Solid Approximation
For the Solid approximation on interior thin layers, the displacements between the two sides of the boundary are decoupled. The two boundaries are then connected by elastic and viscous forces proportional to the relative displacements and the corresponding material model.
By using the notation u1 = down(u) and u2 = up(u), so u1 represents the displacement of the downside (bottom), and u2 is the displacement of the upside (top) of the thin layer. The average displacement of the midsurface reads
and the extension of the thin layer reads
With this notation, the normal vector N points from the downside (bottom) towards the upside (top).
The gradient through the normal direction is approximated as
where d is the thickness of the thin layer, and the quantity ue/d represents the strain in the layer.
The deformation gradient in the layer is then approximated as
The tangential deformation gradient on the midsurface is calculated as
and the normal deformation gradient as
The normal deformation gradient contains both normal as well as transverse shear information, and it is constant through the thickness of the layer. The tangential deformation gradient, Ft, contains information about the stretching in the midsurface, the so-called membrane deformation.
When Layered is selected in the Thin layer type list, the gradient through the normal direction is approximated as
where z is the extra dimension coordinate, that varies between 0 and the layer thickness d.
The deformation gradient in the layered thin structure is then approximated as
The weak contributions are scaled by a correcting factor that accounts for the curvature of the thin layer.
The layered formulation can handle composites and laminates, see the Theory for the Layered Shell Interface section for details.