Skip to main content
SpringerLink
Log in

Mechanical characterization and constitutive modeling of aluminum AA1050 subjected to high strain-rates

  • Published:
Mechanics of Time-Dependent Materials Aims and scope Submit manuscript

Abstract

This work develops an elastic-viscoplastic constitutive formulation to the mechanical behavior of polycrystalline FCC metals subjected to high-strain-rate cold deformation, with application to aluminum AA1050. Following a phenomenological approach, the model can account for important macroscopic features related to the mechanical response of ductile metals subjected to high-velocity plastic deformations, such as: (i) strain-hardening; (ii) strain-rate-hardening; and (iii) instantaneous flow stress rate-sensitivity. As a novelty of this proposal, memory effects corresponding to past strain-rate history are considered to affect both the level and rate of material hardening. Another new aspect is the phenomenological representation of both the thermal activation and dislocation drag mechanisms influencing the instantaneous flow stress response. The model calibration and its validation are performed based upon experimental data for commercial pure aluminum AA1050 for wide strain and strain-rate ranges. In comparison with experiments, the proposed model provides suitable predictions for both continuous and sequential compression tests. Overall, due to its relative simplicity and constitutive capabilities, the proposed model proves to be a potential constitutive alternative to simulate engineering problems involving dynamic plastic deformations.

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

Access this article

Log in via an institution

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

Notes

  1. It is worth mentioning that the present approach extends the formulation given by dos Santos et al. (2016), in which only parameter \(A_{\infty }\) was rate-dependent.

  2. We remark that to obtain adequate stress versus strain results from force–displacement data, careful corrections, considering the time delay between displacement and load data acquisition during high-velocity tests, have to be performed. See, for instance, the numerical analysis reported by dos Santos (2016).

  3. While \(\beta _{T}=0.9\) is a common value used for the Taylor–Quinney coefficient in the literature, detailed investigations on its variation in terms of temperature and strain-rate have been reported for instance by Macdougall (2000), Ristinmaa et al. (2007), and Rittel et al. (2017).

References

  • Anand, L.: Constitutive equations for the rate-dependent deformation of metals at elevated temperatures. J. Eng. Mater. Technol. 104, 12–17 (1982)

    Article  Google Scholar 

  • Andrade-Campos, A., Menezes, L.F., Teixeira-Dias, F.: Large deformation processes on AA1050-O aluminium at elevated temperatures. In: Advanced Materials Forum II. Trans Tech Publications., pp. 723–727 (2004)

    Google Scholar 

  • Bambach, M., Heppner, S., Steinmetz, D., Roters, F.: Assessing and ensuring parameter identifiability for a physically-based strain hardening model for twinning-induced plasticity. Mech. Mater. 84, 127–139 (2015)

    Article  Google Scholar 

  • Bamman, D.J., Chiesa, M.L., Johnson, G.C.: Modeling large deformation and failure in manufacturing process. In: Tatsumi, T., Wannabe, E., Kambe, T. (eds.) Theoretical and Applied Mechanics, pp. 359–376 (1996)

    Google Scholar 

  • Bell, J.F.: Plane stress, plane strain, and pure shear at large finite strain. Int. J. Plast. 4, 127–148 (1988)

    Article  Google Scholar 

  • Beyerlein, I.J., Tomé, C.N.: Modeling transients in the mechanical response of copper due to strain path changes. Int. J. Plast. 23, 640–664 (2007)

    Article  MATH  Google Scholar 

  • Bodner, S.R., Partom, Y.: Constitutive equations for elastic-viscoplastic strain-hardening materials. J. Appl. Mech. 42, 385 (1975)

    Article  Google Scholar 

  • Bodner, S.R., Rubin, M.B.: Modeling of hardening at very high strain rates. J. Appl. Phys. 76, 2742–2747 (1994)

    Article  Google Scholar 

  • Brown, S.B., Kim, K.H., Anand, L.: An internal variable constitutive model for hot working of metals. Int. J. Plast. 5, 95–130 (1989)

    Article  MATH  Google Scholar 

  • Chaboche, J.L.: Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plast. 5, 247–302 (1989)

