On the effective elastic properties of a material with mutually perpendicular systems of parallel cracks
DOI:
https://doi.org/10.21638/11701/spbu10.2022.109Abstract
The effective properties of cracked solids are often expressed in terms of the crack density parameter or its tensor generalization, using the approximation of noninteracting cracks. This approximation remains accurate at sufficiently high crack densities, provided the location of cracks are random. The presented analysis confirms that the effective elastic moduli of a material with ordered fracture structures strongly depend on the linear dimensions of cracks and their mutual arrangement even at a constant crack density. A change in these parameters can cause a noticeable anisotropy of the effective properties of the material even when the eigenvalues of the crack density tensor are equal to each other. The effective elastic characteristics of a material with one doubly periodic system of parallel cracks are compared with those for a material with two mutually perpendicular systems of such cracks in a two-dimensional formulation. The calculations are carried out using the approximate method of M. Kachanov for determining the mean stresses at the cracks edges, applicable for large systems of interacting cracks. Analysis of the obtained results showed that the effective compliance of the material in a certain direction is largely determined by the effects of interaction (shielding and amplification) within a system of parallel cracks perpendicular to this direction. The interaction of this system of cracks with the perpendicular system has a weak effect on the indicated properties in the case of rectangular symmetry of the system. In this case, the interaction of mutually perpendicular systems of cracks leads to a violation of the symmetry of the tensor of effective elastic constants.
Keywords:
crack density, crack interaction, effective elastic properties
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References
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Doan T., Le-Quang H., To Q.-D. Effective elastic stiffness of 2D materials containing nanovoids of arbitrary shape // Intern. J. of Engineering Science. 2020. Vol. 150. N 103234. https://doi.org/10.1016/j.ijengsci.2020.103234
Du K., Cheng L., Barthelemy J. F., Sevostianov I., Giraud A., Adessina A. Numerical computation of compliance contribution tensor of a concave pore embedded in a transversely isotropic matrix // Intern. J. of Engineering Science. 2020. Vol. 152. N 103306. https://doi.org/10.1016/j.ijengsci.2020.103306
Markov A., Trofimov A., Sevostianov I. A unified methodology for calculation of compliance and stiffness contribution tensors of inhomogeneities of arbitrary 2D and 3D shapes embedded in isotropic matrix — open access software // Intern. J. of Engineering Science. 2020. Vol. 157. N 103390. https://doi.org/10.1016/j.ijengsci.2020.103390
Sevostianov I., Kushch V. I. Compliance contribution tensor of an arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic elastic material // Intern. J. of Engineering Science. 2020. Vol. 149. N 103222. https://doi.org/10.1016/j.ijengsci.2020.103222
Kachanov M. Elastic solids with many cracks and related problems // Advances in Applied Mechanics. 1993. Vol. 30(C). P. 259–445. https://doi.org/10.1016/S0065-2156(08)70176-5
Kachanov M., Mishakin V. V. On crack density, crack porosity, and the possibility to interrelate them // Intern. J. of Engineering Science. 2019. Vol. 142. P. 185–189. https://doi.org/10.1016/j.ijengsci.2019.06.010
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Vakulenko A., Kachanov M. Continuum theory of medium with cracks // Mech. of Solids. 1971. Vol. 6(4). P. 145–151.
Kachanov M. On the problems of crack interactions and crack coalescence // Intern. J. of Fracture. 2003. Vol. 120(3). P. 537–543.
Sevostianov I., Kachanov M. On approximate symmetries of the elastic properties and elliptic orthotropy // Intern. J. of Engineering Science. 2008. Vol. 46(3). P. 211–223.
