Topics on Unilateral Contact Problems of Elasticity and Inelasticity
These lectures present selected unilateral boundary and initial value problems of elasticity and inelasticity. Particularly, for elastic Signorini’s problem with general subdifferential sliding rule two dual problems are derived as well as the bidual one.
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J.J. Telega Polish Academy of Sciences, Warsaw, Poland
ABS'IRACT These lectures present selected unilateral boundary and initial value problems of elasticity and inelasticity. Particularly, for elastic Signorini's problem with general subdifferential sliding rule t\~O dual problems are derived as well as the bidual one. Some historical and modern views on friction phenomenon are briefly presented. The next chapter is concerned with unilateral frictionless and frictional problems for viscoelastic, plastic and viscoplastic bodies. In the last chapter homogenization of microfissured elastic solids and plates is studied. It is assumed that microfissures behave unilaterally.
J. J. Moreau et al. (eds.), Nonsmooth Mechanics and Applications © Springer-Verlag Wien 1988
J.J. Telega
342 1 • IN'IRODUCTION
These lectures are focussed on variational approach to unilateral contact problems for elastic and inelastic bodies. Non-smooth effects are imposed by unilateral conditions at the boundary. In the case of perfect plasticity /Chapter 3/ and fissured solids and plates /Chapter 4/ certain functions like displacements and velocities are also non-smooth /discontinuous/. In my opinion it is covenient to classify contact problems of solid and structural mechanics as follows. / i / External and intern~l contact problems. In the first case the body considered is in contact with another rigid or deformable body /elastic or inelastic/, or with a system of such bodies. Internal contact problems are posed by fracture and damage mechanics, see Chapter 4 of these lectures. /ii/ Contact of bodies made of: inorganic materials, for instance metal - metal contact;organic materials, for example cartilage - cartilage contact in human and animal joints, see Refs [1,9,10,22,26,27,36,37;38,40,52] and Fig.1; bone synovial fluid
synovial membrune
Fig.1, Diagrammatic representation of a human joint
artic ular cartilage
organic and inorganic materials /in inplants, for instance bone-polymethyl methacrylate cement 1 bone-metal contact/, of. Refs [2,6,12,13,16,17,28-31 ,48,49,57] and Fig.2. /iii/ Dependence upon friction: frictionless contact, contact with friction but without lubrication /dry contact/, lubricated contact , see Refs [4,15,20,32,42,45,46,53,56]. /iv/ Unilateral or non-unilateral conditions. /v/ Presence or absence of adhesion /see the lectures by Fremond/ and wear. Obviously, in real situations various combinations of points / i / - /v/ occur. Now I shall briefly present the subject of my lectures. Since I largely deal with unilateral contact problems with friction, therefore some historical and modern
Topics on Unilateral Contact Problems
343
Fig.2, Schematic illustration of a Charnley total hip prosthesis /acetabular and femoral component/ fixated in the bone with acrylic cement • [0J
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cortical bo ne spongeous bone acrylic cement
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polyethylene metol
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concepts of the friction phenomenon are first sucointly presented. Various phenomenological friction laws or sliding rules are next introduced. Both asso
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