack path.Next, carbon fiber/epoxy specimens were studied after developing an experimental setup capable of obtaining dynamic mode-II loading conditions. The effect of moisture absorption on the fracture behavior of unidirectional carbon fiber/epoxy samples was studied. Two different methods were used to obtain significant moisture contents, first, samples were hygrothermally aged at 65°C for 28 days and second, submersion in room temperature distilled water for 400 days. Additionally, a control group of samples with no moisture content was used to compare results. This study determined that moisture absorption decreased the fracture toughness of the material, however no differences were observed between the different aging methods.Last, unidirectional and woven carbon fiber samples were subject to

Composite materials are becoming increasingly widespread across applications such as aerospace, automotive, wind and solar energy systems, on account of their superior specific strength and stiffness. There are still unresolved challenges in the structural analysis of composite materials in practical engineering applications, owing to their complex structures. Peridynamic theory is a robust technique for the analysis of composite structures; however, it is computationally costly. Structural idealizations are commonly applied in engineering applications to represent structures with a reasonable simplicity, thus reducing the computational cost. Idealization of a planar structure, such as a plate, reduces the number of peridynamic interactions to be solved. In this study, a peridynamic plate formulation of an orthotropic plate with transverse shear deformation is proposed. Peridynamic bond parameters are derived by localizing the peridynamic equations of motion and comparing them with equations of motion obtained using classical continuum mechanics. A failure theory is implemented in the formulation in order to perform crack propagation simulations. A MATLAB script is written using the proposed formulation for the solution of orthotropic plates under bending. The peridynamic solutions are obtained for both elastic deformations and failure prediction of pre-cracked plates, and are validated using finite-element analysis solutions.

Composite materials are becoming increasingly widespread across applications such as aerospace, automotive, wind and solar energy systems, on account of their superior specific strength and stiffness. There are still unresolved challenges in the structural analysis of composite materials in practical engineering applications, owing to their complex structures. Peridynamic theory is a robust technique for the analysis of composite structures; however, it is computationally costly. Structural idealizations are commonly applied in engineering applications to represent structures with a reasonable simplicity, thus reducing the computational cost. Idealization of a planar structure, such as a plate, reduces the number of peridynamic interactions to be solved. In this study, a peridynamic plate formulation of an orthotropic plate with transverse shear deformation is proposed. Peridynamic bond parameters are derived by localizing the peridynamic equations of motion and comparing them with equations of motion obtained using classical continuum mechanics. A failure theory is implemented in the formulation in order to perform crack propagation simulations. A MATLAB script is written using the proposed formulation for the solution of orthotropic plates under bending. The peridynamic solutions are obtained for both elastic deformations and failure prediction of pre-cracked plates, and are validated using finite-element analysis solutions.

[EN] In this work, the influence of crack propagation velocity in the stress intensity factor has been studied. The analysis is performed with a lattice method and a linear elastic constitutive model. Numerous researchers determined the relationship between the dynamic stress intensity factor and crack propagation velocity with experimental and analytical results. They showed that toughness increases asymptotically when the crack tip velocity is near to a critical. However, these methods are very complex and computationally expensive; furthermore, the model requires the use of several parameters that are not easily obtained. Moreover, its practical implementation is not always feasible. Hence, it is usually omitted. This paper aims to capture the physics of this complex problem with a simple fracture criterion. The selected criterion is based on the maximum principal strain implemented in a lattice model. The method used to calculate the stress intensity factor is validated with other numerical methods. The selected example is a finite 2D notched under mode I fracture and different loads rates. Results show that the proposed model captures the asymptotic behaviour of the SIF in function of crack speed, as reported in the aforementioned models.

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

[EN] In this work, the influence of crack propagation velocity in the stress intensity factor has been studied. The analysis is performed with a lattice method and a linear elastic constitutive model. Numerous researchers determined the relationship between the dynamic stress intensity factor and crack propagation velocity with experimental and analytical results. They showed that toughness increases asymptotically when the crack tip velocity is near to a critical. However, these methods are very complex and computationally expensive; furthermore, the model requires the use of several parameters that are not easily obtained. Moreover, its practical implementation is not always feasible. Hence, it is usually omitted. This paper aims to capture the physics of this complex problem with a simple fracture criterion. The selected criterion is based on the maximum principal strain implemented in a lattice model. The method used to calculate the stress intensity factor is validated with other numerical methods. The selected example is a finite 2D notched under mode I fracture and different loads rates. Results show that the proposed model captures the asymptotic behaviour of the SIF in function of crack speed, as reported in the aforementioned models.

