The effect of holes and free edges on the stress in laminated plates
This work is concerned with the study of the mechanical behaviour of elastic laminated plates subjected to different boundary conditions. For the most part, each lamina is taken to be a fibre-reinforced material which contains a family of straight, continuously distributed fibres. When the modulus for extension in the fibre direction of each lamina is large compared with the other moduli, the laminate is termed 'highly anisotropic' and in such cases, approximate solutions can be obtained by treating the individual laminae as 'ideal' materials in the sense that they are inextensible in the fibre direction and also incompressible. In the context of the plane strain bending of a laminated cantilever, we show that the theory for ideal materials predicts the occurrence of singular fibres at the lateral surfaces of the laminate and at the interfaces between the individual laminae. In a highly anisotropic cantilever these fibres correspond to regions of high stress and accordingly a boundary layer theory is developed for these regions. The boundary layer solution, together with the ideal solution, provide a good approximation to the description of the response of the cantilever, but it is found to be inadequate near the intersections of edges and interfaces, and at corners. A separate investigation is made into the asymptotic behaviour of the stress in these regions. The major part of this thesis is concerned with the development of a general theory for laminated plates in stretching or bending. Given a laminate subject to specified boundary conditions, we define a single homogeneous equivalent plate which has material properties obtained. by an appropriate averaging of the material properties of each lamina. The equivalent plate is subjected to the same boundary conditions as the laminate and the equivalent displacements are determined by classical thin plate theory. The theory then assumes that the displacement components in each lamina can be expressed as the sum of the equivalent displacements and correction displacements. The correction solutions satisfy the conditions of displacement and traction continuity across the inter-laminar boundaries and the condition that the lateral surfaces of the plate are free from traction. In the special case of the laminae being isotropic, the solutions given by the theory exactly satisfy the full three-dimensional equations of linear elasticity. When the equivalent displacements are known, the complete solution in each lamina is readily determined and this is illustrated by examples. At the edges of the laminated plate, the prescribed boundary conditions are satisfied only in an average sense and therefore in these regions, an additional correction is required. The deviation of the calculated boundary condition from the specified boundary condition is used to determine the magnitude of this further correction.