Homogenization of Surface Energy and Elasticity for Highly Rough Surfaces

Neffati, Dajla; Kulkarni, Yashashree

doi:10.1115/1.4053081

Abstract

Surface energy plays a central role in several phenomena pertaining to nearly all aspects of materials science. This includes phenomena such as self-assembly, catalysis, fracture, void growth, and microstructural evolution among others. In particular, due to the large surface-to-volume ratio, the impact of surface energy on the physical response of nanostructures is nothing short of dramatic. How does the roughness of a surface renormalize the surface energy and associated quantities such as surface stress and surface elasticity? In this work, we attempt to address this question by using a multi-scale asymptotic homogenization approach. In particular, the novelty of our work is that we consider highly rough surfaces, reminiscent of experimental observations, as opposed to gentle roughness that is often treated by using a perturbation approach. We find that softening of a rough surface is significantly underestimated by conventional approaches. In addition, our approach naturally permits the consideration of bending resistance of a surface, consistent with the Steigmann–Ogden theory, in sharp contrast to the surfaces in the Gurtin–Murdoch surface elasticity theory that do not offer flexural resistance.

1 Introduction

From an atomistic perspective, a surface is essentially a defect as it disrupts the atomic order in the material. Atoms on a free surface have different arrangement of neighbors, or coordination number, than their counterparts in the bulk. Therefore, if we regard a surface as layer(s) of atoms that differ from atoms in the interior, it is reasonable to expect that the physical properties associated with surfaces would be distinct from those of the bulk of the same material. By virtue of these atomistic underpinnings, surfaces play a vital role in several phenomena pertaining to nearly all aspects of material behavior. Furthermore, as the surface-to-volume ratio becomes significantly large at small scales, the impact of surfaces on the physical response of nanostructures is nothing short of dramatic. With the advent of nanotechnology and the drive toward miniaturization, surface energy-related effects have gained prominence in recent decades and have been studied in a wide range of contexts such as catalysis [1–3], sensing and vibrations [4–6], composites [7–10], self-assembly [11], phase transformation [12,13], fracture [14], nanostructures [15–23], and even fluid mechanics [24], soft materials [25–27], and biology [28,29].

In continuum mechanics, surface effects are usually modeled following the theoretical framework for surface elasticity pioneered by Gurtin and Murdoch [30,31,28,9]. The surfaces are regarded as elastic membrane-like entities with zero thickness attached to the bulk and endowed with a non-trivial excess energy referred to as the surface energy (see Ref. [32]). In the case of solids, the surface energy consists of two primary contributions. The first contribution is akin to capillarity or surface tension in fluids but is known as surface stress in the context of solids. The second contribution comes from surface elasticity as we need to account for the energetic cost associated with the elastic response of a solid surface. Since this area has been very well-studied, we refer to a few recent articles that provide an excellent literature review [2,23,32–36]. Some studies have also investigated the role of surface energy in the context of imperfect interfaces by treating them as a thin elastic membrane [37,38].

The importance of curvature-dependence of surface energy was first propounded by Steigmann and Ogden [39,40]. They demonstrated that the theoretical framework of Gurtin–Murdoch is incapable of modeling equilibrium deformations involving compressive surface stress fields as it does not account for elastic resistance to flexure. Naturally then, they argued, the Gurtin–Murdoch theory cannot be used to simulate local surface features, such as in wrinkling or roughening, which result from compressive surface stresses. Steigmann and Ogden proposed a modified framework which resolved this issue by incorporating curvature-dependent surface energy. However, due to the complexity of the Steigmann–Ogden curvature-dependent surface elasticity, there are very few studies on this topic till date [41–43]. Fried and Todres [41] analyzed wrinkling instability of a soft film subjected to the action of van der Waals force whereas Schiavone and Ru [42] examined the solvability of Steigmann–Ogden theory applied to plane-strain boundary value problems. Chhapadia et al. [43] applied the Steigmann–Ogden theory as well as atomistic simulations to provide a resolution to the apparent anomalous bending behavior of nanostructures.

The continuum theory of surfaces assumes a smooth, flat surface attached to the bulk. However, the surfaces of most real materials, even highly polished ones, exhibit roughness across various length scales. This raises an intriguing question: what is the impact of roughness on surface energy and its role in mechanical and other properties? This question was investigated in a few studies [44,45,23,46]. In particular, Mohammadi et al. [46] approached this problem using an elegant homogenization scheme based on energetics. They considered gently rough surfaces (that is, surfaces with small amplitude waviness) and, hence, were able to use a perturbation method for their homogenization approach. They concluded that even gentle roughness dramatically alters surface elastic properties, although it has a negligible effect on residual surface stress. They also observed that even if the bare surface has a zero surface elasticity modulus, roughness endows it with a finite modulus, and more importantly, some moduli may also change sign.

If gentle roughness can drastically renormalize the surface properties, what is the impact of highly rough surfaces? Addressing this question is the focus of our study. Using a multi-scale homogenization approach inspired by the work of Nevard and Keller [47], we seek to elucidate the effect of a very rough surface possessing surface elasticity. The central assumption underlying this method is that the wavelength of the roughness is small compared with the roughness amplitude which results in a highly rough surface. The significance of the multi-scale homogenization is that it replaces the highly rough surface with an equivalent layer of finite thickness with effective bulk properties that depend on the roughness as well as surface energy (see schematic in Fig. 1). This is in contrast to homogenization based on perturbation theory which yields an equivalent system consisting of a bulk and a flat (zero-thickness) surface with effective surface properties. In other words, the homogenized system furnished by multi-scale asympototics is a composite with distinct properties in the bulk and the effective layer. A unique advantage of the method is that it allows us to invoke the Steigmann–Ogden theory to examine the renormalized curvature-dependent surface elastic constants for a highly rough surface. Due to the small wavelength assumption, this method may not be suitable for modeling weak roughness where the wavelength is much larger than the roughness amplitude.

Fig. 1

Homogenization process of (a) a domain B with a stiffness tensor C and a rough surface ∂B with a stiffness tensor Cs and residual stress τ0, an amplitude δ and period ε in the x1x2x3 domain, (b) homogenized structure consisting of a domain B1 with a stiffness tensor C topped by a layer B2 of thickness δ characterized by an effective tensor Ceff varying along x3 and subject to body forces

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Homogenization process of (a) a domain B with a stiffness tensor C and a rough surface ∂B with a stiffness tensor C_s and residual stress τ₀, an amplitude δ and period ε in the x₁x₂x₃ domain, (b) homogenized structure consisting of a domain B₁ with a stiffness tensor C topped by a layer B₂ of thickness δ characterized by an effective tensor C^eff varying along x₃ and subject to body forces

Some prior studies on homogenization of rough surfaces have been particularly crucial for the development of our work. Our proposed model differs from them in the following aspects:

In the framework of Mohammadi et al. [46], the amplitude of the rough surface is small compared to the wavelength. Thus, the roughness is gentle and can be treated as a perturbation about an effective flat surface. In our model, the wavelength of the roughness is very small compared to the amplitude. Thus, the surface is considered very rough as its slope is large. Hence, it cannot be treated as a perturbation about a flat surface. Taken together, the two studies capture the two limits of surface roughness.
Mohammadi et al. [46] use a perturbation method for homogenization treating the amplitude as a perturbation parameter. They obtain a homogenized system consisting of a flat surface with effective surface stress and effective surface elasticity. In our work, we employ a multi-scale homogenization method which replaces the highly rough surface with an equivalent layer or film of finite thickness with effective bulk elasticity moduli.
Since the model proposed by Mohammadi et al. [46] is based on the Gurtin–Murdoch theory, the effective flat surface obtained by them does not possess curvature-dependence of the surface energy. Our model, based on multi-scale homogenization, furnishes a natural way to draw connection with the Steigmann–Ogden theory to provide estimates for curvature-dependent surface energy constants. This is possible because our homogenized system replaces the rough surface with an effective layer of finite thickness which inherently possesses bending rigidity even though we do not consider curvature-dependent surface energy to begin with.
Although highly rough surfaces have received little attention, there a few notable exceptions [48–51]. However, like Ref. [47], all these studies use the multi-scale homogenization approach to study highly rough interfaces without considering interfacial energy. In contrast, we specifically study the homogenization of highly rough surfaces endowed with surface elasticity.

The outline of the rest of the paper is as follows. Section 2 sets the general mathematical framework for incorporating surface elasticity theory and formulates the homogenization problem. Section 3 describes our multiscale homogenization scheme. In Sec. 4, we derive the effective elastic properties and the effective equations of equilibrium in the homogenized layer. Section 5 presents our explicit calculations for the case of a sinusoidal roughness. We specialize our results to a case of a thin film of finite thickness with a highly rough surface. The discussion includes a comparison of our results with those of Nevard and Keller [47] and Mohammadi et al. [46]. Our results and concluding remarks are summarized in Sec. 6.

2 General Framework and Problem Formulation

2.1 Useful Notations.

Let

B \subset R^{3}

be a regular simply connected domain,

\hat{n}

be the unit outward normal on ∂B at a point P, and I be the identity mapping from

R^{3} to R^{3}

⁠. Then, the surface projection tensor

P

from

R^{3}

to the tangent surface

Γ = {a \in R^{3} : a \cdot \hat{n} = 0}

at point P on ∂B is defined as,

P = I - \hat{n} \otimes \hat{n}

(1)

Let v and T be smooth vector and second-order tensor fields, respectively. Their projection on a smooth surface with outward normal

\hat{n}

is given by

v_{s} = P v and T_{s} = P T P

(2)

Let

ϕ : B \to R

be a smooth scalar field and

v : B \to R^{3}

be a smooth vector field. Then, using Eq. , the surface gradients of these fields are defined as

\nabla_{s} ϕ = P \nabla ϕ, and \nabla_{s} v = (\nabla v) P

(3)

In addition, the surface divergence of a vector field v and tensor field T are defined as

{div}_{s} v = tr (\nabla_{s} v), and a \cdot {div}_{s} T = {div}_{s} (T^{T} a)

(4)

where “tr” denotes the trace and

a \in R^{3}

is an arbitrary constant vector.

2.2 Surface Elasticity.

Consider a linear elastic body occupying a regular domain

B \subset R^{3}

and let

C : R^{3 \times 3} \to R^{3 \times 3}

be a fourth-order stiffness tensor. In the absence of body forces, the displacement

u : B \to R^{3}

satisfies the following equilibrium equation of linear elasticity,

div (C \nabla u) = 0 in B

(5)

To specify the boundary conditions on the surface, we take recourse in the linearized surface elasticity theory [30,31]. Treating the surface as a deformable elastic membrane attached to the bulk without slipping, the surface stress S_s is given by

S_{s} = C_{s} \nabla u + S_{s}^{0} on \partial B

(6)

where

S_{s}^{0} = τ^{0} I_{s}

is the isotropic residual surface stress tensor and τ⁰ is its magnitude. I_s is called the surface identity tensor which is the projection of the identity tensor on the surface calculated using Eq. .

C_{s} : R^{3 \times 3} \to R^{3 \times 3}

is the fourth-order surface stiffness tensor that can be defined as

C_{s} (ϵ_{s}) = 2 μ^{s} ϵ_{s} + λ^{s} Tr (ϵ_{s}) I_{s}, \forall ϵ_{s} \in R^{3 \times 3} .

