Saint-Venant Problem Research Articles

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.

Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.

The Saint-Venant problem of the bending of a cylinder with helical anisotropy

Abstract The Saint-Venant problems of pure bending and the bending of a cylinder with helical anisotropy by a transverse force are reduced to boundary-value problems for systems of ordinary differential equations with variable coefficients. The problems are solved by two methods – the small-parameter method and numerical methods. The behaviour of the stiffnesses and the stress-strain state is investigated as a function of the parameters of the problem.

On the deformation of inhomogeneous orthotropic elastic cylinders

This paper is concerned with the deformation of inhomogeneous orthotropic cylinders. The work is motivated by the recent interest on the functionally graded materials. We assume that the elastic coefficients are independent of the axial coordinate. First, we study the Saint-Venant problem. Then we consider the case when the cylinder is subjected to body forces and surfaces tractions on the lateral surface. The results are applied to study the deformation of a circular cylinder with a given inhomogeneity.

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.

On solution strategies to Saint-Venant problem

Different solution strategies to the relaxed Saint-Venant problem are presented and comparatively discussed from a mechanical and computational point of view. Three approaches are considered; namely, the displacement approach, the mixed approach, and the modified potential stress approach. The different solution strategies lead to the formulation of two-dimensional Neumann and Dirichlet boundary-value problems. Several solution strategies are discussed in general, namely, the series approach, the reformulation of the boundary-value problems for the Laplace's equations as integral boundary equations, and the finite-element approach. In particular, the signatures of the finite-element weak solutions—the computational costs, the convergence, the accuracy—are discussed considering elastic cylinders whose cross sections are represented by piece-wise smooth domains.

The non-linear Saint-Venant problem of the torsion, stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.

Saint venant problem for a naturally twisted rod in nonlinear moment elasticity theory

Saint venant problem for a naturally twisted rod in nonlinear moment elasticity theory

Minimum Energy Characterizations for the Solution of Saint-Venant's Problem in the Theory of Shells

The linear theory of Cosserat surfaces is employed to study Saint-Venant's problem for cylindrical shells of arbitrary cross-section. We prove minimum energy characterizations for the solution of the relaxed Saint-Venant's problem previously obtained. Then, we determine the global measures of strain appropriate to extension, bending, torsion and flexure for certain classes of solutions to the relaxed problem.

Torsion and flexure analysis of orthotropic beams by a boundary element model

This paper deals with a boundary element model, realised for analysing the coupled torsion and flexure Saint-Venant problem of orthotropic beams having a polygonal cross-section. A quadratic B-spline approximation represents the boundary variables. All the integrals are transferred onto the boundary and analytically computed using hierarchically structured formulae, suitable for computer implementation. The model furnishes the complete shear stress field, through the evaluation of section properties and the shear centre location. Numerical results allow an analysis of the performances of the model.

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.

State space elastostatics of prismatic structures

This paper provides an exposition of the problem of a prismatic elastic rod or beam subject to static end loading only, using a state space formulation of the linear theory of elasticity. The approach, which employs the machinery of eigenanalysis, provides a logical and complete resolution of the transmission (Saint-Venant's) problem for arbitrary cross-section, subject to determination of the Saint-Venant torsion and flexure functions which are cross-section specific. For the decay problem (Saint-Venant's principle), the approach is applied to the plane stress elastic strip, but in the transverse rather than the axial direction, leading to the well-known Papkovitch–Fadle eigenequations, which determine the decay rates of self-equilibrated loading; however, extension to other cross-sections appears unlikely. It is shown that only a repeating zero eigenvalue can lead to a non-trivial Jordan block; thus degenerate decay modes cannot exist for a prismatic structure.

A formula for the generalized twist

In this paper, a new formula for the generalized twist is presented which is valid for linearly elastic, nonhomogeneous and anisotropic beams of solid cross section. The generalized twist is expressed in terms of axial component of the infinitesimal rotation vector weighted by the stress function of Saint-Venant's torsional problem. Characterization of a torsion-free bending problem is also presented by the use of the generalized twist.

