International Journal of Engineering Science
Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects
Introduction
There is growing interest in the study of the mechanics of nano-scale structures and devices. The natural approach is to consider atomistic modeling techniques for nano-scale domains but such techniques require a very large computational effort. The application of continuum-based approaches is considered attractive due to their lesser complexity and computational efficiency. The surface-to-volume ratio of a nano-scale domain is relatively high compared to that of macro-scale domains. The energy associated with atoms at or near a free surface is different from that of atoms in the bulk. The effect of surface free energy therefore becomes important in the case of nano-scale problems [1]. Povstenko [2] observed that the stress field caused by heterogeneous surface tension in a solid half-space can be used to explain the high stresses in the surface layer that cause a zone with high dislocation density when a surface-active melt interacts with a metal. In addition, for some soft solids, such as polymer gels, the surface energy (hence surface stresses) has an important influence on surface topographical patterns that are used for applications in surface self-assembly regulation, micro-fluidic flow control and direction, etc. [3], [4]. Consequently, the study of the elastic field of a solid with surface energy effects is of interest to many current technological developments.
Surface energy effects are generally ignored in traditional continuum mechanics. This is not the case for nano-scale structures due to their high surface/volume ratio, soft materials where the ratio of surface energy per unit area to the bulk Young’s modulus is comparable to the characteristic size of the material element and other situations where surface tension gradients and other surface energy driven effects have a significant influence on the response. Gurtin and Murdoch [5], [6] developed a theoretical framework based on continuum mechanics concepts that included the effects of surface and interfacial energy, in which the surface is modeled as a mathematical layer of zero thickness perfectly bonded to an underlying bulk. The surface (interface) has its own properties and processes that are different from the bulk. Miller and Shenoy [7] and Shenoy [8] demonstrated that size-dependent behaviour of nano-scale structural elements can be modeled by applying the Gurtin–Murdoch continuum model with surface properties determined from atomistic modeling [9]. Tian and Rajapakse [10] examined the size-dependent elastic field due to a nano-scale elliptical defect in an isotropic matrix and observed unstable defect geometries.
The elastic field of a surface-loaded layer of nano-scale thickness bonded to a rigid base has important applications in the study of nano-electronics devices, coatings and films, deformations due to quantum dots, etc. Similarly, the response of a soft elastic layer with surface energy effects can be used in the study of adsorption of molecules/cells into thin layers and their interaction energies, micro-fluidic devices, etc. The classical elasticity solution of a layer of finite thickness bonded to a rigid base was given by Pickett [11] which has found extensive applications in tribology, geomechanics, biomechanics, etc. Povstenko [2] derived the elastic field of a half-space cased by a jump in the surface tension over a circular area by neglecting the bulk properties He and Lim [12] derived the surface Green’s functions of a soft incompressible isotropic elastic half-space with surface energy effects by using the Gurtin–Murdoch model. In addition to the incompressibility, they further restricted their derivation to the special case where the surface elastic properties are same as the bulk properties. Wang and Feng [13] studied the response of a half-plane subjected to surface pressures by neglecting the surface elastic constants and considering only the influence of constant surface tension. Huang and Yu [14] considered a surface-loaded half-plane with non-zero surface elastic constants in the absence of any surface tension.
In this paper, the fundamental problem of a compressible isotropic elastic layer with complete surface stress effects (non-zero surface tension and surface elastic properties) that is bonded to a rigid base and subjected to surface loading is considered. Both two-dimensional plane and axisymmetric problems are considered. The Gurtin–Murdoch model is applied to derive the elastic field of the layer. Fourier and Hankel integral transforms are used to solve the boundary-value problems involving non-classical boundary conditions associated with the generalized Young–Laplace equation. Closed-form analytical solutions are presented for the case of a layer of infinite thickness (half-plane/space) and in this case the influence of surface energy effects can be explicitly identified. For a layer of finite thickness, the elastic field is examined numerically to assess the influence of surface energy effects and layer thickness.
Section snippets
Governing equations and general solutions
Consider an elastic layer of finite thickness bonded to a rigid base as shown in Fig. 1. The layer is subjected to surface loading and its response is modeled by using the Gurtin–Murdoch continuum model [5], [6]. According to this model, the surface energy effects are accounted for by considering the surface as a mathematical layer of zero thickness with relevant elastic properties and residual surface tension, that is perfectly bonded to the underlying bulk material. In the bulk, the governing
Plane problems
The following equations can be established by using the boundary conditions corresponding to the system shown in Fig. 1 and Eqs. (4), (5), (6).where p(x) and t(x) denotes the applied surface loading in the z- and x-directions, respectively; and κs = 2μs + λs, is a surface material constant.
Shenoy [5] and Cammarata [17] have reported the values of the surface stress at zero strain (τs) for different surface orientations of pure metals and
Solutions for a semi-infinite medium
Although a closed-form solution for the elastic field of a layer cannot be obtained due to the complexity of the integrals involved in the solution, the results of Section 3 can be specialized for the case where h approaches infinity to obtain a set of closed-form solution.
Numerical results and discussion
In this section, selected numerical results are presented to demonstrate the salient features of the elastic field. The surface elastic constants can be obtained from atomistic simulations [7], [8], [9]. First, consider the case of an elastic half-plane subjected to vertical and horizontal point loads at the surface. The solution for the stress field is given in Section 4.1. A close examination of the present solutions indicates that it is convenient to introduce non-dimensional coordinates, x0 =
Conclusions
The plane and axisymmetric problems for an elastic layer of finite thickness subjected to surface loading are investigated by using the Gurtin–Murdoch continuum model that accounts for surface energy effects. A set of analytical solutions are presented by using Fourier and Hankel Transform techniques. The solutions are expressed in terms of semi-infinite integrals that cannot be evaluated in closed-form and numerical quadrature is used to evaluate the integrals. However, closed-form solutions
Acknowledgement
The work presented in this paper was supported by a grant from the Natural Science and Engineering Research Council of Canada.
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