A tonneau cover for covering a cargo box of a vehicle. The tonneau cover includes a plurality of flexible hinges, each of which is between two rigid members. Each flexible hinge includes an elastomeric material having a first portion connected to a first one of the rigid members, a second portion connected to a second one of the rigid members, and a web between the first portion and the second portion. A cut-resistant barrier is associated with the flexible web and bridges between the first portion and the second portion. The cut-resistant barrier is effective to resist penetration by a sharp edge.