Book Review: Théorie des Corps déformables

Wilson, Edwin Bidwell

doi:10.1090/s0002-9904-1913-02334-6

Cited by 10 publications

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“…The 2D elastic continuum model with bending elasticity is a special case of the general concept of Cosserat continuum, which generalizes the conventional theory of elasticity for media of pronounced microstructure. Helpful comments on the physical meaning of the Cosserat model can be found in ref . Eringen , contributed to the further development of this theoretical approach, which was comprehensively presented in several review articles. − In the linear case, the medium is characterized by three lateral and three flexural elasticities, which have been experimentally determined for different media. , Original algorithms have been developed for numerical solution of the respective mathematical problems. − The Cosserat model has been applied for theoretical description of thin shells, − cell membranes, , and large deformations of linear continua .…”

Section: Introductionmentioning

confidence: 99%

Elastic Langmuir Layers and Membranes Subjected to Unidirectional Compression: Wrinkling and Collapse

2009

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We apply the two-dimensional elastic continuum model to describe the wrinkling of elastic Langmuir layers (membranes) subjected to unidirectional compression. The effects of the dilatational, shear, and bending elasticities are taken into account. Among the numerous solutions of the generalized Laplace equation, corresponding to different membrane tensions, we determine the membrane shape as the profile that minimizes the energy of the system. In the case of small deformations, the problem can be linearized. Its solution predicts a wavelike shape of the compressed membrane. At negligibly small bending elasticity, the energy of the system is minimal for a sinusoidal profile, whose amplitude and wavelength tend to zero. In the opposite limiting case, where the effect of bending elasticity prevails over the effect of gravity, the membrane has a half-wave profile. When the two effects are comparable, the membrane shape exhibits multiple periodic wrinkles (ripples). An expression is derived for calculating the bending elasticity (rigidity) from the wavelength, and reasonable values are obtained from available experimental data. To determine the membrane shape at larger out-of-plane deformations, we solved numerically the respective nonlinear problem. Depending on the values of the physical parameters, the theory predicts various shapes: nonharmonic oscillations, toothed profiles, and profiles with two characteristic wavelengths. The results can be used for determining the bending elastic modulus of Langmuir films (membranes) as well as for the interpretation of buckling and collapse of monolayers.

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