T.W. Wright

On Stress Collapse in Adiabatic Shear Bands.

Abstract T he dynamics of adiabatic shear band formation is considered making use of a simplified thermo/visco/plastic flow law. A new numerical solution is used to follow the growth of a perturbation from initiation, through early growth and severe localization, to a slowly varying terminal configuration. Asymptotic analyses predict the early and late stage patterns, but the timing and structure of the abrupt transition to severe localization can only be studied numerically, to date. A characteristic feature of the process is that temperature and plastic strain rate begin to localize immediately, but only slowly, whereas the stress first evolves almost as if there were no perturbation, but then collapses rapidly when severe localization occurs.

The initiation and growth of adiabatic shear bands

Abstract A simple version of thermo/viscoplasticity theory is used to model the formation of adiabatic shear bands in high rate deformation of solids. The one dimensional shearing deformation of a finite slab is considered. For the constitutive assumptions made in this paper, homogeneous shearing produces a stress/strain response curve that always has a maximum when strain and rate hardening, plastic heating, and thermal softening are taken into account. Shear bands form if a perturbation is added to the homogeneous fields just before peak stress is obtained with these new fields being used as initial conditions. The resulting initial/boundary value problem is solved by the finite element method for one set of material parameters. The shear band grows slowly at first, then accelerates sharply, until finally the plastic strain rate in the center reaches a maximum, followed by a slow decline. Stress drops rapidly throughout the slab, and the central temperature increases rapidly as the peak in strain rate develops.

A scaling law for the effect of inertia on the formation of adiabatic shear bands

For a rigid/perfectly plastic material with linear thermal softening and power law rate hardening there is a competition between heat conduction and inertia in determining the time of shear band formation. In a finite specimen the nominal strain rate that produces the fastest growth of perturbations corresponds to the minimum critical strain. Similarly for a fixed strain rate in an infinite specimen, there is a finite wavelength with the maximum growth rate. It is argued that this wavelength should correspond to the most probable minimum spacing for shear bands.

Approximate analysis for the formation of adiabatic shear bands

Abstract Parametric solutions are given for the formation of adiabatic shear bands in the context of the onedimensional nonlinear theory where inertia and elasticity are ignored. When heat conduction is also ignored, the exact solution reduces completely to a sequence of quadratures. For a perfectly plastic material with heat conduction, an implicit parametric solution is also constructed. This is similar to the previous one in many ways, but now it involves two quadratures, a single nonautonomous first-order ODE. and two functions that obey heat equations. This solution appears to be very accurate (compared to the full finite element solution) until the time of stress collapse. Results indicate that for weak rate hardening of the power law type, intense localization depends strongly on the initial characteristics of thermal softening and not at all on the high temperature characteristics. Within the context of rigid/perfect plasticity, a scaling law for the critical strain is given, and a figure of merit is defined that ranks materials according to their tendency to form adiabatic shear bands.

Steady shearing in a viscoplastic solid

Abstract S teady shearing solutions are found as quadratures within the context of a simple theory of viscoplasticity, which includes thermal softening and heat conduction. The solutions are illustrated by numerical examples for four commonly used versions of viscoplasticity, where each version has first been calibrated against the same hypothetical data set. It is found that, although they all predict qualitatively similar morphology, the four flow laws give results that differ in detail and one in particular differs substantially from the other three at the more extreme conditions. Although definitive data do not exist, there appears to be rough agreement with physical measurements of adiabatic shear bands. The conjecture is made that steady solutions correspond to central boundary layers for the full unsteady theory.

A continuum framework for finite viscoplasticity

Abstract A continuum framework for finite viscoplasticity is developed based on Lees multiplicative decomposition with internal variables. Noteworthy features include a thermodynamically consistent treatment of the storage of cold work and plastic volume change and a careful examination of the restrictions imposed by the entropy inequality and the property of instantaneous thermoelastic response.

Steady state penetration of rigid perfectly plastic targets

Abstract The problem of steady penetration by a semi-infinite, rigid penetrator into an infinite, rigid/perfectly plastic target has been studied. The rod is assumed to be cylindrical, with a hemispherical nose, and the target is assumed to obey the Von-Mises yield criterion with the associated flow rule. Contact between target and penetrator has been assumed to be smooth and frictionless. Results computed and presented graphically include the velocity field in the target, the tangential velocity of target particles on the penetrator nose, normal pressure over the penetrator nose, and the dependence of the axial resisting force on penetrator speed and target strength.

Adiabatic Shear Bands in Simple and Dipolar Plastic Materials

A simple version of thermo/viscoplasticity is used to model the formation of adiabatic shear bands in high rate deformation of solids. The one dimensional shearing deformation of a finite slab is considered. Equations are formulated and numerical solutions are found for dipolar plastic materials. These solutions are contrasted and compared with previous solutions for simple materials.

The asymptotic structure of an adiabatic shear band in antiplane motion

A two parameter solution for the fields near the tip of a propagating, antiplane shear band are found. The material model used here represents a rigid-plastic solid with linear thermal softening and power law rate hardening, but without work hardening.

Elastic wave propagation through a material with voids

Abstract An exact mathematical analogy exists between plane wave propagation through a material with voids and axial wave propagation along a circular cylindrical rod with radial shear and inertia. In both cases the internal energy can be regarded as a function of a displacement gradient, an internal variable, and the gradient of the internal variable. In the rod the internal variable represents radial strain, and in the material with voids it is related to changes in void volume fraction. In both cases kinetic energy is associated not only with particle translation, but also with the internal variable. In the rod this microkinetic energy represents radial inertia ; in the material with voids it represents dilitational inertia around the voids. Thus, the basis for the analogy is that in both cases there are two kinematic degrees of freedom, the Lagrangians are identical in form, and therefore, the Euler–Lagrange equations are also identical in form. Of course, the constitutive details and the internal length scales for the two cases are very different, but insight into the behavior of rods can be transferred directly to interpreting the effects of wave propagation in a material with voids. The main result is that just as impact on the end of a rod produces a pulse that first travels with the longitudinal wave speed and then transfers the bulk of its energy into a dispersive wave that travels with the bar speed (calculated using Youngs modulus), so impact on the material with voids produces a pulse that also begins with the longitudinal speed but then transfers to a slower dispersive wave whose speed is determined by an effective longitudinal modulus. The rate of transfer and the strength of the dispersive effect depend on the details in the two cases.