Small deformations of a nonlinear viscoelastic tube: theory and application to polypropylene

The response of an isotropic, nonlinear viscoelastic, thin-walled tube to combinations of axial force F , axial couple G and pressure difference p is considered theoretically and experimentally. Theory is based on the membrane theory of thin shells, applied to a thin-walled circular cylindrical tube. The components of two dimensional stress and strain in the wall of the tube are derived, allowing for arbitrarily large deformations; but restriction to small deformations is shown to be necessary if the history of stress is to be controlled at will through F , G and p . For arbitrary choice of F , G and p as functions of time the strain is shown to depend on three stress tensors P , Q , R independent of time, and three scalar functions of time. An expression for the linear strain tensor in terms of P , Q , R is obtained which involves four scalar functions ϕ 0 , ϕ 1 , ϕ 2 , ϕ 3 . These functions depend on the invariants of P , Q , R and on the three scalar functions of time. If any one of P , G , p is always zero then R = 0 and only ϕ 0 , ϕ 1 , ϕ 2 are required. In the case of proportional loading ( Q = R = 0 ) only ϕ 0 and ϕ 1 are required and any one of the three strain components can be calculated from the remaining two. Creep and recovery experiments under simultaneous axial force and couple were conducted on a thin-walled tube of polypropylene at 65.5 °C. Theory was used to calculate the circumferential tensile strain from the measured shear strain and longitudinal tensile strain. For this particular tube ϕ 0 and ϕ 1 were found to be related in a special manner, implying that nonlinearity can be adcquatcly described by allowing the shear creep compliance to change with stress history. By varying separately combinations of the invariants of P , ϕ 1 was found to depend on both hydrostatic and deviatoric components ofthe applied stress.