1 Introduction

The present work is inspired by the fact that thin films bonded to a gradient substrate form a local buckle. Hence, the surface patterns can be tuned by changing the properties of gradient substrate. As we know, the wrinkle patterns of thin films on compliant substrates have been widely observed in nature, such as plant leaves and skins^[1-3]. More importantly, the wrinkles are useful in a wide range of applications, including stretchable electronics, bioengineering, and micro-nanofabrication^[4-5].

Consider thin films bonded to homogenous substrates. The films form various global wrinkle patterns such as stripes, chessboards, herringbones, and labyrinths^[6]. The herringbone pattern has proved to have the minimum energy among the several patterns^[7-10]. Importantly, the wrinkle patterns are uniform, and the critical membrane force, wavelength, and amplitude are constants.

More novel patterns can be realized by changing sizes or material properties in the film or substrate. By considering gradient films, the polydimethylsiloxane (PDMS) gradient embedded into a matrix of PDMS-hard resulted in a continuously changing wrinkle wavelength^[11], and the changing thickness in film showed non-uniform wrinkle patterns^[12]. Using a functionally graded material (FGM) thin film or changing thickness of film results in localization of wrinkles^[13-14]. In the above studies, the amplitude and wavelength are non-uniform. Yu et al.^[15] tuned the in-plane temperature gradient during the curing process to obtain the elasticity-gradient PDMS samples. The film deposited on elasticity-gradient PDMS formed the continuously changing wrinkle patterns. However, the mechanism of thin films bonded to gradient substrates remains uncertain.

In this paper, the substrate is modeled as a Winkler foundation. The critical membrane force, wavelength, and normalized amplitude are obtained using the Galerkin truncation method. The effects of material gradients on the wrinkle patterns are discussed. This work can promote a better understanding of controlling surface patterns by tuning the substrate property.

2 Governing equation

Consider a thin film bonded to a gradient elastic substrate, as shown in Fig. 1, where h and H represent the thicknesses of the film and substrate, respectively. The wrinkled film remains bonded to the substrate. The substrate is modeled as the Winkler model^{[9, 16]}. As we know, the Winkler model represents the substrate as a set of parallel springs, and the individual spring is in a state of uniaxial strain with the stiffness K. The shear stress at the film-substrate interface is assumed to be zero^[9]. Hence, the force equilibrium for uniaxial deformation of film requires

(1)

where W is the deflection. D and N refer to the bending stiffness and the membrane force of film, respectively. By introducing the dimensionless variable x=X /L and the normalized deflection w=W /h, the equilibrium equation in the dimensionless form is

(2)

where L represents the length of film. K(x)=K(X)L⁴/D is defined as the wrinkle stiffness, and N=NL²/ D represents the dimensionless membrane force.

3 Galerkin discretization

The governing equation is a forth-order ordinary differential equation. The analytical solution is difficult to obtain even for simple function of wrinkle stiffness. Hence, the Galerkin truncation method is used to discretize the film-substrate system^[17]. The n terms of the series expansion form are considered in order to determine the deflection as

(3)

where are trial functions, and c_k are unknown coefficients. Multiplying Eq. (2) by the weight functions and integrating it over the interval of 0 and 1, the equilibrium equation is written as

(4)

Applying the Galerkin truncation, an algebraic system of equations can be expressed as

(5)

where A and B are n×n matrices. The eigenvalues correspond to the critical membrane force, and the eigenvectors correspond to the patterns of wrinkle. In the present investigation, a simply supported boundary condition is considered. The trial function is chosen as

(6)

4 Results and discussion 4.1 Homogenous model

In this part, the substrate is assumed to be homogenous. In other words, the winkle stiffness K is a constant. Hence, the critical membrane force N can be obtained analytically by minimizing the energy as^[9]

(7)

In contrast to the analytical solution, the results of a few different numbers of Galerkin basis functions, namely, 10-term Galerkin truncation, 20-term Galerkin truncation, and 30-term Galerkin truncation, are shown in Fig. 2(a). The comparisons indicate that the 10-term Galerkin truncation is not accurate enough for the wrinkle analysis. To make the investigation reliable, according to Fig. 2(b), the 30-term Galerkin truncation is used in the next investigation.