    Article  MATH  Google Scholar 

  • Chaboche, J.L.: A review of some plasticity and viscoplasticity constitutive theories. Int. J. Plast. 24, 1642–1693 (2008)

    Article  MATH  Google Scholar 

  • Chen, W., Wang, F., Kitamura, T., Feng, M.: A modified unified viscoplasticity model considering time-dependent kinematic hardening for stress relaxation with effect of loading history. Int. J. Mech. Sci. 133, 883–892 (2017)

    Article  Google Scholar 

  • Chiem, C., Duffy, J.: Strain rate history effects and observations of dislocation substructure in aluminum single crystals following dynamic deformation. Mater. Sci. Eng. 57, 233–247 (1983)

    Article  Google Scholar 

  • Coleman, B.D., Gurtin, M.E.: Thermodynamics with internal state variables. Rep. Prog. Phys., 47, 597–613 (1967)

    Google Scholar 

  • dos Santos, T.: Experimental characterization and constitutive modeling of viscoplastic effects in high strain-rate deformation of polycrystalline FCC metals. PhD thesis, Universidade Federal do Rio Grande do Sul, Brasil (2016)

  • dos Santos, T., Alves, M.K., Rossi, R.: A constitutive formulation and numerical procedure to model rate effects on porous materials at finite strains. Int. J. Mech. Sci. 93, 166–180 (2015a)

    Article  Google Scholar 

  • dos Santos, T., Ramos, G.R., Rossi, R.: A note on overstress and over-thermodynamic forces derivation for elasto-viscoplastic media: thermodynamic analysis of an isothermal relaxation process. Int. J. Eng. Sci. 93, 13–30 (2015b)

    Article  Google Scholar 

  • dos Santos, T., Rosa, P.A., Maghous, S., Rossi, R.: A simplified approach to high strain rate effects in cold deformation of polycrystalline FCC metals: constitutive formulation and model calibration. Int. J. Plast. 82, 76–96 (2016)

    Article  Google Scholar 

  • dos Santos, T., Outeiro, J.C., Rossi, R., Rosa, P.: A new methodology for evaluation of mechanical properties of materials at very high rates of loading. Proc. CIRP 58, 481–486 (2017). 16th CIRP Conference on Modelling of Machining Operations (16th CIRP CMMO)

    Article  Google Scholar 

  • dos Santos, T., Rossi, R., Maghous, S., Rosa, P.A.: Experimental procedure and simplified modeling for the high strain-rate and transient hardness evolution of aluminum aa1050. Mech. Mater. 122, 42–57 (2018)

    Article  Google Scholar 

  • Duvaut, G., Lions, J.: Inequalities in Mechanics and Physics. Grundlehren der mathematischen. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  • Estrin, Y., Mecking, H.: A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall. 32, 57–70 (1984)

    Article  Google Scholar 

  • Eterovic, A.L., Bathe, K.J.: A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Numer. Methods Eng. 30, 1099–1114 (1990)

    Article  MATH  Google Scholar 

  • Follansbee, P.S.: High strain rate deformation of fcc metals and alloys. In: Murr, L.E., Staudhammer, K.P., Meyers, M.A. (eds.) Metallurgical Applications of Shock-Wave and High-Strain Rate Phenomena, pp. 451–479. Dekker, New York (1986)

    Google Scholar 

  • Follansbee, P., Kocks, U.: A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall. 36, 81–93 (1988)

    Article  Google Scholar 

  • Fourmeau, M., Børvik, T., Benallal, A., Lademo, O., Hopperstad, O.: On the plastic anisotropy of an aluminium alloy and its influence on constrained multiaxial flow. Int. J. Plast. 27, 2005–2025 (2011)

    Article  MATH  Google Scholar 

  • Freed, A., Sandor, B.: The plastic compressibility of 7075-t651 aluminum-alloy plate. Exp. Mech. 26, 119–121 (1986)

    Article  Google Scholar 

  • Gao, C., Zhang, L.: Constitutive modelling of plasticity of fcc metals under extremely high strain rates. Int. J. Plast. 32–33, 121–133 (2012)

