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Localized fracture phenomena in thermo-visco-plastic flow processes under cyclic dynamic loadings

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Summary

The main objective of the paper is the investigation of localized fatigue fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings. Recent experimental observations for cycle fatigue damage mechanics at high temperature and dynamic loadings of metals suggest that the intrinsic microdamage process does very much depend on the strain rate and the wave shape effects and is mostly developed in the regions where the plastic deformation is localized. The microdamage kinetics interacts with thermal and load changes to make failure of solids a highly rate, temperature and history dependent, nonlinear process.

A general constitutive model of elasto-viscoplastic damaged polycrystalline solids developed within the thermodynamic framework of the rate type covariance structure with a finite set of the internal state variables is used (cf. Dornowski and Perzyna [16], [17], [18]). A set of the internal state variables is assumed and interpreted such that the theory developed takes account of the effects as follows: (i) plastic nonnormality; (ii) plastic strain induced anisotropy (kinematic hardening); (iii) softening generated by microdamage mechanisms (nucleation, growth and coalescence of microcracks); (iv) thermomechanical coupling (thermal plastic softening and thermal expansion); (v) rate sensitivity; (vi) plastic spin.

To describe suitably the time and temperature dependent effects observed experimentally and the accumulation of the plastic deformation and damage during a dynamic cyclic loading process the kinetics of microdamage and the kinematic hardening law have been modified. The relaxation time is used as a regularization parameter. By assuming that the relaxation time tends to zero, the rate independent elasticplastic response can be obtained. The viscoplastic regularization procedure assures the stable integration algorithm by using the finite difference method. Particular attention is focussed on the well-posedness of the evolution problem (the initial-boundary value problem) as well as on its numerical solutions. The Lax-Richtmyer equivalence theorem is formulated, and conditions under which this theory is valid are examined. Utilizing the finite difference method for a regularized elasto-viscoplastic model, the numerical investigation of the three-dimensional dynamic adiabatic deformation in a particular body under cyclic loading condition is presented.

Particular examples have been considered, namely a dynamic adiabatic cyclic loading process for a thin plate with sharp notch. To the upper edge of the plate is applied a cyclic constraint realized by rigid rotation of the edge of the plate while the lower edge is supported rigidly. A small localized region, distributed asymmetrically near the tip of the notch, which undergoes significant deformation and temperature rise, has been determined. Its evolution until occurrence of fatigue fracture has been simulated.

The propagation of the macroscopic fatigue damage crack within the material of the plate is investigated. It has been found that the length of the macroscopic fatigue damage crack distinctly depends on the wave shape of the assumed loading cycle.

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References

  1. Abraham, R., Marsden, J. E., Ratiu, T.: Manifolds, tensor analysis and applications. Berlin: Springer 1988.

    Google Scholar 

  2. Agah-Tehrani, A., Lee, E. H., Malett, R. L., Onat, E. T.: The theory of elastic-plastic deformation at finite strain with induced anisotropy modelled isotropic-kinematic hardening. J. Mech. Phys. Solids35, 43–60 (1987).

    Google Scholar 

  3. Armstron, P. J., Frederick, C. O.: A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N731, Central Electricity Generating Board, 1966.

  4. Auricchio, F., Taylor, R. L., Lubliner, J.: Application of a return map algorithm to plasticity models. In: COMPLAS Computational Plasticity: Fundamentals and Applications (Owen, D. R. J. and Onate, E. eds.), pp. 2229–2248. Barcelona 1992.

  5. Auricchio, F., Taylor, R. L.: Two material models for cyclic plasticity models: Nonlinear kinematic hardening and generalized plasticity. Int. J. Plasticity11, 65–98 (1995).

    Google Scholar 

  6. Chaboche, J. L.: Time—independent constitutive theories for cyclic plasticity. Int. J. Plasticity2, 149–188 (1986).

    Google Scholar 

  7. Chakrabarti, A. K., Spretnak, J. W.: Instability of plastic flow in the direction of pure shear. Metall. Trans.6A, 733–747 (1975).

    Google Scholar 

  8. Courant, R., Friedrichs, K. O., Lewy, H.: Über die partiellen Differenzgleichungen der Mathematischen Physik. Math. Ann.100, 32–74 (1928).

    Google Scholar 

  9. Curran, D. R., Seaman, L., Shockey, D. A.: Dynamic failure of solids. Physics Reports147, 253–388 (1987).

    Google Scholar 

  10. Dafalias, Y. F.: Corotational rates for kinematic hardening at large plastic deformations. J. Appl. Mech.50, 561–565 (1983).

    Google Scholar 

  11. Dafalias, Y. F., Popov, E. P.: A model of nonlinearly hardening materials for complex loading. Acta Mech.21, 173–192 (1975).

    Google Scholar 

  12. Dafalias, Y. F., Popov, E. P.: Plastic internal variable formalism of cyclic plasticity. J. Appl. Mech.43, 645–651 (1976).

    Google Scholar 

  13. Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology. Vol. 6: Evolution problems II. Berlin: Springer 1993.

