The concepts of damage effective stress and damage equivalence I sis play an important role in the development of continuum damage mechanics. Based on a generalization of the damage equivalence hypothesis, the so-called damage isotropy principle, it is found that the effective stress as a second-order tensor-valued function of the usual stress tensor and the damage tensor (s) has to be isotropic. Particularly. this prop-erty is regardless of the initial material symmetry (isotropy or anisotropy) and the type of damage variables; and thus. it allows general invariant modeling of the effective stress by the use of theory of tensor function representations. Damage material constants are then consistently introduced to the invariant models oftha effective stress and the damage effect tensors. Three new models of the damage effect tensor capable of including realistic dimensionless damage material constants are proposed. The significance of the damage material constants is examined by micromechanical analysis and computer experiments on effective elastic moduli.

The material symmetry of the constitutive law of a continuum material is described by the Kronecker powers of the orthogonal tensors which belong to the so-called material symmetry group, a subgroup of the full orthogonal tensor group, of the material. The properties, especially the canonical representations, of Kronecker powers of orthogonal tensors may be applied to deal with material symmetry problems. In this paper, we obtain the basic recurrence formulae in order to determine the canonical representations for finite order Kronecker powers of any given orthogonal tensor; and by usingthe recurrence formulae we derive the canonical representations for first, second, third and fourth order Kronecker powers of any two or three-dimensional orthogonai tensor. Finally, we apply these results to construct the micropolar elasticity matrices for micropolar elastic tensors under the 13 anisotropic mechanics symmetry groups Cn=1, 2....,13 as well as the isotropic symmetry group Co; and we also explain how to find an appropriate orthogonal tensor subgroup which may be regarded as the idealized material symmetry group for a given tensor.

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.

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.

The theory of tensor function representations constitutes a rational basis for a consistentma them atical modelling of complex mechanical behaviour of anisotropic materials. The so-ca Uedstructural tensors, which characterize the symmetry group of anisotropy of concern, play a key role inobtaining irreducible and coordinate-free representations for anisotropic tensor functions. In thispaper, based on available properties of Kronecker products of orthogonal transformations, a simplemethod of determining the structural tensors with respect to any given symmetry group is developed.As its application, the structural tensors corresponding to the five transverse isotropy groups, all oftheir finite subgroups, and the symmetry group of the 32 crystal classes, which present the most usualand worthwhile anisotropic symmetry groups, are constructed. In particular, we also show that each ofthese anisotropic symmetry groups can be characterized by only one simple structural tensor.

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).

We reconsider the definitions of both material symmetries and physical symmetries which aredescribed in terms of point groups, i.e. subgroups of the full orthogonal group, because these two conceptsare often confused and the classical descriptions of physical symmetry for inelastic behaviour of materials areimpracticable. All two-and three-dimensional point groups are classified into two types: compact andnon-compact. The reality of every compact point group in the description of a material or a physicalsymmetry is justified in four aspects, that is: (i) point groups characterized by a finite set of tensors, (ii)Hilbert's theorem for integrity bases, (iii) correlation between integrity bases and function bases (generalization of Wineman and Pipkin's theorem), and (iv) physical reality. The unreality of anynon-compact point group in the description of a material or a physical symmetry is proposed as a newprinciple of continuum physics. As applications, the complete sets of all classes of two-and three-dimensional point groups which describe physical symmetries for linear physical properties (such as thermalexpansion, piezoelectricity, elasticity, etc.) and for more general mechanical constitutive laws are given.

We reconsider the definitions of both material symmetries and physical symmetries which aredescribed in terms of point groups, i.e. subgroups of the full orthogonal group, because these two conceptsare often confused and the classical descriptions of physical symmetry for inelastic behaviour of materials areimpracticable. All two-and three-dimensional point groups are classified into two types: compact andnon-compact. The reality of every compact point group in the description of a material or a physicalsymmetry is justified in four aspects, that is: (i) point groups characterized by a finite set of tensors, (ii) Hilbert's theorem for integrity bases, (iii) correlation between integrity bases and function bases (generalization of Wineman and Pipkin's theorem), and (iv) physical reality. The unreality of anynon-compact point group in the description of a material or a physical symmetry is proposed as a newprinciple of continuum physics. As applications, the complete sets of all classes of two- and three-dimensional point groups which describe physical symmetries for linear physical properties (such as thermalexpansion, piezoelectricity, elasticity, etc.) and for more general mechanical constitutive laws are given.

In this paper, we give a new derivation procedure for determining the representations for3-and 2-dimensional isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions of vectors, symmetric tensors and skew-symmetric tensors. Simplified representationsfor isotropic scalar-valued and vector-valued functions are suggested.

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)