Strain, definition

The original length 6f the specimen Is LO and its stretched length is L. At very small deformations, all the strain definitions of Table 2 are equivalent, For shear tests (see Figure 2)... 

If a crystal is exposed to stress in such a way that the strain is kept constant, the stress will decrease with time as shown in Figure 14-4. One concludes that stress relaxation has occurred. Conversely, strain does not remain constant under constant load. Time dependent (i.e., plastic) strain in stressed crystals is called creep. It was already mentioned that elastic strain due to the applied stress is usually less than 1%. Plastic strain definitely dominates beyond the elastic limit which, to a large extent, is due to dislocation formation and motion. Since the crystal lattice is conserved during this... 

When using the generalised Hooke s law strain energy function there are a number of possible strain definitions that can be used depending on the situation. When material deformation is very small the infinitesimal strain approach is a valid approximation with the strain defined as... 

The first one is the small strain definition, or Cauchy strain tensor. Equation (4.1) can be rewritten in tensor form as... 

In strength of materials text, it is weU known that Cauchy stress, small strain because the area in the undeformed body and deformed body is almost the same. For a large strain, however, we generally do not know the area of the deformed configuration. Thus we need to define a stress measure that we can use in the reference configuration. However, it is noted that Cauchy stress is still the most used stress definition or tme stress because the equilibrium is about the deformed body but not the undeformed body. Therefore, other definitions of stress are for convenience of mathematical operation. Stress is generally reported as the Cauchy stress. This shows a distinct departure from the strain definitions. 

Stress and Strain Definitions. As remarked, the material functions relate the stress and strain responses of the material through a constitutive equation. For the elastic material, there is no time dependence and the relationships are relatively simple. In the case of linear viscoelasticity, equations that take into accoimt the time history of the stresses or strains are required. First, stress and strain are defined. 

Extension ratios are most often used in cases of large strains or large deformations and can be found by examining the basic tensile strain definition as follows. 

Yield Criteria. A practical 3ueld criterion needs to describe the conditions under which jdeld will occur for a general stress state (eg tension, compression, shear, or some combination). Before we consider the various 3deld criteria which have been proposed, it is useful to very briefly recap the definition of the principal stress components as these are generally the terms which are used to define the yield criteria (see Viscoelasticity for a more thorough introduction to the generalized stress and strain definitions). 

In addition to the trne strain definition in Equation 2.2, there are other large strain definitions that are used in nonlinear continuum mechanics. These are the Lagrangian and Eulerian definitions depending on whether the original coordinates or the deformed coordinates jc, are used as the reference coordinates. The Lagrangian strain is defined by... 

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