Subloading1D

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.

References

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.

Theory

Subloading Surface

The subloading surface is defined as

fs=ηzσyf_s=|\eta|-z\sigma^y

where η=σayα+(z1)σyd\eta=\sigma-a^y\alpha+\left(z-1\right)\sigma^yd is the shifted stress, shifted from the centre defined by ayα+(1z)σyda^y\alpha+\left(1-z\right)\sigma^yd. The scalar 0z10\leqslant{}z\leqslant{}1 is the normal yield ratio that provides a smooth transition from the interior to the normal yield surface. The scalar σy\sigma^y is the yield stress, that is affected by isotropic hardening.

Isotropic Hardening

The isotropic hardening combines linear hardening and exponential saturation.

σy=σi+kisoq+σisos(1emisosq)\sigma^y=\sigma^i+k_\text{iso}q+\sigma^s_\text{iso}\left(1-e^{-m^s_\text{iso}q}\right)

where σi\sigma^i is the initial yield stress, kisok_\text{iso} is the linear hardening modulus, σisos\sigma^s_\text{iso} is the saturation stress and misosm^s_\text{iso} is the hardening rate.

The history variable qq is the accumulated plastic strain, conventionally, it is

q˙=γ\dot{q}=\gamma

where γ\gamma is the plasticity multiplier.

Kinematic Hardening

A modified Armstrong--Frederick rule is adopted for the normalised back stress α\alpha.

α˙=bγ(nα)\dot{\alpha}=b\gamma\left(n-\alpha\right)

with

ay=ai+kkinq+akins(1emkinsq)a^y=a^i+k_\text{kin}q+a^s_\text{kin}\left(1-e^{-m^s_\text{kin}q}\right)

where bb 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 aya^y mimics σy\sigma^y. The parameters aia^i, kkink_\text{kin}, akinsa^s_\text{kin} and mkinsm^s_\text{kin} share similar implications compared to their counterparts.

Evolution of zz

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 zz (RR in the references). We do not adopt such a limit.

z˙=uln(z)γ.\dot{z}=-u\ln\left(z\right)\gamma.

In which, uu is a constant that controls the rate of transition.

Evolution of dd

The evolution of dd resembles that of α\alpha.

d˙=ceγ(zend)\dot{d}=c_e\gamma\left(z_en-d\right)

in which cec_e and ze<1z_e<1 are two constants.

Syntax

material Subloading1D (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) [15]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, initial isotropic stress, \sigma^i
# (4) double, linear isotropic hardening modulus, k_{iso}
# (5) double, isotropic saturation stress, \sigma^s
# (6) double, isotropic saturation rate, m^s_{iso}
# (7) double, initial kinematic stress, a^i
# (8) double, linear kinematic hardening modulus, k_{kin}
# (9) double, kinematic saturation stress, a^s
# (10) double, kinematic saturation rate, m^s_{kin}
# (11) double, yield ratio evolution rate, u
# (12) double, kinematic hardening rate, b
# (13) double, elastic core evolution rate, c_e
# (14) double, limit elastic core ratio, z_e
# [15] double, density, default: 0.0

History Layout

location
parameter

initial_history(0)

iteration counter

initial_history(1)

accumulated plastic strain qq

initial_history(2)

normal yield ratio zz

initial_history(3)

normalised back stress α\alpha

initial_history(4)

normalised elastic core dd

Example

See this example.

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