Orientation of the localized damage zone in brittle solid under true triaxial compression

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Abstract

The problem of finding the optimal orientation of the localized damage zone in a brittle body under triaxial compression with intermediate stress varying from the minimum (Karman scheme) to the maximum (Becker scheme) principal stress is considered in the thin weakened layer approximation. The undamaged material is described by the relations of the linear-elastic isotropic body, the weakened zone is described by the model of nonlinear elasticity of Academician of the Russian Academy of Sciences V.P. Myasnikov with elastic moduli linearly dependent on the scalar parameter of the damage. The orientation of the weakened zone is given by two angles relative to the direction of action of the two main stresses, and the degree of weakening is given by the value of the damage parameter. The search for the optimal orientation of the zone for fixed values of the control parameters consists in maximizing the function that determines the rate of damage growth in this zone. As a result of the solution of the problem, the optimal orientations of the localized damage zone have been established for different ratios of principal stresses and damage level. It is shown that as the intermediate stress increases, there is a decrease in the angle of inclination of the zone relative to the direction of action of the maximum principal stress, as well as a narrowing of the interval of possible orientations of the zone relative to the direction of action of the intermediate principal stress. Based on the analysis of the ratio of the values of the shear components of the stress tensor in the plane of the localized damage zone, the possible shear directions along this zone are determined.

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About the authors

I. A. Panteleev

Institute of Continuous media Mechanics UR RAS

Author for correspondence.
Email: pia@icmm.ru
Russian Federation, Perm

D. V. Lozhkin

Institute of Continuous media Mechanics UR RAS

Email: lozhkin.d@icmm.ru
Russian Federation, Perm

V. A. Lyakhovsky

Geological Survey of Israel

Email: vladimir.lyakhovsky@gmail.com
Israel, Jerusalem

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Supplementary files

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2. Fig. 1. Orientation of localized damage zones under conventional triaxial compression.

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3. Fig. 2. Geometry of a representative volume with a localized damage zone under true triaxial compression (XYZ - global coordinate system, X*Y*Z* - coordinate system of the localized damage zone).

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4. Fig. 3. Dependence of the functional on the orientation angles of the localized damage zone for the parameter k = 0 (a), k = 0.5 (b), k = 1 (c).

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5. Fig. 4. Dependence of the normalized functional at angle for different values of the parameter k.

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6. Fig. 5. Dependence of the angle max, delivering the maximum value to the functional f (k, 0), on the parameter k for two values of the damage parameter (the black line corresponds to the Coulomb-Mohr angle).

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7. Fig. 6. Schematic representation of the optimal orientation angles of the localized damage zone for the case of

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8. Fig. 7. Maps of shear direction (I - shear, II - reset-shift, III - reset) in the plane of the localized damage zone for different angles of its orientation at k = 0.5 (a) and k = 1 (b) (dots indicate the considered cases of localized damage zone orientations).

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9. Fig. 8. Schemes of shear displacements of the medium blocks along the plane of the localized damage zone for cases A1 (reset), A2 (reset-shift) and A3 (horizontal shear).

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