The major objective of this paper is to give a rational approach to identify the dam-age parameters in tensorial form, based on the microstructure of the defects (voids,cavities, microcracks) and the overall physical properties of a solid materiah First,the general representations for the effective linear and nonlinear reversible physicalproperties (e.g. elastic and piezoelectric moduli) of a material representative volumeelement (RVE) containing a single defect of any shape are investigated. Second, itis emphasized that the orientation distribution functions (ODFs) of shapes, separa-tions, etc., of defects constitute the dominant and physically measurable signaturesof the changing microstructure of a damaged material, by assuming that the matrixbehaves reversibly during damage nucleation and growth. Third, it is shown thatthe ODFs can be expanded as absolutely convergent Fourier series with irreducibletensorial coefficients, and that these infinitely many irreducible tensorial coefficient sconstitute the full list of generic damage variables. Furthermore, to specify any givenphysical property, only certain leading tensorial coefficients in these series are need-ed and, therefore, these leading coefficients act as the real damage tensors for sucha physical property. These physically based damage tensors vary significantly fromone physical property to another. For example, it is shown that only the second andfourth tensorial coefficients effect the linear elastic moduli, while only the first andthird tensorial coefficients are needed in linear piezoelectricity theory. Finally, it isobserved from a simple example that the evolution equations for these physicallybased damage tensors normally involve tensorial terms of higher order than thoseneeded to define the considered physically based damage tensors. Consequently, theevolution equations are not self-closed.

Im Hinblick auf Materialsymmetrien kann man ffir den praktischcn Einsatz transversal-isotrope, orthotrope, tetratrope, hexatrope oder auch oktotrope faserverst/irkte Verbundwerkstoffe unterscheiden. Firr Anwendungenim Ingenieurbereich sind sie von groBter Bedeutung. Die Darstellungstheorie tensorwertiger Funktionen bildet die Grundlagezur Formulierung von Stoffgleichungen anisotroper Stoffe, zu denen auch die oben erwiihnten Verbundwerkstoffe geh6ren.Tensorfunktionen speziell ffir transversal-isotrope und orthotrope Stoffe sind hinreichend bekannt. Hingegen fehlenentsprechende Untersuchungenzum tetratropen und hexatropen Verhalten. Daher sollen im folgenden vollstfindige undirreduzible Darstellungen fiir derartige Materialorientierungen entwickelt werden. Dazu miissen vektor-undtensor wertige Funktionenmit unterschiedlichen Tensor-und Vektorargumenten in Betracht gezogen werden. Eine weiter fiihrende Untersuchung (Teil II), in deru. a. auch anisotrope Werks to ffschiidigungen berficksichtigt werden, ist in Vorbereitung.

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: transvers eisotropy, 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)

The recently proposed effective self-consistent method (ESCM) is a quite simple andpowerful micromechanical approach of estimating the interaction effect for the overall physicalproperties of composite materials with multi-phase and multi-shape inclusions (involving holes andmicrocracks). The ESCM was developed by making use for reference the generalised self-consis-tent method (GSCM), while it is conveniently applicable in a similar way to the widely used Mori-Tanaka method. In this paper, as an application of the ESCM, the closed-form interacting solutionsfor the overall elastic moduli of an isotropic matrix with various multi-phase and multi-shape iso-tropic inclusions are derived. These closed-form interacting solutions for the overall elastic modu-lus are compared with other micromechanical estimations and are shown to be perfect predictions ofthe up-to-date numerical and experimental observations.