    Article  Google Scholar 

  • Geier, M., José, M., Rossi, R., Rosa, P., Martins, P.: Interference-fit joining of aluminium tubes by electromagnetic forming. Adv. Mater. Res. 853, 488–493 (2014)

    Article  Google Scholar 

  • Germain, P., Son Nguyen, Q., Suquet, P.: Continuum thermodynamics. J. Appl. Mech. 50, 1010–1020 (1983)

    Article  MATH  Google Scholar 

  • Hall, R.B.: Combined thermodynamics approach for anisotropic, finite deformation overstress models of viscoplasticity. Int. J. Eng. Sci. 46, 119–130 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Halphen, B., Son Nguyen, Q.: Sur les matériaux standard généralisés. J. Méc. 14, 39–63 (1975)

    MATH  Google Scholar 

  • Hansen, N., Barlow, C.: Plastic deformation of metals and alloys. In: Hono, D.E.L. (ed.) Physical Metallurgy 5th edn. pp. 1681–1764. Elsevier, Oxford (2014)

    Chapter  Google Scholar 

  • Hansen, N., Jensen, D.: Flow stress anisotropy caused by geometrically necessary boundaries. Acta Metall. Mater. 40, 3265–3275 (1992)

    Article  Google Scholar 

  • Higashi, K., Mukai, T., Kaizu, K., Tsuchida, S., Tanimura, S.: Strain rate dependence on mechanical properties in some commercial aluminum alloys. J. Phys. IV Fr. 01, 341–346 (1991)

    Google Scholar 

  • Ho, K.: Effect of the rate dependence of nonlinear kinematic hardening rule on relaxation behavior. Int. J. Solids Struct. 45, 821–839 (2008)

    Article  MATH  Google Scholar 

  • Ho, K.: Thermodynamic formulation of a viscoplastic constitutive model capturing unusual loading rate sensitivity. Int. J. Eng. Sci. 100, 162–170 (2016)

    Article  MATH  Google Scholar 

  • Horstemeyer, M.F., Bammann, D.J.: Historical review of internal state variable theory for inelasticity. Int. J. Plast. 26, 1310–1334 (2010). Special Issue in Honor of David L. McDowell

    Article  MATH  Google Scholar 

  • Huang, F., Tao, N.: Effects of strain rate and deformation temperature on microstructures and hardness in plastically deformed pure aluminum. J. Mater. Sci. Technol. 27, 1–7 (2011)

    Article  Google Scholar 

  • Jordan, J., Siviour, C., Sunny, G., Bramlette, C., Spowart, J.: Strain rate-dependant mechanical properties of OFHC copper. J. Mater. Sci. 48, 7134–7141 (2013)

    Article  Google Scholar 

  • Kestin, J., Rice, J.: Paradoxes in the Application of Thermodynamics to Strained Solids. Technical report, Division of Engineering, Brown University (1969)

  • Kistler-Data-Sheet: SlimLine sensors – measurement of dynamic and quasistatic forces, \(0\dots 3\) kN up to \(0\dots 80\) kN. Kistler Group (2016)

  • Klepaczko, J.: Thermally activated flow and strain rate history effects for some polycrystalline f.c.c. metals. Mater. Sci. Eng. 18, 121–135 (1975)

    Article  Google Scholar 

  • Klepaczko, J., Chiem, C.: On rate sensitivity of f.c.c. metals, instantaneous rate sensitivity and rate sensitivity of strain hardening. J. Mech. Phys. Solids 34, 29–54 (1986)

    Article  Google Scholar 

  • Kocks, U.: Laws for work-hardening and low-temperature creep. J. Eng. Mater. Technol. 98 Ser H, 76–85 (1976)

    Article  Google Scholar 

  • Kocks, U., Mecking, H.: Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, 171–273 (2003)

    Article  Google Scholar 

  • Kocks, U.F., Argon, A.S., Ashby, M.F.: Thermodynamics and kinetics of slip. Prog. Mater. Sci. 19, 1–271 (1975)

