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 t1, t2, and t3, 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 = y0 and y = y3 are free boundaries.
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In the absence of body forces and under the condition of infinitesimal deformation, the geometric equations, constitutive relations, and equilibrium equations of every layer are as follows:where superscript “i” denotes the ith layer; , , and are the components of the displacement vector, strain tensor, and stress tensor, respectively. , , and are the elastic modulus, shear modulus, and Poisson’s ratio of the ith layer. For the infinitesimal sections of layers 1–3 (Figure 1(b)), equilibrium (7) is integrated along the thickness direction of every layer. We then obtainwhere indicates averaging over the thickness of the ith layer, such aswhich represents the normal stress of the ith layer. and are the interfacial shear stress at the upper and lower interfaces of the interlayer.
Substituting (1) into (4) and averaging the thickness of the ith layer provideFor stress transfer problems during loading in the direction, can be eliminated. Equation (13) is reduced to 1D Hooke’s law as follows:Tsai et al. have performed experimental analysis to verify the nonlinear deformation and adhesive stress distribution of interlayers in joints [7]. For the isotropic and unidirectional laminated structure, the transferred force is not altered by the interlayers with relatively low shear modulus and thickness. An appropriate assumption about interlayers has been used in Nairn’s planar shear-lag analysis which is employed in this paper. Assuming that the shear stress in each layer is a linear distribution along the thickness, the shear stress in the ith layer can be written as follows:We need to introduce the general shear-lag assumption, , to proceed. Equation (3) can be simplified as . Combining (6) and (15) providesEquation (16) is averaged over the thickness of each layer by using the transform method proposed by McCartuay and developed by Nairn and then expressed as follows [6]:At the and interfaces, the boundary conditions of displacement continuity can be expressed as follows:After subtracting (17) from (18) and (14) from (19), differentiating the differences with respect to x and using (21) lead to
Noticing the stress jump cross the interface induced by surface effects, the equilibrium condition of the Gurtin–Murdoch elasticity model at and interfaces is written as follows [9–11]:where represents the surface stresses; is the interface curvature. and denote the stress jump along the tangential and normal directions, respectively. For the plane interface , no stress jump is found in the direction. Moreover, the surface stress at an isotropic interface can be provided as follows [9–11]:where is the Kronecker delta. is the residual interface tension, and and are the surface elastic constants. The surface stress at and interfaces can be expressed as follows using (4) and (5):The tangential stress jump can be obtained by combining (23), (25), and (26) and using (6):The interlayer can be assumed to only carry shear stress because of the small thickness and stiffness [1–3] and can be neglected. Substituting (25) and (26) into (27) and (28) and differentiating the two equations with respect to x lead to
Furthermore, the normal stress of the two hard layers can be deduced as follows by substituting (29) into (22) and then using (14):Solving (30), the interfacial shear stress at the upper and lower interlayer interfaces can be expressed as follows:where .
Moreover, differentiating (9) with respect to x and using (27) and (31) lead toWe assume that stress carried by the structure is “.” Hence, the equilibrium equation on every section can be expressed as follows:Finally, (33) can be rewritten using the above equation as follows:where ,where is related to stress at the ends and interfaces. Equation (35) is a normal stress dominant governing equation of layer 1 and uses the boundary conditions at the ends of the structure; the solution of (35) can then be obtained. Shear stress distribution can be obtained by substituting the solution to (9), (10), (27), and (28).
2.2. Results and Discussion
The influence of the surface effect of nanoscale interlayers on stress transfer is quantitatively illustrated by a few examples. Equation (35) is a periodic second-order linear differential equation. For stretched structures, (35) has the same general solution. The complete solution depends on a particular solution, which determines the stress distribution of the structure with different residual interface tensions. Specially, stress transfer is changed by the tangential stress jump at the interface of the interlayer, demonstrating the surface effects. For simplicity, we assume that the geometry and material properties of layers 1 and 3 are the same (i.e., , , and ). Stress transfer in simple sandwich laminate structure has been analyzed by Kamat et al. and Nairn and Mendels (see [1, 6]) based on shear-lag mode without consideration of surface effects. The relevant material properties are adopted in our manuscript, GPa, GPa, GPa, GPa, , . The applied pressure is 10 MPa. For the determination of surface effects, the geometries of interlayers and hard layers are decreased, mm, μm, nm (these changes have no effects on the forms of stress distribution). The residual interface stress induced initial mismatch between the hard phase and interlayers have strong influence on the natural frequency of microbeams, pullout force of nanofibers, and stress transfer [10, 11]. The influences of the three typical residual interface tension distributions on stress transfer are elucidated as follows:(1): the influence of the surface effects will not be included in a particular solution, where the interface tension is a linear distribution. The stress distributions of the structure are as follows: where A1, B1, and C1 are known constants provided in Appendix. The surface effects do not directly influence the normal stress of hard layers. Figure 2 shows the stress distributions of the multilayered structure. The structure is essentially symmetric about (asymmetric along the thickness direction is allowed) but the mechanical boundary condition is asymmetric at ends. As shown in Figure 2, the normal stress distribution is asymmetric and interfacial shear stress is symmetric along the length direction (-axis). For the symmetric mechanical boundary