The relationship between damage variables and their evolution laws and microstructural and physical properties

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 1998-05, Vol.454 (1973), p.1469-1498
• Main Authors: Zheng, Q.-S., Collins, I. F.
• Format: Article
• Language: English
• Subjects: Coefficients; Damage Tensors; Elasticity; Evolution Laws; Materials; Matrix materials; Microcracks; Microstructure; Orientation Distribution Function; Physical damage; Piezoelectricity; Point groups; Property damage; Rotation; Symmetry; Tensors
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.1998.0217

The major objective of this paper is to give a rational approach to identify the damage parameters in tensorial form, based on the microstructure of the defects (voids, cavities, microcracks) and the overall physical properties of a solid material. First, the general representations for the effective linear and nonlinear reversible physical properties (e.g. elastic and piezoelectric moduli) of a material representative volume element (RVE) containing a single defect of any shape are investigated. Second, it is emphasized that the orientation distribution functions (ODFs) of shapes, separations, etc., of defects constitute the dominant and physically measurable signatures of the changing microstructure of a damaged material, by assuming that the matrix behaves reversibly during damage nucleation and growth. Third, it is shown that the ODFs can be expanded as absolutely convergent Fourier series with irreducible tensorial coefficients, and that these infinitely many irreducible tensorial coefficients constitute the full list of generic damage variables. Furthermore, to specify any given physical property, only certain leading tensorial coefficients in these series are needed and, therefore, these leading coefficients act as the real damage tensors for such a physical property. These physically based damage tensors vary significantly from one physical property to another. For example, it is shown that only the second and fourth tensorial coefficients effect the linear elastic moduli, while only the first and third tensorial coefficients are needed in linear piezoelectricity theory. Finally, it is observed from a simple example that the evolution equations for these physically based damage tensors normally involved tensorial terms of a higher order than those needed to define the considered physically based damage tensors. Consequently, the evolution equations are not self-closed.