Physics-Based Simulation

Discretization of Plastic Flow

Plasticity is introduced into MPM by multiplicatively decomposing the deformation gradient $\mathbf{F}$ into elastic and plastic parts:

$$\mathbf{F}={\mathbf{F}}_{\mathrm{E}}{\mathbf{F}}_{\mathrm{P}}.$$

Here, ${\mathbf{F}}_{\mathrm{P}}$ represents the accumulated irreversible deformation, while ${\mathbf{F}}_{\mathrm{E}}$ captures the recoverable elastic deformation from the plastically deformed configuration.

This decomposition separates material behavior into two parts:

• The plastic part ${\mathbf{F}}_{\mathrm{P}}$ stores permanent changes (e.g., bending a metal rod into a spring),
• The elastic part ${\mathbf{F}}_{\mathrm{E}}$ stores current deformation relative to that shape (e.g., compressing the spring slightly).

Stress is computed solely from ${\mathbf{F}}_{\mathrm{E}}$ using the hyperelastic constitutive model. Plastic flow is triggered when stress exceeds a material-specific limit and updates ${\mathbf{F}}_{\mathrm{P}}$ to ensure the stress stays within the yield surface.

Definition 28.1.1 (Yield Surface). We define a Yield Condition $y(\mathbf{\tau })\le 0$ on the Kirchhoff stress $\mathbf{\tau }$ derived from ${\mathbf{F}}_{\mathrm{E}}$. The boundary $y(\mathbf{\tau })=0$ is known as the Yield Surface. When elastic stress exceeds this surface, plastic flow is triggered to restore admissibility.

This framework cleanly separates recoverable and permanent deformation by computing stress from ${\mathbf{F}}_{\mathrm{E}}$ and evolving ${\mathbf{F}}_{\mathrm{P}}$ under plastic flow.