    Google Scholar 

  • Krempl, E., Gleason, J.M.: Isotropic viscoplasticity theory based on overstress (VBO). The influence of the direction of the dynamic recovery term in the growth law of the equilibrium stress. Int. J. Plast. 12, 719–735 (1996)

    Article  MATH  Google Scholar 

  • Lea, L., Jardine, A.: Characterisation of high rate plasticity in the uniaxial deformation of high purity copper at elevated temperatures. Int. J. Plast. 102, 41–52 (2018)

    Article  Google Scholar 

  • Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969)

    Article  MATH  Google Scholar 

  • Lubliner, J.: A maximum-dissipation principle in generalized plasticity. Acta Mech. 52, 225–237 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, Z., Zhang, H., Hansen, N., Lu, K.: Quantification of the microstructures of high purity nickel subjected to dynamic plastic deformation. Acta Mater. 60, 1322–1333 (2012)

    Article  Google Scholar 

  • Ma, A., Roters, F.: A constitutive model for fcc single crystals based on dislocation densities and its application to uniaxial compression of aluminium single crystals. Acta Mater. 52, 3603–3612 (2004)

    Article  Google Scholar 

  • Macdougall, D.: Determination of the plastic work converted to heat using radiometry. Exp. Mech. 40, 298–306 (2000)

    Article  Google Scholar 

  • Mamalis, A., Manolakos, D., Kladas, A., Koumoutsos, A.: Electromagnetic forming and powder processing: trends and developments. Appl. Mech. Rev. 57, 299–324 (2004)

    Article  Google Scholar 

  • Mandel, J.: Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Struct. 9, 725–740 (1973)

    Article  MATH  Google Scholar 

  • Mao, Z., An, X., Liao, X., Wang, J.: Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain rates. Mater. Sci. Eng. A 738, 430–438 (2018)

    Article  Google Scholar 

  • Miguélez, M., Soldani, X., Molinari, A.: Analysis of adiabatic shear banding in orthogonal cutting of Ti alloy. Int. J. Mech. Sci. 75, 212–222 (2013)

    Article  Google Scholar 

  • Molinari, A., Ravichandran, G.: Constitutive modeling of high-strain-rate deformation in metals based on the evolution of an effective microstructural length. Mech. Mater. 37, 737–752 (2005)

    Article  Google Scholar 

  • Nemat-Nasser, S., Isaacs, J.: Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta–W alloys. Acta Mater. 45, 907–919 (1997)

    Article  Google Scholar 

  • Nemat-Nasser, S., Li, Y.: Flow stress of f.c.c. polycrystals with application to OFHC Cu. Acta Mater. 46, 565–577 (1998)

    Article  Google Scholar 

  • Nemat-Nasser, S., Ni, L., Okinaka, T.: A constitutive model for fcc crystals with application to polycrystalline OFHC copper. Mech. Mater. 30, 325–341 (1998)

    Article  MATH  Google Scholar 

  • Nes, E.: Modelling of work hardening and stress saturation in FCC metals. Prog. Mater. Sci. 41, 129–193 (1997)

    Article  Google Scholar 

  • Neugebauer, R., Bouzakis, K.D., Denkena, B., Klocke, F., Sterzing, A., Tekkaya, A., Wertheim, R.: Velocity effects in metal forming and machining processes. CIRP Ann. 60, 627–650 (2011)

    Article  Google Scholar 

  • Perić, D.: On a class of constitutive equations in viscoplasticity: formulation and computational issues. Int. J. Numer. Methods Eng. 36, 1365–1393 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Perzyna, P.: The constitutive equations for rate sensitive plastic materials. Q. Appl. Math. 20, 321–332 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • Perzyna, P.: Fundamental Problems in Viscoplasticity. Advances in Applied Mechanics, vol. 9, pp. 243–377. Elsevier, Amsterdam (1966)

    Google Scholar 

  • Perzyna, P.: Thermodynamic Theory of Viscoplasticity. Advances in Applied Mechanics, vol. 11, pp. 313–354. Elsevier, Amsterdam (1971)

    MATH  Google Scholar 

  • Rashid, M.M., Gray, G.T. III, Nemat-Nasser, S.: Heterogeneous deformations in copper single crystals at high and low strain rates. Philos. Mag. A 65, 707–735 (1992)

