Abstract
The probabilistic aspects of multiscale modeling of the fracture of heterogeneous structures are considered. An approach combining homogenization methods with phenomenological and numerical models of fracture mechanics is proposed to solve the problems of assessing the probabilities of destruction of structurally heterogeneous materials. A model of a generalized heterogeneous structure consisting of heterogeneous materials and regions of different scales containing cracks and cracklike defects is formulated. Linking of scales is carried out using the kinematic conditions and the multiscale principle of virtual forces. The probability of destruction is formulated as the conditional probability of successive nested fracture events of different scales. Cracks and cracklike defects are regarded as the main sources of fracture. The distribution of defects is represented in the form of Poisson ensembles. Critical stresses at the crack tips are described by the Weibull model. Analytical expressions for the fracture probabilities of multiscale heterogeneous structures with multilevel limit states are derived. An approach based on a modified Monte Carlo method of statistical modeling is proposed to assess the fracture probabilities with allowance for the real morphology of heterogeneous structures. The characteristic feature of the proposed method is the use of a three-level fracture scheme with numerical solution to the problems at the micro-, meso-, and macroscale. The main variables are the generalized crack driving force and crack growth resistance. Crack sizes are regarded as the generalized coordinates. To reduce the dimension, the problem of fracture mechanics is reformulated to the problem of stability of a heterogeneous structure under loading with variations of the generalized coordinates with the analysis of the virtual work of generalized forces. Expressions for estimating the fracture probabilities for multiscale heterogeneous structures are obtained using the modified Monte Carlo method. The prospects of using the developed approaches to assess the fracture probabilities and address the problems of risk analysis of heterogeneous structures are pointed out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Wu, X. and Zhu, Y., Heterogeneous materials: a new class of materials with unprecedented mechanical properties, Mater. Res. Lett., 2017, vol. 5, no. 8, pp. 527–532.
Computational Approaches to Materials Design: Theoretical and Practical Aspects, Datta, S. and Davin, J.P., Eds., Hershey, PA: IGI Global, 2016.
Hashin, Z. and Shtrickman, S., On some variational principles in anisotropic and nonhomogeneous elasticity, J. Mech. Phys. Solids, 1962, vol. 10, pp. 335–342.
Multiscale Modeling of Heterogenous Materials: From Microstructure to Macro-Scale Properties, Cazaku, O., Ed., Chichester: Willey, 2008.
Oliver, J. and Huespe, A.E., Continuum approach to material failure in strong discontinuity settings, Comput. Methods Appl. Mech. Eng., 2004, vol. 193, pp. 3195–3220.
Nguyen, V.P., Lloberas-Valls, O., Stroeven, M., and Sluys, L.J., Computational homogenization for multiscale crack modeling. Implementational and computational aspects, Int. J. Numer. Methods Eng., 2012, vol. 89, no. 2, pp. 192–226.
Taroco, E., Blanco, J., and Feijoo, R., Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications, Chichester: Willey, 2020.
Lepikhin, A.M., Makhutov, N.A., Moskvichev, V.V., and Chernyaev, A.P., Veroyatnostnyi risk-analiz konstruktsii tekhnicheskikh sistem (Probabilistic Risk Analysis of Technical System Designs), Novosibirsk: Nauka, 2003.
Parton, V.Z. and Morozov, E.M., Mekhanika uprugoplaticheskogo razrusheniya (Elastoplastic Fracture Mechanics), Moscow: Nauka, 1985.
Lepikhin, A.M., Moskvichev, V.V., and Chernyaev, A.P., Acoustic-emission control of the deformation and fracture of high-pressure metal-composite tanks, J. Appl. Mech. Tech. Phys., 2018, vol. 5, no. 3, pp. 511–518. https://doi.org/10.15372/PMTF20180316
Matvienko, Yu.G., Vasil’ev, I.E., and Chernov, D.V., Study of the fracture kinetics of a unidirectional laminate using acoustic emission and video recording, Inorg. Mater., 2020, vol. 56, no. 15, pp. 1536–1550. https://doi.org/10.1134/S0020168520150145
Lepikhin, A.M., Moskvichev, V.V., Burov, A.E., et al., Experimental study of the strength and durability of metal-composite high-pressure tanks, Inorg. Mater., 2020, vol. 56, no. 15, pp. 1478–1484. https://doi.org/10.1134/S0020168520150108
Stefanou, G., The stochastic finite element method: past, present and future, Comput. Methods Appl. Mech. Eng., 2009, vol. 198, pp. 1031–1051.
Zhou, X.-Y., Gosling, P.D., Ullah, Z., Kaczmarczyk, L., and Pearce, C.J., Stochastic multi-scale finite element based reliability analysis for laminated composite structures, Appl. Math. Model., 2017, vol. 45, pp. 457–473.
Bourdin, B. and Francfort, G.A., The variational approach to fracture, J. Elast., 2008, vol. 91, pp. 5–148.
Shayanfar, M.A., Barkhordari, M.A., and Roudak, M.A., An adaptive importance sampling-based algorithm using the first order method for structural reliability, Int. J. Optim. Civ. Eng., 2017, vol. 7, no. 1, pp. 93–107.
Au, S.K. and Beck, J.L., Estimation of small failure probabilities in high dimensions by subset simulation, Probab. Eng. Mech., 2001, vol. 16, pp. 263–277.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by E. Oborin
Rights and permissions
About this article
Cite this article
Lepikhin, A.M., Makhutov, N.A. & Shokin, Y.I. Probabilistic Multiscale Modeling of Fracture in Heterogeneous Materials and Structures. Inorg Mater 57, 1511–1518 (2021). https://doi.org/10.1134/S0020168521150115
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0020168521150115