Effective Stress and Vector-valued Orientational Distribution Functions

Yang, Qiang; Chen, Xin; Weiyuan, Zhou

doi:10.1177/1056789506067938

Cited by 6 publications

(14 citation statements)

References 20 publications

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“…Direction tensor C ji 1 ÁÁÁin for nth order approximation has 3 Á (n + 1)(n + 2)/2 independent variables when D = 3, and 2(n + 1) independent variables when D = 2. This is to allow the differences between the directions of m and n. If the coefficient tensor C ji 1 ÁÁÁin is a complete symmetric tensor, the approximation of M(n) can take the form M j ðnÞ ¼ C i 0 i 1 ÁÁÁin d jði 0 n i 1 n i 2 Á Á Á n inÞ (Yang et al, 2008). The number of independent coefficient in C i 0 i 1 ÁÁÁin to be determined is (n + 2) in 2D and (n + 2)(n + 3)/2 in 3D, indicating that there are additional constrains regarding to the components of directional data.…”

Section: Directional Distribution Functionmentioning

confidence: 99%

“…In the later part of the paper, the proposed method has been used to derive the general stress-force-fabric relationship as an example of applying the proposed theory to bridge up the particle-scale contact forces and the macro-scale stresses. Nevertheless, the method presented in this paper has extensive application for the multi-scale investigation on all materials (Odgaard, 1997;Yang et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

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Tensorial characterisation of directional data in micromechanics

2011

International Journal of Solids and Structures

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a b s t r a c tInformation being dealt with in micro-mechanics is massive. Most of them are directional data. Macroscale physical laws embedded with micro-scale fundamentals need to be developed in terms of the statistics of the micro-scale variable in a frame-indifferent form. Mathematical techniques and theories for characterizing the statistics of directional data with tensors are hence demanded. This is the main concern of the current paper. Starting with the general theory established in Kanatani (1984) of describing the directional distributions of orientations, mathematical formulations have been extended to address the directional distributions of vector-valued directional data, which is the most common data type being dealt with in micromechanical investigations. For vector-valued directional data, statistical analyses are required in regarding to both their directional probability density distribution and their representative values along each direction. The technique used here is to approximate these directional distributions by polynomials in unit directional vector n. The coefficients are in tensorial form and determined from observed directional data by applying the least square error criterion. These coefficient tensors serve as macro-scale variables representing the statistics of the micro-scale directional data, and are referred to as direction tensor. Orthogonal decompositions are addressed so that the coefficient tensor of different orders can be determined independently from each other. The coefficient tensors in the orthogonal decompositions are referred to as deviatoric direction tensor. The choice of sufficient approximation order is suggested. As an example, a general form of the stress-force-fabric relationship is derived for demonstrating the application of the proposed mathematical theory in the micro-mechanical investigation of the behaviour of granular materials.

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Section: Directional Distribution Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tensorial characterisation of directional data in micromechanics

2011

International Journal of Solids and Structures

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“…To keep its original connotations, Yang et al. (2005a, 2008) generalized the K–R effective stress as a vector-valued ODF owing to its microplane-based nature. A microplane is a plane of any orientation cutting the material at a given point, which is represented by the unit normal vector n={ni} of that orientation.…”

Section: Introductionmentioning

confidence: 99%

“…Yang et al. (2005a, 2008) introduced a scalar-valued ODF D(n) as an orientation-dependent damage variable to extend the uniaxial K–R effective stress, equation (1), onto each microplane …”

Section: Introductionmentioning

confidence: 99%

A microplane-based anisotropic damage effective stress

Yang

Leng

2013

International Journal of Damage Mechanics

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The Kachanov–Rabotnov damage effective stress is a fundamental concept in continuum damage mechanics. Controversies remain over the multiaxial generalization of the Kachanov–Rabotnov effective stress, especially when anisotropic damage effects occur. From a microplane point of view, we show that an anisotropic Kachanov–Rabotnov effective stress cannot simply be defined as a Cauchy-like stress tensor, namely a symmetric tensor of order two, but is essentially a vector-valued orientation distribution function. The practical effective stress tensor is identified by the second-order fabric tensor of the vector-valued Kachanov–Rabotnov stress orientation distribution function. The proposed model encompasses a wide range of classical models and thus clarifies the presuppositions of the models at a microplane level. We also prove the positive definiteness of the fourth-order damage effect tensor and provide a sufficient condition for its Voigt symmetry.

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“…Statistical characterization of such directional data is essential 1-4 . With regard to physical problems, such characterization must take a frame-indifferent form, or a tensorial form which is invariant to coordinate transformations; see, for example, Kanatani 5 , Advani and Tucker 6 , and Yang et al 7 .…”

Section: Introductionmentioning

confidence: 99%

Fabric Tensor Characterization of Tensor‐Valued Directional Data: Solution, Accuracy, and Symmetrization

Leng

Yang

2012

Journal of Applied Mathematics

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Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame-indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor-valued orientation distribution functions (ODFs). Two characterizations of the tensor-valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respectively. The later proves to be nonconvergent and less accurate but still an available solution for where fabric tensors are required in full symmetry. Analytic solutions of the two types of fabric tensors characterizing centrosymmetric and anticentrosymmetric tensor-valued ODFs are presented in terms of orthogonal irreducible decompositions in both two- and three-dimensional (2D and 3D) spaces. Accuracy analysis is performed on normally distributed random ODFs to evaluate the approximation quality of the two characterizations, where fabric tensors of higher orders are employed. It is shown that the fitness is dominated by the dispersion degree of the original ODFs rather than the orders of fabric tensors. One application of tensor-valued ODF and fabric tensor in continuum damage mechanics is presented.

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Effective Stress and Vector-valued Orientational Distribution Functions

Cited by 6 publications

References 20 publications

Tensorial characterisation of directional data in micromechanics

Tensorial characterisation of directional data in micromechanics

A microplane-based anisotropic damage effective stress

Fabric Tensor Characterization of Tensor‐Valued Directional Data: Solution, Accuracy, and Symmetrization

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