The axisymmetric double contact problem for a frictionless elastic layer

The general axisymmetric double contact problem for an elastic layer pressed against a half space by an elastic stamp is considered. The problem is solved under the assumptions that the three materials have different elastic properties, the contact along the interfaces is frictionless and only compressive normal tractions can be transmitted across the interfaces, and, in the case of the elastic stamp, the local radius of curvature of the stamp is large compared to the stamp-layer contact radius. The problem is reduced to a system of singular integral equations in which the contact pressures are the unknown functions. The solution is obtained and extensive numerical results are given for three stamp geometries, namely, rigid and elastic spherical stamps, and a flat-ended rigid cylindrical stamp. The results show that in the case of a flat-ended rigid cylindrical stamp the radius b of the contact area between the layer and the subspace is independent of the magnitude P of the total transmitted load and in all other cases b will depend on P. Paccмaтpивaeтcя oбщaя пpoблeмa ocecиммeтpичнoгo двoйнoтo кoнтaктa элacтичнoгo cлoя пpижaтoгo к пoлyпpocтpaнcтвy элacтичным пecтикoм. Пpoблтeмa peшaeтcя пpeдпoлoжeниeм, чтo тpи мaтepиaлa имeют paзличныe элacтичныe cвoйcтвa, чтo кoнтaкт пo пoвepчнocти paздeлa лишeн тpeния и, чтo пo пoвepчнocти paздeлa мoгyт пepeдaвaтьcя тoлькo нopмaльныe cдaвливaющиe тягoвыe ycилия. Mecтный paдиyc кpивизны пecтикa пo cpaвнeнию c paдиycoм кoнтaктa пecтикa-cлoя бoлee бoльшoй. Пpoблeмa пpивoдитcя к cиcтeмe cингyляpныч интeгpaльныч ypaвнeний в кoтopыч дaвлeниe кoнтaктa являeтcя нeизвecтнoй фyнкциeй. Дocтигли peшeния и дaютcя чиcдeнныe peзyльтaты тpeч гeoмeтpичecкич cooбpaжeний для пecтикa, a имeннo, для жecткич и элacтичныч cфepичecкич пecтикoв и для жecткoгo цилиндpичecкoгo пecтикa c плocкими кoнцaми. Пo peзyльтaтaм виднo, чтo в cлyчae жecткoгo цилиндpичecкoгo пecтнкa c плocкими кoнцaми paдиyc b—плoщaдь кoнтaктa мeждy cлoeм и вeщecтвoм зaвиcит oт вeличины P—oбщeй пepeдaннoй нaгpyзки, и вo вceч дpyтич cлyчaяч b бyдeт зaвиceть oт P.