On the significance of coherent interface effects for embedded nanoparticles

Abstract

To extend classical micro and nanomechanics of inclusions and inhomogeneities from bulk phase only to interface-featured multi-phase, we formulated a solution procedure for evaluating the significance of interface stress on embedded nanoparticles. The methodology allows, for instance, analytical determination of the influential effects of interface stress on elastic fields of both nanoparticles and matrices within the general framework of continuum theory of bulk and interface elasticity. A thorough curvilinear analysis of a general Euclidean interface is performed with the aid of field theory and applied to facilitate the integration of displacement continuity and traction jump boundary conditions of interface elasticity into the classical formulation of displacement potentials. For illustration purposes, the solution methodology is applied to a spherical nanoparticle embedded in an infinite substrate subjected to most general far-field tension and eigenstrain loads. The developed model extends the solution limit of interface-induced length-scale effects on embedded ellipsoidal nanoparticles to the solution availability of their corresponding classical counterparts, provided that interface material properties are given.

Elastic disturbance due to a nanoparticle near a free surface

“…For the given ϕ the hoop stressσ θθ becomes constant. Although interface elasticity introduces an additional dependence on the radial coordinate R, the magnitude of variation proves to be very small for T/G 1 [18]. This is confirmed by the modified solutions ofσ θθ shown in Figures 4-6.…”

“…The values ofσ RR and σ RR at the pole A (ϕ = 0 and ϕ = π/2) are reasonably represented by equations (18) and (21) because A is the most distant from the free surface. The corresponding values at the points that are far away from A, on the other hand, deviate from equations (18) and (21) due to the disturbance of the free surface. The magnitude of deviation, however, is minor for the present numerical scenarios and is expected to become more prominent for large ratios of a to d.…”

“…As seen from Figure 10, the classical values ofσ RR and σ RR at points close to A can be reasonably approximated by equations (18) and (21), respectively. Additional superpositions with equations (25) and (26) result approximately in the modified solution, provided that the effects of interface elasticity are marginal.…”

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

“…The displacement field k zx u with the remote shear stress 0 zx σ can be written as: ( 1 2 )(2 ) ); (Duan et al, 2009;Mi and Kouris, 2014). This is also demonstrated in the numerical example shown in Fig.2.…”

“…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).…”