The role of material non-homogeneities on the formation and evolution of strain non-uniformities in thermoviscoplastic shearing
Authors:
Theocharis Baxevanis and Nicolas Charalambakis
Journal:
Quart. Appl. Math. 62 (2004), 97-116
MSC:
Primary 74C10; Secondary 35B30, 35B40, 74E05, 74F05
DOI:
https://doi.org/10.1090/qam/2032574
MathSciNet review:
MR2032574
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Abstract: In this paper we present the effect of macroscopic non-homogeneities on the distribution and evolution of strain non-uniformities during the shearing of thermoviscoplastic materials. The thermomechanical parameters (strain hardening, strain rate sensitivity, and thermal softening), as well as all of the material parameters are supposed to depend explicitly on the space variable. We show that, even under stability conditions, the strain exhibits intense, time-increasing non-uniformities, following the non-homogeneities, in a specific rate, which depends on the degree of non-homogeneity exhibited by the thermomechanical parameters. By considering both the isothermal and anisothermal cases, we obtain results indicating that non-uniformity measures, based on the control of the strain gradient, are more suitable to give stability conditions of non-homogeneous materials. Moreover, we present numerical results concerning the interplay between material non-homogeneities, initial defects, and boundary conditions for two specific cases: the shearing of a reinforced slab and the shearing of a plate composed of several thin layers of periodic structure. The results are in complete agreement with the analytical behavior of the model.
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Wright, T.W. and Walter, J.W., (1987) On stress collapse in adiabatic shear bands. J. Mech. Phys. Solids 35, pp 701 - 720.
Bai, Y., (1982) Thermoplastic instability in simple shear. J. Mech. Phys. Solids 30, pp 195 - 207.
Bensoussan, A., Lions, J.L., Papanikolaou, G., 1978. Asymptotic Analysis for Periodic Structures, North Holland.
Burns, T., (1983) SAND83-1907, Sandia National Laboratories, UC-32.
Charalambakis, N., (1984) Adiabatic shearing flow caused by time dependent inertial force. Quart. Appl. Math. 42, pp 275 - 280.
Charalambakis, N., (1985) Time-asymptotic stability of non-Newtonian fluid or plastic solid. Mech. Research Com. 12, pp 311 - 317.
Charalambakis, N. and Murat, F., (1989) Weak solutions to the initial-boundary problem for the shearing of non-homogeneous thermoviscoplastic materials. Proc. Royal Soc. Edinburgh 113A, pp 257 - 265.
Charalambakis, N. and Murat, F., (1990) Homogenization of stratified thermoviscoplastic materials, unpublished.
Charalambakis, N., (2001) Shear stability and strain, strain-rate and temperature dependent cold work. Int. J. Eng. Sci. 39, pp 1899 - 1911.
Chen, H.T., Douglas, A.S., and Malek-Madani, R., (1989) An asymptotic stability condition for inhomogeneous simple shear. Quart. Appl. Math. 47, pp 247 - 262.
Clifton, R.J., Duffy, J., Hartley, K.A., and Shawki, T.G., (1984) On critical conditions for shear band formation at high strain-rates. Scripta Met. 18, pp 443 - 448.
Dafermos, C.M., (1982) Global smooth solutions to the initial boundary value problem of one dimensional nonlinear thermoviscoelasticity. SIAM J. Math. Analysis 13, pp 397 - 408.
Dafermos, C.M. and Hsiao, L., (1983) Adiabatic shearing of incompressible fluids with temperature dependent viscosity. Quart. Appl. Math 41, pp 45 - 58.
Dafermos, C.M., (1985) Contemporary Issues in the Dynamic Behavior of Continuous Media, LCDS Lecture Notes # 85-1.
Dodd, B. and Bai, Y., (1987) Ductile Fracture and Ductility, Academic Press.
Fressengeas, C. and Molinari, A., (1987) Instability and localization of plastic flow in shear at high strain rates. J. Mech. Phys. Solids 35, pp 185 - 211.
Hodowany, J., Ravichandran, G., Rosakis, A.J., and Rosakis, P., (2000) Partition of plastic work into heat and stored energy in metals. J. Exper. Mech. 40, pp 113 - 123.
Houstis, E. N., (1977) Application of method of collocation on lines for solving hyperbolic systems, Math. Comp. 31, pp 443 - 456.
Leroy, Y. and Molinari, A., (1992) Stability of steady states in shear zones. J. Mech. Phys. Solids 40, pp 181 - 212.
Liu, Q. and Jiang, S., (2003) Global existence and numerical approximation in adiabatic shearing of incompressible fluids. J. Comput. Appl. Math. 154(2), pp. 303–317.
Madsen, N. K. and Sincovec, R. F., (1979) PDECOL : General collocation software for PDE, ACN. Trans. Math. Software 5.
Maddoks, J.H. and Malek-Madani, R., (1992) Steady-state shear-bands in thermoplasticity, Part I: Vanishing yield stress. Int. J. Solids Structures 29, pp 2039 - 2061.
Molinari, A., (1985) Instabilite thermoviscoplastique en cisaillement simple. J. Mec. Theor. Appl. 4, pp 659 - 684.
Molinari, A. and Clifton, R.J., (1987) Analytical characterization of shear localization in thermoviscoplastic materials. Journal of Applied Mechanics54, pp 806 - 812.
Protter, M.H. and Weinberger, H.F., (1967) Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey.
Rigatos, A. and Charalambakis, N., (2001) Two-dimensional adiabatic newtonian flow with temperature dependent viscosity. Int. J. Eng. Sci. 39/10, pp 1143 - 1165.
Rosakis, P., Rosakis, A.J., Ravichandran, G., and Hodowany, J., (2000) A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals. J. Mech. Phys. Solids 48, pp 582 - 607.
Shawki, T.G. and Clifton, R.J., (1989) Shear band formation in thermal viscoplastic materials. Mech. Materials 8, pp 13 - 43.
Suquet, P., (1982) Plasticite et homogeneisation, These de Doctorat dÉtat, Univ. Pierre et Marie Curie, Paris.
Tzavaras, A.E., (1986) Plastic shearing of materials exhibiting strain hardening or strain softening. Arch. Rat. Mech. Anal. 94, pp 39 - 58.
Tzavaras, A.E., (1987) Effect of thermal softening in shearing of strain-rate dependent materials. Arch. Rat. Mech. Anal. 99, pp 349 - 374.f
Tzavaras, A.E., (1991) Strain softening in viscoelasticity of the rate type. J. Integr. Eq. Appl. 3, pp 195 - 238.
Tzavaras, A.E., (1992) Nonlinear analysis techniques for shear band formation at high strain rates. Appl. Mech. Reviews 45, pp 82 - 94.
Wright, T.W. and Batra, R.C., (1985) The initiation and growth of adiabatic shear bands. Int. J. Plasticity 1, pp 205 - 212.
Wright, T.W. and Walter, J.W., (1987) On stress collapse in adiabatic shear bands. J. Mech. Phys. Solids 35, pp 701 - 720.
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