Size effects in two-dimensional layered materials modeled by couple stress elasticity

Abstract

In the present study, the effect of material microstructure on the mechanical response of a two-dimensional elastic layer perfectly bonded to a substrate is examined under surface loadings. In the current model, the substrate is treated as an elastic half plane as opposed to a rigid base, and this enables its applications in practical cases when the modulus of the layer (e.g., the coating material) and substrate (e.g., the coated surface) are comparable. The material microstructure is modeled using the generalized continuum theory of couple stress elasticity. The boundary value problems are formulated in terms of the displacement field and solved in an analytical manner via the Fourier transform and stiffness matrix method. The results demonstrate the capability of the present continuum theory to efficiently model the size-dependency of the response of the material when the external and internal length scales are comparable. Furthermore, the results indicated that the material mismatch and substrate stiffness play a crucial role in the predicted elastic field. Specifically, the study also addresses significant discrepancy of the response for the case of a layer resting on a rigid substrate.

Graphical abstract

Keywords

Cosserat / layered materials / size effects / microstructure

Introduction

Micro- and nanotechnologies play a crucial role in various disciplines ranging from biophysics to the design of next-generation multifunctional materials (e.g., [1–7]). In the past decades, nanomaterials (e.g., carbon nanotubes, lithium ions, nano-clusters, and nano-crystals) and various small-scale devices (e.g., MEMS, NEMS, sensors and actuators, and chips) have attracted significant interest due to their advanced electro-mechanical properties (e.g., [8–17]). At such small scales, their properties are observed as significantly different from those observed at macroscales and evidently become size-dependent (e.g., [18]). The study of the size-dependent behavior of such microstructures is the focus of numerous studies by various investigators with the primary objective of providing an in-depth understanding of size effects (e.g., [19–23]).

Nano-scale and nanostructured materials are well-recognized in surface coating applications. Both single-layer and multi-layer surface coatings are commonly utilized to improve the surface properties and overall performance of the components. For example, a surface coated by nanolayers of carbon nanotubes can enable absorption of light waves, reduce glare on computer screens, and potentially make an object seemingly invisible. A reflective color filter device fabricated from a photonic nanostructure can be used to replace the conventional color filter on the LCD screen to harvest the wasted energy of the absorbed light and convert it into electricity. Similarly, a compact multi-layer film structure can be applied to solar-thermal harvesting, thermoelectric detection, and imaging given its potential in ultra-broadband and perfect omnidirectional absorption. Surface coating is also required for quantum dots (QDs) to improve or strengthen their performance with respect to the fluorescence property and long-term stability. Furthermore, nano-sized complex metal boron-carbides (CMBCs) are observed in the application of surface coating for steel substrates to significantly enhance their surface wear resistance (e.g., [1–4,12]). Given the vast number of applications of nanoscale surface coatings, the capability to assess and predict the performance of the surface after coating is of paramount importance for the design process.

While the modeling framework and corresponding solution methodologies are well established for problems at the macroscale, the issue of determining accurate physically admissible mathematical models to predict the material response at micro-to-nano scales and to capture the pertinent size effects still constitute an open and challenging problem. Two paths are usually followed to study the size effects of microstructured materials under various loading conditions. The first path involves considering the distinct morphology of the material via discrete modeling and simultaneously incorporating the details of the material microstructure directly into the model. This approach, although very detailed and accurate, suffers because the computational cost becomes increasingly high with increasing material complexity. The alternative involves the use of a generalized continuum theory based on which the microstructural characteristics are smeared out, but the characteristic microstructural length is retained. The generalized continuum approach is very powerful because it can be efficiently incorporated into large computations; however, it lacks a detailed description of a discrete representation and treats the microstructural length in an average sense. One of the most effective generalized continuum theories is the so-called couple stress elasticity, which is also known as the Cosserat theory with constrained rotations [24–26]. This theory is the simplest gradient theory in which couple stresses are observed. The couple stress theory may be viewed as a first-step extension of classical elasticity theory and differs from the latter in several significant respects. Specifically, modified strain-energy density and the resulting constitutive relations involve besides the usual infinitesimal strains, the gradients of the rotation vector. Additionally, the generalized stress–strain relations for the isotropic case include, in addition to the conventional pair of elastic constants, two new elastic constants wherein one can be expressed in terms of a material parameter \ell that has dimensions of length. Other versions of the couple stress theory resulting from posing additional assumptions are also recognized and employed in the literature [27–32].

One of the earliest works to study the effects of couple stresses in the mechanical response of a two-dimensional half plane under surface tractions is the work by Muki and Sternberg [33]. In their study, the elastic half plane was subjected to concentrated and distributed surface loadings, and the method of Fourier transform was utilized to solve the boundary value problems. More recently, Zisis et al. [34] and Gourgiotis and Zisis [21] examined the 2D indentation problems by rigid, frictionless indenters with different profiles in the context of couple stress theory. The contact problems were formulated in terms of singular integral equations and solved via the collocation method. Green’s functions were obtained utilizing Mindlin’s stress function and Fourier transform analysis. Karuriya and Bhandakkar [35] extended the solution scheme proposed by Zisis et al. [34] to develop solutions for a finite thickness elastic layer fully bonded to a rigid base and indented by a rigid frictionless punch. Subsequently, Zisis [36] investigated the problem of a single layer fully bonded to a rigid substrate subjected to a normal point load on the surface. The problem was solved by the method of Fourier integral transforms by employing a stress function approach. In the same year, Song et al. [23] investigated size-dependent elastic response (via couple stress theory) of a coated half plane indented by a rigid and frictionless indenter with flat, parabolic, cylindrical, and wedge profiles. A numerical procedure based on the Gauss–Chebyshev quadrature was implemented to solve a Cauchy singular integral equation governing the contact problem. The treatment of more practical contact conditions over the elastic half plane was also considered and included sliding frictional contact by Song et al. [22] and partial slip contact by Wang et al. [37].

The aforementioned studies evidently indicated the strong dependence of the mechanical response on the characteristic material length scales related to the material microstructure and significant departure from the predictions of the classical continuum theories. However, most of the existing studies were performed within the limited context of two-dimensional settings associated only with half-plane or single-layer media resting on rigid substrates. The extension of existing work to treat more practical scenarios, such as a layer on half plane or multi-layered media, is considered challenging and is the focus of the present study. One of the key merits and contributions of current study is in the area of surface coatings. Specifically, the current model considers the elasticity of the substrate and can be employed in various practical cases. Indeed, in many cases, inaccurate or erroneous predictions can occur because the coated surface is treated as rigid (compared to the coating material). The analytical solution of the boundary value problems is established via Fourier transform analysis and the stiffness matrix method. An analytical solution, such as that proposed in the study exhibits an advantage over numerical solutions and especially in new areas of research where benchmark solutions do not exist.

