Skip to main content
Log in

Extension of the Mechanical Threshold Stress Model to Static and Dynamic Strain Aging: Application to AA5754-O

  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

Based on the mechanical threshold stress model and the visco-plastic self-consistent algorithm, a modified constitutive model is developed to model static strain aging and dynamic strain aging for application to a non-heat treatable aluminum alloy, AA5754-O. The implementation is based on a combination of the evolution of dislocation density and the effect of solutes on both mobile dislocations and forest dislocations. Using this model, the stress–strain behavior of AA5754-O is simulated in multi-path, multi-temperature, and variable strain rate tensile tests. The low temperature and strain rate sensitivities of the modified mechanical threshold stress model in the dynamic strain aging regime are successfully accounted for. The results show quantitative agreement with experimental data from multiple sources.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. H. Lim, C. C. Battaile; J. L. Brown and C. R. Weinberger: Model. Simul. Mater. Sci. Eng., 2016, vol. 24, p. 55018.

    Article  Google Scholar 

  2. U. F. Kocks: Philos. Mag., 1966, vol. 13, pp. 541–566.

    Article  Google Scholar 

  3. J.L. Amorós, A.H. Cottrell, A. Seeger: Deformation and Flow of Solids/Verformung Und Fliessen Des Festkörpers, Springer, Berlin (1956).

    Google Scholar 

  4. P. S. Follansbee: Fundamentals of Strength, 1st ed., John Wiley & Sons, Inc., Hoboken, NJ, USA, 2013.

    Google Scholar 

  5. A. H. Cottrell and B. A. Bilby: Proc. Phys. Soc. Sect. A, 1949, vol. 62, p. 49.

    Article  Google Scholar 

  6. H. Mecking and U. F. Kocks: Acta Metall., 1981, vol. 29, pp. 1865–75.

    Article  Google Scholar 

  7. A. Van Den Beukel and U. F. Kocks: Acta Metall., 1982, vol. 30, pp. 1027–34.

    Article  Google Scholar 

  8. U.F. Kocks, A.S. Argon, and M.F. Ashby: Thermodynamics and Kinetics of Slip. Pergamon Press, Oxford, 1975.

    Google Scholar 

  9. P. S. Follansbee and U. F. Kocks: Acta Metall., 1988, vol. 36, pp. 81–93.

    Article  Google Scholar 

  10. A.J. Beaudoin, U.F. Kocks, S.R. MacEwen, S.R. Chen, and M.G. Stout: Hot Deformation of Aluminum Alloys II Symposium, 1999.

  11. S. Kok, A. J. Beaudoin, and D. A. Tortorelli: Int. J. Plast., 2002, vol. 18, pp. 715–41.

    Article  Google Scholar 

  12. B. Banerjee and A. Bhawalkar: J. Mech. Mater. Struct., 2008, vol. 3, pp. 391–424.

    Article  Google Scholar 

  13. M. A. Iadicola, L. Hu, A. D. Rollett, and T. Foecke: Int. J. Solids Struct., 2012, vol. 49, pp. 3507–16.

    Article  Google Scholar 

  14. A. van den Beukel: Scr. Metall., 1983, vol. 17, pp. 659–63.

    Article  Google Scholar 

  15. N. Louat: Scr. Metall., 1981, vol. 15, pp. 1167–70.

    Article  Google Scholar 

  16. P. Franciosi: Acta Metall., 1985, vol. 33, pp. 1601–12.

    Article  Google Scholar 

  17. P. S. Follansbee: Metall. Mater. Trans. A, 2016, vol. 47, pp. 4455–66.

    Article  Google Scholar 

  18. Y. Estrin and L.P. Kubin: Acta Metall., 1986, vol. 34, pp. 2455–64.

    Article  Google Scholar 

  19. L. P. Kubin, Y. Estrin, and C. Perrier: Acta Metall. Mater., 1992, vol. 40, pp. 1037–44.

    Article  Google Scholar 

  20. C. Fressengeas, A. J. Beaudoin, M. Lebyodkin, L. P. Kubin, and Y. Estrin: Mater. Sci. Eng. A, 2005, vol. 400–401, pp. 226–30.

