# 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
The generation of the bending moment can then be derived from the copper foil's structural parameters, and the expression is as follows:  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]3  
In the equation, \(U_{bc}\) represents the bending strain energy introduced by the copper foil, and \(M_{i},\kappa_{i}(i=x,y,xy)\) denote the bending moments and curvatures in the respective directions. The curvature can be obtained from the thin-plate deflection formula in elasticity theory, i.e., by taking the second derivative of the assumed deflection function. In summary, the energy-based wrinkling analysis model for the stiffness-stepped film structure can be expressed as follows:  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]4  
In the above equation, \(w_{c}(x,y)\) denotes the out-of-plane deflection distribution function arising from bending, which does not contain the influence of wrinkling (\(\lambda\) and \(A\)). For the stiffness-stepped film, the copper foil is only effectively subjected to bending moments (under small loading), while the four edges carry asymmetric in-plane loads, so the bending moment magnitudes at each edge differ. This spatial variation in moment makes it challenging to obtain the numerical solution of the model. For such complex cases, the finite element method is recommended for analysis. \[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]5 Uniform wrinkling analysis of an infinitely long thin-film structure (**Left**: out-of-plane displacement U, U3. **Right**: minimum in-plane principal stress (SNEG)). For an infinitely long film, assuming that the bending moment \(M_{x}\) induces curvature deformations \(\kappa_{xb}\), \(\kappa_{yb}\), the new strain energy can be expressed as  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]6  
By taking the partial derivatives, one obtains  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]7  
By setting Equation (15) equal to zero, the wrinkle parameter for the wrinkled region of an infinitely long film under the influence of a bending moment is given by  
\[U_{b}=\frac{1}{2}\int\limits_{\Omega}D\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+\left(\frac{\partial^{2}w}{\partial y^{2}}\right)^ {2}+2v\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2} }\right]d\Omega \tag{1}\]8  
In the above expression, \(\kappa_{yb}\) denotes the curvature in the y-direction obtained from the independent bending-coupled strain calculation. The amplitude \(A\) is determined by substituting the deflection function \(w\) into the potential-energy expression for an infinitely long film without bending moments and finding its extremum. Analysis of Equation (16) shows that as the denominator decreases, the wavelength \(\lambda\) increases; when the denominator reaches zero, wrinkle deformation disappears, and the structural response degenerates to a pure bending, biaxially coupled deformation mode without wrinkling.