Problem formulation

A two-dimensional, infinite, and elastic layer resting on a half plane is considered, as shown schematically in Fig. 1. The layer is perfectly bonded to the half plane and has a constant thickness h. A two-dimensional Cartesian coordinate system \{x,y;O\} is chosen where the origin O is placed at the top surface, and the x-axis lies along the infinite direction of the layer and y-axis points downwards. The surface of the layer is subjected to arbitrarily distributed normal and shear surface tractions over the region x\in [-a,a], while the couple stress tractions are assumed to vanish over the entire bounding edge. The layer and the half plane are composed of different homogeneous, isotropic, and linearly elastic materials that obey the constitutive laws of the couple stress theory. In the present study, it is assumed that body forces and body couples are absent, and the thickness of the medium in the direction perpendicular to the x-y plane is significantly large such that plane strain conditions prevail.

Basic field equations

In this section, we briefly recall certain pertinent elements of the linearized plane-strain theory of couple stress elasticity for homogeneous and isotropic elastic solids. Detailed presentations of the couple stress theory can be found in fundamental papers by Mindlin and Tiersten [24], Mindlin [25], and Koiter [26].

The displacement field of a body under conditions of plane strain is defined as follows:

(1)u_x\equiv u_x(x,y),u_y\equiv u_y(x,y),u_z\equiv \mathrm{0}，

where \{u_x,u_y,u_z\} denote components of the displacement in a reference Catesian coordinate system \{x,y,z\}. Thus, the in-plane strain components \{{\epsilon }_{xx},{\epsilon }_{xy},{\epsilon }_{yx},{\epsilon }_{yy}\}, rotation {\omega }_z, and non-vanishing components of the curvature tensor {\mathit{\kappa }}_{xz} and {\mathit{\kappa }}_{yz} are related by the following linearized kinematics:

(2){\epsilon }_{xx}={\displaystyle \frac{\partial u_x}{\partial x}},{\epsilon }_{yy}={\displaystyle \frac{\partial u_y}{\partial y}},

(3){\epsilon }_{xy}={\epsilon }_{yx}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}}\right),

(4){\omega }_z={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{\partial u_y}{\partial x}}-{\displaystyle \frac{\partial u_x}{\partial y}}\right),

(5){\mathit{\kappa }}_{yz}={\displaystyle \frac{\partial {\omega }_z}{\partial y}},

(6){\mathit{\kappa }}_{xz}={\displaystyle \frac{\partial {\omega }_z}{\partial x}},

Furthermore, the force and moment equilibrium equations are given as follows:

(7){\displaystyle \frac{\partial {\sigma }_{xx}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yx}}{\partial y}}=\mathrm{0},

(8){\displaystyle \frac{\partial {\sigma }_{xy}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yy}}{\partial y}}=\mathrm{0},

(9){\displaystyle \frac{\partial {\mu }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\mu }_{yz}}{\partial y}}+{\sigma }_{xy}-{\sigma }_{yx}=\mathrm{0},

where \{{\mathit{\sigma }}_{xx},{\mathit{\sigma }}_{xy},{\mathit{\sigma }}_{yx},{\mathit{\sigma }}_{yy}\} denote the in-plane components of the asymmetric stress tensor and \{{\mathit{\mu }}_{xz},{\mathit{\mu }}_{yz}\} denote the non-zero components of the couple stress tensor.

The constitutive equations that relate the stresses and couple stresses with the infinitesimal strains and rotation are given by

(10){\mu }_{yz}=\mathrm{4}\eta {\kappa }_{yz}=\mathrm{4}\eta {\displaystyle \frac{\partial {\omega }_z}{\partial y}},

(11){\mu }_{xz}=\mathrm{4}\eta {\kappa }_{xz}=\mathrm{4}\eta {\displaystyle \frac{\partial {\omega }_z}{\partial x}},

(12){\sigma }_{xx}={\displaystyle \frac{\mathrm{2}\mu }{\mathrm{1}-\mathrm{2}\nu }}[(\mathrm{1}-\nu ){\epsilon }_{xx}+\nu {\epsilon }_{yy}],

(13){\sigma }_{yy}={\displaystyle \frac{\mathrm{2}\mu }{\mathrm{1}-\mathrm{2}\nu }}[\nu {\epsilon }_{xx}+(\mathrm{1}-\nu ){\epsilon }_{yy}],

(14){\sigma }_{xy}=\mathrm{2}\mu {\epsilon }_{xy}-\mathrm{2}\eta \mathrm{\Delta }{\omega }_z,

(15){\sigma }_{yx}=\mathrm{2}\mu {\epsilon }_{xy}+\mathrm{2}\eta \mathrm{\Delta }{\omega }_z,

where \lambda and \mu denote the Lamé-type constants defined in the same fashion as that in classical continuum mechanics, and \eta denotes the material constant that accounts for the presence of couple stresses. Additionally, \mathrm{\Delta } denotes the two-dimensional Laplacian operator, and \nu denotes Poisson’s ratio. The additional material parameter \eta is responsible for the length-scale effect, and the couple stress plane strain theory reduces to the classical theory of elasticity if this parameter is set to zero. Finally, it should be noted that the moment equilibrium equation, Eq. (9), is employed to determine the skew-symmetric part of the stress tensor as given in Eqs. (14) and (15) and also implies the symmetry of the stress tensor in the absence of couple stresses and body couples.

When we combine Eqs. (1)–(9) with Eqs. (10)–(15), the expressions for the stresses and couple stresses in terms of the in-plane displacements are obtained as follows:

(16){\sigma }_{xx}={\displaystyle \frac{\mu }{\mathrm{1}-\mathrm{2}\nu }}\left((\mathrm{1}-\nu ){\displaystyle \frac{\partial u_x}{\partial x}}+\nu {\displaystyle \frac{\partial u_y}{\partial y}}\right),

(17){\sigma }_{yy}={\displaystyle \frac{\mathrm{2}\mu }{\mathrm{1}-\mathrm{2}\nu }}\left(\nu {\displaystyle \frac{\partial u_x}{\partial x}}+(\mathrm{1}-\nu ){\displaystyle \frac{\partial u_y}{\partial y}}\right),

(18){\sigma }_{xy}=\mu \left({\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}}\right)-\mu {\ell }^{\mathrm{2}}\mathrm{\Delta }\left({\displaystyle \frac{\partial u_y}{\partial x}}-{\displaystyle \frac{\partial u_x}{\partial y}}\right),

(19){\sigma }_{yx}=\mu \left({\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}}\right)+\mu {\ell }^{\mathrm{2}}\mathrm{\Delta }\left({\displaystyle \frac{\partial u_y}{\partial x}}-{\displaystyle \frac{\partial u_x}{\partial y}}\right),

(20){\mu }_{xz}=\mathrm{2}\mu {\ell }^{\mathrm{2}}\left({\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_y}{\partial x^{\mathrm{2}}}}-{\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_x}{\partial x\partial y}}\right),

(21){\mu }_{yz}=\mathrm{2}\mu {\ell }^{\mathrm{2}}\left({\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_y}{\partial x\partial y}}-{\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_x}{\partial y^{\mathrm{2}}}}\right)\mathrm{.}

where \ell =\sqrt{\eta /\mu } denotes the characteristic material length of couple stress elasticity. We substitute Eqs. (16)-(21) into the force equilibrium equations, Eqs. (7) and (8), to obtain the following pair of equilibrium equations in terms of the displacements:

(22){\displaystyle \frac{\partial }{\partial x}}\left[{\displaystyle \frac{\mathrm{2}\mu }{\mathrm{1}-\mathrm{2}\nu }}\left((\mathrm{1}-\nu ){\displaystyle \frac{\partial u_x}{\partial x}}+\nu {\displaystyle \frac{\partial u_y}{\partial y}}\right)\right]+\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\partial }{\partial y}}\left[\mu \left({\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}}\right)+\mu {\ell }^{\mathrm{2}}\mathrm{\Delta }\left({\displaystyle \frac{\partial u_y}{\partial x}}-{\displaystyle \frac{\partial u_x}{\partial y}}\right)\right]=\mathrm{0},

(23){\displaystyle \frac{\partial }{\partial y}}\left[{\displaystyle \frac{\mathrm{2}\mu }{\mathrm{1}-\mathrm{2}\nu }}\left(\nu {\displaystyle \frac{\partial u_x}{\partial x}}+(\mathrm{1}-\nu ){\displaystyle \frac{\partial u_y}{\partial y}}\right)\right]+{\displaystyle \frac{\partial }{\partial x}}\left[\mu \left({\displaystyle \frac{\partial u_x}{\partial y}}+{\displaystyle \frac{\partial u_y}{\partial x}}\right)-\mu {\ell }^{\mathrm{2}}\mathrm{\Delta }\left({\displaystyle \frac{\partial u_y}{\partial x}}-{\displaystyle \frac{\partial u_x}{\partial y}}\right)\right]=\mathrm{0}

A two-dimensional version of the equivalent Navier–Cauchy equations for the couple stress theory is provided in a concise form:

(24)\left[\begin{array}{ll}{D}_{\mathrm{11}}& D_{\mathrm{12}}\\ D_{\mathrm{21}}& D_{\mathrm{22}}\end{array}\right]\left\{\begin{array}{l}{u}_x\\ u_y\end{array}\right\}=\left\{\begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array}\right\},

where (D_{\mathrm{11}},\hspace{0.17em}{D}_{\mathrm{12}},\hspace{0.17em}{D}_{\mathrm{21}},\hspace{0.17em}{D}_{\mathrm{22}}) are linear differential operators that are defined as follows:

(25)D_{\mathrm{11}}=\alpha {\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial y^{\mathrm{2}}}}-{\ell }^{\mathrm{2}}\left({\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial x^{\mathrm{2}}\partial y^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial y^{\mathrm{4}}}}\right),

(26)D_{\mathrm{12}}=D_{\mathrm{21}}=\beta {\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x\partial y}}+{\ell }^{\mathrm{2}}\left({\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial x^{\mathrm{3}}\partial y}}+{\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial y^{\mathrm{3}}\partial x}}\right),

(27)D_{\mathrm{22}}={\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}+\alpha {\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial y^{\mathrm{2}}}}-{\ell }^{\mathrm{2}}\left({\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial x^{\mathrm{2}}\partial y^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{4}}}{\partial x^{\mathrm{4}}}}\right),

with \alpha =\mathrm{2}(\mathrm{1}-\nu )/(\mathrm{1}-\mathrm{2}\nu ) and \beta =\mathrm{1}/(\mathrm{1}-\mathrm{2}\nu ).

Solution procedure

To construct the general solution of Eq. (24) for the layer and half plane, we apply the Fourier integral transform method (e.g., [38]). The direct and inverse Fourier transforms of a function f are defined, respectively, by

(28)\tilde{f}(\xi )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x)e^{i\xi x}dx},

(29)f(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\tilde{f}(\xi )e^{-i\xi x}d\xi },

where \xi is a transform parameter, and i denotes the imaginary unit. The application of the direct Fourier transform to the system in Eq. (24) leads to a homogeneous system of linear ordinary differential equations with respect to the coordinate y as follows:

(30)\left[\begin{array}{ll}{\tilde{D}}_{\mathrm{11}}& {\tilde{D}}_{\mathrm{12}}\\ {\tilde{D}}_{\mathrm{21}}& {\tilde{D}}_{\mathrm{22}}\end{array}\right]\left\{\begin{array}{l}{\tilde{u}}_x\\ {\tilde{u}}_y\end{array}\right\}=\left\{\begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array}\right\},

where ({\tilde{D}}_{\mathrm{11}},{\tilde{D}}_{\mathrm{12}},{\tilde{D}}_{\mathrm{21}},{\tilde{D}}_{\mathrm{22}}) are given by

(31){\tilde{D}}_{\mathrm{11}}=-\alpha {\xi }^{\mathrm{2}}+(\mathrm{1}+{\ell }^{\mathrm{2}}{\xi }^{\mathrm{2}}){\displaystyle \frac{d^{\mathrm{2}}}{d{y}^{\mathrm{2}}}}-{\ell }^{\mathrm{2}}{\displaystyle \frac{d^{\mathrm{4}}}{d{y}^{\mathrm{4}}}},

(32){\tilde{D}}_{\mathrm{12}}={\tilde{D}}_{\mathrm{21}}=-i\xi \left[(\beta -{\ell }^{\mathrm{2}}{\xi }^{\mathrm{2}}){\displaystyle \frac{d}{dy}}+{\ell }^{\mathrm{2}}{\displaystyle \frac{d^{\mathrm{3}}}{d{y}^{\mathrm{3}}}}\right],

(33){\tilde{D}}_{\mathrm{22}}=-{\xi }^{\mathrm{2}}(\mathrm{1}+{\ell }^{\mathrm{2}}{\xi }^{\mathrm{2}})+(\alpha +{\ell }^{\mathrm{2}}{\xi }^{\mathrm{2}}){\displaystyle \frac{d^{\mathrm{2}}}{d{y}^{\mathrm{2}}}}\mathrm{.}

The general solution of (30) for the layer occupying the region x\in (-\infty ,\infty ),y\in [\mathrm{0},h] assumes the following form:

