Equation 6-1 is exact for flat laminates. For curved laminates, the deformation gradient expression must account for the surface area of each layer. The deformation gradient in a product geometry of a curved layered shell can be written as
In some applications, it is required to model variable thickness layers. This is achieved by scaling the constant thickness of the layer (d_layer) using a thickness scale factor (
lsc), which could be a function of surface coordinates. The deformation gradient in a scaled product geometry of a curved layered shell can be written as
where t is the tangent to the edge. For an internal edge, it is possible that there is a discontinuity in thickness or offset. In such a case, the line scale factor will be an average. Edge conditions are not well defined in such situations because the position of the midsurface can be discontinuous. In practice, errors caused by such effects are small.
The layer thickness scale factor (lsc) is also accounted in the integrations when variable thickness layers are present in the model.
This is automatically handled by the program. The automatic search for these fold lines compares the normals of all the layered shell surfaces sharing an edge. If the angle between the normals is larger than a certain angle (default 3°) it is considered as a fold line.
where ub is the displacement vector at the reference surface location in the through-thickness direction.
where ur is the displacement vector in the through-thickness direction relative to the displacement vector at the reference surface location.
Sometimes, you want to write expressions that are functions of the coordinates in the thickness direction of the layered shell. If you write expressions based on the usual coordinates, like X,
Y, and
Z, such an expression will be evaluated on the reference surface (the meshed boundaries). In addition to this, you can access locations in the through-thickness direction by making explicit or implicit use of the coordinates in the extra dimension.
The extra dimension coordinate has a name like x_llmat1_xdim. The middle part of the coordinate name is derived from the tag of the layered material definition where it is created; in this example a
Layered Material Link.
Finally, the coordinates in 3D space are available using the physics scoped variables lshell.X,
lshell.Y, and
lshell.Z. These coordinates vary also in the thickness direction of the layered shell.
The default value for the through-thickness location is given in the Default through-thickness result location section of the Layered Shell interface.
The Layered Material dataset allows the display of results in 3D solid even though the equations are solved on a 2D surface.