Can a moment-curvature relationship describe the flexion of softening beams? (Q5931915)

The author describes the flexure of a nonlinear elastic beam through a relationship between the bending moment \(M\) and the corresponding curvature \(\lambda\) of the centroid line, neglecting shear deformations. The author uses an odd softening \(M-\lambda\) law with horizontal asymptotes \(M=\pm M_0\) for \(\lambda\to \pm\infty\). The equilibrium configurations of the beam are discussed, and it is shown that solutions, in general, are not unique. The possible difficulties arising from the above-mentioned softening \(M-\lambda\) relationship are summarized.