Bending strength degradation of a cantilever plate with surface energy due to partial debonding at the clamped boundary

• Published in: Applied mathematics and mechanics Vol. 45; no. 9; pp. 1573 - 1594
• Main Authors: Hu, Zhenliang, Zhang, Xueyang, Li, Xianfang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024; Springer Nature B.V; School of Civil Engineering,University of South China,Hengyang 421001,Hunan Province,China%School of Civil Engineering,Central South University,Changsha 410075,China
• Edition: English ed.
• Subjects: Applications of Mathematics; Bend strength; Bending moments; Boundary value problems; Cantilever plates; Clamping; Classical Mechanics; Debonding; Exact solutions; Fluid- and Aerodynamics; Integral transforms; Kirchhoff theory; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Shear forces; Singular integral equations; Singularity (mathematics); Surface energy; Thickness; Thin plates; micro/nanoscale cantilever plate; partially debonded; 74A60; 74K20; bending fracture; oscillatory singularity; 74R10; O346.1; singular integral equation
• DOI: 10.1007/s10483-024-3140-7
• ISSN: 0253-4827

This paper investigates the bending fracture problem of a micro/nanoscale cantilever thin plate with surface energy, where the clamped boundary is partially debonded along the thickness direction. Some fundamental mechanical equations for the bending problem of micro/nanoscale plates are given by the Kirchhoff theory of thin plates, incorporating the Gurtin-Murdoch surface elasticity theory. For two typical cases of constant bending moment and uniform shear force in the debonded segment, the associated problems are reduced to two mixed boundary value problems. By solving the resulting mixed boundary value problems using the Fourier integral transform, a new type of singular integral equation with two Cauchy kernels is obtained for each case, and the exact solutions in terms of the fundamental functions are determined using the Poincare-Bertrand formula. Asymptotic elastic fields near the debonded tips including the bending moment, effective shear force, and bulk stress components exhibit the oscillatory singularity. The dependence relations among the singular fields, the material constants, and the plate’s thickness are analyzed for partially debonded cantilever micro-plates. If surface energy is neglected, these results reduce the bending fracture of a macroscale partially debonded cantilever plate, which has not been previously reported.