DafalisaManzari
Last updated
Last updated
The modified Dafalias--Manzari sand model.
The original model 10.1061/(ASCE)0733-9399(2004)130:6(622) is modified slightly. Readers can also refer to the corresponding section in Constitutive Modelling Cookbook for details on the theory.
The modifications can be summaries as follows.
The continuum mechanics convention (tension positive) is used. All volumetric strain and hydrostatic stress related quantities shall flip their signs.
The Lode angle dependency is removed, which is equivalent to set in Eq. (19).
Constants such as and are removed. They can be combined with the model parameters.
The hardening related parameter as defined in Eq. (24) causes numerical issues under small cyclic loads. Hence, it is changed to a similar form.
The algorithm is correct but the theory needs validation. Your collaboration would be much appreciated.
The hyperelastic response is defined as
The void ratio can be associated to strain so that
The strain increment can be decomposed into elastic and plastic parts.
As such, the stress increment can be expressed accordingly,
In deviatoric and spherical components,
with
The critical state parameter is chosen as
The dilatancy surface is defined as
The bounding surface is defined as
A wedge-like function is chosen to be the yield surface.
A non-associated plastic flow is used, the corresponding flow rule is defined as follows.
To improve numerical stability, is bounded by where is a constant can be chosen as for example . This is equivalent to define an elastic response for .
where is the deviatoric stress, is the hydrostatic stress.
where is the so called back stress ratio and characterises the size of the wedge. For simplicity, is assumed to be a constant in this model.
By denoting , the directional unit tensor is defined as
where is the dilatancy parameter.
Due to the change of sign convention, a negative gives contractive response, thus often needs to be negative.
The evolution rate of the back stress ratio is defined in terms of a proper distance measure from the bounding surface,
where controls the hardening rate,
The parameter is defined as a function of current state,
is updated whenever load reversal occurs.
The fabric tensor changes when is positive,