In the Johnson-Kendall-Roberts (JKR) approximation, adhesion forces outside the area of contact are neglected and elastic stresses at the edge of the contact are infinite, as in linear elastic fracture mechanics. On the other hand, in the Derjaguin-Muller-Toporov (DMT) approximation, the adhesion forces are taken into account, but the profile is assumed to be Hertzian, as if adhesion forces Could not deform the surfaces. To avoid self consistent numerical calculations based on a specific interaction model (Lennard-Jones potential for example) we have used a Dugdale model, which allows analytical solutions. The adhesion forces are assumed to have a constant value σO, the theoretical stress, over a length d at the crack tip. This internal loading acting in the air gap (the external crack) leads to a stress intensity factor Km, which is cancelled with the stress intensity factor KI due to the external loading. This cancellation suppresses the stress singularities, ensures the continuity of stresses, and fixes the radius c and the crack opening displacement δt. The energy release rate G is computed by the J-integral and the equilibrium is given by G = w. The equilibrium curves a(P), a(δ), and P(σ), the adherence forces at fixed load or fixed grips, the profiles, and the stress distributions can therefore be drawn as a function of a single parameter λ. When λ increases from zero to infinity there is a continuous transition from the DMT approximation to the JKR approximation. Furthermore the value of G for the DMT approximation is derived. It is shown that it is not physically consistent to have tensile stresses in the area of contact and no adhesion forces outside or no tensile stresses in the area of contact and adhesion forces outside. In the JKR approximation the distribution of adhesion forces is reduced to a singular stress at r = a+. The total attraction force outside the contact being zero, the integral of stresses in the contact is equal to the applied load P and negative applied loads are supported by the elastic restoring forces. In the DMT approximation the adhesion stresses tend toward zero to have a continuity with the stress at r = a−, but their integral is finite and the total attraction force outside the contact is 2πwR. In the area of contact the distribution of stresses is Hertzian, and their integral is P + 27πwR. Negative applied loads are sustained by adhesion forces outside the contact.

Adhesion of spheres : the JKR-DMT transition using a dugdale model

Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

Compliant mechanisms are mechanical devices that achieve motion via elastic deformation. A new method for topological synthesis of single-piece compliant mechanisms is presented, using a “design for required deflection” approach. A simple beam example is used to illustrate this concept and to provide the motivation for a new multi-criteria approach for compliant mechanism design. This new approach handles motion and loading requirements simultaneously for a given set of input force and output deflection specifications. Both a truss ground structure and a two-dimensional continuum are used in the implementation which is illustrated with design examples.

Topological synthesis of compliant mechanisms using multi-criteria optimization

This chapter provides an overview of the recent developments concerning Saint-Venant's principle The task of determining, within the framework of the linear theory of elasticity, the stresses and displacements in an elastic cylinder in equilibrium, under the action of loads that arise solely from tractions applied to its plane ends has come to be called Saint- Venant's problem Saint-Venant's construction does not permit the arbitrary preassignment of the point-by-point variation of the end tractions giving rise to these forces and moments; indeed, this variation is essentially determined as a consequence of the special assumptions made in connection with his so-called semi-inverse procedure The early work of Saint-Venant and Boussinesq furnished the seeds from which grew a large number of more general assertions, most referring to elastic solids of arbitrary shape and many being rather imprecise, concerning the effect on stresses within the body of replacing the tractions acting over a portion of its surface by statically equivalent ones Such propositions usually went by the name of Saint-Venunt's principle, despite the fact that Saint-Venant's original conjecture was intended to apply only to cylinders This chapter discusses in detail about flow in a cylinder, a representation for the exact solution, and energy decay for other linear elliptic second-order problem Linear elastostatic problems are also stated in the chapter

Recent Developments Concerning Saint-Venant's Principle

A procedure to obtain a topology of an optimal structure considering flexibility is presented. The methodology is based on a mutual energy concept for formulation of flexibility and the homogenization method. A multi-objective optimization problem is formulated as an application of compliant mechanism design. Some examples of the design of compliant mechanisms for plane structures are presented. ( 1998 John Wiley & Sons, Ltd.

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Topology optimization of compliant mechanisms using the homogenization method

A General Theory of Optimal Plastic Design

Es wird ein Prinzip der stationaren gegenseitigen potentiellen Energie aufgestellt fur zwei Belastungssysteme eines elastischen Balkens veranderlicher Biegesteifigkeit. Aus diesem Prinzip wird eine hinreichende Bedingung fur stationares Gewicht eines Sandwichbalkens abgeleitet, wenn die von einer Belastung an einem bestimmten Querschnitt erzeugte Durchbiegung vorgeschrieben ist. Fur statisch bestimmte Balken wird gezeigt, dass diese Bedingung ein globales Minimum des Gewichts sicherstellt. Anwendungsbeipiele und Erweiterungen werden besprochen.

Optimal structural design for given deflection

Large plane deformations of thin elastic sheets of neo-Hookean material are considered and a method of successive substitutions is developed to solve problems within the two-dimensional theory of finite plane stress. The first approximation is determined by linear boundary value problems on two harmonic functions, and it is approached asymptotically at very large extensions in the plane of the sheet. The second and higher approximations are obtained by solving
Poisson equations. The method requires modification when the membrane has a traction-free edge. Several problems are treated involving infinite sheets under uniform biaxial stretching at infinity. First approximations are obtained when a circular or elliptic inclusion is present and when the sheet has a circular or elliptic hole, including the limiting cases of a line inclusion and a straight crack or slit. Good agreement with exact solutions is found for circularly symmetric deformations. Other examples discuss the stretching of a short wide strip, the deformation near a boundary corner which is traction-free, and the application of a concentrated load to a boundary point.

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Large plane deformations of thin elastic sheets of neo-hookean material

Optimal design of multi-purpose structures*

Es werden endliche elastische Deformationen bei ebenen Verzerrungszustanden betrachtet, in welchen die Spannungen nur vom Azimut abhangen. Der allgemeine Charakter der Deformationen wird fur inkompressible isotrope Materialien diskutiert; insbesondere werden zwei partikulare Losungen aufgestellt, die fur alle Formen der Verzerrungsenergie gultig sind. Die Losungen der ersten Klasse sind die erwarteten homogenen Deformationen, wahrend diejenigen der zweiten Klasse in rotatorischen, von einem einzigen Parameter abhangigen Verformungen bestehen. Es wird die Losung von Randwertproblemen fur die Spannungen bei keilformigen Gebieten im einzelnen fur ein Mooney-Material untersucht und dabei gezeigt, dass gewisse besondere Eigenschaften der infinitesimalen Theorie auch in der nichtlinearen auftreten.

Richard T. Shield

Papers

A General Theory of Optimal Plastic Design

Optimal structural design for given deflection

Large plane deformations of thin elastic sheets of neo-hookean material

Optimal design of multi-purpose structures*

On a class of solutions in plane finite elasticity