Hardening parameter

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. 

It is usual in the classical theory to assume that the stress rate is independent of the hardening parameters, since the elastic behavior is expected to be unaffected by plastic deformation. Consequently, the stress rate relation (5.23) reduces to... 

In the classical theory of plasticity, constitutive equations for the evolution of the isotropic and kinematic hardening parameters are usually expressed as... 

Using (5.77), the evolution equations for the hardening parameter (5.76) becomes... 

The yield stress cy increases with rising plastic deformation, which is called strain-hardening of the material. The mathematical description of this effect follows e.g. by means of a strain-hardening parameter fc with Uy = py(fc) and, therefore, O( ) For isotropic... 

Norm of deviatoric plastic strain increment Hardening parameter... 

We now need to determine the hardening parameter h. For simplicity we use the isotropic hardening model (2.307), and from (2.308) we have... 

For simplicity, we introduce the following isotropic hardening rule with the strain hardening parameter ... 

The hardening rule is introduced under an isotropic hardening model with a strain hardening parameter, which is the same as the original Cam clay model ... 

Here, is the current plastic deformation, and ki is a set of hardening parameters which may depend on the deformation history, the strain rate, or the temperature. As in the original yield criterion, the material deforms elastically if the stress state hes with the surface = 0 i. e., if < 0 holds. 

So far, we have not specified how the yield surface changes with increasing plastic deformation. This is done by evolution equations for the hardening parameters, so-called hardening laws. 

Here, C is a hardening parameter and ag a constant reference stress which is equal to the yield strength at = 0. When plastic deformation starts,... 

Consider a tensile specimen of an isotropic metal with elastic parameters E = 210 000 MPa and v = 0.3, and a yield strength (Tf = 210 MPa. The material hardens linearly and isotropically according to equation (3.50), with hardening parameter H = 10 000 MPa. The tensile specimen is elongated, starting with an unloaded state, at a constant strain rate of n = 0.001 s . We want to determine the time-dependence of stresses and strains. 

As can be seen in the figure, the plastic strain increases markedly stronger than the elastic strain after yielding. The slope of the stress-strain curve is 9546 MPa. It is slightly smaller than the hardening parameter H. This is due to the elastic strains which slightly increase with the stress. 

Here, Wo is a characteristic level of back stress that primarily affects the initial slope of the uniaxial stress versus remanent strain curve, and m is another hardening parameter that controls how abruptly the strain saturation conditions are reached. Figure 2a illustrates the predictions of the effective stress versus the effective remanent strain from the constitutive law for uniaxial compression, pure shear strain, pure shear stress and uniaxial tension. It is interesting to note that the shear strain and shear stress curves do not coincide. This feature is due to the fact that the material can strain more in tension than in compression, and has been confirmed in micromechanical simulations. Figure 2b illustrates the uniaxial stress versus remanent strain hysteresis curves for two sets of the material parameters Wq and m. 

Figure 2. (a) Effective stress versus effective remanent strain curves for the model material described in Section 2 in uniaxial compression, pure shear strain, pure shear stress and uniaxial tension tests, (b) Uniaxial stress versus remanent strain hysteresis loops for the model material illustrating the effect of the hardening parameter In both cases notice the asymmetry in the remanent strains that can be achieved in tension versus compression. 

The primary goal of this fracture model is to determine how the steady state toughness enhancement in ferroelastic materials, Ggs/Go, depends on the material properties. Eq. (3.9) identifies the material properties in question and ranks them in order of significance. Poisson s ratio v will be shown to have a very weak influence over the toughness enhancement. The two hardening parameters, Hq/o-q and m, have a much stronger influence on Gss/Gq, and it will be shown that as the hardness of the... 

Figure 6 plots the toughening parameter a as a function of //o/[Pg.373]

Effect of current density (a), kind of electrolytes and heat treatment (b) on hardening parameters of 18-10 type stainless steel [531. 

In order to simplify the exposition, we consider that the process is isothermic, and we assume that the evolution of the internal structure can be described with the aid of a scalar-hardening parameter (the density of dislocations or the equivalent plastic strain) and of an internal tensorial parameter (back stress). These restrictions are eliminated in the authors... 

E = Young s modulus p = strain hardening parameter Oy = initial yield surface E = current strain a = current stress a = current center of elastic range Subscript 0 denotes values at the start of an increment... 

---