Modelling of dynamic ductile fracture and application to the simulation of plate impact tests on tantalum

A model of dynamic damage by void nucleation and growth is proposed for elastic-viscoplastic materials sustaining intense loading. The model is dedicated to ductile materials for which fracture is caused by microvoiding. The material contains potential nucleation sites where microvoids are generated when the local pressure overcomes the nucleation pressure. A probability density function is adopted to describe the fluctuation of the nucleation pressure within the material. The void growth is described by using a hollow sphere model where micro-inertia effects are accounted for. The matrix weakening due to void growth is also included.The model has been first tested under uniaxial deformation. When the strain rate is assumed constant, the pressure inside the material has nearly a linear response up to a maximum. An analytical expression for the maximum pressure is proposed.Finite element simulations of plate impact tests have been carried out and compared to experiments on tantalum. From simulations based on the proposed model, an increase of the spall strength is observed with higher shock intensities. Therefore, the relationship between the velocity pullback and spall strength usually assumed in the literature (based on the acoustic approach) seems to be inadequate. Velocity profiles are simulated for different flyer thicknesses and different impact velocities with close agreement with experiments.     

A void–crack nucleation model for ductile metals

A phenomenological void–crack nucleation model for ductile metals with secondphases is described which is motivated from fracture mechanics and microscale physicalobservations. The void–crack nucleation model is a function of the fracture toughness of theaggregate material, length scale parameter (taken to be the average size of the second phaseparticles in the examples shown in this writing) , the volume fraction of the second phase, strainlevel, and stress state. These parameters are varied to explore their effects upon the nucleationand damage rates. Examples of correlating the void–crack nucleation model to tension data in theliterature illustrate the utility of the model for several ductile metals. Furthermore, compression,tension, and torsion experiments on a cast Al–Si–Mg alloy were conducted to determinevoid–crack nucleation rates under different loading conditions. The nucleation model was thencorrelated to the cast Al–Si–Mg data as well.     

The dynamic growth of a void in a plastic material and an application to fracture

The problem of the dynamic growth of a spherical void in applied stress and strain-rate fields is solved approximately using a variational method. The solution is applied to a model of fracture by void growth from brittle or incoherent inclusions. It is shown that the speed of fracture is limited by inertia near the crack tip. Some numerical results are obtained from data for high strength steels.     

Size effects due to secondary voids during ductile crack propagation

In the present study the size-effect due to a secondary void population during ductile fracture is investigated. Discrete primary voids are resolved in the process zone at the crack tip. A non-local GTN model is employed to describe the evolution of the secondary voids in the intervoid ligaments. The non-local GTN model contains an intrinsic length scale related to the size of the secondary voids. Hence, the ratio of the size of the primary and that of the secondary voids can be varied. The results show that small secondary voids can toughen the material. Such a behavior is in contrast to the prediction of cell model simulations. A theoretical reasoning of this effect and conclusions are given.     

The effects of thermal softening and heat conduction on the dynamic growth of voids

This paper seeks to examine the dynamic growth of a single void in an elastic–plastic medium through analytical and numerical approaches. Particular attention is paid to the instability of void growth, and to the effects of inertia, thermal softening and heat conduction. A critical stress is known to exist for the unstable growth of voids. The dependence of this critical stress on material properties is examined, and this critical stress is demonstrated to correspond to the lower limit for the ductile spall strength in many materials. The effects of heat conduction on the dynamic growth of voids strongly depend on the time and length scales in the early stages of the dynamic void growth.     

Rate effects on toughness in elastic nonlinear viscous solids

A micromechanics-based constitutive relation for void growth in a nonlinear viscous solid is proposed to study rate effects on fracture toughness. This relation is incorporated into a microporous strip of cell elements embedded in a computational model for crack growth. The microporous strip is surrounded by an elastic nonlinear viscous solid referred to as the background material. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness—the work of separation in the strip of cell elements and energy dissipation by inelastic deformation in the background material. As the crack velocity increases, voids grow in the strain-rate strengthened microporous strip, thereby elevating the work of separation. In contrast, the energy dissipation in the background material decreases as the crack velocity increases. In the regime where the work of separation dominates energy dissipation, toughness increases with crack velocity. In the regime where energy dissipation is dominant, toughness decreases with crack velocity. Computational simulations show that the two regimes can exist in certain range of crack velocities for a given material. The existence of these regimes is greatly influenced by the rate dependence of the void growth mechanism (and the initial void size) as well as that of the bulk material. This competition between the two dissipative processes produces a U-shaped toughness-crack velocity curve. Our computational simulations predict trends that agree with fracture toughness vs. crack velocity data reported in several experimental studies for glassy polymers and rubber-modified epoxies.     

