This paper presents applications of accurate orthogonal function approximation methods to the two-dimensional problems of brittle solids containing many randomly located (and possibly randomly oriented) and strongly interacting microcracks within the framework of micromechanics and ensemble-volume average approach. The randomly located and oriented two-crack interaction problems are solved by using the highly accurate Legendre and Tchebycheff orthogonal polynomials to any desired order. The complex stress potential method is subsequently employed to micromechanically derive microcrack opening displacements under complicated loadings due to microcrack interaction effects-including concentrated loadings, arbitrary loadings and polynomial loadings. Improved local ensemble-averaged and overall effective elastic compliances of brittle solids due to microcracks and their interactions are systematically constructed by using the pairwise microcrack interaction mechanism and the ensemble-volume average approach. A number of interesting analytical-numerical examples containing different random microcrack configurations are also presented to illustrate the capabilities of the proposed framework.
1. Benveniste, Y.1986. "On the Mori-Tanakas Method in Cracked Bodies,"Mech. Res. Comm., 13:193-201.
2.
2. Benveniste, Y., J. Z. Dvorak, J. Zarzour and E. C. J. Wung. 1989. "On Interacting Cracks and Complex Crack Configurations in Linear Elastic Media,"Int. J. Solids & Struct., 25:1279-1293.
3.
3. Budiansky, B. and R. J. O'Connell. 1976. "Elastic Moduli of a Cracked Solid,"Int. J. Solids & Struct., 12:81-97.
4.
4. Char, B. W., K. O. Geddes, G. H. Gonnet, M. B. Monagan and S. M. Watt. 1988. MAPLE Reference Manual, 5th Ed., Symbolic Computation Group, Department of Computer Science, University of Waterloo, 1988, and Waterloo Maple Publishing, Waterloo, Ontario, Canada, 1990.
5.
5. Chen, Y. Z.1984. "General Case of Multiple Crack Problems in an Infinite Plate,"Eng. Fract. Mech., 20(4):519-597.
6.
6. Christensen, R. M. and K. H. Lo. 1979. "Solutions for Effective Shear Properties in Three Phase Sphere and Cylinder Models,"J. Mech. Phys. Solids, 27:315-330.
7.
7. Deng, H. and S. Nemat-Nasser. 1994. "Microcrack Interaction and Shear Fault Failure,"Int. J. Damage Mech., 3:3-37.
8.
8. Erdogan, F. 1962. "On the Stress Distribution in Plates with Collinear Cuts under Arbitrary Loads,"Proceedings of the Fourth US. National Congress of Applied Mechanics, pp. 547-553.
9.
9. Fanella, D. and D. Krajcinovic. 1988. "A Micromechanical Model for Concrete in Compression,"Eng. Fract. Mech., 29(1):49-66.
10.
10. Gottesman, T., Z. Hashin and M. A. Brull. 1980. "Effective Elastic Moduli of Cracked Fiber Composites,"Advances in Composites Materials, Proc. ICCM 3, Oxford: Pergamon Press.
11.
11. Gross, D.1982. "Spannungsintensitatsfaktoren von ribsystemen (Stress Intensity Factors of Systems of Cracks),"Ing.-Arch., 51:301-310 (in German).
12.
12. Hashin, Z.1988. "The Differential Scheme and Its Application to Cracked Materials,"J. Mech. Phys. Solids, 36:719-734.
13.
13. Hinch, E. J.1977. "An Averaged-Equation Approach to Particle Interactions in a Fluid Suspension,"J. Fluid Mech., 83:695-720.
14.
14 Horii, H. and S. Nemat-Nasser. 1983. "Overall Moduli of Solids with Microcracks: Load-Induced Anisotropy,"J. Mech. Phys. Solids, 31:155-171.
15.
15. Horii, H. and S. Nemat-Nasser. 1985. "Elastic Fields of Interacting Inhomogeneities,"Int. J. Solids & Struct., 21:731-745.
16.
16. Hudson, J. A.1980. "Overall Properties of a Cracked Solid,"Math. Proc. Camb. Phil. Soc., 88:371-384.
17.
17. Ioakimidis, N. I.1990. "Symbolic Computations: A Powerful Method for the Solution of Crack Problems in Fracture Mechanics,"Int. J. Franc., 43:R39-R42.
18.
18. Ioakimidis, N. I.1992. "Application of Computer Algebra to the Iterative Solution of Singular Equations,"Comp. Meth. Appl. Mech. Eng., 94:229-237.
19.
19. Ju, J. W.1991a. "On Two-Dimensional Self-Consistent Micromechanical Damage Models for Brittle Solids,"Int. J. Solids & Struct., 27(2):227-258.
20.
