Strains uniform extension strain

This is the well-known Considere (1885) criterion for impending necking for a rigid plastic, rate-insensitive material. It is expressed graphically with a simple construction in Fig. 10.1 showing the intersection of the stress-strain curve with the curve of the decreasing slope of that curve, where the point of intersection is at u, gives the uniform extension strain. In Fig. 10.1, for strains e < u,dr/d > t e)... 

This gives immediately an expression for the uniform extension strain s at impending localization beyond which straining ceases to be uniform, i.e.. 

Thus, in a rate-insensitive plastic solid s 0) the uniform extension strain is N and is affected only in a very minor way by strain-rate sensitivity, showing a small decrease in uniform extension strain s - The more dramatic effect of increasing strain-rate sensitivity on stability in axial flow occurs in the post-necking region, where necking localization is importantly counteracted by an increase in flow stress with increasing local strain rate even when there is no strain-hardening rate. 

Another modification to the slow strain-rate test involves the superimposition of a low amplitude sine wave ripple on the slow uniform extension (Fig. 8.47). In effect this produces higher strain rates (which appear to be more damaging for hydrogen embrittlement), while still giving a long test duration, with adequate time for the accumulation of hydrogen in the steeps. 

The phenomena of brittle and tough fracture give rise to fairly characteristic stress-strain curves. Brittle fracture in materials leads to the kind of behaviour illustrated in Figure 7.1 fairly uniform extension is observed with increasing stress, there is minimal yield, and then fracture occurs close to the maximum on this graph. 

Changes in the morphology and the degree of crystallinity have considerable effects on the mechanical properties of polymers. In particular, the deformation mode of polyethylene changes from uniform extension to necking behaviour above about 50% crystallinity, and while the yield-stresses increase with density, the strain at fracture apparently decreases. In understanding deformation mechanism, and so mechanical properties, morphological studies play a central part,... 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

These findings show clearly that even when the animal diets are uniform, highly distinctive urinary excretion patterns are exhibited. The results with different inbred strains show that inheritance is the basic reason for the differences in pattern. It would require extremely extensive genetic studies to demonstrate the inheritance process for each item, but this does not seem crucially important from the standpoint of elucidating the phenomenon of individuality. 

In most of these unit cell analyses the assumptions are made that a) strains are small, so that infinitesimal elasticity theory applies, b) the phases are of uniform density and elastic properties and c) the phases are isotropic. Some extensions to the cases of anisotropic phases have been made54 . 

Fig. 6b illustrates the type of simple interpretation of the fault structure often used as a basis of fault seal analysis. This interpretation is based on representation of the fault as a single fault plane, where the offset is assumed to be equal to the seismic (cumulative) offset. This is clearly not valid as a representation of throw distributions for use in fault seal analysis. Fig. 6 also illustrates that sub-seismic fault populations can be clustered around larger faults with extensive areas of low fault densities away from the large faults. This has important implications for the spatial distribution of sub-seismic faults and emphasises that uniform distributions of small faults are not always applicable (except perhaps in areas where more uniform straining is associated with doming). The information contained in Fig. 6 reinforces the results of other recent studies which have noted different fault... 

An extensive homogeneous sample contains a homogeneous stress field with principal values cr, and (cr, + The material is Newtonian and deforms in a steady manner with principal strain rates e and —e. Energy is dissipated at a rate — a )e per second per unit volume, and energy is withdrawn from the sample at this rate uniformly throughout its volume by some unspecified process, so that the sample s temperature is also steady through time. We identify a portion in the interior of the sample that at some moment is a cube with linear dimension k. We ask where is the source of supply of energy to this cube, with power — cr )e. 

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