Stanley E. Jones

A revised form for the Johnson-Cook strength model

Abstract Strength models play a key role in the numerical simulation of impact events. A revised form of the Johnson–Cook strength model is proposed in this paper. The revised model treats the sudden strengthening that many ductile metals exhibit at strain rates greater than 104/s. Strain rates of this magnitude are generally considered to be beyond the capability of the split-Hopkinson pressure bar and so such abrupt strengthening behavior is often not observed and reported. A method to economically estimate all eight coefficients of the revised strength model using quasi-static tension data and Taylor impact test data reduced with a modified version of the EPIC finite element code is also described. Revised strength model coefficients were determined for: 7075-T6 aluminum, OFHC copper, wrought iron, and a high-strength steel (Astralloy-V®). A good fit to the quasi-static tension data and Taylor impact test results was obtained for these four different metals. The behavior of the revised strength model at high strain rates also compared favorably with independent predictions from an analytical model calibrated with the Taylor impact data.

An elementary theory for the Taylor impact test

Abstract There is a linear relationship between certain normalized lengths in recovered Taylor Impact specimens. This observation seems to have first been made by J. W. House, although G. I. Taylor and A. C. Whiffen both produced graphs involving these same scaled variables. However, no explanation was given for this scaling and it was not used for any specific purpose. In this paper, a theoretical basis for the linearity is established and the slope and intercept are used to determine several important physical parameters. These parameters are then used to determine the state of stress at strain-rates exceeding 104/s. This information is useful because it helps to bridge the strain-rate gap between Split-Hopkinson pressure bar testing and the ultra-high rates achieved with plate impact experiments.

An engineering analysis of plastic wave propagation in the Taylor test

A new one-dimensional analysis of the Taylor impact test is presented. This analysis differs from any previously presented in that the wave mechanics are separated from the calculation of dynamic stress. The new results utilize post test measurements to estimate key parameters in the plastic wave propagation. However, these measurements are incorporated into the analysis in a very unconventional way. A comparison with continuum code calculations shows very good agreement has been achieved.

An Estimate for Mass Loss From High Velocity Steel Penetrators

Analytical models of the penetration process focus on estimating depth of penetration based on target density, target strength (sometimes associated with the unconfined compressive strength of the target for geological targets), the areal density of the penetrator (W/A), and the impact velocity. In this paper, an expression for work is used in conjunction with thermodynamic considerations to devise a simple estimate for mass lost by a high velocity projectile during the penetration process. The result shows that the mass loss is directly proportional to the tunnel length and the target shear strength. The constant of proportionality is not easy to deduce, however, in that it contains an unusual factor from the work analysis. A method for estimating target shear under high pressure from penetration experiments is introduced.Copyright

Calculation of the forming limit diagram

A mathematical model is presented to help understand sheet metal deformation during forming. The particular purpose of this model is to predict the forming limit diagram (FLD). The present model is an extension of a previous analysis by Jones and Gillis (JG)[1] in which the deformation is idealized into three phases: (I) homogeneous deformation up to maximum load; (II) deformation localization under constant load; (III) local necking with a precipitous drop in load. In phase III, the neck geometry is described by a Bridgman-type neck. The present model extends the JG theory, which was applied only to the right-hand side (RHS) of the FLD. The main difference in treating the two different sides of the FLD lies in the assumptions regarding the width direction deformations. For biaxial stretching (the RHS), the minor strain rate is assumed to be homogeneous throughout the process. However, for the left-hand side (LHS) of the FLD in the critical cross section, the minor strain rate is taken to be proportional to major strain rate. This is a critical difference from the JG approach and permits the LHS to be computed with good accuracy. Another important difference between this and the JG analysis is a more realistic neck geometry. At the inception of phase III, JG matched the phase II sheet thickness at the center of the neck, that is, at its minimum cross section. Here, the phase III neck matches the phase II sheet thickness at its ends, that is, at its maximum cross section. Although this may seem a minor point, it greatly improves the geometrical concept involved. Both the actual neck geometry and the criterion for determining the limit strain are modified from the earlier analysis in order to agree more closely with actual press shop practice. Results from this analysis are compared with the experimental ones for aluminum-killed (AK) steel and three aluminum alloys. These results are also compared to other theoretical calculations of the forming limit for AK steel. It is apparent that the present model is best. Unlike the other types of analyses, the present model predicts the limiting strain states for several materials very accurately without any adjustable parameters. This is certainly an unprecedented result. Using the mathematical model, the effects of varying material properties are studied. The properties considered are the strain-hardening exponent,n, the strain-rate sensitivity parameter,m, and the plastic anisotropy ratio,r, The important influence of these material properties upon the formability (level of the FLD) is affirmed.

