scholarly journals An Axisymmetric Contact Problem of an Elastic Layer on a Rigid Circular Base

2020 ◽  
Vol 22 (1) ◽  
pp. 221-238
Author(s):  
B. Kebli ◽  
S. Berkane ◽  
F. Guerrache
Keyword(s):  
Elastic Layer ◽  
Mixed Boundary ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Integral Transforms ◽  
Form Solution ◽  
Rigid Base ◽  
Dual Integral Equations ◽  
Numerical Application ◽  
Stress Functions

AbstractAn analytical solution is presented to a doubly mixed boundary value problem of an elastic layer partially resting on a rigid smooth base. A circular rigid punch is applied to the upper surface of the medium where the contact is supposed to be smooth. The case of the layer with a cylindrical hole was considered by Toshiaki and all [5]. The studied problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer formula we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. The truncation method is used for getting the system coefficients. A closed form solution is given for the displacements, stresses and the stress singularity factors. The stresses and displacements are then obtained as Bessel function series. For the numerical application we give some conclusions on the effects of the radius of the punch with the rigid base and the layer thickness on the displacements, stresses, the load and the stress singularity factors are discussed.

Two dimensional mixed boundary-value problems in a wedge-shaped region

1968 ◽  
Vol 64 (2) ◽  
pp. 503-505 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Mellin Transforms

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.

Compressibility of Two-Dimensional Cavities of Various Shapes

1986 ◽  
Vol 53 (3) ◽  
pp. 500-504 ◽  
Author(s):  
R. W. Zimmerman
Keyword(s):  
Closed Form Solution ◽  
Mapping Function ◽  
Complex Stress ◽  
Form Solution ◽  
Hydrostatic Compression ◽  
Biaxial Compression ◽  
Cavity Surface ◽  
Uniform Pressure ◽  
Two Dimensional ◽  
Stress Functions

Muskhelishvili-Kolosov complex stress functions are used to find the stresses and displacements around two-dimensional cavities under plane strain or plane stress. The boundary conditions considered are either uniform pressure at the cavity surface with vanishing stresses at infinity, or a traction-free cavity surface with uniform biaxial compression at infinity. A closed-form solution is obtained for the case where the mapping function from the interior of the unit circle to the region outside of the cavity has a finite number of terms. The area change of the cavity due to hydrostatic compression at infinity is examined for a variety of shapes, and is found to correlate closely with the square of the perimeter of the hole.

The solution of Bessel function dual integral equations by converting to the first kind Fredholm Integral Equation: an extension of Noble’s solution

2018 ◽  
Vol 24 (8) ◽  
pp. 2536-2557
Author(s):  
S Cheshmehkani ◽  
M Eskandari-Ghadi
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Integral Transforms ◽  
Original Method ◽  
Fredholm Integral Equations ◽  
Dual Integral Equations ◽  
Mixed Boundary Value Problems ◽  
Factor Method

In certain mixed boundary value problems, Hankel integral transforms are applied and subsequently dual integral equations involving Bessel functions have to be solved. In the literature, if possible by employing the Noble’s multiplying factor method, these dual integral equations are usually converted to the second kind Fredholm Integral Equations (FIEs) and solved either analytically or numerically, respectively, for simple or complicated kernels. In this study, the multiplying factor method is extended to convert the dual integral equations both to the first and the second kind FIEs, and the conditions for converting to each kind of FIE are discussed. Furthermore, it is shown that under some simple circumstances, many mixed boundary value problems arising from either elastostatics or elastodynamics can be converted to the well-posed first kind FIE, which may be solved analytically or numerically. Main criteria for well-posedness of FIEs of the first kind in such problems are also presented. Noble’s original method is restricted to some limited conditions, which are extended here for both first and second kind FIEs to cover a wider range of dual integral equations encountered in engineering mixed boundary value problems.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Thermal and Mechanical Model for Rigid Cylinder Indenting an Elastic Layer Resting on Rigid Base: Application to Turned Surfaces

2003 ◽  
Vol 125 (2) ◽  
pp. 186-191
Author(s):  
Zhe Zhang ◽  
E. E. Marotta ◽  
J. M. Ochterbeck
Keyword(s):  
Mechanical Properties ◽  
Mechanical Model ◽  
Elastic Layer ◽  
Mixed Boundary ◽  
Linear Equations ◽  
Finite Thickness ◽  
Half Width ◽  
Rigid Base ◽  
Wide Range ◽  
Analytical Thermal Model

Models are presented for the solution of the thermal and mechanical problem of a rigid metallic cylinder indenting an elastic layer with finite thickness which rests on a rigid substrate without friction. The models were extended to turned surfaces applications. With introduction of an equivalent isothermal flux distribution for the mixed boundary problem—constant temperature over the contact area while adiabatic elsewhere along the top surface—an approximate analytical thermal model was developed. The solution was compared to a numerical solution under certain cases. Both solutions in turn compare very well with the generalized three-dimensional expression proposed by prior investigators. The mechanical model predicts the contact half-width under varying mechanical properties, layer dimensions, and applied load. The mechanical contact problem was solved numerically by substituting the displacement variable with a truncated polynomial to get a system of linear equations from which the dimensionless contact half-width was derived. The model is valid throughout a wide range of parameters, including mechanical properties and geometric dimensions. To explicitly predict the dimensionless contact half-width as a function of dimensionless load, a curve was fitted to the numerically obtained solution.

