Hardening model isotropic

The stress-strain relationship for FGMs is assumed to be a bilinear form and can be described by the isotropic hardening model and kinematic hardening model as ... 

We illustrate the isotropic and kinematic hardening models for the onedimensional problem in Fig. 2.18a, b, and for the two dimensional problem in Fig. 2.19. Note that the behavior shown in Fig. 2.18b is known as the Bauschinger effect. In geotechnical engineering practice the isotropic hardening model is widely used because, except in earthquake situations, it is rare that the loading direction is completely reversed. 

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... 

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 ... 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

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. 

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... 

Usually, the material starts with a kinematic behavior. With increasing strain level, the kinematic behavior saturates and isotropic hardening is taken place. The problem now is to describe the behavior of a as a function of the strain and strain-rate level. Several models are available but only the models from Armstrong and Frederick (Armstrong and Frederick 1966) and the Yoshida model (Yoshida and Uemori 2002) are described in the following because of their popularity in the last decades. 

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. 

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). 

For example, Lemaitre s model with isotropic hardening is based on the following dissipation potential ... 

The original Cam clay model employs isotropic hardening with strain hardening such as... 

The cyclic stress-strain curve can be used, for example, to perform finite element simulations of cyclic loadings. To simulate the complete experiment in the computer, it would be necessary to obtain information on the hardening of the material (isotropic and kinematic hardening) and to determine a material model that correctly describes it. This would be an extremely comphcated procedure. Furthermore, the entire number of cycles would have to be calculated, which would require an immense amount of computing time. Instead, the flow curve, taken by the finite element software to be monotonous, can be replaced by the cyclic stress-strain curve. A single, monotonous loading of the component is then simulated. Stresses and strains calculated in this way correspond well with those in the cyclically loaded component. 

Fig. 7.17. Variation of normalized equi-biaxial film stress as a function of normalized temperature, for the example considered in Section 7.5.2 where the film material is modeled as an isotropically hardening solid. The solid lines denote the response obtained from the numerical integration of (7.65) for the first three thermal cycles where it is assumed that the material properties do not vary with temperature, over the range considered. The dashed lines denote the corresponding behavior for the case where the thin film plastic response is taken to be temperature-dependent.

In an isotropic polycrystalline polymer whose microstructure consists of stacked lamellae arranged in the form of spherolites, the slip systems activated depend on the local orientation of the lamellae with respect to the applied stress and, as deformation proceeds, these orientations are modified. To calculate the evolution of the crystalline texture, one can consider the polymer to behave as a crystalline aggregate. Although the entropic contribution of chain orientation in the amorphous regions may also need to be considered, the major contribution to work hardening in tension is rotation of the slip planes toward the tensile axis, so that the resolved shear stress in the slip direction diminishes. This results in a fiber texture in the limit of large deformations, such that the crystallites are oriented with their c axis (the chain axis) parallel to the stretch direction. Despite the relative success of such models, they do not explicitly address the micro-mechanisms involved in the transformation of the spherulitic texture into a fiber texture. One possibility is that the... 

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 ... 

The clay platelets (Cloisite 30B) are assumed to be elastic and isotropic with tensile modulus E=200 GPa and Poisson ratio v= 0.2 [22]. The epoxy system (DGEBA epoxy base + amminic hardener) is modelled by a nonlinear Ramberg-Osgood (RO) law whose parameters are obtained by fitting to available experimental results - Young s modulus E=3230 MPa, Poisson s ratio v= 0.34, yield stress osn = 30 MPa, and two additional coefficients n= 4.5, a= 0.0767. 

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. 

To account for isotropic hardening in the Bouc-Wen model, the mathematical formulation developed by Karavasilis et al. (2012b) is described below ... 

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. 

The constitutive law describing the behavior of the steel material is the uniaxial Menegotto-Pinto model (1973). This computationally efficient nonlinear law is capable to model both kinematic and isotropic hardening as well as the Bauschinger effect, allowing for accurate simulation and reproduction of experimental results. The response of the steel material is defined by the following nonlinear equation ... 

The matrix material was chosen to be a polyester resin for both the inner and outer layer. The FEA model consisted of four parts namely the helmet shell, the foam liner, the head-form and the hemispherical test anvil. The foam liner was modeled using a set of 8500 solid elements, the woven ply was modeled as an orthotropic material with damage and the glass mat polyester layer was modeled as an isotropic elastic-plastic material with kinematic hardening. 

---