Assessment of Saint-Venant Principle (SVP) in Biological Tissues: Radial Decay of Stress Concentration in Soft Tissues

SVP is commonly accepted as a central idea in structural mechanics. While considerable research on the validity of this principle is available for standard structural materials, few studies have examined the validity of SVP for biological tissues. This is surprising since there are situations, like defects in arteries walls, aneurysm, stent insertion, skin injuries and membrane perforation, which impose local self-equilibrating loads on bio-tissues. There is a need to understand how stress and strain fields induced decay with distance from loaded zone. Unlike metals, bio-tissues admit large strains and become progressively convex as stretches increase. Earlier studies noted that decay of local effects depends on material orthotropic axes orientation. It has also been reported that decay length increase in presence of anisotropy in bio-tissues. In a different direction, work by Humphrey and co-workers on redistribution of localized stresses and strains near circular irregularities indicates sensitivity to anisotropy and initial stretch.

The present research aims at an initial theoretical analysis of diffusion with distance of self-equilibrating loads applied to bio-tissues. Material response is assumed to be hyperelastic, incompressible, and anisotropic in the unloaded reference state. While aspects like growth, tissue pathologies, remodeling and morphogenesis are not considered, we attempt at exposing decay rate sensitivities to level of stretch, constitutive parameters and loading direction.

Formulation is within the framework of finite strain continuum mechanics, employing laboratory verified hyperelastic constitutive relations for representative biological tissues like the aorta, brain, fat tissues, liver and skin. We focus on local irregularities modelled by internally pressurized small spherical or circular cylindrical cavities embedded in large tissues. We investigate the redistribution of stress and strain, in the vicinity of such irregularities, with increase of loading, enhancing earlier studies for circular irregularities in plane membranes. Results for decay of local self-equilibrating fields will be compared for bio-tissues. The load induced intensity of near wall boundary layer buildup, as deformation progresses, will be determined.

Decay rates for spherical and cylindrical fields confirm the validity of SVP for these configurations with considerable sensitivity to loading direction, deformation level and constitutive parameters. Results of the radial decay of the Neo-Hookean and Fung material models are similar for both spherical and cylindrical patterns. A common conclusion from the examples considered here is that the validity of SVP strengthens with increasing stretch. It is seen that there is a build up of a boundary layer with strong gradients near cavity wall at high levels of internal pressure. That layer, (Saint-Venant zone), depends on loading direction and on strain energy function convexity. Key findings are supported by asymptotic expansions.

Exposing the nature of strain redistribution near local irregularities can help with understanding of solid tumors mechanics. While this research is only an initial step in assessing the validity of SVP in soft bio-tissues, it provides new and challenging observations that call for further study on the applicability of that fundamental principle in biomechanics. In particular, the role of strain energy function convexity deserves a comprehensive investigation in context of SVP validity for soft bio-tissues.