(34)\begin{array}{l}{\tilde{u}}_x=-i\left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}-i(\kappa +|\xi |y)e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}+i\left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{3}}-i(\kappa -|\xi |y)e^{-\left|\xi \right|y}{C}_{\mathrm{4}}\\ -{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}+{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}\end{array}

(35){\tilde{u}}_y=\xi e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}+\xi y{e}^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}+\xi e^{-\left|\xi \right|y}{C}_{\mathrm{3}}+\xi y{e}^{-\left|\xi \right|y}{C}_{\mathrm{4}}+\xi e^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}+\xi e^{-\zeta y/\ell }{C}_{\mathrm{6}}

where C_i(i=\mathrm{1},\mathrm{2},\mathrm{...},\mathrm{6}) denote independent unknown functions of \xi with \mathit{\kappa }=\mathrm{3}-\mathrm{4}\nu, and

(36)\zeta =\sqrt{\mathrm{1}+{\ell }^{\mathrm{2}}{\xi }^{\mathrm{2}}}\mathrm{.}

Applying the Fourier transform to Eqs. (16)-(21) and Eq. (4) in combination with the general solution Eqs. (34)–(35) yields the general solution for the rotation, stresses, and couple stresses in the transform space as follows:

(37){\tilde{\omega }}_z={\displaystyle \frac{i}{\mathrm{2}}}(\kappa +\mathrm{1})\left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}-{\displaystyle \frac{i}{\mathrm{2}}}(\kappa +\mathrm{1})\left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{4}}+{\displaystyle \frac{i}{\mathrm{2}{\ell }^{\mathrm{2}}}}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}+{\displaystyle \frac{i}{\mathrm{2}{\ell }^{\mathrm{2}}}}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}

(38)\begin{array}{l}{\tilde{\sigma }}_{xx}=\mathrm{2}\mu [-\xi \left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}-(\mathrm{3}-\mathrm{2}\nu +\left|\xi \right|y)\xi e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}\\ +\xi \left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{3}}-(\mathrm{3}-\mathrm{2}\nu -\left|\xi \right|y)\xi e^{-\left|\xi \right|y}{C}_{\mathrm{4}}-{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}+{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}]\end{array}

(39)\begin{array}{l}{\tilde{\sigma }}_{yy}=\mathrm{2}\mu [\xi \left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}+(\mathrm{1}-\mathrm{2}\nu +\left|\xi \right|y)\xi e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}\\ -\xi \left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{3}}+(\mathrm{1}-\mathrm{2}\nu -\left|\xi \right|y)\xi e^{-\left|\xi \right|y}{C}_{\mathrm{4}}+{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}-{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}]\end{array}

(40)\begin{array}{l}{\tilde{\sigma }}_{xy}=\mathrm{2}\mu [-i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}-i(\mathrm{2}-\mathrm{2}\nu +\left|\xi \right|y)\left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}\\ -i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y}{C}_{\mathrm{3}}+i(\mathrm{2}-\mathrm{2}\nu -\left|\xi \right|y)\left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{4}}-{\displaystyle \frac{i{\zeta }^{\mathrm{2}}}{{\ell }^{\mathrm{2}}}}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}-{\displaystyle \frac{i{\zeta }^{\mathrm{2}}}{{\ell }^{\mathrm{2}}}}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}]\end{array}

(41)\begin{array}{l}{\tilde{\sigma }}_{yx}=\mathrm{2}\mu [-i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|(h-y)}{C}_{\mathrm{1}}-i(\mathrm{2}-\mathrm{2}\nu +\left|\xi \right|y)\left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}\\ -i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y}{C}_{\mathrm{3}}+i(\mathrm{2}-\mathrm{2}\nu -\left|\xi \right|y)\left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{4}}-i{\xi }^{\mathrm{2}}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}-i{\xi }^{\mathrm{2}}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}]\end{array}

(42){\tilde{\mu }}_{xz}=\mathrm{2}\mu [{\ell }^{\mathrm{2}}(\kappa +\mathrm{1})\xi \left|\xi \right|e^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}-{\ell }^{\mathrm{2}}(\kappa +\mathrm{1})\xi \left|\xi \right|e^{-\left|\xi \right|y}{C}_{\mathrm{4}}+\xi e^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}+\xi e^{-\zeta y/\ell }{C}_{\mathrm{6}}]

(43){\tilde{\mu }}_{yz}=\mathrm{2}\mu [i{\ell }^{\mathrm{2}}(\kappa +\mathrm{1}){\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|(h-y)}{C}_{\mathrm{2}}+i{\ell }^{\mathrm{2}}(\kappa +\mathrm{1}){\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y}{C}_{\mathrm{4}}+{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta (h-y)/\ell }{C}_{\mathrm{5}}-{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta y/\ell }{C}_{\mathrm{6}}]

The transformed displacements and tractions at the surface of the layer can be related to the displacements at the interface of the layer with the half-plane using the general solutions given by Eqs. (34)–(35) and Eqs. (37)–(43). This leads to the following compact form:

(44)\left\{\begin{array}{l}{{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}\\ {{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=h}\end{array}\right\}={\mathit{M}}^{(\mathrm{1})}(\xi ){\mathit{C}}^{(\mathrm{1})};\left\{\begin{array}{l}{{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}\\ {{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=h}\end{array}\right\}={\mathit{N}}^{(\mathrm{1})}(\xi ){\mathit{C}}^{(\mathrm{1})},

where {\tilde{\mathit{U}}}^{(\mathrm{1})}={\{{\tilde{u}}_x,{\tilde{u}}_y,{\tilde{\omega }}_z\}}^T; {\tilde{\mathit{P}}}^{(\mathrm{1})}={\{{\tilde{\sigma }}_{yx},{\tilde{\sigma }}_{yy},{\tilde{\mu }}_{yz}\}}^T; {\mathit{C}}^{(\mathrm{1})}={\{C_{\mathrm{1}},C_{\mathrm{2}},C_{\mathrm{3}},C_{\mathrm{4}},C_{\mathrm{5}},C_{\mathrm{6}}\}}^T; {\mathit{M}}^{(\mathrm{1})} and {\mathit{N}}^{(\mathrm{1})} denote 6x6 matrices whose entries are functions of \xi and can be obtained in a closed form from the general solutions of the displacements in Eqs. (34)–(35), rotation in Eq. (37), stresses in Eqs. (39)–(41), and couple stress in Eq. (43); and the superscript “(1)” is used to designate quantities associated with the elastic layer. Combining the two sets of equations in Eq. (44) leads to the following expression:

(45)\left\{\begin{array}{l}{{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}\\ {{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=h}\end{array}\right\}={\mathit{K}}^{(\mathrm{1})}(\xi )\left\{\begin{array}{l}{{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}\\ {{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=h}\end{array}\right\},

where {\mathit{K}}^{(\mathrm{1})} is defined by

(46){\mathit{K}}^{(\mathrm{1})}(\xi )={\mathit{N}}^{(\mathrm{1})}(\xi ){[{\mathit{M}}^{(\mathrm{1})}(\xi )]}^{-\mathrm{1}},

The stiffness matrix {\mathit{K}}^{(\mathrm{1})} for the elastic layer can be further partitioned as follows:

(47){\mathit{K}}^{(\mathrm{1})}(\xi )=\left[\begin{array}{ll}{\mathit{K}}_{tt}^{(\mathrm{1})}(\xi )& {\mathit{K}}_{tb}^{(\mathrm{1})}(\xi )\\ {\mathit{K}}_{bt}^{(\mathrm{1})}(\xi )& {\mathit{K}}_{bb}^{(\mathrm{1})}(\xi )\end{array}\right],

where all the sub-matrices are of dimension \mathrm{3}\times \mathrm{3}.

