Asymptotically Self-consistent Dynamic Shell Models for Incompressible Biological Materials: Theory and Applications
DescriptionAny object is always three-dimensional. However, many structures in nature and engineering applications are often thin, that is, the thickness is much smaller than the length scales of the other two dimensions. In this case, one may call them to be shells. For example, skins, heart membranes, leaves, petals, arteries, airplane wings and cooling towers are shells. Due to the thinness, one can use a two-dimensional model to study the behaviors of shells, instead of a three-dimensional one (whose computation is very costly due to the small aspect ratio). On the other hand, biological materials are often very soft. One characteristic of this type of materials is that they can undergo large deformations withfinite strains.Also, this type of materials is incompressible in general (during the deformation the volume does not change). Usually, the mechanical behaviors of biological materials are modelled as incompressible hyperelastic materials through a strain energy function with incompressibility constraint. There are manysmall-strainshell theories available (e.g. Donell shell theory and Koiter shell theory), originally developed for hard materials such as steel, but relatively speaking,finite-strainshell theories with incompressibility constraint are much fewer. In particular, a derivedconsistentdynamic finite-strain shell theory for incompressible hyperelastic materials with both bending and stretching effects are not available. The main aim of the present project is to derive a dynamic finite-strain shell theory for biological materials with a strain energy function and incompressibility constraint. Our derivation will be for a general strain energy function, and for a particular biological material one can adopt a particular strain energy function form (a few of them are available in literature). The derived shell theory will be shown to be asymptotically consistent with the three-dimensional Hamilton’s principle. The novelty of the derivation technique, which is based on series expansions about the bottom surface, is the finding of certain linear relations between stress, Lagrange multiplier and displacement/position coefficients (although the original problem is highly nonlinear). This enables the establishment of the recurrence relations for the expansion coefficients, leading to a final system with only three unknowns. Proper edge boundary conditions will be proposed. For the numerical computation, associated weak formulations will also be derived. The shell theory will be used to study several problems associated with human arteries. The first one is the natural frequency. If the environment has a frequency close to that of an artery, one may feel uncomfortable. Another application is to study the bifurcations of solutions for understanding aneurysm formation in human arteries, which has been a hot topic of recent research. Previous researchers usually used only a membrane tube model to conduct post-bifurcation analysis without considering the bending or thickness effect. With our derived shell theory, both stretching and bending effects will be taken into account. Also, shear stress due to the blood flow, which is routinely neglected in previous studies and whose effect may be important, will be considered. It is hoped that the derived dynamic shell theory for incompressible hyperelastic materials could have broad applications for thin biological structures and the study on aneurysm formation can shed further light on this important phenomenon.
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