Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects

Shen, Jingjin

doi:10.1177/1081286518797387

Cited by 11 publications

(4 citation statements)

References 30 publications

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“…In fact, within the general context of nanomechanics, the contact properties of a half-plane and a half-space have aroused considerable interests in the literature. The contact loads induced by both surface tractions [20][21][22][23][24][25][26] and rigid indenters [27][28][29][30][31][32] have been considered. In these studies, the curvature-dependence of half-plane or half-space boundaries was not taken into account.…”

Section: Introductionmentioning

confidence: 99%

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

Jiang

2019

Mathematics and Mechanics of Solids

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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.

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Section: Introductionmentioning

confidence: 99%

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

Jiang

2019

Mathematics and Mechanics of Solids

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“…The constitutive relations and equilibrium conditions of the material surface led to non-classical stress-boundary conditions of the surfacedependent contact problem [12]. Until now, using surface elasticity theory, research on micro-/nano-scale contact has mainly focused on dry contact [12,13,18,19]. In addition, due to the widespread appearance of EHL contact in micro/nano bearings and gears [20,21], several researchers have investigated micro-/nano-scale EHL contact problems [22,23].…”

Section: Introductionmentioning

confidence: 99%

Surface effects on elastohydrodynamic lubrication contact of piezoelectric materials with non-Newtonian fluid

Su,

Song,

2023

Smart Mater. Struct.

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By using surface piezoelectricity theory, this article investigates the elastohydrodynamic lubrication (EHL) line contact of a transversely isotropic piezoelectric half-plane with consideration of the surface effect under a rigid cylindrical punch. The surface effect in the surface piezoelectricity theory is mainly described by the following parameters: surface piezoelectric constant, surface dielectric constant, surface elastic constant, and residual surface stress. The punch is treated as the electrical insulator. The lubricant, whose viscosity and density are dependent on fluid pressure, is chosen as non-Newtonian fluid. Firstly, by analyzing the frictionless dry contact of piezoelectric materials with surface effect, dry contact pressure distribution and EHL film thickness equation are obtained. Then, an iterative method is proposed to get fluid pressure and film thickness at lubricant contact region by calculating the fluid-solid coupled nonlinear equations. Effects of surface dielectric constant, surface piezoelectric constant, surface elastic constant, residual surface stress, punch radius, entraining velocity, and slide/roll ratio on the film thickness and fluid pressure are examined. Our analysis indicates that surface effect has an essential effect on the EHL contact behaviors of piezoelectric materials at micro-/nano-scales.

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“…Nanoindentation of a cylindrical roller onto a semi-plane with the Steigmann-Ogden model has been explored in [30]. Axisymmetric problems for transversely isotropic materials with the Gurtin-Murdoch surface elasticity have been investigated in [31,32]. Contact problems with a curvature-dependent surface tension have been studied in [33].…”

Section: Introductionmentioning

confidence: 99%

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

Zemlyanova

White

2021

Mathematics and Mechanics of Solids

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An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

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Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects

Cited by 11 publications

References 30 publications

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

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

Surface effects on elastohydrodynamic lubrication contact of piezoelectric materials with non-Newtonian fluid

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

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