Читать книгу Modern Trends in Structural and Solid Mechanics 1 - Группа авторов - Страница 14

Accurate solutions of the linear elasticity equations governing three-dimensional static deformations of fiber-reinforced composite laminates are needed to efficiently design them for structural applications, by considering failure modes in them. In this chapter, the governing equations are written as first-order partial differential equations for three displacements, three transverse stresses and three in-plane strains. The mixed formulation facilitates the satisfaction of continuity conditions at an interface between two adjacent plies. These nine equations are numerically solved by minimizing residuals in them and in the boundary conditions by using the least-squares method. The functional of the residuals is evaluated by expressing the unknown quantities as the product of complete polynomials of different orders in the three independent coordinates and using appropriate quadrature rules. Minimization of the functional, with respect to coefficients appearing in the polynomials for the solution variables, provides a system of simultaneous linear algebraic equations that are numerically solved. It is shown that polynomial functions of degree, at most, 8 in the in-plane coordinates and 3 in the thickness coordinate for each layer of a laminate, provide accurate solutions for stresses and displacements when compared with the analytical solutions of problems. Through-the-thickness stress distributions are found to agree very well with those found analytically. It provides an efficient numerical scheme for accurately finding stresses and displacements in a laminate.