Plastic strain rate

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... 

If the material is perfectly plastic, i.e., if the yield function is independent of k and a, then = 0 and the magnitude of the plastic strain rate cannot be determined from (5.81). Only its direction is determined by the normality condition (5.80), its magnitude being determined by kinematical constraints on the local motion. 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

An explicit relation for the plastic strain rate may be obtained by using (5.80) through (5.82). The partial derivatives of/are, from (5.92)... 

Naghdi, P.M. and Trapp, J.A., On the Nature of Normality of Plastic Strain Rate and Convexity of Yield Surfaces in Plasticity, J. Appl. Mech. 62, 61-66 (1975). 

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]... 

For example, a 10 GPa (total strain = 0.06) shock wave in copper has a maximum total strain rate 10 s [21] the risetime would thus be (eje) 0.6 ns. For uniaxial-strain compression, y averaged over the entire shock front. The resolution of the shock wave in a large-scale, multidimensional finite-difference code would be computationally expensive, but necessary to get the correct strength f behind the shock. An estimate of the error made in not resolving the shock wave can be obtained by calculating dt/dy)o with y 10 s (the actual plastic strain rate) and y 10 s (the plastic strain rate within the computed shock wave due to a time step of 0.06 qs). From (7.41) with y = 10 s (actual shock wave) and y = 10 s (computation) ... 

The formulation above is assumed to hold for temperatures up to the glass transition Tg. For T > Tg, most studies found in the literature focus on the description of the molten state [14] due to its practical importance, while little attention is paid to the response of glassy polymers in the rubbery state, near Tg. For strain rates larger than 1 s 1, the mechanical response of the molten material is non-Newtonian for most polymers and described by r = qym, where q and m are material parameters. We assume that this non-Newtonian response prevails as soon as Tg is exceeded. Hence, within the same framework as used below Tg, the equivalent plastic strain rate (Eq. 3) is replaced by... 

To illustrate the influence of the craze thickening kinetics on fracture, two sets of craze parameters are used and listed in Table 3. The two sets are borrowed from [22] (cases 8 and 1) and named here A and B. In Fig. 8, we report the plastic strain rate distribution observed near the crack tip. This variable is suitable to track the development of plasticity and is normalized with To = Ki/soy/Fi as a reference strain rate at the tip of the notch (the radius is rt = 0.1 mm and T = 293 K). We compare the cases for which no crazing is considered (Fig. 8a) to those where crazing is accounted for (Fig. 8b), with the set A of craze parameters in Table 3. When crazing is not present and at the particular loading rate considered, plasticity develops in the form of shear bands which originate from the tip of the notch, where the stress concentrates. 

Fig. 8 Distribution of the instantaneous plastic strain rate distribution yP under mode I loading at Rj 3 x 10 2 MPa m/s when K / (so /rt) 1.71 a without crazing, b with crazing...

The constitutive model used to describe large plastic deformations of glassy polymers involves a separate formulation for temperatures above and below the glass transition Tg, since the underlying deformation mechanisms are different. In either regime, the formulation is based on the decomposition of the rate of deformation into an elastic part and a plastic part Z)P so that 0 = 0° + D. By assuming an isotropic yield stress, the isochoric plastic strain rate is given by the flow rule... 

For T < Tg, the viscoplastic model used here accounts for intrinsic softening upon yielding followed by progressive orientational hardening. Rate dependent flow is taken to be governed by Argon s formulation [5] of the equivalent plastic strain rate... 

Shear yielding in the bulk of the material is incorporated through the constitutive law presented in the previous section. The definition of the plastic strain rate CP together with the expression for the driving stress 6 specifies the energy dissipation rate per unit volume G D< = /2tY . The energy balance in the material can then be written as... 

For perfectly plastic materials, post>yieldtng the strain rate is a constant function of the stress and the stress is constant and never exceeds ay (Fig. ISA). The extent of pla.stic deformation p depends upon the proportionality between plastic strain rate and the stress and the how long the. stress is applied as shown in Figure 17A. The elastic perfectly plastic material is highly idealized and not many materials exhibit this type of behavior. 

As a result of these arguments, within the confines of continuum models of single crystal plastic deformation, the total plastic strain rate can be written as... 

The parameter r) is the slope of the boundary line between contractance - dilatance zones. The plastic strain rates are given by ... 

This implies that the volumetric component of the plastic strain rate is zero. There is no hardening/softening for this part of the model. 

Figure 3. Yield curve and plastic strain rate vector in the plane deviatoric stress vs effective isotropic stress.

Based on the theory of damage mechanics, when the material is in one-dimensional state, the relationship between the damage development rate and the cumulative plastic strain rate satisfies the following equation (Li, S.C et al. 2009). 

Where D = damage development rate Y = damage energy consumption rate P = cumulative plastic strain rate Q and q = parameters. 

The different terms in the model assume that the flow stress curve is independently affected by equivalent plastic strain, equivalent plastic strain rate, and temperature. In this model, the temperature is influenced at high equivalent strain rates due to the adiabatic deformation. As a result of the complex interrelations between the material law parameters, the existing approaches show major... 

While the kinetics of plastic response in metallic glasses in the low-temperature realm exhibits a remarkable mechanistic universality, where the nucleation of STs occurs in a substantially frozen structure, the steep decrease in the temperature dependence of the yield stress and the stress exponent m of the plastic strain rate above 0.62rg signifies the onset of a fundamental change in the mechanism. Thus, the universal response at low temperature, with the slow decrease of plastic resistance with temperature, extrapolating to a vanishing level at a universal temperature of... 

While the power-law form of the kinetic flow expression of eq. (7.43) is quite satisfactory for large-strain plastic behavior in the fully developed plastic-flow range where the accompanying elastic response is largely unchanging and the total strain rate y is substantially the same as the plastic-strain rate p, this is not the case for the initial rising portion of the stress-strain response in the transition zone, where the elastic-strain rate y needs to be considered too. 

Then, the plastic strain rate tensor D° resulting from the contributions of all the K active slip systems in the crystalline lamella is given by... 

Fig. 10.5 Plastic-strain-rate contours in three stages of localization at a neck at elongation increments Af/fo of 0.1, 0.25, and 0.5 for an imposed elongation rate dlnf/dt of 1.0 s (from Boyce et al. (1992) courtesy of the SPE).

---