Beam theory models for thin film segments cohesively bonded to an elastic half space
An elastic stiffener of height \(h\) and length \(2L\) is bonded to the boundary \(y=0\) of an elastic half-plane along the interval \(-L\leq x\leq L\). The complementary part of the boundary, \(| x|> L\), being free. The half-plane is loaded by a uniform compressive strain acting parallel to the boundary \(y=0\). If the stiffener were not bonded to the half-plane, this latter would be subject to a uniform compression along the \(x\)-axis, while the stiffener would remain unstressed. The bonding , however, introduces stress interaction between the two bodies, created in the interval of mutual contact, \(-L\leq x\leq L\), \(y=0\). These stresses consist of a normal component \(p\) and a shear component \(q\). These components are determined by imposing the condition that the elastic displacements are continuous among the line of contact. These conditions of continuity give two linear integral equations in the functions \(p\) and \(q\) defined on \(-L\leq x\leq L\). The system can be solved by expanding the unknowns in orthogonal Chebyshev polynomials, and determining the coefficients of the expansions by requiring that the equations should be satisfied at a finite number of points of the interval, equal to the number of unknown coefficients. The same technique is applied to solve the case in which a tangential slippage occurs along the interface. Since the tangential stress \(q\) becomes singular at the points \(x=\pm L\), \(y=0\), a more realistic model is that of assuming \(q\) to be everywhere less than or equal to a limiting cohesive force \(\tau_ y\). The procedure is again the same, but now the points \(x=\pm a\) \((a< L)\) at which \(q\) attains the maximum allowed value \(\tau_ y\) must be determined. The advantage of the method is that all the required integrations can be carried out explicitly and the only numerical procedures rest on the solution of a set of linear equations and in the solution of an algebraic equation to determine \(a\).
scientific article; zbMATH DE number 97979