    Google Scholar 

  14. Dornowski, W.: Influence of finite deformation on the growth mechanism of microvoids contained in structural metals. Arch. Mech.51, 71–86 (1999).

    Google Scholar 

  15. Dornowski, W.: An algorithm of numerical integration for thermo-elasto-viscoplastic constitutive equations. Biul. WAT3, 31–49 (2000).

    Google Scholar 

  16. Dornowski, W., Perzyna, P.: Constitutive modelling of inelastic solids for plastic flow processes under cyclic dynamic loadings. ASME J. Eng. Materials Technology121, 210–220 (1999).

    Google Scholar 

  17. Dornowski, W., Perzyna, P.: Numerical solutions of thermo-viscoplastic flow processes under cyclic dynamic loadings. In: Proc. Euromech Colloquium 383, Inelastic analysis of structures under variable loads: theory and engineering applications (Weichert, D., ed.) Dordrecht: Kluwer Academic Publishers 1999.

    Google Scholar 

  18. Dornowski, W., Perzyna, P.: Localization phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings. Computer Ass. Mech. Engng Sci.7, 117–160 (2000).

    Google Scholar 

  19. Durran, D. R.: Numerical methods for wave equations in geophysical fluid dynamics. New York: Springer 1999.

    Google Scholar 

  20. Duszek, M. K., Perzyna, P.: On combinded isotropic and kinematic hardening effects in plastic flow processes. Int. J. Plasticity7, 351–363 (1991).

    Google Scholar 

  21. Duszek, M. K., Perzyna, P.: The localization of plastic deformation in thermoplastic solids. Int. J. Solids Struct.27, 1419–1443 (1991).

    Google Scholar 

  22. Duszek-Perzyna, M. K., Perzyna, P.: Analysis of the influence of different effects on criteria for adiabatic shear band localization in inelastic solids. In: Material instabilities: theory and applications, ASME Congress, Chicago 9–11 November 1994 (Batra, R. C., Zbib, H. M., eds.), AMD-Vol. 183/MD-Vol. 50, pp. 59–85. New York: ASME 1994.

    Google Scholar 

  23. Duszek-Perzyna, M. K., Perzyna, P.: Analysis of anisotropy and plastic spin on localization phenomena. Arch. Appl. Mech.68, 352–374 (1998).

    Google Scholar 

  24. Eftis, J., Nemes, J. A.: Constitutive modelling of spall fracture. Arch. Mech.43, 399–435 (1991).

    Google Scholar 

  25. Gustafsson, B., Kreiss, H. O., Oliger, J.: Time dependent problems and difference methods. New York Wiley 1995.

    Google Scholar 

  26. Hughes, T. J. R., Kato, T., Marsden, J. E.: Well-posed quasi-linear second order hyperbolic system with application to nonlinear elastodynamics and general relativity. Arch. Rat. Mech. Anal.63, 273–294 (1977).

    Google Scholar 

  27. Ionescu, I. R.: Sofonea, M.: Functional and numerical methods in viscoplasticity. Oxford: Oxford University Press 1993.

    Google Scholar 

  28. Johnson, J. N.: Dynamic fracture and spallation in ductile solids. J. Appl. Phys.52, 2812–2825 (1981).

    Google Scholar 

  29. Khan, A. S., Cheng, P.: Study of thress elastic-plastic constitutive models by non-proportional finite deformations of OFHC copper. Int. J. Plasticity6, 737–759 (1996).

    Google Scholar 

  30. Loret, B.: On the effect of plastic rotation in the finite deformation of anisotropic elastoplastic materials. Mech. Mater.2, 287–304 (1983).

    Google Scholar 

  31. Loret, B.: On the effects of plastic rotation on the localization of anisotropic elastoplastic solids. In: Plastic instability, (Salencon J. et al., eds.), pp. 89–100. Paris: Presses Ponts et Chausees 1985.

    Google Scholar 

  32. Lodygowski, T., Perzyna, P.: Localized fracture of inelastic polycrystalline solids under dynamic loading processes. Int. J. Damage Mech.6, 364–407 (1997).

    Google Scholar 

  33. Lodygowski, T., Perzyna, P.: Numerical modelling of localized fracture of inelastic solids in dynamic loading processes. Int. J. Num. Meth. Engng40, 4137–4158 (1997).

    Google Scholar 

  34. Marsden, J. E., Hughes, T. J. R.: Mathematical foundations of elasticity. Englewood Cliffs: Prentice-Hall 1983.

    Google Scholar 

  35. Mroz, Z.: On the description of anisotropic workhardening. J. Mech. Phys. Solids15, 163–175 (1967).

    Google Scholar 

  36. Nemes, J. A., Eftis, J.: Constitutive modelling on the dynamic fracture of smooth tensile bars. Int. J. Plasticity9, 243–270 (1993).

    Google Scholar 

  37. Oldroyd, J.: On the formulation of rheological equations of state. Proc. Roy. Soc. (London)A 200, 523–541 (1950).

    Google Scholar 

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

    Google Scholar 

  39. Perzyna, P.: Fundamental problems in viscoplasticity. Adv. Appl. Mech.9, 343–377 (1966).

    Google Scholar 

  40. Perzyna, P.: Thermodynamic theory of viscoplasticity. Adv. Appl. Mech.11, 313–354 (1971).

    Google Scholar 

  41. Perzyna, P.: Constitutive modelling of dissipative solids for postcritical behaviour and fracture. ASME J. Eng. Mater. Technol.106, 410–419 (1984).