(7)

μ^s and λ^s are the surface elastic properties analogous to the bulk Lamé constants. ε_s is the infinitesimal surface strain and is defined as

ϵ_{s} = P ϵ P = \frac{1}{2} [P \nabla_{s} u + (P \nabla_{s} u)^{T}]

(8)

Following Mohammadi et al. [46], we obtain the boundary condition as

(C \nabla u) \hat{n} = {div}_{s} (C_{s} \nabla u + S_{s}^{0}) on \partial B

(9)

It is important to bear in mind that the reference configuration taken here is the initial configuration. Since the equations are derived in the reference configuration,

S = C \nabla u

is the first Piola–Kirchhoff bulk stress tensor and S_s defined in Eq. is the first Piola–Kirchhoff surface stress tensor. In linearized elasticity, the different measures of stress are equivalent in the absence of residual stresses. However, in the presence of surface stress, which is a residual stress present even in the reference configuration, the equivalence of the different stress measures—first Piola-Kirchhoff, second Piola-Kirchhoff, and Cauchy stress tensors—no longer holds (see [32,52]). The Cauchy surface stress tensor is then given by

σ_{s} = [\det (I_{s} + \nabla_{s} u)]^{- 1} S_{s} (I_{s} + \nabla_{s} u)^{T}

(10)

2.3 Problem Formulation.

As depicted in Fig. 1(a), we consider a linear elastic body that occupies a three-dimensional domain B = {(x₁, x₂, x₃) : x₃ < f(x₁, x₂)} where f(x₁, x₂) denotes a very rough surface in the reference configuration. Thus, f(x₁, x₂) varies rapidly with a large amplitude δ, and a small wavelength ɛ, i.e., δ ≫ ɛ. Then, our original boundary value problem is given by Eqs. and :

{\begin{matrix} div (C \nabla u) = 0 in B, \\ (C \nabla u) \hat{n} = {div}_{s} (C_{s} \nabla u + S_{s}^{0}) on \partial B \end{matrix}

(11)

The objective of this work is to use multi-scale homogenization to reduce the original problem to an equivalent boundary value problem of the following form and illustrated by Fig. 1(b):

{\begin{matrix} div (C \nabla u) = 0 in B_{1}, \\ div (C^{e f f} \nabla u) + f = 0 in B_{2} \end{matrix}

(12)

System (12) describes the boundary value problem of a linear elastic composite body consisting of regions B₁ and B₂. Region B₂ denotes the upper layer of thickness δ corresponding to the amplitude of the rough surface, and is governed by the effective equilibrium equation (12)₂ with C^eff and f being the effective elastic moduli and forcing term respectively. B₁ corresponds to the bulk which obeys the original equilibrium equation established in (5).

3 Multiscale Homogenization Approach

We follow the multiscale homogenization method proposed by Nevard and Keller [47]. They use it to solve a general boundary value problem in three-dimensions with a very rough interface separating two linear elastic bodies. The explicit solution for the homogenization problem entails solving the boundary value problem over a periodic cell, also known as the “unit cell problem.” To solve the unit cell problem in our case, we follow the notation of Vinh and Tung [48]. They use the multi-scale homogenization approach of Ref. [47] to derive explicit results for a two-dimensional elastic domain with a very rough interface. We also reduce the problem to two dimensions as it not only makes it amenable to explicit calculations, but it also covers a variety of practical problems that can be represented by two dimensional domains, such as biological membranes and thin films. To this end, we consider our domain B to be a half-plane with a very rough surface defined as $\partial B = {(x_{1}, x_{3}) : x_{3} = \bar{f} (x_{1})}$ ⁠. The function $\bar{f} (x_{1})$ has a small period ɛ and oscillates between x₃ = −(δ/2) and x₃ = δ/2 with ɛ/δ ≪ 1. We require $\bar{f} (x_{1})$ to be periodic in x₁ (see Fig. 2).

Fig. 2

The surface ∂B in the yx3 domain where y varies from 0 to 1. y1 and y2 are two root of the equation f(y) = constant in the interval [0, 1]. n denotes the normal.

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The surface ∂B in the yx₃ domain where y varies from 0 to 1. y₁ and y₂ are two root of the equation f(y) = constant in the interval [0, 1]. n denotes the normal.

3.1 Introduction of Separation of Scales.

The central ansatz underlying multiscale homogenization is that the wavelength ɛ of the surface roughness is very small compared to other relevant lengths, specifically the amplitude δ of the surface profile. In other words, as ɛ → 0, we have a very rough surface as its slope is very large. This inherent separation of scales is made explicit by defining y such that y = x₁/ɛ [53]. Thus, y is a fast variable and x₁ is the macroscopic or slow variable. As illustrated in Fig. 2, let

f (y) = \bar{f} (ε y)

be the mathematical function describing the surface profile in terms of the fast variable y. Thus, f(y) is a periodic function with period 1 and amplitude δ. We assume that the equation x₃ = f(y) = constant has exactly two roots, y₁ and y₂ in the interval [0,1]. The outward unit normal to the boundary

\partial B = {x_{3} = \bar{f} (x_{1})}

is expressed as

\hat{n} = \frac{1}{\sqrt{1 + ε^{- 2} f_{y}^{2}}} [\begin{matrix} - ε^{- 1} f_{y} \\ 1 \end{matrix}] = \frac{n}{‖ n ‖}

(13)

where f_y = df/dy and

‖ n ‖ = \sqrt{1 + ε^{- 2} f_{y}^{2}}

is the magnitude of the outward normal vector, n.

To make the dependence of f(y) on δ explicit, we define f(y) = δf₀(y). Note that since δ is the amplitude, f₀(y) is simply a periodic function describing the roughness profile. f₀(y) is a continuously differentiable albeit arbitrary functions of y. Now, using the large amplitude, small wavelength approximation, ɛ/δ ≪ 1, the Taylor expansion of ‖n‖ yields,

‖ n ‖ = ε^{- 1} f_{y} \sqrt{1 + \frac{ε^{2}}{δ^{2}} \frac{1}{f_{0 y}^{2}}} = ε^{- 1} f_{y} + \frac{1}{2} \frac{ε}{f_{y}} - \frac{1}{8} \frac{ε^{3}}{f_{y}^{3}} + o (\frac{ε^{5}}{δ^{5}})

(14)

Similarly, using the expression for

\hat{n}

in Eq. , the projection tensor is obtained as,

P = I - \frac{n}{‖ n ‖} \otimes \frac{n}{‖ n ‖} ≃ [\begin{matrix} \frac{ε^{2}}{f_{y}^{2}} & \frac{ε}{f_{y}} - \frac{ε^{3}}{f_{y}^{3}} \\ \frac{ε}{f_{y}} - \frac{ε^{3}}{f_{y}^{3}} & 1 - \frac{ε^{2}}{f_{y}^{2}} \end{matrix}] + o (\frac{ε^{3}}{δ^{3}})

(15)

where we again invoke the ɛ/δ ≪ 1 approximation and use the Taylor expansion.

Our goal is to determine the displacement u(x₁, x₃, ɛ) for small values of ɛ. The in-plane component u₂ as well as the dependence on x₂ are neglected, since we are working with a two-dimensional domain. The displacement u satisfies the equilibrium equation and the boundary condition in Eq. . Following [47], we define

U (x_{1}, x_{3}, y, ε) = u (x_{1}, x_{3}, ε)

(16)

to make the dependence on the fast variable explicit. Then,

u_{, 1} = U_{, 1} + ε^{- 1} U_{, y} and u_{, 3} = U_{, 3}

(17)

where ϕ_,1 = ∂ϕ/∂x₁, ϕ_,3 = ∂ϕ/∂x₃, and ϕ_,y = ∂ϕ/∂y for a scalar function ϕ(x₁, x₃, y). To express the boundary value problem in terms of U(x₁, x₃, y, ɛ), we employ the notation used by [48] for our subsequent calculations. Assuming isotropic linear elastic material, the equilibrium equation ₁ yields a system of equations that can be written in the following compact form [48,54].

A_{h k} u_{, k h} = 0, h, k = 1, 3

(18)

where u = [u₁ u₃]^T and the A matrices are as follows:

\begin{aligned} A_{11} & = [\begin{matrix} C_{1111} & C_{1131} \\ C_{3111} & C_{3131} \end{matrix}] = [\begin{matrix} λ + 2 μ & 0 \\ 0 & μ \end{matrix}] A_{13} = [\begin{matrix} C_{1113} & C_{1133} \\ C_{3113} & C_{3133} \end{matrix}] = [\begin{matrix} 0 & λ \\ μ & 0 \end{matrix}] \\ A_{31} & = [\begin{matrix} C_{1311} & C_{1331} \\ C_{3311} & C_{3331} \end{matrix}] = [\begin{matrix} 0 & μ \\ λ & 0 \end{matrix}] A_{33} = [\begin{matrix} C_{1313} & C_{1333} \\ C_{3313} & C_{3333} \end{matrix}] = [\begin{matrix} μ & 0 \\ 0 & λ + 2 μ \end{matrix}] \end{aligned}

(19)

Note that since the Lamé constants, λ and μ, are positive, the A matrices are invertible. In order to express the boundary condition ₂ in explicit notation, we first multiply both sides with ‖n‖ to get

(C \nabla u) n = ‖ n ‖ {div}_{s} (C_{s} \nabla u + S_{s}^{0})

(20)

In terms of the matrices introduced in Eq. , the left-hand side of Eq. becomes,

(A_{h k} u_{, k}) n_{h} = - ε^{- 1} f_{y} (A_{11} u_{, 1} + A_{13} u_{, 3}) + A_{31} u_{, 1} + A_{33} u_{, 3}

(21)

while the right-hand side of Eq. can be expressed in the following compact form,

‖ n ‖ {div}_{s} (C_{s} \nabla u + S_{s}^{0}) = B_{h k} u_{, h k} + D_{j} u_{, j} + T, h, k = 1, 3

(22)

The expressions for B, D and T matrices (keeping only the orders that will contribute to the BVPs up to ɛ²) are given in Appendix A. After substituting Eqs. and in Eqs. , , and , and some tedious calculations, we can express the equilibrium equation and the boundary condition in terms of U(x₁, x₃, y, ɛ) (see Appendix A for details).

3.2 Asympotic Expansion.

We now assume that the displacement U(x₁, x₃, y) can be expressed for small values of the wavelength ɛ as an asymptotic expansion,

U (x_{1}, x_{3}, y, ε) = u^{(0)} (x_{1}, x_{3}) + ε u^{(1)} (x_{1}, x_{3}, y) + ε^{2} u^{(2)} (x_{1}, x_{3}, y) + O (ε^{2})

(23)

Note that u⁽⁰⁾(x₁, x₃) does not depend on the slow variable y. In fact, it is the macroscopic displacement that is observed at length scales larger than that of the surface roughness. Since U is periodic in y, it follows that u⁽¹⁾, and u⁽²⁾ are periodic in y with period 1. We now substitute the asymptotic expansion of the displacement in the boundary value problem derived after introduction of the separation of scales (Appendix A) to express the equilibrium equation and boundary condition in terms of u⁽⁰⁾, u⁽¹⁾, and u⁽²⁾.

3.3 Boundary Value Problems for Perturbed Solution.

The equilibrium equation and surface equation obtained after incorporating the asymptotic expansion (23) include terms with different orders of ɛ. Gathering the coefficients of the same power of ɛ yields the following set of boundary value problems.

