Plasticity Theory Based Models

The plasticity theory based models require the definition of yield surface, plastic potential, isotropic hardening laws, and simplified accumulation rules (Bonaquist & Witczak, 1997 and Desai, 2002), or kinematic hardening laws (Chazallon et al., 2006). Some of these models have been used for finite element modelling of pave­ment. Now, the main concepts of these models are presented.

The model developed by Bonaquist is based on the plasticity model developed by Desai et al. (1986). These two models differ from each other by the simplified calculation method for large number of cycles. Consequently, the basis of the Desai model (Desai, 2002) is first presented and the different accelerated analysis proce­dures of each model are introduced.

The Desai formulation is based on the disturbed state concept which provides a unified model that includes various responses such as elastic, plastic, creep and micro-cracking. The idea is that the behaviour of a deforming material can be ex­pressed in terms of the behaviour of the relatively intact or continuum part and of the micro-cracked part. During the deformation, the initial material transforms continuously into the micro-cracked state and, at the limiting load, the entire ma­terial element approaches the fully micro-cracked state. The transformation of the material from one state to another occurs due to the micro-structural changes caused by relative motion such as translation, rotation and interpenetration of the particles and softening or healing at the microscopic scale. The disturbance expresses such micro-structural motions. Under repetitive loading, an accelerated procedure exists. From experimental cyclic tests, the relation between the deviatoric plastic strain trajectory and the number of loading cycles can be expressed as a power function of the number of cycle. Pavement finite element modelling feasibility has been carried out with this model (Desai, 2002). The к-0 model was used for the elastic part and the model requires the following parameters:

• Elastic behaviour: кь к2, v.

• Plastic behaviour:

о Rupture and characteristic parameters: 3R, n, y.

о Plasticity parameters: (3, ab nb Nr, b.

Bonaquist’s approximate accelerated analysis is based on the total plastic strain at the end of each cycle and defined by a power function of the cycle number which depends on the ratio: maximum deviatoric stress and the corresponding deviatoric stress at rupture (for the same q/p ratio). Instead of the Nr and b parameters a parameter %b is required.

The model developed by Chazallon and Hornych (Chazallon et al., 2006) is based on the model of Hujeux (1985) in its simplest formulation. This formulation is a non-associated elasto-plastic model and reproduces the saturated monotonic behaviour of sand and clay. A kinematic hardening has been added to reproduce the accumulation of plastic strains under repeated loadings. Each cycle is calculated, nevertheless, by a simplified approach based on the decoupling of the calculation of the elastic strains and the plastic strains. A pavement finite element modelling feasibility has been carried out with this model, see Hornych et al. (2007).

The elastic part is computed with the anisotropic Boyce model and the model requires the following parameters:

• Elastic behaviour: Ka, Ga, у and n.

• Plastic behaviour:

о Rupture and dilatancy parameters: C0, M, and Mc. о Monotonic plasticity parameters: в, PC0, a, b.

о Cyclic plasticity parameters: rf, Puc, Plc.(where the subscript l and u indi­cate loading and unloading, respectively — see Figs. 9.5 and 9.6).

Fig. 9.6 Representation of the influence of the Puc/ic parameters on plastic strains when an un­loading and a reloading occur (Chazallon et al., 2006) With permission from ASCE