    Google Scholar 

  42. Perzyna, P.: Internal state variable description of dynamic fracture of ductile solids. Int. J. Solids Struct.22, 797–818 (1986).

    Google Scholar 

  43. Perzyna, P.: Constitutive modelling for brittle dynamic fracture in dissipative solids. Arch. Mech.38, 725–738 (1986).

    Google Scholar 

  44. Perzyna, P.: Influence of anisotropic effects on micro-damage process in dissipative solids. In: Yielding, damage and failure of anisotropic solids., Proc. IUTAM/ICM Symposium Villerd-de-Lance, August 1987, (Boehler, J. P., ed.), pp. 483–507. London: Mech. Eng. Publ. 1990.

    Google Scholar 

  45. Perzyna, P.: Instability phenomena and adiabatic shear band localization in thermoplastic flow processes. Acta Mech.106, 173–205 (1994).

    Google Scholar 

  46. Perzyna, P.: Interactions of elastic-viscoplastic waves and localization phenomena in solids. In: Nonlinear Waves in Solids. Proc. IUTAM Symposium, August 15–20, 1993, Victoria, Canada, (Wegner, L. J., Norwood, F. R., eds.), pp. 114–121. ASME Book No AMP 137, 1995.

  47. Perzyna, P., Drabik, A.: Influence of thermal effects on micro-damage solids. Arch. Mech.40, 795–805 (1988).

    Google Scholar 

  48. Perzyna, P., Drabik, A.: Description of micro-damage process by porosity parameter for nonlinear viscoplasticity. Arch. Mech.41, 895–908 (1989).

    Google Scholar 

  49. Perzyna, P., Drabik, A.: Micro-damage mechanism in adiabatic processes. Engineering Transactions (forthcoming).

  50. Prager, W.: The theory of plasticity: A survey of recent achievements. (J. Clayton Lecture). Proc. Inst. Mech. Eng.169, 41–57 (1955).

    Google Scholar 

  51. Richtmyer, R. D.: Principles of advance mathematical physics, vol. I. New York: Springer 1978.

    Google Scholar 

  52. Richtmyer, R. D., Morton, K. W.: Difference methods for initial value problems. New York: Wiley 1967.

    Google Scholar 

  53. Ristinmaa, M.: Cyclic plasticity model using one yield surface only. Int. J. Plasticity11, 163–181 (1995).

    Google Scholar 

  54. Shima, S., Oyane, M.: Plasticity for porous solids. Int. J. Mech. Sci.18, 285–291 (1976).

    Google Scholar 

  55. Shockey, D. A., Seaman, L., Curran, D. R.: The microstatistical fracture mechanics approach to dynamic fracture problem. Int. J. Fracture27, 145–157 (1985).

    Google Scholar 

  56. Sidey, D., Coffin, L. F.: Low-cycle fatigue damage mechanisms at high temperature. In: Fatique Mechanisms, Proc. ASTM STP 675 Symposium, Kansas City, Mo., May 1978 (Fong, J. T., ed.), pp. 528–568. ASTM, Baltimore 1979.

    Google Scholar 

  57. Strang, G., Fix, G. J.: An analysis of the finite element method. Englewood Cliffs: Prentice-Hall 1973.

    Google Scholar 

  58. Van der Giessen, E.: Continuum models of large deformation plasticity, Part I: Large deformation plasticity and the concept of a natural reference state. Eur. J. Mech., A/Solids8, 15–34 (1989).

    Google Scholar 

  59. Van der Giessen, E.: Continuum models of large deformation plasticity, Part II. A kinematic hardening model and the concept of a plastically induced orientational structure. Eur. J. Mech., A/Solids8, 89–108 (1989).

    Google Scholar 

  60. Van der Giessen, E.: Micromechanical and thermodynamic aspects of the plastic spin. Int. J. Plasticity7, 365–386 (1991).

    Google Scholar 

  61. Wang, J.-D., Ohno, N.: Two equivalent forms of nonlinear kinematic hardening: application to nonisothermal plasticity. Int. J. Plasticity7, 637–650 (1991).

    Google Scholar 

  62. Ziegler, H.: A modification of Prager's hardening rule. Quart. Appl. Math.17, 55–65 (1959).

    Google Scholar 

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Author notes

  1. W. Dornowski

    Present address: Military University of Technology, Kaliskiego 2, 01-489, Warsaw, Poland

  2. P. Perzyna

    Present address: Institute of Fundamental Technological Research, Polish Academy of Sciences, Swiętokrzyska 21, 00-049, Warsaw, Poland

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    Dornowski, W., Perzyna, P. Localized fracture phenomena in thermo-visco-plastic flow processes under cyclic dynamic loadings. Acta Mechanica 155, 233–255 (2002). https://doi.org/10.1007/BF01176245

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