The Theoretical and experimental study of the bending of a thin substrate during electrolytic deposition
The present paper is aimed at the theoretical and experimental study of the shape distortion of thin substrates during electrolytic deposition and gaccumulation of residual stresses in them. The theoretical modeling is provided in the framework of the theory of solids with variable material composition. The result of the deposition process is modeled with a continuous family of elastic bodies, which local deformations are incompatible. These deformations act as internal sources for stresses. Formally they are equivalent to the field of distributed defects. Unlike the classical approach adopted in nonlinear elasticity, the elements of the family which present a body with a variable material composition don’t have a global reference natural (free of stresses) form. Instead we used the continuous family being only locally free from stresses. To formulate the boundary value problem, continuous families of reference, intermediate and actual forms and corresponding families of deformations are defined. The deformations, belonging to these families, locally represent implants (local deformations of reference forms into intermediate ones) and deformations that bring intermediate forms into actual ones. Relations for stresses and strains in such bodies are obtained under the assumption that the displacement gradients are small with respect to unity and satisfy the kinematic hypothesis of the technical plate theory. Under these assumptions the equilibrium equations are derived. They include specific terms which determine formal loading that is caused by incompatible deformations. Axisymmetric problems for a circular substrate under various types of fixing and tension on the boundary, which characterize the conditions of the experiment, are obtained. The theoretical distribution for displacements of the substrate surface is formulated upon the obtained solution. They are intended to identify incompatible deformations that cause bending during the deposition process. The experimental measuring setup is constructed according to a holographic scheme of displacement measurements in real time. The deposition process is carried out in a cylindrical chamber with flange fastening of the cathode. The electrochemical process is implemented in sulphate electrolyte. As a result of comparing the theoretically obtained relations for bending surfaces of the substrate with the experimental results, the parameters that characterize the substrate shrinkage and tension are estimated.