In hypoplasticity a direct relation is used between strain rates and stress rates. Specifically:
Here the part with gives a linear relation between strain rates and stress rates and the part with gives a nonlinear relation. The constitutive tensors and are functions of the effective stress tensor and void ratio . The effective stress tensor follows from the total stress tensor minus any pore pressures (see groundflow). Rigid body rotations (objectivity) are treated elsewhere (see the section on memory).
Basic law, Wolffersdorff
The law proposed by WOLFFERSDORFF  is used.
The scalar factors and take into account the influence of mean pressure and density:
Three characteristic void ratios - (during isotropic compression at the
minimum density), (critical void ratio) and (maximum density) -
decrease with mean stress:
The range of admissible void ratios is limited by and . The model parameters can be found in Tab. 1. They correspond to Hochstetten sand from the vicinity of Karlsruhe, Germany .
The basic law parameters should be specified in group_materi_plasti_hypo_wolffersdorff.
A simplistic appraoch to include cohesion is used here. Instead of feeding the real effective stress state into the hypoplastic law, an alternative effective stress state is used. Cohesion is modelled by subtracting in each of the normal stress components a value representing cohesion: , and . The shear stresses are not altered: , etc.
The cohesion value should be specified in group_materi_plasti_hypo_cohesion.
In order to take into account the recent deformation history, an additional tensorial state variable 1 is introduced.
Denoting the normalized magnitude of
where is the objective rate of intergranular strain. Rigid body rotations are treated elsewhere (see the section on memory). From the evolution equation (2.2.4) it follows that must remain between 0 and 1.
The general stress-strain relation is now written as
The fourth order tensor represents the incremental stiffness and is calculated from the hypoplastic tensors and which may be modified by scalar multipliers and , depending on and on the product :
is an additional material parameter.
An example intergranular parameters can be found in Tab. 2.
The intergranular parameters should be specified in group_materi_plasti_hypo_intergranularstrain. Also you need to include materi_strain_intergranular in the initialisation part.
Pressure dependent initial void ratio
You can correct the initial void ratio , as specified in the initial value
for the history variable in the node_dof records, for the initial pressure
to obtain a corrected initial void ratio .
See the basic law description for the parameters and . The denotes the effective stress tensor (total stresses minus any groundflow pressure). This pressure dependent initial void ratio correction can be activated by group_materi_plasti_hypo_pressuredependentvoidratio. After the initial void ratio has been established, the development of the void ratio is governed by volumetric compression or extension of the granular skeleton.