Constitutive-microdamage equations are developed that are capable of simulating thermo-mechanical material behavior following high shock compression, dilatation, microdamage evolution and fracture, caused by projectile-target impact at hypervelocity.

Inconel 718 is a nickle-based superalloy that possesses several outstanding elevated-temperature mechanical properties. It has been widely used for the manufacture of critical compressor and turbine engine components requiring relatively long service lives. A large number of tests, including tensile, creep, fatigue, and creep-fatigue have been perfgrmed to characterize the mechanical properties of Inconel 718 at 1200 F, the operating temperature for turbine blades. In additiog, a few attempts have been made to model the behavior of Inconel 718 at 1200 F using viscoplastic theories.

A constitutive micro-damage model is presented capable of describing material behavior under the extreme states of stress, strain and adiabatic heating generated by hypervelocity impact. The model implements the Mie-Gruneisen equation of state along with expressions of non-linear elastic moduli as functions of volume strain, temperature and microdamage. The viscoplastic material response includes strain and strain rate hardening and temperature and microdamage softening. The microdamage evolution model is based on the micromechanics of an expanding void, and is capable of modeling void compaction and expansion that leads to spall-fracture as an evolutionary time dependent process. A series of computer simulations using the microdamage constitutive model are compared against hypervelocity impact experiments conducted on 1100 Aluminum plates with soda-lime glass spherical projectiles in terms of crater and back-wall spallation geometry of the target plate.

A set of constitutive-microdamage equations are presented that can model shock compression and the microdamage and fracture that can evolve following hypervelocity impact. The equations are appropriate for polycrystalline metals. For impact at a projectile velocity of 6.0 km/s, numerical simulations are preformed that describe the impact of spherical soda-lime glass projectiles with aluminum 1100 rectangular target plates. Three ratios of the projectile diameter to the target thickness are chosen for the simulations, providing a wide range of damage features. The simulated impact damage is compared with experimental damage of corresponding test specimens, illustrating the capability of the model.

Constitutive-microdamage equations are developed that are capable of simulating thermo-mechanical material behavior following high shock compression, dilatation, microdamage evolution and fracture, caused by projectile-target impact at hypervelocity.

Several measurable aspects of the fracture behaviour of a cracked body such as the fracture stress, the maximum shear stress at the ends of the crack, the angle at which an extending crack begins to turn, and the rate of crack growth under cyclic loading, have all been shown experimentally to be affected by load biaxiality. Nevertheless, it has been (and perhaps still is) the view of many engaged in the development of fracture mechanics that fracture behaviour is conditioned only by loads applied normal to the plane of the crack, while indifferent to any loads that may be applied parallel to this plane. This paper is devoted to briefly outlining how this view came to be through a confluence of misunderstandings and omissions in the work of Griffith and MacGregor-Westergaard, that had the effect of precluding from their results and equations the presence of load applied parallel to the plane of the crack in biaxial load situations. The combined effect of these miss-steps was to compel the early theoretical developments in fracture mechanics to ingest and adopt a similar view.

A viscoplastic constitutive theory containing an internal damage variable, representing void volume fraction, is used to model deformation and ductile failure of smooth tensile bars under high rate loading. In the simulations the position of the neck is not always along the centerline, but instead is found to vary as a function of the loading rate. Computed elongation at instability and elongation at fracture is compared to data found in the literature. The theory correctly predicts the increase in elongation with increasing rate of deformation.

Spall type fracture of a circular plate caused by mechanical impact and high strain rate fracture of a tensile bar caused by dynamic load application are described using a viscoplastic-damage type of constitutive model. Identification of the microvoid volume fraction of polycrystalline materials as a microdamage scalar field variable, and inclusion into an elastic-viscoplastic constitutive theory enables description of rate-dependent compressible inelastic deformation that includes material degradation culminating in fracture. Calculations simulating the spall fracture and fragmentation of a circular plate induced by high velocity impact, and the necking localization and fracture at high strain rate of a dynamically loaded smooth tensile bar, are shown.

The ductile spall fracture and post-spall behavior of a circular target plate after impact with a flyer plate having small diameter, is modeled by means of a viscoplastic constitutive theory that includes the microvoid volume fraction as a scalar material damage variable. Incorporation of the damage parameter permits description of rate-dependent, hardening, compressible inelastic deformation and ductile fracture, where local fracture is defined in terms of a critical microvoid volume fraction.

