Luận án Phân tích dao động dầm, tấm Sandwich 2D-FGM hai và ba pha bằng phương pháp phần tử hữu hạn

1. Luận án phát triển các mô hình phần tử dầm cho phân tích dao động tự do và cưỡng
bức của dầm sandwich 2D-FGM hai pha và ba pha chịu tải trong di động trên cơ
sở sử dụng các lý thuyết biến dạng trượt bậc ba cải tiến và lý thuyết biến dạng trượt
lượng giác. Đặc biệt, phần tử dầm sử dụng lý thuyết biến dạng trượt lượng giác với
các hàm nội suy làm giàu để cải tiến sự hộ tụ của phần tử được phát triển lần đầu
tiên trong luận án.
2. Bên cạnh mô hình Voigt, luận án còn sử dụng mô hình Maxwell để đánh giá các
tính chất hiệu dụng của vật liệu FGM của dầm sandwich 2D-FGM. Ảnh hưởng
của mô hình đồng nhất hóa vật liệu tới các đặc trưng dao động của dầm sandwich
2D-FGM được quan tâm nghiên cứu chi tiết trong luận án.
3. Nhằm khắc phục hiện tượng nghẽn trượt và cải tiến sự hội tụ, luận án sử dụng
phương pháp nội suy hỗn hợp để xây dựng phần tử tấm Q9 dùng trong phân tích
dao động tự do của tấm sandwich 2D-FGM ba pha. Ảnh hưởng của hai mô hình
đồng nhất hóa vật liệu, mô hình Voigt và mô hình Maxwell tới tần số dao động
riêng của tấm sandwich 2D-FGM ba pha được nghiên cứu chi tiết lần đầu tiên
trong luận án. 
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  8. 134 for i=1:nG r=PT2(i); x=le*(1+r)/2; % in x direction x1=(ne-1)*le+x; % x measured from beam left end for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; % in [h0,h1] z2=(h2+h1)/2 + (h2-h1)*t/2; % in [h1,h2] z3=(h3+h2)/2 + (h3-h2)*t/2; %in [h2,h3] % V1, V2 of layers 1,2,3 hs=(1-x1/(2*L))^nx; V21=((z1-h1)/(h0-h1))^nz*hs; V11=1-V21; V22=0; V12=1-V22; V23=((z3-h2)/(h3-h2))^nz*hs; V13=1-V23; % switch Model case 1 %Voigt model % EL1,EL2,EL3: E of layers 1,2,3 EL1=E1*V11+E2*V21; EL2=E1*V12+E2*V22; EL3=E1*V13+E2*V23; % nuL1,nuL2,nuL3: G layers 1,2,3 nuL1=nu1*V11+nu2*V21; nuL2=nu1*V12+nu2*V22; nuL3=nu1*V13+nu2*V23; % roL1,roL2,roL3: ro layers 1,2,3 roL1=ro1*V11+ro2*V21; roL2=ro1*V12+ro2*V22; roL3=ro1*V13+ro2*V23; case 2 % Mori-Tanaka scheme [EL1,nuL1]=Enu_MoriT(V11,VL); [EL2,nuL2]=Enu_MoriT(V12,VL); [EL3,nuL3]=Enu_MoriT(V13,VL); % roL1,roL2,roL3: layers 1,2,3 roL1=ro1*V11+ro2*V21; roL2=ro1*V12+ro2*V22; roL3=ro1*V13+ro2*V23;
