Interface stresses in fiber-reinforced materials with regular fiber arrangements

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Residual stresses at fiber/matrix interfaces are frequently responsible for crack nucleation in composite materials: interface failure can occur in the case of a weak interface and a higher thermal expansion coefficient of the fiber. Radial matrix cracking may be observed if the matrix is brittle and possesses the higher thermal expansion coefficient. The quantitative assessment of interface stresses is a key factor in developing reliable fiber reinforced composites. Of further special interest is the state of stress at the surface of the composite where interface-nucleated cracks may originate and propagate into the composite. In this paper the theory of linear elasticity is used to analyze the stresses inside and at the surface of fiber-reinforced composites. All three states of plane deformation are considered: plane strain, plane stress and generalized plane strain. They are investigated analytically using the so-called shell-model (for plane strain and plane stress) and the BHE-model (for generalized plane strain). Moreover, they are treated numerically by means of a finite element analysis. In the shell-model a single fiber in a finite matrix under an additional external pressure is considered. First, Duhamel-Neumann's form of Hooke's law is applied and a general expression for the thermomechanical stresses in cylindrical systems is obtained. The result reduces to Lamé's solution in the absence of thermal stresses. Next, this solution is evaluated by means of the appropriate boundary conditions to give explicit, analytical expressions for the thermomechanical stresses as functions of thermal and elastic mismatch as well as of external pressure. Numerical results of the shell- and the BHE-model are compared with interface stresses obtained from the finite element method for regular cubic and hexagonal fiber arrangements: interface stresses are shown to depend weakly on Poisson's ratio. For equal values of Poisson's ratio, generalized plane strain and plane strain results are identical. The dependence of interface stresses on volume fraction and on the arrangement of fibers is examined for the range of practically important elastic mismatches in composites. For small volume fractions up to 40 vol.% of fibers the shell- and the BHE-model are shown to predict the interface stresses very well over a wide range of elastic mismatches and for different fiber arrangements. At higher volume fractions, however, the stresses are influenced by the interaction with neighboring fibers. An external pressure, which is nearly independent of the elastic mismatch, is introduced into the shell-model to take this influence into account. This allows the prediction of interface stresses in real composites with isolated or regularly arranged fibers. The analytical formulae can be used to assess the influence of residual stresses for the design of new fiber-reinforced composites.