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.

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.

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 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.

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.