The integration schemes of the Preston-Tonks-Wallace (PTW) viscoplasticity model

Abstract

The Preston-Tonks-Wallace (PTW) viscoplasticity model is valid in the wide range of strain, strain rate, and temperature, and therefore has been used in various applications. In particular, it has incorporated the flow properties in the strong shock limit, where the strain rates span the range of 10−3−1012 sec−1 [1].In this paper, we examine how the Preston-Tonks-Wallace (PTW) flow stress was constructed: the closed form expression of the flow stress, which is known as the PTW model was obtained by integrating the differential form of the hardening law under the assumption that the strain rate is constant. We then consider there are cases, like explosively driven deformation and high-velocity impacts, where this is not true and how to use this model. As a case study, we choose a gaussian function as a strain rate history and compare two different ways to use the PTW model. First, we integrate the differential form of the PTW model numerically, coupled with this non-constant strain rate function. That way, we let the integral, and therefore the resulting flow stress path-dependent. Second, we use the closed form of the PTW, which was already integrated for the fixed strain rate and plug in the strain rate value at each discretized time. As will be seen, the discrepancy between the two methods is clear when the strain rate is decreasing.We draw the conclusion that based on the physical and mathematical arguments, one should solve the system of ODEs consisting of the differential form of the PTW model and the particular strain rate history of interest rather than using the closed form expression when the strain rate variation is large.