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

In this paper, the thick-level-set method is used to model stable and unstable (stick/slip) crack propagation in the dynamic double cantilever beam (DCB) test for unidirectional composite laminates. The thick-level-set method uses a predefined damage profile to describe the fracture process zone and allows for accurate evaluation of the global energy release rate. A phenomenological model is introduced to calculate the crack speed as a function of the energy release rate. The potential capability of the proposed approach is demonstrated by simulating a series of dynamic DCB tests under variable test rates., Applied Mechanics, Materials- Mechanics- Management & Design

2017 ISRM Young Scholars’ Symposium on Rock Mechanics (YSRM 2017) & 2017 International Conference on New Development in Rock Mechanics and Geotechnical Engineering (NDRMGE 2017).10-13 May 2017,Jeju,Korea, Smeared crack approach coupled with cohesive zone model has been an attractive way for the realistic simulations of dynamic fracturing of rocks and frequently utilized in the framework of the Finite Element Method (FEM). In many cases, cohesive elements (CEs) with initially-zero-thickness are inserted at the onset of numerical simulations and these are used to express the dynamic fracturing, which is called “intrinsic cohesive zone model (ICZM)”. However, since the ICZM must introduce penalty terms to express the intact behavior of the CEs, this tends to render higher compliance of bulk rock, resulting in smaller wave propagation speed. In this paper, by introducing a so-called “extrinsic cohesive zone model (ECZM)”, which adaptively inserts the CEs, we compared the results of 3-D dynamic fracturing simulations by the ICZM and ECZM using the experimental data obtained from the dynamic spalling test for rocks. Using the same Young’s modulus, Poisson’s ratio, density and strengths estimated from the experiments, our results suggest that the outcome of the ICZM and experiment showed large discrepancy especially for the intact stress wave propagation while the ECZM showed good agreement with the experiment. Therefore, our results could have some implications on the current situation in which more and more simulations using the ICZM such as in hybrid FEM-DEM have been applied to rock fracture mechanics problems.

Smeared crack approach coupled with cohesive zone model has been an attractive way for the realistic simulations of dynamic fracturing of rocks and frequently utilized in the framework of the Finite Element Method (FEM). In many cases, cohesive elements (CEs) with initially-zero-thickness are inserted at the onset of numerical simulations and these are used to express the dynamic fracturing, which is called “intrinsic cohesive zone model (ICZM)”. However, since the ICZM must introduce penalty terms to express the intact behavior of the CEs, this tends to render higher compliance of bulk rock, resulting in smaller wave propagation speed. In this paper, by introducing a so-called “extrinsic cohesive zone model (ECZM)”, which adaptively inserts the CEs, we compared the results of 3-D dynamic fracturing simulations by the ICZM and ECZM using the experimental data obtained from the dynamic spalling test for rocks. Using the same Young’s modulus, Poisson’s ratio, density and strengths estimated from the experiments, our results suggest that the outcome of the ICZM and experiment showed large discrepancy especially for the intact stress wave propagation while the ECZM showed good agreement with the experiment. Therefore, our results could have some implications on the current situation in which more and more simulations using the ICZM such as in hybrid FEM-DEM have been applied to rock fracture mechanics problems., 2017 ISRM Young Scholars’ Symposium on Rock Mechanics (YSRM 2017) & 2017 International Conference on New Development in Rock Mechanics and Geotechnical Engineering (NDRMGE 2017).10-13 May 2017,Jeju,Korea

In this paper, the thick-level-set method is used to model stable and unstable (stick/slip) crack propagation in the dynamic double cantilever beam (DCB) test for unidirectional composite laminates. The thick-level-set method uses a predefined damage profile to describe the fracture process zone and allows for accurate evaluation of the global energy release rate. A phenomenological model is introduced to calculate the crack speed as a function of the energy release rate. The potential capability of the proposed approach is demonstrated by simulating a series of dynamic DCB tests under variable test rates., Applied Mechanics, Materials- Mechanics- Management & Design

