Surface effects in an elastic solid with nanosized surface asperities

Abstract

a b s t r a c tThe effects of surface elasticity and surface tension on the stress field near nanosized surface asperities having at least one dimension in the range 1-100 nm is investigated. The general two-dimensional problem for an isotropic stressed solid with an arbitrary roughened surface at the nanoscale is considered. The bulk material is idealized as an elastic semi-infinite continuum. In accordance with the Gurtin-Murdoch model, the surface is represented as a coherently bonded elastic membrane. The surface properties are characterized by the residual surface stress (surface tension) and the surface Lame constants, which differ from those of the bulk. The boundary conditions at the curved surface are described by the generalized Young-Laplace equation. Using a specific approach to the boundary perturbation technique, GoursatKolosov complex potentials, and Muskhelishvili representations, the boundary value problem is reduced to the solution of a hypersingular integral equation. Based on the first-order approximation, some numerical results in the case of a periodic shape of the surface and the analysis of the influence of surface stress, surface tension, the surface shape, and the size of the asperity on the hoop stress at the surface are presented. It is found that the surface tension alone produces a high level of stress concentration, much more than can be reduced by surface stress arising as a result of deformation. The stress formula obtained by Gao (1991) for sinusoidal surfaces at the macrolevel is extended to nanosized surface asperities.

Spherical nano-inhomogeneity with the Steigmann–Ogden interface model under general uniform far-field stress loading

“…Gurtin et al (1998) generalized the original model by allowing all the components of the displacement vector to undergo a jump across the interface. The Gurtin-Murdoch model has been used to study nanosized rod (Altenbach et al, 2013;Grekov and Kostyrko, 2016), beams (Miller and Shenoy, 2000a;Eltaher et al, 2013;Ansari et al, 2015;Youcef et al, 2018), plates (Eremeyev et al, 2009;Ansari and Sahmani, 2011;Altenbach et al, 2012;Ansari and Norouzzadeh, 2016), shells (Altenbach et al, 2010;Altenbach and Eremeyev, 2011;Rouhi et al, 2016;Sahmani et al, 2016), films (Lu et al, 2011;Zhao and Rajapakse, 2013), wires (Diao et al, 2003;He and Lilley, 2008;Yvonnet et al, 2011), and inhomogeneities (Sharma et al, 2003;Duan et al, 2005a, b;Duan et al, 2005c;He and Li, 2006;Lim et al, 2006;Kushch et al, 2011;Kushch et al, 2013;Mi and Kouris, 2014;Nazarenko et al, 2016;Chen et al, 2018;Wang et al, 2018a), and much progress has been made in both analytical methods (Duan et al, 2009;Altenbach et al, 2013;Kushch et al, 2013;Dong et al, 2018) and numerical methods (Tian and Rajapakse, 2007;Feng et al, 2010;Dong and Pan, 2011).…”

The model of surface nanorelief within continuum mechanics

“…2b were obtained by solving Eq. 7in terms of the Fourier series (see [26]). The stress fields around plain nanobores are studied for the surface relief described by the function…”

“…Basically, the continuum surface stress model assumes that a nanostructure is made of the bulk and some surfaces with the surface module of the nanostructure being different from those of the bulk. Within the simplified Gurtin and Murdoch model of the surface/interface elasticity, a number of classical problems related to elastic phenomena on the nanoscale were studied analytically, and some size effects were found for relevant nanoscale materials (e.g., [12][13][14][15][16][19][20][21][22][23][24][25][26][27]).…”

“…x , a bar over a symbol denotes the complex conjugation, and a prime denotes the derivative with respect to the argument. If the suface nanorelief has a small deviation from the initial surface, then one can solve the corresponding problem by the boundary perturbation technique as in the previous works [24][25][26][27]. In this case, each function of the problem is represented as a power series in the small parameter 1, A a ε = < where A is the amplitude of the deviation and a is the tipical linear parameter of the suface.…”

The effect of nonlinear terms in boundary perturbation method on stress concentration near the nanopatterned bimaterial interface

“…However, as it was mentioned above, it is very important to analyze the stress state around the surface topological defects to predict the material failure. For this purpose, the boundary perturbation method (BPM) has been used extensively [39][40][41]. It should be noted that only the linear approximation of the BPM was analysed in these works.…”