Joseph J. Roseman

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.

Integral estimates for the displacement and strain energy in nonlinear elasticity in terms of the body force

Abstract The boundary value problem of place in nonlinear hyperelastostatics is considered. Integral estimates for the displacement and the strain energy are obtained for the weak (or classical) solution in terms of the given body force. These estimates then imply continuous dependence of the solution on the given data in the appropriate norm.

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.

The effect of nonlinear transformations on the computation of weak solutions

For a nonlinear hyperbolic system, computational methods yield different weak solutions for different forms of the system. An explanation is given of the numerical mechanism by which a scheme selects a particular weak solution and why this mechanism depends not only on the scheme but also on the form of the equations. For the Lax-Friedrichs and Lax-Wendroff schemes, it is shown how a correction term can be added to a transformed system so as to preserve the weak solution. This analysis is illustrated by numerical shock-like solutions of the equations of shallow fluid flow over a ridge.

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

An integral bound for the strain energy in nonlinear elasticity in terms of the boundary displacements

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}},

An integral bound on the strain energy for the traction problem in non-linear elasticity with sufficiently small strains

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.