An Educational Elasticity Problem With Friction, Part 1: Loading, and Unloading for Weak Friction

1981 ◽  
Vol 48 (4) ◽  
pp. 841-845 ◽  
Author(s):  
John Dundurs ◽  
Maria Comninou
Keyword(s):  
Closed Form ◽  
Elasticity Problem ◽  
Concentrated Force ◽  
Initial Loading ◽  
Loading And Unloading

The paper treats an elasticity problem with friction that can be solved in closed form. It involves an unbounded solid with a semi-infinite cut. The solid is compressed in a direction perpendicular to the cut, but the cut is induced to separate and slip locally by applying a concentrated force at the tip of the cut. The first part deals with the initial loading phase and includes unloading for weak friction.

The Elastic Plane With a Circular Insert, Loaded by a Tangentially Directed Force

1962 ◽  
Vol 29 (2) ◽  
pp. 362-368 ◽  
Author(s):  
M. Hete´nyi ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Elastic Plane ◽  
Elementary Functions ◽  
Explicit Formulas ◽  
Concentrated Force

The problem treated is that of a plate of unlimited extent containing a circular insert and subjected to a concentrated force in the plane of the plate and in a direction tangential to the circle. The elastic properties of the insert are different from those of the plate, and a perfect bond is assumed between the two materials. The solution is exact within the classical theory of elasticity, and is in a closed form in terms of elementary functions. Explicit formulas are given for the components of stress in Cartesian co-ordinates, and also in polar co-ordinates at the circumference of the insert.

Unconventional Internal Cracks, Part 2: Method of Generating Simple Cracks

1982 ◽  
Vol 49 (2) ◽  
pp. 383-388 ◽  
Author(s):  
C. H. Wu
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elasticity Problem

The meaning of the word crack is extended to include holes with cusps of vanishing cusp angle. A crack is said to be simple if the associated elasticity problem has a closed-form solution. Many classes of simple cracks are constructed and solved in this two-part paper. In particular, a method of constructing very sharp cusps is described. These cusps possess not only a zero slope (zero cusp angle) but a vanishing curvature as well. In fact, a crack may be constructed in such a way that the first N derivatives are all zero.

Melan’s Problems With Weak Interface

1999 ◽  
Vol 67 (1) ◽  
pp. 22-28 ◽  
Author(s):  
S. Lenci
Keyword(s):  
Closed Form ◽  
Applied Load ◽  
Interface Stress ◽  
Interfacial Stress ◽  
Weak Interface ◽  
Concentrated Force ◽  
Zone Of Influence ◽  
Two Bodies

The problem of a fiber attached to an infinite sheet (Melan’s problem) has been reconsidered under the hypothesis that the adherence between the two bodies is not perfect. We have assumed that the link is guaranteed by the so-called “weak interface,” i.e., we have supposed that the jump of the displacement is linearly proportional to the interface stress. The solutions of (i) the case with a concentrated force acting on the fiber and (ii) the case of the redistribution of stresses as a consequence of the rupture of the fiber have been obtained in closed form. We have discussed how the interface stiffness k influences the solutions and, in particular, the interfacial stress. Emphasis is placed on determining how the zone of influence of the applied load is modified by k. Approximate (though accurate) simple expressions for the length of the zone of influence are given and discussed. [S0021-8936(00)01001-1]

An Educational Elasticity Problem With Friction, Part 3: General Load Paths

1983 ◽  
Vol 50 (1) ◽  
pp. 77-84 ◽  
Author(s):  
J. Dundurs ◽  
M. Comninou
Keyword(s):  
Elasticity Problem ◽  
Exact Nature ◽  
Load Path ◽  
Initial State ◽  
Concentrated Force ◽  
Load Paths

This concluding paper treats general load paths when the two components of the concentrated force are allowed to change independently with time. It is shown that there are two kinds of dependence on the load path. For certain directions of the forward tangent, the dependence is strict in that the deformations depend on the full details of the path. For other directions, however, the dependence is loose, and the deformations do not depend on the exact nature of the path as long as the forward tangent falls within given bounds. The problem also shows that, given an initial state, the load space can be subdivided into different regions each corresponding to a certain mode of deformations.