Order ɛ⁻¹:

{\begin{matrix} [A_{11} u_{, y}^{(1)} + A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)}]_{, y} = 0, x_{3} < f (y) \\ f_{y} (- A_{11} u_{, y}^{(1)} - A_{11} u_{, 1}^{(0)} - A_{13} u_{, 3}^{(0)}) = B_{33}^{(- 1)} u_{, 33}^{(0)} + \frac{f_{y y}}{f_{y}} D_{3}^{(- 1)} u_{, 3}^{(0)} + \frac{f_{y y}}{f_{y}} T^{(- 1)}, x_{3} = f (y) \end{matrix}

(24)

Order ɛ⁰:

{\begin{matrix} [A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)} + A_{11} u_{, y}^{(1)}]_{, 1} + [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} + A_{31} u_{, y}^{(1)}]_{, 3} + [A_{11} u_{, y}^{(2)} + A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)}]_{, y} = 0, x_{3} < f (y) \\ - f_{y} A_{11} u_{y}^{(2)} - f_{y} A_{11} u_{, 1}^{(1)} - f_{y} A_{13} u_{, 3}^{(1)} + A_{31} u_{, y}^{(1)} + A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} = f_{y} B_{33}^{(- 1)} u_{, 33}^{(1)} + B_{13}^{(0)} u_{, 13}^{(0)} + B_{31}^{(0)} u_{, 31}^{(0)} + B_{33}^{(0)} u_{, 33}^{(0)} + B_{13}^{(0)} u_{, y 3}^{(1)} + B_{31}^{(0)} u_{, 3 y}^{(1)} \\ + \frac{B_{11}^{(1)}}{f_{y}} u_{, y y}^{(1)} + \frac{f_{y y}}{f_{y}^{2}} (T^{(0)} + D_{1}^{(0)} u_{, 1}^{(0)} + D_{3}^{(0)} u_{, 3}^{(0)} + D_{1}^{(0)} u_{, y}^{(1)}) + \frac{f_{y y}}{f_{y}} D_{3}^{(- 1)} u_{, 3}^{(1)}, x_{3} = f (y) \end{matrix}

(25)

Order ɛ¹:

{\begin{matrix} [A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)} + A_{11} u_{, y}^{(2)}]_{, 1} + [A_{31} u_{, 1}^{(1)} + A_{31} u_{, y}^{(2)} + A_{33} u_{, 3}^{(1)}]_{, 3} + [A_{11} u_{, 1}^{(2)} + A_{13} u_{, 3}^{(2)}]_{, y} = 0, x_{3} < f (y) \\ - f_{y} A_{11} u_{, 1}^{(2)} - f_{y} A_{13} u_{, 3}^{(2)} + A_{31} u_{, y}^{(2)} + A_{31} u_{, 1}^{(1)} + A_{33} u_{, 3}^{(1)} = \frac{B_{11}^{(2)}}{f_{y}^{2}} u_{, y y}^{(1)} + \frac{B_{11}^{(1)}}{f_{y}} u_{, 11}^{(0)} + \frac{B_{13}^{(1)}}{f_{y}} u_{, 13}^{(0)} + \frac{B_{31}^{(1)}}{f_{y}} u_{, 31}^{(0)} + \frac{B_{33}^{(1)}}{f_{y}} u_{, 33}^{(0)} + \frac{B_{11}^{(1)}}{f_{y}} (u_{, 1 y}^{(1)} + u_{, y 1}^{(1)}) + \frac{B_{13}^{(1)}}{f_{y}} u_{, y 3}^{(1)} \\ + \frac{B_{31}^{(1)}}{f_{y}} u_{, 3 y}^{(1)} + \frac{B_{11}^{(1)}}{f_{y}} u_{, y y}^{(2)} + B_{13}^{(0)} u_{, 13}^{(1)} + B_{31}^{(0)} u_{, 31}^{(1)} + B_{33}^{(0)} u_{, 33}^{(1)} + B_{13}^{(0)} u_{, y 3}^{(2)} + B_{31}^{(0)} u_{, 3 y}^{(2)} + f_{y} B^{(- 1)} u_{, 33}^{(2)} + \frac{f_{y y}}{f_{y}^{3}} (D_{1}^{(1)} u_{, 1}^{(0)} + D_{3}^{(1)} u_{, 3}^{(0)} \\ + D_{1}^{(1)} u_{, y}^{(1)} + T^{(1)}) + \frac{f_{y y}}{f_{y}^{2}} (D_{1}^{(0)} u_{, 1}^{(1)} + D_{3}^{(0)} u_{, 3}^{(1)} + D_{1}^{(0)} u_{, y}^{(2)}) + \frac{f_{y y}}{f_{y}} D_{3}^{(- 1)} u_{, 3}^{(2)}, x_{3} = f (y) \end{matrix}

(26)

Order ɛ²:

{\begin{matrix} A_{11} u_{, 11}^{(2)} + A_{13} u_{, 31}^{(2)} + A_{31} u_{, 13}^{(2)} + A_{33} u_{, 33}^{(2)} = 0, x_{3} < f (y) \\ A_{31} u_{, 1}^{(2)} + A_{33} u_{, 3}^{(2)} = B_{13}^{(0)} u_{, 13}^{(2)} + B_{31}^{(0)} u_{, 31}^{(2)} + B_{33}^{(0)} u_{, 33}^{(2)} + \frac{B_{11}^{(1)}}{f_{y}} u_{, 11}^{(1)} + \frac{B_{13}^{(1)}}{f_{y}} u_{, 13}^{(1)} + \frac{B_{31}^{(1)}}{f_{y}} u_{, 31}^{(1)} + \frac{B_{33}^{(1)}}{f_{y}} u_{, 33}^{(1)} \\ + \frac{B_{11}^{(1)}}{f_{y}} (u_{, 1 y}^{(2)} + u_{, y 1}^{(2)}) + \frac{B_{13}^{(1)}}{f_{y}} u_{, y 3}^{(2)} + \frac{B_{31}^{(1)}}{f_{y}} u_{, 3 y}^{(2)} + \frac{B_{11}^{(2)}}{f_{y}^{2}} u_{, 11}^{(0)} + \frac{B_{13}^{(2)}}{f_{y}^{2}} u_{, 13}^{(0)} + \frac{B_{31}^{(2)}}{f_{y}^{2}} u_{, 31}^{(0)} + \frac{B_{33}^{(2)}}{f_{y}^{2}} u_{, 33}^{(0)} + \frac{B_{11}^{(2)}}{f_{y}^{2}} (u_{, 1 y}^{(1)} \\ + u_{, y 1}^{(1)}) + \frac{B_{13}^{(2)}}{f_{y}^{2}} u_{, y 3}^{(1)} + \frac{B_{31}^{(2)}}{f_{y}^{2}} u_{, 3 y}^{(1)} + \frac{B_{11}^{(2)}}{f_{y}^{2}} u_{, y y}^{(2)} + \frac{B_{11}^{(3)}}{f_{y}^{3}} u_{, y y}^{(1)} + \frac{f_{y y}}{f_{y}^{4}} (D_{1}^{(2)} u_{, 1}^{(0)} + D_{3}^{(2)} u_{, 3}^{(0)} + D_{1}^{(2)} u_{, y}^{(1)} + T^{(2)}) \\ + \frac{f_{y y}}{f_{y}^{3}} (D_{1}^{(1)} u_{, 1}^{(1)} + D_{3}^{(1)} u_{, 3}^{(1)} + D_{1}^{(1)} u_{, y}^{(2)}) + \frac{f_{y y}}{f_{y}^{2}} (D_{1}^{(0)} u_{, 1}^{(2)} + D_{3}^{(0)} u_{, 3}^{(2)}), x_{3} = f (y) \end{matrix}

(27)

4 Homogenized Equations and Effective Properties

Based on the boundary value problem for the order ɛ⁰, we wish to derive homogenized equilibrium equations in terms of only the macroscopic displacement u⁽⁰⁾(x₁, x₃). To this end, we first seek to express the derivatives of u⁽¹⁾ and u⁽²⁾ that appear in Eq. in terms of u⁽⁰⁾. Based on boundary value problem for the order ɛ⁻¹, we suggest the following form for u⁽¹⁾ [48],

u^{(1)} = N^{1} u^{(0)} + N^{11} u_{, 1}^{(0)} + N^{13} u_{, 3}^{(0)} + N^{133} u_{, 33}^{(0)} + N^{0}

(28)

where N¹, N¹¹, N¹³, N¹³³, and N⁰ are 2 × 2 matrices that are functions of x₃ and y and are periodic in y with a period of 1. Substituting Eq. in and gathering the coefficients of u⁽⁰⁾,

u_{, 1}^{(0)}

⁠,

u_{, 3}^{(0)}

⁠, and

u_{, 33}^{(0)}

and the remaining constants, we get the following systems of equations:

\begin{aligned} {\begin{matrix} (A_{11} N_{, y}^{1})_{, y} = 0, x_{3} < f (y) \\ - f_{y} A_{11} N_{, y}^{1} = 0, x_{3} = f (y) \end{matrix} \end{aligned}

(29)

\begin{aligned} {\begin{matrix} (A_{11} N_{, y}^{11} + A_{11})_{, y} = 0, x_{3} < f (y) \\ A_{11} N_{, y}^{11} + A_{11} = 0, x_{3} = f (y) \end{matrix} \end{aligned}

(30)

\begin{aligned} {\begin{matrix} (A_{11} N_{, y}^{13} + A_{13})_{, y} = 0, x_{3} < f (y) \\ - f_{y} (A_{11} N_{, y}^{13} + A_{13}) = \frac{f_{y y}}{f_{y}} D_{3}^{(- 1)}, x_{3} = f (y) \end{matrix} \end{aligned}

(31)

\begin{aligned} {\begin{matrix} (A_{11} N_{, y}^{133})_{, y} = 0, x_{3} < f (y) \\ - f_{y} A_{11} N_{, y}^{133} = f_{y} B_{33}^{(- 1)}, x_{3} = f (y) \end{matrix} \end{aligned}

(32)

\begin{aligned} {\begin{matrix} (A_{11} N_{, y}^{0})_{, y} = 0, x_{3} < f (y) \\ - f_{y} A_{11} N_{, y}^{0} = \frac{f_{y y}}{f_{y}} T^{(- 1)}, x_{3} = f (y) \end{matrix} \end{aligned}

(33)

Solving these equations yields the following expressions for the derivatives of the N matrices,

N_{, y}^{1} = 0, x_{3} \leq f (y)

(34)

\begin{aligned} N_{, y}^{11} = {\begin{matrix} 0, x_{3} < f (y) \\ - I, x_{3} = f (y) \end{matrix} \end{aligned}

(35)

\begin{aligned} N_{, y}^{13} = {\begin{matrix} A_{11}^{- 1} < A_{11}^{- 1} >^{- 1} < A_{11}^{- 1} A_{13} > - A_{11}^{- 1} A_{13}, x_{3} < f (y) \\ - A_{11}^{- 1} A_{13} - A_{11}^{- 1} D_{3}^{(- 1)} \frac{f_{y y}}{f_{y}^{2}} |_{y_{1}}, y = y_{1} \\ - A_{11}^{- 1} A_{13} - A_{11}^{- 1} D_{3}^{(- 1)} \frac{f_{y y}}{f_{y}^{2}} |_{y_{2}}, y = y_{2} \end{matrix} \end{aligned}

(36)

\begin{aligned} N_{, y}^{133} = {\begin{matrix} 0, x_{3} < f (y) \\ - A_{11}^{- 1} B_{33}^{(- 1)}, x_{3} = f (y) \end{matrix} \end{aligned}

(37)

\begin{aligned} N_{, y}^{0} = {\begin{matrix} 0, x_{3} < f (y) \\ - A_{11}^{- 1} T^{(- 1)} \frac{f_{y y}}{f_{y}^{2}} |_{y_{1}}, y = y_{1} \\ - A_{11}^{- 1} T^{(0)} \frac{f_{y y}}{f_{y}^{2}} |_{y_{2}}, y = y_{2} \end{matrix} \end{aligned}

(38)

where

< ϕ > = \int_{0}^{1} ϕ d y = \int_{0}^{y_{1}} ϕ d y + \int_{y_{2}}^{1} ϕ d y = (1 - y_{2} + y_{1}) ϕ

for a constant scalar function ϕ.