Symplectic solution system for reissner plate bending

Based on the Hellinger-Reissner variatonal principle for Reissner plate bending and introducing dual variables, Hamiltonian dual equations for Reissner plate bending were presented. Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem, and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized. So in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction-vector expansion. All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail, and their physical meanings are showed clearly. The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed. It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint-Venant problem and they form a perfect symplectic subspace for zero eigenvalue. And the eigensolutions for nonzero eigenvalue are covered by the Saint-Venant theorem. The symplectic solution method is not the same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution. The symplectic solution method will have vast application.

Solutions of the Saint Venant problem for a cylinder with helical anisotropy

Solutions of the Saint-Venant problem for a cylinder with helical anisotropy are presented in the form of a linear combination of elementary homogeneous solutions, the construction of which is reduced to boundary-value problems for ordinary differential equations with variable coefficients. These problems are integrated using analytical and numerical methods, and the elements of the stiffness matrix are investigated over a wide range of parameters. It is established that when the cylinder is stretched the sign and value of the torsional deformation depends considerably on the value of the relative angle of twist of the helices.

Elastic equations for a cylindrical section of a tree

Considering a cylindrical section of a tree subjected to loads independent of x 3 as a relaxed Saint-Venant's problem, it was shown that plane sections remain plane. Since plane sections remain plane, the displacement equations for the neutral fiber derived using either the relaxed Saint-Venant's problem or elementary beam theory are equivalent. The stresses in the plane of the transverse cross-section were found to equal to zero. Therefore, it is appropriate to use elementary beam theory to estimate the three-dimensional stress functions when the wood is considered to be homogeneous. In addition the three-dimensional displacement equations allow the required elastic coefficients in cylindrical coordinates to be measured from full size samples.

Stress functions for a heterogeneous section of a tree

Consider a tree that is composed of wood, which is orthotropic with respect to the cylindrical coordinate axes, where the z-axis is directed up the center of the tree. When a section of a tree is considered as a Relaxed Saint-Venant's Problem the stresses in the plane of a transverse cross section will equal zero. By allowing some of, or all of the compliance coefficients to be functions of the radial coordinate r, the structure of the stress functions can be fundamentally altered. If the compliance coefficient in the z-direction ( S 3333) is allowed to remain constant, the S rz and S θz shear stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients, while the other stresses will only be functions of the coordinates, dimensions and applied loads. If S 3333 is also allowed to be a function of r, then all the nonzero stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients.

Considering heterogeneity in a cylindrical section of a tree

Published values for the elastic coefficients of wood indicate that this material may be considered orthotropic with respect to the cylindrical coordinates. This indicates that simplifying the elasticity tensor to allow for the non-unique strains at r=0, is a simplification that may ignore important structural characteristics of a tree. The constitutive equations for a cylindrical section of a tree were posed in cylindrical coordinates as a linear function of the radial coordinate r. The constitutive equations were transformed to a Cartesian basis so that a solution to Saint-Venant's Problem, proposed by Iesan (Lecture Notes in Mathematics (1987) 161) could be employed for a cylindrical section of a tree. From Iesan's solution it was possible to determine that the auxiliary generalized plane strain stresses can only be a function of r, and that the total stresses (in cylindrical coordinates) in the plane of the transverse cross-section must be equal to zero.

Saint-Venant problems for a bar with a screw anisotropy

Saint-Venant problems for a bar with a screw anisotropy

Torsion of compound cross-sections with imperfect interface

The Saint-Venant torsion problem of compound sections with imperfect interfaces is studied. Two kinds of an imperfect interface are considered: an imperfect interface which models a thin interphase of low shear modulus and an interface which models a thin interphase of high shear modulus. At the former kind, the tractions are continuous but the warping displacement undergoes a discontinuity; at the latter kind the warping displacement is continuous but the shear traction undergoes a discontinuity. These imperfect interface conditions have been derived in a companion study [1]. The present paper is concerned with deriving benchmark solutions for the Saint-Venant torsion problem of compound sections with imperfect interfaces. Specifically, analytical solutions are given for a) a two-phase rectangular section, b) a two-phase section in the shape of a circular sector with an imperfect interface located along a circular arc, c) a two-phase circular sector with an imperfect interface along a radial line. The effect of imperfect bonding on the torsional rigidity of the compound bar is examined.