Elastic limit

Elastic-Element Methods Elastic-element pressure-measuring devices are those in which the measured pressure deforms some elastic material (usually metallic) within its elastic limit, the magnitude of the deformation being approximately proportional to the applied pressure. These devices may be loosely classified into three types Bourdon tube, bellows, and diaphragm. 

The development of flaws and the loss of interparticle bonding during decompression substantially weaken compacts (see breakage subsection). Delamination during load removal involves the fracture of the compact into layers, and it is induced by strain recovery in excess of the elastic limit of the material which cannot be accommodated by... 

Elastic limit the maximum stress a test specimen may be subjected to and which may return to its original length when the stress is released. 

Yield strength or tensile proof stress the maximum stress that can be applied without permanent deformation of the test specimen. For the materials that have an elastic limit (some materials may not have an elastic region) this may be expressed as the value of the stress on... 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

Corresponding referential, current spatial, and unrotated spatial inelastic constitutive equations are equivalent, and identical results are obtained from their use, if corresponding moduli and elastic limit functions are used. In the current spatial constitutive equations, if the dependence of the spatial moduli... 

Figure 5.1. Particle history and elastic limit surface in strain space.

In six-dimensional strain space, may be viewed as the inner produet of the normal to the elastic limit surface and the tangent to the strain history , see Fig. 5.1. Its value is negative, zero, or positive depending on whether i, points inward, along the tangent, or outward to the elastic limit surface. Four cases may be distinguished. 

Elastic. The strain lies within the elastic limit surface < 0. In this case, it is assumed that k = 0. The material is said to be elastic and the elastic limit surface is stationary. 

Inelastic Loading. The strain lies on the elastic limit surface = 0, and the tangent to the strain history points in a direction outward from the elastic limit surface > 0. The material is said to be undergoing inelastic loading, and k is assumed to be a function of the strain s, the internal state variables k, and the strain rate k... 

The elastic limit surface now moves outward so that the strain remains on it and f continues to be zero. 

Unloading. The stress lies on the elastic limit surface and the tangent to the stress history points inward into the elastic region/= 0,/< 0. Then ic = 0 and the elastic limit surface is stationary. 

Inelastic Loading. The stress lies on the elastic limit surface / = 0 but the conditions on / depend on whether the material is hardening, perfectly inelastic, or softening. 

A similar argument leads to the result that, in stress spaee, the normal veloeity of the elastie limit surfaee is given by = fjn where is the magnitude of the normal veetor = d//ds. Consequently, = R(vJvJ where R = njn is a positive sealar, and the hardening index A has the same sign as the ratio of the outward normal veloeities of the elastic limit surfaees in stress spaee and strain spaee, respeetively. 

Softening. When /I < 0 then / < 0. The elastic limit surface in stress space is moving inward, and the material is said to be softening. 

This ambiguity in the stress space loading criterion may be illustrated by considering a stress-strain plot corresponding to simple tension, as shown schematically in Fig. 5.3. With each point on the stress-strain curve past the initial elastic limit point A, there is associated a point on the elastic limit surface in stress space and a point on the elastic limit surface in strain space. On the hardening portion of the stress strain curve AB, both the stress and the strain are increasing, and the respective elastic limit surfaces are moving... 

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