We now integrate the equilibrium equation ₁ with respect to y over [0, 1] for a constant x₃ [47]. To perform this integration, we first define y₁ and y₂ as the roots of the equation x₃ = h(y) at a constant x₃ (see Fig. 2).

\begin{aligned} \int_{0}^{1} ([A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)} + A_{11} u_{, y}^{(1)}]_{, 1} + [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} + A_{31} u_{, y}^{(1)}]_{, 3} \\ + [A_{11} u_{, y}^{(2)} + A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)}]_{, y}) d y = 0 \end{aligned}

(39)

For convenience, we define

\begin{aligned} e_{1} & = \int_{0}^{1} [A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)} + A_{11} u_{, y}^{(1)}]_{, 1} d y \\ e_{2} & = \int_{0}^{1} [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} + A_{31} u_{, y}^{(1)}]_{, 3} d y \\ e_{3} & = \int_{0}^{1} [A_{11} u_{, y}^{(2)} + A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)}]_{, y} d y \end{aligned}

We now note that ‖C‖ ≫ ‖C_s‖ɛ for typical solids where ɛ is the wavelength and typically on the order of 10 nm. For instance, the inequality is valid for copper since it has ‖C‖ ∼ 10¹¹ Pa while ‖C_s‖ ∼ 1 N/m. Then, enforcing this inequality and neglecting lower order terms on the right-hand side of Eq. ₂, we get,

\begin{aligned} - f_{y} (A_{11} u_{y}^{(2)} + A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)}) + A_{31} u_{, y}^{(1)} + A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} \\ = B_{13}^{(0)} u_{, 13}^{(0)} + B_{31}^{(0)} u_{, 31}^{(0)} + B_{33}^{(0)} u_{, 33}^{(0)} + B_{13}^{(0)} u_{, y 3}^{(1)} + \frac{f_{y y}}{f_{y}^{2}} (T^{(0)} \\ + D_{1}^{(0)} u_{, 1}^{(0)} + D_{3}^{(0)} u_{, 3}^{(0)}), x_{3} = f (y) \end{aligned}

(40)

We can now evaluate e₃ using Eq. to get,

\begin{aligned} e_{3} & = \frac{1}{f_{y} (y_{1})} A_{31} u_{, y}^{(1)} |_{y_{1}} - \frac{1}{f_{y} (y_{2})} A_{31} u_{, y}^{(1)} |_{y_{2}} + (\frac{1}{f_{y} (y_{1})} - \frac{1}{f_{y} (y_{2})}) \\ \times [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} - B_{13}^{(0)} u_{, 13}^{(0)} + B_{31}^{(0)} u_{, 31}^{(0)} + B_{33}^{(0)} u_{, 33}^{(0)}] \\ + (\frac{f_{y y}}{f_{y}^{3}} |_{y_{2}} - \frac{f_{y y}}{f_{y}^{3}} |_{y_{1}}) [D_{1}^{(1)} u_{, 1}^{(0)} + D_{3}^{(1)} u_{, 3}^{(0)} + T^{(0)}] \end{aligned}

(41)

By substituting for

u_{, y}^{(1)} |_{y_{i}}

in terms of the expressions – for the N_,y matrices we have,

\begin{aligned} e_{3} & = (\frac{1}{f_{y} (y_{1})} - \frac{1}{f_{y} (y_{2})}) [(A_{33} - A_{31} A_{11}^{- 1} A_{13}) u_{, 3}^{(0)} \\ - (A_{31} A_{11}^{- 1} B_{33}^{(- 1)} + B_{33}^{(0)}) u_{, 33}^{(0)} - B_{13}^{(0)} u_{, 13}^{(0)} \\ - B_{31}^{(0)} u_{, 31}^{(0)}] + (\frac{f_{y y}}{f_{y}^{3}} |_{y_{2}} - \frac{f_{y y}}{f_{y}^{3}} |_{y_{1}}) [D_{1}^{(0)} u_{, 1}^{(0)} \\ + D_{3}^{(0)} u_{, 3}^{(0)} + T^{(0)}] - (\frac{f_{y y}}{f_{y}^{3}} |_{y_{1}} - \frac{f_{y y}}{f_{y}^{3}} |_{y_{2}}) \\ \times [A_{31} A_{11}^{- 1} D_{3}^{(- 1)} u_{, 3}^{(0)} + A_{31} A_{11}^{- 1} T^{(- 1)}] \end{aligned}

(42)

We now proceed to evaluate e₁ and e₂ as follows.

\begin{aligned} e_{1} & = \int_{0}^{1} [A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)} + A_{11} (N_{, y}^{11} u_{, 1}^{(0)} + N_{, y}^{13} u_{, 3}^{(0)} + N_{, y}^{133} u_{, 33}^{(0)} + N_{, y}^{0})]_{, 1} d y \\ = ⟨ A_{11} + A_{11} N_{, y}^{11} ⟩ u_{, 11}^{(0)} + ⟨ A_{13} + A_{11} N_{, y}^{13} ⟩ u_{, 31}^{(0)} + ⟨ A_{11} N_{, y}^{133} ⟩ u_{, 33}^{(0)} \end{aligned}

(43)

Using Eqs. –, the terms between angular brackets can be evaluated as

⟨ A_{11} + A_{11} N_{, y}^{11} ⟩ = ⟨ A_{11} ⟩

(44)

⟨ A_{13} + A_{11} N_{, y}^{13} ⟩ = ⟨ A_{11}^{- 1} ⟩^{- 1} ⟨ A_{11}^{- 1} A_{13} ⟩ = A_{13}

(45)

⟨ A_{11} N_{, y}^{133} ⟩ = 0

(46)

Similarly,

\begin{aligned} e_{2} & = \int_{0}^{1} [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} + A_{31} (N_{, y}^{11} u_{, 1}^{(0)} + N_{, y}^{13} u_{, 13}^{(0)} + N_{, y}^{133} u_{, 33}^{(0)} + N_{, y}^{0})]_{, 3} d y \\ = [⟨ A_{31} + A_{31} N_{, y}^{11} ⟩ u_{, 1}^{(0)}]_{, 3} + [⟨ A_{33} + A_{31} N_{, y}^{13} ⟩ u_{, 3}^{(0)}]_{, 3} \\ + [⟨ A_{31} N_{, y}^{133} ⟩ u_{, 33}^{(0)}]_{, 3} \end{aligned}

(47)

where the terms between angular brackets can be evaluated as

⟨ A_{31} + A_{31} N_{, y}^{11} ⟩ = ⟨ A_{31} ⟩

(48)

⟨ A_{33} + A_{31} N_{, y}^{13} ⟩ = ⟨ A_{33} ⟩

(49)

⟨ A_{31} N_{, y}^{133} ⟩ = 0

(50)

Thus, we get the following equilibrium equation in terms of u⁽⁰⁾ which is valid in the equivalent effective layer:

\begin{aligned} ⟨ A_{11} ⟩ u_{, 11}^{(0)} + [A_{13} - (y_{1}^{'} - y_{2}^{'}) B_{31}^{(0)}] u_{, 31}^{(0)} + [(⟨ A_{31} ⟩ - (y_{1}^{'} - y_{2}^{'}) B_{13}^{(0)}) u_{, 1}^{(0)}]_{, 3} \\ + [(⟨ A_{33} ⟩ - (y_{1}^{'} - y_{2}^{'}) [A_{31} A_{11}^{- 1} B_{33}^{(- 1)} + B_{33}^{(0)}]) u_{, 3}^{(0)}]_{, 3} \\ + (y_{1}^{″} - y_{2}^{″}) [A_{31} A_{11}^{- 1} T^{(- 1)} + T^{(0)}] = (y_{2}^{″} - y_{1}^{″}) [B_{13}^{(0)} + D_{1}^{(0)}] u_{, 1}^{(0)} \\ + [(y_{2}^{″} - y_{1}^{″}) (B_{33}^{(0)} + D_{3}^{(0)}) + (y_{2}^{'} - y_{1}^{'}) (A_{33} \\ - A_{31} A_{11}^{- 1} A_{13})] u_{, 3}^{(0)}, - \frac{δ}{2} < x_{3} < \frac{δ}{2} \end{aligned}

(51)

and where we have used the following identities:

\frac{1}{f_{y}} (y_{i}) = \frac{d y_{i}}{d x_{3}} = y_{i}^{'} and - \frac{f_{y y}}{f_{y}^{3}} (y_{i}) = \frac{d^{2} y_{i}}{d x_{3}^{2}} = y_{i}^{″}, i = {1, 2}

(52)

Introducing the separation of scales and the asymptotic expansion in the effective boundary value problem stated in Eq. , we can write the boundary value problem statement for u⁽⁰⁾ as

{\begin{matrix} A_{11} u_{, 11}^{(0)} + A_{13} u_{, 31}^{(0)} + A_{31} u_{, 13}^{(0)} + A_{33} u_{, 33}^{(0)} = 0, x_{3} < - \frac{δ}{2} \\ A_{11}^{e f f} u_{, 11}^{(0)} + A_{13}^{e f f} u_{, 31}^{(0)} + [A_{31}^{e f f} u_{, 1}^{(0)}]_{, 3} + [A_{33}^{e f f} u_{, 3}^{(0)}]_{, 3} + \nabla \cdot σ^{*} = D_{1}^{e f f} u_{, 1}^{(0)} + D_{3}^{e f f} u_{, 3}^{(0)}, - \frac{δ}{2} \leq x_{3} \leq \frac{δ}{2} \end{matrix}

(53)

Comparing Eqs. and ₂ and introducing k_s = λ_s + 2μ_s, we obtain the effective (bulk) properties in the equivalent layer (−(δ/2) ≤ x₃ ≤ δ/2),

\begin{aligned} A_{11}^{e f f} & = [\begin{matrix} (λ + 2 μ) (1 - y_{2} + y_{1}) & 0 \\ 0 & μ (1 - y_{2} + y_{1}) \end{matrix}] \\ A_{13}^{e f f} & = [\begin{matrix} 0 & λ \\ μ & k_{s} (y_{2}^{'} - y_{1}^{'}) \end{matrix}] \\ A_{31}^{e f f} & = [\begin{matrix} 0 & μ (1 - y_{2} + y_{1}) \\ λ (1 - y_{2} + y_{1}) & k_{s} (y_{2}^{'} - y_{1}^{'}) \end{matrix}] \\ A_{33}^{e f f} & = [\begin{matrix} (1 - y_{2} + y_{1}) μ & 2 k_{s} (y_{2}^{'} - y_{1}^{'}) \\ k_{s} (y_{2}^{'} - y_{1}^{'}) & (1 - y_{2} + y_{1}) (λ + 2 μ) \end{matrix}] \end{aligned}

(54)

D_{1}^{e f f} = [\begin{matrix} 0 & 0 \\ 0 & - k_{s} (y_{2}^{″} - y_{1}^{″}) \end{matrix}], D_{3}^{e f f} = [\begin{matrix} 0 & 0 \\ - k_{s} (y_{2}^{″} - y_{1}^{″}) & \frac{4 μ (λ + μ)}{λ + 2 μ} (y_{2}^{'} - y_{1}^{'}) \end{matrix}]

(55)

\nabla \cdot σ^{*} = [\begin{matrix} - 2 τ^{0} (y_{1}^{″} - y_{2}^{″}) \\ 0 \end{matrix}]

(56)

There are several observations that emerge from this homogenization process. First, the surface roughness seems to endow the body with finite elastic moduli even when those elastic moduli are zero in the original material which is rather unexpected. Specifically,

C_{3331}^{e f f}

⁠,

C_{3133}^{e f f}

⁠,

C_{3313}^{e f f}

⁠, and

C_{1333}^{e f f}

are no longer null but are functions of the surface roughness as well as the surface elastic properties. Second, both the surface roughness and the surface elasticity disrupt the major and minor symmetries of the effective elasticity tensor. Specifically, since

C_{3133}^{e f f} \neq C_{1333}^{e f f}

and

C_{1313}^{e f f} \neq C_{3113}^{e f f}

⁠, it results in a non-symmetric stress tensor in the effective layer. This implies that the rough surface with elasticity endows the body with a body moment in the effective layer which breaks the symmetry of the stress tensor. Third, we observe that although the bulk material is homogeneous, some of the components of the effective elasticity tensor vary along the x₃ direction, since y₁ and y₂ are functions of x₃. This is a direct consequence of averaging between the two limits of the surface roughness profile. Finally, the residual stress of the surface τ⁰ is manifested as a residual stress σ* in the effective layer after homogenization. In Eq. ₂, the two terms on the right-hand side involve first derivatives of the macroscopic displacement but do not contribute to the divergence of the stress term. Taken together, they contribute to a force term in the equilibrium equation in the effective layer. This is in contrast to the equilibrium equation in the bulk material ₁ which does not involve a force term.

5 Numerical Calculations and Discussion

To gain insights into the implications of a highly rough surface on the mechanical response of materials, especially, nanostructures, we evaluate our results numerically for the case of uniaxial tensile deformation of a thin film, albeit with a very rough surface. We also examine by comparison the effect of high degree of roughness (based on the present work) versus gentle roughness (based on the work of Mohammadi et al. [46]). To this end, we start by considering a sinusoidal profile for the surface roughness.

