Lamellar bone is known to be the most typical structure of cortical bone in large mammals including humans. This type of tissue provides a good combination of strength and fracture toughness. As has been shown by John D Currey and other researchers, large deformations are associated with the appearance of microdamage that optically whitens the tissue, a process that has been identified as a contribution to bone toughness. Using finite-element modelling, we study crack propagation in a material with periodic variation of mechanical parameters, such as elastic modulus and strength, chosen to represent lamellar bone. We show that a multitude of microcracks appears in the region ahead of the initial crack tip, thus dissipating energy even without a progression of the initial crack tip. Strength and toughness are shown to be both larger for the (notched) lamellar material than for a homogeneous material with the same average properties and the same initial notch. The length of the microcracks typically corresponds to the width of a lamella, that is, to several microns. This simultaneous improvement of strength and toughness may explain the ubiquity of lamellar plywood structures not just in bone but also in plants and in chitin-based cuticles of insects and arthropods.