Large 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 

This is called the Kirchoff strain by Nielsen. The Eulerian strain is defined by 

It can be shown for uniaxial strain that the Eulerian strain is 

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Secondly, the Llo structure must be stable with respect to both small and large strains. The former is assured by the positive definiteness of elastic moduli. Clearly, an exhaustive test of the stability with respect to large strains cannot be made and the following two types of tests were performed. First, the energy of the structure was calculated as a function of strains corresponding to 20% changes in the lattice parameters a and... 

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... 

It is now easy to understand why the viscoelastic behavior of filled vulcanizates at large strains is so complex and why it is so difficult to investigate in definitive manner. None of the processes responsible for stress-softening occur instantaneously (16,182), nor can they generally be expected to obey the time-temperature relationships of linear viscoelasticity. Since time-temperature superposition is no longer applicable,... 

The specific material constants were introduced earlier through eqs. (7.24c), with 9 = T/Tg and/(yP) being given by eqs. (7.41). In eq. (7.46) is an empirical rate parameter, to be chosen as a best fit to experimental results to complete the definition, and to replace the mechanistic expressions with the more convenient equivalent phenomenological forms introduced here. Through these choices, the power-law form of the kinetic law of eq. (7.43) becomes a flexible representation of the large-strain plastic flow of metallic glasses near Jg. 

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. 

The large strain response in the glassy or semicrystalline state is that of a nonlinear viscoelastic solid. However, both engineering and theoretical approaches to plasticity in polymers have largely developed as an independent discipline, in which (Ty plays a central role, in spite of its somewhat arbitrary definition (indeed it is not always possible to associate cty with a maximum in the force-deformation curve [5]). This is because in practice the yield point, rather than the ultimate strength, is usually considered to be the failure criterion for ductile materials. 

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. 

The non-linearity may arise for a variety of reasons. First, the linear theory has been developed for small strains, and to generalise it to large strain requires decisions on the appropriate definitions of both strain and stress, in effect making it necessary to create a new theory. Typical polymer applications may require the material to operate at strains in excess of 10%, and for elastomers the strains may be up to several hundred percent. Secondly, even at small strains linear behaviour may not be obtained. The behaviour may be quite rich, with the possibility of the polymer being initially linear but becoming non-linear at large times. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

It follows that s = In Aq/A for large plastic strains. The quantity A0/A is a measure of the reduction in area due to deformation. One can define a quantity the true reduction in area (f), in a manner analogous to the definition of true strain ... 

There are many deposit-substrate combinations where the basic lattice mismatch is very large, such as when a compound is formed on an elemental substrate, but where excessive strain does not necessarily result. Frequently a non one-to-one lattice match can be formed. If a material can match up every two or three substrate surface unit cells, it may still form a reasonable film [16]. In many cases the depositing lattices are rotated from the substrate unit cells, as well. In a strict definition of epitaxy, these may not be considered, however, it is not clear why high quality devices and materials could not be formed. 

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