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Update README.md
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ZAARAOUI999 authored Dec 18, 2023
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[![Generic badge](https://img.shields.io/badge/pypi-v0.1.1-<COLOR>.svg)](https://pypi.org/project/hypermat/) [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.8285247.svg)](https://doi.org/10.5281/zenodo.8285247) [![Downloads](https://static.pepy.tech/badge/hypermat/week)](https://pepy.tech/project/hypermat)


HyperMAT is based on the definitions of the second Piola-Kirchhoff stress $\large S$ and the material tangent modulus $\large \hat{C}$ given below:
HyperMAT is based on the definitions of the first Piola-Kirchhoff stress $\large P$ and the tangent modulus $\large \hat{A}$ given below:

<p align="center">
$\large S = 2 \frac{\partial{W}}{\partial{C}}$ <br> $\large \hat{C} = 4 \frac{\partial^2{W}}{\partial{C}^2}$
$\large P = \frac{\partial{W}}{\partial{F}}$ <br> $\large \hat{A} = \frac{\partial^2{W}}{\partial{F}^2}$
</p>

<h2>How to use</h2>
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F[...,2,1] = 0.3

# Get stress tensor
S = umat.jacobian(F)
P = umat.jacobian(F)

# Get material tangent modulus tensor
C = umat.hessian(F)
# Get tangent modulus tensor
A = umat.hessian(F)
```

Sometimes a lucky engineer will have some tension or compression stress-strain test data, or simple shear test data. Processing and applying these data is a critical step to analyze the hyperelastic models. HyperMAT has a calibration module that can help to get the best fitted model parameters. Let's take a look on how are things going on:
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