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## Über dieses Buch

This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated.

The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students.

These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.

## Inhaltsverzeichnis

### Chapter 1. Compensated Compactness with More Geometry

Abstract
The theory of compensated compactness, partly developed in collaboration with François Murat, proved useful for attacking questions in the nonlinear PDE (partial differential equations) of continuum mechanics and physics. Many years ago, I believed an improvement could be achieved that included more geometrical ideas. I can only conjecture what should be done, but I consider it useful to present an historical perspective, in order to place ideas in their context, and indicate where geometry may be appropriate.
Luc Tartar

### Chapter 2. Global Isometric Embedding of Surfaces in $$\mathbb R^3$$ R 3

Abstract
In this note, we give a short survey on the global isometric embedding of surfaces (2-dimensional Riemannian manifolds) in $$\mathbb R^3$$. We will present associated partial differential equations for the isometric embedding and discuss their solvability. We will illustrate the important role of Gauss curvature in solving these equations.
Qing Han

### Chapter 3. Singular Perturbation Problems Involving Curvature

Abstract
Consider an anisotropic area functional, giving rise to a variational principle for the shape of crystal surfaces. Sometimes such a functional is regularised with an additional curvature term to avoid difficulties coming from a lack of convexity. We study the asymptotic behaviour of the resulting functional as the strength of the regularisation tends to 0. We consider two cases. The first corresponds to a cubic crystal structure. The expected shapes of the crystal surfaces are polyhedra with faces parallel to the coordinate planes, and for the regularised functionals, we discover a limiting energy depending on the lengths of the edges. In the second case, we have a uniaxial anisotropy. We calculate the limiting energy for surfaces of revolution and give a lower bound for topological spheres.
Roger Moser

### Chapter 4. Lectures on the Isometric Embedding Problem $$(M^{n},g)\rightarrow \mathrm {I\!R\!}^{m},\,m=\frac{n}{2}(n+1)$$

Abstract
This work derives the basic balance laws of Codazzi, Ricci, and Gauss for the isometric embedding of an n-dimensional Riemannian manifold into the $$m=\frac{n}{2}\left( n+1\right)$$-dimensional Euclidean space. It is shown how the balance laws can be expressed in quasi-linear symmetric form and how weak solutions for the linearized problem can be established from the Lax-Milgram theorem.
Marshall Slemrod

### Chapter 5. Continuum Mechanics of the Interaction of Phase Boundaries and Dislocations in Solids

Abstract
The continuum mechanics of line defects representing singularities due to terminating discontinuities of the elastic displacement and its gradient field is developed. The development is intended for application to coupled phase transformation, grain boundary, and plasticity-related phenomena at the level of individual line defects and domain walls. The continuously distributed defect approach is developed as a generalization of the discrete, isolated defect case. Constitutive guidance for equilibrium response and dissipative driving forces respecting frame-indifference and non-negative mechanical dissipation is derived. A differential geometric interpretation of the defect kinematics is developed, and the relative simplicity of the actual adopted kinematics is pointed out. The kinematic structure of the theory strongly points to the incompatibility of dissipation with strict deformation compatibility.
Amit Acharya, Claude Fressengeas

### Chapter 6. Manifolds in a Theory of Microstructures

Abstract
A synopsis, broadly based on contributions by Capriz and co-workers, is presented of a model for a body with microstructure that employs the Cartesian product of a Euclidean space (a fit set part of which is instantaneously occupied by the gross image of the body) and a Riemannian manifold each of whose members specifies a microstructure. Motivation is provided by known special theories. Macro and micro kinetic energy, kinetic coenergy, and inertia are discussed preparatory to the derivation of the governing nonlinear partial differential equations from the Lagrangian action principle, Noether’a theorem, and a Hamiltonian formulation. Precise mathematical specification of initial and boundary conditions remains fragmentary.
G. Capriz, R. J. Knops

### Chapter 7. On the Geometry and Kinematics of Smoothly Distributed and Singular Defects

Abstract
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current $$T$$ and the associated defects are given by its boundary $$\partial T$$. For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.
Marcelo Epstein, Reuven Segev

