Simple Uniaxial and Uniform Biaxial Deformation of Nearly Isotropic Incompressible Tissues

J. Hildebrandt, H. Fukaya and C. J. Martin

ABSTRACT

A method is developed for analyzing in a unified manner bothuniaxial and uniform biaxial strain data obtained from nearlyisotropic tissues. The formulation is a direct application ofnonlinear elasticity theory pertaining to large deformations.The general relation between Eulerian stress ( ${sigma}$ ) and extensionratio ( ${lambda}$ ) in soft isotropic elastic bodies undergoing uniformdeformation takes the simple form: ${sigma}$ = (( ${lambda}$ ³ - 1)/ ${lambda}$ ) f( ${lambda}$ ), wheref( ${lambda}$ ) must be determined for each material. The extension ratiomay be either greater than 1.0 (uniaxial elongation), or liebetween zero and 1.0 (uniform biaxial extension). Simple analyticalfunctions for f( ${lambda}$ ) are most readily found for each tissue byplotting all data as ( ${lambda}$ ³ - 1)/ ${lambda}$ ${sigma}$ vs. ${lambda}$ . Of those tissues investigatedin this way (dog pericardium and pleura, and cat mesentery anddura), all but pleura could be adequately described by a parabola:1/f( ${lambda}$ ) = 1/k{[( ${lambda}$ _M - ${lambda}$ )( ${lambda}$ - ${lambda}$ _m)]/[ ${lambda}$ _M - ${lambda}$ _m}. In these instances, threematerial constants per tissue (K, ${lambda}$ _M, ${lambda}$ _m) served to predict approximatelythe stresses attained during both small and large deformations,in strips and sheets alike. It was further found that the uniaxialstrain asymptote ( ${lambda}$ _M) was linearly related to the biaxial strainasymptote ( ${Lambda}$ _M), thus effectively reducing the number of constantsby one.