Elastic-Plastic Loading

Alternatively, if detachment is associated with a brittle failure, then one must first determine if the fracture followed an elastic loading where an elastic model such as the JKR theory is appropriate or if it follows a plastic or elastic-plastic loading. In this latter case, the force needed to detach the particle from the substrate depends on the specific properties of the materials and the details of the deformations [63]. 

In the model it was assumed that the upper pipe was half full and the lower pipe full of liquid cyclohexane. It was also assumed that the scaffolds were made of normal straetural steel these were simulated hy beam elements with contact elements to prevent any loading when the pipe eontact was lost. Elastic-plastic load-deflection relationships were used to simulate the pipe, the bellows, and the scaffolding. 

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. 

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

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... 

As with the purely elastic case, the energy values associated with elastic-plastic fracture may be ascertained from the load versus load-point deflection diagram for a cracked body as shown in Fig. 5. 

Fig. 5. Diagram of load versus load-point displacement for an elastic-plastic body experiencing stable crack extension [48],...

Fig. 2.2. The characteristic stress pulses produced by shock loading differ considerably depending upon the stress range of the loading. The first-order features of the stress pulses can be anticipated from critical features of the stress-volume relation. In the figure, P is the applied pressure and HEL is the Hugoniot elastic limit. Characteristic regimes of materials response can be categorized as elastic, elastic-plastic, or strong shock.

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

Fig. 2.5. The idealized elastic/perfectly plastic behavior results in a well defined, two-step wave form propagating in response to a loading within the elastic-plastic regime. Such behavior is seldom, if ever, observed.

The test can provide compressive stress, compressive yield, and modulus. Many plastics do not show a true compressive modulus of elasticity. When loaded in compression, they display a deformation, but show almost no elastic portion on a stress-strain curve those types of materials should be compressed with light loads. The data are derived in the same manner as in the tensile test. Compression test specimen usually requires careful edge loading of the test specimens otherwise the edges tend to flour/spread out resulting in inacturate test result readings (2-19). 

Let us analyse the above data on the basis of Lawn and Howes analysis 29). Based on the mechanics of hardness identation - assuming the loading cycle to be elastic-plastic and unloading to be elastic — these authors have recently derived an interesting expression of the residual impression parameter (relative depth recovery) as function of the ratio MH/E. Accordingly ... 

The two mechanical properties measured most frequently using indentation techniques are the hardness, H, and the elastic modulus, E. A t5pical load-displacement curve of an elastic-plastic sample during and after indentation is presented in Fig. 30, which also serves to define some of the experimental quantities involved in the measurement. 

Figure 20. Elastic-Plastic Solution for Bending of Blast Loaded Beams. (Reprinted with permission from ref. 15. Copyright 1983 Elsevier Science.)...

Although in practice the formation of inner cracks requires a certain threshold loading, for most brittle ceramic materials this threshold is negligibly small (usually less than 1 newton, seen clearly in hardness tests). It is thought that cracks make well defined spheres entirely beneath the contact zone, and that they grow downwards as the load is applied. Such a system presents a complicated elastic-plastic problem. 

Extensive theoretical investigations devoted to calculation of residual stresses have been carried out for metals. The principal theme of this work is assumption that residual stresses and strains are the result of differences between pure elastic and elastic-plastic deformations under fixed loading.127 128 The same mechanism, i.e., the appearance of plastic deformed zones, is responsible for the residual stresses arising during crystallization of metals, which occurs on quenching from the melt or cooling after welding. 

G. Harkegard Application of the finite element method to cyclic loading of elastic-plastic structures containing effects. Int. J. Fract. 9, 322 (1973)... 

Consider a continuous fiber-reinforced ceramic as a multiphase system where the individual phases are parallel to one another and to the uniaxial loading direction. The fibers (or fiber bundles), matrix, and interface zone are treated as individual phases. In general, each phase undergoes elastic-plastic (creep) deformation. In the present analysis, the creep rate of each phase, e is assumed to obey a general creep law of the following form... 

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