Note on the deformation-induced change in the curvature of a material surface in plane deformations

This article presents a semianalytical solution to a half-plane contact problem subjected to an arbitrarily distributed surface traction. The half-plane boundary is treated as a material surface of the Steigmann–Ogden type. Under the assumption of plane strain condition, the problem is formulated by coupling the methods of an Airy stress function and Fourier integral transforms. Stresses and displacements in the form of semi-infinite integrals are derived. A non-classical Flamant solution that is able to simultaneously account for the surface tension, membrane stiffness, and bending rigidity of the half-plane boundary is derived through limit analysis on the half-plane contact problem owing to a uniform surface traction. The fundamental Flamant solution is further integrated for tackling two half-plane contact problems owing to classical contact pressures corresponding to a rigid cylindrical roller and a rigid flat-ended punch. The resultant semi-infinite integrals are integrated by the joint use of the Gauss–Legendre numerical quadrature and the Euler transformation algorithm. Extensive parametric studies are conducted for comparing and contrasting the effects of Gurtin–Murdoch and Steigmann–Ogden surface mechanical models. The major observations and conclusions are two-fold. First, the introduction of either surface mechanical model results in size-dependent elastic fields. Second, the incorporation of the curvature-dependent nature of the half-plane boundary leads to bounded stresses and displacements in the fundamental Flamant solution. This is in contrast to the otherwise singular classical and Gurtin–Murdoch solutions. For all four case studies, the Steigmann–Ogden surface model also results in much smoother displacement and stress variations, indicating the significance of surface bending rigidity in nanoscale contact problems.

Section: Introductionmentioning

confidence: 99%

Flamant solution of a half-plane with surface flexural resistibility and its applications to nanocontact mechanics

Jiang

2019

“…The first-order expression for the deformation-induced change in the curvature of a material surface has attracted some attention in the literature (see, e.g., Appendix A in Wu et al [9] and the discussions in Dai et al [7] and Ti et al [10]). As clarified in Dai et al [7], d ω / ds fails to be the exact first-order expression for representing the change in the curvature of L under plane deformation unless L is assumed to be unstretchable. In this case, a modified boundary condition (hereinafter referred to as the modified boundary condition I) is suggested as [7],…”

Section: Boundary Conditions With Surface Tensionmentioning

confidence: 99%

“…For example, an analytic procedure, based on the Gurtin–Murdoch model and the complex variable method, was proposed in Dai et al [6] to determine the quantitative influence of surface tension on the stress concentration around a small elliptical hole located in an elastic gel–like material under remote loadings. As mentioned in Dai et al [7], however, the boundary condition arising from the Gurtin–Murdoch model fails to capture the influence of the stretch of a material surface on the surface tension–induced traction imposed on the material. As a consequence, some researchers, when studying the surface tension effects on the local deformation and overall properties of liquid inclusion–filled soft solids [8–10] which exhibit significant configurational changes under moderate external loadings, resorted to certain geometry formulae describing the change in the curvature of the liquid–solid interface during deformation instead of the Gurtin–Murdoch model to form the surface tension–related boundary condition on the liquid–solid interface.…”

Section: Introductionmentioning

confidence: 99%

Consistency of the boundary value problem of an elastic body involving surface tension in small deformations

Pei

Yang

Dai

2022

Self Cite

Surface tension has a significant influence on the mechanical response of solids at small scales to external loadings. Conventional small deformation theories remain effective in capturing such an influence provided the corresponding boundary condition incorporates some measure of the deformed configuration of the solid’s surface in evaluating the surface tension–induced traction acting on the solid. Since the governing equations arising from conventional small deformation theories for the bulk of the solid are defined strictly in the undeformed configuration, they may be sometimes incompatible with the surface tension–related boundary condition incorporating certain measure of the deformed configuration, leading to an inconsistent boundary value problem. To clarify how to pose a self-consistent boundary value problem involving surface tension, we consider the plane deformation of a perforated medium with its internal boundary endowed with surface tension and surface elasticity and its external boundary subjected to a mechanical loading. We first review three kinds of related boundary conditions on the internal boundary of the medium: the Gurtin–Murdoch-type boundary condition and two modified boundary conditions based on the change-in-curvature formula. We then derive the resultant force and moment of the traction caused by surface tension and surface elasticity for each of the boundary conditions. We show that for the Gurtin–Murdoch-type boundary condition, the resultant force vanishes unconditionally but the resultant moment depends on the rigid-body rotation of the medium, while for the two modified boundary conditions, both the resultant force and resultant moment (about a certain point) rely on local deformation gradients and the curvature of the internal boundary of the medium. Consequently, for the Gurtin–Murdoch-type boundary condition, one may easily establish a self-consistent boundary value problem by specifying an appropriate mechanical loading, while for the two modified boundary conditions, the boundary value problem is only occasionally self-consistent for some special and compatible combinations of the geometry of the internal boundary and the mechanical loading. Nevertheless, if the medium becomes infinitely large with its external boundary extremely distant from the internal boundary, then the boundary value problem corresponding to either of the three boundary conditions is always self-consistent for an arbitrary configuration of the internal boundary and an arbitrary far-field loading.

“…In all the works reviewed so far, the flexural resistibility of a solid surface or interface was neglected. Nonetheless, for a solid surface or interface that is likely to experience appreciable curvature changes during the loading and equilibrium process, the elastic energy stored in the surface or interface depends strongly on its curvature changes [35, 36]. In terms of molecular dynamics simulations, Chhapadia et al [35] unraveled the discrepancy between the Young moduli of an identical nanowire under bending and tensile deformation.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

Ban

2020

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.