On mathematical models of stress-strain relationship for living soft tissues

How detailed must a mathematical expression of the mechanical properties of a tissue be? The answer depends on the use intended for the constitutive equation. With the objective of application to physiology and medicine, a simplified approach is recommended. We point out that all biological tissues are composites and have complex behavior. In general, there is no "natural" state. Preconditioning is necessary to obtain repeatable experimental results. The viscoelastic properties of several tissues are examined and a unique feature is pointed out in that the stress-strain relationship in cyclic loading and unloading is not very sensitive to strain rate, and that the hysteresis loop is virtually constant for a wide range of frequencies. This is interpreted as being due to a continuously distributed spectrum of relaxation times which spread over a very wide range. This unique feature justifies the assumption of "pseudoelasticity," and the use of a pseudo strain-energy function to derive the stress-strain relationship in specific loading processes. Basic materials and tissues of higher structures may need other considerations. For example, in the lung the contribution of the surface tension between the air-liquid interface on the interalveolar septa is a major factor. The derivation of the stress-strain relationship for the lung tissue is presented.