Nonlinear correction to the bending stiffness of a damaged composite beam

When studying damage in composite materials, the classic beam theory is sometimes used in a modified manner to calculate the bending response of a damaged composite laminate. The longitudinal stiffness E0 is then replaced by a field variable E(x,y)=E0×[1−D(x,y)]. The damage distribution D(x,y) affects the calculation of stresses and strains, and requires a modified calculation of the neutral fibre y0(x) and the bending stiffness EI(x). It seems obvious to postulate that the bending stiffness EI(x) is also multiplied with [1−D(x,y)], as was proposed by, amongst others, Sidoroff and Subagio [Fatigue damage modelling of composite materials from bending tests, in: F.L. Matthews, N.C.R. Buskell, J.M. Hodgkinson, J. Morton, editors. Proceedings of the Sixth International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II), 20–24 July 1987, vol. 4, Elsevier, London, UK, p. 4.32]. It is shown here that this formulation is not correct and that a nonlinear correction term to the bending stiffness degradation is necessary. This term is especially important when the damage D(x,y) is reaching higher values. The results of a semi-analytical simulation for cantilever beams is shown to support the findings.