Stress estimation of joints having adherends with different curvatures bonded with viscoelastic adhesives

The residual stresses in an adhesive layer between adherends with different curvatures were investigated analytically assuming the adherends as linear elastic beams and the adhesive layer as linear viscoelastic springs. A governing equation giving the time variation of adhesive thickness caused by the viscoelastic deformation is presented. The equation is an ordinary differential equation of the 4th order including a convolution equation of the creep compliance or the relaxation function equivalent to the linear viscoelastic characteristics of the adhesive. The equation gives also the distribution of a normal stress perpendicular to the interface between the adhesive and the adherends.The governing equation can be transformed into a complex number domain s by the Laplace transform, but it is difficult to solve generally the transformed equation in the s domain. However, when the adherends are long enough, the transformed equation at the end of the joint can be expressed in a simple form that can be treated analytically. In this case, the autoconvolution of the stress variation with time at the joint end is in proportion with the time integrals of the relaxation function of the adhesive. Therefore, the residual stress in the adhesive layer can be calculated by a numerical integration and a numerical auto-deconvolution, which is easier than other numerical calculations.This method has also another advantage in which the relaxation function or the creep compliance of the adhesive can be applied directly to the calculation without any mechanical analogy expressed with springs or dashpots of the adhesive's characteristics. The applicable region of the method, however, is limited by the necessary condition of long adherends.