Cohesive Elements Research Articles

Dynamic impact experimental and global cohesive element method to shale fracture characterization

The fracture behavior and various of mechanical mechanisms of shale are greatly influenced by the bedding surface. To study the effects of bedding angles on the crack propagation patterns and dynamic fracture properties, the shale notched semi-circular bend (NSCB) specimens were studied by the modified split Hopkinson pressure bar (SHPB) device, three-dimensional digital image correlation (3D-DIC) system with the high-speed (HS) photography and ABAQUS finite element software. The research results showed that at the same impact velocity, the 30°, 45° and 60° beddings played an obvious deflexion effect on the shale crack propagation. However, the 0° and 90° bedding did not deflect the crack propagation. The dynamic fracture toughness of shale NSCB specimens increased with the increase of impact loading rate but the increase magnitude decreased with the higher impact rate. The kinetic energy during specimen fracture was calculated by the extracting displacement data from the 3D-DIC, and it was found that the kinetic energy accounted for less than 6% of the absorbed energy during the dynamic compression of the shale. The crack propagation characteristics of ABAQUS simulation was approximately consistent with the experimental results. It is conducive to the study of shale gas fracturing and the formation of complex seam networks in shale and it is of great research significance for the efficient extraction of shale gas.

Influence of Failure-Load Prediction in Composite Single-Lap Joints with Brittle and Ductile Adhesives Using Different Progressive-Damage Techniques.

The usage of adhesively bonded joints, such as single-lap and double-lap joints, is increasing rapidly in aerospace composite structures as a popular alternative to bolts and rivets. Compared to the conventional joining methods such as fastening and riveting, adhesive-bonding technology better prevents damage to composite structures due to the smooth configuration and the mitigation of stress concentration around holes. In this work, the built-in progressive-damage-modeling techniques in Abaqus, including the cohesive zone model (CZM) and the virtual crack closure technique (VCCT), are used to predict the strength and progressive failure of composite single-lap joints subjected to tensile loading. Modeling of an adhesive layer by using a zero/non-zero-thickness cohesive element, cohesive surface, and VCCT is investigated, as is the effect of brittle and ductile adhesives. Two-dimensional finite-element models with different damage-modeling strategies are performed in this study. The failure-load predictions are compared with the experimental results obtained from the literature. For the ductile adhesive, the predicted failure loads using a zero/non-zero-thickness cohesive elements and a cohesive surface are all shown to be in good agreement with the experiments. However, the VCCT technique predicts higher failure loads. For a brittle adhesive, on the other hand, the predictions by zero/non-zero-thickness cohesive elements and cohesive surfaces reveal notable deviations compared to the experimental results. In contrast to the ductile adhesive, the VCCT technique is revealed to be accurate in predicting the brittle adhesive.

A 3D discrete model for soil desiccation cracking in consideration of moisture diffusion

Soil desiccation cracking is a common phenomenon on the earth surface. Numerical modeling is an effective approach to study the desiccation cracking mechanism of soil. However, no three-dimensional (3D) model in the literature can consider the influence of desiccation cracking on moisture migration. To fill this gap, this work develops a novel 3D moisture diffusion discrete model that is capable of dynamically assessing the effect of cracking on moisture diffusion and allowing moisture to be discontinuous on both sides of the cracks. Then, the parametric analysis of the moisture exchange coefficient in the 3D moisture diffusion discrete model is carried out for moisture diffusion in continuous media, and the selection criterion of the moisture exchange coefficient for the unbroken cohesive element is given. Subsequently, an example of moisture migration in a medium with one crack is provided to illustrate the crack hindering effect on moisture migration. Finally, combining the 3D moisture diffusion discrete model with the finite-discrete element method (FDEM), the moisture diffusion-fracture coupling model is built to study the desiccation cracking in a strip soil and the crack pattern of a rectangular soil. The evolution of crack area and volume with moisture content is quantitatively analyzed. The modeling number and average width of cracks in the strip soil show a good consistency with the experimental results, and the crack pattern of the rectangular soil matches well with the existing numerical results, validating the coupled moisture diffusion-fracture model. Additionally, the parametric study of soil desiccation cracking is performed. The developed model offers a powerful tool for exploring soil desiccation cracking.