The Axially Loaded Circular Cylinder

1962 ◽  
Vol 29 (2) ◽  
pp. 318-320
Author(s):  
H. D. Conway
Keyword(s):  
Exact Solution ◽  
Fourier Transform ◽  
Circular Cylinder ◽  
Closed Form ◽  
Cross Sections ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Transform Method ◽  
Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

scholarly journals Permeability of the Hydrated Shale under Cyclic Loading and Unloading Conditions

Geofluids ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Liu Xianshan ◽  
Li Man ◽  
Xu Ming ◽  
Kang Zhiyong
Keyword(s):  
Cyclic Loading ◽  
Confining Pressure ◽  
Loading Conditions ◽  
Permeability Evolution ◽  
Loading Process ◽  
Cyclic Loading And Unloading ◽  
Initial Loading ◽  
Cyclic Processes ◽  
Loading And Unloading ◽  
Shale Permeability

The hydrated shales under cyclic loading and unloading conditions are common for the shale reservoir development; corresponding mechanical properties and permeability evolution are very significant and should be deeply researched. Firstly, the experiments of the hydrated shales under the above conditions are discussed, showing that the peak strength is lower and corresponding permeability is higher for more days of hydrating treatment. Secondly, the damage theory is proposed to analyze the shale permeability evolution due to hydromechanical damage and get permeability variation under initial loading and unloading conditions, observing that the permeability in the loading process decreases with increasing confining pressure and increases in the unloading process with decreasing confining pressure; however, the former changes much greater than the latter considering the same confining pressure, indicating that the irreversible damage for the hydrated shales in this cyclic condition has resulted in obvious difference of the permeability. Furthermore, the curves between the permeability and confining pressure based on the experimental data are fitted as negative exponential functions under initial loading conditions and power functions under more cyclic loading conditions, showing that more loading process will change the permeability evolution model. However, the permeability while unloading changes smoothly and can be fitted as a power function with the confining pressure. And in addition, the loss ratio and recovery ratio of the permeability have been deeply researched under five cyclic loading and unloading conditions, thoroughly explaining the permeability decreasing variation with more cyclic processes. Finally, the sensitive coefficients of the permeability have been investigated to observe the largest coefficients under initial cyclic conditions and less and less with more cyclic processes, especially the coefficients while loading which are more sensitive to lower confining pressure and smaller while unloading, which is in accordance with the shale permeability loss and recovery variation, revealing the permeability evolution of the hydrated shale under complex extracted environment.

Some Closed-Form Results for Adhesive Rough Contacts Near Complete Contact on Loading and Unloading in the Johnson, Kendall, and Roberts Regime

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Michele Ciavarella ◽  
Yang Xu ◽  
Robert L. Jackson
Keyword(s):  
Fractal Dimension ◽  
Closed Form ◽  
Adhesive Contact ◽  
Full Contact ◽  
Loading And Unloading ◽  
Complete Contact

Recently, generalizing the solution of the adhesiveless random rough contact proposed by Xu, Jackson, and Marghitu (XJM model), the first author has obtained a model for adhesive contact near full contact, under the Johnson, Kendall, and Roberts (JKR) assumptions, which leads to quite strong effect of the fractal dimension. We extend here the results with closed-form equations, including both loading and unloading which were not previously discussed, showing that the conclusions are confirmed. A large effect of hysteresis is found, as was expected. The solution is therefore competitive with Persson's JKR solution, at least in the range of nearly full contact, with an enormous advantage in terms of simplicity. Two examples of real surfaces are discussed.

An Educational Elasticity Problem With Friction, Part 2: Unloading for Strong Friction and Reloading

1982 ◽  
Vol 49 (1) ◽  
pp. 47-51 ◽  
Author(s):  
Maria Comninou ◽  
John Dundurs
Keyword(s):  
Elasticity Problem ◽  
Concentrated Force ◽  
Frictional Slip

This paper examines the case of unloading with strong friction, and also treats reloading of the unbounded solid with a semi-infinite cut. The solid is compressed uniformly normal to the cut, and a concentrated force acts on the tip of the cut causing localized separation and frictional slip.

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