Controlled elastic deformations

Figure: Graphical overview of the inverse design framework of spatial geodesic strips.

Making double-curved timber structures by moulding, gluing or subtractive milling takes considerable material and effort. Elastic bending offers a resource-efficient alternative: initially straight laths are bent into spatial configurations, so curvature is achieved through the flexibility of the material itself.

Geodesic curves are particularly well suited to this approach. Because they develop straight, a flat strip can in principle be bent onto a geodesic on any surface without material strain. Geodesic strips also remain tangent to the surface. This allows them to be stacked in superposed layers and mechanically joined into compound members of greater stiffness. Unlike Chebyshev nets or asymptotic gridshells, geodesic networks are not restricted to particular classes of surface.

In practice, however, an elastically bent strip does not adopt a geodesic configuration by itself. It deforms into the elastica, the state of minimal elastic energy for the given boundary conditions. Constraining the strip to a different path introduces residual stresses and spring-back, which cause deviations from the intended geometry.

The research addresses this discrepancy as an inverse problem: determining the distribution of mechanical properties for which a strip naturally assumes a prescribed geodesic shape. Local stiffness is controlled through incisions aligned with the conjugate direction of the surface. Restricting the target to geodesics aligns the material frame with the Darboux frame, which reduces the inverse constitutive problem within Kirchhoff rod theory to two stiffness unknowns. The required stiffness distribution follows from moment equilibrium and can be mapped explicitly to incision dimensions in conventional sawn sections.

The research combines differential geometry, nonlinear rod mechanics and fabrication-aware design. It aims to broaden the design space of elastically bent grid structures and to make them viable as load-bearing systems made from renewable, adhesive-free and separable materials.

Figure: Target configuration and numerical forward simulation of the tailored strip.