Constitutive Equations and Failure Criteria for Amorphous Polymeric Solids by BARKER

Anand & Gurtin (2002) have recently developed a continuum theory for the elasticviscoplastic deformation of amorphous solids. Their theory is motivated by and builds on the work of Parks, Argon, Boyce, Arruda, and their co-workers (e.g. Boyce et al., 1988; Arruda & Boyce, 1993) on modeling the plastic deformation of amorphous polymers. The theory of Anand & Gurtin (2002) carefully accounts for restrictions placed on constitutive assumptions by frame-indifference and by a new mathematical definition of an amorphous material based on the notion that the constitutive relations for such materials should be invariant under all rotations of the reference configuration and, independently, all rotations of the relaxed configuration. Also, they explicitly account for the dependence of the Helmholtz free energy on the plastic deformation in a thermodynamically consistent manner, a dependence which leads directly to a backstress in the underlying flow rule. In addition to the standard kinematic and stress variables, their theory contains two internal variables: a variable s > 0 that represents an isotropic intermolecular resistance to plastic flow; and an unsigned variable rq that represents the local free-volume. In this thesis, we extend the work of Anand & Gurtin (2002) to model the deformation and fracture response of amorphous glassy polymers which exhibit both a ductile mechanism of fracture associated with large plastic stretches and subsequent chain scission and a brittle mode of fracture. For polymers such as polycarbonate (PC), the brittle fracture mode is characterized by a mechanism of elastic cavitational failure, which results in cleavage-type fracture similar to that observed in brittle fracture of metals. In contrast, polymers such as polymethylmethacrylate (PMMA) and polystyrene (PS) exhibit a brittle mode of fracture characterized by craze initiation, flow, and breakdown. To model crazing, we introduce a continuum constitutive relation which contains the three ingredients of crazing initiation, widening, and breakdown in a suitable statistically-averaged sense. We allow for local inelastic deformation due to shear yielding in possible concurrence with that due to crazing, and introduce a craze initiation criterion based on the local maximum principal tensile stress reaching a critical 2 value which depends on the local mean normal stress. After crazing has initiated, our continuum model represents the transition from shear-flow to craze-flow by a change in the viscoplastic flow rule, in which the dilational inelastic deformation associated with craze-plasticity is taken to occur in the direction of the local maximum principal stress. Finally, for situations in which the local maximum tensile stress is positive, craze-breakdown and fracture is taken to occur when a local tensile plastic craze strain reaches a critical value. We apply our model to the techologically important polymer, polymethylmethacrylate. We show that the constitutive model and fracture criteria, when properly calibrated, are able to reasonably-well predict the macroscopic loaddisplacement curves, and local aspects of the craze-flow and fracture processes in (a) a thin plate with a circular hole under tension, and (b) notched-beams in four-point bending. For amorphous glassy polymers which show ductile tearing and brittle cavitational fracture, as in the case of the fracture behavior of PC, we modify the model of Anand & Gurtin (2002) for large elastic volume changes to accommodate the experimental observation of the key role of the hydrostatic tension in the nucleation of internal cracks. To model fracture we introduce two simple local fracture criteria: (i) Brittle fracture is taken to occur when a local elastic volumetric strain reaches a failure value; (ii) Ductile fracture will be taken to occur when a measure of the plastic stretch of the polymer chains reaches a critical value. We show that the constitutive model and fracture criteria, when suitably calibrated, are able to quantitatively capture the notch-sensitive fracture response exhibited by notched-beams of polycarbonate in bending. To further demonstrate the capabilities of the constitutive model, we study the micro-indentation behavior of PC and PMMA. Our work in this area builds upon the development of novel flexure-based apparatuses for mechanical testing at the small scale. Details of our nanoand micro-indentation apparatuses as well as a biaxial compression/shear apparatus are presented. We analyze Berkovich and conical microindentation and perform parametric studies with dimensional analysis to elucidate the key material parameters that determine the indentation response. Our study of the forward problem in indentation motivates a framework for the reverse problem in indentation of amorphous glassy polymers. We show that an applicaton of our proposed reverse approach is able to reasonably-well predict the macroscopic stressstrain behavior of polystyrene (PS) in simple compression. Thesis Supervisor: Lallit Anand Title: Professor of Mechanical Engineering