The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...

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The Periodic Unfolding Method in Homogenization

The Physics of Rubber Elasticity (J. Meixner)

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with non homogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding.

/pdf/the-periodic-unfolding-method-in-domains-with-holes-45n90n5w50.pdf

The periodic unfolding method in domains with holes

We propose an approximation scheme for a variational theory of brittle fracture. In this scheme, the energy functional is approximated by a family of functionals depending on a small parameter and on two fields: the displacement field and an eigendeformation field that describes the fractures that occur in the body. Specifically, the eigendeformations allow the displacement field to develop jumps that cost no local elastic energy. However, this local relaxation requires the expenditure of a certain amount of fracture energy. We provide a construction, based on the con- sideration of e-neighborhoods of the support of the eigendeformation field, for calculating the right amount of fracture energy associated with the eigendeformation field. We prove the Γ-convergence of the eigendeformation functional sequence, and of finite element approximations of the eigende- formation functionals, to the Griffith-type energy functional introduced in Francfort and Marigo (J. Mech. Phys. Solids, 46 (1998), pp. 1319-1342). This type of convergence ensures the convergence of eigendeformation solutions, and of finite element approximations thereof, to brittle-fracture solu- tions. Numerical examples concerned with quasi-static mixed-mode crack propagation illustrate the versatility and robustness of the approach and its ability to predict crack-growth patterns in brittle solids.

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Eigenfracture: An Eigendeformation Approach to Variational Fracture

/pdf/junction-of-a-periodic-family-of-elastic-rods-with-a-3d-53gqmi18zd.pdf

Junction of a periodic family of elastic rods with a 3d plate. Part I

We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of whose boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities whose size depends on a small parameter, $ \varepsilon >0.$ The assumption of sharp asperities is made; that is, the height of the asperities is fixed. Using a boundary layer corrector, we derive and analyze a nonoscillating approximation of the solution at order ${\cal O}(\varepsilon^{3/2})$ for the H1 -norm.

Asymptotic Approximation of the Solution of the Laplace Equation in a Domain with Highly Oscillating Boundary

We investigate the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains $\Omega_\varepsilon^+\cup\Omega^-_\varepsil...

/pdf/boundary-homogenization-and-reduction-of-dimension-in-a-44bu7tzntm.pdf

Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary

We investigate the asymptotic behaviour, as e → 0, of a class of monotone nonlinear Neumann problems, with growth p -1 (p ∈]1, +∞[), on a bounded multidomain (N ≥ 2). The multidomain ΩE is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hE in the x N direction, as e → 0. The second one is a “forest" of cylinders distributed with e -periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small cross section of size e and fixed height (for the case N=3 , see the figure). We identify the limit problem, under the assumption: . After rescaling the equation, with respect to hE , on the plate, we prove that, in the limit domain corresponding to the “forest" of cylinders, the limit problem identifies with a diffusion operator with respect to x N , coupled with an algebraic system. Moreover, the limit solution is independent of x N in the rescaled plate and meets a Dirichlet transmission condition between the limit domain of the “forest" of cylinders and the upper boundary of the plate.

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Antonio Gaudiello

Papers

Junction of a periodic family of elastic rods with a 3d plate. Part I

Asymptotic Approximation of the Solution of the Laplace Equation in a Domain with Highly Oscillating Boundary

Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary

Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem