Elastic Strain Rates Research Articles

Dynamic crack-tip fields in rate-sensitive solids

T he asymptotic stress and strain fields near the tip of a crack which propagates dynamically in a rate-sensitive solid are obtained under anti-plane shear and plane strain conditions. The problem is formulated within the context of a small-strain theory for a solid whose mechanical behavior under high strain rates is described by an elastic-viscoplastic constitutive relation. It is shown that, if the stresses are singular at the crack-tip, the viscoplastic relation is equivalent asymptotically to an elastic-non-linear viscous relation. Furthermore, for a certain range of the material parameter which characterizes the rate-sensitivity of the material, the elastic strain-rates near the propagating crack tip are shown to have the same asymptotic radial dependence near the propagating crack-tip as the inelastic strain-rates. This determines the order of the stress singularity uniquely. The governing equations for anti-plane shear and plane strain are then derived. The numerical results for the stress and strain fields are presented for anti-plane shear and plane strain. For the present model, the results suggest that under small-scale yielding conditions, there exists a minimum velocity for stable steady crack propagation. The implication that a terminal velocity for a running crack may exist is also discussed.

Numerical solutions for stresses near crack tips in time-dependent inelastic fracture mechanics

The boundary element method is used in this paper to calculate the stresses in a finite planar body with a traction free crack present in it. The body is assumed to undergo time-dependent inelastic deformation and is described by an elastic-power law creep constitutive model. A stationary crack is modelled as a very thin ellipse and numerical solutions to the complete transient problem are obtained by step-wise integration in time. The numerical results for stresses near a crack tip are compared with the asymptotic analytical results obtained by other researchers. These asymptotic results are those for a plate with a line crack where the elastic strain rates are neglected relative to the nonelastic strain rates near the crack tips. Both Mode III and Mode I deformation, under time-independent and time-dependent remote loading, are considered in the calculations.

Inclusion of elastic strain rate in the analysis of viscoplastic flow during rolling

Elastic strain rates are accounted for in a steady state, isothermal, analysis of viscoplastic flow of metal during rolling. The inclusion of the elastic strain rates allows evaluation of the residual stresses in the product, not possible from analyses based on purely viscoplastic flows.

On finite deformation elasto-plasticity

Some fundamental kinematical and kinetical results in finite elasto-plastic deformations of crystalline solids are reviewed. It is shown that essentially all existing elasto-plasticity concepts lead rigorously to total strain rate measures which are additively decomposed to an elastic and a plastic contribution, provided that the corresponding total, elastic and plastic strain rates are conjugate to the same stress measure. Furthermore, this additivity follows from the conservation of energy which further shows that, under the most general setting, the plastic strain rate may include a “workless” additive part which renders the Eulerian plastic strain rate tensor noncoaxial with the Cauchy stress tensor even when an isotropic yield function is assumed. Various Lagrangian and Eulerian strain measures, their rates and the corresponding conjugate stress measures are examined, and it is established that the additive decomposition of the strain rate holds independently of the particular choice of the strain measure or the ground state. Finally, a conflicting theory by Lee [33,36], who claims to have shown that the usual additive decomposition of the strain rate to an elastic and a platic part is in “error”, is reviewed, and it is shown that this theory also leads to an additive strain rate decomposition, and that Lee's conflicting conclusion stems from misinterpretation. Certain undesirable features of this theory, which emerge from a decomposition of the “total” deformation gradient into an elastic and a plastic part, are discussed and it is concluded that the commonly accepted physically based incremental theories of elasto-plasticity, either phenomenological or microscopically based, present distinct advantages.

Some comments on elastic-plastic analysis

Definitions of plastic and elastic deformation are discussed which are directly related to the physical characteristics of the phenomena and which provide essential uncoupling of elastic and plastic variables. Using finite deformation kinematics and appropriate strain-rate variables in order to carry out total incremental theory, analysis is presented concerning the fact that the total strain rate is not equal to the sum of elastic and plastic rates as is universally assumed. The consequences of this circumstance are examined.It is shown that a conflicting approach to the finite-deformation kinematics of elastic-plastic theory by Nemat-Nasser[4], which generates summable elastic and plastic strain rates, exhibits undesirable features and in particular the associated kinetics does not permit completion of the objective of developing an elastic-plastic constitutive relation on this basis.

Variational principles in uncoupled thermoplasticity

Stress-rate and strain-rate variational principles are derived for a viscoplastic material characterized by additive elastic and inelastic strain rates. Elastic, viscous and plastic properties are assumed to be temperature dependent but all coupling effects are neglected.

The Effect of Strain Rate and Temperature on Yielding in Steels

The effect of elastic strain rates ranging from 10−14 to 10 sec−1 and temperatures ranging from 200 K (−100 F) to 590 K (600 F) on the yield strength of several steels is reported. The steels utilized are a 1018 mild steel, 4340 steel, H-11 tool steel, and 300 grade maraging steel. The results are interpreted in terms of the Cottrell-Bilby yielding model based on release of dislocations from locking carbon atmospheres. The results for all of the materials except the maraging steel are consistent with this model if it is modified to account for relocking of dislocations by migration of carbon atoms. The maraging steel shows a constant strain rate sensitivity at a constant temperature, over the range of strain rates investigated. This rate sensitivity decreases with increasing temperature and at 590 K (600 F) a decreasing strength with increasing strain rate is found. This is attributed to stress aging effects.

On the stability of elastic-plastic and rigid-plastic plates of arbitrary thickness, and flat bars of arbitrary width

Constitutive relations in a form suitable for use in stability problems are derived for an elastic-plastic material undergoing a perturbation from a state of uniaxial stress, based on Green and Naghdi's general theory of an elastic-plastic continuum [1], The deformation is considered to be quasi-static and isothermal, the material is assumed to yield according to the von Mises criterion, and to flow plastically according to the Reuss equations. The elastic strain rates are determined from Hooke's law. These relations are then applied to study the influence of plate thickness on the instability of a rectangular plate under uniform uniaxial compression. The instability of the surface of a compressed semi-infinite elastic-plastic continuum is then considered as the limiting case of infinite plate thickness. It is found that plastic flow has little influence on surface instability which is attributed to elastic deformation. Finally, two cases are considered for a rigid-plastic material, plane strain and plane stress, in order to compare the present treatment with previous work.

The Penetration of Hard-Steel Spheres Into Plane Metal Surfaces

An experimental investigation was undertaken to determine the force-indentation relations governing the contact of hard-steel balls and plane surfaces of various metals both under static and dynamic conditions, the latter involving the Hopkinson-bar technique, with maximum elastic strain rates of 500 sec−1. Excellent correlation was obtained between the measured permanent crater diameter at the contact point and that calculated from strain-gage data by means of an equation treating the bar as a one-dimensional member. A comparison also was effected between the static and dynamic force-indentation curves, the Hertz law of contact, and a relation based upon the concept of a constant flow pressure in the plastic regime.