> For the complete documentation index, see [llms.txt](https://tlcfem.gitbook.io/suanpan-manual/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://tlcfem.gitbook.io/suanpan-manual/material/material1d/vonmises/armstrongfrederick1d.md).

# ArmstrongFrederick1D

1D Armstrong-Frederick Steel Model

This model is a uniaxial version of the [`ArmstrongFrederick`](/suanpan-manual/material/material3d/vonmises/armstrongfrederick.md) steel model. Readers can also refer to the corresponding section in [Constitutive Modelling Cookbook](https://github.com/TLCFEM/constitutive-modelling-cookbook/releases/download/latest/COOKBOOK.pdf) for details on the theory.

## Theory

A von Mises type yield function is used. The associated plasticity is assumed. Both isotropic and kinematic hardening rules are employed.

### Isotropic Hardening

An exponential function is added to the linear hardening law.

$$
\sigma^y=\sigma^i+Kq+\sigma^s(1-e^{-m^sq}),
$$

where $$\sigma^i$$ is the initial elastic limit (yielding stress), $$\sigma^s$$ is the saturation stress, $$K$$ is the linear hardening modulus, $$m^s$$ is a constant that controls the speed of hardening, $$\mathrm{d}q=|\mathrm{d}\varepsilon^p|$$ is the rate of accumulated plastic strain $$q$$.

### Kinematic Hardening

The Armstrong-Frederick type rule is used. Multiple back stresses are defined,

$$
\beta=\sum\beta\_i
$$

in which

$$
\mathrm{d}\beta\_i=a\_i\~\mathrm{d}\varepsilon^p-b\_i\beta\~\mathrm{d}q,
$$

where $$a\_i$$ and $$b\_i$$ are material constants.

## Syntax

The following applies to `v3.6` and later. Check the older syntax in the older version of the documentation.

```
material ArmstrongFrederick1D (1) (2) (3) (4) (5) (6) [(7) (8)...] [9]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, yield stress, \sigma^i
# (4) double, linear hardening modulus, K
# (5) double, saturation stress, \sigma^s
# (6) double, m^s, saturation rate
# (7) double, a_i, kinematic hardening
# (8) double, b_i, kinematic hardening
# [9] double, density, default: 0.0
```

## Example

### Kinematic Hardening Only With No Elastic Range

```
material ArmstrongFrederick1D 1 2E2 0. 0. 0. 0. 50 500.
```

The maximum stress can be computed as

$$
\sigma\_{\text{max}}=\sigma^i+\sum\dfrac{a\_i}{b\_i}=\dfrac{50}{500}=0.1.
$$

![Example 1](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-3437c9b2a64e9f05343d5a87274ee50169da2f20%2FArmstrongFrederick1D.EX1.svg?alt=media)

## 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 ArmstrongFrederick1D 1 {young_modulus} {yield_stress} {hardening_ratio * young_modulus} 500 5 500 5",
    ref_strain=yield_stress / young_modulus,
    ref_stress=yield_stress,
) as error_map:
    error_map.contour("af1d", center=-5, size=10, type={"abs"})
```

![accuracy analysis](https://1124082496-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FEquxr3H9WpPg4iPvT2mB%2Fuploads%2Fgit-blob-6f5223e1467a81690b1f8e1d7a197bc9cbac8cf3%2Faf1d.abs.error.svg?alt=media)


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation by asking a question.

Perform an HTTP GET request on the following URL with the `ask` and `goal` query parameters:

```
GET https://tlcfem.gitbook.io/suanpan-manual/material/material1d/vonmises/armstrongfrederick1d.md?ask=<question>&goal=<user_goal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is what the user is ultimately trying to achieve, the reason they need the answer. Sharing it helps GitBook give you a better, more relevant answer. A goal is most helpful when it describes the outcome the user wants rather than restating the question. For example, with `ask=how do I create an API token`, a goal like `automate deployments from our CI pipeline` lets GitBook tailor the answer to that use case.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.