The paper deals with degradation of the structural and mechanical parameters of DOMEX 700MC steel due to welding process. It is focused on evaluation of the mechanical heterogeneities influence at higher strain rate. DOMEX 700MC presents the microalloyed steel with the increasing utilization mainly for the lightweight construction of the transportation means. The precise characterization of material behavior at high strain rate is of great importance for reliable prediction of welded parts in that application. Evaluation of the local mechanical parameters of experimental welding joints was performed. The three-point instrumented dynamic fracture analyses and instrumented indentation tests were employed to study the elastic-plastic response of critical weldments sublayers. The static vs. dynamic yield strength and energy consumption to fracture was compared in different stages of temper influence. Structural analyses brought the information about the conducted structural changes in tempered zone. Fractography analyses were focused on evaluation of local differences in fracture behavior influenced by real metallurgy quality. Based on the experimental results, the used combination of loading method is suitable to determine the local mechanical properties and enable the estimation of temper sensitivity of analyzed steel., Článek je věnován degradaci strukturních a mechanických parametrů oceli DOMEX700MC vlivem svařování. K hodnocení kritické subvrstvy svarem ovlivněných oblastí byly použity dynamické rázové zkoušky, instrumentované indentační zkoušky, strukturní a fraktografické analýzy.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material's microstructure is described through the configuration and connectivity of the particles and the model's sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

In this report, we present peridynamic simulation results and experimental evidence for the origin and evolution of cracks in a thin glass plate that is impacted near its center, at moderate speeds (150m/s), by a small steel projectile. A polycarbonate thin backing plate is used to preserve location of glass fragments. In the computational model the tape used on the sides to hold the two plates together is absent. Upon impact, a series of circular cracks form on the strike face, followed by radial cracks advancing from the impact site towards the boundaries, driven by elastic deformations induced behind the Rayleigh wave. At the same time, a through-thickness Hertz-crack grows, as well as cracks parallel to the sides of the plates, induced by waves reflected from the boundaries reinforcing surface waves emanated from the impact region. We present evidence gathered from our simulations for the origin and evolution of these cracks, and confirm these results with fractography experiments of post-mortem samples. The results provide evidence of the predictive capabilities of the peridynamic model for simulating complex dynamic fracture behavior in brittle targets and offer arguments for what the sufficient ingredients in a model for dynamic brittle fracture should be., Presented at the U.S. National Congress on Theoretical and Applied Mechanics, June 17 2014.

The purpose of this project has been to develop continuum phase-field models in concert with numerical methods for their solution to study dynamic brittle fracture. In contrast to discrete descriptions of fracture, phase-field descriptions do not require numerical tracking of discontinuities in the displacement field. This greatly reduces implementation complexity. During this project we have studied the basic formulations of the phase-field fracture theory, leading to second order partial differential equations (PDEs), along with the effect of adding higher-order gradients to the standard phase-field theory, leading to fourth and higher order PDEs. We have derived the thermodynamically consistent governing equations for the phase-field models by way of both balance law approaches and variational principles based on energy balance assumptions. We have completed studies on the implementation of second and fourth order phase-field methods for fracture brittle elastic materials as well as for fracture in brittle piezoelectric materials. We found that the fourth order phase-field model leads to higher regularity in the exact phase-field solution, which is exploited by the smooth function spaces utilized in isogeometric analysis. This increased regularity improves the convergence rate of the numerical solution and opens the door to higher-order convergence rates for fracture problems.

This paper proposes a computational framework for dynamic crack propagation in rubber in which a nonlinear finite element analysis using cohesive zone modeling approach is used. A suddenly initiated crack at the center of biaxially stretched sheet problem is studied under plane stress conditions. A transient dynamic analysis using implicit time integration scheme is performed. In the constitutive modeling, the continuum is characterized by finite-viscoelasticity theory and coupled with the fracture processes using a cohesive zone model. This computational framework was introduced previously by the present authors (Elmukashfi and Kroon, 2012). In the current work, the use of a rate-dependent cohesive model is examined in addition to investigation of generalized biaxial loading cases. A Kelvin-Voigt element is used to describe the rate-dependent cohesive model wherein the spring is described by a bilinear law and dashpot with a constant viscosity is adopted. An explicit integration is used to incorporate the rate-dependent cohesive model in the finite element environment. A parametric study over the cohesive viscosity is performed and the steady crack propagation velocity is evaluated and compared with experimental data. It appears that the viscosity varies with the crack speed. Further, the total work of fracture is estimated using rate-independent cohesive law such that the strength of the cohesive zone is assumed to be constant and the separation work per unit area is determined form the experimental data. The results show that fracture-related processes, i.e. creation of new surfaces, cavitation and crystallization; contribute to the total work of fracture in a contradictory manner., QC 20140811

