Normal Surface Tractions Research Articles

Nonhomogeneous nonsymmetrical plane problems with several griffith cracks one or two partially closed

The objective of this paper is to modify and generalize the results of a previous study carried out by the author for a single Griffith crack in a homogeneous infinite plane under normal surface tractions [1], to cases in which a nonhomogeneous nonsymmetrical plane problem of several cracks with partial closures, placed and loaded in an arbitrary manner, is considered. Several special examples are treated in detail and the numerical results are plotted in the figures.

On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function—II. The analysis, solution and conclusions

The articulating process of synovial joints is represented by a spatially fixed cyclically time varying normal surface traction applied over a layered model of the cartilage-subchondral bone system. A two-phase mechanically interacting mixture is used to represent the solid matrix and the interstitial fluid of the cartilage continuum. The resulting equations of motion may be reduced to the currently employed physical laws used to describe the transport of fluid through the tissue, i.e. Darcy's law and Biot's consolidation equations. These equations were simplified by an order of magnitude analysis, and the resulting equations were solved by a double Fourier-Laplace transform procedure. The analytical solution shows that the fluid transport mechanism is strongly dependent upon a nondimensional parameter defined by ϵ 12 = Nk h 2 ω . This parameter is the ratio of the force required to deform the tissue as a whole to the force of frictional resistance due to the rate of movement of the interstitial fluid relative to the solid matrix. It is found that during normal function of healthy articular cartilage the consolidation effects will dominate the movement of interstitial fluid. In degenerative cartilage, as characterized by increasing the surface porosity and permeability and a decrease in tissue stiffness, the direct pressure effects, i.e. Darcy's law, becomes comparable to those of consolidation.

Mechanics of bilayer membranes

The general mechanics of elastic deformations in bilayer membranes is developed, resulting in three elastic constants characterizing a bilayer surface. The theoretical analysis is applied to bilayer systems containing organic solvents (referred to as “solvent bilayers”) and to planar bilayer membranes free of solvent (referred to as “nonsolvent bilayers”). The relationship between a normal surface traction (e.g. electrocompression resulting from transmembrane potential difference) and change in membrane thickness is investigated for the two types of bilayers.

The effect of surface finish on the creep and wear of a rolling ball subjected to normal and tangential surface tractions

Experimental results have been obtained for the creep of a relatively smooth steel ball rolling along planes of various surface finish. The ball was subjected to combined normal and tangential surface tractions over a wide range of loading. It is found that the measured creep is only marginally dependent on the degree of surface roughness. The coefficient of sliding friction derived from the rolling tests, and also as measured by independent sliding tests, shows no significant variation with surface texture. These results are considered in relation to other workers'evidence relating to the effects of surface finish and it is concluded that theoretical predictions of these phenomena based on the assumption of ideal smooth surfaces may generally be used for real engineering surfaces. It is suggested that this assumption may be invalid at small normal loads. The wear of a rolling ball subjected to these loading conditions was also studied by using radioactive tracer techniques. These results again show virtually no dependence on the surface texture except in such cases as when the surface is modified by the removal of the hard amorphous surface layer created by the machining process.

Thermoelastic stresses and deformations in reactor fuel plates

The thermoelastic stresses and displacements in a reactor fuel plate are determined for arbitrary temperature distributions and normal surface tractions which do not vary across the width of the plate and are symmetrical about the mid-plane of the thickness. It is assumed that cross sections perpendicular to the width dimension are in a state of plane strain. The exact solution involving no thin plate approximations is obtained in part A through the use of the Papkovich stress functions and Fourier series methods. The determination of the coefficients of the resulting series for the stresses and displacements requires the numerical solution of infinite sets of simultaneous linear algebraic equations. In part B are derived two approximate solutions based on small thickness-to-length ratios; quantitative estimates of the errors involved in these approximate solutions are obtained by comparison with the exact solution of part A.

The contact problem in the linear theory of viscoelasticity

Expressions are given for the displacement stress field produced at any point of a linear homogeneous and isotropic viscoelastic half space by an arbitrary time dependent distribution of pressure acting on its boundary. The distribution of normal surface traction prevailing when a rigid indentor of arbitrary profile is pressed against the surface of a viscoelastic half space is determined in terms of a one parameter family of solutions to the corresponding elastic problem. This viscoelastic solution which is valid if the contact area possesses a single maximum is applied to the particular cases where the indentor is a sphere and a right circular cone. Finally Hertz's theory of the contact of elastic bodies is generalised to viscoelasticity theory and the equations governing the viscoelastic counterpart of Hertz's theory of elastic impact are given.

A note on the torsion of an incompressible highly-elastic cylinder

In a previous paper (1), it has been shown that a state of simple torsion in a cylinder of incompressible highly elastic material, isotropic in its un-deformed state, may be produced by the application of surface tractions to the plane ends of the cylinder. In the deformation, each circular section of the cylinder normal to its axis is rotated, in its own plane, through an angle ψz, where z is the distance of the plane from one end of the cylinder. W is the stored-energy function for the material, given in terms of two strain invariants I 1 and I 2. The system of surface tractions which must be applied to each plane end of the cylinder in order to maintain the state of deformation is (i) a tangential surface traction Θ v , azimuthal in the deformed state, and (ii) a normal surface traction Z v , taken positive when in the direction of the outward drawn normal to the surface.

Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates

Expressions for the components of strain and the incompressibility condition, for large deformations, are obtained in a cylindrical polar co-ordinate system. The stress-strain relations, equations of motion and boundary conditions for an incompressible, neo-Hookean material, in such a coordinate system, are also obtained and specialized to the case of cylindrical symmetry. These results are applied to the special cases of the simple torsion of a solid cylinder and of a hollow, cylindrical tube and to their combined simple extension and simple torsion. In the case of a solid cylinder, it is found that a state of simple torsion can be maintained by surface tractions applied to the ends of the cylinder only, and these consist of a torsional couple together with a compressive force. The necessary torsional couple is proportional to the amount of torsion and the compressive force to the square of the torsion. In the case of a hollow, cylindrical tube, it is again necessary to exert a torsional couple, proportional to the torsion, and a compressive force, proportional to the square of the torsion, on the plane ends, but it is also necessary to exert a normal surface traction, acting in a positive radial direction, on one or other of the curved surfaces of the tube and proportional to the square of the torsion.

Torsion of a rubber cylinder

It has been predicted theoretically that, in general, a right‐circular cylinder of incompressible, highly elastic material, which is isotropic in its undeformed state, cannot be held in a state of pure torsional deformation by means of a torsional couple alone. In addition, normal surface tractions must be exerted over the plane ends of the cylinder. These normal surface tractions depend on the amount of torsion and on position on the plane ends of the cylinder. Experiments are reported here in which this phenomenon is observed in a right‐circular cylinder of pure gum compound. The dependence of the surface traction on amount of torsion and its distribution over the surface of the cylinder is studied by measuring the bulging of the rubber into small holes in a metal plate on one end of the cylinder.

Torsion of a Rubber Cylinder

It has been predicted theoretically that, in general, a right-circular cylinder of incompressible, highly elastic material, which is isotropic in its undeformed state, cannot be held in a state of pure torsional deformation by means of a torsional couple alone. In addition, normal surface tractions must be exerted over the plane ends of the cylinder. These normal surface tractions depend on the amount of torsion and on position on the plane ends of the cylinder. Experiments are reported here in which this phenomenon is observed in a right-circular cylinder of pure gum compound. The dependence of the surface traction on amount of torsion and its distribution over the surface of the cylinder is studied by measuring the bulging of the rubber into small holes in a metal plate on one end of the cylinder.