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  1. Material
  2. Material1D
  3. vonMises

Subloading1D

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Last updated 1 month ago

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

The implementation is based on the following paper.

Additional material on the same topic can be found in the following literature.

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 for implementation details.

Theory

Subloading Surface

The subloading surface is defined as

fs=∣η∣−zσyf_s=|\eta|-z\sigma^yfs​=∣η∣−zσy

where η=σ−ayα+(z−1)σyd\eta=\sigma-a^y\alpha+\left(z-1\right)\sigma^ydη=σ−ayα+(z−1)σyd is the shifted stress, shifted from the centre defined by ayα+(1−z)σyda^y\alpha+\left(1-z\right)\sigma^ydayα+(1−z)σyd. The scalar 0⩽z⩽10\leqslant{}z\leqslant{}10⩽z⩽1 is the normal yield ratio that provides a smooth transition from the interior to the normal yield surface. The scalar σy\sigma^yσy 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(1−e−misosq)\sigma^y=\sigma^i+k_\text{iso}q+\sigma^s_\text{iso}\left(1-e^{-m^s_\text{iso}q}\right)σy=σi+kiso​q+σisos​(1−e−misos​q)

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

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

q˙=γ\dot{q}=\gammaq˙​=γ

where γ\gammaγ is the plasticity multiplier.

Kinematic Hardening

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

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

with

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

where bbb 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^yay mimics σy\sigma^yσy. The parameters aia^iai, kkink_\text{kin}kkin​, akinsa^s_\text{kin}akins​ and mkinsm^s_\text{kin}mkins​ share similar implications compared to their counterparts.

Evolution of zzz

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 zzz (RRR in the references). We do not adopt such a limit.

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

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

Evolution of ddd

The evolution of ddd resembles that of α\alphaα.

d˙=ceγ(zen−d)\dot{d}=c_e\gamma\left(z_en-d\right)d˙=ce​γ(ze​n−d)

in which cec_ece​ and ze<1z_e<1ze​<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)

initial_history(2)

initial_history(3)

initial_history(4)

Example

accumulated plastic strain

normal yield ratio

normalised back stress

normalised elastic core

See example.

qqq
zzz
α\alphaα
ddd
10.1007/s00707-025-04339-0
10.1007/978-3-030-93138-4
10.1007/s11831-023-10022-1
10.1007/s11831-022-09880-y
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