To construct the general solution for the elastic field of the half plane (occupying the regionx\in (-\infty ,\infty ),y\in [h,\infty ]), the same procedure is applied together with the additional requirement of the boundedness of the solution at infinity. The final explicit solutions are given by

(48){\tilde{u}}_x=i\left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}-i(\kappa -|\xi |y^{*})e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}+{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}},

(49){\tilde{u}}_y=\xi e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}+\xi y^{*}{e}^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}+\xi e^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}},

(50){\tilde{\omega }}_x=-{\displaystyle \frac{i}{\mathrm{2}}}(\kappa +\mathrm{1})\left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}+{\displaystyle \frac{i}{\mathrm{2}{\ell }^{\mathrm{2}}}}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}},

(51){\tilde{\sigma }}_{xx}=\mathrm{2}\mu [\xi \left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}-(\mathrm{3}-\mathrm{2}\nu -\left|\xi \right|y^{*})\xi e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}+{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}],

(52){\tilde{\sigma }}_{yy}=\mathrm{2}\mu [-\xi \left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}+(\mathrm{1}-\mathrm{2}\nu -\left|\xi \right|y^{*})\xi e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}-{\displaystyle \frac{\zeta \xi }{\ell }}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}],

(53){\tilde{\sigma }}_{xy}=\mathrm{2}\mu [-i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}+i(\mathrm{2}-\mathrm{2}\nu -\left|\xi \right|y^{*})\left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}-{\displaystyle \frac{i{\zeta }^{\mathrm{2}}}{{\ell }^{\mathrm{2}}}}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}],

(54){\tilde{\sigma }}_{yx}=\mathrm{2}\mu [-i{\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y^{*}}{C}_{\mathrm{7}}+i(\mathrm{2}-\mathrm{2}\nu -\left|\xi \right|y^{*})\left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}-i{\xi }^{\mathrm{2}}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}],

(55){\tilde{\mu }}_{xz}=\mathrm{2}\mu [-{\ell }^{\mathrm{2}}(\kappa +\mathrm{1})\xi \left|\xi \right|e^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}+\xi e^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}],

(56){\tilde{\mu }}_{yz}=\mathrm{2}\mu \left[i{\ell }^{\mathrm{2}}(\kappa +\mathrm{1}){\xi }^{\mathrm{2}}{e}^{-\left|\xi \right|y^{*}}{C}_{\mathrm{8}}-{\displaystyle \frac{i\zeta }{\ell }}{e}^{-\zeta y^{*}/\ell }{C}_{\mathrm{9}}\right],

where y^{*}=y-h and the material properties \mu, \nu, \ell involved in the general solutions (48)–(56) are now those of the half plane.

Similarly, the general solutions in Eqs. (48)–(56) can be utilized to establish the following relations

(57){{\tilde{\mathit{U}}}^{(\mathrm{2})}|}_{y^{*}=\mathrm{0}}={\mathit{M}}^{(\mathrm{2})}(\xi ){\mathit{C}}^{(\mathrm{2})};{{\tilde{\mathit{P}}}^{(\mathrm{2})}|}_{y^{*}=\mathrm{0}}={\mathit{N}}^{(\mathrm{2})}(\xi ){\mathit{C}}^{(\mathrm{2})},

where {\tilde{\mathit{U}}}^{(\mathrm{2})}={\{{\tilde{u}}_x,{\tilde{u}}_y,{\tilde{\omega }}_z\}}^T; {\tilde{\mathit{P}}}^{(\mathrm{2})}={\{{\tilde{\sigma }}_{yx},{\tilde{\sigma }}_{yy},{\tilde{\mu }}_{yz}\}}^T; {\mathit{C}}^{(\mathrm{2})}={\{C_{\mathrm{7}},C_{\mathrm{8}},C_{\mathrm{9}}\}}^T; {\mathit{M}}^{(\mathrm{2})} and {\mathit{N}}^{(\mathrm{2})} are \mathrm{3}\times \mathrm{3}-matrices whose elements are explicit functions of \xi; and the superscript “(2)” is used to designate quantities associated with the half plane. We combine two sets of equations in Eq. (57) to yield the following expression:

(58){{\tilde{\mathit{P}}}^{(\mathrm{2})}|}_{y^{*}=\mathrm{0}}={\mathit{K}}^{(\mathrm{2})}(\xi ){{\tilde{\mathit{U}}}^{(\mathrm{2})}|}_{y^{*}=\mathrm{0}},

where {\mathit{K}}^{(\mathrm{2})} denotes the stiffness matrix of the half plane defined by

(59){\mathit{K}}^{(\mathrm{2})}(\xi )={\mathit{N}}^{(\mathrm{2})}(\xi ){[{\mathit{M}}^{(\mathrm{2})}(\xi )]}^{-\mathrm{1}}\mathrm{.}

Boundary conditions at the top surface of the elastic layer and the continuity conditions along the interface of the layer and half plane must be appropriately enforced to determine all the unknown functions {\mathit{C}}^{(\mathrm{1})} for the elastic layer and {\mathit{C}}^{(\mathrm{2})} for the half plane. Force- and couple-tractions are fully prescribed at the top surface of the layer (i.e., at y=\mathrm{0}), and the boundary conditions in the transform space are given as follows:

(60){{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}=\left\{\begin{array}{l}{\tilde{q}}_{\mathrm{0}}(\xi )\\ {\tilde{p}}_{\mathrm{0}}(\xi )\\ \mathrm{0}\end{array}\right\}\equiv {\tilde{\mathit{P}}}^{\mathrm{0}},

where {\tilde{p}}_{\mathrm{0}} and {\tilde{q}}_{\mathrm{0}} denote the transformed boundary tractions. Furthermore, the elastic layer is perfectly bonded to the half plane, and thus the continuity of the displacements, rotation, and tractions at the material interface leads to the following expressions:

