Dynamical Asymptotic Model for Soft Material Plates and Its Applications

Project: Research

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Many natural and man-made materials are very soft, e.g. polymer, elastomer, polymer gel andbiological tissue, etc. In terms of mechanics, it means that these soft materials can undergo largeelastic deformations with finite strains. In recent years, there have been increasing researchinterests in modeling behaviors of this type of materials due to their wide engineering andscientific applications. A classical example is tire and a more recent example is microlense madeof a ph-sensitive gel. Also, it is now understood that mechanical forces alone can play a veryimportant role for determining the pattern formations in biological objects (e.g. buckledcucumbers and wrinkled pumpkins). On the other hand, plates are very common structures (e.g.plant leaves, thin gel films and biological membranes). Thus, there is a great need to study softmaterial plates. Any object is always three-dimensional. However, for a plate structure thethickness is much less than the in-plane length scale. As a result, instead of using a three-dimensionaltheory, which is difficult to tackle both numerically and analytically, one canproceed to investigate the behaviors of soft material plates through proper reduced two-dimensionalplate theories. There are many small-strain plate theories available (e.g. thecelebrated von Karman plate equations), and relatively speaking plate theories for finite-straindeformations (a character of soft materials) are much fewer. In particular, consistent dynamicalfinite-strain plate theories are not available. From the point of view of mathematical modeling,constitutively the mechanical behaviors of soft materials can be described through a strain (free)energy function. One aim of the present project is to derive a consistent dynamical finite-strainplate theory for constitutively nonlinear materials with a general strain energy function. Tovalidate the correctness of this new theory, a linearization will be used to obtain thecorresponding small-strain plate theory. Then, several benchmark problems will be solved bythis linear plate model and the solutions will be compared with the exact ones and those obtainedby other well-known small-strain plate models. It is expected that this can demonstrate thesuperiority of the present plate theory due to its consistency with the 3-D formulation. Severalapplications of the derived finite-strain plate theory will be considered. One is the swelling of apolymer gel layer, which has been studied by many people experimentally and numerically. Awell-observed phenomenon is crease formation on the gel surface. The aim here is to carry outan analytical and numerical study based on our derived asymptotic model to understand themechanism leading to crease formation and provide interpretations to some key experimentalobservations. Actually, crease formation is a widely-spread phenomenon, e.g., it is also found inour brain and skin. It is hoped that the results may also shed some lights in those contexts.Another problem is the stretching of an elastic sheet leading to a wrinkled pattern, which has itsrelevance in applications in biological tissues and solar sails, etc. In recent years, this problemhas been studied in literature by using three different plate models (in which the finite-straineffects were not fully taken into account). Here, our asymptotic model will be used to investigatesuch a wrinkled pattern and comparisons with other three studies will be made. The von Karmanplate theory for small strain and large deflections has been used to study many problems, and itmight be expected the derived dynamical plate model for finite-strain problems could also havebroad applications for soft materials.


Project number9042236
Grant typeGRF
Effective start/end date1/09/1519/02/20

    Research areas

  • Soft materials,Instability,Asymptotic analysis,Large deformations,