E32 Stress distribution

Topic

The distribution of stress within an insect's exoskeleton—specifically within the cuticle—determines which region is most susceptible to failure and where, consequently, greater thickness or additional reinforcement is required.

Cauchy's equilibrium equations state that, in the absence of acceleration, the internal stress gradient of the exoskeleton balances any external force applied to it.

Laplace's solution for a thin membrane relates the pressure of the insect's hemolymph—its circulating fluid—to the stress within the cuticle: a significantly distended abdominal segment experiences correspondingly higher circumferential stress. A further increase in hemolymph pressure within the abdominal segment results in a corresponding increase in circumferential stress on the segment's cuticle.

The phenomenon of stress concentration around an opening in the exoskeleton—described by the Kirsch solution—explains why openings such as spiracles or joints are surrounded by thickened cuticular reinforcement. Consequently, in this model—featuring the specific exoskeleton region most prone to the type of failure observed upon opening—Hertzian contact theory characterizes the extremely high local stress that arises when the mandible (or a specific claw of the insect) presses against a hard surface; indeed, the cuticle must withstand considerable final contact pressure in certain predatory beetles. As the cuticle of the mandible or claw approaches the region most susceptible to failure, the final contact pressure it must endure becomes even greater.

ID:2524