Abakarov A., Pronina Y., Kachanov M. Symmetric arrangements of cracks with perturbed symmetry: extremal properties of perturbed configurations // Intern. J. of Engineering Science. 2021. Vol. 171(4). N 103617. https://doi.org/10.1016/j.ijengsci.2021.103617
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Pronina Yu., Maksimov A., Kachanov M. Crack approaching a domain having the same elastic properties but different fracture toughness: Crack deflection vs penetration // Intern. J. of Engineering Science. 2020. Vol. 156. N 103374. https://doi.org/10.1016/j.ijengsci.2020.103374
Shuvalov G. M., Vakaeva A. B., Shamsutdinov D. A., Kostyrko S. A. The effect of nonlinear terms in boundary perturbation method on stress concentration near the nanopatterned bimaterial interface // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2020. Т. 16. Вып. 2. С. 165–176. https://doi.org/10.21638/11701/spbu10.2020.208
References
Kachanov M. On the effective moduli of solids with cavities and cracks. Intern. J. of Fracture, 1993, vol. 59(1), pp. R17–R21. https://doi.org/10.1007/BF00032223
Doan T., Le-Quang H., To Q.-D. Effective elastic stiffness of 2D materials containing nanovoids of arbitrary shape. Intern. J. of Engineering Science, 2020, vol. 150, no. 103234. https://doi.org/10.1016/j.ijengsci.2020.103234
Du K., Cheng L., Barthelemy J. F., Sevostianov I., Giraud A., Adessina A. Numerical computation of compliance contribution tensor of a concave pore embedded in a transversely isotropic matrix. Intern. J. of Engineering Science, 2020, vol. 152, no. 103306. https://doi.org/10.1016/j.ijengsci.2020.103306
Markov A., Trofimov A., Sevostianov I. A unified methodology for calculation of compliance and stiffness contribution tensors of inhomogeneities of arbitrary 2D and 3D shapes embedded in isotropic matrix — open access software. Intern. J. of Engineering Science, 2020, vol. 157, no. 103390. https://doi.org/10.1016/j.ijengsci.2020.103390
Sevostianov I., Kushch V. I. Compliance contribution tensor of an arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic elastic material. Intern. J. of Engineering Science, 2020, vol. 149, no. 103222. https://doi.org/10.1016/j.ijengsci.2020.103222
Kachanov M. Elastic solids with many cracks and related problems. Advances in Applied Mechanics, 1993, vol. 30(C), pp. 259–445. https://doi.org/10.1016/S0065-2156(08)70176-5
Kachanov M., Mishakin V. V. On crack density, crack porosity, and the possibility to interrelate them. Intern. J. of Engineering Science, 2019, vol. 142, pp. 185–189. https://doi.org/10.1016/j.ijengsci.2019.06.010
Bristow J. R. Microcracks and the static and dynamic elastic constants of annealed and heavily cold-worked metals. British J. Appl. Phys., 1960, vol. 11, pp. 81–85.
Vakulenko A., Kachanov M. Continuum theory of medium with cracks. Mech. of Solids, 1971, vol. 6(4), pp. 145–151.
Kachanov M. On the problems of crack interactions and crack coalescence. Intern. J. of Fracture, 2003, vol. 120(3), pp. 537–543.
Sevostianov I., Kachanov M. On approximate symmetries of the elastic properties and elliptic orthotropy. Intern. J. of Engineering Science, 2008, vol. 46(3), pp. 211–223.
Abakarov A., Pronina Y., Kachanov M. Symmetric arrangements of cracks with perturbed symmetry: extremal properties of perturbed configurations. Intern. J. of Engineering Science, 2021, vol. 171(4), no. 103617. http://doi.org/10.1016/j.ijengsci.2021.103617
Grekov M. A., Sergeeva T. S. Interaction of edge dislocation array with bimaterial interface incorporating interface elasticity. Intern. J. of Engineering Science, 2020, vol. 149, no. 103233. https://doi.org/10.1016/j.ijengsci.2020.103233
Pronina Yu., Maksimov A., Kachanov M. Crack approaching a domain having the same elastic properties but different fracture toughness: Crack deflection vs penetration. Intern. J. of Engineering Science, 2020, vol. 156, no. 103374. https://doi.org/10.1016/j.ijengsci.2020.103374
Shuvalov G. M., Vakaeva A. B., Shamsutdinov D. A., Kostyrko S. A. The effect of nonlinear terms in boundary perturbation method on stress concentration near the nanopatterned bimaterial interface. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2020, vol. 16, iss. 2, pp. 165–176. https://doi.org/10.21638/11701/spbu10.2020.208
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