Constraining the lateral dimensions of uniaxially loaded materials increases the calculated strength and stiffness: application to muscle and bone

Abstract

If a solid body is deformed along one direction, by a uniaxial applied stress for instance, then strains will also be induced in perpendicular directions. The negative ratio of the induced strain to the applied strain is known as the Poisson ratio. Analysis of the elasticity tensor relating stress and strain within a solid shows that if the induced strain is restricted, then a greater stress is required to produce the same strain; it appears stiffer. Many biological materials with a mechanical function are subject to forces which are primarily uniaxial. This mechanism appears to be used to maximize the uniaxial load-bearing properties of some of these materials. Muscles are commonly surrounded by strong sheets of connective tissue which will constrain the lateral expansion of the muscle as it contracts. This increases the stress in the muscle for a given strain, and hence the load it can support. Similarly, cancellous bone is normally surrounded by a shell of much stronger compact bone and this effectively increases the stiffness of the cancellous bone without the penalty of increasing the mass, as would be the case if the same stiffening was produced by increasing the degree of calcification. It also has important implications for the failure of bone, which is largely a function of strain rather than stress.