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Material Damage Models for Creep Failure Analysis and Design of Structures

  • J. J. Skrzypek
Part of the International Centre for Mechanical Sciences book series (CISM, volume 399)

Abstract

A concise review of one and three-dimensional theories of isotropic or anisotropic damage coupled constitutive equations of time-dependent elastic or inelastic materials is systematically presented. When damage is considered as isotropic phenomenon both phenomenologically-based damage-creep-plasticity models (Kachanov, Rabotnov, Hayhurst, Leckie, Kowalewski, Dunne, etc.) and unified irreversible thermodynamics formulation of coupled isotropic damage-thermoelastic-creep-plastic materials (Lemaitre and Chaboche, Mou and Han, Saanouni, Foster and Ben Hatira) are reported. In case when anisotropic nature of damage is described in frame of the continuum damage mechanics (CDM) approach, a concept of the fourth-rank damage effect tensor M is introduced in order to define the constitutive tensors of damaged materials, stiffness or compliance
$$\tilde \Lambda \,or\,{\tilde \Lambda ^{ - 1}}$$
in terms of those of virgin isotropic materials. Matrix representation of constitutive tensors is reviewed in case of energy based damage coupled constitutive model of elastic-brittle (Litewka, Murakami and Kamiya) or elastic-plastic engineering materials (Hayakawa and Murakami). Particular attention is paid to the orthotropic creep-damage model and its computer applications to the case of non-proportional loading conditions, when the objective damage rate is applied. A non-classical problem of thermo-damage coupling is developed, when the second-rank tensors of thermal conductivity
$$\tilde L$$
and radiation
$$\tilde \Gamma $$
in the extended heat transfer equation are defined for damaged material in terms of the damage tensor D.

The CDM based finite difference method (FDM) and finite element method (FEM) computer applications to the analysis and design of simple engineering structures under damage conditions are developed. Structures of uniform creep damage strength are examined from the point of view of maximum lifetime prediction when the equality and inequality constraints are imposed, and the thickness and initial prestressing are chosen as design variables.

Keywords

Finite Difference Method Creep Rupture Creep Crack Growth Tertiary Creep Continuum Damage Mechanic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Skrzypek J. and Ganczarski A.: Application of the orthotropic damage growth rule to variable principal directions, Int. J. Damage Mech., 7 (1998), pp. 180–206.CrossRefGoogle Scholar
  2. 2.
    Kachanov L.M.: Time of the rupture process under creep conditions, lzv. AN SSR, Otd. Tekh. Nauk, 8 (1958), pp. 26–31.Google Scholar
  3. 3.
    Kachanov L.M.: Foundations of Fracture Mechanics, Nauka, Moscow, 1974, in Russian.Google Scholar
  4. 4.
    Rabotnov Ju.N.: Creep rupture, in: Proc. 12 Int. Congr. Appl. Mech., Stanford, Calif., 1968 pp. 342–349.Google Scholar
  5. 5.
    Martin J.B. and Leckie F.A.: J. Mech. Phys. Solids, 20 (1972), pp. 223.CrossRefMATHGoogle Scholar
  6. 6.
    Hayhurst D.R. and Leckie F.A.: The effect of creep constitutive and damage relationships upon rupture time of solid circular torsion bar, J. Mech. Phys. Solids, 21 (1973), pp. 431–446.CrossRefGoogle Scholar
  7. 7.
    Leckie F.A. and Hayhurst D.R.: Creep rupture of structures, Proc. Roy. Soc. London, A 340 (1974), pp. 323–347.CrossRefMATHGoogle Scholar
  8. 8.
    Hayhurst D.R.: Creep rupture under multiaxial state of stress, J. Mech. Phys. Solids, 20 (1972), pp. 381–390.CrossRefGoogle Scholar
  9. 9.
    Hayhurst D.R.: On the role of creep continuum damage in structural mechanics, in: Engineering Approaches to High Temperature Design (Edited by Wilshire and D. Owen ), Pineridge Press, Swansea, 1983.Google Scholar
  10. 10.
