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We derive, via simultaneous homogenization and dimension reduction, the Gamma-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates.

 This book is based on acourse of lectures given by the author in 1947 in the Mathematical-Mechantical Department of the Leningrad National University. It is devoted to the exposition of the theory of elasticity without any assumptions restricting the magnitude of elongations, displacements, or angles of rotation. It also examines, in a general formulation, the connection between stresses and strains in an isotropic elastic body. The equations of the classical (linear) theory of elasticity can be obtained from the equations of the general theory which is to be presented, by assuming that   a) the elongations and shears are negligibly small compared to unity;   b) the squares and products of the angles of rotation are negligibly small compared to the elongations and shears;   c) the connection between the stresses and strains is expressible by Hooke’s law.   The nonlinear theory of elasticity, being an essential generalization of the classical theory, permits an approach to the solution of a series of important problems which do not arise in the latter theory because of its limitations. Such problems are, in particular:   1. The stability of elastic equilibrium.   2. The deformation of bodies having initial stresses.   3. The large deflection of rods.   4. The problems of torsion and bending complicated by the presence of axial forces.   5. The bending of plates and shells under deflections of the order of magnitude of the thickness.   6. The deformation of elastic bodies which do not obey Hooke’s law.   Finally, it has been shown recently that the equilibrium of elastic -p1astic bodies (with certain restrictions) can be examined on the basis of general principles of the theory of elasticity. It is due to this fact, that the problem of the equilibrium of an elastic-plastic medium is, to a certain extent, included in the circle of problems belonging to the nonlinear theory of elasticity . The problems listed above are very much in the forefront, and that is why ever increasing attention is being devoted to the nonlinear theory of elasticity by the scientists in the Soviet Union and other countries (see the Bibliography). To make the book accessible to as wide a circle of readers as possible, the author has attempted to carry out all deductions in the simplest and most intuitive manner, avoiding, in particular, tensor calculus and the complicated symbolism connected with it (or, to speak more precisely, applying only the small amount of it which is given in books on the classical theory of elasticity). In conclusion the author wishes to thank Prof. A.I. Lurye , L.M. Kachanov , Docent at Leningrad National University, and A.I. Chekmarev, the editor of this book, for some valuable critical remarks which they made when they read the manuscript. 

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.

We consider vectorial variational problems in nonlinear elasticity of the form $I[u]=\int W(Du)dx$, where $W$ is continuous on matrices with positive determinant and diverges to infinity long sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional $\int W^{qc}(Du)dx$ is an upper bound on the relaxation of $I$, and coincides with the relaxation if the quasiconvex envelope $W^{qc}$ of $W$ is polyconvex and has $p$-growth from below with $p\ge n$. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending-torsion theory for rods are deduced.

We study the problem of minimizing the functional $$ I(\varphi)=\int\limits_{\Omega} W(x,D\varphi)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $\varphi\in W^1_n(\Omega)$ and growth conditions on the integrand $W(x,F)$. To compensate for that, we impose the finite distortion condition and the condition $\frac{|D\varphi(x)|^n}{J(x,\varphi)} \leq M(x) \in L_{s}(\Omega)$, $s>n-1$, on the characteristic of distortion. On assuming that the integrand $W(x,F)$ is polyconvex and coercive, we obtain an~existence theorem for the problem of minimizing the functional $I(\varphi)$ on a new family of admissible deformations. KEYWORDS: functional minimization problem, nonlinear elasticity, mapping with finite distortion, polyconvexity.

A micromechanical model for the analysis of particulate mechanical behavior is presented. Nonlinear effects are introduced in the model by a nonlinear elastic description of the matrix and through a modulus degradation routine. The first part of the study uses the experimental data from a range of glass bead/HTPB composites to back calculate the model parameters. The results showed that the model gave a good representation of the processes believed to control mechanical behavior. These processes Include partial particle deboning and progressive debonding from the largest to smallest particles throughout the strain history. The second part of the study examines the sensitivity of the model results to small changes In the adjustable input parameters. The residual bond in a debonded particle was found to have a dominating effect on the calculated results. Based on the sensitivity results, best guess? Interaction and debonding parameters were selected to examine the predictive capability of the model. For glass bead!HTPB composites, the predicted composite stresses were within 1 0% of the experimental data. Dilatation was usually over-predicted. For glass bead/polyethylene and glass bead/polyurethane data found in the literature, predicted composite stresses were within 15% to 24%, respectively. The results showed that the model was capable of predicting the mechanical behavior of composites comprised of glass beads in HTPB, PU or HOPE matrices as long as characteristic adhesive parameters were available for each system.

1Jabberwocky of Authors

By Harry Persons Taber

LibriVox volunteers offer you 12 different recordings of <i>The Jabberwocky of Authors</i> by Harry Persons Taber. This parody of Carroll's <i>Jabberwocky</i> consists almost entirely of authors' names. See how many you can spot! <br /><br />This was the weekly poetry project for the week of April 4th, 2010.