# Predicting Electromagnetic Performance Under Wrinkling in Thin-Film Phased Arrays
## 2 Wrinkling of Step-Stiffness Structure and Dual-Domain Displacement Mod  
### Wrinkling Mechanism of Step-Stiffness Structure  
Several methods are widely used to analyze wrinkling in uniform thin films, including stress analysis, initial imperfection buckling simulations, and energy competition mechanisms. In this section, the wrinkling deformation of stiffness-stepped film structures is analyzed based on the energy competition mechanism [12]. To analyze wrinkling deformation induced by buckling, the film should not be treated as a membrane with zero bending stiffness, but rather as a thin plate considering bending stiffness, the film as a thin-plate (its thickness being much smaller than the other dimensions), and satisfying the Kirchhoff-Love hypotheses and the small-deflection assumption. The analysis of a single wrinkle stripe is performed using an energy-based formulation, calculating the energy associated with stretching along the wrinkle direction and bending along the wrinkle wavelength direction. Considering wrinkle patterns in Figure 1.  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]In the equation, \(U_{b}\) denotes the bending energy of the film, \(w\) denotes the out-of-plane deformation (which can often be expressed as a polynomial or trigonometric function; for the wrinkle problem, \(w\) is also a function of the wrinkle wavelength \(\lambda\) and amplitude \(A\), and the function mode of \(w\) could be \(w(x,y,\lambda,A)\)), \(D\) denotes the stiffness matrix \(D=\frac{E^{3}}{12(1-\
u^{2})}\), \(l,\lambda\) represent the length and width of the wrinkle strip, \(v\) is [PERSON]'s ratio, \(E\) is [PERSON]'s modulus, and t is the structure thickness. \[U_{s}=\int\limits_{\Omega}\sigma_{x}\varepsilon_{x}+\sigma_{y}\varepsilon_{y}d \Omega=\frac{1}{2}\int\limits_{\Omega}T_{x}\left(\frac{\partial w}{\partial x} \right)^{2}+T_{y}\left(\frac{\partial w}{\partial y}\right)^{2}d\Omega \tag{2}\]  
In the equation, \(U_{s}\) represents the bending energy of the film, \(T_{x}\) is the tensile force in the x-direction, and \(T_{y}\) is the tensile force in the y-direction. Based on Equations (1) and (2), one obtains  
\[\Pi=U_{s}+U_{b} \tag{3}\]  
In the equation, \(\Pi\) denotes the total potential energy of the film. Based on the energy method, to solve for the unknowns in the above expression, one takes its partial derivatives and sets them equal to zero. For example, let \(\frac{\partial\Pi}{\partial\lambda}=0\). Because the tensile load in an infinitely long film structure is uniformly distributed, it can be assumed that there is no bending strain along the long-edge direction, then the out-of-plane deformation trial function can be assumed as follows:  
\[w(x,y,A,\lambda)=A\sin(\frac{2\pi y}{\lambda}) \tag{4}\]  
Assume an infinitely long film structure whose wrinkling can be considered as strips of wavelength \(\lambda\) uniformly distributed along the short-edge direction. The corresponding energy can then be calculated using the following formula:  
\[U_{b}=\frac{1}{2}\!\!\int\limits_{\Omega}D\!\left(\frac{\partial^{2}w}{ \partial y^{2}}\right)^{2}d\Omega=\int\limits_{0}^{L}\int\limits_{0}^{W} \frac{1}{2}D\!\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^{2}\!dxdy \tag{5}\]  
\[U_{s}=\frac{1}{2}\!\!\int\limits_{\Omega}T_{y}\!\left(\frac{\partial w}{ \partial y}\right)^{2}\!d\Omega \tag{6}\]  
Figure 1: Definition of the wrinkling coordinate system. Therefore, the total potential energy can be expressed as  
\[\prod=\int\limits_{0}^{L}\int\limits_{0}^{W}\Biggl{[}\frac{1}{2}D\biggl{(}\frac{ \partial^{2}w}{\partial y^{2}}\biggr{)}^{2}+T_{y}\biggl{(}\frac{\partial w}{ \partial y}\biggr{)}^{2}\Biggr{]}dydx \tag{7}\]  
In this expression, \(A\) denotes the wrinkle amplitude, and \(\lambda\) denotes the wavelength, which can also be written as \(\lambda=\frac{W}{n}\).