Refined theory of composite beams: The role of short-wavelength extrapolation

The present paper presents an asymptotically-correct beam theory with nonclassical sectional degrees of freedom. The basis for the theory is the variational-asymptotical method, a mathematical technique by which the three-dimensional analysis of composite beam deformation can be split into a linear, two-dimensional, cross-sectional analysis and a nonlinear, one-dimensional, beam analysis. The elastic constants used in the beam analysis are obtained from the cross-sectional analysis, which also yields approximate, closed-form expressions for three-dimensional distributions of displacement, strain, and stress. Such theories are known to be valid when a characteristic dimension of the cross section is small relative to the wavelength of the deformation. However, asymptotically-correct refined theories may differ according to how they are extrapolated into the short-wavelength regime. Thus, the re is no unique asymptotically-correct refined theory of higher order than classical (Euler-Bernoulli-like) theory. Different short-wavelength extrapolations can be obtained by changing the meaning of the theory's one-dimensional variables. Numerical results for the stiffness constants of a refined beam theory and for deformations from the corresponding one-dimensional theory are presented. It is shown that a theory can be asymptotically correct and still have non-positive-definite strain energy density, which is completely inappropriate mathematically and physically. A refined beam theory, which appropriately possesses a positive-definite strain energy density and agrees quite well with experimental results, is constructed by using a certain short-wavelength extrapolation.