4.2 Gradient model

In contrast to commonly used homogenous substrates, the wrinkle patterns of film bonded to gradient substrate are investigated. For simplicity, let the reference wrinkle stiffness be constant K₀ =10⁷. The wrinkle patterns of the exponential model, the power-law model, and the symmetric model are investigated. The wrinkle stiffness is defined by the exponential function as

(8)

the power-law function as

(9)

and the symmetric function as

(10)

where α, β, and γ are the material gradients. To illustrate the effects of material gradient on wrinkle patterns, the wrinkle stiffness and normalized amplitude with various material gradients are plotted in Fig. 3.

For a gradient substrate, the wrinkle patterns from global to local can be observed with the increasing material gradient. More importantly, the wrinkle patterns strongly depend on the wrinkle stiffness. As shown in Fig. 3(a), the wrinkle stiffness decreases steadily with the increasing distance. As α increases, the wrinkle stiffness decays quickly, and the area of wrinkle region is smaller. The transition of wrinkle patterns from global to local can be realized. As shown in Fig. 3(c), as β increases, the wrinkle stiffness increases quickly with the increasing distance, and the area of wrinkle region is also smaller. Hence, it can be concluded that localization of wrinkle patterns strongly depends on the material gradient. The effects of material gradient on wrinkle patterns based on the symmetric model are investigated in Fig. 3(e). For the symmetric wrinkle stiffness, it can be obtained that the wrinkle patterns are perfectly symmetric.

The effects of the material gradient on the critical membrane force with various gradient models are plotted in Fig. 4. For the exponential model, the critical membrane force decreases steadily with the increasing material gradient. As we know, according to the homogenous model, N ∝ K^1/2, the critical membrane force can be calculated by the minimum value of wrinkle stiffness. In contrast to the results of exponential model, the error is attributed to the gradient of wrinkle stiffness.

For the power-law and symmetric models, the minimum values of wrinkle stiffness in both models are zero. However, when the wrinkle stiffness is at its minimum, the gradient of wrinkle stiffness is uniform for the symmetric model and non-uniform for the power-law model. Hence, the non-uniform critical membrane force in the power-law model is attributed to the gradient of wrinkle stiffness. The critical membrane force depends on both the minimum values of wrinkle stiffness and the gradient of wrinkle stiffness when the wrinkle stiffness is at its minimum.

4.3 Numerical simulation

In contrast with the results of Galerkin method, the numerical simulation based on the finite element method (FEM) is investigated. In the simulation, we keep h=0.2 mm, H=1 mm, and L=25 mm fixed. Young's modulus of film is E_f =80 GPa. Poisson's ratios of film and substrate are 0.3. Three gradient models including the exponential model, the power-law model, and the symmetry model are considered. The film is discretized by 4-node general purpose shell elements (S4R) with reduced integration and accounting for large rotation. The gradient elastic substrate is meshed by hexahedron solid continuum elements (C3D8R)^{[7, 13]}. The results are consistent with the Galerkin method, as shown in Fig. 5.

5 Conclusions

In this paper, the Galerkin truncation method is used to analyze the wrinkling behavior of the film-gradient substrate system. In contrast to commonly used homogenous substrates, three types of gradient substrates, including the exponential, power-law, and symmetry models, are considered. The wrinkle pattern is non-uniform. The evolution of wrinkle patterns from global to local can be observed with the increasing material gradient. The wavelength and the normalized amplitude of the wrinkles for substrates of various material gradients are obtained. It is concluded that localization of wrinkle patterns strongly depends on the material gradient. In addition, the numerical simulation based on the FEM is also used to evolve the wrinkle patterns. The critical membrane force depends on both the minimum values of wrinkle stiffness and the gradient of wrinkle stiffness when the wrinkle stiffness is at its minimum.