Configurational mechanics of necking phenomena in engineering thermoplastics

Necking is a significant part of the yielding process in many thermoplastics. It starts as strain localization associated with microshear banding and/or cavitations and appears as a domain of oriented (drawn) material, i.e., a “neck”, separated from the domain of original (isotropic) material by a narrow transition zone, which appears as a distinct boundary of the neck region. On further increase of displacement, the neck propagates through the test specimen under constant draw stress. Strain localization such as crazing and shear bending is associated with necking on micro- and sub-microscales. As a result material toughness, i.e., resistance to cracking, as well as durability, i.e., service lifetime under various service conditions, are related to the material ability to necking and specific characteristics of necking process. Necking is manifested in significant changes in a characteristic length scale, e.g., the distance between equally spaced marks in the reference state may increases by factor of 2 in amorphous polymers and up to a factor of 10 in some semicrystalline thermoplastics. There is also a characteristic relaxation time change during the necking. Thus from continuum mechanics viewpoint, the changes of intrinsic material space-time metric are the most fundamental manifestation of necking. Therefore we model necking phenomena as space-time scales transformation and introduce a four-dimensional (4D) Riemannian metric tensor of a material space-time imbedded into 4D Newtonian (laboratory) space-time with a Euclidean metric. Kinetic equation of necking, i.e., evolution equation for material metric tensor is derived using extremal action principle. An example of traveling wave solution for neck propagation in a tensile bar is presented. Analysis of the solution and comparison with experimental observations are discussed.