5.1 Explicit Effective Properties for a Sinusoidal Roughness.

First, we obtain explicit expressions for effective elastic properties for a rough surface with a sinusoidal roughness of the form,

f (x_{1}) = \frac{δ}{2} \cos (2 π \frac{x_{1}}{ε})

(57)

Introducing the slow variable y,

f (y) = \frac{δ}{2} \cos (2 π y)

(58)

the two inverses of the function f(y) in the [0,1] interval, for x₃ = constant, are

y_{1} = \frac{1}{2 π} \arccos (2 \frac{x_{3}}{δ}) and y_{2} = \frac{1}{2 π} \arcsin (2 \frac{x_{3}}{δ}) + \frac{3}{4}

(59)

Using Eq. , the effective matrices

A_{h k}^{e f f}

⁠, {h, k} = {1, 3} become,

\begin{aligned} A_{11}^{e f f} & = [\begin{matrix} (λ + 2 μ) p (x_{3}) & 0 \\ 0 & μ p (x_{3}) \end{matrix}], A_{13}^{e f f} = [\begin{matrix} 0 & λ \\ μ & \frac{k_{s}}{δ} g (x_{3}) \end{matrix}] \\ A_{31}^{e f f} & = [\begin{matrix} 0 & μ p (x_{3}) \\ λ p (x_{3}) & \frac{k_{s}}{δ} g (x_{3}) \end{matrix}], A_{33}^{e f f} = [\begin{matrix} μ p (x_{3}) & 2 \frac{k_{s}}{δ} g (x_{3}) \\ \frac{k_{s}}{δ} g (x_{3}) & (λ + 2 μ) p (x_{3}) \end{matrix}] \end{aligned}

(60)

where we have introduced the following for simplification,

p (x_{3}) = \frac{1}{π} \arccos (2 \frac{x_{3}}{δ}) and g (x_{3}) = \frac{2}{π \sqrt{1 - {(2 \frac{x_{3}}{δ})}^{2}}}

(61)

5.1 Extending these results to random roughness.

The results for sinusoidal roughness can be extended to a general roughness by assuming that the roughness is statistically invariant over a length scale Λ. Then, without loss of generality, f₀(x₁) can be considered to be a periodic function with a large enough period Λ where f(x₁) = δf₀(x₁). Applying Fourier analysis, f₀(x₁) can be expressed as a superposition of sinusoidal waves. Furthermore, for random roughness, we can consider an ensemble of general surfaces with specified autocorrelation function and correlation length, which are statistical properties of the random roughness. We then assume that for a random roughness, f₀(x₁) is even and periodic with the period being larger than the correlation length. Expressing the profile for a general roughness as a Fourier series and applying the statistical properties for random surfaces should provide an avenue for extending the results for the sinusoidal roughness to the case of random roughness. However, such an analysis, although possible, is beyond the scope of this work. Moreover, we emphasize that such an analysis will necessarily have to be numerical and will be an interesting future study.

5.2 The Effective Young’s Modulus of a Thin Film.

We now consider a thin film of thickness h with a surface roughness profile given by Eq. . We apply our homogenization scheme and then study the response of the effective thin film under plane strain conditions subjected to uniaxial tensile deformation along the x₁ direction. Knowing that the traction along e₃ is zero, gives the following two equations in the effective layer:

\begin{aligned} σ_{13} & = 2 μ p (x_{3}) ε_{13} + \frac{k_{s}}{δ} g (x_{3}) ε_{33} = 0 \\ σ_{33} & = λ p (x_{3}) ε_{11} + 2 \frac{k_{s}}{δ} g (x_{3}) ε_{13} + p (x_{3}) (λ + 2 μ) ε_{33} = 0 \end{aligned}

(62)

Thus, ɛ₁₃ and ɛ₃₃ can be written in terms of ɛ₁₁ as

ε_{33} = \frac{λ μ p (x_{3})^{2}}{J (x_{3})} ε_{11} and ε_{13} = \frac{- λ \frac{k_{s}}{δ} p (x_{3}) g (x_{3})}{J (x_{3})} ε_{11}

(63)

where we have defined

J (x_{3}) = \frac{2 k_{s}^{2}}{δ^{2}} g (x_{3})^{2} - μ (λ + 2 μ) p (x_{3})^{2}

(64)

From equations , we get the remaining stress components in terms of ɛ₁₁ as,

\begin{aligned} σ_{11} & = [(λ + 2 μ) p (x_{3}) + \frac{λ^{2} μ p (x_{3})^{2}}{J (x_{3})}] ε_{11} and \\ σ_{31} & = - \frac{λ μ \frac{k_{s}}{δ} g (x_{3}) p (x_{3})}{J (x_{3})} ε_{11} \end{aligned}

(65)

Therefore, the strain energy density of the effective layer (region B₂ in Fig. 1(b)) is

\begin{aligned} W_{layer} & = \frac{1}{2} A ε_{11}^{2} \int_{- (δ / 2)}^{δ / 2} [(λ + μ) p (x_{3}) + \frac{λ^{2} μ p (x_{3})^{2}}{J (x_{3})} \\ + \frac{λ^{2} μ \frac{k_{s}^{2}}{δ^{2}} g (x_{3})^{2} p (x_{3})^{2}}{J (x_{3})^{2}} d x_{3} \end{aligned}

(66)

Using Taylor series expansion of the integrand, we obtain

\begin{aligned} W_{layer} & = \frac{1}{2} A ε_{11}^{2} [\frac{δ λ}{2} + δ μ - \frac{16 π^{4} (12 + π^{2}) δ^{7} λ^{2} μ^{3} (λ + 2 μ)^{2} k_{s}^{2}}{3 (π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})^{4}} + \frac{2048 π^{2} (12 + π^{2}) δ^{5} λ^{2} μ^{2} (λ + 2 μ) k_{s}^{4}}{3 (π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})^{4}} \\ - \frac{π^{2} δ^{3} λ^{2} μ}{π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2}} + \frac{16 π^{2} δ^{3} λ^{2} μ k_{s}^{2}}{(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})^{2}} - \frac{16384 (π^{2} - 4) δ^{3} λ^{2} μ k_{s}^{6}}{(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})^{4}}] \end{aligned}

(67)

The strain energy density in the bulk (region B₁ in Fig. 1(b)) with thickness h − δ is obtained as,

W_{bulk} = \frac{1}{2} A (h - δ) \frac{4 μ (λ + μ)}{λ + 2 μ} ε_{11}^{2}

(68)

Denoting the total strain energy density as

W_{total}^{1}

and the effective Young’s modulus as

E_{1}^{e f f}

⁠, we have,

\frac{\partial^{2} W_{total}^{1}}{\partial ε_{11}^{2}} = \frac{\partial^{2} (W_{bulk} + W_{layer})}{\partial ε_{11}^{2}} = A h E_{1}^{e f f}

(69)

Thus, for a thin film with a rough surface, multiscale homogenization yields the following effective Young’s modulus,

\begin{aligned} E_{1}^{e f f} & = (1 - \frac{δ}{h}) \frac{4 μ (λ + μ)}{λ + 2 μ} + \frac{δ}{h} (\frac{1}{6 {(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})}^{4}}) \\ \times [- 32 π^{4} (12 + π^{2}) δ^{6} λ^{2} μ^{3} (λ + 2 μ)^{2} k_{s}^{2} \\ + 4096 π^{2} (12 + π^{2}) δ^{4} λ^{2} μ^{2} (λ + 2 μ) k_{s}^{4} \\ - 98304 (π^{2} - 4) δ^{2} λ^{2} μ k_{s}^{6} - 6 π^{2} δ^{2} λ^{2} μ {(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})}^{3} \\ + 96 π^{2} δ^{2} λ^{2} μ k_{s}^{2} {(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})}^{2} \\ + 3 λ {(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})}^{4} \\ + 6 μ {(π^{2} δ^{2} μ (λ + 2 μ) - 32 k_{s}^{2})}^{4}] \end{aligned}

(70)

We compare this expression with the results of Mohammadi et al. [46] for a thin film with sinusoidal roughness under uniaxial tension. If the effective surface elastic constant obtained by them has the following expression,

k_{s}^{e f f} = k_{s} - \frac{2 π}{ε} δ^{2} \frac{4 μ (λ + μ)}{λ + 2 μ}

(71)

the strain energy for a thin film of thickness h with a homgenized flat surface characterized by

k_{s}^{e f f}

is given as

W_{total}^{2} = \frac{1}{2} A k_{s}^{e f f} ε_{11}^{2} + A τ^{0} ε + \frac{1}{2} A h \frac{4 μ (λ + μ)}{λ + 2 μ} ε_{11}^{2}

(72)

where the first two terms arise from the homogenized surface and the last term is the contribution from the bulk. Then, the effective Young’s modulus is obtained as

E_{2}^{e f f} = \frac{k_{s}}{h} + \frac{4 μ (λ + μ)}{λ + 2 μ} [1 - \frac{2 π}{ε h} δ^{2}]

(73)

We wish to emphasize that the effective Young’s modulus derived from the work of Ref. [46] (Eq. ) depends on the wavelength whereas in our work (Eq. ) it does not. While the wavelength is a characteristic of the surface profile, the separation of scales in multiscale homogenization is enforced by assuming ɛ to be very small compared to the amplitude δ or rather δ/ɛ ≫ 1. Although ɛ does not enter explicitly in our final results, we will work within the limit that ɛ/δ ≪ 1 which corresponds to a very rough surface. In that sense, our work and that of Mohammadi et al. [46] present two extreme cases for surfaces with varying degree of roughness.

We now examine $E_{1}^{e f f}$ and $E_{2}^{e f f}$ numerically to gain more insights. We consider a thin film of Cu with λ = 91 GPa and μ = 43 GPa, and k_s = λ_s + 2μ_s = −3.16 N/m. We use these material constants for a (001) Cu surface taken from Ref. [55] and take the wavelength, ɛ, to be 10 nm to facilitate comparison with numerical calculations of Ref. [46]. We note that our theoretical model is applicable for any wavelength as long as the high roughness approximation is satisfied. Furthermore, although the present numerical example uses material parameters derived from atomistic simulations using embedded-atom method (EAM) potentials [55], our model can be used with material parameters obtained from experiments or computed from more sophisticated methods such as density functional theory (DFT) for more accurate predictions of surface properties. Since this work is focused on very rough surfaces with large amplitude while in Ref. [46], the roughness amplitude is smaller than the wavelength, we plot the E^eff normalized with respect to E_Cu = 4μ(λ + μ)/(λ + 2μ) as a function of the film thickness h ranging from δ to 10δ.

Figures 3(a) and 3(b) reveal that while in both cases the film has a softer elastic behavior as the amplitude of the roughness becomes comparable to the film thickness, a highly rough surface has a more dramatic impact on the effective Young’s modulus than a surface with mild roughness. Specifically, the Young’s modulus of the film is reduced to 30% for high roughness while it only reduces to 70% for mild or gentle roughness. When the film thickness becomes much larger than the roughness amplitude, the rough surface has a negligible effect on the Young’s modulus.

Fig. 3

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(a) $E_{1}^{e f f} / E_{C u}$ as a function of film thickness for δ = 10ε (high roughness). (b) $E_{2}^{e f f} / E_{C u}$ as a function of film thickness for δ = 0.01ε (mild roughness). ɛ = 10 nm. h varies from δ to 10δ. Here, δ is the amplitude and ɛ is the wavelength of the roughness. $E_{1}^{e f f}$ and $E_{1}^{e f f}$ are the effective Young’s modulus of a thin film with a rough surface using our method and the method of Mohammadi et al. [46] respectively while E_Cu is the Young’s modulus of Cu.

In order to understand the contribution of surface energy and surface roughness on the effective properties of the homogenized thin film, we compare the effective Young’s modulus of the thin film with and without surface energy effects. To this end, we first derive explicitly the homogenization results of Nevard and Keller [47] for a very rough surface and use them to obtain an expression for the Young’s modulus of a thin film with sinusoidal roughness (Appendix B).

