Non-developable surfaces are those which cannot be formed from sheet material without the material shearing in its plane. Surfaces of this type have curvatures in two different directions (they are therefore often called doubly-curved surfaces). Although these surfaces are usually more difficult to manufacture than flat or developable surfaces, their form gives them great stiffness and strength.

Because composite materials can often shear during the manufacturing process, they are more suitable for manufacturing these shapes than conventional materials like aluminium. This magnifies the advantage of laminated composite materials for many classes of structure.

The extent of double curvature at any point is reflected in a value called the Gaussian curvature. This is the product of the curvatures in the principal directions at any point on a surface. The Gaussian curvature reveals many of the characteristics of a surface. Positive Gaussian curvature means that the surface is locally dome-shaped, with the curvatures in the same direction with respect to the surface normal. In contrast, negative Gaussian curvature implies saddle-shaped topology, with curvatures in opposite directions. Finally, zero Gaussian curvature is characteristic of a developable area.

Note that surfaces often have varying Gaussian curvature over their extent. As an example, a torus (donut) is saddle-shaped on the inside (negative Gaussian curvature) but dome-shaped on the outside (positive Gaussian curvature).

Gaussian curvature can be given a physical significance by drawing geodesic lines on a surface. (Geodesic lines are straight in the plane of the surface at any point; meridians are geodesic, but lines of latitude are constantly turning in one direction with respect to the surface.) A pair of lines which are initially parallel will tend to converge on surfaces of positive Gaussian curvature, but will diverge on a surface of negative Gaussian curvature. In contrast, the lines will remain parallel on the surface until they reach an edge if the surface is developable.

Another interpretation of Gaussian curvature is the extent of misfit in the surface. Consider a circular disk made up of several flat segments. This necessarily has zero Gaussian curvature even if it is bent along the joints between segments. However, if one of the segments is removed and the neighboring segments joined, the disk will adopt a dome-like shape which is indicative of positive Gaussian curvature. In contrast, adding a segment will result in the disk forming a saddle-like shape with negative Gaussian curvature.

Draping of non-developable surfaces is an extremely difficult task. Essentially, this process involves extremely large geometric and, perhaps, material nonlinearities. A direct consequence of this is that there is no unique solution for the draping process. The draped shape is highly dependent on the point at which the draping starts, the directions in which the draping proceeds and the properties of the material itself. In addition, if there is interaction between different layers, friction between them would have a significant effect. A detailed analysis of the draping process for arbitrary geometries is therefore a considerable analysis task in itself.

This difficulty in analysis reflects a real-world difficulty in manufacturing complex composite components consistently. Engineering drawings of composite components typically specify fibre angles within a tolerance of 3 degrees. In practice, if there is significant curvature in a surface, the manufacturing tolerance could easily reach 15 degrees or more.

These problems can be mitigated to a large extent by limiting the degree of shear developed within reinforcing layers during the manufacturing process. The degree of shear is primarily dependent on the Gaussian curvature and the area of a layer. Therefore, a design incorporating two layers of excessive shear can be replaced by three smaller layers with less shear and greater quality. The MSC.Laminate Modeler employs a rapid draping module which allows the designer to investigate the likely degree of shear, and make rational engineering decisions on the basis of manufacturing simulations.

Whatever simulation process is used, two different levels of draping should be considered. Local Draping reflects the behavior of an infinitesimal material element applied to a point on a surface having general curvature. This is a material characteristic and is determined from tests on materials. In contrast, Global Draping considers how the many material elements are placed on a surface, and is dependent on the manufacturing process used.

Local draping is concerned with fitting a small section of material to a generally‑curved surface. If the surface has nonzero Gaussian curvature, the material element must shear in its plane to conform to the surface. This deformation is highly dependent on the microstructure of the material. As a result, local shearing behavior can be regarded as a layer material property.

MSC.Laminate Modeler currently supports two local draping algorithms: scissor and slide draping. For scissor draping (Figure 2‑5), an element of material which is originally square shears in a trellis-like mode about its vertices to form a rhombus. In particular, the sides of the material element remain of constant length. This type of deformation behavior is characteristic of woven fabrics which are widely used to manufacture highly-curved composite components.

For slide draping (Figure 2‑6), two opposite sides of a square material element can slide parallel to each other while their separation remains constant. This is intended to model the application of parallel strips of material to a surface. It can also model, very simply, the relative sliding of adjacent tows making up a strip of unidirectional material.

When draping a given surface using the two different local draping algorithms, the shear in the layer builds up far more rapidly for the slide draping mechanism than for the scissor draping mechanism. This observation is compatible with actual manufacturing experience that woven fabrics are more suitable for draping curved surfaces than unidirectional pre-pregs.

For small deformations, the predictions of the different algorithms are practically identical. Therefore, it is suggested that the scissor draping algorithm be used in the first instance.

Global draping is concerned with placing a real sheet of material onto a surface of general curvature. This is not a trivial task as there are infinite ways of doing this if the surface has nonzero Gaussian curvature at any point. Therefore, it is important to define procedures for the global draping simulation which are reproducible and reflect what can be manufactured in a production situation. As a result, global draping behavior can be regarded as a manufacturing, rather than material, property.

The MSC.Laminate Modeler currently supports three different global draping algorithms: Geodesic, Planar and Energy. For the Geodesic global draping option, principal axes are drawn away from the starting point along geodesic paths on the surface (i.e., the lines are always straight with respect to the surface). Once these principal axes are defined, there is then a unique solution for draping the remainder of the surface. This may be considered the most “natural” method and appropriate for conventional laminating methods. However, for highly-curved surfaces, the paths of geodesic lines are highly dependent on initial conditions and so the drape simulation must be handled with care.

For the Planar global draping option, the principal axes may be defined by the intersection of warp (and weft for scissor draping) planes which pass through the viewing direction. This method is appropriate where the body has some symmetry, or where the layup is defined on a space-centered rather than a surface‑centered basis.

Finally, the Energy global draping option is provided for draping highly-curved surfaces where the manufacturing tolerances are necessarily greater. Here, the draping proceeds outwards from the start point, while the direction of draping is controlled by minimizing the shear strain energy along each edge.

Many pressure vessels are made of composite materials, particularly via the filament winding process. However, it is often necessary to add woven reinforcements to the shell. In this case, it is vital to understand the mechanics of the draping process because the curvature of the surface varies so much. In particular, the body of a pressure vessel is developable and has zero gaussian curvature. In contrast, the ends have constant positive Gaussian curvature.

If draping begins at the pole of the vessel (Figure 2‑7), the shear in the material increases rapidly away from the start point due to the severe curvature. The amount of shearing is indicated by the color of the draping pattern lines. Note that the degree of shear is zero along the principal axes, which are defined by geodesic lines.

The draping algorithm stops where the shear reaches the cutoff value for the material, or the override value defined in the Additional Controls form. This gives an indication of where the material would fold when being formed.

To cover the same area, it is also possible to begin draping on the cylindrical part of the surface (Figure 2‑8). Because this region is developable, there is no shear deformation until the end cap is reached. This means that the average degree of shear on the surface is much lower, which should lead to better quality and better mechanical performance.