Strain rate equation

The principal strain rates are a similar set again, and the strain-rate equations are exact. 

We have already remarked that the shear-strain-rate equation (7.10) is a flux/driving-gradient relation, of the same type as eqn. (8.3). If a first-derivative equation (8.3) plus a conservation statement leads to second-derivative equations (8.6) and (8.7), we can enquire whether eqn. (7.10) has consequences of a parallel kind. 

As before, a situation can be chosen for study so that in the strain-rate equation (12.8), the five terms reduce to two terms—an orientation term d jda and a spatial position term d /dx. Suppose varies with composition and stress as just noted in the previous paragraph then the strain rate of component A gains in principle four contributions, from change of stress with position, stress with orientation, composition with position, and composition with orientation. But the last is disregarded, so that three contributions to strain rate remain. 

Integrating this equation with respect to time, assuming a constant strain rate, equating it to the decrease in potential energy of the system, and ignoring surface energy changes, one can show that... 

Q.J. Peng, J. Kwon, T. Shoji, Development of a fundamental crack tip strain rate equation and its application to quantitative prediction of stress corrosion cracking of stainless steels in high temperatme oxygenated water, J. Nucl. Mater. 324 (2004) 52-61. 

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

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 remaining equations of the referential formulation may be translated into spatial terms by proceeding in the same way. Using (A.36) and (A.45) in the strain rate relation (5.131)... 

Similarly, the same equations combine to provide a strain-rate-dependent fracture stress criterion... 

Figure 8.14 shows that a transition in spall mechanism is predicted to occur at a critical strain rate e, which corresponds to the intersection of the brittle and ductile fragmentation energy curves. By equating (8.39) and (8.43)... 

To tackle any of these we need constitutive equations which relate the strain-rate e or time-to-failure tf for the material to the stress ct and temperature T to which it is exposed. These come next. 

When the stress is removed there is an instantaneous recovery of the elastic strain, e, and then, as shown by equation (2.31), the strain rate is zero so that there is no further recovery (see Fig. 2.35). 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

Equations (5.21), (5.22) and (5.23) are useful for the high strain rates experienced in injection moulding or extrusion but unfortunately they do not predict the low strain rate situation very well where plastic melts tend towards Newtonian behaviour (ie n -) 1). This is illustrated in Fig. 5.7. 

However, for the high strain rates appropriate for the analysis of typical extrusion and injection moulding situations it is often found that the simple Power Law is perfectly adequate. Thus equations (5.22), (5.23) and (5.27) are important for most design situations relating to polymer melt flow. 

Various workers have used equation 8.8, or some modified version thereof, to compare observed with calculated crack velocities as a function of strain rate, but Fig 8.8 shows results from tests on a ferritic steel exposed to a carbonate-bicarbonate solution. The calculated lines move nearer to the experimental data as the number of cracks in equation 8.9 is increased, while the numbers of cracks observed varied with the applied strain rate, being about 100 for 4pp 10 s , but larger at slower 4pp and smaller at higher 4pp. 

If crack propagation occurs by dissolution at an active crack tip, with the crack sides rendered inactive by filming, the maintenance of film-free conditions may be dependent not only upon the electrochemical conditions but also upon the rate at which metal is exposed at the crack tip by plastic strain. Thus, it may not be stress, per se, but the strain rate that it produces, that is important, as indicated in equation (8.8). Clearly, at sufficiently high strain rates a ductile fracture may be propagated faster than the electrochemical reactions can occur whereby a stress-corrosion crack is propagated, but as the strain rate is decreased so will stress-corrosion crack propagation be facilitated. However, further decreases in strain rate will eventually result in a situation where the rate at which new surface is created by straining does not exceed the rate at which the surface is rendered inactive and hence stress corrosion may effectively cease. 

Static and dynamic property The uses of these foams or porous solids are used in a variety of applications such as energy absorbers in addition to buoyant products. Properties of these materials such as a compressive constitutive law or equation of state is needed in the calculation of the dynamic response of the material to suddenly applied loads. Static testing to provide such data is appealing because of its simplicity, however, the importance of rate effects cannot be determined by this one method alone. Therefore, additional but numerically limited elevated strain-rate tests must be run for this purpose. 

For the simplest case of a linear dumbbell in a homogeneous velocity gradient of strain rate s, the force balance equation is the following ... 

In flow-induced degradation, K is strongly dependent on the chain length and on the fluid strain-rate (e). According to the rate theory of molecular fracture (Eqs. 70 and 73), the scission rate constant K can be described by the following equation [155]... 

The Stokes viscous drag equation predicts a proportionality between the molecular stress ( / ) with the product of solvent viscosity (qs) and fluid strain-rate... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

By using the kinetic equations developed in Sect. 5.2, the degradation yield as a function of strain rate and temperature can be calculated. The results, with different values of the temperature and preexponential factor, are reported in Fig. 51 where it can be seen that increasing the reaction temperature from 280 K to 413 K merely shifts the critical strain rate for chain scission by <6%. 

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