Experiments and modelling of competitive failure behaviour of CFRP stepped-lap repairs with different design parameters

Stepped-lap bonding has been preferably applied in composite repairing. Nevertheless, the design of repairs presents challenges due to the complex failure mechanism. This paper aims to investigate the mechanical performance and failure mechanism of Carbon Fibre Reinforced Polymer (CFRP) repairs with stepped-lap under tensile loading. Two design parameters are studied, i.e. step numbers and slope combinations. Furthermore, the two different configuration methods of Cohesive Zone Model (CZM) are creatively employed in the developed numerical models, i.e. the adhesive interface adopts cohesive contact, while zero-thickness cohesive elements are inserted in the interlayer between the first two plies of CFRP to simulate possible delamination. Simulations are verified against experimental failure loads, as well as damage evolutions in the repaired adhesive interface, ply interlayer and CFRP laminate. Results showed that the tensile strength of stepped-lap repairs increased with step numbers. The three-step stepped-lap repair with a slope combination of 1/20–1/20–1/20 bears the strongest tensile strength and greatest load-bearing capacity. This work reveals the competitive failure mechanism in adhesively repaired composites and offers a reliable prediction approach for the design of bonded CFRP repairs in practical aeronautic applications.

FDEM analysis of deep rock mass failure and its impact on horizontal bearing capacity in offshore wind turbine piles

Rock-socketed single piles are integral components of offshore wind turbine foundations in rock-based sea areas. Traditional numerical models have been employed to analyze the failure modes and lateral interactions between piles and the rock medium. However, these do not simulate the decline in the horizontal bearing capacity due to rock breakage, posing potential safety risks. In this study, the combined finite-discrete element method (FDEM) was used to simulate the deformation process of submarine rock masses transitioning from an intact to a fractured state. A notable innovation in the proposed approach is the consideration of the effects of the rock mass fracture and pile-rock cohesion on the load-bearing behavior of single-pile foundations. The FDEM model was validated through full-scale lateral load tests. The results showed a distinctive sequence in failure mechanisms, with the tensile failure of the rock mass around the pile at significant depths preceding the compression failure. Local failure at the contact interface significantly reduced the horizontal bearing capacity—a phenomenon that is inadequately represented by conventional finite element models. Furthermore, the cohesion element embedding method was introduced and used to simulate the influence of the tensile properties inherent in grouting materials on the failure strength and mode of the pile-rock interface. This study highlighted the importance of considering the nuanced effects of rock failure in the design of horizontally loaded piles. Failure to do so could lead to nonconservative designs and compromise the safety and efficacy of offshore wind turbine foundations. By pushing the boundaries of numerical modeling with the FDEM, the study aims to provide a more accurate and reliable framework for the design and assessment of rock-socketed single piles for offshore wind energy applications.

Seepage propagation simulation of a tunnel gasketed joint using the cohesive zone model

The gasketed joint plays a critical role in the waterproof or water-leakage resistance performance of tunnel segmental linings. This study presents a novel numerical simulation method based on the cohesive zone model (CZM) to accurately analyze the seepage phenomena along the interface of elastic sealing gaskets in shield tunnel joints. While traditional finite element methods are widely used, they cannot simulate interface opening, limiting their applicability. To overcome this challenge, the proposed method leverages the concept of hydraulic fracturing and redefines the seepage problem as a seepage problem of elastic impermeable media. By establishing a two-dimensional hydraulic fracturing model using the CZM, this study conducts a comprehensive sensitivity analysis of the cohesive element parameters and validates the accuracy and effectiveness of simulating seepage along the interface of elastic impermeable media. Based on the insights gained from the sensitivity analysis, a seepage calculation model is successfully developed, considering the unique characteristics of elastic sealing gaskets. The model is applied to three cross-sectional shapes of elastic sealing gaskets, and their seepage characteristics are analyzed based on the interface opening displacement and liquid pressure distributions during seepage. The results highlight the SC3 cross-sectional shape, which exhibits minimal opening displacement and the highest net water pressure at the moment of seepage occurrence, demonstrating superior waterproof performance compared to other forms. Through the introduction of the CZM, comprehensive parameter analysis, validation, and practical application, this study overcomes the limitations of traditional finite element methods in simulating interface opening. The proposed model, which more accurately captures seepage characteristics than other methods, provides an optimized simulation method for analyzing hydraulic coupling issues related to seepage at shield tunnel joints and is a significant contribution to the field, enhancing our understanding and enabling more accurate analyses of seepage phenomena in shield tunnel joints using elastic sealing gaskets.

