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

By G. Friesecke, Richard D. James, Stefan Müller. 2002

849 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/c844e470d0fedac3e15f7d432b63273a76fdf35e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks (2010, neutral): “In that regard, it would be interesting to see what is the limiting form of our results as one of the dimensions approaches zero, as in [37].”
Regularity properties of isometric immersions (2005, methodology): “From this one easily sees that finite energy implies y ∈ W 2,2 (see [2], Remark (vii) after Theorem 6.”
The uniform Korn–Poincaré inequality in thin domains (2008, methodology): “In [5], the L2 distance of ∇u from a single proper rotation is estimated in terms of the L2 norm of the pointwise distance of ∇u from the space of proper rotations SO(n).”
On generalized Cosserat-type theories of plates and shells: a short review and bibliography (2010, positive): “After the pioneering work of Kirchhoff [160] thousands of publications are presented, which try to give the foundations and the methods of deduction of the equations of the Kirchhoff–Love theory, but also of improvements, see, for example, [45,46,110–114,158,159,227,249] among others.”
On the Sobolev Space of Isometric Immersions (2004, background): “Γ-convergence of the standard 3-dimensional nonlinear elasticity was proven later on by G. Friesecke, R.D. James and S. M¨ uller [7], [ 8 ].”
A Proof of Crystallization in Two Dimensions (2006, background): “A theorem by Müller, Friesecke and James in [8] states that there is a universal constant C which does not depend on u such that for each u we can find a rotation matrix R ∈ SO(2) with the property ∫ |∇u− R|2 ≤ C ∫ dist(∇u, SO(d))2.”
Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma$-convergence (2003, background): “Proof of Theorem 2.1 The argument follows closely the proof of Theorem 4.1 in [ 6 ].”
Linear elasticity obtained from finite elasticity by Γ -convergence under weak coerciveness conditions (2012, methodology): “The compactness result requires the following extension of the well-known geometric rigidity result of [8], where a power of d(∇v, SO(n)) is replaced by gp(d(∇v, SO(n))).”
Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, background): “Friesecke, James and Müller [78], for example, show that if there exists a constant C > 0 such that W (F ) ≥ C · dist(2)Euclid(F,SO(3)) (2.”
Approximation of Flat W2,2 Isometric Immersions by Smooth Ones (2011, background): “The relation (13) is proven in [2] Proposition 6.”
Korn-Poincaré inequalities for functions with a small jump set (2016, background): “This has been generalized to many different settings, including the geometrically nonlinear case [FJM02], mixed growth [CDM14] and incompatible strain fields which contain dislocations [NPW12, MSZ14].”
Korn’s second inequality and geometric rigidity with mixed growth conditions (2012, methodology): “1 in [8] and follows from the same proof.”