The modeling of high velocity impact into brittle or quasibrittle solids is hampered by the unavailability of a constitutive model capturing the effects of material comminution into very fine particles. The present objective is to develop such a model, usable in finite element programs. The comminution at very high strain rates can dissipate a large portion of the kinetic energy Of an impacting missile. The spatial derivative of the energy dissipated by comminution gives a force resisting the penetration, which is superposed on the nodal forces obtained from the static constitutive model in a finite element program. The present theory is inspired partly by Grady's model for expansive comminution due to explosion inside a hollow sphere, and partly by analogy with turbulence. In high velocity turbulent flow, the energy dissipation rate gets enhanced by the formation of micro-vortices (eddies) which dissipate energy by viscous shear stress. Similarly, here it is assumed that the energy dissipation at fast deformation of a confined solid gets enhanced by the release of kinetic energy of the motion associated with a high-rate shear strain of forming particles. For simplicity, the shape of these particles in the plane of maximum shear rate is considered to be regular hexagons. The particle sizes are assumed to be distributed according to the Schuhmann power law. The condition that the rate of release of the local kinetic energy must be equal to the interface fracture energy yields a relation between the particle size, the shear strain rate, the fracture energy and the mass density. As one experimental justification, the present theory agrees with Grady's empirical observation that, in impact events, the average particle size is proportional to the (-2/3) power of the shear strain rate. The main characteristic of the comminution process is a dimensionless number B-a (Eq. (37)) representing the ratio of the local kinetic energy of shear strain rate to the maximum possible s, Postprint (published version)

The new theory presented in the preceding paper, which models the dynamic comminution of concrete due to very high shear strain rate, is now compared to recent test data on the penetration of projectiles through concrete walls of different thicknesses, ranging from 127 to 254 mm. These data are analyzed by an explicit finite element code using the new microplane constitutive model M7 for concrete, which was previously shown to provide the most realistic description of the quasi-static uni-, bi- and tri-axial test data with complex loading path and unloading. Model M7 incorporates the quasi-static strain rate effects due viscoelasticity and to the rate of cohesive crack debonding based on activation energy of bond ruptures, which are expected to extend to very high rates. Here model M7 is further enhanced by apparent viscosity capturing the energy dissipation due to the strain-rate effect of comminution. The maximum shear strain rates in the computations are of the order of 10(5) s(-1). The simulations document that, within the inevitable uncertainties, the measured exit velocities of the projectiles can be matched quite satisfactorily and the observed shapes of the entry and exit craters can be reproduced correctly. (C) 2014 Published by Elsevier Ltd., Postprint (published version)

Dynamic crack propagation in rubber is modeled and analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the cohesive strength has been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with the experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth such that the strength and the surface energy vary with the crack speed., QC 20140626

The behavior of concrete structures is strongly influenced by the loading rate. Compared to quasi-static loading, on meso and macro-scale concrete loaded by impact loading acts in a different way. First, there is a strain-rate influence on strength, stiffness, ductility, and, second, there are inertia forces activated which influence the resistance and failure mode of concrete structure. The experimental and theoretical studies show that the influence of loading rate on tensile behavior of concrete is relatively strong. In dynamic testing the split Hopkinson bar (SHB) is used to measure concrete tensile resistance. The results of the experimental measurements show that after reaching some critical strain rate tensile resistance progressively increases with increase of strain rate. The questions discussed in the paper are: (i) what is the reason for progressive increase of tensile resistance? and (ii) can the resistance be attributed only to material strength or are there some other effects? To answer these questions the numerical analysis on a simple elastic-cohesive FE model is carried out. Moreover, simulation of the compressive pulse in a concrete bar, which is reflected from the free end-surface of the bar and causes tensile fracture, is carried out for different loading rates. The evaluation of the results clearly shows that the progressive increase of tensile resistance (apparent strength) can be attributed to structural inertia of the fracture zone, which is invoked by cracking of concrete and is not to the true material strength. It is shown that the size of the fracture process zone significantly influence apparent strength. Similar as the true strength it is also discussed that with the increase of strain rate concrete fracture energy does not increase progressively.

