Stress and Its Derivatives

Having introduced standard strain energies, we now proceed to their differentiation with respect to the world space coordinates, $x$, to simulate realistic elastic behaviors. However, it's important to first establish the explicit relationship between these coordinates $x$ and the deformation gradient $\mathbf{F}$. This relationship heavily depends on specific discretization choices.

Before we explore discretization in depth, we should understand how to compute the derivatives of the strain energy function, $\Psi$, with respect to $\mathbf{F}$. These derivatives are fundamentally linked to the concept of stress, a critical element in understanding material behavior under deformation.