Mathematics and Mechanics of Solids

The instability of a cylindrical tube of uniform crosssection, made of a homogeneous, isotropic Bell constrained material and subjected to end thrust parallel to its axis, is examined. The theory of small deformations superimposed on an assigned finite homogeneous deformation is applied to derive equations for the stability analysis. The solution presented by Wilkes for the incompressible case is extended to study the system of equations for the infinitesimal superimposed displacements. In contrast to Wilkes, the roots of the characteristic equation are examined for three possible situations in which these roots may be real, pure imaginary, or complex conjugate. The solutions of the system of ordinary differential equations corresponding to these cases are discussed. The buckling equation is thus derived, and conditions imposed by the nature of the characteristic roots are presented.

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming either a plastic potential or dilatant double shearing, a set of three first-order partial differential equations is obtained. These equations turn out to be deceptive, because although they are simple in appearance, the determination of simple exact solutions is nontrivial. For one of the known families of solutions, the authors present a simple and straightforward asymptotic expansion for the stress angle and show that in a restricted range, this expansion provides an accurate solution of the nonlinear ordinary differential equation. In addition, for the same family of solutions, the authors examine a special case that permits one integration and enables the velocity components to be given explicitly in terms of the stress angle. This partial analytical solution may then be coupled either with the asymptotic expansion or with a full numerical solution of the governing first-order differential equation for the stress angle. In addition, the authors show that a special case of the nonlinear ordinary differential equation admits a simple first integral that has not been given previously. Finally, a detailed numerical comparison is made for various parameter values of the asymptotic expansion, the full numerical solution, and with known exact analytical solutions.

It is known that all transverse plane acceleration waves that propagate through a homogeneously deformed Mooney Rivlin material do so without a change in amplitude. Here it is shown that the MooneyRivlin material is the only incompressible isotropic hyperelastic material that permits transverse plane acceleration waves to propagate in any direction without a change in amplitude, regardless of the homogeneous prestrain.

Under a simple nondegeneracy condition, the displacement and edge traction of a planar, isotropic, linearly elastic solid determine its Lame moduli. When these moduli are constant, they can be recovered exactly; this is demonstrated by a specific traction satisfying the nondegeneracy condition. Spatially varying moduli can be computed numerically by considering the equations of linear elasticity as a hyperbolic system for the unknown moduli. A stable finite difference scheme for solving this system is given; synthetic experiments demonstrate its efficacy.

We deal with the flexure (flexion inegale) of a Saint Venant cylinder whose sections we call Bredtlike with variable thickness. We consider a family of sections Dc whose thickness is scaled by a parameter c. This scaling allows for the construction of an soneparameter family of coordinate mappings from a fixedplane domain D onto ZDe. We represent the Helmholtz operator in D) in terms of a fixed system of coordinates in D and represent the shear stress field in what we call the Bredt basis field, which is not the natural basis associated with any coordinate system. Assuming that the shear stress admits a formal C power series expansion, we obtain a hierarchy of perturbation problems for its coefficients, finding the wellknown Jouravski formula at the lowest iterative step and obtaining its generalization at higher stepsthat is, when the section becomes thick. Similar results are obtained for the warping, the resultant shear stress, and the shear shape factors.

This article concerns the construction of a posteriori error estimates for the boundary element methods in plate-bending problems. The proposed approach allows one to assess the size of the error of a numerical solution through an accurate satisfaction of the prescribed boundary conditions. This error estimate analysis can also be used for boundary-type method formulations such as the Trefftz method, the boundary collocation method, and so on.

An analytical solution is obtained in this article for the axisymmetric shrink fit problem with a thin strain hardening hub and an elastic solid shaft. The solution is based on the deformation theory of Hencky, the yield criterion of von Mises, and the assumption of infinitesimal deformations. An elastic power law plastic material model is employed to represent the stressstrain relation of the hub material, with the compressibility of material being included. The solution is derived by using a stress formulation and with the help of a modified Nadai's auxiliary variable method and the extended Michell theorem. All expressions for the stress, strain, and displacement components are derived in explicit forms in terms of an auxiliary variable and four constant parameters that are determined from given boundary conditions by an iterative process. Three specific solutions are presented as limiting cases of the solution. Numerical results are also provided to show quantitatively applications of the solution. This solution, together with the related specific solutions and numerical results, furnishes a new theoretical basis for the mechanical design of shrink-fit sets.

In this article, the authors consider the linearized dynamic theory of a binary mixture consisting of an elastic solid and a gas. First, the continuous dependence of solutions on initial data and body forces is established. Then, a spatial decay estimate of an energetic measure associated to an admissible process is derived.