    Article  Google Scholar 

  • Regazzoni, G., Kocks, U., Follansbee, P.: Dislocation kinetics at high strain rates. Acta Metall. 35, 2865–2875 (1987)

    Article  Google Scholar 

  • Rice, J.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)

    Article  MATH  Google Scholar 

  • Ristinmaa, M., Wallin, M., Ottosen, N.S.: Thermodynamic format and heat generation of isotropic hardening plasticity. Acta Mech. 194, 103–121 (2007)

    Article  MATH  Google Scholar 

  • Rittel, D., Wang, Z.G., Merzer, M.: Adiabatic shear failure and dynamic stored energy of cold work. Phys. Rev. Lett. 96, 075502 (2006)

    Article  Google Scholar 

  • Rittel, D., Landau, P., Venkert, A.: Dynamic recrystallization as a potential cause for adiabatic shear failure. Phys. Rev. Lett. 101, 165501 (2008)

    Article  Google Scholar 

  • Rittel, D., Zhang, L., Osovski, S.: The dependence of the Taylor–Quinney coefficient on the dynamic loading mode. J. Mech. Phys. Solids 107, 96–114 (2017)

    Article  Google Scholar 

  • Rodríguez-Martínez, J.: Advanced Constitutive Relations for Modeling Thermo-viscoplastic Behaviour of Metallic Alloys Subjected to Impact Loading. PhD thesis (2010)

  • Rodríguez-Martínez, J., Rodríguez-Millán, M., Rusinek, A., Arias, A.: A dislocation-based constitutive description for modeling the behavior of FCC metals within wide ranges of strain rate and temperature. Mech. Mater. 43, 901–912 (2011)

    Article  Google Scholar 

  • Rollett, A., Kocks, U., Stout, M., Embury, J., Doherty, R.: Strain hardening at large strains. In: Kettunen, P., Lepisto, T., Lehtonen, M. (eds.) Strength of Metals and Alloys (ICSMA 8), pp. 433–438. Pergamon, Oxford (1989)

    Chapter  Google Scholar 

  • Rusinek, A., Jankowiak, T.: Constitutive relations under impact loadings: experiments, theoretical and numerical aspects. In: Dynamic Behavior of Materials. Constitutive Relations and Applications, pp. 87–135. Springer, Vienna (2014)

    Google Scholar 

  • Rusinek, A., Rodríguez-Martínez, J.: Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects. Mater. Des. 30, 4377–4390 (2009)

    Article  Google Scholar 

  • Salvado, F.C., Teixeira-Dias, F., Walley, S.M., Lea, L.J., Cardoso, J.B.: A review on the strain rate dependency of the dynamic viscoplastic response of FCC metals. Prog. Mater. Sci. 88, 186–231 (2017)

    Article  Google Scholar 

  • Silva, C., Rosa, P., Martins, P.: Innovative testing machines and methodologies for the mechanical characterization of materials. Exp. Tech. 40, 569–581 (2016)

    Article  Google Scholar 

  • Tanner, A.B., McDowell, D.L.: Deformation, temperature and strain rate sequence experiments on OFHC Cu. Int. J. Plast. 15, 375–399 (1999)

    Article  Google Scholar 

  • Tjøtta, S., Mo, A.: A constitutive model for cold deformation of aluminium at large strains and high strain rates. Int. J. Plast. 9, 461–478 (1993)

    Article  Google Scholar 

  • Tong, W., Clifton, R.J., Huang, S.: Pressure-shear impact investigation of strain rate history effects in oxygen-free high-conductivity copper. J. Mech. Phys. Solids 40, 1251–1294 (1992)

    Article  Google Scholar 

  • Voce, E.: The relationship between stress and strain for homogeneous deformation. J. Inst. Met. 74, 537–562 (1948)

    Google Scholar 

  • Wang, W.M., Sluys, L.J., de Borst, R.: Viscoplasticity for instabilities due to strain softening and strain-rate softening. Int. J. Numer. Methods Eng. 40, 3839–3864 (1997)