    Article  Google Scholar 

  21. R. A. Lebensohn and C. N. Tomé: Acta Metall. Mater., 1993, vol. 41, pp. 2611–24.

    Article  Google Scholar 

  22. A. H. Cottrell: Dislocations and Plastic Flow in Crystals, 1st ed., Oxford University Press, New York, 1953.

    Google Scholar 

  23. J. Friedel: Dislocations, 1st ed., Addison-Wesley Publishing Company Inc, Massachusetts, 1964.

    Google Scholar 

  24. E. Nes: Acta Metall. Mater., 1995, vol. 43, pp. 2189–2207.

    Article  Google Scholar 

  25. M. Pham, M. Iadicola, A. Creuziger, L. Hu, and A. D. Rollett: Int. J. Plast., 2015, vol. 75, pp. 226–43.

    Article  Google Scholar 

  26. N. Abedrabbo, F. Pourboghrat, and J. Carsley: Int. J. Plast., 2007, vol. 23, pp. 841–75.

    Article  Google Scholar 

  27. F. Ozturk, S. Toros, and H. Pekel: Mater. Sci. Technol., 2009, vol. 25, pp. 919–24.

    Article  Google Scholar 

  28. K. Zhao, R. Fan and L. Wang: J. Mater. Eng. Perform., 2016, vol. 25, pp. 781–89.

    Article  Google Scholar 

  29. J. Cheng and S. Nemat-Nasser: Acta Mater., 2000, vol. 48, pp. 3131–44.

    Article  Google Scholar 

  30. M. A. Soare and W. A. Curtin: Acta Mater., 2008, vol. 56, pp. 4046–61.

    Article  Google Scholar 

  31. M. A. Soare and W. A. Curtin: Acta Mater., 2008, vol. 56, pp. 4091–4101.

    Article  Google Scholar 

  32. F. Zhang, A. F. Bower, and W. A. Curtin: Int. J. Numer. Methods Eng., 2011, vol. 86, pp. 70–92.

    Article  Google Scholar 

  33. J. Cheng, S. Nemat-Nasser, and W. Guo: Mech. Mater., 2001, vol. 33, pp. 603–16.

    Article  Google Scholar 

  34. S. Nemat-Nasser and Y. Li: Acta Mater., 1998, vol. 46, pp. 565–77.

    Article  Google Scholar 

  35. W. A. Curtin, D. L. Olmsted, and L. G. Hector: Nat. Mater., 2006, vol. 5, pp. 875–80.

    Article  Google Scholar 

  36. S. M. Keralavarma, A. F. Bower, and W. A. Curtin: Nat. Commun., 2014, vol. 5, pp. 4604–12.

    Article  Google Scholar 

  37. J. D. Eshelby: Ann. Phys., 1957, vol. 456, pp. 116–21.

    Article  Google Scholar 

  38. P. Franciosi, M. Berveiller, and A. Zaoui: Acta Metall., 1980, vol. 28, pp. 273–83.

    Article  Google Scholar 

  39. A. S. Argon and E. Orowan: Physics of Strength and Plasticity, Cambridge, M.I.T. Press, 1969.

    Google Scholar 

  40. U. Essmann and H. Mughrabi: Philos. Mag. A, 1979, vol. 40, pp. 731–56.

    Article  Google Scholar 

  41. L.P. Kubin and Y. Estrin: Acta Metall. Mater., 1990, vol. 38, pp. 697–708.

    Article  Google Scholar 

  42. B. S. Anglin, B. T. Gockel, and A. D. Rollett: Integr. Mater. Manuf. Innov., 2016, vol. 5, p. 11.

    Article  Google Scholar 

  43. L. Hu, A. D. Rollett, M. Iadicola, T. Foecke, and S. Banovic: Metall. Mater. Trans. A, 2012, vol. 43, pp. 854–69.

    Article  Google Scholar 

  44. MATLAB: Version 8.5.0 (R2015a), The MathWorks Inc., Natick, 2015.

  45. M.S. Pham, A. Creuziger, M. Iadicola, and A.D. Rollett: J. Mech. Phys. Solids, 2016, DOI:10.1016/j.jmps.2016.08.011.

Download references

Acknowledgments

ADR and SM acknowledge support from the Boeing company for this work. Discussions with M.S. Pham of Imperial College, London, Paul Follansbee of St. Vincent College, and W.A. Curtin of the Ecole Polytechnique Federale de Lausanne (EPFL) are gratefully acknowledged. Support by the NIST Center for Automotive Lightweighting is also acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu Feng.

