10.24423/engtrans.140.2011
Micromechanics of Localized Fracture Phenomena in Inelastic Solids Generated by Impact-Loaded Adiabatic Processes
The main objective of the present paper is to discuss very efficient procedure of the numerical investigation of localized fracture in inelastic solids generated by impact-loaded adiabatic processes. Particular attention is focused on the proper description of a ductile mode of fracture propagating along the shear band for high impact velocities. This procedure of investigation is based on utilization the finite difference method for regularized thermo-elasto-viscoplastic constitutive model of damaged material. A general constitutive model of thermo-elasto-viscoplastic damaged polycrystalline solids with a finite set of internal state variables is used. The set of internal state variables consists of two scalars, namely equivalent inelastic deformation and volume fraction porosity. The equivalent inelastic deformation can describe the dissipation effects generated by viscoplastic flow phenomena and the volume fraction porosity takes into account the microdamage evolution effects. The relaxation time is used as a regularization parameter. Fracture criterion based on the evolution of microdamage is assumed.
As a numerical example we consider dynamic shear band propagation and localized fracture in an asymmetrically impact-loaded prenotched thin plate. The impact loading is simulated by a velocity boundary condition which are the results of dynamic contact problem. The separation of the projectile from the specimen, resulting from wave reflections within the projectile and the specimen, occurs in the phenomenon.
A thin shear band region of finite width which undergoes significant deformation and temperature rise has been determined. Its evolution until occurrence of final fracture has been simulated. Shear band advance, microdamage and the development of the temperature field as a function of time have been determined. Qualitative comparison of numerical results with experimental observation data has been presented. The numerical results obtained have proven the usefulness of the thermo-elasto-viscoplastic theory in the investigation of dynamic shear band propagations and localized fracture.
References
R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, Tensor Analysis and Applications, Springer, Berlin, 1988.
A. K. Chakrabarti, J. W. Spretnak, Instability of plastic flow in the direction of pure shear, Metallurgical Transactions, 6A, 733–747, 1975.
Y. C. Chi, S. H. Lee, K. Cho, J. Duffy, The effects of tempering and test temperatures on the dynamic fracture initiation behaviour of an AISI 4340 VAR steel, Brown University Technical Report, August 1988.
K. Cho, Y. C. Chi, J. Duffy, Microscopic observations of adiabatic shear bands in three different steels, Brown University Report No DAAL03-88-K-0015/3, September 1988.
B. D. Coleman, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal., 13, 167–178, 1963.
R. Courant, K. O. Friedrichs, H. Lewy, Uber die Partiellen Differenzgleichungen der Mathematischen Physik, Math. Ann., 100, 32–74, 1928.
D. R. Curran, L. Seaman, D. A. Shockey, Dynamic failure of solids, Physics Reports, 147, 253–388, 1987.
R. Dautray, J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6. Evolution Problems II, Springer, Berlin 1993.
W. Dornowski, Influence of finite deformation on the growth mechanism of microvoids contained in structural metals, Arch. Mech., 51, 71–86, 1999.
W. Dornowski, P. Perzyna, Constitutive modelling of inelastic solids for plastic flow processes under cyclic dynamic loadings, In: Transaction of the ASME, J. Eng. Materials and Technology, 121, 210–220, 1999.
W. Dornowski, P. Perzyna, Localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings, CAMES, 7, 117–160, 2000.
W. Dornowski, P. Perzyna, Localized fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings, Acta Mechanica, 155, 233–255, 2002.
W. Dornowski, P. Perzyna, Numerical investigation of localized fracture phenomena in inclastic solids, Foundations of Civil and Environmental Engineering, 7, 79–116, 2006.
M. K. Duszek, P. Perzyna, The localization of plastic deformation in thermoplastic solids, Int. J. Solids Structures, 27, 1419–1443, 1991.
M. K. Duszek–Perzyna, P. Perzyna, Analysis of the influence of different effects on criteria for adiabatic shear band localization in inelastic solids [in:] Proceedings Material Instabilities: Theory and Applications, ASME Congress, Chicago, 9–11 November 1994, R. C. Batra and H. M. Zbib [Eds.], AMD-Vol. 183/MD-Vol.50, ASME, New York, pp. 59–85, 1994.
D. R. Durran,, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer, New York, 1999.
A. Glema, T. Łodygowski, P. Perzyna, Interaction of deformation waves and localization phenomen in inelastic solids, Comput. Methods Appl. Mech. Engng., 183, 123–140, 2000.
A. Glema, T. Łodygowski, P. Perzyna, The role of dispersion for the description of strain localization in materials under impact loading, [in:] European Conference on Computational Mechanics, June 26–29, 2001, Cracow, Poland.
A. Glema, T. Łodygowski, P. Perzyna, Localization of plastic deformations as a result of wave interaction, CAMES, 10, 81–91, 2003.
A. Glema, T. Łodygowski, P. Perzyna, Numerical investigation of dynamic shear bands in inelastic solids as a problem of mesomechanics, Comput. Mech., 41, 219–229, 2008.
P. R. Guduru, A. J. Rosakis, G. Ravichandran, Dynamic shear bands: an investigation using high speed optical and infrared diagnostic, Mechanics of Materials, 33, 371–402, 2001.
P. R. Guduru, A. T. Zehnder, A. J. Rosakis, G. Ravichandran, Dynamic full field measurements of crack tip temperatures, Engineering Fracture Mechanics, 68, 1535–1556, 2001.