    Trgpczyriski W.A., Hayhurst D.R. and Leckie F.A.: Creep rupture of copper and aluminium under non—proportional loading, J. Mech. Phys. Solids, 29(1981), pp. 353374.Google Scholar
  11. 11.
    Lemaitre J. and Chaboche J.L.: A non—linear model of creep fatigue damage cumulation and interaction, in: Proc. of IUTAM Symp. Mechanics of Visco—Elastic Media and Bodies (Edited by J. Hult ), Springer, Gothenburg, Sweden, 1975 pp. 291–301.CrossRefGoogle Scholar
  12. 12.
    Lemaitre J. and Chaboche J.L.: Aspect phenomenologique de la rapture per endommagement, J. de Méchanique applique, 2 (1978), pp. 317–365.Google Scholar
  13. 13.
    Lemaitre J. and Chaboche J.L.: Méchanique des Matériaux Solides, Dunod Publ., Paris, 1985.Google Scholar
  14. 14.
    Chaboche J.L.: Continuum damage mechanics: Part I: General concepts, Part II: Damage growth, crack initiation, and crack growth, J. Appl. Mech., 55 (1988), pp. 5971.Google Scholar
  15. 15.
    Dunne F.P.E. and Hayhurst D.R.: Continuum damage based constitutive eqaution for copper under high temperature creep and cyclic plasticity, Proc. R. Soc. Lond., A 437 (1992), pp. 545–566.CrossRefGoogle Scholar
  16. 16.
    Dunne F.P.E. and Hayhurst D.R.: Modelling of combined high—temperature creep and cyclic plasticity in components using Continuum Damage Mechanics, Proc. R. Soc. Lond., A 437 (1992), pp. 567–589.CrossRefGoogle Scholar
  17. 17.
    Dunne F.P.E. and Hayhurst D.R.: Efficient cycle jumping techniques for the modelling of materials and structures under cyclic mechanical and thermal loadings, Eur. J. Mech., A/Solids, 13 (1994), pp. 639–660.MATHGoogle Scholar
  18. 18.
    Dunne F.P.E. and Hayhurst D.R.: Physically based temperature dependence of elastic—viscoplastic constitutive equations for copper between 20 and 500°C, Philosophical Mag., A 74 (1995), pp. 359–382.Google Scholar
  19. 19.
    Othman A.M., Hayhurst D.R. and Dyson B.F.: Skeletal point stresses in circumferentially notched tension bars undergoing tertiary creep modelled with physically based constitutive equations, Proc. R. Soc. London, 441 (1993), pp. 343–358.CrossRefGoogle Scholar
  20. 20.
    Germain P., Nguyen Q.S. and Suquet P.: Continuum Thermodynamics, ASME J. Appl. Mech., 50 (1983), pp. 1010–1020.CrossRefMATHGoogle Scholar
  21. 21.
    Dufailly J. and Lemaitre J.: Modeling very low cycle fatigue, Int. J. Damage Mech., 4 (1995), pp. 153–170.CrossRefGoogle Scholar
  22. 22.
    Mou Y.H. and Han R.P.S.: Damage evolution in ductile materials, Int. J. Damage Mech., 5 (1996), pp. 241–258.CrossRefGoogle Scholar
  23. 23.
    Saanouni K., Forster C.H. and Hatira F. Ben: On the anelastic flow with damage, Int. J. Damage Mech., 3 (1994), pp. 140–169.CrossRefGoogle Scholar
  24. 24.
    Davison L. and Stevens A.L.: Thermodynamical constitution of spalling elastic bodies, J. Appl. Phys., 44 (1973), pp. 668–674.CrossRefGoogle Scholar
  25. 25.
    Kachanov L.M.: Introduction to Continuum Damage Mechanics, Martinus Nijhoff, The Netherlands, 1986.CrossRefMATHGoogle Scholar
  26. 26.