Additional information

Manuscript submitted December 2, 2016.

Appendices

Appendix I: The Derivation of \( \tau_{\text{e}} \) in the Modified MTS Model

In the main paper, a brief derivation of \( \tau_{\text{e}} \) was described. Here, we provide a more complete description of the definition of \( \tau_{\text{e}} \). For more details, the reader is referred to Curtin et al.[30,31,32,35]

$$ \tau_{\text{es}} = \left( {1 + \delta } \right) \tau_{{{\text{es}}0}} \left( {\frac{{\dot{\gamma }}}{{\dot{\gamma }_{{{\text{es}}0}} }}} \right)^{{\left( {\frac{kT}{{g_{{{\text{es}}0}} \mu b^{3} }}} \right)}} $$
(A1.1)

In the expression of \( \tau_{\text{es}} \), we introduce an extra variable, \( \delta \), in our equation. As mentioned before, \( \delta \) is described by the equation given below:

$$ \delta = R^{\prime}\times\tanh \left( { \frac{{g_{{0{\text{w}}}} \mu b^{3} }}{2kT} } \right)\times\left\{ { 1 - \exp \left\{ {\frac{{\dot{\gamma }_{{ 0 {\text{c}}}} }}{{\dot{\gamma }}}\exp \left( { - \frac{{g_{{0{\text{c}}}} \mu b^{3} }}{kT}} \right)} \right\}^{1/3} } \right\}. $$
(A1.2)

Here \( R' \) is a dimensionless pre-factor. In Curtin et al.’s, \( R' \) was described as

$$ R^{\prime} = 8.7\frac{{2c\bar{w}}}{{\sqrt {3b^{2} } }}\frac{{\Delta \bar{G}_{\text{W}} }}{A} $$
(A1.3)

where \( c \), \( \bar{w} \), \( b \), and \( \Delta \bar{G}_{\text{W}} \) were introduced previously. A is the line tension energy per unit length of dislocation, which can be estimated by

$$ A = \frac{{3\mu b^{2} }}{{4\pi \left( {1 - v} \right)}}, $$
(A1.4)

where \( v \) is the Poisson ratio of the material. Here both \( A \times b \) and \( \Delta \bar{G}_{\text{W}} \) can be normalized by \( \mu b^{3} \). Only \( 8.7\frac{{2c\bar{w}}}{\sqrt 3 b} \) is left on the right of the expression. Curtin et al. showed that \( \bar{w} \) is on the scale of 7.5 \( b \), and they estimated \( R^{\prime} \) is 0.4 for Al-Mg dilute alloys. Thus, for simplicity in our model, we assume a fixed value of \( R^{\prime} \) as a characteristic constant. Note that we also include the bulk solute concentration in \( R' \) for simplification. In order to investigate a related alloy with different bulk solute concentrations, \( R' \) can be further expanded to be a function of solute concentration, \( C \).

Figure A1 shows two curves calculated from the two characteristic functions, \( \left[ {1 - \exp \left( {1 - \exp \left( x \right)} \right) } \right] \) (Figure A1(a)) and \( \tanh \left( x \right) \) (Figure A1(b)), in a DSA simulation. From a mathematical perspective, the first one gives a temperature-dependent critical point above which DSA will be invoked at a given strain rate. The second function, \( \tanh \left( x \right) \), washes away the DSA effect at high temperatures. Note that both the X and Y axes in the two plots are normalized.

Fig. A1
figure 13

Two characteristic functions, \( \left[ {1 - \exp \left( {1 - \exp \left( x \right)} \right)} \right] \) (a) and \( \tanh \left( x \right) \) (b), are plotted. Note that, both temperature and energy scale are normalized in two plots

Finally, the overall expression of \( \tau_{\text{e}} \) is shown below:

