Kinematic 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... 

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

Prager s rule of kinematic hardening is expressed by a = ce where c is a constant. Generalizing these concepts, the evolution equations for the internal state variables will be taken in the form... 

Simple kinematic hardening can be accommodated by assuming instead that n = c3 and A = 0, so that... 

Atluri, S.N., On Constitutive Relations at Finite Strain Hypo-Elasticity and Elasto-Plasticity with Isotropic or Kinematic Hardening, Comput. Methods Appl. Mech. Engrg. 43, 137-171 (1984). 

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... 

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 ... 

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. 

De Angelis F. (2012)—A comparative analysis of hnear and nonhnear kinematic hardening rules in computational elastoplastidty, Technische Mechanik, Vol. 32 (2-5), pp. 164-173. 

Chaboche, J.L. (1991) On some modifications of kinematic hardening to improve the description of ratchetting effects. International Journal of Plasticity, 7, 661-678. 

As with common paving bitumens, the tests conducted on hard paving bitumens aim at determining their consistency at intermediate service temperatures (penetration test) and at elevated service temperatures (softening point and dynamic viscosity test) and their durability (resistance to hardening test). Kinematic viscosity, Fraass breaking point, flash point and solubility are also properties considered useful in the specification of hard paving bitumens. 

For the purpose of this simplistic calculation it is assumed that the structure fails when the stress utilisation factor becomes equal to 1. This is conservative because the structure retains some strength after the first yield through kinematic hardening. The kinematic hardening however also reduces with rising temperature. 

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). 

CD inequality will no longer be automatically verified. This means that thermodynamic principles may then be violated in some evolutions. Note that in order to describe isotropac and kinematic hardening, the thermodynamic flux is often decomposed into a tensor a and a scalar r, associated with thermodynamic forces X and R. We would then have to write ... 

Figure 2 Sensitivities of roof displacement to kinematic hardening modulus for different FE meshes (a) using force-based frame elements and (b) using displacement-based frame elements.

The total hardening is obtained by means of a mixed rule consisting of both a kinematic and an isotropic component. A linear combination of two of the hardening types is assumed using a weighing coefficient P, which is calibrated experimentally ... 

The response obtained using three different reinforcing steel models is shown in Figure 10. The widely used bilinear model with kinematic hardening and the model... 

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