.. _cauchy_strain:

Cauchy Strain (Infinitesimal)
=============================

The Cauchy strain tensor (also known as the engineering strain or infinitesimal strain tensor) is used for analyses where deformations and rotations are assumed to be very small (typically < 5%). It is a linear approximation of the deformation.

Theory
------

In the limit of small deformations, the difference between the reference and current configuration is negligible. The strain tensor :math:`\boldsymbol{\varepsilon}` is the symmetric part of the displacement gradient.

.. math::

    \boldsymbol{\varepsilon} = \frac{1}{2} (\nabla \mathbf{u} + (\nabla \mathbf{u})^T)

Component Formulation
---------------------

For a 2D displacement field :math:`\mathbf{u} = (u, v)`:

.. math::

   \varepsilon_{xx} = \frac{\partial u}{\partial x}

.. math::

   \varepsilon_{yy} = \frac{\partial v}{\partial y}

.. math::

   \varepsilon_{xy} = \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)

Python Implementation
---------------------

The implementation uses `numpy.gradient` to compute the partial derivatives. Note that for Cauchy strain, second-order terms are discarded.

.. code-block:: python

    import numpy as np

    def compute_strain_cauchy(disp_x, disp_y, dx, dy):
        """
        Compute Infinitesimal (Cauchy) Strain.
        
        Assumes indexing='ij' (axis 0 is x, axis 1 is y).
        """
        # 1. Compute Gradients
        # gradient returns [d/dx (axis 0), d/dy (axis 1)]
        grad_u = np.gradient(disp_x, dx, dy)
        du_dx, du_dy = grad_u[0], grad_u[1]

        grad_v = np.gradient(disp_y, dx, dy)
        dv_dx, dv_dy = grad_v[0], grad_v[1]

        # 2. Compute Strain Components
        epsilon_xx = du_dx
        epsilon_yy = dv_dy
        epsilon_xy = 0.5 * (du_dy + dv_dx)

        return epsilon_xx, epsilon_yy, epsilon_xy

References
----------

* `Wikipedia: Infinitesimal Strain Theory <https://en.wikipedia.org/wiki/Infinitesimal_strain_theory>`_
* *Continuum Mechanics*, A.J.M. Spencer (Dover Publications).