Quasiperiodic contact problem for an elastic half-plane

Chuvash State University, Cheboksary. Translated from Prikladnaya Mekhanika, Vol. 28, No. 6, pp. 22–28, June, 1992.     

Variational approach to the solution of a contact problem for an elastic half-plane

Belorussian Polytechnic Institute, Minsk. Translated from Prikladnaya Mekhanika, Vol. 30, No. 7, pp. 70–73, July, 1994.     

The hertz contact problem with finite friction

The indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient in the region of contact. Within an inner circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. The slip boundary between the two regions depends on and the Poisson ratio v, and is found uniquely as an eigenvalue of a certain integral equation.For power law indentors of the form zr ⁿ , a group property of the integral operator connecting stresses and displacements makes it possible to derive the contact stress distributions from those under a flat punch by a simple quadrature, and shows that the slip radius is the same in all such cases.An iterative numerical solution using a dual system of Volterra equations is described, and calculated distributions of surface stress presented for the cases of indentation by a flat punch and by a sphere.

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Zusammenfassung Das Eindringen eines axial-symetrischen Stempels in einen elastischen unendlichen Halbraum, unter Einwirkung einer monotonischen senkrechten Kraft, wird dargestellt als ein gemischtes Grenzwert problem, wobei ein Coulombscher Reibungskoeffizient im Kontaktvolumen angenommen wird. Innerhalb eines inneren Kreises der Kontakt is haftend während in dem umgebenden Kreisring eine nach innen gerichtete Bewegung der Ebene stattfindet, die mit der Kraft wächst. Die Gleitgrenze zwischen diesen beiden Gebieten hängt von und dem Poisson Verhältnis ab und ist ein eindeutiger Eigenwert eines Integralgleichung.Für Stempel, deren Form zr ⁿ gehorcht, wird gezeigt, dass eine Gruppeneigenschaft des Integral Operators, welche Spannungen und Verschiebungen verknüpft, ermöglicht, die Kontaktspannungsverteilung in der Umgebung eines flachen Stempels durch einfache Quadratur abzuleiten und zu zeigen, dass der Gleitradius in allen Fällen der gleiche ist.Es wird eine iterative numerische Lösung beschrieben, die Volterra-Gleichungen benutzt. Die Berechnungen der Oberflächenspannungsverteilung für das Eindringen eines ebenen Stempels in eine Kugel wird gegeben.

     

Existence results for unilateral contact problem with friction of thermo-electro-elasticity

This work studies a mathematical model describing the static process of contact between a piezoelectric body and a thermally-electrically conductive foundation. The behavior of the material is modeled with a thermo-electro-elastic constitutive law. The contact is described by Signorini's conditions and Tresca's friction law including the electrical and thermal conductivity conditions. A variational formulation of the model in the form of a coupled system for displacements, electric potential, and temperature is de- rived. Existence and uniqueness of the solution are proved using the results of variational inequalities and a fixed point theorem.     

Approximation of thermoelasticity contact problem with nonmonotone friction

The paper presents the formulation and approximation of a static thermoelasticity problem that describes bilateral frictional contact between a deformable body and a rigid foundation. The friction is in the form of a nonmonotone and multivalued law. The coupling effect of the problem is neglected. Therefore, the thermic part of the problem is considered independently on the elasticity problem. For the displacement vector, we formulate one substationary problem for a non-convex, locally Lipschitz continuous functional representing the total potential energy of the body. All problems formulated in the paper are approximated with the finite element method.     

Plane contact problem for an elastic rectangular punch and a half-plane with initial stresses

International Applied Mechanics -     

The contact problem for an elastic half-plane in the case of a transonic load velocity

The stress-strain state of an elastic half-space under the action of a moving load is studied. The problem is solved analytically. The solution obtained is analyzed for the case of a transonic load velocity.     

The axisymmetric viscoelastic contact problem with rotational friction

The boundary value problem that arises when a mechanically rough rigid punch of arbitrary axisymmetric profile is pressed against the surface of a linear aging viscoelastic half space and is also made to rotate about its axis, so that there is total slip between the contacting surfaces is analysed and solved. The moment required to make the punch rotate, on the assumption that the coefficient of friction obeys a power law or is a constant, and the total normal pressure acting on the punch may each be evaluated in terms of the history of the radius of the contact area. Application is made to the special cases where the punch has the form of (i) a cone, (ii) a paraboloid of revolution and (iii) a flat ended cylinder. Apart from case (iii) where the contract area is constant we can only find an explicit expression for the moment in terms of total pressure so long as the contact area is increasing. The case of constant total pressure and Maxwell viscoelastic material is examined in more detail.     

Effect of friction in a contact problem for a plate with a pin

A solution is obtained for a contact problem concerning the tension of a rectangular elastic plate with a circular hole into which a rigid stationary pin has been inserted. There is a small gap between the hole and the pin, which is of circular cross section. Friction acts in the contact region in accordance with the Coulomb law. The finite-element method and the Boussinesq principle are used to determine the load that realizes a specified contact region. Two variants of boundary conditions on the contour of the hole are examined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 184–192, July–August, 1998.