Variable strain rate

Other test modes, such as constant loading rate and variable strain rate, have been used on a limited basis as research techniques to investigate such phenomena as the path dependence of failure, but no general description of these tests can be provided. Of course the entire area of dynamic testing and fatigue uses various specialized test conditions, but these are discussed later. 

The mechanisms in steady-state creep are usually studied experimentally using Eq. (7.6). There are four main variables strain rate stress temperature and... 

Table 4.6 Mullins number (M) data at 300% maximum strain tests conducted at variable strain rate for PUll (DBD1 BG PTHF)...

Tensile and notch-tensile tests of AISI 304 were conducted at the four test temperatures listed above. In addition, two other variables, strain rate and thermal cycling, were introduced. The strain rates employed were those resulting from crosshead speeds of 0.005, 0.02 and 0.2 in. per min. at each test temperature. To determine the effect of thermal cycling on mechanical properties of AISI 304 in the absence of spontaneous martensitic transformation, one half of all the specimens were subjected to temperature cycling between 300° and 76°K (holding at 76°K) for a period of one year [2]. 

J. Liang, N. Dariavach, and D. Shangguan, Deformation Behavior of Solder Alloys under Variable Strain Rate Shearing and Creep Conditions, Proceedings of Tenth International Symposium and Exhibition on Ad-... 

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 normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

As a starting point it is useful to plot the relationship between shear stress and shear rate as shown in Fig. 5.1 since this is similar to the stress-strain characteristics for a solid. However, in practice it is often more convenient to rearrange the variables and plot viscosity against strain rate as shown in Fig. 5.2. Logarithmic scales are common so that several decades of stress and viscosity can be included. Fig. 5.2 also illustrates the effect of temperature on the viscosity of polymer melts. 

The sometimes contradictory results from different workers in relation to the elements mentioned above extends to other elements . Some of these differences probably arise from variations in test methods, differences in the amounts of alloying additions made, variations in the amounts of other elements in the steel and the differing structural conditions of the latter. Moreover, the tests were mostly conducted at the free corrosion potential, and that can introduce further variability between apparently similar experiments. In an attempt to overcome some of these difficulties, slow strain-rate tests were conducted on some 45 annealed steels at various controlled potentials in three very different cracking environments since, if macroscopic... 

According to some recent results (Sect. 5.5), the dependence of K on e and qs is more involved than suggested by Eq. (94). The dependence of K on ris is much weaker than a direct proportionality and the correct flow parameter to be used in Eq. (94) should be the local fluid kinetic energy ( v2) rather than the strain rate (e). For a constant flow geometry, however, the two variables v and e are interchangeable. At the present stage and in order not to complicate unduly the kinetic scheme, it will be assumed that the rate constant K varies with the MW and strain rate as ... 

In flame extinction studies the maximum temperature is used often as the ordinate in bifurcation curves. In the counterflowing premixed flames we consider here, the maximum temperature is attained at the symmetry plane y = 0. Hence, it is natural to introduce the temperature at the first grid point along with the reciprocal of the strain rate or the equivalence ratio as the dependent variables in the normalization condition. In this way the block tridiagonal structure of the Jacobian can be maintained. The flnal form of the governing equations we solve is given by (2.8)-(2.18), (4.6) and the normalization condition... 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

It is possible to determine the actual strain rate of a material during calculation of dynamic response using an iterative procedure. A rate must be assumed and a DIF selected. The dynamic strength is determined by multiplying the static strength (increased by the strength increase factor) by the DIF. The time required to reach maximum response can be used to determine a revised strain rate and a revised DIF. This process is repealed until the computed strain rate matches the assumed value. There are uncertainties in many of the variables used to calculate this response and determination of strain rates with great accuracy is not warranted. 

The evaluation of viscosity is similar to the evaluation of elastic deformation except the stress in the element changes due to the local velocity gradient. The time variable is defined as the strain rate. The element changes its shape as a function of time, as long as the strain-rate-induced stress is present. Viscosity is the local slope of the function relating stress in the element to strain rate. The usual functionality is found in Fig. 3.3. The process can be visualized by a constant force on the top of the element that creates a strain rate throughout the element. This strain rate causes each molecular layer of the material to move relative to the adjacent layer continuously. Obviously the element is suspended in a continuum and material flows into and out of the geometric element. 

One of the simplest criteria specific to the internal port cracking failure mode is based on the uniaxial strain capability in simple tension. Since the material properties are known to be strain rate- and temperature-dependent, tests are conducted under various conditions, and a failure strain boundary is generated. Strain at rupture is plotted against a variable such as reduced time, and any strain requirement which falls outside of the boundary will lead to rupture, and any condition inside will be considered safe. Ad hoc criteria have been proposed, such as that of Landel (55) in which the failure strain eL is defined as the ratio of the maximum true stress to the initial modulus, where the true stress is defined as the product of the extension ratio and the engineering stress —i.e., breaks down at low strain rates and higher temperatures. Milloway and Wiegand (68) suggested that motor strain should be less than half of the uniaxial tensile strain at failure at 0.74 min.-1. This criterion was based on 41 small motor tests. 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

In earlier sections the fluid strain rate was described in terms of the velocity field. Up to this point, however, the stress has not been related to the underlying flow field. It is the quantitative relationship between fluid strain rate and stress that permits the momentum-conservation equations (Navier-Stokes equations) to be written with the velocity field as the dependent variable. 

In these equations /,- are the components of the volumetric body forces in the i coordinate direction. At this point the stresses are still written as r. The next step is to use the stress-strain-rate expressions developed in Chapter 2 to write the stresses in terms of the velocities, thus deriving systems of equations in which the velocities are the dependent variables. 

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