Errol Montes-Pizarro

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter 0$" align="middle" border="0">. These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small . We used these expressions to show that in the limit as the characteristic equation for 0$" align="middle" border="0"> converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for . When the base material () is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with 0$" align="middle" border="0"> and small. Our numerical results in this case indicate that the number of bifurcation points is finite when 0$" align="middle" border="0"> and that this number monotonically increases as . For the problem with 0$" align="middle" border="0"> we get conditions for the existence of local branches of non-trivial solutions.

We consider the nonlinear boundary value problem of specifying the displacement of the lateral surface of a cylindrical body subject to zero normal stresses on the top and the bottom and sliding conditions (i.e. no tangential components of the surface traction) at the lateral surface. We restrict our analysis to study the existence of axisymmetric deformations assuming that the material

Abstract. The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic partial differential operator and the principal part of the corresponding boundary operators. When the CC holds the linear boundary value problem has many important functional analytic properties. Recently it has been found that the CC plays a very important role in the construction of a generalized degree, with all the properties of the Leray-Schauder degree, applicable to a general class of problems in nonlinear elasticity. In this paper we discuss the role of the CC in such development and present some examples of boundary value problems in nonlinear elasticity to which this new degree is applicable. In addition, when the CC fails we present some recent results on the implications of this failure in the context of nonlinear elasticity.

From Vodou to Zonk: A Bibliographic Guide to Music of the French-Speaking Caribbean and its Diaspora. By John Gray, foreword by Julian Gerstein. Nyack, NY: African Diaspora Press, 2010. ISBN 978-0-9844134-0-9. 237 pp. $79.95 cloth. This book provides an annotated bibliography on musical traditions of the French and Creole speaking Caribbean islands and of some of their diasporic connections with musical traditions of other Caribbean islands and immigrant communities in the United States, France, and Canada. It contains around 1,140 citations on works dating from 1698 to 2008, with the majority of them published from 1930 to 2008. Most of the works cited are in English, French, and Spanish, but there are also some in Creole. There are citations of books, anthology chapters, PhD dissertations, periodical and newspapers articles, films, videos, field recordings, and digital resources. The material is organized in four sections: "Cultural History and the Arts," "Festivals...

The complementing condition (CC) is an algebraic compat-ibility requirement between the principal part of a linear elliptic partial differential operator and the principal part of the corresponding bound-ary operators. When the CC holds the linear boundary value problem has many important functional analytic properties. Recently it has been found that the CC plays a very important role in the construction of a generalized degree, with all the properties of the Leray–Schauder de-gree, applicable to a general class of problems in nonlinear elasticity. In this paper we discuss the role of the CC in such development and present some examples of boundary value problems in nonlinear elastic-ity to which this new degree is applicable. In addition, when the CC fails we present some recent results on the implications of this failure in the context of nonlinear elasticity.

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics — the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics - the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter $\varepsilon>0$ . These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small $\varepsilon$ . We used these expressions to show that in the limit as $\varepsilon \to 0$ the characteristic equation for $\varepsilon>0$ converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for $\varepsilon=0$ . When the base material ( $\varepsilon=0$ ) is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with $\varepsilon>0$ and small. Our numerical results in this case indicate that the number of bifurcation points is finite when $\varepsilon>0$ and that this number monotonically increases as $\varepsilon \to 0$ . For the problem with $\varepsilon>0$ we get conditions for the existence of local branches of non-trivial solutions.

The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic differential operator and the principal part of the corresponding boundary operators. We study the implications of failure of the CC in the context of nonlinear elasticity. In particular we show that for axisymmetric deformations of cylinders and for any homogeneous isotropic material, failure of the CC is equivalent to the existence of sequences of possible bifurcation points accumulating at the point where the CC fails. For non axisymmetric deformations and for Hadamard–Green type materials, we show for axial compressions of the cylinder that the CC fails on a full interval of values of the loading parameter, and for the lateral compression problem it fails at least once.

We consider the nonlinear boundary value problem of specifying the displacement of the lateral surface of a cylindrical body subject to zero normal stresses on the top and the bottom and sliding conditions (i.e. no tangential components of the surface traction) at the lateral surface. We restrict our analysis to study the existence of axisymmetric deformations assuming that the material of the body is homogeneous, isotropic and hyperelastic. We study the linearization of the nonlinear equations about a trivial solution and show that smooth solutions of the linear problem must be separable. We classify the nontrivial axisymmetric solutions of the linearized problem in two types that we call buckling and barrelling like solutions. We characterize the eigenvalues for both solutions types as well as those displacements of the lateral surface at which the complementing condition for the linearized equations fails to be satisfied. For a class of Blatz–Ko type materials we give a complete characterization of the existence, multiplicity and disposition of the corresponding eigenvalues. We show, for such material, that the eigenvalues of buckling and barrelling types are simple, and that they form monotone sequences (decreasing for the former and increasing for the latter) both of which converge to a value at which the complementing condition fails. Moreover, it is shown that the cylinder looses stability first to buckling rather than to barrelling.

Resumen en español

En los discursos de formación de una identidad nacional tanto en Puerto Rico como en Colombia, así como en muchos otros países de las Américas y del Caribe, ha sido crucial una narrativa de mezcla racial y cultural llamada mestizaje. Sin embargo, en esas narrativas no se le adjudica la misma valoración a todos los elementos de la supuesta “mezcla de razas”. En la mayoría de los casos la herencia cultural afrodescendiente se desvaloriza promulgando un ideal que muchos(as) estudiosos(as) han llamado “de blanqueamiento”. En Puerto Rico, la historia de la música jíbara se ha construido desde esa óptica como una tradición descendiente casi exclusivamente de tradiciones hispanas y sus elementos afrodescendientes, con contadas excepciones, se han obviado o minimizado sistemáticamente. Por su parte en Colombia, los habitantes afrodescendientes de la costa caribeña del país han luchado historicamente por tener reconocimiento en un discurso que construye la cultura nacional como una mezcla racial que valoriza lo “blanco” sobre la “negritud” o lo “europeo” sobre lo indígena y lo afrodescendiente. A partir principalmente de los años sesenta del siglo pasado diversos géneros musicales de las Antillas no hispanoparlantes y del continente africano comenzaron a ganar popularidad entre la población de la costa caribeña de Colombia. Simultáneamente, la música jíbara de Puerto Rico comenzó a popularizarse en esa región de Colombia al extremo de que canciones intepretadas por músicos puertorriqueños como Odilio González, entre otros, se han convertido en himnos no oficiales de ciudades como Baranquilla y Cartagena de Indías. En este contexto colombiano la música jíbara es considerada música afrocaribeña en claro contraste con la interpretación mayoritaria que sobre esa misma música prevalece en Puerto Rico. En esta artículo se analiza cómo dos ideologías nacionales semejantes pueden, paradójicamente, promover interpretaciones opuestas de una misma tradición musical.