$$ \tau_{\text{es}} = \left( {1 + \delta } \right)\tau_{{{\text{es}}0}} \left( { \frac{{\dot{\gamma }}}{{\dot{\gamma }_{{{\text{es}}0}} }} } \right)^{{\left( {\frac{kT}{{g_{{0{\text{es}}}} \mu b^{3} }}} \right)}}, $$
(A1.5)
$$ \delta = R^{\prime}\; \times \;\tanh \left( {\frac{{g_{{ 0 {\text{w}}}} \mu b^{3} }}{2kT}} \right)\; \times \;\left\{ {1 - \exp \left\{ {\frac{{\dot{\gamma }_{{ 0 {\text{c}}}} }}{{\dot{\gamma }}}\exp \left( { - \frac{{g_{{ 0 {\text{c}}}} \mu b^{3} }}{kT}} \right)} \right\}^{1/3} } \right\}, $$
(A1.6)
$$ \frac{{{\text{d}}\hat{\tau }_{\text{e}} }}{{{\text{d}}\gamma }} = \left( {\frac{{{\text{d}}\hat{\tau }_{\text{e}} }}{{{\text{d}}\gamma }}} \right)_{0} F\left( {\frac{{\hat{\tau }_{\text{e}} }}{{\hat{\tau }_{\text{es}} }}} \right) = \theta_{0} \left\{ { 1 - \frac{{\hat{\tau }_{\text{e}} }}{{\hat{\tau }_{\text{es}} }} } \right\}^{k}. $$
(A1.7)

Appendix II: Matlab Routine for SSA and DSA Simulation

By using MATLAB[44] as a platform, similar optimization routines are applied to optimize the MTS parameters in SSA and DSA. In DSA modeling, the simulation routine is more straightforward. After reading the experimental data and the initial estimation of MTS, the main function calls the VPSC program and reads back the simulation data from VPSC outputs, then the main function (including Eqs. [2] through [7] and [21] through [29]) calculates the difference between the two sets of data and loops again until the optimization criterion is satisfied as mentioned above. The DSA simulation routine is shown in Figure A2.

Fig. A2
figure 14

Block diagram for the parameter fitting procedure in DSA modeling

However, the SSA simulation routine needs two segments to finish one loop. In SSA simulation routine, the main loop also starts with reading the experimental data into MATLAB. After the experimental data have been read, the MATLAB routine calls the VPSC program and specifies the explicit strain path, prescribed strain and initial MTS parameters as input for VPSC program. Once the VPSC has read those inputs, it starts calculating (including Eqs. [2] through [7]) until the end of simulation in segment 1. Then, the program outputs the results in segment 1 and preserves the present states for the flow stress and texture. After that, MATLAB calculates the updated MTS parameters, here, the new value of \( \Delta \tau_{\text{SSA}} \), \( \tau_{\text{eini}} \), \( \tau_{\text{es0}}, \) and \( \theta_{0} \) by using the SSA model (Eqs. [8] through [15]) and SSR (Eqs. [16] through [20]). The newly calculated MTS parameters are written in the input file for the next VPSC simulation in segment 2. Once the updating finishes, the MATLAB routine calls VPSC again, and this time, the VPSC program reads back in the updated input file with the preserved output file and completes the rest of the simulation. The optimization routine is aimed at generating the best fit and optimizing the value of the unknown parameters. The schematic of this fitting routine is shown in Figure A3.

Fig. A3
figure 15

Block diagram for the parameter fitting procedure in SSA modeling

Note that the input requirement of MTS parameters in the VPSC program is at the “slip system” level (microscopic level). When calculating the hardening rate for the second stage, we arbitrarily chose the Taylor factor equal to 2.8 for converting between the value of the initial evolution structure, \( \tau_{\text{eini}} \) (microscopic level) to \( \sigma_{\text{eini}} \) (macroscopic level). The next section shows the explicit values of the parameters in the SSA and DSA simulation.

Appendix III: The Tables of Relevant MTS Parameters for SSA and DSA

Note that, because of the discrepancies between the different data sources, the yield stress and UTS are quite different as Table II shows. Comparing Figures 1 and 8, the evolution in flow stress of the different source materials is also different at equivalent test conditions. Such stress differences will affect the MTS parameters, e.g., \( \hat{\tau }_{\text{a}} \), \( \hat{\tau }_{{{\text{es}}0}} \). Tables VI and VII show more details. Therefore, we used two different sets of MTS parameters in order to obtain acceptable fits for the SSA and DSA fittings.

Table VI Table of the Relevant Parameters in the SSA Model
Table VII Table of the Relevant Parameters in the DSA Model

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, Y., Mandal, S., Gockel, B. et al. Extension of the Mechanical Threshold Stress Model to Static and Dynamic Strain Aging: Application to AA5754-O. Metall Mater Trans A 48, 5591–5607 (2017). https://doi.org/10.1007/s11661-017-4276-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-017-4276-6

Navigation