condition, the normal stress distribution is symmetric but the zero stress at center of the structure is not certainly reached. One of the distributions is the result of our model, and the other two are calculated through classical shear-lag theory without considering the surface effects. Figure 2(a) shows that the normal stress is smaller and smoother than those calculated using the models of Kotha et al. and Tsai et al. [2, 7]. This difference could be attributed to two reasons: uniform deformation of the interlayer over the thickness direction is assumed in their models but is not necessarily limited in our work; and stress jump across the interface is emphasized in our model. Figure 2(b) shows the shear stress of layer 1. The stress obtained using the models of Kotha et al. and Tsai et al. is zygomorphic. The calculated result is smaller at the left end and larger at the right end because surface effect increases the efficiency of stress transfer with increasing normal stress. However, the shear stress of the interlayer at the and interface differs by a residual surface tension, , compared with the one in layers 1 and 3. The influence of the geometry and material properties of the interlayer on stress transfer can be studied under linear surface stress distribution. Figure 3(a) shows the plot of normal and shear stress distributions versus the ratio of the thickness of layer 1 to the interlayer. The normal stress in the left region is slightly influenced by the ratio. Accordingly, the transferred stress in the right region increases with decreasing interlayer thickness. The surface effects are enhanced by the nanoscale interlayer. As shown in Figure 3(b), the shear stress also increases with decreasing ratio. Overall, the surface effect influences stress transfer efficiency under the decrease of the interlayer scale. Figure 4 presents the stress distributions of layer 1 versus the ratio of the elastic modulus of layer 1 to that of the interlayer. Similarly, the surface effect definitely increases stress transfer efficiency under the decrease of the interlayer elastic modulus. However, the influence on the shear stress is not significant, except for the right edge (Figure 4(b)).(2): the influence of the residual interface tension will appear in a particular solution due to the nonzero second derivative of , where interface tension is a parabolic distribution. The stress distributions of the structure are as follows: where , , and are known constants and provided in Appendix. The stress transfer of the structure differs from that without the surface effect, where the residual surface stress is a parabolic distribution along the interface. Figure 5 presents the stress distributions. Figure 5(a) shows that the normal stress curve exhibits the same shape with that under the linear surface tension because of the same general solution of (35). The difference in the shear stress of the interlayer at the interfaces and the tangential stress jump is smaller when the residual surface tension amplitude is smaller. Figure 5(b) shows the shear stress distributions versus the tension amplitude of the interlayer. The maximum shear stress at the left and right ends increases with increasing amplitude, except for the central region (). The nonuniform residual surface stress distribution will amplify the influence of the surface effect on stress concentration.(3), : where , , , , , and are known constants and provided in Appendix. The surface effect definitely influences the stress transfer when the residual surface stress is an exponential distribution along the interface. Only the shear stress of the interlayer is studied in light of the same general solution and normal stress distribution (Figure 6(a)). Figure 6(b) shows that the shear stress distributions in layer 1 and the interlayer are similar. The difference of the two is also an exponential distribution. The surface effect influence enhances with the transferring distance increase. The shear stress at the interface increases with the surface tension amplitude increase, except for the central region. The larger surface stress distribution will amplify the stress concentration.
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3. Conclusions
An improved shear-lag model considering Gurtin–Murdoch elasticity theory is constructed to focus on the stress transfer of nanoscale interlayer with surface effects. The two roles played by the nanoscale interlayer, bounding hard layers, and stress transfer are related to displacement continuity and stress jump at an isotropic interface in this study. For biomaterials, the residual interface tension has a strong influence on the stress transfer of the interlayer. The influences of the three typical residual interface tension distributions on stress transfer are in detail elucidated. The stress distributions of the multilayered structure with the three surface stress situations are obtained. The shear-lag analysis shows that stress transfer of the multilayered structure differs from that of interlayer without the surface effect, where the transferred stress increases with decreasing interlayer thickness and amplified by the surface tension amplitude. Finally, the 2D shear-lag model (see [6]) and interfacial continuity condition and stress jump can be combined to extend our model’s application which may consider the mechanical anisotropy and bending of multilayered structure.
Appendix
For differential (35),The general solution of the associated homogenous equation iswhere A and B are integration constants determined by the following boundary conditions:The complete solution depends on the distribution of the residual interface tension.(1), : the particular and complete solutions are The constants are as follows when the boundary conditions (A.2) are used:(2), : the particular and complete solutions are The constants are as follows when the boundary conditions (A.2) are used:(3), where , : the particular and complete solutions are The constants are as follows when the boundary conditions (A.2) are used:
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the financial supports to this work from the Natural Science Foundations of China (Grant no. 11272367), the Chunhui Project of Ministry of Education of the People’s Republic of China (Grant no. z2014040), the Research Project of Sichuan Provincial Department of Education (Grant no. 15ZA0138), and the Key Research Project of Xihua University (Grant no. z1320608).