  9. 136 le=HH(1);h0=HH(2);h1=HH(3); h2=HH(4);h3=HH(5);b=HH(6); L=HH(9);h=HH(10); E1=VL(1);E2=VL(2);E3=VL(3); ro1=VL(7);ro2=VL(8);ro3=VL(9); [PT2,WT2] = GaussRule(nG); Ke1=zeros(22,22); Ke2=zeros(22,22); Ke3=zeros(22,22); Me1=zeros(22,22); Me2=zeros(22,22); Me3=zeros(22,22); % Determinant of Jacobian matrix det1=0.25*le*(h1-h0); det2=0.25*le*(h2-h1); det3=0.25*le*(h3-h2); for i=1:nG r=PT2(i); x=le*(1+r)/2; % Phep doi can theo phuong x x1=(ne-1)*le+x; %x trong V tinh tu dau trai dam for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; %Doi can in [h0,h1] z2=(h2+h1)/2 + (h2-h1)*t/2; %Doi can in [z1,z2] z3=(h3+h2)/2 + (h3-h2)*t/2; %Doi can in [z2,z3] %EL1,EL2,EL3: E tren lop 1,2,3 [EL1,E12,E13]=pro(z1,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); [E21,EL2,E23]=pro(z2,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); [E31,E32,EL3]=pro(z3,x1,E1,E2,E3,h0,h1,h2,h3,nz,nx,L); % below matrix k [hsE1,hsT1]=hesoKeTe_Voigt(x,z1,VL,HH); [hsE2,hsT2]=hesoKeTe_Voigt(x,z2,VL,HH); [hsE3,hsT3]=hesoKeTe_Voigt(x,z3,VL,HH); Ke1=Ke1 + WT2(i)*WT2(j)*EL1*hsE1*det1;
  10. 138 % below shape functions for Wb H0=1-3*x^2/le^2+2*x^3/le^3; H1=x-2*x^2/le+x^3/le^2; H2=3*x^2/le^2-2*x^3/le^3; H3=-x^2/le+x^3/le^2; H4=sqrt(5/128)*(1-xi^2)^2; H5=sqrt(7/128)*(1-xi^2)^2*xi; H6=1/6*sqrt(9/128)*(1-xi^2)^2*(-7*xi^2+1); H7=1/2*sqrt(11/128)*(1-xi^2)^2*(3*xi^2-1)*xi; Nwb = [0 H0 H1 zeros(1,6) H4 H5 H6 H7 zeros(1,5) H2 H3 zeros(1,2)]; % 22 d.o.f Nws = [zeros(1,3) H0 H1 zeros(1,8) H4 H5 H6 H7 zeros(1,3) H2 H3]; % 22 d.o.f % xi1=2/le; % Derivative of Nu respect to x N0x=-1/le; N1x=1/le; N2x=1/2*sqrt(3/2)*2*xi1*xi; N3x=1/2*sqrt(5/2)*xi1*(3*xi^2-1); N4x=1/8*sqrt(7/2)*xi1*(20*xi^3-12*xi); N5x=1/8*sqrt(9/2)*xi1*(35*xi^4-30*xi^2+3); Nux=[N0x zeros(1,4) N2x N3x N4x N5x zeros(1,8) N1x zeros(1,4)]; % 22 d.o.f % Dao ham bac nhat ham dang Nwb theo x H0x=-6*x/le^2+6*x^2/le^3; H1x=1-4*x/le+3*x^2/le^2; H2x=6*x/le^2-6*x^2/le^3; H3x=-2*x/le+3*x^2/le^2; H4x=-4*sqrt(5/128)*xi1*xi*(1-xi^2); H5x=sqrt(7/128)*xi1*(5*xi^4-6*xi^2+1); H6x=1/6*sqrt(9/128)*xi1*(-42*xi^5+60*xi^3-18*xi);
  11. 140 A2.2. Function tính ma trận k và m tính theo mô hình Maxwell function [k,m]=KeMesandwichbeam_MoriT(HH,VL,nG,nx,nz,ne) % Stiffness and mass matrices using enrichment % Sinusoidal theory with Maxwell m odel % nG - number of Gauss points % HH=[le h0 h1 h2 h3 b A I LT h nel]; % VL=[E1 E2 E3 G1 G2 G3 ro1 ro2 ro3 psi nu1 nu2 nu3]; % ne- element number ne