B. Gustafsson, H. O. Kreiss, J. Oliger, Time Dependent Problems and Difference Methods, John Wiley, New York, 1995.
J. W. Hutchinson, Plasticity at the micron scale, Int. J. Solids and Structures, 37, 225–238, 2000.
I. R. Ionescu, M. Sofonea, Functional and Numerical Methods in Viscoplasticity, Oxford 1993.
J. N. Johnson, Dynamic fracture and spallation in ductile solids, J. Appl. Phys., 52, 2812–2825, 1981.
S. Li, W.-K. Liu, D. Qian, P. R. Guduru, A. J. Rosakis, Dynamic shear band propagation and micro-structure of adiabatic shear band, Comput. Methods Appl. Mech. Engng., 191, 73–92, 2001.
T. Łodygowski, P. Perzyna, Numerical modelling of localized fracture of inelastic solids in dynamic loading processes, Int. J. Num. Meth. Engng., 40, 4137–4158, 1997.
J. E. Marsden, T. J. R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, New York, 1983.
H. C. Meyers, Dynamic Behaviour of Materials, John Wiley, New York 1994.
A. Needleman, Computational mechanics at the mesoscale, Acta Materialia, 48, 105–124, 2000.
J. A. Nemes, J. Eftis, Constitutive modelling of the dynamic fracture of smooth tensile bars, Int. J. Plasticity, 9, 243–270, 1993.
J. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. (London), A 200, 523–541, 1950.
P. Perzyna, The constitutive equations for rate sensitive plastic materials, Quart. Appl. Math., 20, 321–332, 1963.
P. Perzyna, Fundamental problems in viscoplasticity, Advances in Applied Mechanics, 9, 243–377, 1966.
P. Perzyna, Thermodynamic theory of viscoplasticity, Advances in Applied Mechanics, 11, 313–354, 1971.
P. Perzyna, Constitutive modelling of dissipative solids for postcritical behaviour and fracture, ASME J. Eng. Materials and Technology, 106, 410–419, 1984.
P. Perzyna, Internal state variable description of dynamic fracture of ductile solids, Int. J. Solids Structures, 22, 797–818, 1986a.
P. Perzyna, Constitutive modelling for brittle dynamic fracture in dissipative solids, Arch. Mechanics, 38, 725–738, 1986b.
P. Perzyna, Instability phenomena and adiabatic shear band localization in thermoplastic flow processes, Acta Mechanica, 106, 173–205, 1994.
P. Perzyna, Interactions of elastic-viscoplastic waves and localization phenomena in solids, [in:] Proceedings IUTAM Symposium on Nonlinear Waves in Solids, August 15–20, 1993, Victoria, Canada, J. L. Wegner and F. R. Norwood [Eds.], ASME 1995, pp. 114–121, 1995.
P. Perzyna, Thermo-elasto-viscoplasticity and damage, [in:] Handbook of Materials Behaviour Models, J. Lemaitre [Ed.], Academic Press, New York, pp. 821–834, 2001.
P. Perzyna, The thermodynamical theory of elasto-viscoplasticity, Engineering Transactions, 53, 235–316, 2005.
P. Perzyna, Application of the thermodynamical theory of elasto-viscoplasticity in modern manufacturing processes, [in:] Damage Mechanics and Micromechanics of Localized Fracture Phenomena in Inelastic Solids, CISM Courses and Lectures, G. Z. Voyiadjis [Ed.], Vol. 525, Springer, Wien New York, pp. 225–376, 2011.
P. Perzyna, A. Drabik, Description of micro-damage process by porosity parameter for nonlinear viscoplasticity, Arch. Mech., 41, 895–908, 1989.
R. D. Richtmyer, Principles of Advance Mathematical Physics, Vol. I, Springer, New York 1978.
R. D. Richtmyer, K. W. Morton, Difference Methods for Initial-Value Problems, John Wiley, New York 1967.
S. Shima, M. Oyane, Plasticity for porous solids, Int. J. Mech. Sci., 18, 285–291, 1976.
D. A. Shockey, L. Seaman, D. R. Curran, The microstatistical fracture mechanics approach to dynamic fracture problem, Int. J. Fracture, 27, 145–157, 1985.
D. Sidey, L. F. Coffin, Low-cycle fatigue damage mechemism at high temperature, [in:] Fatigue Mechanism, Proc. ASTM STP 675 Symposium, Kansas City, Mo., May 1978, J. T. Fong [Ed.], Baltimore, pp. 528–568, 1979.
L. J. Sluys, Wave propagation, localization and dispersion in softening solids, Doctoral thesis, Delft Uniwersity Press, Delft, 1992.
G. Strang, G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs 1973.
S. A. Thau, Linear dispersive waves, [in:] Nonlinear Waves, S. Leibovich and A. R. Seebass [Eds.], Cornell University Press, Ithaca, 1974, pp. 44-81.
C. Truesdell, W. Noll, The nonlinear field theories, Handbuch der Physik, Band III/3, pp. 1–579, Springer, Berlin, 1965.
G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974a.
G. B. Whitham, Dispersive waves and variational principles, [in:] Nonlinear Waves, S. Leibovich and A. R. Seebass [Eds.], Cornell University Press, Ithaca, 1974b, pp. 139–169.
M. Zhou, A. J. Rosakis, G. Ravichandran, Dynamic propagating shear band in impactloaded prenotched plates. I. Experimental investigations of temperature signatures and propagation speed, J. Mech. Phys. Solids, 44, 981–1006, 1996.
M. Zhou, G. Ravichandran, A. J. Rosakis, Dynamic propagating shear band in impactloaded prenotched plates. II. Numerical simulations, J. Mech. Phys. Solids, 44, 1007–1032, 1996.
DOI: 10.24423/engtrans.140.2011