A contact problem for bending of two-leaf spring with the leaves curved along the circular arc

The unbonded contact problem for bending of two-leaf spring is considered; the leaves are curved along the circular arc in their natural states. The lengths of the leaves are different; each leaf has one end clamped and the other free. The angle formed by the long leaf is less than the right one. The cross-sections of leaves are the rectangles of the same width but of the different thickness. These thicknesses are considered to vanish while the elastic lines of the leaves are analyzed geometrically. The real thicknesses affect only the bending stiffness of the leaves. The given loading is applied transversely to the leaves. There is no friction between the leaves. The bending is described by Bernoulli - Euler model. The problem is reduced to finding the density of the leaves interacting forces. This density is the sum of the piecewise-continuous part and the concentrated forces. The rigorous problem statement is formulated, the uniqueness of solution is established and the complete analytical solution of the problem is provided. This construction also proves the existence of the solution. The substantiation of the solution includes proving of the non-negativity of the contact forces and contact distances and the proving of the existence of the root of transcendental equation that gives the length of the contact segment. The proving of the non-negativity of the contact distances uses a new approach that is based on the fact that these distances can be regarded as the solutions of some variational problems. It is shown that three patterns of the leaves contact are possible: the contact along the whole short leaf; the contact at the point on the end of the short leaf; the contact along the part of the short leaf and at the point. The pattern kind depends on the given loading. The obtained results generalize the known sufficient condition for the pointwise contact of the leaves.