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and therefore the normality rule assures that no further plastic volumetric strain (dilatency) is
created. Adjustment of the parameters that control the rate of cap contractions permits
experimentally observed amounts of dilatency to be incorporated into the cap model, thus producing
a constitutive law which better represents the physics to be modeled.
Another advantage of the cap model over other models such as the Drucker-Prager and
Mohr-Coulomb is the ability to model plastic compaction. In these models all purely volumetric
response is elastic. In the cap model, volumetric response is elastic until the stress point hits the cap
surface. Therefore, plastic volumetric strain (compaction) is generated at a rate controlled by the
hardening law. Thus, in addition to controlling the amount of dilatency, the introduction of the cap
surface adds another experimentally observed response characteristic of geological material into the
model.
The inclusion of kinematic hardening results in hysteretic energy dissipation under cyclic
loading conditions. Following the approach of Isenberg, et. al. [1978] a nonlinear kinematic
hardening law is used for the failure envelope surface when nonzero values of and N are specified.
In this case, the failure envelope surface is replaced by a family of yield surfaces bounded by an
initial yield surface and a limiting failure envelope surface. Thus, the shape of the yield surfaces
described above remains unchanged, but they may translate in a plane orthogonal to the J axis,
Translation of the yield surfaces is permitted through the introduction of a “back stress”
tensor, α . The formulation including kinematic hardening is obtained by replacing the stress σ with
the translated stress tensor η ≡ σ − α in all of the above equation. The history tensor α is assumed
deviatoric, and therefore has only 5 unique components. The evolution of the back stress tensor is
governed by the nonlinear hardening law
⋅p
α = cF (σ, α) e
⋅p
where c is a constant, F is a scalar function of σ and α and e is the rate of deviator plastic
strain. The constant may be estimated from the slope of the shear stress - plastic shear strain curve
at low levels of shear stress.
The function F is defined as

(σ − α ) • α 
F ≡ max 0,1 −
2 NFe ( J1 ) 

where N is a constant defining the size of the yield surface. The value of N may be interpreted as
the radial distant between the outside of the initial yield surface and the inside of the limit surface.
In order for the limit surface of the kinematic hardening cap model to correspond with the failure
19.76 (MAT)
LS-DYNA3D Version 936