# 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

The implementation is based on the following paper.

1. [10.1007/s00707-025-04339-0](https://doi.org/10.1007/s00707-025-04339-0)

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

1. [10.1007/978-3-030-93138-4](https://doi.org/10.1007/978-3-030-93138-4)
2. [10.1007/s11831-023-10022-1](https://doi.org/10.1007/s11831-023-10022-1)
3. [10.1007/s11831-022-09880-y](https://doi.org/10.1007/s11831-022-09880-y)

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](https://github.com/TLCFEM/constitutive-modelling-cookbook/releases/download/latest/COOKBOOK.pdf) for implementation details.

## Theory

### Subloading Surface

The subloading surface is defined as

$$
f\_s=|\eta|-z\sigma^y
$$

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

### Isotropic Hardening

The isotropic hardening combines linear hardening and exponential saturation.

$$
\sigma^y=\sigma^i+k\_\text{iso}q+\sigma^s\_\text{iso}\left(1-e^{-m^s\_\text{iso}q}\right)
$$

where $$\sigma^i$$ is the initial yield stress, $$k\_\text{iso}$$ is the linear hardening modulus, $$\sigma^s\_\text{iso}$$ is the saturation stress and $$m^s\_\text{iso}$$ is the hardening rate.

The history variable $$q$$ is the accumulated plastic strain, conventionally, it is

$$
\dot{q}=\gamma
$$

where $$\gamma$$ is the plasticity multiplier.

### Kinematic Hardening

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

$$
\dot{\alpha}=b\gamma\left(n-\alpha\right)
$$

with

$$
a^y=a^i+k\_\text{kin}q+a^s\_\text{kin}\left(1-e^{-m^s\_\text{kin}q}\right)
$$

where $$b$$ 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 $$a^y$$ mimics $$\sigma^y$$. The parameters $$a^i$$, $$k\_\text{kin}$$, $$a^s\_\text{kin}$$ and $$m^s\_\text{kin}$$ share similar implications compared to their counterparts.

### Evolution of $$z$$

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 $$z$$ ($$R$$ in the references). We do not adopt such a limit.

$$
\dot{z}=-u\ln\left(z\right)\gamma.
$$

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

### Evolution of $$d$$

The evolution of $$d$$ resembles that of $$\alpha$$.

$$
\dot{d}=c\_e\gamma\left(z\_en-d\right)
$$

in which $$c\_e$$ and $$z\_e<1$$ 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>=0
# (4) double, linear isotropic hardening modulus, k_{iso}>=0
# (5) double, isotropic saturation stress, \sigma^s>=0
# (6) double, isotropic saturation rate, m^s_{iso}>=0
# (7) double, initial kinematic stress, a^i>=0
# (8) double, linear kinematic hardening modulus, k_{kin}>=0
# (9) double, kinematic saturation stress, a^s>=0
# (10) double, kinematic saturation rate, m^s_{kin}>=0
# (11) double, yield ratio evolution rate, u>=0
# (12) double, kinematic hardening rate, b>=0
# (13) double, elastic core evolution rate, c_e>=0
# (14) double, limit elastic core ratio, 1>z_e>=0
# [15] double, density, default: 0.0
```

## History Layout

| location             | parameter                         |
| -------------------- | --------------------------------- |
| `initial_history(0)` | iteration counter                 |
| `initial_history(1)` | accumulated plastic strain $$q$$  |
| `initial_history(2)` | normal yield ratio $$z$$          |
| `initial_history(3)` | normalised back stress $$\alpha$$ |
| `initial_history(4)` | normalised elastic core $$d$$     |

## Example

See [this](https://tlcfem.gitbook.io/suanpan-manual/example/structural/statics/calibration-subloading) example.

## Accuracy

```py
from plugins import ErrorLine
# note: the dependency `ErrorLine` can be found in the following link
# https://github.com/TLCFEM/suanPan-manual/blob/dev/plugins/scripts/ErrorLine.py

young_modulus = 2e5
yield_stress = 4e2
hardening_ratio = 0.01

with ErrorLine(
    f"""material Subloading1D 1 {young_modulus} \
{yield_stress} 10 50 10 \
200 10 50 10 \
4E2 10 10 0.3""",
    ref_strain=yield_stress / young_modulus,
    ref_stress=yield_stress,
) as error_map:
    error_map.contour("subloading", center=-5, size=5, type={"abs"})
```

![accuracy analysis](https://4006314410-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MQ5rMqEBbzA9NgGETGX%2Fuploads%2Fgit-blob-73ce0d8543212126bb1a307d5084eefaa29862e9%2Fsubloading.abs.error.svg?alt=media)


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