le=HH(1);h0=HH(2);h1=HH(3); h2=HH(4);h3=HH(5);b=HH(6); L=HH(9);h=HH(10); E1=VL(1);E2=VL(2);E3=VL(3); ro1=VL(7);ro2=VL(8);ro3=VL(9); [PT2,WT2] = GaussRule(nG); Ke1=zeros(22,22); Ke2=zeros(22,22); Ke3=zeros(22,22); Me1=zeros(22,22); Me2=zeros(22,22); Me3=zeros(22,22); % Determinant of Jacobian matrix det1=0.25*le*(h1-h0);det2=0.25*le*(h2-h1);det3=0.25*le*(h3-h2); for i=1:nG r=PT2(i); x=le*(1+r)/2; % change integral in x direction x1=(ne-1)*le+x; % x in potential V for j=1:nG t=PT2(j); z1=(h1+h0)/2 + (h1-h0)*t/2; % change integrals limits z2=(h2+h1)/2 + (h2-h1)*t/2; % Change integrals limits z3=(h3+h2)/2 + (h3-h2)*t/2; % Change integrals limits % EL1,EL2,EL3; nuL1,nuL2,nuL3: E and nu of layers 1,2,3 [V11,V21,V31]=V123_layer1_MoriT(z1,x1,h0,h1,h2,h3,nz,nx,L);
  12. 142 Matlab function tính các hệ số "hsE" và "hsT" trong function tính k và m theo mô hình Maxwell function [hsE,hsT]=hesoKeTe_MoriT(x,z,nu,HH) % HH=[le h0 h1 h2 h3 b A I L h] le=HH(1); h=HH(10); xi=2*x/le-1; % change variable % below shape functions for u N0=(le-x)/le; N1=x/le; N2=1/2*sqrt(3/2)*(xi^2-1); N3=1/2*sqrt(5/2)*(xi^2-1)*xi; N4=1/8*sqrt(7/2)*(xi^2-1)*(5*xi^2-1); N5=1/8*sqrt(9/2)*(xi^2-1)*(7*xi^2-3)*xi; Nu=[N0 zeros(1,4) N2 N3 N4 N5 zeros(1,8) N1 zeros(1,4)]; % below shape functions for w H0=1-3*x^2/le^2+2*x^3/le^3; H1=x-2*x^2/le+x^3/le^2; H2=3*x^2/le^2-2*x^3/le^3; H3=-x^2/le+x^3/le^2; H4=sqrt(5/128)*(1-xi^2)^2; H5=sqrt(7/128)*(1-xi^2)^2*xi; H6=1/6*sqrt(9/128)*(1-xi^2)^2*(-7*xi^2+1); H7=1/2*sqrt(11/128)*(1-xi^2)^2*(3*xi^2-1)*xi; Nwb = [0 H0 H1 zeros(1,6) H4 H5 H6 H7 zeros(1,5) H2 H3 zeros(1,2)]; % 22 d.o.f Nws = [zeros(1,3) H0 H1 zeros(1,8) H4 H5 H6 H7 zeros(1,3) H2 H3]; % 22 d.o.f % xi1=2/le; % Derivative of Nu respect to x
  13. 144 H6xx=1/6*sqrt(9/128)*xi1^2*(-210*xi^4+180*xi^2-18); H7xx=1/2*sqrt(11/128)*xi1^2*(126*xi^5-140*xi^3+30*xi); Nwbxx=[0 H0xx H1xx zeros(1,6) H4xx H5xx H6xx H7xx zeros(1,5) H2xx H3xx zeros(1,2)]; % 22 d.o.f % Second order derivative of Nws respect to x Nwsxx=[zeros(1,3) H0xx H1xx zeros(1,8) H4xx H5xx H6xx H7xx zeros(1,3) H2xx H3xx]; % 22 d.o.f % fz=-z+(h/pi)*sin(pi*z/h); gz=-1+cos(pi*z/h); % Sinusoidal Theory hsE=Nux’*Nux+z^2*Nwbxx’*Nwbxx+fz^2*Nwsxx’*Nwsxx -z*Nux’*Nwbxx-z*Nwbxx’*Nux+fz*Nux’*Nwsxx+fz*Nwsxx’*Nux -z*fz*Nwbxx’*Nwsxx-z*fz*Nwsxx’*Nwbxx +(1+gz)^2/(2*(1+nu))*Nwsx’*Nwsx; hsT=Nu’*Nu+z^2*Nwbx’*Nwbx+fz^2*Nwsx’*Nwsx -z*Nu’*Nwbx-z*Nwbx’*Nu+fz*Nu’*Nwsx+fz*Nwsx’*Nu -z*fz*Nwbx’*Nwsx-z*fz*Nwsx’*Nwbx+Nwb’*Nwb+Nws’*Nws +Nwb’*Nws+Nws’*Nwb;