In this part, the irreducibility of the representations derived in Part IV for threedimensional orthotropic scalar-valued, vector-valued, symmetric tensor-valued and skew-symmetrictensor-valued functions of symmetric tensors, skew-symmetric tensors and vectors is proved. Finally, asummary for all of this five-part paper is also given.

All kinds of physically possible material symmetry in two-dimensional space were investigated in a recent work of Q.-S. Zheng and J. P. Boehler. In this paper, we establish the complete and irreducible representations with respect to every kind of material symmetry for scalar-, vector-, and second-order tensor-valued functions in two-dimensional space of any finite number of vectors and second-order tensors. These representations allow general invariant forms of physical and constitutive laws of anisotropic materials to be developed in plane problems.

Summary From the continuum mechanics points ofview, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverseisotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropictypes of FRC has been found in the literature. In this paper, the restriction to the admissible deformationof a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducibletwo-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derivedin Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yieldingand failure in the present work, and of heat conduction, continuum damage and asymmetric elasticityin a continued work (Part III, forthcoming).

In our another recent work (Zheng & Hwang 1996), we established the explicit relationsbetween the effective and the matrix elastic properties in two-dimensional (2D) elasticity, as the effective medium is comprised of an isotropic and homogeous matrix, microcracks,and holes of various shapes. The present paper extends the above result to the case for acomposite material comprising either isotropic or anisotropic matrix and inclusions whichhave respectively continuously varying compliances, and for a variety of interfaceconditions. It is shown that the effective compliance depends upon a reduced list of thematrix and inclusion material constants. In particular, when the matrix and inclusions areall isotropic and respectively homogeneous, then Em, H (i.e., the defect compliance H whichis the difference S-Sm, between the effective compliance S and the matrix compliance S, multiplied by the matrix Young's modulus Em) has a reduced dependence upon the Em/EIand vm-viEm/EI for the inclusion Young's moduli E1 and Poisson's ratios v1 and the MatrixPoisson's ratio vu, rather than the full material constant list of EI, vI, Em, vm. Applicationsto various important types of matrix and inclusion materials are given. We also show thatthe generalized self-consistent method (Christenson & Lo, 1979) provides solutions of theeffective moduli for unidirectional fibre composites in compatible forms with the generaldependence relations proved in the present paper.

Recent work has reported the independence of the effective Young's modulus, E, ofa two-dimensional (2D) isotropic matrix containing holes from the matrix materialPoisson's ratio, Ym, if the effective medium remains isotropic. The present workgives the explicit relations between the effective and matrix elastic properties foran isotropic 2D matrix containing given holes and microcracks of any density, size, shape, distribution, and orientation in either isotropic or anisotropic arrangement.To this general case, it is rigorously proved that Emil, i.e. the damage complianceH (which is the difference S-Sm between the effective compliance S and the matrixcompliance Sm) multiplied by the matrix Young's modulus Em is independent fromboth Em and the matrix material Poisson's ratio Ym. Consequently, the effectiveYoung's modulus is independent of Ym, and the dependent relations of the effectiveshear and area bulk moduli on the matrix material Young's shear, area bulk moduliand Poisson's ratio are given. We also show that the 2D elastic properties of anisotropic or anisotropic solid can be written in terms of the area bulk modulus andthe orientation distribution function of the extension modulus.