Figures 4(a) and 4(b) show the results for our model and the model by Mohammadi et al. [46] for roughness with small amplitude (δ = 0.01ɛ) whereas Figs. 4(c) and 4(d) compare their results for roughness with large amplitude (δ = 10ɛ). As seen in Fig. 4(b), the model in [46] predicts that a gentle rough surface without surface elasticity does not change the Young’s modulus considerably. This is expected since it is very close to a flat surface. However, including surface energy effects even with gentle roughness results in a noticeable softening. Comparing this to Fig. 4(a), we note that our approach predicts a significant reduction in the Young’s modulus even when the roughness is very small. An even more curious outcome is that including surface elasticity results in a slight increase in the Young’s modulus. This is physically unreasonable since the numerical calculations based on Ref. [46] study shows that surface effects should cause softening for Cu. This implies that since the multi-scale homogenization treats the roughness as an equivalent layer of finite thickness, it may not be suitable for modeling rough surfaces with very small amplitudes. In the case of very rough surfaces, Fig. 4(c) based on our homogenization method reveals that roughness has a dominant effect, and hence, curves with and without surface energy effects are identical. Figure 4(d) shows that the model by Ref. [46] gives physically unreasonable results for the case of very rough surfaces. Taken together, Fig. 4 provides clear evidence that the multi-scale homogenization and conventional homogenization based on perturbation method yield two limiting cases for rough surfaces with large amplitudes and rough surfaces with very small amplitudes respectively. We wish to note that we do not know the range of validity of either of these approaches for intermediate values of surface roughness and hence a direct quantitative comparison of our results obtained for high roughness with those presented for weak roughness in prior studies [44,46] is not possible. A theoretical analysis of the limits of validity of our method for intermediate surface roughness is beyond the scope of the present study.

Fig. 4

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(a) $E_{1}^{e f f} / E_{C u}$ with and without surface effects as a function of the film thickness for gentle roughness (δ = 0.01ɛ). (b) $E_{2}^{e f f} / E_{C u}$ with and without surface effects as a function of the film thickness for gentle roughness (δ = 0.01ɛ). (c) $E_{1}^{e f f} / E_{C u}$ with and without surface effects as a function of the film thickness for high roughness (δ = 10ɛ). (d) $E_{2}^{e f f} / E_{C u}$ with and without surface effects as a function of the film thickness for high roughness (δ = 10ɛ). Film thickness h varies from δ to 10δ and ɛ = 10 nm. Here, δ is the amplitude and ɛ is the wavelength of the roughness. $E_{1}^{e f f}$ and $E_{1}^{e f f}$ are the effective Young’s modulus of a thin film with a rough surface using our method and the method of [46] respectively while E_Cu is the Young’s modulus of Cu.

5.3 Curvature-dependent Surface Energy.

Here, we demonstrate that the multi-scale homogenization approach furnishes a way to establish a connection with the Steigmann-Ogden theory for curvature-dependent surface elasticity [40]. The homogenized system obtained by way of multi-scale homogenization method replaces the surface with a roughness of amplitude δ with an equivalent layer of thickness δ with effective bulk properties. This endows the equivalent layer with a bending stiffness even if we do not consider curvature-dependent surface energy in our original model. As shown by Chhapadia et al. [43], the Steigmann-Ogden surface energy constants can be used to define the thickness of a surface t as a ratio of the energy contribution due to bending and stretching, and is given by

t = \sqrt{\frac{C_{1}}{C_{0}}}

(74)

where C₀ and C₁ are surface elastic modulus and Steigmann-Ogden constant, respectively. Interestingly, our method naturally yields that the thickness of the rough surface is the amplitude δ. Furthermore, according to [43], based on the Gurtin-Murdoch theory, the renormalized effective elastic modulus in tension can be written in the following form:

\frac{E^{e f f}}{E} = 1 + \frac{4 C_{0}}{h E}

(75)

In order to compare our results with , we can write as a function of film thickness h by substituting the elastic moduli for Cu mentioned earlier, and specifying a value for δ. The equation has the form:

\frac{E^{e f f}}{E} (h) = 1 - \frac{α}{h}

(76)

where α is some numerical constant. Comparing and , we can obtain an estimate for C₀. Then, substituting δ for the thickness of the surface in , we can provide a rough estimate for C₁.

For numerical calculations, we use two values of δ – 1 nm and 10 nm – to compute the estimates for C₀ and C₀ for rough surfaces with small amplitude and large amplitude. For δ = 1 nm, we get C₀ = −9.0571 J/m² ( = −0.5653 eV/Å²) and C₁ = −1.963 × 10⁻¹⁷ J ( = −122.528 eV). The values are comparable to those obtained by Chhapadia et al. [43]. For δ = 10 nm, we get C₀ = −221.261 J/m² ( = −13.81 eV/Å²) and C₁ = −2.2169 × 10⁻¹⁴ J (= −138369.27 eV). The large value for C₁ for large amplitude is expected since a layer of larger thickness would have greater bending stiffness. However, we do emphasize that these are rough estimates simply to demonstrate that multi-scale homogenization provides a simple way to determine curvature-dependent surface energy constants. For more accurate estimates, one would need to equate the homogenized thin film to an equivalent thin film with a flat surface including curvature-dependent surface energy and would be an interesting future study.

6 Conclusion

In this work, we have extended the multi-scale homogenization method of Nevard and Keller [47] for highly rough surfaces to take into account the effect of surface energy. The novelty of the method is that it replaces the highly rough surface by a layer of finite thickness with effective bulk elastic properties as opposed to a flat surface with effective surface properties furnished by conventional perturbation methods. We obtain analytical expressions for a general highly rough surface and specialize our results for the case of a thin film with a sinusoidal roughness. Using numerical calculations, we compare our results with those based on the work of Mohammadi et al. [46] which treats the rough surface as a perturbation about a flat surface. Our study shows that the two methods provide two limiting cases for surface roughness. Specifically, homogenization based on perturbation methods are appropriate for gentle roughness with small amplitude but high roughness with large amplitude necessitates the use of multi-scale homogenization. We also show that the multi-scale homogenization approach furnishes a natural way for estimating curvature-dependent surface energy constants based on the Steigmann–Ogden theory [40].

Acknowledgment

We gratefully acknowledge the support of NSF under grant DMR-1508484. We also acknowledge the use of the Maxwell cluster and the support from the Research Computing Data Core at the University of Houston. We thank Professor Pradeep Sharma and Dr. Sana Krichen for insightful discussions.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper. Data provided by a third party listed in Acknowledgment.

Appendix A: Introduction of Separation of Scales

Using the relation between u(x₁, x₃) and U(x₁, x₃, y) (Eqs. and ), the equilibrium equation becomes,

A_{11} (U_{, 11} + ε^{- 1} U_{, 1 y} + ε^{- 1} U_{, y 1} + ε^{- 2} U_{, y y}) + A_{13} (U_{, 31} + ε^{- 1} U_{, 3 y}) + A_{31} (U_{, 13} + ε^{- 1} U_{, y 3}) + A_{33} U_{, 33} = 0, x_{3} < h (y)

(A1)

The left-hand side and right-hand side of the boundary condition are obtained, respectively, as

(A_{h k} u_{, k}) n_{h} = - ε^{- 2} (h_{y} A_{11} U_{, y}) + ε^{- 1} (- h_{y} A_{11} U_{, 1} - h_{y} A_{13} U_{, 3} + A_{31} U_{, y}) + A_{31} U_{, 1} + A_{33} U_{, 3}

(A2)

\begin{aligned} ‖ n ‖ {div}_{s} S_{s} & = ε^{2} [\frac{B_{11}^{(4)}}{h_{y}^{4}} U_{, y y} + \frac{B_{11}^{(3)}}{h_{y}^{3}} (U_{, 1 y} + U_{, y 1}) + \frac{B_{11}^{(2)}}{h_{y}^{2}} U_{, 11} + \frac{B_{13}^{(2)}}{h_{y}^{2}} U_{, 13} + \frac{B_{31}^{(2)}}{h_{y}^{2}} U_{, 31} + \frac{B_{13}^{(3)}}{h_{y}^{3}} U_{, y 3} + \frac{B_{31}^{(3)}}{h_{y}^{3}} U_{, 3 y} + \frac{B_{33}^{(2)}}{h_{y}^{2}} U_{, 33} \\ + \frac{h_{y y}}{h_{y}^{4}} D_{1}^{(2)} U_{, 1} + \frac{h_{y y}}{h_{y}^{5}} D_{1}^{(3)} U_{, y} + \frac{h_{y y}}{h_{y}^{4}} D_{3}^{(2)} U_{, 3} + \frac{h_{y y}}{h_{y}^{4}} T^{(2)}] + ε [\frac{B_{11}^{(3)}}{h_{y}^{3}} U_{, y y} + \frac{B_{11}^{(2)}}{h_{y}^{2}} (U_{, 1 y} + U_{, y 1}) + \frac{B_{11}^{(1)}}{h_{y}} U_{, 11} + \frac{B_{13}^{(1)}}{h_{y}} U_{, 13} \\ + \frac{B_{13}^{(2)}}{h_{y}^{2}} U_{, y 3} + \frac{B_{31}^{(1)}}{h_{y}} U_{, 31} + \frac{B_{31}^{(2)}}{h_{y}^{2}} U_{, 3 y} + \frac{B_{33}^{(1)}}{h_{y}} U_{, 33} + \frac{h_{y y}}{h_{y}^{3}} D_{1}^{(1)} U_{, 1} + \frac{h_{y y}}{h_{y}^{4}} D_{1}^{(2)} U_{, y} + \frac{h_{y y}}{h_{y}^{3}} D_{3}^{(1)} U_{, 3} + \frac{h_{y y}}{h_{y}^{3}} T^{(1)}] \\ + ε^{0} [\frac{B_{11}^{(2)}}{h_{y}^{2}} U_{, y y} + \frac{B_{11}^{(1)}}{h_{y}} (U_{, y 1} + U_{, 1 y}) + B_{13}^{(0)} U_{, 13} + \frac{B_{13}^{(1)}}{h_{y}} U_{, y 3} + B_{31}^{(0)} U_{, 31} + \frac{B_{31}^{(1)}}{h_{y}} U_{, 3 y} + B_{33}^{(0)} U_{, 33} \\ + \frac{h_{y y}}{h_{y}^{3}} D_{1}^{(1)} U_{, y} + \frac{h_{y y}}{h_{y}^{2}} D_{1}^{(0)} U_{, 1} + \frac{h_{y y}}{h_{y}^{2}} D_{3}^{(0)} U_{, 3} + \frac{h_{y y}}{h_{y}^{2}} T^{(0)}] + ε^{- 1} [\frac{B_{11}^{(1)}}{h_{y}} U_{, y y} + B_{13}^{(0)} U_{, y 3} + B_{31}^{(0)} U_{, 3 y} + h_{y} B_{33}^{(- 1)} U_{, 33} + \frac{h_{y y}}{h_{y}^{2}} D_{1}^{(0)} U_{, y} \\ + ε^{- 1} \frac{h_{y y}}{h_{y}} D_{3}^{(- 1)} U_{, y} + \frac{h_{y y}}{h_{y}} T^{(- 1)}] + o (ε^{2}) \end{aligned}

(A3)

where h_y = dh/dy and h_yy = d²h/dy². The B matrices have the following expressions:

\begin{aligned} B_{11} & = [\begin{matrix} - \frac{ϵ^{3}}{h_{y}^{3}} (λ_{s} + 2 μ_{s}) & - \frac{ϵ^{4}}{h_{y}^{4}} (\frac{3}{2} λ_{s} + 2 μ_{s}) + \frac{ε^{2}}{h_{y}^{2}} (λ_{s} + 2 μ_{s}) \\ - \frac{ϵ^{4}}{h_{y}^{4}} (\frac{3}{2} λ_{s} + μ_{s}) + \frac{ε^{2}}{h_{y}^{2}} (λ_{s} + 2 μ_{s}) & - \frac{ϵ^{3}}{h_{y}^{3}} (\frac{3}{2} λ_{s} + 2 μ_{s}) + \frac{ε}{h_{y}} (λ_{s} + 2 μ_{s}) \end{matrix}] \\ = \frac{ε^{4}}{h_{y}^{4}} B_{11}^{(4)} + \frac{ε^{3}}{h_{y}^{3}} B_{11}^{(3)} + \frac{ε^{2}}{h_{y}^{2}} B_{11}^{(2)} + \frac{ε}{h_{y}} B_{11}^{(1)} \end{aligned}

