On Euler’s observation and Ericksen’s class of controllable deformations in isotropic incompressible elasticity including conservative body forces

Abstract

Conservative body force loads are special in homogeneous incompressible finite elasticity since they do not change the displacement solution (Euler 1757). According to Euler they change only the pressure reaction solution. This has several interesting theoretical and numerical implications. One almost trivial implication is that a conservative body force can be added to all problems in Ericksen’s class of controllable deformations in homogeneous incompressible finite elasticity without harm to the deformation part. In this work three boundary value problems corresponding to controllable deformations are solved numerically without and with conservative body forces using the standard displacement - pressure form of the Principle of Virtual Work for (near) incompressible finite elasticity. The numerical results support Euler’s observation on the whole, provided an appropriate pressure-like correction is added on the Neumann boundary whenever the body force potential does not vanish there.