The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a 0-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 V U ! R n , U R n . We show that the L 2 -distance of rv from a single rotation matrix is bounded by a multiple of the L 2 -distance from

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A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity

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the derivation of nonlinear plate theory from three dimensional elasticity

We study the nonlinear monotone elliptic problem  −div (a(x,∇u)) = μ on Ω, u = 0 on ∂Ω, when Ω ⊂ R , μ is a Radon measure with bounded total variation on Ω, 1 < p ≤ N , and u 7→ −div (a(x,∇u)) is a monotone operator acting on W 1,p 0 (Ω). We introduce a new definition of solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates). We also prove partial uniqueness results. SISSA, via Beirut 4, 34014 Trieste, Italy Laboratoire d’Analyse Numerique, Universite Paris VI, Case 187, 75252 Paris cedex 05, France Dipartimento di Matematica, Universita di Roma I, P.le A. Moro 2, 00185 Roma, Italy Universite d’Orleans, rue de Chartres, B.P. 6759, 45067 Orleans cedex 2, France

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Renormalized solutions of elliptic equations with general measure data

Material properties emerge from phenomena on scales ranging from Angstroms to millimeters, and only a multiscale treatment can provide a complete understanding. Materials researchers must therefore understand fundamental concepts and techniques from different fields, and these are presented in a comprehensive and integrated fashion for the first time in this book. Incorporating continuum mechanics, quantum mechanics, statistical mechanics, atomistic simulations and multiscale techniques, the book explains many of the key theoretical ideas behind multiscale modeling. Classical topics are blended with new techniques to demonstrate the connections between different fields and highlight current research trends. Example applications drawn from modern research on the thermo-mechanical properties of crystalline solids are used as a unifying focus throughout the text. Together with its companion book, Continuum Mechanics and Thermodynamics (Cambridge University Press, 2011), this work presents the complete fundamentals of materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.

Modeling Materials: Continuum, Atomistic and Multiscale Techniques

Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20


Published: SIAM Review, 44 (2002) 631--658.

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Convergence of Adaptive Finite Element Methods

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Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side

Geometric rigidity states that a gradient field which is $L^p$-close to the set of proper rotations is necessarily $L^p$-close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in $L^p+L^q$ and in $L^{p,q}$ interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn’s inequality to these spaces.

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Korn’s second inequality and geometric rigidity with mixed growth conditions

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier--Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the nonlinear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.

Well-Posedness of a Fully Coupled Navier--Stokes/Q-tensor System with Inhomogeneous Boundary Data

The Cauchy-Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy-Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke and Theil [J. Nonlin. Sci. {\bf 12} (2002), 445--478] for a two-dimensional model. As in their work the key idea is to use a discrete version of polyconvexity (ordinary convexity of the elastic energy as a function of the atomic positions is ruled out by frame-indifference). The main point is the construction of a suitable {\discrete} null Lagrangian which allows one to separate rigid motions. To do so we observe a simple identity for the determinant function on $\SO(n)$ and use interpolation to convert ordinary null Lagrangians into {\discrete} ones.

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Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)

We derive a two-dimensional model for elastic plates as a Γ-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The resulting model describes plate bending, and is determined from the isochoric elastic moduli of the three-dimensional problem. Without the constraint of incompressibility, a plate theory was first derived by Friesecke et al. (Comm Pure Appl Math 55:1461–1506, 2002). We extend their result to the case of p growth at infinity with p ϵ [1, 2), and to the case of incompressible materials. The main difficulty is the construction of a recovery sequence which satisfies the nonlinear constraint pointwise. One main ingredient is the density of smooth isometries in W
 2,2 isometries, which was obtained by Pakzad (J Differ Geom 66:47–69, 2004) for convex domains and by Hornung (Comptes Rendus Mathematique 346:189–192, 2008) for piecewise C
 1 domains.

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Georg Dolzmann

Papers

Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side

Korn’s second inequality and geometric rigidity with mixed growth conditions

Well-Posedness of a Fully Coupled Navier--Stokes/Q-tensor System with Inhomogeneous Boundary Data

Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)

Γ-convergence for incompressible elastic plates