(A4)

\begin{aligned} B_{13} & = [\begin{matrix} (λ_{s} + 2 μ_{s}) \frac{ε^{2}}{h_{y}^{2}} & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{3}}{h_{y}^{3}} + (2 μ_{s} + λ_{s}) \frac{ε}{h_{y}} \\ - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{3}}{h_{y}^{3}} + (2 μ_{s} + λ_{s}) \frac{ε}{h_{y}} & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{2}}{h_{y}^{2}} + (2 μ_{s} + λ_{s}) \end{matrix}] \\ = B_{13}^{(0)} + \frac{ε}{h_{y}} B_{13}^{(1)} + \frac{ε^{2}}{h_{y}^{2}} B_{13}^{(2)} + \frac{ε^{3}}{h_{y}^{3}} B_{13}^{(3)} \end{aligned}

(A5)

\begin{aligned} B_{31} & = [\begin{matrix} (λ_{s} + 2 μ_{s}) \frac{ε^{2}}{h_{y}^{2}} & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{3}}{h_{y}^{3}} + (2 μ_{s} + λ_{s}) \frac{ε}{h_{y}} \\ - (\frac{3}{2} λ_{s} + 2 μ_{s}) \frac{ε^{3}}{h_{y}^{3}} + (2 μ_{s} + λ_{s}) \frac{ε}{h_{y}} & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{2}}{h_{y}^{2}} + (2 μ_{s} + λ_{s}) \end{matrix}] \\ = B_{31}^{(0)} + \frac{ε}{h_{y}} B_{31}^{(1)} + \frac{ε^{2}}{h_{y}^{2}} B_{31}^{(2)} + \frac{ε^{3}}{h_{y}^{3}} B_{31}^{(3)} \end{aligned}

(A6)

\begin{aligned} B_{33} & = [\begin{matrix} (λ_{s} + 2 μ_{s}) \frac{ε}{h_{y}} & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{2}}{h_{y}^{2}} + (2 μ_{s} + λ_{s}) \\ - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε^{2}}{h_{y}^{2}} + (2 μ_{s} + λ_{s}) & - \frac{3}{2} (2 μ_{s} + λ_{s}) \frac{ε}{h_{y}} + ε^{- 1} h_{y} (2 μ_{s} + λ_{s}) \end{matrix}] \\ = ε^{- 1} h_{y} B_{33}^{(- 1) *} + B_{33}^{(0)} + \frac{ε}{h_{y}} B_{33}^{(1)} + \frac{ε^{2}}{h_{y}^{2}} B_{33}^{(2)} \end{aligned}

(A7)

where

\begin{aligned} B_{11}^{(1)} & = [\begin{matrix} 0 & 0 \\ 0 & λ_{s} + 2 μ_{s} \end{matrix}], B_{11}^{(2)} = [\begin{matrix} 0 & λ_{s} + 2 μ_{s} \\ λ_{s} + 2 μ_{s} & 0 \end{matrix}] \\ B_{11}^{(3)} & = [\begin{matrix} - (λ_{s} + 2 μ_{s}) & 0 \\ 0 & - (\frac{3}{2} λ_{s} + 2 μ_{s}) \end{matrix}], B_{11}^{(4)} = [\begin{matrix} 0 & 0 \\ 0 & λ_{s} + 2 μ_{s} \end{matrix}] \end{aligned}

(A8)

\begin{aligned} B_{13}^{(0)} & = [\begin{matrix} 0 & 0 \\ 0 & λ_{s} + 2 μ_{s} \end{matrix}], B_{13}^{(1)} = [\begin{matrix} 0 & λ_{s} + 2 μ_{s} \\ λ_{s} + 2 μ_{s} & 0 \end{matrix}] \\ B_{13}^{(2)} & = [\begin{matrix} - (λ_{s} + 2 μ_{s}) & 0 \\ 0 & - \frac{3}{2} (λ_{s} + 2 μ_{s}) \end{matrix}], B_{13}^{(3)} = [\begin{matrix} 0 & - \frac{3}{2} (λ_{s} + 2 μ_{s}) \\ - \frac{3}{2} (λ_{s} + 2 μ_{s}) \end{matrix}] \end{aligned}

(A9)

\begin{aligned} B_{31}^{(0)} & = [\begin{matrix} 0 & 0 \\ 0 & λ_{s} + 2 μ_{s} \end{matrix}], B_{31}^{(1)} = [\begin{matrix} 0 & λ_{s} + 2 μ_{s} \\ λ_{s} + 2 μ_{s} & 0 \end{matrix}] \\ B_{31}^{(2)} & = [\begin{matrix} - (λ_{s} + 2 μ_{s}) & 0 \\ 0 & - \frac{3}{2} (λ_{s} + 2 μ_{s}) \end{matrix}], B_{31}^{(3)} = [\begin{matrix} 0 & - (\frac{3}{2} λ_{s} + 2 μ_{s}) \\ - (\frac{3}{2} λ_{s} + 2 μ_{s}) \end{matrix}] \end{aligned}

(A10)

\begin{aligned} B_{33}^{(- 1)} & = [\begin{matrix} 0 & 0 \\ 0 & λ_{s} + 2 μ_{s} \end{matrix}], B_{33}^{(0)} = [\begin{matrix} 0 & λ_{s} + 2 μ_{s} \\ λ_{s} + 2 μ_{s} & 0 \end{matrix}] \\ B_{33}^{(1)} & = [\begin{matrix} λ_{s} + 2 μ_{s} & 0 \\ 0 & - \frac{3}{2} (λ_{s} + 2 μ_{s}) \end{matrix}], B_{33}^{(2)} = [\begin{matrix} 0 & - \frac{3}{2} (λ_{s} + 2 μ_{s}) \\ - \frac{3}{2} (λ_{s} + 2 μ_{s}) & 0 \end{matrix}] \end{aligned}

(A11)

The D matrices have the following expressions:

\begin{aligned} D_{1} & = [\begin{matrix} - ε^{2} \frac{h_{y y}}{h_{y}^{4}} (λ_{s} + 2 μ_{s}) & ε^{3} \frac{h_{y y}}{h_{y}^{5}} (6 λ_{s} + 8 μ_{s}) - 2 ε \frac{h_{y y}}{h_{y}^{3}} (λ_{s} + 2 μ_{s}) \\ ε^{3} \frac{h_{y y}}{h_{y}^{5}} (\frac{15}{2} λ_{s} + 5 μ_{s}) - 3 ε \frac{h_{y y}}{h_{y}^{3}} (λ_{s} + 2 μ_{s}) & ε^{2} \frac{h_{y y}}{h_{y}^{4}} (6 λ_{s} + 8 μ_{s}) - 2 \frac{h_{y y}}{h_{y}^{2}} (λ_{s} + 2 μ_{s}) \end{matrix}] \\ = ε^{3} \frac{h_{y y}}{h_{y}^{5}} D_{1}^{(3)} + ε^{2} \frac{h_{y y}}{h_{y}^{4}} D_{1}^{(2)} + ε \frac{h_{y y}}{h_{y}^{3}} D_{1}^{(1)} + \frac{h_{y y}}{h_{y}^{2}} D_{1}^{(0)} \end{aligned}

(A12)

\begin{aligned} D_{3} & = [\begin{matrix} - 2 ε \frac{h_{y y}}{h_{y}^{3}} (λ_{s} + 2 μ_{s}) & - \frac{9}{2} ε^{2} \frac{h_{y y}}{h_{y}^{4}} (λ_{s} + 2 μ_{s}) - (λ_{s} + 2 μ_{s}) \frac{h_{y y}}{h_{y}^{2}} \\ ε^{2} \frac{h_{y y}}{h_{y}^{4}} (6 λ_{s} + 8 μ_{s}) - 2 (λ_{s} + 2 μ_{s}) \frac{h_{y y}}{h_{y}^{2}} & - \frac{9}{2} ε \frac{h_{y y}}{h_{y}^{3}} (λ_{s} + 2 μ_{s}) - ε^{- 1} (λ_{s} + 2 μ_{s}) \frac{h_{y y}}{h_{y}} \end{matrix}] \\ = ε^{2} \frac{h_{y y}}{h_{y}^{4}} D_{3}^{(2)} + ε \frac{h_{y y}}{h_{y}^{3}} D_{3}^{(1)} + \frac{h_{y y}}{h_{y}^{2}} D_{3}^{(0)} + ε^{- 1} \frac{h_{y y}}{h_{y}} D_{3}^{(- 1) *} \end{aligned}

(A13)

where

\begin{aligned} D_{1}^{(0)} & = [\begin{matrix} 0 & 0 \\ 0 & - 2 (λ_{s} + 2 μ_{s}) \end{matrix}], D_{1}^{(1)} = [\begin{matrix} 0 & - 2 (λ_{s} + 2 μ_{s}) \\ - 3 (λ_{s} + 2 μ_{s}) & 0 \end{matrix}] \\ D_{1}^{(2)} & = [\begin{matrix} - 3 (λ_{s} + 2 μ_{s}) & 0 \\ 0 & (6 λ_{s} + 8 μ_{s}) \end{matrix}], D_{1}^{(3)} = [\begin{matrix} 0 & (6 λ_{s} + 8 μ_{s}) \\ \frac{15}{2} λ_{s} + 5 μ_{s} & 0 \end{matrix}] \end{aligned}

(A14)

\begin{aligned} D_{3}^{(- 1)} & = [\begin{matrix} 0 & 0 \\ 0 & - (λ_{s} + 2 μ_{s}) \end{matrix}], D_{3}^{(0)} = [\begin{matrix} 0 & - (λ_{s} + 2 μ_{s}) \\ - 2 (λ_{s} + 2 μ_{s}) & 0 \end{matrix}] \\ D_{3}^{(1)} & = [\begin{matrix} - 2 (λ_{s} + 2 μ_{s}) & 0 \\ 0 & \frac{9}{2} ((λ_{s} + 2 μ_{s})) \end{matrix}], D_{3}^{(2)} = [\begin{matrix} 0 & \frac{9}{2} (λ_{s} + 2 μ_{s}) \\ 6 λ_{s} + 8 μ_{s} & 0 \end{matrix}] \end{aligned}

(A15)

The T vector has the following expression:

T = [\begin{matrix} \frac{3}{2} ε^{2} \frac{h_{y y}}{h_{y}^{4}} τ^{0} - \frac{h_{y y}}{h_{y}^{2}} τ^{0} \\ \frac{3}{2} ε \frac{h_{y y}}{h_{y}^{3}} τ^{0} - ε^{- 1} \frac{h_{y y}}{h_{y}} τ^{0} \end{matrix}] = ε^{2} \frac{h_{y y}}{h_{y}^{4}} T^{(2)} + ε \frac{h_{y y}}{h_{y}^{3}} T^{(1)} + \frac{h_{y y}}{h_{y}^{2}} T^{(0)} + ε^{- 1} \frac{h_{y y}}{h_{y}} T^{(- 1) *}

(A16)

where

T^{(- 1)} = [\begin{matrix} 0 \\ - τ^{0} \end{matrix}], T^{(0)} = [\begin{matrix} - τ^{0} \\ 0 \end{matrix}], T^{(1)} = [\begin{matrix} 0 \\ \frac{3}{2} τ^{0} \end{matrix}], T^{(2)} = [\begin{matrix} \frac{3}{2} τ^{0} \\ 0 \end{matrix}]

(A17)

Note that the superscript (−1) is used only for notation purposes to differentiate between the orders of ɛ associated with different matrices and doesn’t refer to the inverse of the matrix.