伴随微孔洞生长的裂纹弹塑性定常扩展

裂纹在韧性材料中扩展时,将们随着微孔洞的萌生和生长,孔洞的萌生和深化将直接影响着材料的总体断裂韧性和强度,以往的研究主要集中在将裂纹的扩展刻划为微孔洞的萌生、生长和汇合这样一个过程。从传统的断裂过程区模型出发研究微孔洞的萌生和生长对材料总体断裂韧性的影响,通过采用Ｇｕｒｓｏｎ模型,建立塑性增量本构关系,然后针对定常扩展情况直接进行分析,孔洞对材料断裂韧性的影响由本构关系刻划,而在孔洞汇合模型中,上     

Evaluation of Ductile Damage Parameters under High Stress Triaxiality

The GTN model proposed by Gurson, Tvergaard and Needleman has been widely applied to predict ductile fracture. However, the evaluation of the GTN model under high stress triaxiality has only been reported in a few studies. In this paper, a series of tensile tests on round notched specimens were performed to evaluate the applicability of the GTN model parameters under high stress triaxiality. The evaluation was carried out by comparing the predicted load-displacement curves with experimental results. It was observed the GTN model parameters only depend on the material except the critical void volume fraction. The influence of stress triaxiality on the critical void volume fraction was discussed. A further discussion about the construction of a new void coalescence criterion for the GTN model was also presented in this paper.     

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.     

Algorithm for simulating fracture processes in concrete by lattice modeling

To simulate fracture behaviors in concrete more realistically, a theoretical analysis on the potential question in the quasi-static method is presented, then a novel algorithm is proposed which takes into account the inertia effect due to unstable crack propagation and meanwhile requests much lower computational efforts than purely dynamic method. The inertia effect due to load increasing becomes less important and can be ignored with the loading rate decreasing, but the inertia effect due to unstable crack propagation remains considerable no matter how low the loading rate is. Therefore, results may become questionable if a fracture process including unstable cracking is simulated by the quasi-static procedure excluding completely inertia effects. However, it requires much higher computational effort to simulate experiments with not very high loading rates by the dynamic method. In this investigation which can be taken as a natural continuation, the potential question of quasi-static method is analyzed based on the dynamic equations of motion. One solution to this question is the new algorithm mentioned above. Numerical examples are provided by the generalized beam (GB) lattice model to show both fracture processes under different loading rates and capability of the new algorithm.     

Void nucleation and growth in mineral-filled PVC – An experimental and numerical study

The nucleation and growth of voids in mineral-filled PVC have been investigated through experimental and numerical studies. Uniaxial tensile specimens were deformed in tension to different elongation levels and then unloaded. The macroscopic strain fields were recorded by use of digital image correlation. After the test, the microstructure of the deformed specimens was investigated in a scanning electron microscope. It was found that the observed volume strain on the macroscale is related to void growth on the microscale. In addition, finite element simulations were performed on unit cell models representing the microstructure of the material in a simplified manner. The numerical simulations demonstrate macroscopic dilation as a result of void growth. Moreover, the numerical simulations indicate that the experimentally observed stress-softening response of the PVC composite material may result from matrix-particle debonding.     

Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

Wave dispersion in gradient elastic solids and structures: A unified treatment

Analytical wave propagation studies in gradient elastic solids and structures are presented. These solids and structures involve an infinite space, a simple axial bar, a Bernoulli–Euler flexural beam and a Kirchhoff flexural plate. In all cases wave dispersion is observed as a result of introducing microstructural effects into the classical elastic material behavior through a simple gradient elasticity theory involving both micro-elastic and micro-inertia characteristics. It is observed that the micro-elastic characteristics are not enough for resulting in realistic dispersion curves and that the micro-inertia characteristics are needed in addition for that purpose for all the cases of solids and structures considered here. It is further observed that there exist similarities between the shear and rotary inertia corrections in the governing equations of motion for bars, beams and plates and the additions of micro-elastic (gradient elastic) and micro-inertia terms in the classical elastic material behavior in order to have wave dispersion in the above structures.     

A physical model for nucleation and early growth of voids in ductile materials under dynamic loading

Spall fracture and other rapid tensile failures in ductile materials are often dominated by the rapid growth of voids. Recent research on the mechanics of void growth clearly shows that void nucleation may be represented as a bifurcation phenomenon, wherein a void forms spontaneously followed by highly localized plastic flow around the new void. Although thermal, viscoplastic, and work hardening effects all play an essential role in the earliest stages of nucleation and growth, the flow becomes dominated by spherical radial inertia, which soon causes all voids to grow asymptotically at the same rate, regardless of differences in initial conditions or constitutive details, provided only that there is the same density of matrix material and the same excess loading history beyond the cavitation stress.These two facts, initiation by bifurcation at a cavitation stress, at which a void first appears, and rapid domination by inertia, are used to postulate a simple, but physically realistic, model for nucleation and early growth of voids in a ductile material under rapid tensile loading. A reasonable statistical distribution for the cavitation stress at various nucleation sites and a simple similarity solution for inertially dominated void growth permit a simple calculation of the initiation and early growth of porosity in the material.Parametric analyses are presented to show the effect that loading rate, peak loading stress, density of nucleation sites, physical properties of the material, etc. have on the applied pressure and distribution of void sizes when a critical porosity is reached.     