Numerical simulation study of PDC cutter rock breaking based on CT scanning to construct a rock model and apply a cohesive element to construct a discretization element

To more accurately simulate the interaction between a polycrystalline diamond compact (PDC) cutter and rock, the structural characteristics of basalt were analysed using computed tomography (CT), and a hard particle–matrix–pore three-phase rock model was established due to its structural characteristics. Three-dimensional models of the interaction between PDC cutters and rock were established by using the finite–discrete element method (FDEM). Using Voronoi diagrams to randomize the grid structure and optimize the propagation of cracks. This study revealed that a broken zone near the cutters will form during the rock breaking process, resulting in the affected area being larger than the area in contact with the formation rock. The rock breaking ability of circular cutters is stronger than that of conical cutters. When a conical cutter breaks hard particles, they are mainly striped or forced away from the rock, but circular cutters can divide hard particles into small pieces. The numerical model used in the research institute can effectively calculation results of the process of interaction between PDC cutter and rock, this proves that the application of the cohesive method to simulate rock breaking by PDC cutters is reliable.

Investigation of ice wedge bearing capacity based on an anisotropic beam analogy

Ships and offshore structures operating in ice-covered waters need to be able to break ice. The hulls of ships and offshore structures are often designed to be inclined so that an out-of-plane force can be incurred to initiate flexural failure. During this process, an in-plane force co-exists as the result of the hull inclination, which can affect the bending process and alter the magnitude of interaction force. Additionally, the loading process can be rapid and incur dynamic effect within the ice sheet, resulting in higher peak loads compared to the scenario of quasi-static loading. The effect of in-plane force and loading speed, however, are often neglected in the modelling of the icebreaking process in order to enable simplified solutions of analytical form, which leads to deviations in the estimation of ice loads. This paper aims to investigate the effects of in-plane force and loading speed on the ice bearing capacity during the icebreaking process. This is conducted via the Finite Element Analysis (FEA) with the Cohesive Element Method (CEM). An anisotropic beam analogy concept is proposed to model a 3D ice wedge with a 2D beam which has varying properties along the axial direction. The analogy enables fast and stable simulation of ice crack propagation through its thickness direction, while maintaining properties of the ice wedge in 3D. The analogy is verified with Nevel's closed-form solution for a narrow wedge resting on an elastic foundation. After that, the beam is loaded with a rigid wall with different inclination angles and loading speeds. The results reveal that the in-plane force and loading speed have significant influence on the icebreaking process, resulting in different bearing capacity and overall energy consumption. Based on the numerical results, a regression study is conducted to derive a correction coefficient accounting for the joint effect of in-plane force and loading speed on the bearing capacity, which can be incorporated into ice-structure interaction simulation models.

Computational Evaluation of the Fracture Behavior of Porous Asphalt Concrete Exposed to Moisture and Salt Erosion.

Salt erosion has an adverse impact on the durability of asphalt pavements. Porous asphalt concrete is particularly susceptible to the influence of salt. In this study, a finite element model was developed to investigate the fracture behavior of PAC exposed to salt erosion. The 2D heterogeneous structure of PAC was generated with an image-aided approach to computationally study the fracture behavior of PAC. Laboratory SCB tests were conducted to validate the finite element model. The simulation results of the SCB tests indicate that the peak load of the PAC decreased by 21.8% in dry-wet cycles and 26.1% in freeze-thaw cycles compared to the control group. The salt solution accelerated the degradation of the durability of PAC under both dry-wet cycles and freeze-thaw cycle conditions, which is consistent with laboratory tests. After flushing treatment before the drying phase, the peak load of the PAC in salt environments increased by 5.3% compared to that of the samples with no flushing. Salt erosion also results in a higher average value of scalar stiffness degradation (SDEG), and the damaged elements were primarily the cohesive elements in the fracture of the PAC. Additionally, the influence of crucial factors including the void content, adhesion and cohesion, and loading rate on the fracture behavior of the PAC was analyzed. As the void content increases, the average SDEG value of the cohesive elements increases and surpasses the average SDEG value of the adhesive elements at a void content of approximately 9%. The performance of the fine aggregate matrix (FAM) has a much greater impact than the FAM-aggregate interface on the durability of the PAC. And there were more damaged CZM elements with the increase in the loading rate. Salt erosion results in higher SDEG values and a larger number of cohesive damaged elements at each loading rate.

