Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Approximate linear stability analysis of a model of adiabatic shear band formation

Author: Timothy J. Burns
Journal: Quart. Appl. Math. 43 (1985), 65-84
MSC: Primary 73E50; Secondary 73M15
DOI: https://doi.org/10.1090/qam/782257
MathSciNet review: 782257
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The formation of adiabatic shear bands in ductile metals under dynamic loading conditions is generally thought to result from a material instability, which is associated with a peak in the curve of engineering plastic flow stress vs. engineering shear strain. This instability arises from the effect of thermal softening, caused by irreversible adiabatic heating, which counteracts the tendency of the material to harden with increasing plastic strain. An approximate linear stability analysis of a one-dimensional rigid-thermoviscoplastic model, based on data taken from dynamic torsion experiments on thin-walled tubes of mild steel, shows that shear band formation in this situation can be interpreted as a bifurcation from a homogeneous simple shearing deformation which occurs at the peak in the homogeneous stress vs. strain curve. The asymptotic method of multiple scales is used to show that the growth rate of small perturbations on the homogeneous deformation is controlled by the ratio of the slope of the homogeneous stress vs. strain curve to the material viscosity, i.e., the rate of change of the plastic flow stress with respect to the strain-rate. In addition, it is shown that this growth rate is essentially independent of wavelength in any small perturbation. Numerical methods are used to show that this growth rate beyond the bifurcation point may not be sufficiently large for the model to account for the experimental data, and some suggestions are made on how to modify the constitutive equation so that it better fits the experimental data.

References [Enhancements On Off] (What's this?)

  • [1] Y. L. Bai, Thermo-plastic instability in simple shear, J. Mech. Phys. Solids 30, 195-207 (1982)
  • [2] Timothy J. Burns, Dynamic instability of a homogenous deformation of a thin elastic bar, Quart. Appl. Math. 40 (1982/83), no. 3, 357–361. MR 678208, https://doi.org/10.1090/S0033-569X-1982-0678208-6
  • [3] T. J. Burns and T. G. Trucano, Instability in simple shear deformations of strain-softening materials, Mech. Mat. 1, 313-324 (1982)
  • [4] R. J. Clifton, Adiabatic shear banding, in Materials response to ultra-high loading rates, NMAB-356, National Materials Advisory Board (NRC), Washiangton, D.C., 1980, Chap. 8
  • [5] L. S. Costin, E. E. Crisman, R. H. Hawley, and J. Duffy, On the localisation of plastic flow in mild steel tubes under dynamic torsional loading, in Proc. 2nd conf. on mechanical properties of materials at high rates of strain, (J. Harding, ed.) Inst. Phys. Conf. Ser. No. 47: Chap. 1, p. 90, Adam Hilger, Bristol, 1979
  • [6] C. M. Dafermos and L. Hsiao, Adiabatic shearing of incompressible fluids with temperature-dependent viscosity, Quart. Appl. Math. 41 (1983/84), no. 1, 45–58. MR 700660, https://doi.org/10.1090/S0033-569X-1983-0700660-8
  • [7] D. E. Grady, Shock deformation of brittle solids, J. Geophys. Res. (B2) 85, 913-924 (1980)
  • [8] J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
  • [9] L. D. Landau and E. M. Lifschitz, Fluid mechanics, Pergamon Press, Oxford, 1959, Sec. 27
  • [10] J. Litonski, Plastic flow of a tube under adiabatic torsion, Bull. l'Academie Polonaise des Sciences 25, 7-14 (1977)
  • [11] A. M. Merzer, Modelling of adiabatic shear band development from small imperfections, J. Mech. Phys. Solids 30, 323-338 (1982)
  • [12] E. Orowan, Mechanism of seismic faulting, in Rock deformation (a symposium), D. Griggs & J. Handin, ed., Geological Society of American Memoir 79, Waverly Press, Baltimore, 1960, Chap. 12
  • [13] J. Pan, Perturbation analysis of shear strain localization in rate sensitive materials, Int. J. Solids Structures 19, 153-164 (1983)
  • [14] H. C. Rogers, Adiabatic plastic deformation, Ann. Rev. Mater. Sci. 9, 283-311 (1979)
  • [15] L. F. Shampine and M. K. Gordon, Computer solution of ordinary differential equations, W. H. Freeman and Co., San Francisco, Calif., 1975. The initial value problem. MR 0478627
  • [16] D. Tabor, Gases, liquids and solids, 2nd ed., Cambridge University Press, Cambridge, 1979, Chap. 11
  • [17] R. E. Winter and J. E. Field, The role of localized plastic flow in the impact initiation of explosives, Proc. R. Soc. Lond. A343, 399-413 (1975)
  • [18] C. Zener and J. H. Hollomon, Effect of strain rate upon plastic flow of steel, J. Appl. Phys. 15, 22-32 (1944)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73E50, 73M15

Retrieve articles in all journals with MSC: 73E50, 73M15

Additional Information

DOI: https://doi.org/10.1090/qam/782257
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society