Journal of Nanomaterials

Volume 2018, Article ID 9063609, 9 pages

https://doi.org/10.1155/2018/9063609

## On the Stress Transfer of Nanoscale Interlayer with Surface Effects

Correspondence should be addressed to Mengjun Wu; moc.khmc@nujgnemuw

Received 18 September 2017; Revised 7 December 2017; Accepted 14 December 2017; Published 17 January 2018

Academic Editor: Jim Low

Copyright © 2018 Quan Yuan and Mengjun Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An improved shear-lag model is proposed to investigate the mechanism through which the surface effect influences the stress transfer of multilayered structures. The surface effect of the interlayer is characterized in terms of interfacial stress and surface elasticity by using Gurtin–Murdoch elasticity theory. Our calculation result shows that the surface effect influences the efficiency of stress transfer. The surface effect is enhanced with decreasing interlayer thickness and elastic modulus. Nonuniform and large residual surface stress distribution amplifies the influence of the surface effect on stress concentration.

#### 1. Introduction

Biomaterials and composites, such as nacre, bone, and ceramic, exhibit typical characteristics of a multilayered structure, which is composed of thick platelet-shaped hard phases and thin soft interlayers [1–4]. The discrete hard phases are bounded together but spaced from one another by the interlayers (adhesive layer). The volume fraction of interlayers is much smaller than the one of the large-scale hard phase in biomaterials. Interfaces of multilayered structure are coupled by those interlayers whose thicknesses have to shrink to microns or nanometers. For instance, the protein interlayer in nacre is about 20 nm thick; the matrix layer of bone and dentin is about 240 nm thick [3]. The robustness and stress transfer of the multilayered structure are ensured by the introduction of interlayers. The interlayer morphology determines the stress distribution and failure mode of structures under external forces [5, 6]. Classical shear-lag theory is based on some simple assumptions and continuity condition at the interfaces; this theory is used to effectively study stress distribution in a multilayered structure [6]. In shear-lag analysis, the interlayers are treated as macroscopic continuum matter, regardless of their thickness [2, 6]. Kotha et al. [2] and Tsai et al. [7] assumed that the interlayers only carry shear stress and are uniformly deformed with the stress field of the structure; when the interlayer thickness is reduced to the nanoscale level, the surface effects of the interlayers will influence the stress transfer of the multilayered structure because of the extremely large ratio of the surface area to the interlayer volume. However, the mechanism through which size and surface effects of the nanoscale interlayers influence stress transfer remains unclear.

Two main types of surface elasticity theory can characterize the surface effects for nanoscale objects [8]. The first type is the Gurtin–Murdoch elasticity model and its extension, in which the surface layer is considered as a membrane with vanishing thickness and assumed to perfectly adhere to the bulk substrate [9]. The Gurtin–Murdoch elasticity model has been used to analyze the natural frequency of microbeams [10], bridging mechanism of nanofibers [11], and near-tip stress fields [12]. Continuum quantities, surface free energy, and surface stress are central concepts in these analyses. Analogous to the bulk elastic constants, the introduced surface elastic constants are difficult to measure. An alternative choice of Eulerian surface-energy density is adopted to describe surface traction; the surface elastic constants are then no longer needed [13]. This improved surface elastic theory is further used to study the elastic properties of fcc metallic nanofilms under biaxial tension. The second type is the discrete Frenkel–Kontorova model [14], where the surface layer is represented by atoms connected together by springs or virtual bonds. The Frenkel–Kontorova model is used to calculate the surface stress of nanosheets [8]. The two types of surface elasticity model are both effective in describing the size-dependent mechanical properties of micro-/nanosized materials. Therefore, the macroscopic hard layers of multilayered structures comprise a considerable volume fraction and carry all external loads. The nanoscale interlayers dominate stress transfer, which results in the feasibility of applying the Gurtin–Murdoch model.

This study aims to elucidate the planar stress transfer mechanism of the nanoscale interlayers in a multilayered structure. Considering the geometry of the multilayered structure, the plane strain behavior of the hard layer subjected to tension is formulated using shear-lag theory. The surface effects of the interlayers at the interface coupled with stress transfer are formulated by the Gurtin–Murdoch model. Using the stress jump across the interface and the continuity condition of the interlayers, the global interfacial shear stress dominant governing equations are established. We obtain the stress distributions of the multilayered structure with three surface stress situations.

#### 2. Shear-Lag Analysis of Interlayers with Surface Effects

##### 2.1. Mathematical Formulation

As the main parts of a multilayered structure, hard layers are subjected to external load. The interlayers acting as adhesive layers strongly influence connection and stress transfer in the structure. We consider a simple sandwich laminate structure with one adhesive (layer 2) and two hard layers (layer 1 and layer 3) to estimate how the surface effects of the nanoscale interlayers influence stress transfer (Figure 1(a)). For hard layers, it seems reasonable to utilize approximate planar isotropy and emphasize the mechanical anisotropy along the thickness direction. In order to investigate the stress transfer under tension, the through-thickness normal strain and normal stress (*y* direction) are neglected according to Kamat et al.’s and Nairn and Mendels’s shear-lag model (see [1, 6]). The thickness and elastic modulus of interlayers are much smaller than those of hard layers, and the macroscopic heterogeneity is neglected. The structure length is* l*, whereas the thicknesses of layers 1–3 are* t*_{1},* t*_{2}, and* t*_{3}, respectively, where . Equilibrium tensions are forced on the right edge of layer 1 and left edge of layer 3. and are the upper and lower interfaces of the interlayer. The edges at* y* =* y*_{0} and* y* =* y*_{3} are free boundaries.