论文:Theoretical and Experimental Studies of Bending of Inorganic Electronic Materials on Plastic Substrates
1. 使用材料及结构:Si带,胶层(环氧树脂)、PET基板
2. 弯曲测试:测量原长L、横跨距离L-dL,外部力F
3. 计算理论:
弯曲形状的描述:
其中,dL/L为施加的应变,h为基底的厚度,w为z方向的基底偏角,w0是中心处的w
弯曲半径:
相应的应变为
4.三种失效形式:断裂、滑移、分离
5.机理模型
(1)基于梁理论
轴向压缩应力为
其中,
Es是基底杨氏模量,vs是基底泊松比
弯曲力矩
最大剪应力
其中Ga是粘附层剪切模量
最大剥离应力(薄膜/粘附界面)
其中,
断裂应力:
界面滑移发生在
界面分裂发生在
(2)基于有限元理论
当柔电薄膜宽度相比长度无法被忽略时,
MATLAB代码
clc,clear; % set the parameters Ef=130; vf=0.27; hf=1; Es=4.2; vs=0.44; hs=50; Ea=4.2; va=0.44; ha=1; theta=108; L=78000; tc=0.02; A2_u=-0.02; % set the limit of l l=0:0.01:0.11; % the value of E_(i=f,s,a) E_f=Ef/(1-vf^2); E_s=Es/(1-vs^2); E_a=Ea/(1-va^2); %the value of ki(i=1`7) Ga=E_a*(1-va)/2; k1=4*Ga*ha*(1/(E_f*hf)+1/(E_s*hs)); k2=12*E_a*ha^3*(1/(E_f*hf^3)+1/(E_s*hs^3)); lamda1=sqrt(k1+2*sqrt(k2))/2; lamda2=sqrt(-1*k1+2*sqrt(k2))/2; beta=6*(1/(E_f*hf^2)-1/(E_s*hs^2))/(ha*12*(1/(E_f*hf^3)+1/(E_s^hs^3))+lamda1^4/(E_a*ha^3)); X=(3*E_a*ha^3*(1/(E_f*hf^3)+1/(E_s*hs^3)))^(1/4); p=sin(theta/360*pi); kp=ellipticK(p); % calculation f1=exp(X*l).*(sin(X*l)-cos(X*l)); f2=exp(X*l).*(sin(X*l)+cos(X*l)); k4=(1-lamda1^2/(2*X^2)-2*X/lamda1)*beta; k5=(1/(E_f*hf)+1/(E_s*hs)*tc); k6=1.5*(1/(E_f*hf^2)-1/(E_s*hs^2)); zz=(exp(X*l).*(1-tan(X*l))); z11=(X^2*L*f1.*zz).*(tan(X*l).*f1+f2); z12=ones(12)/z11; z1=4*E_a*ha*p*kp*z12; k72=ones(12)/(X^2*L*exp(X*l).*cos(X*l)); k71= k6*ha^2*E_a*p*kp/k72; k7=z1*k6*ha/(2*X) +k71 -k5+ (lamda1^2+lamda2^2)*A2_u/(4*Ga*ha); k8=lamda1/(2*Ga*ha) + k4*beta*k6*ha* (1-tan(X*l))/(2*X*(tan(X*l).*f1+f2))-k4*beta; A_u=k7/k8; Cs1=k4*beta*A_u-4*E_a*ha*p*kp*f1/(X^2*L*exp(X*l).*cos(X*l)); Cs=Cs1/(tan(X*l).*f1+f2); Bs1=X^2*L*exp(X*l).*cos(X*l); Bs2=Cs*tan(X*l); Bss=ones(12)/Bs1; Bs=4*E_a*ha*p*kp*Bss+Bs2; dQ1=(l.*f1/(2*X)+(f2-f1-2)/(4*X)); dQ2=(l.*f2/(2*X)-(f2+f1)/(4*X)); dQ=dQ1*Bs+dQ2*Cs*ha^2; %plot ls=l*ha; F=A_u/(4*Ga)-(1/(E_f*hf)+1/(E_s*hs))*tc*ls+hs*p*kp/L-1.5*(1/(E_f*hf^2)-1/(E_s*hs^2))*dQ; plot(l,F)
学习笔记(一)| 塑料基板上的无机电子材料的弯曲理论实验研究
原文:https://www.cnblogs.com/Sonny-xby/p/12849072.html