A model to predict material damage and spall fracture is applied to low-angle oblique plate-impact experiments. In this configuration, impact of parallel plates occurs along a direction inclined relative to the flyer velocity direction, resulting in normal and transverse deformations in both plates. The model uses the Perzyna viscoplastic constitutive theory that contains a scalar field variable description of the material damage, which is taken as the void volume fraction of the polycrystalline material. Incorporation of the damage parameter permits description of rate-dependent, compressible, inelastic deformation and ductile fracture. The model for microvoid growth is based upon a random distribution of microvoids idealized as spherical holes of arbitrary size, which, as an approximation, are assumed to remain spherical under the high mean stresses and moderate shear stresses that occur when the angle of inclination of the impact is small. A local spall fracture criterion based on critical microvoid volume is utilized. The constitutive equations are specialized for deformation associated with propagating plane waves of combined pressure and shear. Normal and transverse rear surface velocities are computed for oblique impact of 6061-T6 aluminum and compared to measured velocity histories. Numerical predictions are extended to conditions resulting in spallation under combined pressure-shear waves, using OFHC copper as the plate material. The void volume distribution resulting from the tensile mean stress in the target plate is computed to predict material damage. Computations are performed to illustrate the effect of damage on both the normal and transverse rear surface velocities.

A model to predict material damage and spall fracture under high strain-rate conditions is applied to plate impact experiments. The model uses the Perzyna viscoplastic constitutive theory, appropriately modified to include a nonlinear isotropic hardening law that allows for saturation of the hardening with increase of strain. The constitutive equation contains a scalar variable for description of the material damage, expressed as the void volume fraction of the polycrystalline solid with microvoids. Incorporation of the damage parameter into the completely phenomenological elasto-viscoplastic constitutive equations permits description of rate-dependent, compressible, inelastic deformation, and ductile fracture. The evolution equation for the parameter describes microvoid nucleation and growth. The model for microvoid growth is based upon a random distribution of microvoids, idealized as spherical holes of arbitrary size. Microvoid coalescence is considered by incorporation of void interaction functions to describe enhanced nucleation and growth rates. A local spall fracture criterion based upon the attainment of a critical microvoid volume is utilized.The constitutive equations are specialized to uniaxial deformation with multiaxial stress, which is appropriate for the planar impact experiments. A finite-difference wave propagation computer code is used to solve the equation of motion. The computed stress wave profiles demonstrate the effect of using a viscoplastic material description. Calculations predicting the rear-surface velocity-time profile and the stress-time profiles of the OFHC copper target are compared with measured profiles. The damage (void volume) distribution across the plate thickness is also calculated and compared with experimental data.

A mathematical model for the rate of growth of microvoids under mean tensile stress in dynamic processes is developed, which represents an extension of previous analysis of ductile void growth rates using the hollow sphere model. A viscoplastic material is assumed for which the isotropic hardening saturates as the strain progresses. The microvoid growth model is used as an internal damage variable in Perzyna's elasto-viscoplastic constitutive theory for solids experiencing ductile modes of material, having been taken from our application of the viscoplastic-damage constitutuve theory to model shock-induced high strain-rate deformation and spall fracture in polycrystalline solids.

A recently developed viscoplastic-damage type of constitutive theory for high strain-rate flow processes and ductile fracture is used to model the deformation and fracture of dynamically loaded smooth cylindrical tensile bars. The analysis assumes polycrystalline materials which usually contain microvoids with an average density of the order of 106 per cm3 that are dispersed homogeneously throughout. It is shown that for dynamically imposed loading that produce nominal strain rates ranging between 5 × 102 − 5 × 103 sec −1, the inhomogeneous fields of stress and deformation caused by wave propagation and wave reflection induce necking at different locations along the gauge section, depending upon the strain-rate imposed. This occurs without imposition of any geometrical or material irregularity to preposition the location of the necking. The imposed rate of strain is also shown to affect the magnitude of the strain at which necking initiates, as well as the strain required for fracture.

A viscoplastic constitutive theory that contains a scalar variable description of damage is applied to a two-dimensional axisymmetric analysis of plate-impact spallation. The model uses the Perzyna viscoplastic constitutive formulation with inclusion of a nonlinear isotropic hardening law that saturates with increasing strain. For ductile metals the damage variable is taken to be the void volume fraction of the polycrystalline solid. The evolution equation for damage is based on the nucleation, growth, and eventual coalesence of the microvoids. A spallation criterion based on critical void volume fraction is utilized.The theory is applied to normal impact of circular plates where the diameter of the flyer is smaller than the diameter of the target. Multidimensional axisymmetric strains are developed where, because of the edge effect of the smaller flyer plate, nonplanar as well as planar waves are generated. The equations for balance of mass and momentum are solved using a Lagrangian finite-element computer program with explicit time integration. Four-noded uniform strain quadrilateral elements are used for the spatial discretization. Numerical simulations of impact are performed over a wide range of initial velocities. Contours of the void volume fraction illustrate the damage threshold and the effect of increasing impact stress on damage evolution. Material damage and spall fracture are also illustrated by plots of the deformed geometry, in which the material softening associated with increase in the void volume fraction results in extensive local deformations that, in effect, simulate the openings that are observed in spalled plates. Additional simulations are performed to study the effect of varying parameters that appear in the microvoid evolution equations.

