The mechanical behaviour of anisotropic solids requires a suitable mathe matical modelling. The properties of tensor functions with several argument tensors con stitute a rational basis for a consistent mathematical modelling of complex material be haviour.
This paper presents certain principles, methods, and recent successful applications of tensor functions in continuum mechanics. The rules for specifying irreducible sets of ten sor invariants and tensor generators for material tensors of rank two and four are also dealt with.
Furthermore, it is very important to determine the scalar coefficients in constitutive and evolutional equations. This problem is explained in more detail.
Betten, J.1987. Tensorrechnung für IngenieureStuttgart : B.G. Teubner-Verlag.
2.
Betten, J.1989 "Recent Advances in Mathematical Modelling of Matenals Behaviour," paper presented at the Seventh Intern. Conf. on Mathematical and Computer Modelling, Chicago, 1989; published in Mathem. and Comput. Modelling, 14:37-51
3.
Boehler, J. P., ed. 1987. Applications of Tensor Functions m Solid Mechanics. Wien, New York: Springer-Verlag.
Betten, J., M. Borrmann and T. Butters.1989. "Materialgleichungen zur Beschreibung des @folio val="58">pnmaren Kriechverhaltens innendruckbeanspruchter Zylinderschalen aus isotropem Werkstoff," Ingenieur-Archiv , 60.99-109
6.
Betten, J. and T. Butters.1990. "Rotationssymmetrisches Kriechbeulen dunnwandiger Kreiszylinderschalen im primaren Kriechberiech," Forschung im Ingenieurwesen, 56:84-89.
7.
Rabotnov, Yu. N.1969Creep Problems in Structural MembersAmsterdam, London: North-Holland
8.
Betten, J.1975 "Zur Verallgemeinerung der Invariantentheone in der Kriechmechanik ," Rheol. Acta, 14:715-720
9.
Betten, J.1981. "Creep Theory of Anisotropic Solids," Journal of Rheology, 25:565-581, presented at the Golden Jubilee Meeting of The Society of Rheology, Boston, MA, 1979.
10.
Betten, J.1985. "The Classical Plastic Potential Theory in Comparison with the Tensor Function Theory," Engineering Fracture Mechanics , 21:641-652; presented at the Int Symposium "Plasticity Today," Udme, 1983.
11.
Betten, J.1983. "Damage Tensors in Continuum Mechanics," Journal de Mécanique théorique et appliquée, 2:13-32; presented at Euromech Colloquium 147 on Damage Mechanics, Paris-VI , Cachan, 1981.
12.
Betten, J.1982 "Net-Stress Analysis in Creep Mechanics," Ingenieur Archiv, 52:405-419; presented at the Second Gernian-Polish Symposium on Inelastic Solids and Structures, Bad Honnef, 1981.
13.
Bodner, S. R. and Z. Hashin, eds. 1986. Mechanics of Damage and Fatigue. New York, Toronto: Pergamon Press.
14.
Krajcinovic, D. and J. Lemaitre, eds. 1987. Continuum Damage Mechanics , Wien, New York : Springer-Verlag
15.
Murakami, S. and N. Ohno.1981. "A Continuum Theory of Creep and Creep Damage," in Creep in Structures 1980, A R. S. Ponter and D. R Hayhurst , eds., Berlin, Heidelberg, New York: Spnnger-Verlag, pp. 422-444.
16.
Murakami, S.1983. "Notion of Continuum Damage Mechanics and Its Application to Anisotropic Creep Damage Theory," J. of Engin. Materials and Technology, 105:99-105.
17.
Murakami, S. and A. Sawczuk.1981 "A Unified Approach to Constitutive Equations of Inelasticity Based on Tensor Function Representations ," Nucl. Engin. and Design, 6533-47.
18.
Murakami, S. , Y. Sanomura and K. Saitoh.1986. " Formulation of Cross-Hardening in Creep and Its Effect on the Creep Damage Process of Copper," J. of Engin. Matenals and Technology , 108:167-173.
19.
Murakami, S.1987. "Progress of Continuum Damage Mechanics," JSME Intern. Journal, 30:701-710.
20.
Chaboche, J.L.1984. "Anisotropic Creep Damage in the Framework of Continuum Damage Mechanics," Nucl. Engng. Des. , 79304-319.
21.
Chrzanowski, M.1976. "Use of the Damage Concept in Describing Creep-Fatigue Interaction under Prescnbed Stress," Int. J. Mech. Sci., 18:69-73.