A continuum mechanics code analysis of steady plastic wave propagation in the Taylor test

Simple conservation relationships (jump conditions) in conjunction with postulated material constitutive behavior are applied to steady plastic strain waves propagating in problems of uniaxial stress and Taylor Cylinder Impact. These problems are simulated with a two-dimensional Lagrangian continuum mechanics code for the purpose of numerically validating the jump relationships as an accurate analytical representation of plastic wave propagation. The constitutive behavior used in this effort assumes isotropy and models the thermodynamic response with a Mie-Grunisen Equation-of-State and the mechanical response with the rate-dependent Johnson-Cook and MTS flow stress models. The jump relationships successfully replicate the results produced by continuum code simulations of plastic wave propagation and provide a methodology for constructing mechanical constitutive models from experimental plastic wave speed information. Comparisons are also presented between experimental speeds from Taylor Cylinder Impact tests with jump relationships and continuum code predictions, indicating that the above mentioned flow stress models may not accurately capture plastic wave propagation speeds in annealed and hardened copper.

Experimental rod impact results

Abstract The one-dimensional long rod penetration theory of Tate [ J. Mech. Phys. Solids 15 387 (1967)] has been modified by the authors [ J. Mech. Phys. Solids 35 , 121 (1987)] to include mass transfer into the plastic zone. The mushroom strain at the penetrator tip is incorporated into the analysis. This latter factor has a very substantial effect on calculated penetrations. In this paper, a comparison between the modified Tate theory and a series of penetration experiments is given. The results tend to confirm the validity of the mathematical model for realistic values of the penetrator and target strengths.

On the Taylor test: A continuum analysis of plastic wave propagation

The determination of the mechanical properties of materials is the foundation of many engineering design problems. Numerous test methods have evolved as standards for the determination of these properties. Design problems which require inelastic behavior of the material are unique because the test methods must provide a detailed knowledge of the evolution of the yield behavior. High rate problems represent a special class of inelastic engineering design problems and the interpretation of test methods used to determine material’s behavior for these problems are an important research topic. The Taylor Anvil or Taylor Impact test is a test commonly employed to determine the mechanical properties of materials for this important class of engineering design problems. A continuum approach based on jump discontinuities at the plastic wave front is developed which can be used as the basis for advanced engineering models of the experiment and analysis of the numerical method used to incorporate various constitutive...

An elementary theory of one-dimensional rod penetration using a new estimate for pressure

In this paper, a new pressure law is proposed to replace the modified Bernoulli equation of Tate in 1967 and 1969. It is achieved by decomposing the equation of motion, which was proposed by Jones et al. in 1987, into two parts and incorporating the kinematic equation by Wilson et al. in 1989. The new pressure law takes the effect of mushroom strain into account. From two different considerations, the pressure law is applied to the one-dimensional penetration modeling. First, by assuming that the rod/target interface pressure is approximately constant during the quasi-steady state, the governing equations can be analytically integrated to give a closed form solution for the penetration depth. The prediction is reasonably good in the low velocity regime. Secondly, a velocity-dependent interface pressure is added. A so-called shape factor, which was first introduced without physical interpretation by Alekseevskii in 1966, is substantiated. With this factor, the governing equations can be numerically integrated to give very accurate predictions for the impact velocity range from 1 km/s to 4 km/s.

High-Speed Penetration of Concrete Using a New Analytical Model of Velocity-Dependent Friction

In this paper, a completely analytical formulation of velocity-dependent friction is used to eliminate the analytical complexity associated with the evaluation of velocity-dependent integrals in the penetration model presented in [5]. The simplification employed is an extension of the method used in [6], in which the velocity regime governing the coefficient of sliding friction was divided into two sectors, producing two distinct velocity-dependent friction coefficients, for low-speed and high-speed friction. In this paper, the velocity regime is divided into a number of intervals, thereby replacing the linear velocity-dependent friction relationship at low speeds with a series of steps; the method is flexible enough also to permit a nonlinear profile for low-speed sliding friction. As a byproduct of the work, a completely analytical estimate for the work done by sliding friction in the problem can be found, permitting the evaluation of projectile mass loss.Copyright