    Krajcinovic D. and Fonseka G.U.: The continuous damage theory of brittle materials, Part I and II: General theory, J. Appl. Mech., Trans ASME, 48 (1981), pp. 809–824.CrossRefMATHGoogle Scholar
  27. 27.
    Krajcinovic D.: Constitutive theory of damaging materials, J. Appl. Mech., Trans. ASME, 50 (1983), pp. 355–360.CrossRefMATHGoogle Scholar
  28. 28.
    Krajcinovic D.: Damage Mechanics, North Holland Series in Appl. Math. and Mech., Elsevier, Amsterdam, 1996.Google Scholar
  29. 29.
    Lubarda V.A. and Krajcinovic D.: Damage tensors and the crack density distribution, Int. J. Solids Struct., 30 (1993), pp. 2859–2877.CrossRefMATHGoogle Scholar
  30. 30.
    Rabotnov Ju.N.: Creep Problems in Structural Members, North-Holland, Amsterdam, 1969, engl. trans. by F.A. Leckie.Google Scholar
  31. 31.
    Vakulenko A.A. and Kachanov M. L.: Continuum theory of medium with cracks, lzv. A.N. SSSR, M.T.T., 4 (1971), pp. 159–166, in Russian.Google Scholar
  32. 32.
    Murakami S. and Ohno N.: A continuum theory of creep and creep damage, in: Creep in Structures (Edited by A. Ponter and D. Hayhurst ), Springer, Berlin, 1980 pp. 422–444.Google Scholar
  33. 33.
    Cordebois J.P. and Sidoroff F.: Damage induced elastic anisotropy, in: Col. EUROMECH 115, Villard de Lans, 1979 Also in Mechanical Behavior of Anisotropic Solids (Ed. Boehler, J. P. ), Martinus Nijhoff, Boston 1983, 761–774.Google Scholar
  34. 34.
    Cordebois J.P. and Sidoroff F.: Endommagement anisotrope an élasticité at plasticité, J. Méc. Théor. Appl., Numero Spécial, (1982), pp. 45–60.Google Scholar
  35. 35.
    Betten J.: Damage tensors in continuum mechanics, J. Méc. Théor. Appl., 1 (1983), pp. 13–32.Google Scholar
  36. 36.
    Betten J.: Application of tensor functions in continuum damage mechanics, Int. J. Damage Mech., 1 (1992), pp. 47–59.CrossRefGoogle Scholar
  37. 37.
    Litewka A.: Effective material constants for orthotropically damaged elastic solid, Arch. Mech., 6 (1985), pp. 631–642.Google Scholar
  38. 38.
    Litewka A.: Analytical and experimental study of fracture of damaging solids, in: Proc. of IUTAM/ICM Symp. Yielding, Damage, and Failure of Anisotropic Solids (Edited by J. Boehler ), Mech. Eng. Publ, London, 1987 pp. 655–665.Google Scholar
  39. 39.
    Litewka A.: Creep rupture of metals under multi-axial state of stress, Arch. Mech., 41 (1989), pp. 3–23.Google Scholar
  40. 40.
    Murakami S.: Notion of continuum damage mechanics and its applications to anisotropic creep damage theory, J. Eng. Mater. Technol., 105 (1983), pp. 99–105.CrossRefGoogle Scholar
  41. 41.
    Murakami S.: Failure Criterion of Structural Media, Balkema, 1986.Google Scholar
  42. 42.
    Murakami S.: Progress of continuum mechanics, JSME, Int. J., 30(1987), pp. 701–710.CrossRefGoogle Scholar
  43. 43.
    Murakami S.: Mechanical modelling of material damage, J. Appl. Mech., Trans. ASME, 55 (1988), pp. 280–286.CrossRefGoogle Scholar
  44. 44.
    Chow C.L. and Lu T.J.: An analytical and experimental study of mixed-mode ductile fracture under nonproportional loading, Int. J. Damage Mech., 1 (1992), pp. 191–236.CrossRefGoogle Scholar
  45. 45.
    Chaboche J.L.: Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage, Int. J. Damage Mech., 2(1993), pp. 311–329.CrossRefGoogle Scholar
  46. 46.