22.
Krajcinovic, D.1983. "Constitutive Equations for Damaging Materials," J. Appl. Mech. , 50: 355-360.
23.
Betten, J.1983. "Damage Tensors and Tensors of Continuity in Stress Analysis ," Z. Naturforschung, 38a:1383-1390
24.
Betten, J.1986. "Applications of Tensor Functions to the Formulation of Constitutive Equations Involving Damage and Initial Anisotropy," Engng. Frac. Mech., 25.73-584, presented at the IUTAM Symposium on Mechanics of Damage and Fatigue, Haifa and Tel Aviv, Israel, 1985 .
25.
Onat, E.T.1986. "Representation of Mechanical Behavior in the Presence of Internal Damage," Engng. Frac. Mech. , 25:605-614; presented at the IUTAM Symposium on Mechanics of Damage and Fatigue , Haifa and Tel Aviv, Israel, 1985.
26.
Litewka, A. and J. Morzynska.1989. "Theoretical and Experimental Study of Fracture for a Damaged Solid," Res Mechamca , 27:259-272.
27.
Litewka, A. and J. Hult.1989. "One Parameter CDM Model for Creep Rupture Prediction," Eur. J. Mech. , A/Solids, 8:185-200
28.
Betten, J., E. El-Magd and S.C. Meydanli. "Anisotropic Damage Growth under Multi-Axial Stress (Theory and Experiments," in preparation, financially supported by the German Research Association (DFG).
29.
Kachanov, L.M.1958. "On the Time to Failure under Creep Conditions" (in Russian, Izv. Akad. Nauk USSR Otd. Tekh. Nauk, 8:26-31
30.
Kachanov, L.M.1986. Introduction to Continuum Damage Mechanics. Dordrecht, Boston, Lancaster : Martinus Nijhoff Publishers.
31.
Chrzanowski, M.1973. "The Description of Metallic Creep in the Light of Damage Hypothesis and Strain Hardening," Krakow: Diss. hab., Politechnika Krakowska.
32.
Leckie, F.A. and A.R.S. Ponter.1974. "On the State Vanable Descnption of Creeping Materials," Ing. Archiv, 43:158-167.
33.
Leckie, F.A. and D.R. Hayhurst.1977. "Constitutive Equations for Creep Rupture," Acta Metallurg., 25:1059-1070.
34.
Goel, R.P.1975. "On the Creep Rupture of a Tube and a Sphere," J Appl Mech. , Trans. ASME, 43:625-629.
35.
Hayhurst, D.R. , W.A. Trampczynski and F.A. Leckie.1980. " Creep Rupture and Non-Proportional Loading," Acta Metall. , 28:1171-1183.
36.
Monkman, F.C. and N.J. Grant1956. "An Empirical Relationship between Rupture Life and Minimum Creep Rate in Creep-Rupture Tests," Proc. ASTM, 56:593-620.
37.
Ilschner, B.1973. Hochtemperatur-Plastizitat . Berlin , Heidelberg, New York: Springer-Verlag.
38.
Edward, G.H. and M.F. Ashby.1979. "Intergranular Fracture during Power-Law Creep," Acta Metall. , 27:1505-1518.
39.
Evans, H.E.1984. Mechanisms of Creep Rupture. London /New York. Elsevier Appl. Sci. Publishers.
40.
Riedel, H.1987. Fracture at High Temperatures. Berlin , Tokyo: Springer-Verlag .
41.
Rabotnov, Y.N.1968. "Creep Rupture," in Appl. Mech. Conf, M. Hetenyi and H. Vincenti, eds., Stanford University , pp. 342-349.
42.
Betten, J.1984. "Interpolation Methods for Tensor Functions," in Mathematical Modelling in Science and Technology, J. R. Avula et al , eds , New York: Pergamon Press, pp. 52-57.
43.
Betten, J.1989 "Generalization of Nonlinear Material Laws Found in Experiments to Multi-Axial States of Stress," Eur. J. Mech. , A/Solids , 8 325-339; presented at Euromech Colloquium 224 on Experimental Analysis of Nonlinear Problems in Solid Mechanics , Poznan, Poland, 1988.
44.
Betten, J. "Determination of Scalar Coefficients in Constitutive Equations," Forschung im Ingenieurwesen (in preparation).
45.
Betten, J. and S.C. Meydanli. "Materialgleichungen," DFG-Report1990 (in press).
