Abstract
We study an instability occurring at high strain-rate deformations, induced by thermal softening properties of metals, and leading to the formation of shear bands. We consider adiabatic shear deformations of thermoviscoplastic materials and establish the existence of a family of focusing self-similar solutions that capture this instability. The self-similar solutions emerge as the net response resulting from the competition between Hadamard instability and viscosity. Their existence is turned into a problem of constructing a heteroclinic orbit for an associated dynamical system, which is achieved with the help of geometric singular perturbation theory.
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References
M. Bertsch, L. Peletier, S. Verduyn Lunel, The effect of temperature dependent viscosity on shear flow of incompressible fluids. SIAM J. Math. Anal. 22, 328–343 (1991)
R.J. Clifton, J. Duffy, K.A. Hartley, T.G. Shawki, On critical conditions for shear band formation at high strain rates. Scripta Met. 18, 443–448 (1984)
C.M. Dafermos, L. Hsiao, Adiabatic shearing of incompressible fluids with temperature-dependent viscosity. Q. App. Math. 41, 45–58 (1983)
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)
C. Fressengeas, A. Molinari, Instability and localization of plastic flow in shear at high strain rates. J. Mech. Phys. Solids 35, 185–211 (1987)
Th. Katsaounis, A.E. Tzavaras, Effective equations for localization and shear band formation. SIAM J. Appl. Math. 69, 1618–1643 (2009)
Th. Katsaounis, J. Olivier, A.E. Tzavaras, Emergence of coherent localized structures in shear deformations of temperature dependent fluids. Arch. Ration. Mech. Anal. https://doi.org/10.1007/s00205-016-1071-2. (online)
C. Kuehn, Multiple Time Scale Dynamics. Applied Mathematical Sciences, vol. 191 (Springer, Basel, 2015)
M-G. Lee, A.E. Tzavaras, Existence of localizing solutions in plasticity via the geometric singular perturbation theory. SIAM J. Appl. Dyn. Syst. arXiv:1608.00198. (to appear)
M-G. Lee, Th. Katsaounis, A.E. Tzavaras, Localization and the formation of shear bands in thermoviscoplastic deformations. (in preparation)
Th. Katsaounis, M.-G. Lee, A.E. Tzavaras, Localization in inelastic rate dependent shearing deformations. J. Mech. Phys. Solids 98, 106–125 (2017)
T.G. Shawki, R.J. Clifton, Shear band formation in thermal viscoplastic materials. Mech. Mater. 8, 13–43 (1989)
A.E. Tzavaras, Shearing of materials exhibiting thermal softening or temperature dependent viscosity. Q. Appl. Math. 44, 1–12 (1986)
A.E. Tzavaras, Nonlinear analysis techniques for shear band formation at high strain-rates. Appl. Mech. Rev. 45, S82–S94 (1992)
C. Zener, J.H. Hollomon, Effect of strain rate upon plastic flow of steel. J. Appl. Phys. 15, 22–32 (1944)
Acknowledgements
This research was supported by King Abdullah University of Science and Technology (KAUST).
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Lee, MG., Katsaounis, T., Tzavaras, A.E. (2018). Localization of Adiabatic Deformations in Thermoviscoplastic Materials. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_21
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