Effects of a Change of Poisson’s Ratio Analyzed by Twinned Gradients: Illustrated by Simple Solutions of Boussinesq’s Problem of a Normal Force and Cerruti’s Problem of a Tangential Force Acting on the Surface of a Large Solid, and by a Simple Derivation of the Stresses in a Rotating Thick Disk

1940 ◽  
Vol 7 (3) ◽  
pp. A113-A116
Author(s):  
H. M. Westergaard
Keyword(s):  
Poisson’S Ratio ◽  
Normal Force ◽  
Plane Surface ◽  
Tangential Force ◽  
Thick Disk ◽  
Poisson's Ratio ◽  
Total Force ◽  
Principal Stresses ◽  
Simple Derivation ◽  
Boussinesq’S Problem

Abstract Some problems of elasticity have a simple solution for a particular value of Poisson’s ratio. For example, Boussinesq’s problem of a normal force and Cerruti’s problem of a tangential force, acting on the plane surface of a semi-infinite solid, are solved when Poisson’s ratio is 1/2 by referring to Kelvin’s problem of a force at a point in the interior of an infinite solid. For, when Poisson’s ratio is 1/2, the solution of Kelvin’s problem can be stated in terms of one principal stress at each point, acting along the radial line from the point of the load; the other principal stresses are zero; and one half of the total force may be assigned to one half of the infinite solid. For other values of Poisson’s ratio terms must be added in the formulas for the displacements and stresses. The derivations that have been available are somewhat lengthy, especially for Cerruti’s problem. The difficulties are reduced by a simple analytical device, here called “the twinned gradient.” The displacement to be added by the change of Poisson’s ratio is stated as the gradient of a potential except that one of the components is replaced by its twin, an identical component in reversed direction. This device also lends itself to a simplification of the analysis of stresses in a rotating thick disk.

scholarly journals Rational modelling of elastic soil behaviour in 3D condition

2019 ◽  
Vol 92 ◽  
pp. 15003
Author(s):  
Teruo Nakai ◽  
Hossain Md. Shahin ◽  
Akira Ishikawa
Keyword(s):  
Principal Stress ◽  
Linear Relation ◽  
Poisson’S Ratio ◽  
Elastic Strain ◽  
Volumetric Strain ◽  
Poisson's Ratio ◽  
Elastic Component ◽  
Principal Stresses ◽  
Strain Relation ◽  
Proposed Model

A simple and rigorous formulation of elastic component of elastoplastic model for geomaterials is presented. Although linear relation between elastic volumetric strain and mean principal stress in log scale is assumed in most of the usual models, linear relation between each principal stress and the corresponding principal elastic strain in log scale is assumed. Incorporating Poisson's ratio, three principal stresses vs. three elastic principal strain relation is obtained. Also, assuming coaxially between stresses and elastic strains, this relation can be transformed to stress- elastic strain relation in general coordinate. The material parameters of the proposed model of the elastic component are the same as those of the usual models, i.e., swelling index κ and Poisson's ratio ν. This proposed model can describe typical unloading behaviour of various shear tests and constant stress ratio unloading tests reported before.

On the mechanics of the intrusion of sills

1969 ◽  
Vol 6 (6) ◽  
pp. 1415-1419 ◽  
Author(s):  
P. E. Gretener
Keyword(s):  
Principal Stress ◽  
Upper Mantle ◽  
Poisson’S Ratio ◽  
Vertical Stress ◽  
Poisson's Ratio ◽  
Stress Axis ◽  
Principal Stresses ◽  
State Of Stress ◽  
As Stress

Diabase sills contain material originating from the base of the crust or the upper mantle. As a result they must be fed by dike- or plug-like bodies. The formation of a sill thus represents a major reorientation of the form of the intrusion. Tabular intrusive bodies tend to orient themselves perpendicular to the least compressive principal stress axis as shown by E. M. Anderson. It is suggested that diabase sills form under sedimentary strata in which the two horizontal principal stresses exceed the vertical stress (Sx > Sy > Sz). Such strata act as stress barriers and prevent further ascent of the magma, In order for this situation to occur the sediments must be in compression in the x-direction and confined in the y-direction. The parameter of importance to produce the above state of stress is the effective Poisson's ratio.

scholarly journals Tangential Loading of General Three-Dimensional Contacts

1998 ◽  
Vol 65 (4) ◽  
pp. 998-1003 ◽  
Author(s):  
M. Ciavarella
Keyword(s):  
Pressure Distribution ◽  
Poisson’S Ratio ◽  
Normal Load ◽  
Complete Solution ◽  
Tangential Force ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Partial Slip ◽  
Normal Contact ◽  
Slip Regime

A general three-dimensional contact, between elastically similar half-spaces, is considered. With a fixed normal load, we consider a pure relative tangential translation between the two bodies. We show that, for the case of negligible Poisson’s ratio, an exact solution is given by a single component of shearing traction, in the direction of loading. It is well known that, for full sliding conditions, the tangential force must be applied through the center of the pressure distribution. Instead, for a full stick case the tangential force must be applied through the center of the pressure distribution under a rigid flat indenter whose planform is the contact area of the problem under consideration. Finally, for finite friction a partial slip regime has to be introduced. It is shown that this problem corresponds to a difference between the actual normal contact problem, and a corrective problem corresponding to a lower load, but with same rotation of the actual normal indentation. Therefore for a pure translation to occur in the partial slip regime, the point of application of the tangential load must follow the center of the “difference” pressure. The latter also provides a complete solution of the partial slip problem. In particular, the general solution in quadrature is given for the axisymmetric case, where it is also possible to take into account of the effect of Poisson’s ratio, as shown in the Appendix.

