M.Sc Thesis

M.Sc StudentBlum Neta
SubjectAssessment of Saint-Venant Principle in Biological Tissues
DepartmentDepartment of Mechanical Engineering
Supervisors PROFESSOR EMERITUS David Durban
DR. Baruch Karp
Full Thesis textFull thesis text - English Version


Saint-Venant principle is commonly accepted as a central idea in structural mechanics, particularly in engineering practice. However, while considerable research on the validity of this principle is available for standard structural materials, like metals, relatively few studies have examined the validity of the principle for biological tissues. This lacuna is surprising since there are numerous practical situations, like local defects in arteries walls, onset of aneurysm, stent insertion, skin injuries and membrane perforation, which impose local self-equilibrating loads on bio-tissues. There is a clear need to understand how stress and strain fields induced by such loads decay with distance from the loaded zone. It is noted that unlike common metals, biological tissues admit large strains and are strongly convex as stretches increase.

The present research aims at an initial theoretical analysis of diffusion with distance of self-equilibrating loads applied to biological tissues. 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 assume material incompressibility and isotropy in reference state.

We start with the simple cases of internally pressurized spherical and cylindrical cavities and examine intensity of near wall boundary layer build up, as deformation progresses. Radial decay rates of stresses are sensitive to levels of internal pressure and strong gradients develop within the Saint-Venant zone (with possible influence on cell migration). This decay characteristic depends on loading direction (cavity expanding or contracting) and on SEF convexity. Simple approximate relations are derived for near cavity field.

The second part of the research addresses decay of incremental end loads in an axially pre-stretched strip, in the style of classical linear elastic work by Papkovich and Fadle. Again, considerable sensitivity of exponential decay rates in axial direction is exposed, as influenced by material properties and initial strain. A noteworthy result is that presence of transverse stretch (simulating aorta blood pressure) lowers the axial decay rate. Findings are supported by asymptotic expansions revealing the role of SEF convexity in reducing decay rate.

Exposing the nature of strain redistribution near irregularities can help with understanding of solid tumors mechanics (e.g. a nearly spherical tumor inside a soft tissue or healing of a circular cut in skin). Likewise, understating applicability of SVP in the aorta could help in context of cardiovascular pathologies, possibly improving treatments. Research results can contribute also in fixing boundary conditions of experimental procedures.

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.