### Chapter 8. Non-metricity and the Nonlinear Mechanics of Distributed Point Defects

Abstract
We discuss the relevance of non-metricity in a metric-affine manifold (a manifold equipped with a connection and a metric) and the nonlinear mechanics of distributed point defects. We describe a geometric framework in which one can calculate analytically the residual stress field of nonlinear elastic solids with distributed point defects. In particular, we use Cartan’s machinery of moving frames and construct the material manifold of a finite ball with a spherically-symmetric distribution of point defects. We then calculate the residual stress field when the ball is made of an arbitrary incompressible isotropic solid. We will show that an isotropic distribution of point defects cannot be represented by a distribution of purely dilatational eigenstrains. However, it can be represented by a distribution of radial eigenstrains. We also discuss an analogy between the residual stress field and the gravitational field of a spherical mass.
Arash Yavari, Alain Goriely

### Chapter 9. Are Microcontinuum Field Theories of Elasticity Amenable to Experiments? A Review of Some Recent Results

Abstract
It is well known that the material behavior at the micro- and even more at the nano-scale is size dependent, which is, for example, reflected in a stiffer elastic response. Thus modeling of micro- and nanoelectromechanical systems should be ready to incorporate size dependency as well. However, the classical Boltzmann continuum fails to reproduce the size effect. In this work special attention is paid to higher gradient theories such as the strain gradient theory (of Mindlin’s form-II), the modified strain gradient theory and the couple stress theory for linear elasticity. In particular, the latter will also be investigated in terms of finite elements. A confrontation to the Cosserat- or micropolar theory, the non-local continuum, the fractional calculus and the surface elasticity is carried out.
Christian Liebold, Wolfgang H. Müller

### Chapter 10. On the Variational Limits of Lattice Energies on Prestrained Elastic Bodies

Abstract
We study the asymptotic behavior of the discrete elastic energies in the presence of the prestrain metric G, assigned on the continuum reference configuration $$\Omega$$. When the mesh size of the discrete lattice in $$\Omega$$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $$\Gamma$$-limit) energy. In the case of the nearest-neighbour and next-to-nearest-neighbour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to G.
Marta Lewicka, Pablo Ochoa

### Chapter 11. Static Elasticity in a Riemannian Manifold

Abstract
We discuss the equations of elastostatics in a Riemannian manifold, which generalize those of classical elastostatics in the three-dimensional Euclidean space. Assuming that the deformation of an elastic body arising in response to given loads should minimize over a specific set of admissible deformations the total energy of the elastic body, we derive the equations of elastostatics in a Riemannian manifold first as variational equations, then as a boundary value problem. We then show that this boundary value problem possesses a solution if the loads are sufficiently small in a specific sense. The proof is constructive and provides an estimation for the size of the loads.
Cristinel Mardare

### Chapter 12. Calculating the Bending Moduli of the Canham–Helfrich Free-Energy Density

Abstract
The Canham–Helfrich free-energy density for a lipid bilayer involves the mean and Gaussian curvatures of the midsurface of the bilayer. The splay and saddle-splay moduli $$\kappa$$ and $$\bar{\kappa }$$ regulate the sensitivity of the free-energy density to changes in the values of these curvatures. Seguin and Fried derived the Canham–Helfrich energy by taking into account the interactions between the molecules comprising the bilayer, giving rise to integral representations for the moduli in terms of the interaction potential. In the present work, two potentials are chosen and the integrals are evaluated to yield expressions for the moduli, which are found to depend on parameters associated with each potential. These results are compared with values of the moduli found in the current literature.
Brian Seguin, Eliot Fried

### Chapter 13. Elasticity of Twist-Bend Nematic Phases

Abstract
The ground state of twist-bend nematic liquid crystals is a heliconical molecular arrangement in which the nematic director precesses uniformly about an axis, making a fixed angle with it. Both precession senses are allowed in the ground state of these phases. When one of the two helicities is prescribed, a single helical nematic phase emerges. A quadratic elastic theory is proposed here for each of these phases which features the same elastic constants as the classical theory of the nematic phase, requiring all of them to be positive. To describe the helix axis, it introduces an extra director field which becomes redundant for ordinary nematics. Putting together helical nematics with opposite helicities, we reconstruct a twist-bend nematic, for which the quadratic elastic energies of the two helical variants are combined in a non-convex energy.
Epifanio G. Virga

### Backmatter

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