Physics-Based Simulation

Summary

Stress is a tensor field that quantifies the pressure or tension exerted on a material object. In the context of hyperelastic materials, the first Piola-Kirchhoff stress tensor $\mathbf{P}$ plays a crucial role. It is defined as the derivative of the strain energy density function $\Psi$, with respect to the deformation gradient $\mathbf{F}$, establishing a constitutive relationship between stress and strain.

In practical computations, particularly for the implicit integration of solid dynamics, it is essential to compute $\mathbf{P}$ and its derivative $\frac{\partial \mathbf{P}}{\partial \mathbf{F}}$ efficiently. By leveraging the sparsity structure in diagonal space, these computations become more feasible. Here, differentiations are primarily required for $\Psi$ with respect to the principal stretches ${\sigma }_i$, which simplifies the calculation process.

In the upcoming lecture, we will apply these principles to an inversion-free elasticity model, which will be demonstrated through the compressing square simulation. This application will use the concepts discussed in this chapter to address complex real-world problems in solid mechanics.