Bending, extension, and torsion of naturally twisted rods

Saint-Venant /1/ established that the spatial problem of linear elasticity theory of the deformation of straight rods with a load-free side surface allows of practically complete investigation: the extension problem is solved exactly (if the boundary layer is ignored), and the bending and torsion problems reduce to Neumann problems for the Laplace equation in the region of the rod cross-section (see /2, 3/). It is shown below that an analogous situation holds for a naturally twisted rod: the spatial problem is successfully reduced to a Neumann-type problem for a certain system of second-order elliptic equations in the cross-section. It is essential that this can be done for an arbitrary value of the rod twist. For zero twist the problem in the section reduces to the Saint-Venant problem. In the case of centrally-symmetric sections, the problem decomposes into two independent problems, on bending and on extensiontorsion. Variational principles and certain bilateral estimates of the extension and torsion stiffness are constructed for the latter, and the case of oblong sections is investigated.

The extension-torsion problem for naturally twisted rods was examined earlier in /4/. The difference from this research is discussed in Sect.4.     

Neutral stability of compression solitons in the bending of a non-linear elastic rod

The spectral stability of compression solitons in non-linear elastic rods with respect to perturbations of the flexural mode of the oscillations of the rod is investigated. The system of equations of the isotropic theory of elasticity, taking account of the non-linear corrections corresponding to the interaction being studied, is used to describe the interaction of longitudinal and flexural waves in the rod. This system of equations describes long longitudinal-flexural waves of small but finite amplitude. It is shown that trapped flexural modes exist, which propagate together with a compression soliton. It is established that these modes, which are the least stable, do not increase with time.     

The complex three-dimensional elastoplastic bending of a rod with a square cross-section

The complex three-dimensional bending of a long rod (with square cross-section) made of an isotropic ideally plastic and ideally cyclical material is investigated. The bending of the rod occurs due to the action of two moments, applied to its ends in such a way that the longitudinal deformation of the middle fibres of two neighbouring sides is described by a dashed line.     

The Perron method and the non-linear Plateau problem

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in ${\mathbb{R}^{n+1}}$ .     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.     

Determining knots by optimizing the bending and stretching energies

For a given set of data points in the plane, a new method is presented for computing a parameter value (knot) for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.     

The singular Cauchy problem for a non-linear hyperbolic equation

Summary The author demonstrates the existence of a smooth solution to a singular initial value problem for a quasiliuear hyperbolic equation in two independent variables. The problem is transformed into an equivalent system of integral equations for which a solution is obtained by invoking Schauder’s fixed point theorem. Entrata in Redazione il 19 ottobre 1970.     

The contact problem for a geometrically non-linear Timoshenko-type shell

An algorithm is developed for the numerical solution of the contact problem of an elastic Timoshenko-type shell subjected to arbitrarily large displacements and rotations, using mixed finite-element approximations. It is essential that six displacements of the faces of the shell are chosen as the required functions. This enables one, first, to simplify the formulation of contact problems in the mechanics of thin-walled structures, since functions by means of which the conditions for the non-penetration of the bodies are formulated are chosen as the required functions and, second, to obtain relations for the components of the Green-Lagrange strain tensor in curvilinear, orthogonal coordinates which accurately represent arbitrarily large displacements of a shell as a rigid body.     

The Dirichlet divisor problem,traces and determinants for complex powers of the twisted bi-Laplacian

Estimating the counting function for the eigenvalues of the twisted bi-Laplacian leads to the Dirichlet divisor problem, which is then used to compute the trace of the heat semigroup and the Dixmier trace of the inverse of the twisted bi-Laplacian. The zeta function regularizations of the traces and determinants of complex powers of the twisted bi-Laplacian are computed. A formula for the zeta function regularizations of determinants of heat semigroups of complex powers of the twisted bi-Laplacian is given.     

The dam problem for non-linear Darcy's laws and non-linear leaky boundary conditions

The goal of this paper is to establish that even for strongly non-linear Darcy's laws and strongly non-linear leaky boundary conditions on the bottom of the reservoirs the fluid flow through a porous dam is a well-posed problem for which existence of a solution can be established. In a simple case we provide also the exact solution to the problem. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The non-linear oscillations of a rod with close values of the cross-section axial moments of inertia

An analytical solution of the problem of the forced flexural oscillations of a rod with fixed hinged supports is presented. The rod has close natural frequencies of flexural oscillations in two mutually perpendicular planes due to the close values of the principal axial moments of inertia of the cross-section. The geometrical non-linearity, due to the change in the length of the middle line of the rod when it undergoes three-dimensional motion, is taken into account. The oscillations of the rod in the neighbourhood of the principal and first superharmonic resonances are investigated.     

On subsolutions of a non-linear diffusion problem

In the paper a subsolution of a non-linear diffusion problem in the radial case is constructed. Some integral equation methods are used.