.. _green_lagrange_strain:

Green-Lagrange Strain
=====================

The Green-Lagrange strain tensor (:math:`\mathbf{E}`) is a finite strain measure suitable for large displacements and rotations. It is defined in the **reference (material) configuration**.

Unlike Cauchy strain, it accounts for geometric non-linearities (such as the stretching of a rotated element).

Theory
------

It is derived from the Right Cauchy-Green deformation tensor :math:`\mathbf{C}`:

.. math::

    \mathbf{E} = \frac{1}{2} (\mathbf{C} - \mathbf{I}) = \frac{1}{2} (\mathbf{F}^T \mathbf{F} - \mathbf{I})

Where :math:`\mathbf{F} = \mathbf{I} + \nabla \mathbf{u}` is the deformation gradient.

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

When expanded in 2D cartesian coordinates, the components include the quadratic terms of the displacement gradients:

.. math::

   E_{xx} = \frac{\partial u}{\partial x} + \frac{1}{2} \left[ \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial x}\right)^2 \right]

.. math::

   E_{yy} = \frac{\partial v}{\partial y} + \frac{1}{2} \left[ \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2 \right]

.. math::

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

where 
  - :math:`u`: displacement in x-direction
  - :math:`v`: displacement in y-direction  
  - :math:`\frac{\partial u}{\partial x}`: the partial derivative of the displacement field along x (:math:`\mathbf{u}`) with respect to the spatial coordinate :math:`x`,
  - :math:`\frac{\partial v}{\partial y}`: the partial derivative of the displacement field along y (:math:`\mathbf{v}`) with respect to the spatial coordinate :math:`y`,
  - :math:`\frac{\partial u}{\partial y}`: the partial derivative of the displacement field along y (:math:`\mathbf{u}`) with respect to the spatial coordinate :math:`y`,
  - :math:`\frac{\partial v}{\partial x}`: the partial derivative of the displacement field along x (:math:`\mathbf{v}`) with respect to the spatial coordinate :math:`x`,

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

.. code-block:: python

    import numpy as np

    def compute_strain_green_lagrange(disp_x, disp_y, dx, dy):
        """
        Compute Green-Lagrange Strain (Large Strain).
        
        Assumes indexing='ij' (axis 0 is x, axis 1 is y).
        """
        # 1. Compute Gradients
        grad_u = np.gradient(disp_x, dx, dy)
        du_dx, du_dy = grad_u[0], grad_u[1] # u_x, u_y

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

        # 2. Compute Strain Components (Corrected for cross-terms)
        # Exx uses derivatives w.r.t X only (u_x and v_x)
        E_xx = du_dx + 0.5 * (du_dx**2 + dv_dx**2)
        
        # Eyy uses derivatives w.r.t Y only (u_y and v_y)
        E_yy = dv_dy + 0.5 * (du_dy**2 + dv_dy**2)
        
        # Exy mixes terms
        E_xy = 0.5 * (du_dy + dv_dx + (du_dx * du_dy) + (dv_dx * dv_dy))

        return E_xx, E_yy, E_xy

References
----------

* `Wikipedia: Finite Strain Theory (Green-Lagrangian) <https://en.wikipedia.org/wiki/Finite_strain_theory#The_Green%E2%80%93Lagrangian_strain_tensor>`_
* *Nonlinear Solid Mechanics: A Continuum Approach for Engineering*, G. Holzapfel.