Subloading1D
Last updated
Last updated
The Modified Extended Subloading Surface (Hashiguchi) Model
The subloading surface framework provides a very versatile approach to model cyclic behaviour. It is highly recommended to try it out.
Prof. Koichi Hashiguchi has published a large amount of papers on this topic. To find more references, please refer to the monograph and the references therein.
Alternatively, refer to the corresponding section in Constitutive Modelling Cookbook for implementation details.
The subloading surface is defined as
The isotropic hardening combines linear hardening and exponential saturation.
with
initial_history(0)
iteration counter
initial_history(1)
initial_history(2)
initial_history(3)
initial_history(4)
See this example.
where is the shifted stress, shifted from the centre defined by . The scalar is the normal yield ratio that provides a smooth transition from the interior to the normal yield surface. The scalar is the yield stress, that is affected by isotropic hardening.
where is the initial yield stress, is the linear hardening modulus, is the saturation stress and is the hardening rate.
The history variable is the accumulated plastic strain, conventionally, it is
where is the plasticity multiplier.
A modified Armstrong--Frederick rule is adopted for the normalised back stress .
where is hardening rate. Compared to the conventional AF rule, the saturation bound is not a constant in this model. Instead, it is associated to plasticity. The backbone mimics . The parameters , , and share similar implications compared to their counterparts.
The following rule is used. Noting that the original formulation uses a cotangent function. Here, the logarithmic function is used instead. Also, the original formulation sets a minimum value for ( in the references). We do not adopt such a limit.
In which, is a constant that controls the rate of transition.
The evolution of resembles that of .
in which and are two constants.
accumulated plastic strain
normal yield ratio
normalised back stress
normalised elastic core