Haward-Thackray model

Figure 12.23 Schematic diagram of the Haward-Thackray model.

The Gaussian Haward-Thackray model leads to a uniaxial true stress (o ) true strain (e) relationship... 

Figure 23 The Haward-Thackray model for large deformations in solid polymers. Left Illustration of the stress decomposition on rejuvenated PC. Right Model analog.

Haward and Thackray Model, in 1968 Haward and Thackray (37) developed a one-dimensional model for yield using a spring in series with a parallel arrangement of a spring and dashpot (Fig. 12). The dashpot, rather than being Newtonian as with the standard Maxwell/Kelvin models, was instead an Eyring... 

Equation (12.25) was then integrated numerically, using Equations (12.24) and (12.26), to give results like those shown in Eigure 12.24. It can be seen that the Haward and Thackray model is able to reproduce the main features of the stress-strain curve and provide a semi-quantitative fit to the experimental data. However, it may be recalled that the size of the activation volumes are very large compared with the size of an individual molecular segment. 

In order to investigate the correlation between tensile drawing data and Reversed Charpy results, true stress vs. true strain properties of PE are measured at several strain rates and temperatures [6]. Each stress-strain curve is represented using a Haward-Thackray [7] spring-dashpot model, whose applicability to polyethylene has been established empirically by a number of previous studies [5, 7] although the physical basis for the model remains... 

R. N. Haward and G. Thackray, The Use of a Mathematical Model to Describe Isothermal Stress-Strain Curves in Glassy Thermoplastics, Proc. R. Soc. London, Series A, 302 (1471), 453 (1967). 

Haward, R. N. and Thackray, G. (1968) The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics, Proc. Roy. Soc. London, A302, 453 72. 

In an earlier development, Haward and Thackray [55] had proposed a very similar representation to describe the yield behaviour of polymers. Their model is shown schematically in Figure 12.23. The initial part of the stress-strain curve is modelled by the Hookean spring E and the yield point and subsequent strain hardening by the Eyring dashpot and the Langevin spring. Haward and Thackray relate the total strain e and the plastic strain 6a from the activated dashpot to the nominal stress a (load applied divided by initial cross-sectional area). We have... 

Figure 12.24 Stress-strain curves for cellulose nitrate at 23°C. Experimental curves (0) and calculated curves, Langevin model (e). n 1/2 = 0.30, N = 1.57 x chain m . (Redrawn from Haward, R.N. and Thackray, C. (1968) The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. Roy. Soc. A, 302, 453. Copyright (1968).)...

Fotheringham and Cherry [70] adopted a similar representation to Haward and Thackray and used the stress-transient dip test to determine the internal stress or and hence the effective stress a — or acting on the Eyring dashpot. Fotheringham and Cherry proposed a model based on cooperative Eyring processes with the probability of a successful cooperative event involving the simultaneous occurrence of n transitions. Then... 

As shown by Senden et al. [121], conventional material models that incorporate entropic strain hardening give a qualitatively incorrect prediction of the Bauschinger effect. This can be illustrated by investigating the effects of cyclic loading. For a two-arm model such as that of Haward and Thackray, when loaded in tension and then unloaded, during unloading the... 

Yield is generally taken to be an activated process, and the first three of the following theories address this aspect. Models are then presented which address not only the yield of the material but the subsequent strain-softening and strain-hardening events that are observed. The first of these is the Haward and Thackray (37) one-dimensional model, which, while not physically realistic, laid the groundwork for many of the theories that followed it. This is followed in some... 

Boyce, Parks, and Argon model (BRA Model). In a fashion that is conceptually the same as the Haward and Thackray (37) model, the BPA (39)... 

---