Crack straining-based spall model

Mean void growth-based spall models that avoid the complications of nucleation have been successfully applied to the problem of ductile spallation. However, similar models based on mean crack growth, applicable to brittle spallation, are not promising. This is because it remains to be demonstrated how an appropriate mean crack size is chosen to identify the brittle spall strength as the threshold pressure for crack growth. In the authors' view, the solidity evolution is not merely a consequence of the nucleation and growth of cracks but also determined by the crack straining caused by the relaxed tensile pressure. This paper presents a crack straining-based spall model which assumes that the inelastic volumetric strain caused by the relaxed tensile pressure at a critical fragment volume is the main factor governing the solidity evolution during the process of coalescence and fragmentation. Such an approach makes the modelling of brittle spall compatible with that of ductile. The developed model requires only two additional parameters to those required by the chosen constitutive model and equation of state. We present the comparisons between experimental and computational simulation of the free surface velocity history of the target in a plane impact plate experiment. These show very good agreement.     

A modified Cochran–Banner spall model

The original Cochran–Banner spall model was modified to suit the usual definition of damage and to abandon the simplifying approximation as unnecessary. The strength function given by Cochran–Banner was maintained using the redefined damage and the correction concerning the volume of the mesh cells was realized considering it unnecessary to expect that it is much easier to open microcracks once they are formed than to strain the solid further. In the case of abandoning the simplifying approximation made by Cochran–Banner to calculate the damage, the redefined damage also does not become a new independent variable for which new dynamic laws would need to be specified. Once the spall strength was reached, the damage would be only determined by a series of closed equations including the stress.relaxation relationship given by the strength function, the energy conservation equation, the equation of state and the constitutive equations for the damaged aggregate. The modified Cochran–Banner spall model also included only two parameters: the spall strength and the critical damage, which relate to material properties and specific loading conditions. Comparison of theoretical and experimental results for some spall tests was performed. It was found that the computed free surface velocity profile of target or stress profile of interface between target and soft buffer in plane spall tests was sensitive to the spall strength and the critical damage in the modified Cochran–Banner spall model.     

A variational void coalescence model for ductile metals

We present a variational void coalescence model that includes all the essential ingredients of failure in ductile porous metals. The model is an extension of the variational void growth model by Weinberg et al. (Comput Mech 37:142–152, 2006). The extended model contains all the deformation phases in ductile porous materials, i.e. elastic deformation, plastic deformation including deviatoric and volumetric (void growth) plasticity followed by damage initiation and evolution due to void coalescence. Parametric studies have been performed to assess the model’s dependence on the different input parameters. The model is then validated against uniaxial loading experiments for different materials. We finally show the model’s ability to predict the damage mechanisms and fracture surface profile of a notched round bar under tension as observed in experiments.     

Modelling of nucleation and void growth in dynamic pressure loading, application to spall test on tantalum

Dynamic ductile fracture is a three stages process controlled by nucleation, growth and finally coalescence of voids. In the present work, a theoretical model, dedicated to nucleation and growth of voids during dynamic pressure loading, is developed. Initially, the material is free of voids but has potential sites for nucleation. A void nucleates from an existing site when the cavitation pressure p _c is reached. A Weibull probability law is used to describe the distribution of the cavitation pressure among potential nucleation sites. During the initial growth, the effect of material properties is essentially appearing through the magnitude of p _c. In the later stages, the matrix softening due to the increase of porosity has to be taken into account. In a first step, the response of a sphere made of dense matrix but containing a unique potential site, is investigated. When the applied loading is a pressure ramp, a closed form solution is derived for the evolution of the void that has nucleated from the existing site. The solution appears to be valid up to a porosity of 0.5. In a second part, the dynamic ductile fracture of a high-purity grade tantalum is simulated using the proposed model. Spall stresses for this tantalum are calculated and are in close agreement with experimental levels measured by Roy (2003, Ph.D. Thesis, Ecole Nationale Supérieure de Mécanique et d’Aéronautique, Université de Poitiers, France). Finally, a parametric study is performed to capture the influence of different parameters (mass density of the material, mean spacing between neighboring sites, distribution of nucleation sites...) on the evolution of damage.     

A micromechanical cyclic void growth model for ultra-low cycle fatigue

The influence of stress triaxiality and Lode parameter on microvoid growth phase of ductile fracture under ultra-low cycle fatigue (ULCF) (N_f < 100, N_f = cycles to failure) loading is investigated using micromechanical analyses. A new micromechanical cyclic void growth model (MM-CVGM) to predict the ULCF life of ASTM A992 steels is presented. The MM-CVGM is calibrated and validated from the experiments conducted on axisymmetrically notched specimens. Number of cycles to failure (N_f) and the fracture initiation locations predicted by the model closely matched the experimental observations.     

