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

Abstract

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 -limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → R n , U ⊂ R n . We show that the L 2 -distance of ∇v from a single rotation matrix is bounded by a multiple of the L 2 -distance from the group SO(n) of all rotations.

Rigidity estimate for two incompatible wells

“…Its value can vary from line to line, but each line is valid with C being a pure positive number. By a truncation argument, see Proposition A.1 in [9] it is enough to prove the inequality (10) for maps with ∇u L ∞ (Ω) ≤ M, for some constant M depending only on Ω and the set K. To see this, first observe that…”

“…by Theorem 1 ( Theorem 3.1 in [9]), there exists R ∈ SO(n) satisfying (17) and hence we are done in this case. Let U be a relatively compact subset of Ω.…”

“…Recently, Friesecke, James and Müller [8,9] obtained the following interesting rigidity estimate in connection to their study in nonlinear plate theory.…”

Quasiconvex envelopes in nonlinear elasticity

“…Preliminary results for a rigorous derivation of the bending model by Γ-convergence arguments were given in Pantz (2001). The issue was completely settled in Friesecke et al (2002) where an important rigidity lemma is proved. See also Friesecke et al (2006).…”

Variational Dimension Reduction in Nonlinear Elasticity: A Young Measure Approach

“…The choice of a scaling factor different from 1/ε in front of the 3D energies would lead to other limit problems; 1/ε 3 , for example, corresponds to a plate model (see for instance Friesecke, James and Müller [14]). Under the change of variable x 3 = εx 3 (still called x 3 ) the energy functionals become…”