Analytical solution for the stress field of hierarchical defects: multiscale framework and applications
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-021-2673-9
Analytical solution for the stress field of hierarchical defects: multiscale framework and applications∗ Baijian WU1,2 , Sheng ZHOU1,2 , Zhaoxia LI1,2,† 1. Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China; 2. Department of Engineering Mechanics, Southeast University, Nanjing 210096, China (Received May 3, 2020 / Revised Jul. 6, 2020)
Abstract Hierarchical defects are defined as adjacent defects at different length scales. Involved are the two scales where the stress field distribution is interrelated. Based on the complex variable method and conformal mapping, a multiscale framework for solving the problems of hierarchical defects is formulated. The separated representations of mapping function, the governing equations of potentials, and the stress field are subsequently obtained. The proposed multiscale framework can be used to solve a variety of simplified engineering problems. The case in point is the analytical solution of a macroscopic elliptic hole with a microscopic circular edge defect. The results indicate that the microscopic defect aggregates the stress concentration on the macroscopic defect and likely leads to global propagation and rupture. Multiple micro-defects have interactive effects on the distribution of the stress field. The level of stress concentration may be reduced by the coalescence of micro-defects. This work provides a unified method to analytically investigate the influence of edge micro-defects within the scope of multiscale hierarchy. The formulated multiscale approach can also be potentially applied to materials with hierarchical defects, such as additive manufacturing and bio-inspired materials. Key words hierarchical defect, stress field, multiscale framework, scale separation, complex variable method, elliptic crack, edge defect Chinese Library Classification O302, O343 2010 Mathematics Subject Classification 35Q74, 74S70, 74G10
Nomenclature E, ν, σ, u, ϕ,
Young’s modulus; Poisson’s ratio; stress; displacement; complex potential;
ψ, ω, K, r, ρ,
complex potential; conformal transformation; stress concentration factor (SCF); defect size in z-plane; defect size in ζ-plane;
∗ Citation: WU, B. J., ZHOU, S., and LI, Z. X. Analytical solution for the stress field of hierarchical defects: multiscale framework and applications. Applied Mathematics and Mechanics (English Edition) (2021) https://doi.org/10.1007/s10483-021-2673-9 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 51878154) and the National Program on Major Research Project of China (No. 2016YFC0701301) ©The Author(s) 2021
2 Ω, D,
1
Baijian WU, Sheng ZHOU, and Zhaoxia LI domain in z-plane; unit disk in ζ-plane;
Γ,
unit circle.
Introduction
During the manufacturing process, microscopic edge defects emerge around non-ideal cracks such as elliptic cracks and inclusion. The adjacent defects at different length scales can
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