Plastic Discrete Shell Bending

Plastic Discrete Shell Bending is a constitutive model for simulating the bending behavior of thin shell structures with residual creases. This model captures the dihedral angle-based bending energy between adjacent triangular faces in a shell mesh and allows the bending rest angle to evolve after yielding.

Reference:

Discrete Shell

#31 Plastic Discrete Shell Bending

For a shell bending element defined by four vertices at positions $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, and $\mathbf{x}_3$, where $(\mathbf{x}_1, \mathbf{x}_2)$ forms the shared edge between two adjacent triangular faces, we define:

Dihedral Angle:

The dihedral angle $\theta$ is the angle between the two triangular faces sharing the edge $(\mathbf{x}_1, \mathbf{x}_2)$. The first triangle is formed by vertices $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, and the second triangle is formed by vertices $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$.

Rest Configuration Parameters:

From the reference configuration with rest positions $\bar{\mathbf{x}}_0$, $\bar{\mathbf{x}}_1$, $\bar{\mathbf{x}}_2$, and $\bar{\mathbf{x}}_3$:

Rest Length: $L_0 = \|\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_1\|_2$
Average Height: $\bar{h}$ is computed as: $$ \bar{h} = \frac{A}{3L_0} $$ where $$ A = \frac{1}{2}\left(|(\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_0) \times (\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_0)|_2 + |(\bar{\mathbf{x}}_2 - \bar{\mathbf{x}}_3) \times (\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_3)|_2\right) $$
Rest Angle State: $\bar{\theta}$ is initialized from the rest dihedral angle and then treated as an internal variable.

Define the wrapped angle difference $$ \Delta\theta = \operatorname{wrap}(\theta - \bar{\theta}), $$ where $\operatorname{wrap}(\cdot)$ maps an angle to the principal branch $[-\pi,\pi]$.

Bending Energy Density

The bending energy density is $$ E = \kappa \frac{\Delta\theta^2 L_0}{\bar{h}}, $$ where:

$\kappa$ is the bending stiffness parameter
$\theta$ is the current dihedral angle
$\bar{\theta}$ is the current bending rest angle state
$L_0$ is the rest length of the shared edge
$\bar{h}$ is the average height parameter from the rest configuration

Plastic Evolution

Let $\theta_y$ denote the yield threshold and let $H$ denote the hardening modulus. The yielding condition is written as $$ |\Delta\theta| > \theta_y. $$

The plastic increment is $$ \Delta\theta_p = \max(|\Delta\theta| - \theta_y, 0). $$

When $\Delta\theta_p > 0$, the internal variables evolve as $$ \bar{\theta} \leftarrow \bar{\theta} + \operatorname{sign}(\Delta\theta)\Delta\theta_p, $$ $$ \theta_y \leftarrow \theta_y + H \Delta\theta_p. $$

The evolution of $\bar{\theta}$ stores the residual crease in the bending rest configuration. The case $H=0$ corresponds to perfect plasticity, while $H>0$ increases the admissible elastic bending range as plastic bending accumulates.

Attributes

On edges:

bending_stiffness: $\kappa$ in the energy above
bending_yield_threshold: $\theta_y$ in the yielding condition
bending_hardening_modulus: $H$ in the hardening rule