The behavior of concrete structures is strongly influenced by the loading rate. Compared to quasi-static loading, on meso and macro-scale concrete loaded by impact loading acts in a different way. First, there is a strain-rate influence on strength, stiffness, ductility, and, second, there are inertia forces activated which influence the resistance and failure mode of concrete structure. The experimental and theoretical studies show that the influence of loading rate on tensile behavior of concrete is relatively strong. In dynamic testing the split Hopkinson bar (SHB) is used to measure concrete tensile resistance. The results of the experimental measurements show that after reaching some critical strain rate tensile resistance progressively increases with increase of strain rate. The questions discussed in the paper are: (i) what is the reason for progressive increase of tensile resistance ? and (ii) can the resistance be attributed only to material strength or are there some other effects ? To answer these questions the numerical analysis on a simple elastic-cohesive FE model is carried out. Moreover, simulation of the compressive pulse in a concrete bar, which is reflected from the free end-surface of the bar and causes tensile fracture, is carried out for different loading rates. The evaluation of the results clearly shows that the progressive increase of tensile resistance (apparent strength) can be attributed to structural inertia of the fracture zone, which is invoked by cracking of concrete and is not to the true material strength. It is shown that the size of the fracture process zone significantly influence apparent strength. Similar as the true strength it is also discussed that with the increase of strain rate concrete fracture energy does not increase progressively., Structural Engineering, Civil Engineering and Geosciences

The behavior of concrete structures is strongly influenced by the loading rate. Compared to quasi-static loading, on meso and macro-scale concrete loaded by impact loading acts in a different way. First, there is a strain-rate influence on strength, stiffness, ductility, and, second, there are inertia forces activated which influence the resistance and failure mode of concrete structure. The experimental and theoretical studies show that the influence of loading rate on tensile behavior of concrete is relatively strong. In dynamic testing the split Hopkinson bar (SHB) is used to measure concrete tensile resistance. The results of the experimental measurements show that after reaching some critical strain rate tensile resistance progressively increases with increase of strain rate. The questions discussed in the paper are: (i) what is the reason for progressive increase of tensile resistance ? and (ii) can the resistance be attributed only to material strength or are there some other effects ? To answer these questions the numerical analysis on a simple elastic-cohesive FE model is carried out. Moreover, simulation of the compressive pulse in a concrete bar, which is reflected from the free end-surface of the bar and causes tensile fracture, is carried out for different loading rates. The evaluation of the results clearly shows that the progressive increase of tensile resistance (apparent strength) can be attributed to structural inertia of the fracture zone, which is invoked by cracking of concrete and is not to the true material strength. It is shown that the size of the fracture process zone significantly influence apparent strength. Similar as the true strength it is also discussed that with the increase of strain rate concrete fracture energy does not increase progressively., Structural Engineering, Civil Engineering and Geosciences

In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place., QC 20121109

The semi-circular bend (SCB) specimen was suggested in 1984 for testing mode I fracture toughness of rock and other geo or brittle materials. Since then SCB has been used worldwide, extended and improved for many other applications by various researchers. Formulations for mode I and mixed mode fracture of this specimen proposed by a number of researchers are presented. Methods to determine fracture toughness using both straight notched and chevron-notched specimens have been proposed although the general consensus is that a specimen having a sharp straight notch should yield accurate fracture toughness. Other applications of SCB specimen include testing of rock subjected to in situ conditions such as elevated temperature, confining pressure and pore water pressure. Furthermore it has been proven that it is a suitable specimen to test fracture toughness of rock at very high strain rates. Areas requiring further research to improve the accuracy of formulations are identified.

This report presents a summary of the research supported by the AFOSR during the period June 1 2008 to May 31 2011 on the formulation and analysis of numerical methods for the analysis of failure in solids and structures of interest to the Air Force mission. Of special interest is the consideration of dynamic range, including dynamic fracture and failure. The development of new enhanced finite element methods that incorporate the discontinuous solutions characteristic of the failure of solids and structures has been one of the main outcomes of the project. The new methods include finite element formulations that address the special issues arising in the dynamic range, including for instance brittle/ductile failure mode transitions and crack branching. Similarly, the extension of these finite elements to the general three-dimensional range and the failure of partially saturated poroplastic media have also been addressed in this project., The original document contains color images.

