Shlomo Breuer

On Saint-Venant's Principle in three-dimensional nonlinear elasticity

Consider a long elastic isotropic beam with a convex cross-section and a sufficiently smooth boundary. Suppose that a self-equilibrated load is applied at each end but the sides are stress-free and there are no internal body forces. It is proved in the context of three-dimensional, nonlinear elastostatics that if the first four derivatives of the displacement vector are a priori assumed to be everywhere sufficiently small with respect to the physical constants and the geometry of the cross-section, then the strains at any point decay exponentially with the distance of the point from the nearest end.This result is an extension of known results on Saint-Venants Principle in linear and two-dimensional nonlinear elasticity.

Phragmen-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder

Abstract Classes of nonlinear elliptic equations in a long circular cylinder of radius one are considered. The equations are of the form ▽2u = S(u, u′)u″ + T(u)u′2, where u = u(x1, x2, x3), and u′, u″ represent general partial derivatives of the indicated order. Homogeneous Dirichlet data are prescribed on the long sides of the cylinder, and throughout the cylinder u is a priori assumed to be sufficiently small while u′ (and, for some classes, also u″) is assumed to be bounded in absolute value by one. With the above assumptions, it is proved that every solution u decays exponentially with distance from the nearer end with a decay constant k which depends on the smoothness properties of S and T but is independent of the length of the cylinder.

The reduction of linear ordinary differential equations to equations with constant coefficients

Abstract Necessary and sufficient conditions on the coefficients of the general linear, homogeneous, n-th order, ordinary differential equation are obtained, so that it can be carried, by a transformation of the independent variable, into an equation with constant coefficients. The transformation is determined and the solutions are given explicitly, in terms of the coefficients. Further application is made in the form of oscillation theorems.

Area approximations in the mathematical laboratory

The article discusses the educational potential and special advantages of a mathematical laboratory in the teaching of mathematics at the pre‐calculus level. The laboratory envisaged is equipped with a set of microcomputers or programmable pocket calculators, which play a central role in the teaching process, with particular emphasis on algorithmization. As a typical laboratory subject we concentrate on the approximation of areas to a prescribed accuracy, including the appropriate error analysis. Different types of approximations, with progressively increasing efficiency, are examined and their laboratory implementation discussed. Extensions and generalizations are indicated for more advanced mathematical education.

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y1 < 2L; 0 < y2 < h] with h/L ≪ 1, is considered. There are no body forces and the long sides are stress free. At y1 = 0 and y1 = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if ταβ(α,β=1,2) represents the Cauchy stress tensor and δαβ the Kronecker delta, then |ταβ−cδα1δβ1| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.

Spatial decay theorems for nonlinear parabolic equations in semi-infinite cylinders

AbstractClasses of nonlinear parabolic equations in a semi-infinite cylinder are considered. The equations are of the form

Function approximations in the mathematical laboratory

u,_{jj} + g_{ij} \left( {x,u,p,\partial ^2 u} \right)u,_i u,_j = c\left( {x,u,p} \right)\frac{{\partial u}}{{\partial t}},

A bound on the strain energy for the traction problem in finite elasticity with localized non-zero surface data

wherep=u,ku,k and∂2u represents a general space derivative of second order. Homogeneous Dirichlet data are prescribed on the lateral sides of the cylinder for all time, along with zero initial data. At any fixed timet, the solution is assumed to be bounded throughout the cylinder, as is the corresponding symmetric matrixgij. Under these assumptions, it is proved that each solution decays pointwise exponentially to zero with distance from the face of the cylinder and the exponential decay rate depends only upon the cross-section of the cylinder, but not upon time or the bounds foru andgij. In addition, if the boundary data on the face of the cylinder satisfy certain mild smoothness conditions, one obtains a decay rate equal to the best possible rate for the Laplace equation.