Asymptotically exact theory of fiber-reinforced composite beams

Abstract

An asymptotic analysis of the energy functional of a fiber-reinforced composite beam with a periodic microstructure in its cross section is performed. From this analysis the asymptotically exact energy as well as the 1-D beam theory of first order is derived. The effective stiffnesses of the beam are calculated in the most general case from the numerical solution of the cell and homogenized cross-sectional problems.

Introduction

Fibre reinforced composite (FRC) beams are widely used in civil, mechanical, and aerospace engineering due to their low weight, high strength, and good damping properties [10], [9], [19]. The exact treatment of FRC beams within the 3D elasticity is only possible in a few exceptional cases due to their complicated microstructure (see, e.g., [24], [26] and the references therein). For this reason, different approaches have been developed depending on the type of beams. If FRC beams are thick, no exact one-dimensional theory can be constructed, so only the numerical methods or approximate semi-analytical methods applied to three-dimensional elasticity theory make sense. However, if the FRC beams are thin, the reduction from the three-dimensional to the one-dimensional theory is possible. This dimension reduction can be made asymptotically exact in the limit when the thickness-to-length ratio of the beam goes to zero. The rigorous derivation of the asymptotically exact one-dimensional beam theory based on the variational-asymptotic method (VAM) developed by Berdichevsky [3] was first performed in [4]. His asymptotic analysis shows that the static (or dynamic) three-dimensional problem of the beam can be split into two problems: (i) the two-dimensional cross-sectional problem and (ii) the one-dimensional variational problem whose energy functional should be found from the solution of the cross-sectional problem. The latter has been solved both for anisotropic homogeneous beams and for inhomogeneous beams with the constant Poisson’s ratio [4]. In addition to these findings, Berdichevsky [4] has shown that the average energy as well as the extension and bending stiffnesses of FRC beams with piecewise constant Poisson’s ratio must be larger than those of FRC beams with constant Poisson’s ratio (see also [23]). However, to our knowledge, the question of how the corrections in energy and stiffnesses depend on the difference in Poisson’s ratio of fibers and matrix of FRC beams remains still an issue. It should be noted that VAM has been further developed in connection with the numerical analysis of cross-sectional problems for geometrically nonlinear composite beams by Hodges, Yu and colleagues in a series of papers [11], [28], [29], [30]. Note also that VAM has been used, among others, to derive the 2D theory of homogeneous piezoelectric shells [13], the 2D theory of purely elastic anisotropic and inhomogeneous shells [5], the 2D theory of purely elastic sandwich plates and shells [6], [7], the theory of smart beams [25], the theory of low and high frequency vibrations of laminate composite shells [17], [18], and more recently, the theory of smart sandwich and functionally graded shells [15], [16].

For FRC beams that have the periodic microstructure in the cross section, an additional small parameter appears in the cross-sectional problems: The ratio between the length of the periodic cell and the characteristic size of the cross section. In this case the finite element code VABS developed in the above mentioned papers [11], [28], [29], [30] cannot be applied to the cross-sectional problem because it requires a large computational capacity. The presence of this small parameter allows however an additional asymptotic analysis of the cross-sectional problems to simplify them. By solving the cell problems imposed with the periodic boundary conditions according to the homogenization technique [8], [22], one finds the effective coefficients in the homogenized cross-sectional problems, which can then be solved analytically or numerically (cf. also [20]). The aim of this paper is to derive and solve the cell and homogenized cross-sectional problems for unidirectional FRC beams whose cross section has the periodic microstructure. For simplicity, we will assume that both matrix and fibers are elastically isotropic but have different Poisson’s ratio. The solution of the cell problems found with the finite element method is used to calculate the asymptotically exact energy and the extension and bending stiffnesses in the 1-D theory of FRC beams. Thus, we determine the dependence of the latter quantities on the shape and volume fraction of the fibers and on the difference in Poisson’s ratio of fibers and matrix that solves the above mentioned issue.

The paper is organized as follows. After this short introduction the variational formulation of the problem is given in Section 2. Sections 3 Dimension reduction, 4 Homogenization are devoted to the multi-scaled asymptotic analysis of the energy functional of FRC beams leading to the cross-sectional and cell problems. In Section 5 the cell problems are solved by the finite element method. Section 6 provides the solutions of the homogenized cross-sectional problems. Section 7 present one-dimensional theory of FRC beams. Finally, Section 8 concludes the paper.