Void interaction and coalescence in polymeric materials

Beyond pressure-sensitivity, plastic deformation of glassy polymers exhibits intrinsic softening followed by progressive rehardening at large strains. This highly nonlinear stress–strain behavior is captured by a constitutive model introduced in this work. In the first part of the paper, we focus on void growth and coalescence in an axisymmetric representative material volume consisting of a single large void and a population of discrete microvoids. Our study shows that microvoid cavitation, enhanced by strain softening, accelerates the process of void coalescence resulting in brittle-like failure at lowered stresses and strains. Pressure-sensitivity also reduces stress-carrying capacity as well as influences the strain for void coalescence; plastic dilatancy effects are relatively milder. In the second part of the paper, we introduce a population of discrete spherical voids within a three-dimensional computational model to study void growth and damage ahead of a crack front. Our studies reveal a distinctive change in the deformed void shape from oblate to prolate when strain softening is followed by high rehardening at large plastic strains. By contrast, an extended strain softening regime promotes oblacity and facilitates multiple void interaction and their cooperative growth over large distances ahead of the crack front. This multi-void failure mechanism is exacerbated by pressure-sensitivity.     

Modeling of crack growth in ductile solids: a three-dimensional analysis

A population of several spherical voids is included in a three-dimensional, small scale yielding model. Two distinct void growth mechanisms, put forth by [Int. J. Solids Struct. 39 (2002) 3581] for the case of a two-dimensional model containing cylindrical voids, are well contained in the model developed in this study for spherical voids. A material failure criterion, based on the occurrence of void coalescence in the unit cell model, is established. The critical ligament reduction ratio, which varies with stress triaxiality and initial porosity, is used to determine ligament failure between the crack tip and the nearest void. A comparison of crack initiation toughness of the model containing cylindrical voids with the model containing spherical voids reveals that the material having a sizeable fraction of spherical voids is tougher than the material having cylindrical voids. The proposed material failure determination method is then used to establish the fracture resistance curve (J–R curve) of the material. For a ductile material containing a small volume fraction of microscopic voids initially, the void by void growth mechanism prevails, which results in a J–R curve having steep slope. On the other hand, for a ductile material containing a large volume fraction of initial voids, the multiple voids interaction mechanism prevails, which results in a flat J–R curve. Next, the effect of T-stress on fracture resistance is examined. Finally, nucleation and growth of secondary microvoids and their effects on void coalescence are briefly discussed.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals

The fracture toughness of ductile materials depends upon the ability of the material to resist the growth of microscale voids near a crack tip. Mechanics analyses of the elastic–plastic deformation state around such voids typically assume the surrounding material to be isotropic. However, the voids exist predominantly within a single grain of a polycrystalline material, so it is necessary to account for the anisotropic nature of the surrounding material. In the present work, anisotropic slip line theory is employed to derive the stress and deformation state around a cylindrical void in a single crystal oriented so that plane strain conditions are admitted from three effective in-plane slip systems. The deformation state takes the form of angular sectors around the circumference of the void. Only one of the three effective slip systems is active within each sector. Each slip sector is further subdivided into smaller sectors inside of which it is possible to derive the stress state. Thus the theory predicts a highly heterogeneous stress and deformation state. In addition, it is shown that the in-plane pressure necessary to activate plastic deformation around a cylindrical void in an anisotropic material is significantly higher than that necessary for an isotropic material. Experiments and single crystal plasticity finite element simulations of cylindrical voids in single crystals, both of which exhibit a close correspondence to the analytical theory, are discussed in a companion paper.     

A multiscale model for the ductile fracture of crystalline materials

In this paper, a multiscale model that combines both macroscopic and microscopic analyses is presented for describing the ductile fracture process of crystalline materials. In the macroscopic fracture analysis, the recently developed strain gradient plasticity theory is used to describe the fracture toughness, the shielding effects of plastic deformation on the crack growth, and the crack tip field through the use of an elastic core model. The crack tip field resulting from the macroscopic analysis using the strain gradient plasticity theory displayes the 1/2 singularity of stress within the strain gradient dominated region. In the microscopic fracture analysis, the discrete dislocation theory is used to describe the shielding effects of discrete dislocations on the crack growth. The result of the macroscopic analysis near the crack tip, i.e. a new K-field, is taken as the boundary condition for the microscopic fracture analysis. The equilibrium locations of the discrete dislocations around the crack and the shielding effects of the discrete dislocations on the crack growth at the microscale are calculated. The macroscopic fracture analysis and the microscopic fracture analysis are connected based on the elastic core model. Through a comparison of the shielding effects from plastic deformation and the discrete dislocations, the elastic core size is determined.