    Chaboche J.L.: Thermodynamically founded CDM models for creep and other conditions, in: Creep and Damage in Materials and Structures (Edited by H. Altenbach and J. Skrzypek), Advanced School No. 187, Udine, Sept. 7–11, 1998, Springer Vienna, 1999.Google Scholar
  47. 47.
    Murakami S. and Kamiya K.: Constitutive and damage evolution equations of elastic—brittle materials based on irreversible thermodynamics, Int. J. Solids Struct., 39 (1997), pp. 473–486.MATHGoogle Scholar
  48. 48.
    Hayakawa K. and Murakami S.: Thermodynamical modeling of elastic—plastic damage and experimental validation of damage potential, Int. J. Damage Mech., 6 (1997), pp. 333–362.CrossRefGoogle Scholar
  49. 49.
    Hayakawa K. and Murakami S.: Space of damage conjugate force and damage potential of elastic–plastic damage materials, in: Damage Mechanics in Engineering Materials (Edited by G. Z. Voyiadjis, J.-W. Ju and J.-L. Chaboche ), Elsevier Science, Amsterdam, 1998 pp. 27–44.CrossRefGoogle Scholar
  50. 50.
    Skrzypek J. and Ganczarski A.: Modeling of damage effect on heat transfer in time–dependent non–homogeneous solids, J. Thermal Stresses, 21 (1998), pp. 205–231.CrossRefGoogle Scholar
  51. 51.
    Skrzypek J. and Ganczarski A.: Modeling of Material Damage and Failure of Structures, Springer, Berlin–Heidelberg, 1999.CrossRefMATHGoogle Scholar
  52. 52.
    Leckie F.A. and Onat E.T.: Tensorial nature of damage measuring internal variables, in: Proc. of IUTAM Symp. Physical Non–linearities in Structural Analysis (Edited by J. HuIt and J. Lemaitre ), Springer, Berlin, 1981.Google Scholar
  53. 53.
    Chaboche J.L.: Le concept de contraine appliqué à l’élasticité et la viscoplasticité en présence d’un endommagement anisotrope, in: Mechanical Behaviour of Anisotropic Solids (Edited by J. Boehler), Col. EUROMECH 115, Grenoble 1979, Editions du CNRS No. 295, Paris, 1982 pp. 737–760.Google Scholar
  54. 54.
    Simo J.C. and Ju J.W.: Strain– and stress–based continuum damage models. I — Formulation, Il — Computational aspects, Int. J. Solids Struct., 23(1987), pp. 821–869.CrossRefMATHGoogle Scholar
  55. 55.
    Krajcinovic D.: Damage mechanics, Mech. Mater., 8 (1989), pp. 117–197.CrossRefGoogle Scholar
  56. 56.
    Chen X.F. and Chow C.L.: On damage strain energy release rate Y, Int. J. Damage Mech., 4 (1995), pp. 251–236.CrossRefGoogle Scholar
  57. 57.
    Voyiadjis G.Z. and Park T.: Anisotropic damage for the characterization of the onset of macro–crack initiation in metals, Int. J. Damage Mech., 5 (1996), pp. 68–92.CrossRefGoogle Scholar
  58. 58.
    Voyiadjis G.Z. and Park T.: Kinematics of large elastoplastic damage deformation, in: Damage Mechanics in Engineering Materials (Edited by G. Voyiadjis, J.-W. Ju and J.-L. Chaboche ), Elsevier Science, Amsterdam, 1998 pp. 45–64.CrossRefGoogle Scholar
  59. 59.
    Qi W. and Bertram A.: Anisotropic creep damage modeling of single crystal super-alloys, Techn. Mechanik, 17 (1997), pp. 313–332.Google Scholar
  60. 60.
    Zheng Q.-S. and Betten J.: On damage effective stress and equivalence hypothesis, Int. J. Damage Mech., 5 (1996), pp. 219–240.CrossRefGoogle Scholar
  61. 61.