Numerical Analysis of the Wing Leading Edge Electro-Impulse De-Icing Process Based on Cohesive Zone Model

Aircraft icing has historically been a critical cause of airplane crashes. The electro-impulse de-icing system has a wide range of applications in aircraft de-icing due to its lightweight design, low energy consumption, high efficiency, and other advantages. However, there has been little study into accurate wing electric-impulse de-icing simulation methods and the parameters impacting de-icing efficacy. Based on the damage mechanics principle and considering the influence mechanisms of interface debonding and ice fracture on ice shedding, this paper establishes a more accurate numerical model of wing electric-impulse de-icing using the Cohesive Zone Model (CZM). It simulates the process of electric-impulse de-icing at the leading edge of the NACA 0012 wing. The numerical results are compared to the experimental results, revealing that the constructed wing electro-impulse de-icing numerical model is superior. Lastly, the effects of varying ice–skin interface shear adhesion strengths, doubler loading positions, and impulse sequences on de-icing effectiveness were studied. The de-icing rate is a quantitative description of the electro-impulse’s de-icing action, defined in the numerical model as the ratio of cohesive element deletions to the total elements at the ice–skin interface. The findings reveal that varying shear adhesion strengths at the ice–skin interface significantly impact the de-icing effect. The de-icing rate steadily falls with increasing shear adhesion strength, from 66% to 56%. When two, four, and seven impulses were applied to doubler two, the de-icing rates were 59%, 71%, and 71%, respectively, significantly increasing the de-icing efficiency compared to when impulses were applied to doubler one. Doubler one and two impulse responses are overlaid differently depending on the impulse sequences, resulting in varying de-icing rates. When the impulse sequence is 20 ms, the superposition results are optimal, and the de-icing rate reaches 100%. These studies can guide the development and implementation of a wing electric-impulse de-icing system.

A Detailed Numerical Model for a New Composite Slim-Floor Slab System.

The paper concerns the numerical modelling of a new slim-floor system with innovative steel-concrete composite beams called "hybrid beams". Hybrid beams consist of a high-strength TT inverted cross-section steel profile and a concrete core made of high-performance concrete and are jointed with prestressed hollow core slabs by infill concrete and tie reinforcement. Such systems are gaining popularity since they allow the integration of the main structural members within the ceiling depth, shorten the execution time, and reduce the use of concrete and steel. A three-dimensional finite element model is proposed with all parts of the system taken into account and detailed geometry reproduction. Advanced constitutive models are adopted for steel and concrete. Special attention is paid to the proper characterisation of interfaces. The new approach to calibration of damaged elastic traction-separation constitutive model for cohesive elements is applied to concrete-to-concrete contact zones. The model is validated with outcomes of experimental field tests and analytical calculations. A satisfactory agreement between different assessment methods is obtained. The model can be used in the development phase of a new construction system, for instance, to plan further experimental campaigns or to calibrate simplified design formulas.

Energy release rate in bimaterial specimens tested in pure modes I and II

New closed-form analytical expressions of the energy release rate of a bimaterial cracked specimen are derived, when it experiences pure mode I in an asymmetric double cantilever test, and pure mode II in an asymmetric end-notched flexure test. The analysis is based on the strain coenergy and on the laminated beam theory. The results of the analytical approach are compared with those obtained using the finite element method in two ways: by the virtual crack closure technique at a test point and by cohesive elements during a virtual delamination test. The agreement is excellent in both cases.

