Axially Symmetric Stress Distributions in Elastic Solids Containing Ring-Shaped Cracks Under Torsion

1974 ◽  
Vol 41 (2) ◽  
pp. 516-517 ◽  
Author(s):  
R. P. Kanwal ◽  
M. L. Pasha
Keyword(s):  
Elastic Medium ◽  
Crack Problem ◽  
Outer Radius ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
First Case

We present the axially symmetric stress distributions in elastic solids containing ring-shaped cracks when the solids are kept under torsion. Two cases are analyzed. In the first case we discuss the crack problem in an infinite elastic medium when the ratio of the inner to the outer radius is small. In the second case this ratio is almost equal to unity.

Axially Symmetric Stress Distributions in Elastic Solids Containing Penny-Shaped Cracks Under Torsion

1975 ◽  
Vol 42 (4) ◽  
pp. 896-897 ◽  
Author(s):  
M. L. Pasha
Keyword(s):  
Stress Intensity ◽  
Integral Representation ◽  
Elastic Medium ◽  
Stress Intensity Factors ◽  
Representation Formula ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Intensity Factors ◽  
Elastic Solids ◽  
Integral Representation Formula

We present the axially symmetric stress distributions in elastic solids containing a pair of axially symmetric penny shaped cracks when the infinite elastic medium is kept under torsion. We derive the integral representation formula for the torsion function and the expressions for the stress-intensity factors.

Some axially symmetric stress distributions in elastic solids containing penny-shaped cracks I. Cracks in an infinite solid and a thick plate

1962 ◽  
Vol 266 (1326) ◽  
pp. 359-386 ◽  
Keyword(s):  
Integral Equations ◽  
Interior Point ◽  
Thick Plate ◽  
Elastic Solid ◽  
Point Force ◽  
Fredholm Integral Equations ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Single Crack ◽  
Elastic Solids

Some axially symmetric stress distributions in an infinite elastic solid and in a thick plate containing penny-shaped cracks are considered. It is shown that, by use of a representation for the displacement in an infinite elastic solid containing a single crack, representations for the displacements in an infinite solid containing two or more cracks and in a thick plate containing a single crack can be constructed and used to reduce the problems of determining the stresses in these solids to the solutions of Fredholm integral equations of the second kind. Various stress distributions investigated include those due to the opening of a crack in an infinite solid by a point force acting at an interior point of the solid and the opening of cracks in an infinite solid and a thick plate under the action of constant pressures over the cracks.

scholarly journals Some Axially Symmetric Stress Distributions in Elastic Solids containing Penny-shaped Cracks

1962 ◽  
Vol 13 (1) ◽  
pp. 69-78 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Integral Equation ◽  
Stress Distribution ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
General Representation ◽  
Uniform Tension ◽  
Circular Beam

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.

scholarly journals A mixed boundary value problem of a cracked elastic medium under torsion

2021 ◽  
pp. 10-10
Author(s):  
Belkacem Kebli ◽  
Fateh Madani
Keyword(s):  
Boundary Value Problem ◽  
Elastic Medium ◽  
Integral Equations ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Crack Problem ◽  
Fredholm Integral Equations ◽  
Mixed Boundary Value Problem ◽  
Penny Shaped Crack

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

Rough Contacts Modeled by Nonlinear Coatings

2007 ◽  
Author(s):  
I. I. Kudish
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Experimental Studies ◽  
Nonlinear Function ◽  
Normal Displacement ◽  
Experimental Fact ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
Local Pressure

A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.

Dynamical Behaviour of Orthotropic Micropolar Elastic Medium

2002 ◽  
Vol 8 (8) ◽  
pp. 1053-1069 ◽  
Author(s):  
Rajneesh Kumar ◽  
Suman Choudhary
Keyword(s):  
Elastic Medium ◽  
Couple Stress ◽  
Normal Force ◽  
Numerical Technique ◽  
Integral Transforms ◽  
Dynamical Behaviour ◽  
Normal Displacement ◽  
Physical Domain ◽  
Elastic Solids ◽  
Eigenvalue Approach

The present paper is concerned with the plane strain problem in homogeneous micropolar orthotropic elastic solids. The disturbance due to continuous normal and tangential sources are investigated by employing eigenvalue approach. The integral transforms have been inverted by using a numerical technique to obtain the normal displacement, normal force stress and tangential couple stress in the physical domain. The expressions of these quantities are given and illustrated graphically.

Approximate Solution of High-Frequency-Factor Vibrations of Rigid Bodies on Elastic Media

1971 ◽  
Vol 38 (1) ◽  
pp. 111-117 ◽  
Author(s):  
A. O. Awojobi
Keyword(s):  
Approximate Solution ◽  
Elastic Medium ◽  
Half Space ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Factor ◽  
Rigid Bodies ◽  
Elastic Media ◽  
Stress Distributions ◽  
Rectangular Body

The mixed boundary-value problems of the vibrations of rigid bodies on elastic media are generally considered in the low-frequency-factor range. It is first established that, quite apart from a consideration of resonance, the usual assumption that this range predominates in practice is erroneous. The present work, therefore, is concerned with vibrations at frequency factors which are much greater than unity. Five cases have been considered: torsional vibration of a rigid circular body on a semi-infinite elastic medium and on an infinitely wide elastic stratum on a rigid bed; vertical vibration of a rigid circular body and of an infinitely long rectangular body on a semi-infinite elastic medium; rocking of a long rectangular body on a semi-infinite elastic medium. An estimate of both the unknown dynamic stress distribution under the rigid bodies and their amplitude responses has been obtained by finding an approximate solution to the exact governing dual integral equations. It is shown that at high-frequency factors, stress distributions are approximately constant for vertical vibrations and vary linearly from the center for rotational vibrations as in a Winkler model of theoretical soil statics contrary to increasing stresses with infinite edge stresses for low-frequency and static stress distributions of rigid bodies on elastic half space. We also obtain the important conclusion for amplitude response that it is predominantly governed by the inertia of the bodies because the contribution due to the dispersion of waves in the elastic medium is generally of a lower order of frequency factor than the inertia term except for an incompressible medium which has been analyzed separately and found to be of the same order leading to expressions for equivalent inertia of the vibrating medium. The theoretical results are used to derive the “tails” of resonance curves for both half space and stratum cases where experimental results are available. The agreement is fair and improves with increasing frequency factor.

scholarly journals Relaxation methods applied to engineering problems - Stress distributions in elastic solids of revolution

1945 ◽  
Vol 239 (810) ◽  
pp. 501-537 ◽  
Keyword(s):  
Stress Distributions ◽  
Elastic Solids ◽  
Relaxation Methods ◽  
Solid Of Revolution ◽  
Circular Shaft ◽  
Engineering Problems ◽  
Uniform Diameter ◽  
Solids Of Revolution

Relaxation methods are employed to solve, without restriction on the form of the generating curve , the following problems relating to solids of revolution: (1) torsional stresses in an incomplete tore, (2) torsional stresses in a circular shaft of non-uniform diameter, (3) axially symmetrical stresses in a complete solid of revolution, (4) flexural stresses in an incomplete tore, (5) shearing and flexural stresses in a toroidal ‘hook’. Accuracy sufficient for all practical purposes is attained in every case.

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