Elastic strain increment

We develop first the considerations related to shear response in a ID context of plastic-shear flow to state the basic kinetic response of the solid, where s stands for an applied shear stress, t stands for a threshold plastic-shear resistance, and y is taken to be the plastic-shear strain yP. As a useful simplification, we first consider the material to be rigid on the basis that the plastic-shear increments are large, in comparison with the elastic-strain increments. At temperatures T > OK, for which the elastic moduli of the solid are significantly lower than at 0 K, we expect that the rate-independent plastic-shear resistance z temperature dependence as the elastic-shear modulus (Chapter 4). Then, where the plastic response in a rate-independent manner is initiated when s = z(T), under conditions of s < z T), a plastic response is still possible by thermal assistanee and occurs at a (plastic) shear rate of (Argon 1973)... 

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor... 

Figure 7.3. Determination of elastic and viscous components. Incremental stress-strain curve constructed by stretching a specimen in strain increments of 2 to 5% and allowing the specimen to relax to an equilibrium stress before an additional strain increment is added. The elastic fraction is defined as the equilibrium stress divided by the initial stress. (Adapted from Silver, 1987.)...

At room temperature, most metals have some elasticity, which manifests itself as soon as the slightest stress is applied. Usually, they also possess some plasticity, but this may not become apparent until the stress has been raised appreciably. The magnitude of plastic strain, when it does appear, is likely to be much greater than that of the elastic strain for a given stress increment. Metals are likely to exhibit less elasticity and more plasticity at elevated temperatures. A few pure unalloyed metals (notably aluminum, copper and gold) show little, if any, elasticity when stressed in the annealed (heated and then cooled slowly to prevent brittleness) condition at room temperature, but do exhibit marked plasticity. Some unalloyed metals and many alloys have marked elasticity at room temperature, but no plasticity. 

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

In an elastic solid exhibiting energy-elasticity, the increment of external work done, dW, per unit volume must be equal to the increment of stored elastic strain... 

The term — dUldA) is defined as the strain energy release rate, or the fracture energy. The word rate normally means rate with to time. However in this context, the word rate refers to the rate of releasing elastic strain energy in propagating a fracture over an increment of area 8A, and not the time. 

The first of the above equations can be rewritten to introduce an elastic effecttve stress a[j which determines entirely the strain increments under elastic behaviour ... 

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

Since the plastic compliance tensor of (2.312), determined by the flow rule, is represented by a product of two second-order tensors, the determinant is identically zero (detC = 0, if we set the second-ordCT tensors as vectors as mentioned in (2.310)). Since it is not possible to obtain the inverse of Cp directly, we use the properties of the elastic compliance C, which has the inverse, along with the direct sum of the strain increment given by (2.293). That is. 

Each strain increment has an elastic and a plastic part ... 

However, the lines are not smooth but contain numerous small discontinuities where the system stress decreases precipitously after a small strain increment owing to a plastic response of the inclusion arrows mark the particularly conspicuous drops. The two curves for uniaxial and pure-shear deformations would superpose perfectly if the system behaved totally elastically deviations are evident after a sufficient number of plastic unit events, which are irreversible and dissipative and probably different for... 

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

Figure 2. FEM simulation of the elastic plastic particle constrained by elastic matrix illustrating a layer undergoing a nonzero axial plastic strain increment Asyy.

The relationship between the incremental stress, A[Pg.519]

Answer by Author Yes, However, our testing was not set up with the determination of elastic moduli in mind. The strain increments were too large and the scatter too wide in range to determine an absolute value for the elastic moduli. The values derived, however, did follow the normal tendency, i.e., the elastic modulus at liquid-nitrogen temperatures of the material in the two tested orientations was greater than that at room temperature. 

Another property that determines a plastic s usefulness is its modulus. Modulus is related to stiffness high stiffness corresponds to high modulus. Modulus is defined as the ratio of stress (deforming force per unit cross-sectional area) to strain (increment of deformation) for elastic deformation (Fig. 19.7). Its dimensions are the same as those of pressure or tensile strength. 

As shown in Fig. 2, increment, rises as the drop of the temperature and shrinkage of volume from crystallization. Meanwhile, solidifyed layer of the part develops horizontal strain under the action of packing pressm-e. Under the actions of, and 4, the spring is forced to move and lead to elastic strain 2, which brings an elastic stress field. Furthermore,... 

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

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