Equivalent plastic strain rate

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... 

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... 

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... 

Since the equivalent plastic strain rate defined this way is positive for all plastic strain rates, the equivalent plastic strain increases for any plastic deformation, regardless of the deformation orientation. 

The equivalent plastic strain is calculated from the equivalent plastic strain rate eiq by integrating ... 

The equivalent plastic strain rate for the case [Pg.105]

If the plastic response remains essentially isotropic, then there must be an equivalent plastic strain measure, having the form of some function of the principal values of the plastic strain rate tensor, such that the Mises stress times the equivalent plastic strain rate is the local rate of plastic work per unit volume of material. In terms of the principal plastic strain rates e, 2 3, this equivalent plastic strain rate is defined as... 

The other two principal plastic strain rates are then equal to each other with values eP = eP = ffP for deformation without volume change. The equivalent plastic strain rate is = eP. The product = creP is... 

An important observation, which can be seen from the shapes of the surfaces shown in Figure 4, is the increased strain rate dependence of the flow stress with increasing plastic strain e (or equivalently draw ratio R or X, where e = nR). We have taken as our starting point the Eyring equation for a thermally activated rate process. The plastic strain rate e is given at high stresses by... 

For the simulation of isotropic thermoplastics elasto-viscoplastic material models are used. They are composed of an elastic part consisting of a constant Young s modulus and Poisson s ration and a plastic part being described by true stress/strain-curves depending on the true plastic strain-rate. As a failure criterion a maximal endurable hydrostatic stress, a critical equivalent plastic strain or a combination of these can be used. The strain criterion can also be set as a function of the strain-rate. 

If the maximum resolved shear stress r and the plastic shear strain rate y are defined according to (it is assumed that the Xj and Xj directions are equivalent)... 

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

Material Flow Rule Cutting processes of metallic materials involve highly complex interactions between plastic material behavior, strain rate, temperature, and material microstructure. Even in conventional cutting, very high equivalent strain rates of approximately 10 s are... 

While mechanistic models of plastic flow consider simple shear or pure shear (extensional flow where mean normal stress is absent), in experiments fundamental shear information may need to be extracted from more complex 3D flow fields. This is done through the use of multi-axial flow formalisms that are based on the von Mises approach of relating the 3D response to an equivalent ID response described in Chapter 3. In this formalism the deviatoric shear response of the multi-axial field of stress and plastic strain is taken to represent shear flow in the mechanistic context where the effect of the accompanying mean normal stress is considered through its effect on the plastic resistance. There exists an experimental procedure for extracting deviatoric plastic-response information from a tensile-flow field that accomplishes this through the use of specially contoured bars with pre-machined neck regions where the concentrated extensional flow is monitored under conditions of imposed constant deviatoric strain rates (G Sell et al. 1992). 

Correspondingly, eq. (13.2) representing the strain-rate dependence of the plastic resistance is taken to be given by a standard uniaxial reference experiment at a reference strain rate e gf, typically of magnitude 10 s that evokes a reference tensile uniaxial plastic resistance o-j-gf, which in this case would be the tensile yield stress o-q. The form of the idealized power law relating eg to Ug is given by the exponent m of the equivalent stress, which must be temperature-dependent in a form given in Chapter 8 as... 

A very good way to characterize and differentiate between elastomers and rigid plastics is by the measurement of dynamic mechanical properties. A most convenient method to study dynamic mechanical properties is to impose a small, sinusoidal shear or tensile strain and measure the resulting stress. Dynamic mechanical properties are most simply determined for a small sinusoidally varying strain, for which the response is a sinusoidally varying stress. An increase in frequency of the sinusoidal deformation is equivalent to an increase in strain rate. 

Based on the mathematical theory of plasticity, the plastic deformation behavior of the material can be described by the three components of the rate-independent plasticity model, namely yield criterion, flow rule, and hardening rule. The yield criterion determines the stress level at which yielding is initiated. This is represented by the equivalent stress Ueq, which is a function of the individual stress vector components a. Plastic strain is developed in the metal parts when the equivalent stress is equal to a material yield parameter ay finally, the flow rule determines the direction of plastic straining ... 

The range of material behavior considered next is broadened significantly by appeal to the notion of a plastic rate equation as a model for any possible physical mechanism of deformation that may be operative. The ideas will be developed for general states of stress, but will be applied primarily for the case of thin films in equi-biaxial tension. Constitutive relationships that serve as models for inelastic response of materials for a wide variety of physical mechanisms of deformation have been compiled by Frost and Ashby (1982). These constitutive equations are represented as scalar equations expressing the inelastic equivalent strain rate /3e in terms of the effective stress (Tm/ /3 and temperature T. These strain rate and stress measures are denoted by 7 and as by Frost and Ashby (1982), and the rate equations representing models of material behavior all take the form... 

A similar approach was proposed by Khoramishad et al. (2010a) in which the rate of damage was related to the maximum principal strain rather, than the equivalent strain and a threshold strain, Sth, was used rather than a plastic strain in order to define a strain helow which fatigue damage does not occur. 

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