(61){{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=h}={{\tilde{\mathit{U}}}^{(\mathrm{2})}|}_{y=h},

(62){{\tilde{\mathit{P}}}^{(\mathrm{1})}|}_{y=h}={{\tilde{\mathit{P}}}^{(\mathrm{2})}|}_{y=h}

By combining the stiffness equation for the elastic layer given by Eq. (45), the stiffness equation for the half plane given by Eq. (58), the boundary conditions in Eq. (60), and continuity conditions in Eqs. (61)–(62) leads to a system of six linear algebraic equations as follows:

(63)\left[\begin{array}{ll}{\mathit{K}}_{tt}^{(\mathrm{1})}(\xi )& {\mathit{K}}_{tb}^{(\mathrm{1})}(\xi )\\ {\mathit{K}}_{bt}^{(\mathrm{1})}(\xi )& {\mathit{K}}_{bb}^{(\mathrm{1})}(\xi )-{\mathit{K}}^{(\mathrm{2})}(\xi )\end{array}\right]\left\{\begin{array}{l}{{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=\mathrm{0}}\\ {{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=h}\end{array}\right\}=\left\{\begin{array}{l}{\tilde{\mathit{P}}}^{\mathrm{0}}\\ \mathrm{0}\end{array}\right\}\mathrm{.}

After the system (63) is solved for {{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=\mathrm{0}} and {{\tilde{\mathit{U}}}^{(\mathrm{1})}|}_{y=h} for any value of the transform parameter \xi, the unknown functions {\mathit{C}}^{(\mathrm{1})} and {\mathit{C}}^{(\mathrm{2})} can be readily obtained from the first system of Eq. (44) and the first system of Eq. (57) together with the continuity conditions in Eq. (61), respectively. When {\mathit{C}}^{(\mathrm{1})} and {\mathit{C}}^{(\mathrm{2})} are determined, the elastic fields within the elastic layer and half plane are obtained via Fourier integral transform inversion in Eq. (29). A numerical quadrature based on the Gauss–Legendre is employed for the numerical evaluation of inversion integrals.

Results and discussion

To verify the solution procedure and resulting elastic fields, a benchmark case corresponding to a layer that is fully bonded to a rigid base and subjected to a concentrated normal force, as reported by Zisis [36], is considered first. When the validity of the proposed solution is confirmed, the elastic response of an elastic layer perfectly bonded to an elastic half plane is investigated in the context of couple stress theory. The aim of the study is two-fold: (i) to examine the effects of the material mismatch of the layer and elastic half plane on the macroscopic response of the layer and (ii) to highlight pertinent size effects when the microstructural length of the couple stress theory is comparable to the geometric lengths of the problem. Thus, two characteristic surface-type loadings are considered:

Case A (uniformly distributed normal traction p_{\mathrm{0}}):

p_{\mathrm{0}}(x)=p_{\mathrm{0}},q_{\mathrm{0}}(x)=\mathrm{0}，\mathrm{f}\mathrm{o}\mathrm{r}\left|x\right|\le a,

Case B (uniformly distributed shear traction q_{\mathrm{0}}):

p_{\mathrm{0}}(x)=\mathrm{0},q_{\mathrm{0}}(x)=q_{\mathrm{0}}，\mathrm{f}\mathrm{o}\mathrm{r}\left|x\right|\le a\mathrm{.}

The two representative surface tractions can be viewed as those resulting from an approximate estimation of the distributed forces transmitted to the surface by an indenter with a contact width corresponding to \mathrm{2}a. This type of an estimation renders the contact problem statically determinate in the sense that the contact forces are known a priori and treated as prescribed surface loadings. The results for the selected stress and couple stress components and equivalent (von Mises) stress {\sigma }_{eq} at the material interface (i.e., at y=h) are reported for the two loading cases. It should be noted that the equivalent stress {\sigma }_{eq} is defined in the couple stress theory as [39]

(64){\sigma }_{eq}={\displaystyle \frac{\sqrt{\mathrm{3}}}{\mathrm{2}}}[\mathrm{2}(s_{xx}^{\mathrm{2}}+s_{yy}^{\mathrm{2}}+s_{zz}^{\mathrm{2}})+{\sigma }_{xy}^{\mathrm{2}}+\mathrm{2}{\sigma }_{xy}{\sigma }_{yx}{+{\sigma }_{yx}^{\mathrm{2}}+\mathrm{2}{\ell }^{-\mathrm{2}}({\mu }_{xz}^{\mathrm{2}}+{\mu }_{yz}^{\mathrm{2}})]}^{\mathrm{1}/\mathrm{2}},

where the deviatoric stresses s_{xx}, s_{yy} , and s_{zz} are defined in the standard way as follows:

(65)\begin{array}{l}{s}_{xx}={\sigma }_{xx}-({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz})/\mathrm{3},\\ s_{yy}={\sigma }_{yy}-({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz})/\mathrm{3},\\ s_{zz}={\sigma }_{zz}-({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz})/\mathrm{3},\end{array}

with {\sigma }_{zz}=\nu ({\sigma }_{xx}+{\sigma }_{yy}) ( [33]). As in the classical theory, the equivalent stress provides a measure for the onset of material yielding and an estimation of the plastic zone size.

In the following, the superscripts “(1)” and “(2)” are used throughout to designate material properties of the elastic layer and the half plane, respectively. Specifically, \{{\mu }^{(\mathrm{1})},{\nu }^{(\mathrm{1})},{\ell }^{(\mathrm{1})}\}and \{{\mu }^{(\mathrm{2})},{\nu }^{(\mathrm{2})},{\ell }^{(\mathrm{2})}\} represent the shear modulus, Poisson’s ratio, and the material length scale of couple stress theory. For all numerical computations in the study, we assume that {\nu }^{(\mathrm{1})}={\nu }^{(\mathrm{2})}=\nu. Furthermore, to facilitate the numerical computations, we introduce the following ratios that measure the mismatch in the material properties of the layer and half-plane:

(66)\gamma ={\displaystyle \frac{{\mu }^{(\mathrm{1})}}{{\mu }^{(\mathrm{2})}}}\mathrm{a}\mathrm{n}\mathrm{d}\rho ={\displaystyle \frac{{\ell }^{(\mathrm{1})}}{{\ell }^{(\mathrm{2})}}}\mathrm{.}

Finally, it should be noted that the formulation shown in Section 3 is general and can be used to solve traction boundary value problems involving more complicated surface-type loadings. The results for an elastic half-plane or an elastic layer on a rigid substrate under normal and tangential concentrated loads [21,36] can be easily deduced from the present approach by the usual limit process [33].

