S.R. Bodner

A constitutive model for inelastic flow and damage evolution in solids under triaxial compression

Abstract A constitutive model for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids under non-hydrostatic compression has been developed on the basis that the relevant damage and dislocation flow processes both contribute to the overall inelastic strain rate. A damage-based kinetic equation is first formulated using the work-conjugate approach and the continuum damage concept. That relation is then added to the dislocation-based kinetic equation of a multi-mechanism deformation (M-D) model to obtain the macroscopic inelastic strain rate. The proposed kinetic relation for the overall inelastic strain rate is shown to be derivable from a flow potential. The kinetic equation indicates plastic dilatancy under triaxial compression when the damage term is activated, and leads to plastic incompressibility when inelastic straining is primarily provided by dislocation flow mechanisms. The dependence of creep rate and plastic dilatancy on confining pressure shown by model calculations for rock salt is in accordance with experimental observations.

Damage-induced nonassociated inelastic flow in rock salt

Abstract The multimechanism deformation coupled fracture model recently developed by Chan et al. [1992], for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids was evaluated against triaxial creep experiments on rock salt. Guided by experimental observations, the kinetic equation and the flow law for damage-induced inelastic flow in the model were modified to account for the development of damage and inelastic dillation in the transient creep regime. The revised model was then utilized to obtain the creep response and damage evolution in rock salt as a function of confining pressure and stress difference. Comparison between model calculation and experiment revealed that damage-induced inelastic flow is nonassociated, dilational, and contributes significantly to the macroscopic strain rate observed in rock salt deformed at low confining pressures. The inelastic strain rate and volumetric strain due to damage decrease with increasing confining pressures, and all are suppressed at sufficiently high confining pressures.

Analysis of very high speed impact

Abstract Investigations of the very high speed impact (> 1000 m/s) of cylindrical rods on target plates have generally concentrated on the numerical solution of the governing field and material equations at mesh points subdividing the structural components. In the present paper, an approximate two-dimensional analytical solution is obtained for the response to normal impact in the regions of influence. The analysis considers shock waves generated in the structural components and their subsequent reduction due to rarefaction waves. This, in turn, leads to zones of plastic flow in adjacent target regions and in the mushroomed frontal section of the projectile. An overall work rate equality that accounts for strain rate dependent plastic flow and also for inertial forces due to local and convective accelerations enables determination of the target cavity diameter. Possible failure of projectile material is also considered. The analysis is carried out up to the time that shock wave effects in the rod become relatively unimportant. During this period, the analytical model determines the interactive forces, the penetration velocity, and the deformation geometry, strain rates, and flow stresses of the projectile and target.

Modeling of hardening at very high strain rates

A modification of the Bodner–Partom elastic‐viscoplastic constitutive model is proposed to account for strain rate dependence of the evolution of hardening. The suggested procedure is for the rate of hardening in the hardening evolution equation to be a direct function of total strain rate. Material constants are determined for oxygen‐free‐electronic copper and simulations of response behavior at very high rates of straining appear to be consistent with test observations.

Cleavage and creep fracture of rock salt

Abstract The dominant failure mechanism in rock salt at ambient temperature is either cleavage or creep fracture. Since the transition of creep fracture to cleavage in a compressive stress field is not well understood, failure of rock salt by cleavage and creep fracture is analyzed in this paper to elucidate the effect of stress state on the competition between these two fracture mechanisms. For cleavage fracture, a shear crack is assumed to cause the formation and growth of a symmetric pair of wing cracks in a predominantly compressive stress field. The conditions for wing-crack instability are derived and presented as the cleavage fracture boundary in the fracture mechanism map. Using an existing creep fracture model, stress conditions for the onset of creep fracture and isochronous failure curves of specified times-to-rupture are calculated and incorporated into the fracture mechanism map. The regimes of dominance by cleavage and creep fracture are established and compared with experimental data. The result indicates that unstable propagation of cleavage cracks occurs only in the presence of tensile stress. The onset of creep fracture is promoted by a tensile stress, but can be totally suppressed by a high confining pressure. Transition of creep fracture to cleavage occurs when critical conditions of stress difference and tensile stress for crack instability are exceeded.

