For small displacements ( w s( x, y) ≪ L, b), the constraints are mathematically expressed by applying Pythagorean theorem to infinitesimal lengths in the chosen Cartesian coordinate system ( Fig. These constraints are consequence of that no in-plane stresses are applied to the sheet. The original flat structure must then transform into a structure with out-of-plane displacement w s( x, y) that obeys the following three topological constraints the volume, contour length and contour width must remain constant. The structure is unrestrained at the edges and it is only subject to bending moments. We start with a thin sheet with length, width and thickness referred to as L, b and h, respectively, ( Fig. First, we mathematically demonstrate by appealing to simple geometry arguments this third property. We will show that this property is the origin of the stiffening of the paper cantilever when it is transversally bent. This experiment reveals a third property: in-plane stresses are necessarily developed when the sheet is subject to biaxial bending. Far from achieving a parabolic-like surface 8, our sheet exhibits a characteristic wrinkle pattern that impedes biaxial bending. 2a,b), biaxial bending is not so straightforward ( Fig. We observe that whereas the sheet can be easily bent in one direction within the elastic regime ( Fig. To understand why the paper cantilever stiffens when it is transversally curved, we make a second experiment with the paper sheet. This experiment shows a second property referred to as bending asymmetry: the upward and downward bending stiffness of a cantilever sheet is different when the cantilever is transversally curved. The cantilever deflects downwards due to the gravity, more than before rotation, but substantially less than in the flat configuration ( Fig. We now rotate 180 degrees the transversally curved cantilever. This experiment reveals a first property: the stiffness of cantilever sheets significantly increases with the transversal curvature. If we add a small transversal curvature of 0.0035 mm −1 with positive sign (against gravity) to the clamped cantilever end, the sheet gets almost straight ( Fig. The cantilever largely bends downwards due to gravity ( Fig. The paper cantilever is obtained by fixing one of the ends of the sheet to a desk. It is important to remark that the problem is scalable and the same phenomena can, for instance, be found in a microfabricated thin plate 297 μm long, 210 μm wide and 340 nm thick. Our paper sheet is 297 mm long, 210 mm wide and 340 μm thick ( Fig.
We show here simple experiments with a cantilever made of a paper sheet that question this widely accepted assumption and reveal new fundamental properties of thin sheets. Under this prism, the stiffness of a thin sheet not subjected to stress is determined by its mechanical properties and its dimensions. In fact, it is generally assumed that the deflection of thin sheets does not affect their stiffness within the framework of linear elasticity 6, 7. Whereas the role of internal stress in the three dimensional configurations of thin sheets is starting to be understood, the reverse effect has received little attention. Small amounts of in-plane stress in thin elastic sheets give rise to a host of different and complex patterns 1, 2, 3, 4, 5. The results shed new light on plant and insect wing biomechanics and provide an easy route to engineer micro- and nanomechanical structures based on thin materials with extraordinary stiffness tunability. The theory predicts experimental results with a macroscopic cantilever sheet as well as numerical simulations by the finite element method. We develop a theory that describes the stiffness of curved thin sheets with simple equations in terms of the longitudinal and transversal curvatures. The coupling between the internal stresses and the bending moments can increase the stiffness of the plate by several times. We here demonstrate by using simple geometric arguments that thin sheets subject to two-dimensional bending necessarily develop internal stresses. Simple experiments with a cantilever sheet made of paper show that the cantilever stiffness largely increases with small amounts of transversal curvature. This assumption, however, goes against intuition.
Within the framework of classical thin plate theory, the stiffness of thin sheets is independent of its bending state for small deflections. Curved thin sheets are ubiquitously found in nature and manmade structures from macro- to nanoscale.