The approach to systematically eliminate members from the infinite list can best be illustrated with a simple example. Consider (2.1.11) By the Cayley-Hamilton theorem, the second tensor A will satisfy its own characteristic polynomial, i.e., (2.1.12) where (2.1.13) (2.1.14) (2.1.15) (2.1.16) and (2.1.17) Multiplying the characteristic polynomial equation by A gives (2.1.18) Taking the trace of this last expression yields (2.1.19) and this shows that since k1, k2 and k3 are functions of tr(A), tr(A2), and tr(A3), then (2.1.20) Is a function of only these three invariants as well. Indeed repeated applications of the preceding argument would demonstrate that tr(A5), tr(A6), … , can be written in terms of a linear combination of the first three traces of A. Therefore, by induction (2.1.21)
Table 2.1 Grade H-451 Graphite: Load Paths and Corresponding Failure Data Data Set Ratio  Failure Stresses (MPa）  # B-1 1 : 0 10.97 0 9.90 0 9.08 0 9.22 0 12.19 0 11.51 0 # B-2 0 : 1 0 15.87 0 12.83 0 18.06 0 20.29 0 14.32 0 14.22 # B-3 0 : - 1 0 -47.55 0 -50.63 0 -59.72 0 -56.22 0 -48.19 0 -51.54 # B-4 1 : - 1 9.01 -8.94 7.68 -7.68 14.34 -14.16 8.93 -8.78 13.23 -13.14 9.21 -9.11 Data Set Ratio  Failure Stresses (MPa） 
Data Set Ratio  Failure Stresses (MPa）  # B-8 1 : 1.5 6.69 10.03 6.51 9.78 8.07 12.11 9.13 13.74 6.11 9.19 9.24 13.91 9.93 14.93 8.93 13.41 7.20 10.79 # B-9 1 : - 5 6.35 -31.61 8.69 -43.44 7.40 -36.86 7.09 -35.30 5.94 -29.50 6.83 -32.83 8.06 -40.21 7.75 -38.58
MPa and a ratio of compressive strength to tensile strength of (C T) = 2, then C =- 31.86 MPa and the parameters A and B are (3.1.38) and (3.1.39 ) Equation (3.1.37) becomes (3.1.40 ) This expression is plotted in the 11 – 22 stress plane depicted in Figure 3.1.5 Figure 3.1.5 The Drucker-Prager (1952) criterion projected onto the 1 -2 principle stress plane (T = 15.93 MPa, C = 31.86 MPa) and compared with the Burchell’s data This combination of strength parameters leads to a biaxial strength of well over 900 MPa. If Burchell’s (2007) compressive strength of C = 52.93 MPa is utilized from along with a stress ratio (C T = 2), then the tensile strength is T =26.465 MPa. The parameters A and B are (3.1.41) and (3.1.42) Now (3.1.44) This expression is plotted in the 11 – 22 stress depicted in Figure 3.1.6 NEED A THIRD FIGURE, 3.1.7 WHERE A NUMBER OF ELLIPSES ARE SUPERIMPOSED IN THE SAME FIGURE. EACH ELLIPSE WOULD HAVE A
and in Figure 3.2.5 (3.2.32) In these figures the Willam-Warnke (1975) criterion presents as slices through a right triangular pyramid. Figure 3.2.4 The Willam-Warnke criterion projected onto a deviatoric parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa, BC = 61.40 MPa Figure 3.2.5 The Willam-Warnke criterion projected onto a deviatoric parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa,
Figure 4.4.3 The Green-Mkrtichian criterion projected onto a deviatoric plane parallel to the -plane with T = 15.93 MPa, C = 52.93MPa, BC = 61.40 MPa The meridian of the isotropic Green-Mkrtichian (1977) failure surface corresponding to and are depicted on Figure 4.4.4. Obviously the meridian lines are not linear. The meridian line goes through point defined by 9.02 MPa and r = 13.01 MPa. The meridian line goes through the point defined by  = 30.56 MPa and r = 43.22 MPa. Figure 4.4.4 The Green-Mkrtichian criterion projected onto the meridian plane for a material strength parameter of T = 15.93 MPa, C = 52.93 MPa, BC = 61.40 MPa As the value of the I1 stress invariant associated with the hydrostatic stress increases in the negative direction, failure surfaces perpendicular to the hydrostatic stress line become circular again. The model suggests that as hydrostatic compression stress increases the difference between tensile strength and compressive strength diminishes and approach each other asymptotically. This is a material behavior that should be verified experimentally in a manner similar to Bridgman’s (1953) bend bar experiments conducted in hyperbaric chambers on cast metal alloys. Balzer (1998) provides an excellent overview of Bridgman’s experimental efforts, as well as others and their accomplishments in the field of high pressure testing.
