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Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen’s extension of Noll’s theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips.
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A midsurface elasticity model for a thin, nonlinear, gradient elastic plate
In this paper, we derive a dynamic surface elasticity model for the two-dimensional midsurface of a thin, three-dimensional, homogeneous, isotropic, nonlinear gradient elastic plate of thickness ℎ. The resulting model is parameterized by five, conceivably measurable, physical properties of the plate, and the stored surface energy reduces to Koiter’s plate energy in a singular limiting case. The model corrects a theoretical issue found in wave propagation in thin sheets and, when combined with the author’s theory of Green elastic bodies possessing gradient elastic material boundary surfaces, removes the singularities present in fracture within traditional/classical models. Our approach diverges from previous research on thin shells and plates, which primarily concentrated on deriving elasticity theories for material surfaces from classical three-dimensional Green elasticity. This work is the first in rigorously developing a surface elasticity model based on a parent nonlinear gradient elasticity theory.
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- PAR ID:
- 10528138
- Publisher / Repository:
- IJES24_Rodriguez
- Date Published:
- Journal Name:
- International Journal of Engineering Science
- Volume:
- 197
- Issue:
- C
- ISSN:
- 0020-7225
- Page Range / eLocation ID:
- 104026
- Subject(s) / Keyword(s):
- Elasticity Thin plates Koiter’s energy Surface elasticity Gradient elasticity Wave propagation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ijengsci.2024.104026