Rate-dependent augmented finite element method for fast arbitrary crack growth

Experimental observations indicate that fast crack growth often exhibits a rate dependency, which is caused by the viscosity of the material and the dissipation of energy resulting from the branching of microcracks. To address this phenomenon in simulation modeling, this study introduces a rate-dependent augmented finite element method designed for analyzing fast arbitrary crack growth. The method utilizes the maximum principal stress criterion to simulate arbitrary crack growth and predict the path of crack growth. The elements where crack growth is expected are divided into subelements, and they are re-assembled by incorporating rate-dependent cohesive elements between them to construct augmented finite elements. The cohesive elements are based on a rate-dependent trapezoid traction–separation relationship, where the critical traction and fracture toughness are modeled as functions of the separation rate and crack growth speed. The augmented finite elements only contain external nodal degrees of freedom (DoFs), while internal nodal DoFs are condensed. They can be utilized as conventional elements. A numerical algorithm based on the proposed method was developed in this article, and it was implemented in ABAQUS via user element subroutine. The effectiveness of this method and algorithm is validated through an example of dynamic crack growth in a double cantilever beam. The results demonstrate agreement with experimental and computational findings reported in the literature. Additionally, the method was applied to other structures, yielding reasonable outcomes. Analysis of these examples confirms the method’s utility in predicting fast arbitrary crack growth under dynamic load conditions.

Numerical Investigation of the R-Curve Effect in Delamination of Composite Materials Using Cohesive Elements

This paper presents a numerical investigation of the R-curve effect in delamination propagation in composite materials. The R-curve effect refers to the phenomenon whereby resistance to crack propagation increases with the advancement of the delamination, due to toughening mechanisms, such as fiber bridging. Numerical models often neglect this effect assuming a constant value of the fracture toughness. A numerical approach based on cohesive elements and on the superposition of two bilinear traction-separation laws is adopted here to accurately predict the R-curve effect in skin-doubler composite specimens subjected to three-point bending tests. The carbon-epoxy material presents two different sensitivities to the fiber bridging phenomenon resulting in two different R-curves. Comparisons with literature experimental data, in terms of load and delaminated area vs. applied displacement, and ultrasonic C-scan images show the effectiveness of the adopted approach in simulating the R-curve effect. The predicted numerical stiffness aligns with the experimental scatter, although the maximum load is slightly underestimated by approximately 15% compared with the average experimental results. The numerical model accurately predict the R-curve effect observed in the experimental data, demonstrating a 31% increase in the maximum load for the material configuration exhibiting greater sensitivity to fiber bridging.

Applicability evaluation of damage evolution models in simulating repeated low-velocity impact behavior of composite laminates

This study aims to provide a valuable reference for selecting appropriate damage evolution models (DEMs) when simulating the behavior of repeated low-velocity impacts (LVIs). Four DEMs are systematically evaluated in simulating mechanical response, dissipated energy, damage evolution, and accumulation under three repeated impacts. 3D Puck failure criteria and various DEMs combined with element characteristic lengths associated with physical fracture modes are used to predict repeated LVI damage behavior. Delamination behavior is simulated by zero-thickness cohesive elements. Two damage indexes are proposed to quantify and compare the damage distribution and accumulation. The post-peak softening and LVI behavior predicted by four DEMs are compared and verified through representative volume elements simulation and LVI experiments. The findings reveal that DEMs have varying degrees of influence on the mechanical response, damage evolution, damage distribution, and accumulation under repeated impacts. The exponential models proposed by P. Maimı et al. and Matzenmiller et al. are more suitable to simulate mechanical response or damage accumulation behavior under repeated impacts than the linear damage evolution model due to their non-linear post-peak softening characteristics. In contrast, Linde model shows brittle behavior due to the rapid damage accumulation during repeated impacts, rendering it unsuitable for repeated impact simulation.

A combined ALE-cohesive fracture approach for the arbitrary crack growth analysis

This work introduces an advanced numerical model, based on a cohesive zone approach and the moving mesh technique, for simulating fracture propagation in quasi-brittle materials. The proposed procedure involves two stages: first, a mesh boundary representing the crack is selected and aligned with the crack growth direction by using the Arbitrary Lagrangian-Eulerian (ALE) methodology; next, a zero-thickness interface cohesive element, equipped with a traction-separation law, is adaptively inserted along the previously selected mesh boundary, in order to describe the nonlinear fracture process. The proposed model allows for multiple crack onset and propagation without requiring mesh-updated procedures and sensibly reduces the well-known mesh dependency issues of the standard discrete fracture approaches. Numerical analyses of mixed-mode fracture in concrete specimens are performed and suitable comparisons with experiments show the effectiveness and reliability of the proposed model in predicting arbitrary crack growth.