    Article  MATH  Google Scholar 

  • Wang, B., Liu, Z., Su, G., Song, Q., Ai, X.: Investigations of critical cutting speed and ductile-to-brittle transition mechanism for workpiece material in ultra-high speed machining. Int. J. Mech. Sci. 104, 44–59 (2015)

    Article  Google Scholar 

  • Yan, S., Yang, H., Li, H., Yao, X.: A unified model for coupling constitutive behavior and micro-defects evolution of aluminum alloys under high-strain-rate deformation. Int. J. Plast. 85, 203–229 (2016)

    Article  Google Scholar 

  • Zenasni, Z., Haterbouch, M., Atmani, Z., Atlati, S., Zenasni, M., Nasri, K., Oussouaddi, O.: Physics-based plasticity model incorporating microstructure changes for severe plastic deformation. C. R., Méc. 347, 601–614 (2019)

    Article  Google Scholar 

  • Zener, C., Hollomon, J.H.: Effect of strain rate upon plastic flow of steel. J. Appl. Phys. 15, 22–32 (1944)

    Article  Google Scholar 

  • Zhang, B., Shim, V.: Effect of strain rate on microstructure of polycrystalline oxygen-free high conductivity copper severely deformed at liquid nitrogen temperature. Acta Mater. 58, 6810–6827 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

TdS wishes to acknowledge the doctoral scholarship support of CAPES, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior of Brazil. Process number BEX 7023/15-4. RR wishes to acknowledge the support of CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico of Brazil, grant number 306058/2018-9.

Author information

Authors and Affiliations

  1. Departamento de Engenharia Mecânica, Universidade Federal de Santa Maria, Av. Roraima, 1000, Prédio 7, Santa Maria, RS, 97105-900, Brazil

    Tiago dos Santos

  2. Departamento de Engenharia Mecânica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil

    Rodrigo Rossi

  3. Departamento de Engenharia Civil, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil

    Samir Maghous

  4. Instituto de Engenharia Mecânica, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    Pedro A. R. Rosa

Authors
  1. Tiago dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rodrigo Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samir Maghous
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pedro A. R. Rosa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tiago dos Santos.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Illustration of the strain-rate-dependent hardening evolution

Appendix: Illustration of the strain-rate-dependent hardening evolution

Graphical illustrations of the strain-rate influence on the material hardening evolution are schematically shown in Fig. 13. Figure 13(a) shows the strain-rate influence on the flow stress versus strain response by means of a rate-dependent hardening saturation \(A_{\infty }\), for \(\delta \) constant and \(c=0\). This figure shows a sequential test in which a high-strain-rate loading is followed by unloading and subsequent quasistatic reloading. This curve (solid-line) is compared with quasistatic loading (dashed-line). The comparison shows the past strain-rate influence on the hardening saturation \(A_{\infty }\), where \(A_{\infty }\left (\dot{\varepsilon }\right )\) is greater than the quasi-static value \(A_{\infty }^{lwr}\). Therefore, the quasistatic curve is not recovered by the quasistatic reloading. Furthermore, since \(\dot{\varepsilon }<\dot{\varepsilon }_{up}\), then \(A_{\infty }\left (\dot{\varepsilon }\right )< A_{\infty }^{up}\). Figure 13(b) shows the role played by the rate-dependent parameter \(\delta \left (\dot{\varepsilon }\right )\) on the hardening evolution, for \(A_{\infty }\) constant and \(c=0\). In summary, the higher the imposed strain-rate, the faster the hardening saturation \(A_{\infty }\) is reached.

Fig. 13
figure 13

Schematic representation of the strain-rate influence on (a) hardening saturation \(\left (A_{\infty }\right )\) and (b) nonlinear hardening-rate \(\left (\delta \right )\) parameters

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

dos Santos, T., Rossi, R., Maghous, S. et al. Mechanical characterization and constitutive modeling of aluminum AA1050 subjected to high strain-rates. Mech Time-Depend Mater 26, 347–375 (2022). https://doi.org/10.1007/s11043-021-09491-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11043-021-09491-3

Keywords

Search

Navigation