In both Griffith's global energy rate theory for crack instability and Irwin's local crack-tip stress intensity theory for fracture toughness, only the tensile load perpendicular to the crack influences fracture behavior of the body. Thus according to these theories, outer boundary loads applied parallel to the crack have no effect on the fracture process. This viewpoint has been widely held since its inception with the work of Griffith in 1921, and has strongly influenced the development of fracture mechanics. Investigations by the authors have shown the contrary however, in that the load biaxiality strongly affects many aspects of the fracture behavior of a cracked body. Almost all of the characteristics of brittle fracture have been shown theoretically and/or experimentally to be sensitive to load biaxiality. Specifically, this work has shown that the stress and displacement fields, the elastic strain energy density, and the maximum shear stress near the crack tip are all altered by loads applied parallel to the crack, as are the angle of initial crack extension, the strain energy of the entire body, the fracture load, and the rate of fatigue crack growth. Since the results of this research have been published piecemeal over the years, the authors are presenting, herein, a synthesis and summary of this work for the purpose of demonstrating the overall presence and consistency of the biaxial effects.

The Perzyna viscoplastic constitutive theory, which contains a scalar variable for description of material damage, is used to study material behavior at high strain rates. The damage parameter for materials which undergo ductile fracture by nucleation, growth, and coalescence of microvoids, is taken to be the void volume fraction. The linear hardening law in both the constitutive equation and the derivation of the void growth rate equation has been replaced by a nonlinear hardening law that allows for the saturation of the hardening with increase of strain. The modified constitutive equations are then specialized to uniaxial deformation with multiaxial stress, which is typical of that occurring in flyer plate impact experiments. Calculations are performed showing the rate dependence of the material response and the effects of the growth of the void volume (damage). The change in the predicted response due to the modification of the hardening law is illustrated. Ductile spall fracture is modeled by considering the response to a simulated compressive-tensile wave using a critical value of the void volume as the local criteria for fracture.

The titanium alloy Ti-6Al-4V is known to exhibit creep behavior at temperatures as low as room temperature. Consequently, for cyclic loading with hold times it is possible that the rate dependent behavior of Ti-6Al-4V can have negative bearing upon the low cycle fatigue life. If this effect is shown to be present at room temperatures, then it will certainly be magnified and, therefore, very important at elevated temperatures. In order to account for the effects of strain rate dependent deformation in fatigue life prediction methodology, it was considered necessary to incorporate a viscoplastic constitutive equation into the fatigue life calculational algorithm. After critical evaluation of a score of recently proposed viscoplastic constitutive theories, the Chaboche theory, which employs a yield condition, was considered to offer the most promise for description of a wide range of inelastic material behavior characteristics. The six viscoplastic material parameters that are required for nonelevated temperature applications were determined from data of uniaxial tests, conducted elsewhere and made avialable to this study. The fatigue life testing of smooth round bar specimens included load cycles with load hold times. Fatigue life predictions were performed using the equivalent fully reversed symmetric cycle, and the Smith-Watson-Topper parameter, for load cycles having varying stress amplitudes and varying hold times. The predicted fatigue life results indicate that: (i) For a given stress level above the initial yield stress, shorter load hold time periods result in longer fatigue lives. (ii) The higher the stress level (above the initial yield stress) the more pronounced becomes the effect of the load hold time on the fatigue life prediction. (iii) The rate of loading also has an effect on fatigue life. Analysis indicates that the slower the rate of loading, the higher the rate dependent (primary creep) deformation, and consequently, the lower the resulting fatigue life.

Since the year 1919, upwards of a dozen theories for viscoplastic material response have been proposed. These theories can be separated into two categories, depending on whether a yield condition is assumed and incorporated into the structure of the constitutive equations. This article presents a comparative study between a theory from each of the two categories. Specifically, a modified form of the Chaboche theory and the Bodner-Partom theory are examined qualitatively by predicting the uniaxial stress-strain behavior of INCONEL 718 at 1200°F. These comparisons encompass predictions of monotonic tension at different strain and stress rates, jumps in strain rates, primary and secondary creep, stress relaxation, load-unload and load-unload-reload behavior at different strain rates, and controlled cyclic loading.