Elastic plastic

Sadovskii V.M. (1997) Discontinuous solutions in dynamic elastic-plastic problems. Nauka, Moscow (in Russian). 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

Hardness is a measure of a material s resistance to deformation. In this article hardness is taken to be the measure of a material s resistance to indentation by a tool or indenter harder than itself This seems a relatively simple concept until mathematical analysis is attempted the elastic, plastic, and elastic recovery properties of a material are involved, making the relationship quite complex. Further complications are introduced by variations in elastic modulus and frictional coefficients. 

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Assume pressure, needed to take the elastic—plastic boundary to radius r corresponds to point B (see Fig. 3). Then provided the cylinder unloads elasticady when the internal pressure is removed, ie, unloading path BE is paradel to OA, the residual shear stress distribution is as fodows. 

Fig. 12. Uniaxial stress—strain curve for an elastic plastic material. See text.

Plastic Forming. A plastic ceramic body deforms iaelastically without mpture under a compressive load that produces a shear stress ia excess of the shear strength of the body. Plastic forming processes (38,40—42,54—57) iavolve elastic—plastic behavior, whereby measurable elastic respoase occurs before and after plastic yielding. At pressures above the shear strength, the body deforms plastically by shear flow. 

The fact that shock waves continue to steepen until dissipative mechanisms take over means that entropy is generated by the conversion of mechanical energy to heat, so the process is irreversible. By contrast, in a fluid, rarefactions do not usually involve significant energy dissipation, so they can be regarded as reversible, or isentropic, processes. There are circumstances, however, such as in materials with elastic-plastic response, in which plastic deformation during the release process dissipates energy in an irreversible fashion, and the expansion wave is therefore not isentropic. 

Figure 5.5. Elastic-plastic finite closed cycle of homogeneous deformation in strain space.

Casey, J. and Naghdi, P.M., Strain Hardening Response of Elastic Plastic Materials, in Mechanics of Engineering Materials (edited by C.S. Desai and R.H. Gallagher), Wiley, New York, 1984, Chap. 4, pp. 61-89. 

Naghdi, P.M. and Trapp, J.A., Restrictions on Constitutive Equations of Finitely Deformed Elastic-Plastic Materials, Quart. J. Mech. Appl. Math. 28, Part 1,25-46(1975). 

For a given amplitude of the quasi-elastic release wave, the more the release wave approaches the ideal elastic-plastic response the greater the strength at pressure of the material. The lack of an ideally elastic-plastic release wave in copper appears to suggest a limited reversal component, however, this is much less than in the silicon bronze. Collectively, the differences in wave profiles between these two materials are consistent with a micro-structurally controlled Bauschinger component as supported by the shock-recovery results. Further study is required to quantify these findings and... 

Figure 7.2. Response of elastic-plastic solid to planar impact at X = 0 u = longitudinal particle velocity. Measurements are made as a function of time at fixed Lagrangian position X.

Introduction of the surface-nucleation mechanism in numerical computation of elastic-plastic wave evolution leads to enhanced precursor attenuation in thin specimens, but not in thicker ones. Inclusion of dislocation nucleation at subgrain boundaries indicates that a relatively low concentration of subgrain boundaries ( 2/mm) and nucleation density (10"-10 m ) is sufficient to obtain predicted precursor decay rates which are comparable to those obtained from the experiments. These experiments are only slightly above the threshold necessary to produce enhanced elastic-precursor decay. 

W. Herrmann, D.L. Hicks, and E.G. Young, Attenuation of Elastic-Plastic Stress Waves, in Shock Waves and the Mechanical Properties of Solids (edited by J.J. Burke and V. Weiss), Syracuse University Press, Syracuse, 1971, pp. 23-63. 

Romanchenko and Stepanov (1981) recognized that spall in an elastic-plastic material can involve plastic release to the tensile failure stress followed by elastic recovery as stress relaxation at the spall plane proceeds. Thus, the... 

Figure 8.7. Propagation of wave profile in an elastic-plastic material from the spall plane to the monitoring interface. The wave front propagates at a plastic wave speed whereas the wave release propagates at an elastic wave speed and complicates the analysis of the material spall strength.

Figure 8.8. The stress-particle velocity impedance diagram for planar spall in an elastic-plastic material.

Romanchenko and Stepanov (1981) recognized that the impulse imparted at the spall plane to material downstream, because of the elastic-plastic nature of this material, led to an attenuating tensile stress pulse propagating toward the sample-window interface as is illustrated in Fig. 8.9. Thus, the maximum tension inferred from the measured spall signal should be adjusted for this attenuation in estimating a material spall strength at the spall plane. 

Figure 8.9. The effect of attenuation of the pullback wave signal in an elastic-plastic material. Amplitude of the pullback signal at the recording interface will be diminished due to wave attenuation and will not provide an accurate measure of the material spall strength.

Figure 8.10. Approximation of the pullback signal amplitude and shape used to estimate correction to the spall strength due to elastic-plastic attenuation.

Lui, D.T., Longitudinal Wave Propagation in an Elastic-Plastic Medium, Lockheed Missiles and Space Company Technical Report LAC-MSC-2-60-64-39, Sunnyvale, CA, September 1964. 

Herrman, W., Some Recent Results in Elastic-Plastic Wave Propagation, in Propagation of Shock Waves in Solids (edited by Varley, E.), the American Society of Mechanical Engineers, New York, 1976, pp. 1-26. 

Wagner, M.H. and Kreyenhagen, K.N., Review of Hydro-Elastic-Plastic Code Analyses as Related to the Hypervelocity Particle Impact Hazard, European Space Agency Report No. ESA SP-153, Paris Cedex, France, pp. 115-120, October 1979. 

Load-extension curves for non-elastic (plastic) behaviour... 

AqIq = A1 for plastic deformation or for elastic or elastic/plastic deformation when v = 0.5. Hence... 

The utility of K or any elastic plastic fracture mechanics (EPFM) parameter to describe the mechanical driving force for crack growth is based on the ability of that parameter to characterize the stress-strain conditions at the crack tip in a maimer which accounts for a variety of crack lengths, component geometries and loading conditions. Equal values of K should correspond to equal crack tip stress-strain conditions and, consequently, to equivalent crack growth behavior. In such a case we have mechanical similitude. Mechanical similitude implies equivalent crack tip inelastic zones and equivalent elastic stress fields. Fracture mechanics is... 

Elastic-plastic Fracture Mechanics Behavior of Graphite... 

That fraction of the applied work which is not consumed in the elastic-plastic deformation remains to create the new crack surface, i.e., the crack driving force. Therefore, a nonlinear fracture toughness, G, may be defined as follows ... 

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