Raster approach to modelling the failure of arbitrarily inclined interfaces with structured meshes

This paper presents an approach to evaluate the failure of arbitrarily inclined interfaces using FE models with structured spatial discretization, providing accurate prediction of crack propagation along paths known a priori that are not constrained to the element boundaries. The combination of algorithms for the generation of structured discretization of representative polycrystalline microstructures with novel cohesive element formulations allow modelling the failure of complex topologies along rasterised boundaries, with noticeably higher computational efficiency and comparable accuracy. Two formulations of raster cohesive elements are presented, adopting either elastic-brittle or Tvergaard–Hutchinson traction separation laws. The formulations proposed are first validated comparing the failure of the interface within bi-crystal structures discretised using hexahedral elements either within a structured mesh (i.e. with rasterised boundaries) or an unstructured mesh (i.e. with planar boundary). Subsequently, the effectiveness of the formulations is demonstrated comparing the inter-granular crack propagation within complex polycrystalline microstructures. The behaviour of the novel cohesive element formulation in structured meshes consisting of regular hexahedral elements is in excellent agreement with the deformation and failure of classic cohesive element formulations placed along the planar boundaries of unstructured meshes consisting of tetrahedral elements. The higher computational cost of the raster cohesive elements is more than compensated by the increase in computational efficiency of structured meshes when compared to unstructured meshes, leading to a reduction of the simulation time of up to over 200 times for the simulations presented in the paper, thus allowing the simulation of large domains.

Atomic force microscopy, deformation-recovery and numerical study of wheat dough

Experimental and numerical modelling work related to microstructure and deformation of starch, gluten and wheat dough was presented. Atomic force microscopy analyses through force spectroscopy on starch samples showed elastic behaviour of both dried and reconstituted wet starch granules, where wet starch was shown to be more cohesive than dried starch. Deformation-recovery test over time under compression mode showed almost complete recovery to original sample height of gluten, whereas only a slight recovery was observed for dough. Finite element modelling was then commenced to simulate gluten deformation-recovery and starch-gluten dough deformation. The gluten geometry used for the starch-gluten dough model was obtained from cryo-SEM image from the previous work before the starch geometry was then included. The interface between the starch and gluten model was defined using viscoelastic cohesive zone elements. The modelling results suggested significant influence of gluten cellular structure towards starch-gluten dough integrity. In addition, the results from the current work further supported the previous report of the starch-gluten interfacial behaviour using viscoelastic cohesive zone model.

A novel mesoscale modelling method for steel fibre-reinforced concrete with the combined finite-discrete element method

This paper proposes a novel mesoscale modelling method to study the effect of steel fibres on the fracture properties of steel fibre reinforced concrete (SFRC) based on an integrated finite-discrete element method (FDEM) framework. The present method models the steel fibre of SFRC as multiple bar elements, while simulating the cement matrix using triangle elements. A comprehensive bond-slip model is also introduced between the triangle element and bar element, enabling accurate reproduction of the interaction behaviour between the concrete matrix and steel fibre. Additionally, the progressive failure process of cohesive elements in FDEM is considered as crack propagation, confirming the superiority of FDEM from continuum to discontinuum. The results of the Brazilian split test and uniaxial compression test validate the capability of the FDEM framework for mode Ι and mode II fractures, while single fibre pull-out tests and 3-point-bending beam tests provide validation of the effectiveness of the integrated FDEM framework for the mesoscale modelling of SFRC. The discussion also covers employing the mesoscale modelling method to investigate the effect of steel fibre on the fracture properties of SFRC. Owing to its merits in quantitative analysis, the present method can serve as a feasible potential tool for mesoscale modelling analysis and SFRC design.

Modeling the dynamic fracture of concrete — A robust, efficient, and accurate mesoscale description

Dynamic loading of concrete often leads to excessive cracking and fragmentation. Since these processes are massively influenced by the underlying heterogeneity of the material, discrete modeling of the lower-scale features strongly improves the simulation results. At the same time, the computational effort is greatly increased. We therefore propose here a mesomechanical simulation approach which is efficient in terms of model generation and computational efforts even for large 3d models. Furthermore, it is robust by limiting the application of computationally less efficient cohesive zone elements to the weak transition boundary between matrix and inclusions. Intermatrix failure is considered by a smeared crack approach and element removal after a certain crack opening has been reached. We apply an improved version of the widely used RHT concrete model, which is enhanced with a principal stress criterion. The resulting approach leads to an accurate replication of experiments in terms of crack propagation and stress wave induced dynamic fragmentation.