On a Consistent Dynamic Finite-strain Model for Soft-material Shells and its Applications
DescriptionShell-like structures are ubiquitous in nature and engineering applications. Examples include heart valves, blood vessels, epithelial tissues in organisms, plant petals, roofs of infrastructures and cooling towers, etc. On the other hand, many materials, like biological tissues, polymer gels and elastomers, are very soft. One characteristic of this type of materials is that they can undergo large deformations with finite strains. Recently, there have been increasing interests in understanding pattern formations in soft materials due to mechanical forces, e.g., wrinkles in skin and creases in brain. Thus, there is a great need to study mechanical behaviors of softmaterial shells. Any object is always three-dimensional. However, for a shell structure the thickness is much less than the length scales of its base surface. As a result, instead of using a three-dimensional theory, which is difficult to tackle both numerically and analytically, one can proceed to investigate behaviors of soft material shells through proper reduced two-dimensional shell theories. There are many small-strain shell theories for hard materials available (e.g. the celebrated Kirchhoff-Love shell theory and Koiter shell theory), relatively speaking finite-strain shell theories are much fewer. In particular, a consistent dynamic finite-strain shell theory with both bending and stretching effects are not available. Constitutively, one can model mechanical behaviors of soft materials through a strain (free) energy function. The main aim of the present project is to derive a finite-strain shell theory for constitutively nonlinear materials with a general strain energy function, which is asymptotically consistent with the three-dimensional Hamilton’s principle. The derivation will be based on series expansions about the bottom surface of the shell. The novelty is the finding of certain linear relations between stress and displacement/position coefficients (although the problem is highly nonlinear), which enable us to establish the recursion relations for expansion coefficients, leading to a final system with only three unknowns. Edge conditions and associated weak formulations will also be considered to complete the theory. Several benchmark problems will be studied for testing the validity of the shell theory. To demonstrate the powerfulness of this new theory, two applications will be considered. One is the swelling-induced instability in a polymer gel tube, which is relevant to a microlense device. Another is the modeling of skin folding in a turtle’s neck. Based on the shell theory, we will use a combined analytical and numerical approach to tackle these two problems, and the results can help the understanding how instabilities lead to pattern formations. Some classical small-strain shell theories have found broad applications for hard materials, and it is hoped that the derived dynamic finite-strain shell theory could also have broad applications for soft materials.
|Effective start/end date||1/01/18 → …|