On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

@article{Ozenda2020OnTK,
  title={On the Kirchhoff-Love Hypothesis (Revised and Vindicated)},
  author={Olivier Ozenda and Epifanio G. Virga},
  journal={Journal of Elasticity},
  year={2020},
  volume={143},
  pages={359-384}
}
The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard Γ $\varGamma $ -convergence) have proven to be incompatible with it. We find that an… 
View on Springer
link.springer.com
11 Citations
Highly Influential Citations
3
Background Citations
3
Methods Citations
7

11 Citations

Model for a Photoresponsive Nematic Elastomer Ribbon

We study the equilibria of a photoresponsive nematic elastomer ribbon within a continuum theory that builds upon the statistical mechanics model put forward by Corbett and Warner (Phys. Rev. E…

J ul 2 02 1 Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between…

A blend of stretching and bending in nematic polymer networks.

This paper derives the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematics elastomers in three space dimensions via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools.

Quadratic-stretch elasticity

A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between…

Ridge approximation for thin nematic polymer networks

Nematic polymer networks (NPNs) are nematic elastomers within which the nematic director is enslaved to the elastic deformation. The elastic free energy of a NPN sheet of thickness $h$ has both…

Model for Photoresponsive Nematic Elastomers

We study the equilibria of a photoresponsive nematic elastomer ribbon within a continuum theory that builds upon the statistical mechanics model put forward by Corbett and Warner [Phys. Rev. E 78 ,…

Ridge energy for thin nematic polymer networks

The class of admissible spontaneous deformations is broadened: ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces are considered, and a model to compute the extra energy distributed along such ridges is proposed.

A Ribbon Model for Nematic Polymer Networks

We present a theory of deformation of ribbons made of nematic polymer networks (NPNs). These materials exhibit properties of rubber and nematic liquid crystals, and can be activated by external…

Bending and Stretching in a Narrow Ribbon of Nematic Polymer Networks

We study the spontaneous out-of-plane bending of a planar untwisted ribbon composed of nematic polymer networks activated by a change in temperature. Our theory accounts for both stretching and…

Energies for Elastic Plates and Shells from Quadratic-Stretch Elasticity

We derive stretching and bending energies for isotropic elastic plates and shells. Through the dimensional reduction of a bulk elastic energy quadratic in Biot strains, we obtain two-dimensional…

References

SHOWING 1-10 OF 76 REFERENCES

An existence theorem for a two-dimensional nonlinear shell model of Koiter’s type

We propose a minimization problem with a stored energy function that is polyconvex and satisfies all the other assumptions of John Ball’s theorem, while being at the same time well adapted for…

A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume…

Buckling transition and boundary layer in non-Euclidean plates.

In the infinitely thin plate limit, under the assumption that a limiting configuration exists, it is shown that the limit is a configuration that minimizes the bending content, among all configurations with zero stretching content (isometric immersions of the midsurface).

Théorie des Corps déformables

THE authors, who are well known by their writings on general elastic theory, here reprint in separate form an appendix contributed by them to M. Chwolson's “Traité de Physique.” The kinematical and…

A treatise on the mathematical theory of elasticity

The present treatise is the outcome of a suggestion made to me some years ago by Mr R. R. Webb that I should assist him in the preparation of a work on Elasticity. He has unfortunately found himself…

Elastic theory of unconstrained non-Euclidean plates

On the validation of theories of thin elastic structures

This writing is meant to restitute the contents of a lecture I gave in Torino, on September 19, 2013, within the framework of the 21st AIMETA Congress. For this reason, I have made an effort to use…

A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of…

The Theory of Shells and Plates

A plate and more generally a shell is a special three-dimensional body whose boundary surface has special features. Although we defer defining a shell-like body in precise terms until Sect. 4, for…

A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity

Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical…
...