# 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
By substituting (7) into (6), one obtains  
\[\prod=\frac{1}{2}LWA^{2}\Biggl{[}D\biggl{(}\frac{2\pi}{\lambda}\biggr{)}^{4}+ T_{y}\biggl{(}\frac{2\pi}{\lambda}\biggr{)}^{2}\Biggr{]} \tag{8}\]  
To determine the wavelength \(\lambda\), the equation is  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]0  
Moreover, the force in the y-direction is generated by the Poisson effect, yielding \(T_{y}<0\), which could be expressed as \(T_{y}=-vT_{x}=-vF\). If the force in the x-direction is \(F\) and Poisson's ratio of the infinitely long film structure is \(v\), then  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]1  
By setting Equation (10) equal to 0, the wavelength is obtained as  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]2  
The wavenumber can be determined from the ratio of the film structure's width to the wavelength. To validate the analytical results, finite element simulations were conducted. In ABAQUS 2020, a film with a large aspect ratio of 1000:40 was modeled (the length was set to 1000 mm and the width was set to 40 mm), the mesh length was set at 1 mm, element type was set as S4R. Buckle analysis was analyzed firstly, and the keywords \"'imperfection\" were used to set the buckle mode to analyze the wrinkle results. The central region of the film was used to analyze the uniform wrinkle patterns. As shown in Figure 2, three full wrinkle wavelengths appear at the center of the sheets. Figure 2 (left) presents the out-of-plane displacement U, U3 of the tensioned membrane under axial loading, highlighting the wrinkle amplitude across the central region. Figure 2 (right) shows the distribution of the minimum in-plane principal stress (SNEG), which captures the compressive stress concentrations induced by wrinkling. Using the material parameters--a film thickness of 50 \(\upmu\)m and [PERSON]'s modulus of 3 Gpa--as well as the loading boundary, a tensile load of 58 N, Equation (11) predicts a wavelength of 11.89 mm. Since the specimen width is 40 mm, only three complete wrinkle wavelengths can be accommodated across the membrane. This geometric constraint directly fixes the number of wrinkles at \(n=3\). This result matches the simulation results and confirms the validity of the approach (10.96 mm in ABAQUS results). For an infinitely long film, the amplitude A can only be determined by substituting the deflection profile into the energy expression to find its extremum, since in Equation (8), any choice of \(A\) does not affect the derivative's root. Because the tensile load on the thin-film stirp is relatively small, the bending moment induced by the copper foil and the tensile load's effect on the film deformation can be treated independently. So, the step-stiffness structure can be regarded as a pure thin-film under tensile and bending load (ignoring the surface interfacing problem, since the loading leads to a small strain distribution). In other words, the boundary condition of the film can be represented as a superposition of (1) the bending deformation imposed by the copper foil and (2) the original tensile load acting on the film. Summing these two effects yields the wrinkle analysis model for the stiffness-stepped film structure.