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in Engineering, 2004) where the crack is modeled by a set of cracked segments. However, in contrast to the above mentioned paper, we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity. Instead, the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics. We apply this method to several two- and three-dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any "mesh" orientation bias. © 2010 Elsevier B.V.

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in Engineering, 2004) where the crack is modeled by a set of cracked segments. However, in contrast to the above mentioned paper, we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity. Instead, the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics. We apply this method to several two- and three-dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any "mesh" orientation bias. © 2010 Elsevier B.V.

A fundamental study was conducted to evaluate dynamic fracture and failure in functionally graded materials (FGM) at room and high temperatures. Asymptotic analysis in conjunction with displacement potentials was used to develop thermo-mechanical stress fields for a mixed mode propagating crack in homogenous and FGMs. First, asymptotic temperature fields were derived for an exponential variation of thermal conductivity and later these temperature fields were used in deriving stress fields. Using asymptotic thermo-mechanical stress fields the variation of maximum shear stress, circumferential stress and strain energy density as a function of temperature around the crack-tip were generated. Finally, utilizing the minimum strain energy density criterion and the maximum circumferential stress criterion, the crack growth direction for various crack-tip speeds, non-homogeneity coefficients and temperature fields were determined., The original document contains color images.

A very important concern on the dynamic fracture of engineering materials is the initiation and propagation of cracks which are subjected to elastic, plastic, or shock waves. However, this review is limited to elastic stress waves for which the theory of elastodynamics is applicable and thereby minimizing the complexity of the report. Several topics were reviewed for this report and, although they were limited to elastic stress waves. The following areas of interest were covered: the development of an experimental method that uses a disc for loading a half-plane crack with a tensile pulse; the response of a crack under plane-strain loading using a plate-impact setup; and, finally, the effects of dilatational (longitudinal) and transverse (shear) waves interacting with a crack., The original document contains color images.

In ferritic steels a propagating cleavage microcrack changes its propagation direction as it advances from grain to grain. This is due to differences in the orientation of the cleavage planes of two neighboring grains. In order to reach a cleavage plane in a new grain, a microcrack must first penetrate the grain boundary. Grain boundaries therefore act as natural barriers in cleavage fracture. The influence of a grain boundary and the associated misorientation in cleavage planes on crack arrest is here examined using a 3D finite element model with axisymmetric periodicity, representing two grains whose cleavage planes are tilted and twisted relative to each other. The temperature dependent mechanical properties of ferrite are modeled using a temperature dependent viscoplastic response. The development of the crack front as the microcrack penetrates through a grain boundary is here presented. The influence of the twist misorientation on the critical grain size, defined as the largest grain size that can arrest a rapidly propagating microcrack, is examined in a temperature range corresponding to the ductile to brittle transition (DBT) region. It is shown that when both tilt and twist misorientation are present, the influence of tilt and twist, respectively, on crack growth resistance can be decoupled., QC 20100525

Cleavage fracture in ferritic steels is controlled by several critical steps. First a microcrack must nucleate, grow and overcome barriers, such as grain boundaries. The latter is examined here by use of a periodic, axisymmetric model representing two grains. A microcrack nucleated at the center in one grain is driven by a constant remotely applied stress towards the second grain. The cleavage planes of the grain in which the microcrack is nucleated coincide with the principal loading direction. In the adjacent grain, due to misalignment in possible cleavage planes, the propagation direction changes and separation occurs in mixed mode, involving both normal and shear separations. The temperature dependence of the mechanical properties of the material is accounted for by use of a temperature dependent elasto viscoplastic material model. The largest grain size that can arrest a rapidly propagating microcrack is defined as the critical grain size. The effects of stress state and temperature on the critical grain size are examined. The influence of mismatch in lattice orientation between two adjacent grains in terms of a tilt angle is both qualitatively and quantitatively described. It is shown that the critical grain size is influenced by plastic geometry change and prestraining, which depend on the applied stress state. The results also show that a microcrack can be arrested in an adjacent grain under specific conditions., QC 20100525

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.