An Application of the Interferometer Strain Gage in Photoelasticity

1935 ◽  
Vol 2 (3) ◽  
pp. A99-A102
Author(s):  
R. W. Vose
Keyword(s):  
Strain Gage ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Massachusetts Institute ◽  
Massachusetts Institute Of Technology ◽  
Lateral Deformation ◽  
Principal Stresses ◽  
Photoelastic Analysis ◽  
The Difference ◽  
Institute Of Technology

Abstract This paper was written at the suggestion of Mr. Mieth Maeser, in response to numerous inquiries concerning the methods of photoelastic analysis in use at the Massachusetts Institute of Technology. By the use of any of the usual photoelastic methods the difference of the principal stresses and their direction at any point in a suitable loaded specimen are determined, and through a knowledge of Poisson’s ratio their sum is obtained (and a solution made possible) by a measurement of the lateral deformation of the specimen by means of an interferometer strain gage. This instrument, together with its accessories and their use, is illustrated and described in the paper. Examples of the problems solved by the use of the instrument show its accuracy and the consistency of the results obtained by the method.

Compliance of Elastic Bodies in Contact

1949 ◽  
Vol 16 (3) ◽  
pp. 259-268
Author(s):  
R. D. Mindlin
Keyword(s):  
Contact Area ◽  
Contact Surface ◽  
Poisson’S Ratio ◽  
Tangential Force ◽  
Normal Component ◽  
Poisson's Ratio ◽  
Elastic Bodies ◽  
Complete Elliptic Integrals ◽  
Elliptic Contact ◽  
Circular Contact

Abstract A small tangential force and a small torsional couple are applied across the elliptic contact surface of a pair of elastic bodies which have been pressed together. If there is no slip at the contact surface, considerations of symmetry and continuity lead to the conclusion that there is no change in the normal component of traction across the surface and, aside from warping of the surface, there is no relative displacement of points on the contact surface. The problem is thus reduced to a “problem of the plane” in which the tangential displacements and normal component of traction are given over part of the boundary and the three components of traction are given over the remainder. In the case of the tangential force it is observed that, when Poisson’s ratio is zero, the problem is a simple one, in potential theory, which is then generalized by means of a special device. An expression for tangential compliance is found as a linear combination of complete elliptic integrals. In general, the compliance is greater in the direction of the major axis of the elliptic contact surface than in the direction of the minor axis. Both components of tangential compliance increase as Poisson’s ratio decreases and become equal when Poisson’s ratio is zero. Over the practical range of Poisson’s ratio, the tangential compliance is greater than the normal compliance, but never more than twice as great as long as there is no slip. The tangential traction on the contact surface is everywhere parallel to the applied force. Contours of constant traction are ellipses homothetic with the elliptic boundary. The magnitude of the traction rises from one half the average at the center of the contact surface to infinity at the edge. Due to this infinity, there will be slip, the effect of which is studied for the circular contact surface. In the case of the torsional couple, the solution is obtained by generalizing a solution by H. Neuber pertaining to a hyperbolic groove in a twisted shaft. The torsional compliance is expressed in terms of complete elliptic integrals and, for the circular contact area, reduces to that found by E. Reissner and H. F. Sagoci. The resultant traction at a point rises from zero at the center to infinity at the edge of the contact surface, but is constant along and parallel to homothetic ellipses only in the case of the circular contact area.

The Model for Calculating Elastic Modulus and Poisson's Ratio of Coal Body

2013 ◽  
Vol 6 (1) ◽  
pp. 36-43 ◽  
Author(s):  
Ai Chi ◽  
Li Yuwei
Keyword(s):  
Elastic Modulus ◽  
Poisson’S Ratio ◽  
Fractured Rock ◽  
Rock Block ◽  
Poisson's Ratio ◽  
Linear Elastic ◽  
Study Results ◽  
The Face ◽  
Block Face ◽  
Coal Rock

Coal body is a type of fractured rock mass in which lots of cleat fractures developed. Its mechanical properties vary with the parametric variation of coal rock block, face cleat and butt cleat. Based on the linear elastic theory and displacement equivalent principle and simplifying the face cleat and butt cleat as multi-bank penetrating and intermittent cracks, the model was established to calculate the elastic modulus and Poisson's ratio of coal body combined with cleat. By analyzing the model, it also obtained the influence of the parameter variation of coal rock block, face cleat and butt cleat on the elastic modulus and Poisson's ratio of the coal body. Study results showed that the connectivity rate of butt cleat and the distance between face cleats had a weak influence on elastic modulus of coal body. When the inclination of face cleat was 90°, the elastic modulus of coal body reached the maximal value and it equaled to the elastic modulus of coal rock block. When the inclination of face cleat was 0°, the elastic modulus of coal body was exclusively dependent on the elastic modulus of coal rock block, the normal stiffness of face cleat and the distance between them. When the distance between butt cleats or the connectivity rate of butt cleat was fixed, the Poisson's ratio of the coal body initially increased and then decreased with increasing of the face cleat inclination.

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