    Taher S.F., Baluch M.H. and Al-Gadhib A.H.: Towards a canonical elastoplastic damage model, Eng. Fracture Mechanics, 48 (1994), pp. 151–166.CrossRefGoogle Scholar
  62. 62.
    Robinson E.L.: Effect of temperature variation on the long time rupture strength of steel, Trans. ASME, 74 (1952), pp. 777–780.Google Scholar
  63. 63.
    Chrzanowski M. and Madej J.: Construction of the failure curves based on the damage parameter concept, Mech. Teor. Stos., 4 (1980), pp. 587–601, (in Polish).Google Scholar
  64. 64.
    Chaboche J.L.: Une Loi Différentielle d’Endommagement de Fatigue avec Cumulation non Linéaire, Revue Française de Mecanique, (1974), pp. 50–51, english trans. in: Annales de I’IBTP, HS 39, (1977).Google Scholar
  65. 65.
    Othman A.M. and Hayhurst D.R.: Multi–axial creep rupture of a model structure using a two–parameter material model, Int. J. Mech. Sci., 32 (1990), pp. 35–48.CrossRefGoogle Scholar
  66. 66.
    Kowalewski Z.L., Hayhurst D.R. and Dyson B.F.: Mechanisms–based creep constitutive equations for an aluminium alloy, J. Strain Analysis, 29 (1994), pp. 309–316.CrossRefGoogle Scholar
  67. 67.
    Kowalewski Z.L., Lin J. and Hayhurst D.R.: Experimental and theoretical evaluation of a high–accuracy uni–axial creep testpiece with slit extensometers ridges, Int. J. Mech. Sci., 36 (1994), pp. 751–769.CrossRefGoogle Scholar
  68. 68.
    Hayhurst D.R.: Material data bases and mechanisms–based constitutive equations for use in design, in: Creep and Damage in Materials and Structures (Edited by H. Altenbach and J. Skrzypek), Advanced School No. 187, Udine, Sept. 7–11, 1998, Springer Vienna, 1999.Google Scholar
  69. 69.
    Rides M., Cocks A.C. and Hayhurst D.R.: The elastic response of damaged materials, J. Appl. Mech., 56 (1989), pp. 493–498.CrossRefGoogle Scholar
  70. 70.
    Johnson A.E., Henderson J. and Mathur V.D.: Combined stress creep fracture of commercial copper at 250°, The Engineer, 24 (1956), pp. 261–265.Google Scholar
  71. 71.
    Johnson A.E., Henderson J. and Khan B.: Complex-stress creep, relaxation and fracture of metallic alloys, HMSO, Edinbourgh, 1962.Google Scholar
  72. 72.
    Chaboche J.L. and Rousselier G.: On the plastic and viscoplastic constitutive equations–P.1: Rules developed with internal variable concept, P.2: Application of internal variable concepts to the 316 stainless steel, J. Pressure Vessel Technol, 105 (1983), pp. 153–164.CrossRefGoogle Scholar
  73. 73.
    Lemaitre J.: A continuum damage mechanics model for ductile fracture, ASME J. Engng. Mat. and Technology, 107 (1985), pp. 83–89.CrossRefGoogle Scholar
  74. 74.
    Lemaitre J.: Formulation and identification of damage kinetic constitutive equations, in: Continuum damage mechanics — theory and application (Edited by D. Krajcinovic and J. Lemaitre), CISM Courses and Lectures, 295, Springer, Berlin, 1987 pp. 37–89.Google Scholar
  75. 75.
    Broberg H.: Damage measures in creep deformation and rupture, Swedish Solid Mechanics Report, 8 (1974), pp. 100–104.Google Scholar
  76. 76.
    Chow C.L. and Wang L.: An anisotropic theory of elasticity for continuum damage mechanics, Int. J. Fracture, 33 (1987), pp. 3–16.CrossRefGoogle Scholar
  77. 77.
    Chow C.L. and Wang L.: An anisotropic theory of continuum damage mechanics for ductile materials, Eng. Fract. Mech., 27 (1987), pp. 547–558.CrossRefGoogle Scholar
  78. 78.
