Visco-plastic equivalent models based on an equivalent time: number-of-cycles relationship, have been developed by Suiker and de Borst (2003) for the finite element modelling of a railway track structure and by Mayoraz (2002) for a laboratory study of sand.
Suiker has developed a cyclic densification model. It is based on repeated load triaxial tests carried out on two ballast materials. The idea was to develop a model that captures only the envelope of the maximum plastic strain generated during the cyclic loading process. The unloading is considered as elastic. The plastic deformation behaviour is composed of two different mechanisms namely: frictional sliding and volumetric compaction. In general, both mechanisms densify the material. This model is based on the Drucker-Prager yield surface and Cap surface. The stress space is divided into four regimes (see Fig. 9.7):
(i) the shakedown regime, in which the cyclic response of the granular media is fully elastic.
(ii) the cyclic densification regime, in which the cyclic loading submits the granular material to progressive plastic deformations.
(iii) the frictional regime, in which frictional collapse occurs, since the cyclic load level exceeds the static peak strength of the granular material.
(iv) the tensile failure regime, in which the non cohesive granular material instantaneously disintegrates, as it can not sustain tensile stresses.
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Fig. 9.7 Map of various response regimes in (p, q) plane during cyclic loading (Suiker and de Borst, 2003). Copyright John Wiley & Sons Limited. Reproduced with permission |
The model requires the following parameters:
• Elastic behaviour: k1, k2, v; and
• Plastic behaviour:
о Monotonic parameters: pt, hm, h0, p0, d0, dm, Zf, Zc.
о Cyclic plasticity parameters: pt, h0, hm, Zf, af, ac, yc, p0, nf, nc, d0, dm.
The monotonic parameters initialise the state of stress and strains in the railway track structure which are required for the cyclic model.
Mayoraz has developed a visco-plastic equivalent model based on the associated modified Cam-Clay model with no elastic part. Permanent deformations comparisons with the results of repeated load triaxial tests performed on a sand have been carried out. This model requires only 5 parameters and is based on the Perzyna concept (Perzyna, 1966) developed for visco-plastic creep of clay.
Parameters needed for the model are:
• Plastic behaviour:
о Rupture parameters: M; and о Plasticity parameters: n2, Г, J°nit and в*.
