Generalized inner bending continua for linear fiber reinforced materials

Abstract This paper deals with the effective behaviour of elastic materials periodically reinforced by linear slender elastic inclusions. Assuming a small scale ratio e between the cell size and the characteristic size of the macroscopic deformation, the macro-behaviour at the leading order is derived by the homogenization method of periodic media. Different orders of magnitude of the contrast between the shear modulus of the material μ m and of the reinforcement μ p are considered. A contrast μ m / μ p of the order of e 2 leads to a full coupling between the beam behaviour of the inclusions and the elastic behaviour of the matrix. Under transverse motions, the medium behaves at the leading order as a generalized continuum that accounts for the inner bending introduced by the reinforcements and the shear of the matrix. Instead of the second degree balance equation of elastic Cauchy continua usually obtained for homogenized composites, the governing equation is of the fourth degree and the description differs from that of a Cosserat media. This description degenerates into, (i) the usual continua behaviour of elastic composite materials when O ( μ m / μ p ) ⩾  e , (ii) the usual Euler–Bernoulli beam behaviour when O ( μ m / μ p ) ⩽  e 3 . The constitutive parameters are derived and can be computed or estimated from simplified geometries. Simple criteria are given to identify the appropriate model for real reinforcements under given loadings.