    Voyiadjis G.Z. and Kattan P.I.: A plasticity–damage theory for large deformation of solids, Part I: Theoretical formulation, Int. J. Eng. Sci., 30 (1992), pp. 1089–1108.CrossRefMATHGoogle Scholar
  79. 79.
    Chaboche J.L., Lesne P.M. and Moire J.F.: Continuum damage mechanics, anisotropy and damage deactivation for brittle materials like concrete and ceramic composites, Int. J. Damage Mech., 4 (1995), pp. 5–22.CrossRefGoogle Scholar
  80. 80.
    Ganczarski A. and Skrzypek J.: Effect of initial prestressing on the optimal design of plates with respect to orthotropic brittle rupture, Arch. Mech., 46 (1994), pp. 463–483.MATHGoogle Scholar
  81. 81.
    Litewka A. and Hult J.: One parameter CDM model for creep rupture prediction, Eur. J. Mech., A/Solids, 8 (1989), pp. 185–200.Google Scholar
  82. 82.
    Gallagher R.H.: Fully stressed design, in: Optimal Structural Design (Edited by R. Gallagher and O. Zienkiewicz ), John Willey, New York, 1973 pp. 19–23.Google Scholar
  83. 83.
    Zyczkowski M.: Optimal structural design in rheology, J. Appl. Mech., 38(1971), pp. 39–46, proc. 12 Int. Cong. Theor. Appl. Mech., Standford 1968.Google Scholar
  84. 84.
    Zyczkowski M.: Optimal structural design under creep conditions (1), Appl. Mech. Reviews, 41 (1988), pp. 453–461.CrossRefGoogle Scholar
  85. 85.
    Zyczkowski M.: Problems of structural optimization under creep conditions, in: Proc. of IUTAM Symp. Creep in Structures IV, 1990 (Edited by M. Zyczkowski ), Springer, Berlin, 1991 pp. 519–530.Google Scholar
  86. 86.
    Zyczkowski M.: Optimal structural design under creep conditions (2), Appl. Mech. Reviews, 49 (1996), pp. 433–446.CrossRefGoogle Scholar
  87. 87.
    Hayhurst D.R., Dimmer P.R. and Chernuka M.W.: Estimates of the creep rupture lifetimes of structures using the finite element methods, J. Mech. Phys. Solids, 23 (1975), pp. 335–355.CrossRefMATHGoogle Scholar
  88. 88.
    Hayhurst D.R., Dimmer P.R. and Morrison C.J.: Development of continuum damage in the creep rupture of notched bars, Phil. Trans. R. Soc. London, A 311 (1984), pp. 103–129.CrossRefGoogle Scholar
  89. 89.
    Saanouni K., Chaboche J.L. and Bathias C.: On the creep crack growth prediction by a non—local damage formulation, Eur. J. Mech., A/Solids, 8 (1986), pp. 677–691.Google Scholar
  90. 90.
    Liu Y., Murakami S. and Kanagawa Y.: Mesh—dependence and stress singularity in finite element analysis of creep crack growth by Continuum Damage Mechanics, Eur. J. Mech. A/Solids, 13 (1994), pp. 395–417.MATHGoogle Scholar
  91. 91.
    Murakami S., Kawai M. and Rong H.: Finite element analysis of creep crack growth by a local approach, Int. J. Mech. Sci., 30 (1988), pp. 491–502.CrossRefGoogle Scholar
  92. 92.
    Murakami S. and Liu Y.: Mesh—dependence in local approach to creep fracture, Int. J. Damage Mech., 4 (1995), pp. 230–250.CrossRefGoogle Scholar
  93. 93.
    Skrzypek J., Kuna-Ciskal H. and Ganczarski A.: On CDM modelling of pre— and post—critical failure modes in the elastic—brittle structures, in: Beiträge zur Festschrift zum 60 Geburstag von Prof. Dr.—Ing. Peter Gummert, Mechanik, Berlin, 1998 pp. 203–228.Google Scholar
  94. 94.
