Strain rate hardening

Besides frequency effects due to strain-rate hardening and/or specific cyclic damage of the fibrils, there is another typical effect in the case of cyclic loading at high frequencies, namely hysteretic heating. The increase of temperature T near the crack tip, in that case, scales as in Eq. (5)" ... 

C where hysteretic heating occurs and another one at 23 °C where strain rate hardening occurs. The shift between the 50 °C 100 Hz lifetime and the 23 100 Hz... 

As developed in Section 10.3, the stability of extensional flow in visco-plastic solids is governed by intrinsic properties of the solid, such as its plastic resistance, its strain-hardening rate and its strain-rate-hardening rate, through the sensitivity of the plastic resistance to the strain rate. In many instances, however, the deforming bar or fiber contains imperfections that can affect or hasten localization in necks and subsequent rupture. Such perturbations of flow by imperfections and their effect on material stability in extensional flow have been of great interest. A well-defined scenario of this was conceived by Hutchinson and Obrecht (1977) and further developed by Hutchinson and Neale (1977). 

The hemispherical dome (HD) test provides more reproducibility of results than the Olsen and Erichsen cup tests. For low-carbon steels, the dome height, which is determined at the point of maximum load, increases linearly with the n value. For other materials such as brass, aluminum alloys, and zinc, optimum correlation has been found between the dome height and the total elongation, which includes the effects of strain rate hardening and limiting strains. 

Figure 16.25 The effect of shear or strain rate on the elastic viscosity of semicrystalline high-density polyethylene. (Note the upsweep in viscosity is often referred to as strain rate hardening).

Figure 16.26 Time-dependent strain rate behavior of homopolymer polypropylene showing lack of strain-rate hardening leading to inability to stretch into deep cavities.

Figure 16.27 The effect of adding strain-rate hardening polymers to homopolymer polypropylene allowing increasing ability to stretch into deep cavities.

Both % El and % RA are frequendy used as a measure of workabifity. Workabifity information also is obtained from parameters such as strain hardening, yield strength, ultimate tensile strength, area under the stress—strain diagram, and strain-rate sensitivity. 

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... 

The initial strain hardening rate (df/dy)o is given as a function of plastic strain rate for strain rates up to 10 s (which includes shock compression to 5.4 GPa) as [38]... 

A ductile material can be stretched uniformly only when stable flow occurs. The stable flow of materials has been investigated by Hart who described the transition from the stable to unstable flow. The beginning of geometrical instability and localisation of strain is the limit of the stable flow. At temperatures above 0.5 T (at equilibrium between recovery and hardening) the strain rate sensitivity parameter "m" may be derived from the expression ... 

Strain hardening effect, 20 224 Straining efficiency, 77 340 Strain rate, 73 473 Strain recovery rate (Rr), in testing shape-memory polymers, 22 361 Strain sensors, 77 150, 151-152 Strain tensor, for noncentrosymmetry pont group crystals, 77 93-94 Strain versus time curve factors affecting, 73 473 material and microstructure effect on, 73 473-474... 

Strain energy B-3 Strain hardening B-3, 5-14 Strain rates B-3, 5-2, 5-10—5-15 Strehlow Curves, overpressure calculations 3-1 1... 

PP bead foams were subjected to oblique impacts (167), in which the material was compressed and sheared. This strain combination could occur when a cycle helmet hit a road surface. The results were compared with simple shear tests at low strain rates and to uniaxial compressive tests at impact strain rates. The observed shear hardening was greatest when there was no imposed density increase and practically zero when the angle of impact was less than 15 degrees. The shear hardening appeared to be a unique function of the main tensile extension ratio and was a polymer contribution, whereas the volumetric hardening was due to the isothermal compression of the cell gas. Eoam material models for FEA needed to be reformulated to consider the physics of the hardening mechanisms, so their... 

Depending on the material and deformation conditions (strain rate, temperature) other stress-strain curve shapes can be observed (Fig. 2b and c). In Fig. 2b, the plastic flow occurs at the same stress level as that required for the yielding so the strain softening does not exist. In the case shown in Fig. 2c, the strain hardening happens very close to yielding, suppressing both strain softening and plastic flow behaviour. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

In order to start the multiscale modeling, internal state variables were adopted to reflect void/crack nucleation, void growth, and void coalescence from the casting microstructural features (porosity and particles) under different temperatures, strain rates, and deformation paths [115, 116, 221, 283]. Furthermore, internal state variables were used to reflect the dislocation density evolution that affects the work hardening rate and, thus, stress state under different temperatures and strain rates [25, 283-285]. In order to determine the pertinent effects of the microstructural features to be admitted into the internal state variable theory, several different length scale analyses were performed. Once the pertinent microstructural features were determined and included in the macroscale internal state variable model, notch tests [216, 286] and control arm tests were performed to validate the model s precision. After the validation process, optimization studies were performed to reduce the weight of the control arm [287-289]. 

In Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

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