20. Ju, J. W.1991b. "A Micromechanical Damage Model for Uniaxially Reinforced Composites Weakened by Interfacial Arc Microcracks,"J. Appl. Mech., 58(4):923-930.
21.
21. Ju, J. W. and T. M. Chen. 1994a. "Effective Elastic Moduli of Two-Dimensional Brittle Solids with Interacting Microcracks. Part I: Basic Formulations,"J. Appl. Mech., 61:349-357.
22.
22. Ju, J. W. and T. M. Chen. 1994b. "Effective Elastic Moduli of Two-Dimensional Brittle Solids with Interacting Microcracks. Part II: Evolutionary Damage Models; J. Appl. Mech., 61:358-366.
23.
23. Ju, J. W. and X. Lee. 1991. "On Three-Dimensional Self-Consistent Micromechanical Damage Models for Brittle Solids. Part I: Tensile Loadings,"J. Eng. Mech., ASCE, 117(7):1495-1515.
24.
24. Ju, J. W. and K. H. Tseng. 1992a. "A Three-Dimensional Statistical Micromechanical Theory for Brittle Solids with Interacting Microcracks,"Int. J. Damage Mech., 1:102-131.
25.
25. Ju, J. W. and K. H. Tseng. 1992b. "Recent Development in Two-Dimensional Micromechanical Theories for Elastic Solids with Interacting Microcracks," in Recent Advances in Damage Mechanics and Plasticity, J. W. Ju, ed., New York: ASME Press, pp. 91-101.
26.
26. Kachanov, M.1985. "A Simple Technique of Stress Analysis in Elastic Solids with Many Cracks,"Int. J. Fract., 28:R11-R19.
27.
27. Kachanov, M.1987. "Elastic Solids with Many Cracks: A Simple Method of Analysis,"Int. J. Solids & Struct., 23:23-43.
28.
28. Kachanov, M.1992. "Effective Elastic Properties of Cracked Solids: Critical Review of Some Basic Concepts,"Appl. Mech. Rev., 45(8):304-335.
29.
29. Kachanov, M. and J.-P. Laures. 1989. "Three-Dimensional Problems of Strongly Interacting Arbitrarily Located Penny-Shaped Cracks,"Int. J. of Fract., 41:289-313.
30.
30. Krajcinovic, D.1989. "Damage Mechanics,; Mech. of Mater., 8(2-3):117-197.
31.
31. Krajcinovic, D. and D. Fannela. 1986. "A Micromechanical Damage Model for Concrete,"Eng. Fract. Mech., 25:585-596.
32.
32. Krajcinovic, D. and D. Sumarac. 1989. "A Mesomechanical Model for Brittle Deformation Processes: Part I,"J. Appl. Mech., 56:51-62.
33.
33. Lee, X. and J. W. Ju. 1991. "On Three-Dimensional Self-Consistent Micromechanical Damage Models for Brittle Solids. Part II: Compressive Loadings, J. Eng. Mech., ASCE, 117(7):1516-1537.
34.
34. MACSYMA. 1988. MACSYMA Reference Manual, Version 13, 1988, Massachusetts: Symbolics Inc.
35.
35. McLaughlin, R.1977. "A Study of the Differential Scheme for Composite Materials,"Int. J. Engng. Sci., 15:237-244.
36.
36. Mori, T. and K. Tanaka. 1973. "Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions,"Acta Metallurgica, 21:571-574.
37.
37. Roscoe, R. A.1952. "The Viscosity of Suspensions of Rigid Spheres,"Brit. J. Appl. Phys., 3:267-269.
38.
38. Roscoe, R. A.1973. "Isotropic Composites with Elastic or Viscoelastic Phases: General Bounds for the Moduli and Solutions for Special Geometries,"Rheol. Acta, 12:404-411.
39.
39. Sneddon, I. N. and M. Lowengrub. 1969. Crack Problems in the Classical Theory of Elasticity. New York: John Wiley & Sons, Inc.
40.
40. Sumarac, D. and D. Krajcinovic. 1987. "A Self-Consistent Model for Microcrack-Weakened Solids,"Mech. Mater., 6:39-52.
41.
41. Sumarac, D. and D. Krajcinovic. 1989. "A Mesomechanical Model for Brittle Deformation Processes: Part 11,"J. Appl. Mech., 56:57-62.
42.
42. ibda, H., P. Pars and G. Irwin. 1974. The Stress Analysis of Cracks Handbook, Pennsylvania: Del Research Corp., p 5.9.
43.
43. Wolfram, S.1991. Mathematica: A System for Doing Mathematics by Computer, 2nd Edition. California: Addison-Welsley.
44.
44. Zhao, Y. H., G. P. Tendon and G. J. Weng. 1989. "Elastic Moduli for a Class of Porous Materials,"Acta Mechanica, 76:105-130.