    Skrzypek J., Kuna-Ciskal H. and Ganczarski A.: Continuum damage mechanics modeling of creep—damage and elastic—damage—fracture in materials and structures, in: Proc. Workshop on Modeling Damage, Localization and Fracture Process in engineering Materials, Kazimierz Dolny, 1999 (to be published).Google Scholar
  95. 95.
    Ganczarski A. and Skrzypek J.: Optimal prestressing and design of rotating disks against brittle rupture under unsteady creep conditions, Eng. Trans., 37 (1989), pp. 627–649, (in Polish).Google Scholar
  96. 96.
    Ganczarski A. and Skrzypek J.: On optimal design of disks with respect to creep rupture, in: Proc. of IUTAM Symp. Creep in Structures (Edited by M. Zyczkowski ), Cracow, 1990 pp. 571–577.Google Scholar
  97. 97.
    Ganczarski A. and Skrzypek J.: Optimal shape of prestressed disks against brittle rupture under unsteady creep conditions, Struct. Optim., 4 (1992), pp. 47–54.CrossRefGoogle Scholar
  98. 98.
    Ganczarski A. and Skrzypek J.: Optimal design of rotationally symmetric disks in thermo—damage coupling conditions, Techn. Mechanik, 17 (1997), pp. 365–378.Google Scholar
  99. 99.
    Skrzypek J.J.: Plasticity and Creep, Theory, Examples, and Problems, Begell House — CRC Press, Boca Raton, 1993, ed. R. B. Hetnarski.Google Scholar
  100. 100.
    Skrzypek J. and Egner W.: On the optimality of disks of uniform creep strength against brittle rupture, Eng. Opt., 21 (1993), pp. 243–264.CrossRefGoogle Scholar
  101. 101.
    Egner W. and Skrzypek J.: Effect of pre-loading damage on the net—lifetime of optimally prestressed rotating disks, Arch. Appl. Mech., 64 (1994), pp. 447–456.MATHGoogle Scholar
  102. 102.
    Ganczarski A. and Skrzypek J.: Axisymmetric plates optimally designed against brittle rupture, in: Proc. World Congr. on Optimal Design of Structural Systems, Structural Optimization 93 (Edited by J. Herskovits ), Rio de Janeiro 1993, 1993 pp. 197–204.Google Scholar
  103. 103.
    Ganczarski A. and Skrzypek J.: Brittle-rupture mechanisms of axisymmetric plates subject to creep under surface and thermal loadings, in: Proc. of SMIRT-12 (Edited by K. Kussmaul ), Edited by K. 1993, 1993 pp. 263–268.Google Scholar
  104. 104.
    Ganczarski A., Freindl L. and Skrzypek J.: Orthotropic brittle rupture of Reissner’s prestressed plates, in: Proc. 5 Int. Conf. On Computational Plasticity (Edited by D. Owen, E. O. Nate and E. Hinton ), Barcelona, 1997 pp. 1904–1909.Google Scholar
  105. 105.
    Ganczarski A. and Skrzypek J.: Concept of thermo—damage coupling in continuum damage mechanics, in: Proc. First Int. Symp. Thermal Stresses ‘85 (Edited by R. Hetnarski and N. Noda ), Hamamatsu, Japan, Act City, 1995 pp. 83–86.Google Scholar
  106. 106.
    Ganczarski A. and Skrzypek J.: Modeling of damage effect of heat transfer in solids, in: Proc. Second Int. Symp. Thermal Stresses ‘87 (Edited by R. Hetnarski and N. Noda ), Rochester, NY, 1997 pp. 213–216.Google Scholar
  107. 107.
    Holman J.P.: Heat Transfer, McGraw-Hill, 1990.Google Scholar
  108. 108.
    Odqvist F.K.G.: Mathematical Theory of Creep and Creep Rupture, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1966.Google Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • J. J. Skrzypek
    • 1
  1. 1.Cracow University of TechnologyCracowPoland

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