Appendix B: Homogenized System in the Absence of Surface Elasticity

The boundary value problems for order ɛ⁻¹ and ɛ⁰ in the absence of surface elasticity:

Order ɛ⁻¹:

{\begin{matrix} [A_{11} u_{, y}^{(1)} + A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)}]_{, y} = 0, x_{3} < h (y) \\ h_{y} (- A_{11} u_{, y}^{(1)} - A_{11} u_{, 1}^{(0)} - A_{13} u_{, 3}^{(0)}) = 0, x_{3} = h (y) \end{matrix}

(B1)

Order ɛ⁰:

{\begin{matrix} [A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)} + A_{11} u_{, y}^{(1)}]_{, 1} + [A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} + A_{31} u_{, y}^{(1)}]_{, 3} + [A_{11} u_{, y}^{(2)} + A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)}]_{, y} = 0, x_{3} < h (y) \\ - h_{y} A_{11} u_{y}^{(2)} - h_{y} A_{11} u_{, 1}^{(1)} - h_{y} A_{13} u_{, 3}^{(1)} + A_{31} u_{, y}^{(1)} + A_{31} u_{, 1}^{(0)} + A_{33} u_{, 3}^{(0)} = 0, x_{3} = h (y) \end{matrix}

(B2)

Order ɛ¹:

{\begin{matrix} [A_{11} u_{, 1}^{(1)} + A_{13} u_{, 3}^{(1)} + A_{11} u_{, y}^{(2)}]_{, 1} + [A_{31} u_{, 1}^{(1)} + A_{31} u_{, y}^{(2)} + A_{33} u_{, 3}^{(1)}]_{, 3} + [A_{11} u_{, 1}^{(2)} + A_{13} u_{, 3}^{(2)}]_{, y} = 0, x_{3} < h (y) \\ - h_{y} A_{11} u_{, 1}^{(2)} - h_{y} A_{13} u_{, 3}^{(2)} + A_{31} u_{, y}^{(2)} + A_{31} u_{, 1}^{(1)} + A_{33} u_{, 3}^{(1)} = 0, x_{3} = h (y) \end{matrix}

(B3)

order ɛ²:

{\begin{matrix} A_{11} u_{, 11}^{(2)} + A_{13} u_{, 31}^{(2)} + A_{31} u_{, 13}^{(2)} + A_{33} u_{, 33}^{(2)} = 0, x_{3} < h (y) \\ A_{31} u_{, 1}^{(2)} + A_{33} u_{, 3}^{(2)} = 0, x_{3} = h (y) \end{matrix}

(B4)

Based on Eq. , we suggest the following form for u⁽¹⁾:

u^{(1)} = N^{1} u^{(0)} + N^{11} u_{, 1}^{(0)} + N^{13} u_{, 3}^{(0)}

(B5)

Where N¹, N¹¹, N¹³ are 2 × 2 matrices, functions of x₃ and y, periodic in y with period 1.

Implementing Eqs. into yields:

\begin{aligned} [A_{11} u_{, y}^{(1)} + A_{11} u_{, 1}^{(0)} + A_{13} u_{, 3}^{(0)}]_{, y} & = [A_{11} (N_{, y}^{1} u^{(0)} + N_{, y}^{11} u_{, 1}^{(0)} + N_{, y}^{13} u_{, 3}^{(0)} \\ + N_{, y}^{133} u_{, 33}^{(0)} + N_{, y}^{0}) + A_{11} u_{, 1}^{(0)} \\ + A_{13} u_{, 3}^{(0)}]_{, y}, x_{3} < h (y) \end{aligned}

(B6)

h_{y} [- A_{11} (N_{, y}^{1} u^{(0)} + N_{, y}^{11} u_{, 1}^{(0)} + N_{, y}^{13} u_{, 3}^{(0)} + N_{, y}^{133} u_{, 33}^{(0)} + N_{, y}^{0}) - A_{11} u_{, 1}^{(0)} - A_{13} u_{, 3}^{(0)}] = 0, x_{3} = h (y)

(B7)

Gathering the coefficients of u⁽⁰⁾,

u_{, 1}^{(0)}

⁠,

u_{, 3}^{(0)}

⁠, and

u_{, 33}^{(0)}

and from the remaining constants, we get the following systems for N¹, N¹¹, and N¹³:

\begin{aligned} {\begin{matrix} [A_{11} N_{, y}^{1}]_{, y} = 0, x_{3} < h (y) \\ - h_{y} A_{11} N_{, y}^{1} = 0, x_{3} = h (y) \end{matrix} \end{aligned}

(B8)

\begin{aligned} {\begin{matrix} [A_{11} N_{, y}^{11} + A_{11}]_{, y} = 0, x_{3} < h (y) \\ A_{11} N_{, y}^{11} + A_{11} = 0, x_{3} = h (y) \end{matrix} \end{aligned}

(B9)

\begin{aligned} {\begin{matrix} [A_{11} N_{, y}^{13} + A_{13}]_{, y} = 0, x_{3} < h (y) \\ - h_{y} (A_{11} N_{, y}^{13} + A_{13}) =, x_{3} = h (y) \end{matrix} \end{aligned}

(B10)

which can be solved to obtain the following expressions for the

N_{, y}^{i}

⁠, i ∈ {1, 11, 13}:

N_{, y}^{1} = 0, x_{3} \leq h (y)

(B11)

\begin{aligned} N_{, y}^{11} & = {\begin{matrix} 0, x_{3} < h (y) \\ - I, x_{3} = h (y) \end{matrix} \end{aligned}

(B12)

\begin{aligned} N_{, y}^{13} = {\begin{matrix} A_{11}^{- 1} ⟨ A_{11}^{- 1} ⟩^{- 1} ⟨ A_{11}^{- 1} A_{13} ⟩ - A_{11}^{- 1} A_{13}, x_{3} < h (y) \\ - A_{11}^{- 1} A_{13}, x_{3} = h (y) \end{matrix} \end{aligned}

(B13)

where

⟨ \cdot ⟩ = \int_{0}^{1} \cdot d y = \int_{0}^{y_{1}} \cdot d y + \int_{y_{2}}^{1} \cdot d y

⁠.

We integrate the bulk equation in system over y from 0 to 1, and by a process similar to the one presented in Sec. 4, we get the following equilibrium equation in the effective layer

\begin{aligned} ⟨ A_{11} ⟩ u_{, 11}^{(0)} + A_{13} u_{, 31}^{(0)} + [⟨ A_{31} ⟩ u_{, 1}^{(0)}]_{, 3} + [⟨ A_{33} ⟩ u_{, 3}^{(0)}]_{, 3} \\ = (y_{2}^{'} - y_{1}^{'}) (A_{33} - A_{31} A_{11}^{- 1} A_{13})] u_{, 3}^{(0)}, - \frac{δ}{2} < x_{3} < \frac{δ}{2} \end{aligned}

(B14)

Comparing it to

\begin{aligned} A_{11}^{e f f} u_{, 11}^{(0)} & + A_{13}^{e f f} u_{, 31}^{(0)} + [A_{31}^{e f f} u_{, 1}^{(0)}]_{, 3} + [A_{33}^{e f f} u_{, 3}^{(0)}]_{, 3} = D_{3}^{e f f} u_{, 3}^{(0)}, \\ - \frac{δ}{2} < x_{3} < \frac{δ}{2} \end{aligned}

(B15)

we get the effective properties as

A_{11}^{e f f} = [\begin{matrix} (λ + 2 μ) (1 - y_{2} + y_{1}) & 0 \\ 0 & μ (1 - y_{2} + y_{1}) \end{matrix}], A_{13}^{e f f} = [\begin{matrix} 0 & λ \\ μ & 0 \end{matrix}]

(B16)

A_{31}^{e f f} = [\begin{matrix} 0 & μ (1 - y_{2} + y_{1}) \\ λ (1 - y_{2} + y_{1}) & 0 \end{matrix}], A_{33}^{e f f} = [\begin{matrix} (1 - y_{2} + y_{1}) μ & 0 \\ 0 & (1 - y_{2} + y_{1}) (λ + 2 μ) \end{matrix}]

(B17)

D_{3}^{e f f} = [\begin{matrix} 0 & 0 \\ 0 & \frac{4 μ (λ + μ)}{λ + 2 μ} (y_{2}^{'} - y_{1}^{'}) \end{matrix}]

(B18)

Then, we calculate the effective Young’s modulus of a thin film similar to the one presented in Sec. 5.

σ_{13} = C_{1311} ε_{11} + C_{1313} ε_{13} + C_{1331} ε_{31} + C_{1333} ε_{33} = 2 μ f (x_{3}) ε_{13} = 0

(B19)

\begin{aligned} σ_{33} & = C_{3311} ε_{11} + C_{3313} ε_{13} + C_{3331} ε_{31} + C_{3333} ε_{33} \\ = λ f (x_{3}) ε_{11} + f (x_{3}) (λ + 2 μ) ε_{33} = 0 \end{aligned}

(B20)

and give the following relations between the strain tensor components:

ε_{33} = \frac{- λ}{(λ + 2 μ)} ε_{11}

(B21)

ε_{13} = 0

(B22)

From Eqs. and , we get the remaining stress tensor components in terms of ɛ₁₁:

\begin{aligned} σ_{11} & = C_{1111} ε_{11} + C_{1113} ε_{13} + C_{1131} ε_{31} + C_{1133} ε_{33} \\ = [(λ + 2 μ) f (x_{3}) - \frac{λ^{2}}{λ + 2 μ}] ε_{11} \end{aligned}

(B23)

σ_{31} = C_{3111} ε_{11} + C_{3113} ε_{13} + C_{3131} ε_{31} + C_{3133} ε_{33} = 0

(B24)

Therefore, by Eqs. –, the strain energy density in the layer is

W_{l a y e r} = \frac{1}{2} A ε_{11}^{2} \int_{- (δ / 2)}^{δ / 2} [(λ + 2 μ) f (x_{3}) - \frac{λ^{2}}{λ + 2 μ}] d x_{3}

(B25)

Adding the strain energy density in the bulk and then using Eq. , we get the renormalized Young’s modulus for the thin film due to surface roughness in the absence of surface elasticity:

E^{e f f} = \frac{4 μ (λ + μ)}{λ + 2 μ} - \frac{λ + 2 μ}{2} \frac{δ}{h}

(B26)

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Homogenization of Surface Energy and Elasticity for Highly Rough Surfaces

Abstract

1 Introduction

2 General Framework and Problem Formulation

2.1 Useful Notations.

2.2 Surface Elasticity.

2.3 Problem Formulation.

3 Multiscale Homogenization Approach

3.1 Introduction of Separation of Scales.

3.2 Asympotic Expansion.

3.3 Boundary Value Problems for Perturbed Solution.

4 Homogenized Equations and Effective Properties

5 Numerical Calculations and Discussion

5.1 Explicit Effective Properties for a Sinusoidal Roughness.

5.1 Extending these results to random roughness.

5.2 The Effective Young’s Modulus of a Thin Film.

5.3 Curvature-dependent Surface Energy.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

Appendix A: Introduction of Separation of Scales

Appendix B: Homogenized System in the Absence of Surface Elasticity

References

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Homogenization of Surface Energy and Elasticity for Highly Rough Surfaces

Abstract

1 Introduction

2 General Framework and Problem Formulation

2.1 Useful Notations.

2.2 Surface Elasticity.

2.3 Problem Formulation.

3 Multiscale Homogenization Approach

3.1 Introduction of Separation of Scales.

3.2 Asympotic Expansion.

3.3 Boundary Value Problems for Perturbed Solution.

4 Homogenized Equations and Effective Properties

5 Numerical Calculations and Discussion

5.1 Explicit Effective Properties for a Sinusoidal Roughness.

5.1 Extending these results to random roughness.

5.2 The Effective Young’s Modulus of a Thin Film.

5.3 Curvature-dependent Surface Energy.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

Appendix A: Introduction of Separation of Scales

Appendix B: Homogenized System in the Absence of Surface Elasticity

References

Get Email Alerts

Cited By

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