Fig. 1. Cylindrical sample in uniaxial compression: (a) original dimensions; (b) ideal homogeneous compression; (c) real case, showing barrelling, due to friction and a temperature gradient.

Fig. 2. Methodology for correcting true stress-strain response: (a) Calculating correction $\Delta \sigma=f(T, \dot{\varepsilon},\varepsilon)$ from initial $\sigma=f(T, \dot{\varepsilon}, \varepsilon)$ input to the FE and the predicted σ(ε); (b) applying correction $\Delta \sigma=f(T, \dot{\varepsilon},\varepsilon)$ to the initial $\sigma=f(T, \dot{\varepsilon}, \varepsilon)$ ; (c) validation of corrected constitutive data.

Fig. 3. Microstructure and barrelling of a Zr-2.5Nb sample deformed with a nominal temperature of 800 °C and strain-rate of 0.0032 s-1 [44].

Fig. 4. Thermocouple data for the hot compression stage at selected nominal ε˙ and T conditions.

Fig. 5. Notional true stress-strain curves for Zr-2.5Nb calculated directly from experimental data (solid lines), and the smoothed data fit (dashed lines), at selected nominal ε˙ and T conditions.

Fig. 6. Experimental notional true stress of Zr2.5Nb (solid lines) at a strain of 0.05, with the best fit to the Sellars-Tegart equation (dashed lines).

Fig. 7. Projected views of a second order surface fit to log?$\log\sigma=f(T,\log \dot{\varepsilon})$ (dashed lines), compared with experimental notional true stress of Zr2.5Nb (solid lines) at ε = 0.05.

Fig. 8. Flow stress vs. log(strain-rate) experimental data (for strain = 0.05), and the fitted model (dashed lines) extrapolated to lower temperatures and strain-rates.

Fig. 9. Axisymmetric finite element model of the hot compression dilatometer workpiece: undeformed geometry, mesh and boundary conditions.

Fig. 10. Predictions of the FE model using smoothed stress-strain data as input, at Tnominal =750 °C, $\dot{\varepsilon}_{nominal}$ = 0.1 s-1, for selected combinations of friction coefficient and temperature gradient: (a) force vs. displacement; (b) “notional” true stress vs. strain.

Fig. 11. Stress-strain curves used as input to the FE analysis (solid curves) vs. FE predicted output (dashed curves), for nominal temperatures of 700 and 800 °C and strain-rates of: (a) 0.01 s-1; (b) 3.2 s-1.

Fig. 12. Stress correction Δσ for all experimental combinations of temperature T and log(strain rate, $\dot{\varepsilon}$), for strains of: (a) ε = 0.05; (b) ε = 0.5.

Fig. 13. Corrected stress-strain curves for selected nominal test conditions.

Fig. 14. Experimental (smoothed) stress-strain curves, and FE predicted after correction of the input constitutive response, for the same nominal temperatures and strain-rates as Fig. 11.

Fig. 15. FE results obtained with smoothed, corrected stress-strain data at Tnominal =750 °C, $\dot{\varepsilon}_{ nominal }$ = 0.1 s-1, with μ = 0.5 and ΔT = 50 °C: (a) temperature; (b) equivalent plastic strain; (c) von Mises stress; (d) axial strain-rate; (e) radial strain-rate.

Fig. 16. von Mises equivalent plastic strain as a function of temperature and strain-rate, for Tnominal =750 °C, $\dot{\varepsilon}_{ nominal}$ = 0.1s-1 (indicated by the circle), with μ = 0.5 and Δ T = 50 °C.

Fig. 17. FE predicted evolution of von Mises equivalent plastic strain on the sample mid-plane, at the centre and at half the radius, for nominal T =750 °C, $\dot{\varepsilon}$ = 0.1s-1, with μ = 0.5 and ΔT = 50 °C.

Fig. 18. Ratio of the maximum cross-section area to the average cross-section area, as a function of temperature and log(strain rate): (a) experimental data; (b) FE modelling results.

Fig. 19. Processing maps of strain-rate sensitivity m, for ZrNb alloy, calculated from: (a) cubic fit to the raw experimental data; (b) smoothed data corrected using the FE model.