Application of a nonlinear viscoelastic constitutive equation to isothermal pure bending near the material yield point.

Abstract

A constitutive equation for nonlinear viscoelasticity is used to model the mechanical response, near the material yield point, of solid polymers. The constitutive equation had previously been used almost exclusively to study the response of materials subjected to homogeneous strains. The nonlinearity in the constitutive equation arises from a strain dependent reduced time which causes the stress relaxation to accelerate with increasing strain. The constitutive equation is modified to include the spatial variation in the strain dependent reduced time. The mechanical response is focused on one specific design area--isothermal pure bending of cylindrical beams. Since polymers are used in beam designs subjected to isothermal pure bending, results of this study will offer the broadest impact on designs near the material yield. Step and constant rate curvature and moment control histories are considered along with curvature deformation and recovery and moment load and unload histories. Beam responses for cross-sections with a horizontal axis of symmetry (rectangular, circular and I-beam) are studied and unsymmetrical T-sections are also considered. The classical assumption of beam theory, i.e. plane sections remain plane, is made. At a fixed time, the strains vary linearly through the depth of the beam and at a fixed material element, the strain varies in time with curvature. The spatial variation of the strains combined with the nonlinear dependence of the reduced time on strain leads to a significantly different response from that given by traditional elastic and linear viscoelastic beam theory. For example, the stresses at the elements farthest from the neutral axis relax faster than at those elements near the neutral axis. This implies that the stress distribution, at a fixed time, may no longer vary linearly with distance from the neutral axis. Consequently, the maximum stress may occur in the interior of the beam and the neutral axis may no longer coincide with the centroid. The implications of this for the curvature, bending moment and neutral axis responses as well as stress distributions and other factors which relate to beam design are discussed.