A grain-scale study of spall in brittle materials

The evolution of spall for a brittle material is investigated under variance of anisotropy, grain boundary fracture energy, and loading. Because spall occurs in the interior of the specimen, fundamental studies of crack nucleation and growth are needed to better understand surface velocity measurements. Within a cohesive approach to fracture, we illustrate that for anisotropic materials, increases in the fracture energy cause a transition in crack nucleation from triple-points to entire grain boundary facets. Analysis of idealized flaws reveals that while crack initiation and acceleration are strong functions of the fracture energy, flaws soon reach speeds on the order of the Rayleigh wave speed. Finally, simulated surface velocities of spalled configurations are correlated with microstructural evolution. These fundamental studies of nucleation, growth, and spall attempt to link atomic separation to the macroscopic spall strength and provide a computational framework to examine the evolution of spall and the impact on the simulated surface velocity field.     

Models of spall fracture

Strength of Materials -     

A continuum damage mechanics model for void growth and micro crack initiation

A damage mechanics model is proposed to study the void growth and crack initiation. J₂ incremental flow theory along with a damage variable is used to model the material behaviour in elasto-plastic regime. Large deformation (large rotation and finite strain) finite element analysis is carried out for five different cases. In all the cases it is observed that the triaxiality and the plastic strain play an important role in void growth and crack initiation in ductile material. A failure curve is obtained for the material AISI-1090 spheroidised steel. Finally, it is concluded that the critical value of the damage variable can be taken as a crack initiation parameter.     

A non-local void filling model to describe its dynamics during processing thermoplastic composites

A model is developed to describe the void dynamics within thermoplastic composite tape during the tape placement process. The model relates the volatile pressure in voids, the applied compaction load, fiber bed response and the resin pressure due to squeeze-flow of resin from resin-rich regions to fill void regions. This model relies on some geometric simplifications, but incorporates the relevant physical phenomena.This void consolidation model was implemented in a numerical code which predicts the void development during the process. The initial void geometry can be introduced either manually, using a random generation algorithm or from actual processed tape micrographs.The model predicts that the final void content depends on the original void content but also on the initial void distribution. Presented results analyze the influence of void distribution on tape consolidation. Limitations of the consolidation process rate by the resin squeeze flow pressures are clearly demonstrated.     

A constitutive model for porous solids taking into account microscale inertia and progressive void nucleation

In the present paper, a micromechanical constitutive framework for ductile voided solids sustaining dynamic loading is proposed. The micro-inertia based model proposed by Jacques et al. (2012a) has been extended by taking void nucleation into account. To capture the combined effects of microscale inertia and void nucleation, a micro-mechanical model of damage is proposed. This model is based on three physical parameters: the porosity, the number of voids per unit volume and an internal variable aiming to represent the effect of void size heterogeneity. The evolution equations for these internal variables are derived and an application to the failure of an axisymmetric notched bar is presented.     

Three-dimensional cell model analyses of void growth in ductile materials

Three-dimensional micromechanical models were developed to study the damage by void growth in ductile materials. Special emphasis is given to the influence of the spatial arrangement of the voids. Therefore, periodical void arrays of cubic primitive, body centered cubic and hexagonal structure are investigated by analyzing representative unit cells. The isotropic behaviour of the matrix material is modelled using either v. Mises plasticity or the modified Gurson-Tvergaard constitutive law. The cell models are analyzed by the large strain finite element method under monotonic loading while keeping the stress triaxiality constant. The obtained mesoscopic deformation response and the void growth of the unit cells show a high dependence on the value of triaxiality. The spatial arrangement has only a weak influence on the deformation behaviour, whereas the type and onset of the plastic collapse behaviour are strongly affected. The parameters of the Gurson-Tvergaard model can be calibrated to the cell model results even for large porosity, emphasizing its usefulness and justifying its broad applicability.     

Direct numerical simulation of ductile spall failure

Large-scale direct numerical simulations of void growth and coalescence from 3-dimensional distributions of void nucleating particles are used to investigate the effect of material strain hardening and strain rate sensitivity on spall response. The computational model spans multiple particle spacings in the in-plane directions, and several finite elements span the initial particle diameters in the mixed-zone Arbitrary Lagrange–Eulerian (ALE) simulations. The matrix material is represented by traditional plasticity models in which material failure is not permitted. The 1000\(+\) particles are represented by the same material model as the surrounding matrix except the particles have low tensile strength to permit fracture, which is used to simulate particle cracking or decohesion. Voids grow and coalesce naturally in the ALE framework, and the simulations produce dimpled failure surfaces similar to those observed experimentally in spalled samples. The strain hardening and strain rate sensitivity of the matrix material are altered to explore their influence on the void growth and coalescence processes and on the simulated free surface velocity. The details available from the computational model permit association of the longitudinal stress evolution with features on the free surface velocity profile.     