Verification

We consider a single elastic layer fully bonded to a rigid base and subjected to a concentrated normal force P at the surface, as shown in Fig. 2. The problem is examined within the context of couple stress elasticity, and the results obtained are compared with those reported by Zisis [36]. It should be noted that this case can be modeled within the present formulation by considering the shear modulus of the half-plane to be sufficiently higher than that of the elastic layer (i.e., {\mu }^{(\mathrm{2})}/{\mu }^{(\mathrm{1})}\gg \mathrm{1}). The normalized vertical displacement (\mathrm{2}\mu u_y/P) along the surface and normalized force stress and couple stress components (h{\sigma }_{yy}/P, h{\sigma }_{yx}/P, and {\mu }_{yz}/P) at the bottom of the layer are reported as a function of the normalized coordinates in Fig. 3 for {\nu }^{(\mathrm{1})}=\{\mathrm{0},\mathrm{0.49}\} and h/\ell =\mathrm{2}. Evidently, the computed results exhibit excellent agreement with the benchmark solutions. This confirms the validity of the current approach. The results also indicated that Poisson’s ratio plays a significant role in the value and distribution of the elastic field within the layer.

Material mismatch effects

To explore the effect of the material contrast between the layer and the half-plane (substrate) on the predicted elastic fields, two parametric studies were conducted: (i) the shear modulus ratio \gamma was varied with \rho =\mathrm{1} ({\ell }^{(\mathrm{1})}={\ell }^{(\mathrm{2})}=\ell) , and (ii) the ratio of the characteristic material length scales \rho was varied with \gamma =\mathrm{1}.

For case (i), the following parameters \nu =\mathrm{0.3}, h/a=\mathrm{1}, and a/\ell =\mathrm{1} were used in the numerical simulations. It should be noted that in this case, we also obtain h/\ell =\mathrm{1}, which implies that the material microstructure is pronounced in the layer. Figures 4 and 5 illustrate the variation of selected stress and couple stress components at the interface of the layer and half-plane (y=h) for different values of the shear modulus ratio \gamma. Additionally, the results for the equivalent stress {\sigma }_{eq} for both loading cases are shown in Fig. 6. It is observed that the ratio \gamma significantly affects the magnitude and the variation in the elastic fields within the layer for both loading cases. The case \gamma =\mathrm{0} (dashed line) corresponds to an elastic layer bonded on a rigid substrate. When the ratio \gamma departs from zero, the stresses exhibit a significant deviation from those obtained in the case of a rigid substrate. Evidently, this type of discrepancy is still significant when the shear modulus of the half-plane is comparable to that of the layer or the ratio \gamma decreases to that in the practical range of surface coating applications. For example, \gamma =\{\mathrm{0.45},\hspace{0.17em}\hspace{0.17em}\mathrm{0.85}\} corresponds to that of diamond-like carbon coatings of steel substrates and \gamma \approx \mathrm{4.59} corresponds to that of aluminum film coatings of glass and tungsten coatings of glass [40,41]. It should also be noted that \gamma =\mathrm{0}, i.e., rigid substrate solution, does not always constitute an upper or lower bound, which highlights the need to appropriately handle the finite modulus of the substrate below the layer during modeling. Additionally, the information of the shear stress {\sigma }_{yx} along the material interface is crucial in the assessment of interface debonding. Specifically, the debonding failure can be initiated as the maximum shear stress {\sigma }_{yx} that develops as the interface reaches the critical value. For case A, the shear stress attains its maximum near the edges (x\to \pm a) of the uniformly distributed load, p_{\mathrm{0}}, and its maximum value decreases as \gamma increases. In case B, the maximum shear stress is attained directly below the central line of the distributed shear load q_{\mathrm{0}} and again decreases as ratio \gamma increases. An interesting behavior is also exhibited by the equivalent (von Mises) stress, especially in case B, wherein based on the value of the ratio \gamma, {\sigma }_{eq} can assume either its maximum or minimum value at the central line of the distributed loading. In case A, a different trend is observed, i.e., {\sigma }_{eq} always attains its maximum value at the central line of the distributed loading, and its value increases monotonically when the ratio \gamma increases. As the substrate becomes more compliant relative to the elastic layer (\gamma increases), the couple stress component {\mu }_{xz}increases significantly (Fig. 4(c)) and evidently dominates the behavior of the equivalent stress (Fig. 6(a)). The behavior of the force stress and couple stress components for a large ratio \gamma is significantly different from that for the rigid substrate. This observation additionally emphasizes that the elasticity of the substrate should be integrated during modeling.

For case (ii), parameters corresponding to \nu =\mathrm{0.3}, a/h=\mathrm{1}, and a/{\ell }^{(\mathrm{1})}=\mathrm{1} were used in numerical simulations. The results for the selected stress and couple stress components at the interface are reported in Figs. 7–9 for both loading cases. Evidently, the contrast in the material length scale of the elastic layer and half-plane can also significantly affect the response of the layer. Specifically, when {\ell }^{(\mathrm{2})} (half-plane) is either comparable to or exceeds {\ell }^{(\mathrm{1})} (layer), the effect of the microstructure of half-plane material becomes more pronounced and must be considered during modeling. Specifically, as shown in Figs. 7(c), 7(d), 8(c), and 8(d), the couple stress component {\mu }_{xz} of the layer at the interface tends to vanish as \rho approaches zero and increases in magnitude as \rho increases. Conversely, a reverse behavior is observed for the couple stress component {\mu }_{yz}. Additionally, no significant change in the predicted couple stresses is observed when \rho becomes relatively large (i.e., the material length scale of the half plane is significantly less than that of the elastic layer and the size of the loading region). This indicates that classical elasticity can be used during the modeling of the half plane without loss in accuracy. Finally, we note that case \rho =\mathrm{1} corresponds to the homogeneous half-plane case.

