Yield isotropic hardening

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... 

Powder Yield Loci For a given shear step, as the applied shear stress is increased, the powder will reach a maximum sustainable shear stress "U, at which point it yields or flows. The functional relationship between this limit of shear stress "U and applied normal load a is referred to as a yield locus, i.e., a locus of yield stresses that may result in powder failure beyond its elastic limit. This functional relationship can be quite complex for powders, as illustrated in both principal stress space and shear versus normal stress in Fig. 21-36. See Nadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.) for details. Only the most basic features for isotropic hardening of the yield surface are mentioned here. 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

Here, s means the stress tensor, a and fi are the backstress-tensors of the yield surface / and the bounding surface F, respectively. Y is the radius of the yield surface B and R are the initial size of the isotropic hardening. 

An example of a material model based on the physics of material behavior is classical metals plasticity theory. This theory, often referred to as /2-flow theory, is based on a Mises yield surface with an associated flow rule, followed by rate-independent isotropic hardening (Khan and Huang 1995). Physically, plastic flow in metals is a result of dislocation motion, a mechanism known to be driven by shear stresses and to be insensitive to hydrostatic pressure. 

The ore material is elasto-plastic with isotropic hardening mle. The plastic properties are described by yield stress Rp = 9.946 q MPa and fracture stress Rm = 12.661 MPa, which are given by bounded histograms, see Figures 6 and 7. 

In practice, is often the variable which determines the size (isotropic hardening) or the amount of translation (kinematic hardening) of the yield surface and represents in a simplified manner all the effects of the loading history. One particular example is the preconsolidation pressure which determines the current yield envelope of clays (as in Camclay model). 

During isotropic hardening, the yield surface grows symmetrically around the origin as sketched in figure 3.30(a). Mathematically, this imphes that the argument in equation (3.43) plays no role, and the yield criterion is thus... 

If yielding of the material is governed by the von Mises yield criterion, we find the yield criterion for isotropic hardening, using the changing flow stress... 

The stress-strain diagram of a material with isotropic hardening that is deformed by uniaxial tension first and uniaxial compression afterwards can be found in figure 3.30(b). In compression, the material yields at a stress —crpi, given by the absolute value of the maximum stress in tension, api. 

If we deform a kinematically hardening material in uniaxial tension and compression, its behaviour differs drastically from the isotropically hardening material discussed above (figure 3.32). Upon load reversal, the material yields at a stress yield surface remains unchanged. In the extreme case, this may lead to plastic deformation while the stress is still tensile (figure 3.32(b)). 

Equation 5 shows that the Bouc-Wen model accounts for kinematic hardening (i.e., post-yield force increases with increasing deformation) due to the post-yield stiffness ratio p. However, the model does not account for the isotropic hardening (i.e., yield force Fy increases due to cyclic inelastic deformation) in the hysteresis of steel energy dissipation devices. 

Figure 7 shows how the aforementioned modified Bouc-Wen model captures the behavior of low yield strength shear panels showing significant isotropic hardening in their hysteresis as well as the behavior of BRBs showing different isotropic hardening in tension and compression. 

During isotropic loading, plastic deformation takes place when the isotropic stress p reaches the preconsolidation pressure pf. The pressure pf is a measure of the size of the yield surface on the isotropic axis and can be viewed as an hardening/softening parameter (the specific shape of the yield surface is described in the next section). An essential feature of the proposed model is the decrease of pf with respect to an increase in contaminant concentration. This can be expressed as... 

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... 

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. 

A thin film of rate-independent elastic-plastic material is subjected to temperature cycling between some reference absolute temperature and a higher temperature To + T. The film material exhibits isotropic strain hardening, and the quantity (3 in the yield function (7.59) has been determined to be... 

Unified Plasticity Model The time-independent plastic deformation and fee time-dependent creep deformation arise from fee same fundamental mechanism of dislocation motion. Hence, a constitutive model which captures both of these deformation mechanisms is desirable. Such a constitutive model is referred to as a unified plasticity model. A commonly-used unified plasticity model is the Anand s model. This is a rate-dependent phenomenological model (Ref 17 and 18). There are two basic characteristics of fee Anand s model. First, no explicit yield criterion is specified, and second, a single internal state variable (ISV) s, the deformation resistance, represents the isotropic resistance to inelastic strain hardening. Anand s model can represent fee strain rate and temperature sensitivity, strain rate history effects, strain hardening, and fee restoration process of dynamic recovery. Equation 9 shows the functional form of fee flow equation that accommodates fee strain rate dependence on the stress ... 

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