Constitutive equations and failure criteria for amorphous polymeric solids

Abstract

Anand & Gurtin (2002) have recently developed a continuum theory for the elastic-viscoplastic 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 7 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.

(cont.) 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 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 ...