Shakedown in brittle-matrix fiber-reinforced cracked composite beams

The cracking behaviour of a composite beam with multiple reinforcing fibers under periodic traction-flexure is analysed through a fracture mechanics-based model, where the edge-cracked beam section is exposed to external loads and crack bridging reactions due to the fibers. Assuming a rigid-perfectly plastic bridging law for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging forces are obtained from compatibility conditions. Under general load paths, shakedown conditions are explored by making use of the Melan’s theorem, here reformulated for the discrete problem under consideration, where the crack opening displacement at the fiber level plays the role of the plastic strain in the counterpart problem of an elastic-plastic solid. The limit of shakedown is determined through an optimization procedure based on a linear programming technique.