A two-dimensional analysis of penetration by an eroding projectile

Summary A full two-dimensional solution is obtained for the problem of high velocity penetration by an eroding long rod projectile into a thick target. The present analysis is based on the methods of dynamic plasticity in contrast to the 1-D treatments of Alekseevskii [ Combustion, Explosion and Shock Waves , Vol. 2 (1966)] and of Tate [ J. Mech. Phys. Solids 15 , 287–399 (1967) and 17 , 141–150 (1969)] using the modified hydrodynamics approach. The formulation considers fully developed plastic flow fields in the penetrator and in the target. A variational theorem is used to constrain the geometry of each of those flow fields for minimum dissipation conditions. The analysis requires satisfaction of the governing equations of motion and overall force balance at the target/projectile interface. Comparisons of the present 2-D formulation with test data indicate very good agreement.

Penetration into thick targets-refinement of a 2D dynamic plasticity approach

Summary Generalizations have been made to the two-dimensional penetration and perforation analysis of Ravid and Bodner (1983) for blunt nosed, rigid projectiles impacting viscoplastic targets in the ordinary ballistic range. The new developments consist of consideration of ogival, conical, and spherical projectile nose shapes, possible alterations of the plastic flow field due to deep penetrations, thermo-mechanical coupling leading to thermal softening, and the addition of a requirement on the inertial work rate of target material.

A Unified Elastic-Viscoplastic Theory with Large Deformations

A review is presented of a set of elastic-viscoplastic constitutive equations that incorporate isotropic and directional hardening and additional hardening due to nonproportional loading. These equations employ the isotropic form of the flow law in the presence of directional hardening and a physical argument is given to justify this use. An alternative form of the flow law is also suggested that could account for deviations of the directions of stress and plastic strain rate. Generalization of the theory to large deformations has been carried out using Lagrangian quantities and thermodynamic restrictions. Examples of simple tension and simple shear show that the large strain theory produces physically plausible results.

A lower bound on bifurcation buckling of viscoplastic structures

The buckling of structures of elastic-viscoplastic materials is a stability problem that does not admit a realistic bifurcation formulation in the classical manner. In the absence of imperfections and inertial effects, the standard bifurcation criterion leads only to elastic buckling since an instantaneous jump in strain rate would develop at the critical condition. However, an expression for the “short time” inelastic tangent modulus at the pre-buckling strain rate can be developed from an appropriate incremental constitutive theory and this can be used in the quasi-static bifurcation buckling condition appropriate to the structure and loading. Such a buckling value can be interpreted as a lower bound on the actual instability condition. For the case of structures with initial imperfections, the calculation of local inelastic tangent moduli at the current state should lead to close correspondence between bifurcation and instability. Under creep conditions, the procedure gives approximate creep buckling times for both perfect and initially imperfect structures. For situations where the buckling mode generates abrupt changes in the multiaxial stress state, modifications to the reference constitutive theory are required to properly represent the governing physics. In this manner, the procedure seems capable of indicating buckling values consistent with test results without relying on a “deformation” type plasticity theory.

Mechanics applied to the underground storage of radioactive waste materials

An application of Mechanics to an important environmental problem is an investigation on the creep properties of rock salt which was undertaken in relation to the planned encapsulation of transuranic nuclear waste in caverns excavated in bedded salt formations (the WIPP program in the USA). Those caverns are intended to serve as permanent repositories for radioactive waste over an extensive period so that complete isolation is required of the facility including the shafts that are initially connected to the outside. In conjunction with an extensive experimental program, the analytical and numerical studies on creep of rock salt were concerned with the following subjects: creep based on dislocation mechanisms; damage induced creep leading to volumetric changes, pressure dependence, and creep rupture: healing of damage; failure and fracture mechanisms; and structural integrity of underground storage rooms