On the Changing Order of Singularity at a Crack Tip

1977 ◽  
Vol 44 (4) ◽  
pp. 625-630 ◽  
Author(s):  
R. J. Nuismer ◽  
G. P. Sendeckyj
Keyword(s):  
Closed Form ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Crack Tip ◽  
Form Solution ◽  
Square Root ◽  
Rigid Line Inclusion ◽  
Rigid Line ◽  
Line Inclusion ◽  
Crack Tip Stress

The nature of the transition in the crack tip stress singularity from an inverse square root to an inverse fractional power as a crack tip reaches a phase boundary or a geometrical discontinuity for interface cracks is examined. This is done by analyzing the simple closed-form solution to the problem of a rigid line inclusion with one side partially debonded for the case of antiplane deformation. For this example, the crack tip stress singularity changes from an inverse square root to an inverse three-quarters power as the crack tips approach the inclusion tips (i.e., when one face of the rigid line inclusion is completely debonded). A detailed analysis, based on series expansions of the closed-form solution, is used to show how the singularity transition occurs. Moreover, the expansions indicate difficulties that may be encountered when solving such problems by approximate methods.

scholarly journals Axisymmetric Torsion of an Elastic Layer Sandwiched between Two Elastic Half-Spaces with Two Interfaced Cracks

2019 ◽  
Vol 41 (2) ◽  
pp. 57-66
Author(s):  
Fateh Madani ◽  
Belkacem Kebli
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Elastic Layer ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Symmetry Plane ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
The Moment

AbstractThe present article examines the problem related to the axisymmetric torsion of an elastic layer by a circular rigid disc at the symmetry plane. The layer is sandwiched between two similar elastic half-spaces with two penny-shaped cracks symmetrically located at the interfaces between the two bonded dissimilar media. The mixed boundary-value problem is transformed, by means of the Hankel integral transformation, to dual integral equations, that are reduced, to a Fredholm integral equation of the second kind. The numerical methods are used to convert the resulting system to a system of infinite algebraic equations. Some physical quantities such as the stress intensity factor and the moment are calculated and presented numerically according to some relevant parameters. The numerical results show that the discontinuities around the crack and the inclusion cause a large increase in the stresses that decay with distance from the disc-loaded. Furthermore, the dependence of the stress intensity factor on the disc size, the distance between the crack and the disc, and the shear parameter is also observerd.

Adhesive Contact Between the Surface Wave and a Rigid Strip

1995 ◽  
Vol 62 (2) ◽  
pp. 368-372 ◽  
Author(s):  
O. Y. Zharii
Keyword(s):  
Integral Equation ◽  
Surface Wave ◽  
Stress Distribution ◽  
Contact Area ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Adhesive Contact ◽  
Form Solution ◽  
Mixed Boundary Value Problem ◽  
Analysis Of Variation

A problem of adhesive contact between the running surface wave and a rigid strip is investigated. The mixed boundary-value problem of elastodynamics is reduced to a singular integral equation for a complex combination of stresses and an exact closed-form solution of it has been derived. Analysis of variation of contact area dimensions, stress distribution and rotor velocity on the frequency of excitation displayed significant differences between the results corresponding to conditions of adhesion and slipping in contact area. The origin of these differences is discussed.

scholarly journals New Poisson's Type Integral Formula for Thermoelastic Half-Space

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Victor Seremet ◽  
Guy Bonnet ◽  
Tatiana Speianu
Keyword(s):  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Mixed Boundary ◽  
Integral Formula ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Volume Dilatation ◽  
Type Integral

A new Green's function and a new Poisson's type integral formula for a boundary value problem (BVP) in thermoelasticity for a half-space with mixed boundary conditions are derived. The thermoelastic displacements are generated by a heat source, applied in the inner points of the half-space and by temperature, and prescribed on its boundary. All results are obtained in closed forms that are formulated in a special theorem. A closed form solution for a particular BVP of thermoelasticity for a half-space also is included. The main difficulties to obtain these results are in deriving of functions of influence of a unit concentrated force onto elastic volume dilatation and, also, in calculating of a volume integral of the product of function and Green's function in heat conduction. Using the proposed approach, it is possible to extend the obtained results not only for any canonical Cartesian domain, but also for any orthogonal one.

Closed-Form Solution of the Frictional Sliding Contact Problem for an Orthotropic Elastic Half-Plane Indented by a Wedge-Shaped Punch

2014 ◽  
Vol 618 ◽  
pp. 203-225 ◽  
Author(s):  
Aysegul Kucuksucu ◽  
Mehmet A. Guler ◽  
Ahmet Avci
Keyword(s):  
Contact Problem ◽  
Half Plane ◽  
Mixed Boundary ◽  
Closed Form Solution ◽  
Contact Problems ◽  
Form Solution ◽  
Orthotropic Materials ◽  
Principal Axes ◽  
The Coefficient Of Friction ◽  
Processing Techniques

In this paper, the frictional contact problem of a homogeneous orthotropic material in contact with a wedge-shaped punch is considered. Materials can behave anisotropically depending on the nature of the processing techniques; hence it is necessary to develop an efficient method to solve the contact problems for orthotropic materials. The aim of this work is to develop a solution method for the contact mechanics problems arising from a rigid wedge-shaped punch sliding over a homogeneous orthotropic half-plane. In the formulation of the plane contact problem, it is assumed that the principal axes of orthotropy are parallel and perpendicular to the contact. Four independent engineering constants , , , are replaced by a stiffness parameter, , a stiffness ratio, a shear parameter, , and an effective Poisson’s ratio, . The corresponding mixed boundary problem is reduced to a singular integral equation using Fourier transform and solved analytically. In the parametric analysis, the effects of the material orthotropy parameters and the coefficient of friction on the contact stress distributions are investigated.

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