Figure 5.4.4 Anisotropic (1977) failure criterion with Burchell’s (2007) failure data projected onto (= -30.20) parallel to the -plane(TT = 10.48 MPa, TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa) The meridian lines of the anisotropic Green-Mkrtichian (1977) failure surface corresponding to and are depicted on Figure 5.4.5 Obviously the meridian lines are not linear. The meridian line goes through point defined by 6.05MPa and r = 8.56 MPa. meridian line goes through point defined by 9.02 MPa and r = 13.01 MPa. The meridian line goes through the point defined by  = 30.2 MPa and r = 42.17MPa. Figure 5.4.5 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto meridian-plane(TT = 10.48 MPa, TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa) As the value of the I1 stress invariant associated with the hydrostatic stress increases in the negative direction, failure surfaces perpendicular to the hydrostatic stress line become circular again. The model suggests that as hydrostatic compression stress increases the difference between tensile strength and compressive strength diminishes and approach each other asymptotically. This is a material behavior that should be verified experimentally in a manner similar to Bridgman’s (1953) bend bar experiments conducted in hyperbaric chambers on cast metal alloys. Balzer (1998) provides an excellent overview of Bridgman’s experimental efforts, as well as others and their accomplishments in the field of high pressure testing.
Theocaris, P.S., “The Elliptic Paraboloid Failure Criterion for Cellular Solids and Brittle Foams,” Acta Mechanica, Vol. 89, pp. 93-121, 1991. Wetherhold, R.C. and Ucci, A.M., “Probability Techniques for Reliability Analyses of Composite Materials,” NASA CR-195294, National Aeronautics and Space Administration, National Technical Information Service, 1994. US DOE Nuclear Energy Research Advisory Committee and the Generation IV International “A Technology Roadmap for Generation IV Nuclear Energy Systems,” GIF-002-00, December, 2002. Vijavakumar, K. and Ashoka, J.G., “A Bilinear Constitutive Model for Isotropic Bimodulus Materials,” ASME Journal of Engineering Materials and Technology, Vol. 112, pp. 372-379, 1990. Vijayakumar, K., and Rao, K.P., “Stress-Strain Relations for Composites with Different Stiffness in Tension and Compression,” Computational Mechanics, Vol. 2, pp. 167-175, 1987. Von Mises, R., “Mechanik der festen Korper im plastisch deformablen Zustand,” Göttinger Nachrichten, pp. 582–592, 1913. Weibull, W., “A Statistical Distribution Function of Wide Applicability,” Journal of Applied Mechanics, Vol. 18, No. 3, pp. 293-297, 1951. Weibull, Waloddi. Ingeniors Vetenshaps Akadamein, Handlingar, 151(1939) Willam, K.J. and Warnke, E.P., “Constitutive Models for the Triaxial Behavior of Concrete,” in: Proceedings of the International Association for Bridge and Structural Engineer Seminar on Structures Subjected to Triaxial Stresses (Bergamo, Italy), Vol. 19, pp. 1-31, 1974.