Size dependent behavior

The size-dependent behavior of the predicted elastic response due to the presence of couple stresses is examined in further detail for the two loading cases. In the pertinent numerical simulations, the normalized thickness of the layer, shear modulus ratio, ratio of the material length scales, and Poisson’s ratio assume the following fixed values: a/h=\mathrm{1}, \gamma =\mathrm{0.5}, \rho =\mathrm{1}, and \nu =\mathrm{0.3}, while the microstructural ratio a/\ell is varied. The results for the stresses {\sigma }_{yx} and {\sigma }_{yy}, and the couple stress components {\mu }_{xz} and {\mu }_{yz} at the material interface (i.e., at y=h) are shown in Figs. 10 and 11, respectively, while the equivalent stress {\sigma }_{eq} at the interface is shown in Fig. 12. The classical elasticity curves (dashed lines) are also superimposed on the graphs for comparison. Furthermore, this set of results evidently indicates that the stresses predicted by the couple stress theory become size-dependent and deviate from the classical elasticity predictions as the size of the loading region becomes comparable to the material length scale. Conversely, the results converge to classical elasticity as the microstructural ratio a/\ell increases. In the two loading cases, a qualitatively different behavior is observed in Case A (distributed normal traction – Fig. 10) when compared to that in Case B (distributed shear traction, Fig. 11). Specifically, for a distributed normal traction, the stresses at the interface decrease when ratio a/\ell decreases, i.e., the material microstructure becomes more pronounced. This implies that the microstructure shields the layer. Conversely, when a shear traction is applied on the surface of the layer, the shear stress significantly increases with ratio a/\ell increases, thereby implying that the classical theory of elasticity underestimates the magnitude of the true interfacial shear stresses, which can lead to the debonding of the layer. This behavior can be justified by the fact that shear effects are more pronounced when couple stresses are considered. Specifically, couple stress elasticity is based only on the gradient of the rotation and does not include stretch gradients. Hence, couple stress effects are more intense in the (antisymmetric) shear modes of deformation. This behavior has also been corroborated in crack and notch problems within the context of couple stress theory [42,43]. Finally, it should be noted that in the shear loading case (Case B), the location of the maximum equivalent stress shifts from the edges of the distribution (x=\pm a) (classical elasticity) to the center of the load distribution when coupled stress effects are considered, thereby indicating a migration of the location for the onset of plasticity in the layer. Additionally, the couple stress {\mu }_{yz} in Case A (shown in Fig. 10(d)) and vertical stress {\sigma }_{yy} in Case B (shown in Fig. 11(a)) exhibits the same trend as ratio a/\ell is varied. Specifically, their value can change from positive to negative or vice versa as a/\ell increases. This observation contrasts with other reported observations wherein the increase or decrease varies monotonically with respect to ratio a/\ell. Thus, these types of findings additionally confirm the significant role of the couple stress effects in thin layers and films.

To further investigate the size-dependent response of the layer, the same stress and couple stress components are plotted along the line of symmetry (i.e., along x=\mathrm{0}), as shown in Figs. 13 and 14, for the two loading cases. It is observed again that for case A, the normal stresses significantly decreases within the layer with decreasing values of the ratio a/\ell. Conversely, as shown in Figs. 14(a) and 14(b) (case B), there is an aggravation of the shear stress {\sigma }_{yx} in the layer when a/\ell decreases while the shear stress {\sigma }_{xy} decreases significantly. This indicates the strong asymmetry of the stress tensor due to the pronounced couple stress effects. As a/\ell increases, the classical solution is recovered, and the stress tensor becomes symmetric. It should be noted that the recovery of the classical solution near the material interface for certain stress components, such as {\sigma }_{xy} shown in Fig. 14(a), can become very difficult and requires a significantly large value of ratio a/\ell when the material contrast exists. The boundary-layer effect directly results from the enforcement of the continuity of the rotation {\omega }_z along the interface of the elastic layer and half plane and from the fact that {\omega }_z predicted by the classical elasticity is generally discontinuous along the material interface.

Figures 15(a) and 16(a) illustrate the effect of the geometrical ratio a/h on the size-dependent response of the elastic layer due to the presence of couple stresses. The stress components are plotted again as a function of the microstructural length a/\ell albeit now at the middle point of the layer (x=\mathrm{0}, y=h/\mathrm{2}) for all loading cases. It is observed that the solution predicted by the couple stress theory is again strongly size-dependent for a wide range of ratios a/\ell. When a/\ell increases, the predicted solution converges monotonically to the classical case. Additionally, when ratio a/\ell approaches zero, the results obtained from the couple stress theory become finite, thereby evidently indicating size-independent behavior as observed in the classical case. Although the size-dependent characteristic disappears because the size of the loading region is considerably lower than the material length scale, the effect of the couple stresses is still significant (indicated by the discrepancy of the two solutions) and cannot be ignored in the model. It should also be noted that as ratio a/h decreases, the discrepancy between the classical solution and that from couple stress theory increases.

Finally, the effect of the shear modulus mismatch between the layer and elastic half plane on the size-dependent behavior of the solution is shown more evidently by plotting the stresses at a representative point within the layer. Specifically, vertical stress {\sigma }_{yy} (for case A) and shear stress {\sigma }_{yx} (for case B) at the midpoint of the layer (x=\mathrm{0}, y=h/\mathrm{2}) are shown in Figs. 15(b) and 16(b) for a/h=\mathrm{1}, \nu =\mathrm{0.3}, and varying values of shear modulus ratio \gamma. The results indicate that the variation of ratio \gamma does not alter the general characteristics of the size-dependent behavior of the couple stress solution, which follows the same qualitative trend as that in classical elasticity, i.e., the stresses increase with decreasing \gamma values.

Conclusions

In the present work, fundamental solutions for the plane strain problem of an elastic layer perfectly bonded on an elastic half plane were established in the context of the generalized continuum theory of couple stress elasticity. The couple stress elasticity is the simplest gradient type theory that encompasses the analytical possibility of size effects by introducing characteristic length scales that can be related to the material microstructure of the layer and the substrate. An analytical solution procedure based on Fourier integral transforms and direct stiffness approach was adopted to obtain the closed-form general solution in the Fourier transform space. An efficient quadrature was subsequently implemented to carry out all the involved Fourier transform inversions. Numerical results were derived for two characteristic loading cases, and a thorough parametric study was performed to highlight the role of the material mismatch and role of the microstructure of the layer and that of the substrate based on their macroscopic responses. It should be noted that the cases of an elastic half plane [21] and single layer bonded on a rigid-based substrate [36] can be readily recovered from the present solution by adjusting the material constants.

The results indicate that the present mathematical model is evidently more realistic than that of a layer bonded on a rigid substrate. This is due to the fact that the coated surface is generally not rigid and the material contrast relative to the coating surface is significant. Specifically, the results indicate the importance of material mismatch in the mechanical response of the layer and the substrate. This response becomes highly size-dependent when the size of the loading region is comparable to the material length scale of the couple stress theory, thereby revealing the crucial role of the material microstructure in surface coating applications. Nevertheless, as the size of the loading region is sufficiently larger or smaller than the material length scale, the size-dependency characteristics diminish and the predicted solutions converge to their classical elasticity counterparts in the former case. The results of the current model can also provide fundamental information (e.g., von Mises stress and interface shear stress) to assess and predict the onset of material yielding, plastic zone size, and debonding along the material interfaces in surface coating applications.

Finally, it should be noted that the results in the present study can also be applied to construct fundamental solutions that are essential to analyze indentation problems in layered microstructured materials. Additionally, the extension of the current study to treat a multi-layer medium, with possible applications to a multi-layer surface coating, is also of practical interest and will be the subject of a future investigation.

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