Closed-form solution of the deflection of a prestressed multilayer cantilever

Abstract

Abstract In this work, a closed-form solution for the deflection of a multilayer cantilever with initial residual stress in each layer is presented. In particular, this model aims to take into account the relaxation of stresses due to the curvature of the cantilever and to calculate the curvature radius and the bending moment due to stresses. Stress-induced deflections and force & stress-induced deflections of cantilever with different shapes (rectangular, triangular, truncated triangular, V-shaped and truncated V-shaped cantilevers) are obtained from the Euler-Bernoulli beam theory. Taking a SiN/SiO2/Si multilayer as an example, the calculated stress-induced profiles do not depend on the shape of the beam and are in good agreement with finite element results. A comparison with results obtained from a model under generalized plane strain condition is discussed. Stress \& Force-induced deflections are also successfully compared to finite element results. Finally, a new formula is proposed to accurately determine the stress in a thin film material from the study of the curvature of a multilayer cantilever. As illustrated by the example of a virtual SiO2/Si bi-layer rectangular cantilever, the initial stress value which is an input parameter of the finite element analysis is calculated back from the simulated curvature with an accuracy of 0.2%.