[Study] Shell-to-solid coupling

[WeiyiVirtonomy]

The subdomains of a body might have different resolution requirements, so the computational effort could be reduced significantly if the domain could be discretized with coupled solid and shell elements. The coupled bodies should satisfy displacement and traction continuity at the coupling interface.

Example

Problem description

As Prof. Hu suggested, I'll start from the simplest case with a softer solid plate and a stiffer shell coupled at the in-plane surface:

The thickness of the shell is set to 1dp, so that the real geometry of the shell is consistent with the discretized model.
As SPH doesn't have boundary nodes, I extend the solid particles by 1 dp and the extended layer overlaps with the shell particles.

The Young's modulus of solid and shell is 10 MPa and 100 MPa, respectively. A constant pressure of 0.05 MPa is applied to the upper surface of the solid.

Coupling method

The current coupling method treats shell particles as master and solid particles as slave. The shell body is updated first, and its velocity is set as a boundary condition of the coupled solid particles.
On the other hand, the stress of the solid is used to compute the coupling forces on the shell by:

$$\mathbf{F_i} = \mathbf{sigma_j} \cdot \mathbf{n_i} a_i = \mathbf{P_j} \cdot \mathbf{N_i} A_i$$

where $i$ refers to the shell particle id, and $j$ is the id of the solid particle overlapping with it. $\mathbf{P}, \mathbf{N} and A$ denote the corrected PK1 stress, the initial normal and the initial area respectively.

Results

Without coupling, the deformation of the solid under the load is shown below:

With the coupling constraints, the configuration becomes:

It seems that the stiffer shell has reduced the deformation of the solid.

Before exploring how to generalize the method, I need to find a test case for comparison first.

[WeiyiVirtonomy]

I'll compare it with the fully volumetric solid results.

Example: a solid plate is reinforced by two shell plates on top and bottom surface. The stiffness ratio is 0.1.
The 1st load case is extension, while the 2nd load case is shear.
The height of the solid plate is 20mm, while the thickness of shell is selected as 4mm for now.
For the fully volumetric solid test, a composite material is used.

With a resolution of thickness/4.0:

• Extension case
  Left: solid, right: solid-shell coupling

Grey: solid-shell coupling

The position discrepancy is larger near the coupled surfaces:

• Shear case

• Stability
  I started from dp = thickness, but the coupled case soon failed. The stability of the coupling algorithm and the resolution needs further investigation.

• Discussion
  For the next step, I think there are several directions:

1. Replace the lattice particle generator with the relaxed particles and identify the surface layers as shell
2. Test on different stiffness ratios
3. Test on different shell thickness / length ratios
4. Use v_mid+omegaxr_rel instead of the mid-surface velocity as the constraint velocity
5. Discuss how to deal with the normal direction at the corners

I hope we can discuss the priority of these problems tomorrow.