Mathematics & Physics 2015, 8(3), 352–355 УДК 517.958:539.3 Exact Analytical Solution of One Problem on Planar Deformation of Nonlinear-Elastic Media Georgiy M. Sevastyanov∗ Institute of Machine Engineering and Metallurgy FEB RAS Metallurgov, 1, Komsomolsk-on-Amur, 681005 Russia Received 10.05.2015, received in revised form 15.06.2015, accepted 10.07.2015 An analytical solution of a problem on planar deformation of isotropic incompressible nonlinear-elastic (rubber-like) media with a cylindrical cavity is constructed in quasi-static approximation. <...> A contour of the cavity is a smooth symmetrical curve. <...> The special kind of follower load provides purely radial movement of material. <...> A physical model of medium is given by elastic potential, which is analogous of Mooney-Rivlin strain energy potential (with a difference in the used finite strain tensor). <...> The obtained solution is exact since in equations connecting Cauchy stress tensor and Almansi finite strain tensor all nonlinear terms are kept (for accepted medium model the maximum is the fourth power of components of displacement gradient tensor). <...> Keywords: planar deformation, nonlinear elasticity, incompressibility, Mooney-Rivlin solid, finite strain, Almansi strain tensor, exact analytical solution. <...> DOI: 10.17516/1997-1397-2015-8-3-352-355 Let incompressible nonlinear-elastic medium occupies unlimited space and has a cylindrical cutout of infinite length (Fig. 1), on the surface of which the special kind of an uneven follower load (Fig. 2) is given. <...> We assume the contour R0 of the cutout border in undeformed state a smooth curve, symmetric about two perpendicular lines. <...> Georgiy M. Sevastyanov Exact Analytical Solution of One Problem on Planar Deformation . <...> We introduce a cylindrical coordinate system (R, θ,Z) centered at the point of intersection of axes of symmetry of the cutout (an axis Z is perpendicular to the plane of the Fig. 1), counting a polar angle θ will lead from one of the lines of symmetry. <...> Integration of the incompressibility equation relative to displacement (assuming ur/r < 1, ∂ur/∂r < 1) gives a view of the solution: ur(r,φ) = r −r2 +ψ(φ), thus the problem of determining the kinematics of media is reduced to identification of ψ(φ). <...> We define physical relations by elastic potential of isotropic incompressible medium (which is analogous to the Mooney-Rivlin <...>