Copper spall fracture under sub-nanosecond electron irradiation

Ultra-short powerful electron beam is a suitable tool for producing of high rate deformation in substance. In paper we present a new model of high rate fracture and use this model for numerical investigation of fracture of copper target at irradiation by sub-nanosecond electron beam. In this model, fracture is considered as a time-dependent process of nucleation and growth of opening mode cracks. The nucleation and growth rates are controlled by specific free energy of crack surface which is sole fitted parameter. Plastic deformations, both in cracks vicinity and total in substance, are described in frames of dislocation theory. For verification of the model, we performed simulations of spall fracture at plate impact and at irradiation by high-current electron beam with pulse duration of tens of nanoseconds, and reasonable agreement with experimental data has been demonstrated. Simulations of the sub-nanosecond electron beam action on target indicate that spall fracture of the irradiated target surface is possible. This fracture takes place at the enclosed energy density slightly below the value, which is sufficient for melting of irradiated substance. Fracture threshold energy density does not depend on the origin dislocation density and it increases with the increase of pulse duration. As a result, at long pulse durations (more than ten nanoseconds) the substance melts before fracture.     

Comparison of hypervelocity fragmentation and spall experiments with Tuler–Butcher spall and fragment size criteria

The present investigation compares predictive theories of dynamic spall and fragmentation with previously reported experimental data. In the experimental tests, aluminum spheres normally impacted thin aluminum plates at over approximately 4.5–7.5 km/s. Scaling features of the impact breakup phenomenon were explored through selected variation in sphere size and plate thickness. The principal diagnostic was high-resolution flash radiography. Fragment-size features of resulting fragment clouds were determined through detailed analysis of the recorded radiographs. Other investigators have measured the spall strengths for aluminum at comparable ultra-high strain rates. Spall strength amplitude and the corresponding strain rate dependence are principal results of the study. Existing dynamic fracture criteria are specialized here to the sphere impact spall and fragmentation event, and compared with empirical data. Velocity and strain rate scaling relations are developed for fragmentation size in the sphere impact event.     

Mechanics of void growth and void collapse in crystals

The mechanics of void deformation in single crystals is studied in a fully three-dimensional setting, taking into account all twelve slip systems for fcc and bcc crystals. The increments of the local field variables are calculated analytically using Eshelby's approach for the three-dimensional inclusion problem. The collapse or growth of voids in three dimensions is investigated when a rate-dependent elastoplastic material is subjected to compression and tension at a high rate. The deformation of an initially spherical cavity is calculated incrementally, and, it is shown that, even under all-around uniform loading, a void deforms into a complicated shape which is defined by the structure and symmetry of the slip systems. An equivalent ellipsoid is used to approximate the deformed void shape at each incremental step, and the procedure is continued until the equivalent ellipsoid collapses into a crack or a needle, or it expands or shrinks in a self-similar manner. Several numerical examples are presented, and the numerical results are compared with the experimental observations, obtaining good correlation. The significant effects of loading rate on the material response are illustrated. It is shown that the material becomes stronger at higher loading rates. The three-dimensional final void geometry under various loading conditions is studied. The difference between void deformation mechanism under tension and compression is illustrated. The possible overall failure mechanism caused by collapse and/or growth of pre-existing cavities is discussed. From the comparison of the corresponding results with those of the two-dimensional case it is shown that the double-slip system in two dimensions can be used effectively to simulate the three-dimensional problem.     

A unified damage factor model for ductile fracture of steels with different void growth and shrinkage rates

Micromechanical fracture modelling is an effective method to predict ductile fracture in steel structures. This paper aims to establish a simplified and general fracture model for various loading conditions, which is convenient to calculate the instantaneous damage index. With the concept of the ductile damage factor, a unified ductile damage factor model considering the difference of rates between void growth and shrinkage has been proposed. Based on experiment results and finite element analysis, the model parameters for Q235B Chinese structural steel were calibrated. The ductile damage factor and equivalent plastic strain at fracture initiation were investigated. Comparison among the proposed model, experimental results, and the cyclic void growth model demonstrated the effectiveness and accuracy of the proposed model. A parametric study was conducted to investigate the influence of cyclic constitutive parameters on the accuracy of fracture prediction. The predicted results are acceptable while reducing those calibrated cyclic constitutive parameters by 20%.