Santosh RajuME5312

HW#61001030383

1.Comparing values for epsilon[1,1] for both cases, epsillon for case Awill be positive considering E11 > E22 for wood.

Answer: Option (A) is best, assuming E11>E22

Code:s=Table[0,{i,2},{j,2},{k,2},{l,2}];s[[1,1,1,1]]=1/E11;s[[2,2,2,2]]=1/E22;s[[1,1,2,2]]=s[[2,2,1,1]]=-v12/E22;s[[1,2,1,2]]=s[[1,2,2,1]]=s[[2,1,2,1]]=s[[2,1,1,2]]=1/4/G12;beta={{Cos[theta],Sin[theta]},{-Sin[theta],Cos[theta]}};snew=Table[Sum[beta[[i,ip]] beta[[j,jp]] beta[[k,kp]]beta[[l,lp]]s[[ip,jp,kp,lp]],{ip,2},{jp,2},{kp,2},{lp,2}],{i,2},{j,2},{k,2},{l,2}];sigma={{0,-tau},{-tau,0}};epsilon=Table[Sum[snew[[i,j,k,l]]sigma[[k,l]],{k,2},{l,2}],{i,2},{j,2}];epsilon/.theta-Pi/4//Simplify(*theta for case A is -Pi/4*){{((E11-E22) tau)/(2 E11 E22),-((tau (E11+E22+2 E11 v12))/(2E11 E22))},{-((tau (E11+E22+2 E11 v12))/(2 E11 E22)),((E11-E22)tau)/(2 E11 E22)}}epsilon/.thetaPi/4//Simplify(*theta for case B is -Pi/4*){{((-E11+E22) tau)/(2 E11 E22),-((tau (E11+E22+2 E11 v12))/(2 E11E22))},{-((tau (E11+E22+2 E11 v12))/(2 E11 E22)),((-E11+E22) tau)/(2 E11E22)}}

3.ClearAll[u1]; ClearAll[u2]; ClearAll[u3]x={r,theta,phi};R={r Sin[theta] Cos[phi],r Sin[theta]Sin[phi],r Cos[theta]};e=Table[D[R,x[[i]]],{i,3}];g=Table[e[[i]].e[[j]]//Simplify,{i,3},{j,3}];Print["g(",i,",",j,")=",MatrixForm[g]]ginv=Inverse[g];gamma=Table[Sum[1/2 ginv[[i,l]](D[g[[l,j]],x[[k]]]+D[g[[l,k]],x[[j]]]-D[g[[j,k]],x[[l]]]),{l,3}],{i,3},{j,3},{k,3}];Do[If[gamma[[i,j,k]]=!=0,Print["Gamma(",i,j,k,")=",gamma[[i,j,k]]]],{i,3},{j,3},{k,3}]U={u1,u2,u3};Sum[Dt[U[[i]],x[[i]]],{i,3}]+Sum[gamma[[i,i,l]]U[[l]],{l,1,3},{i,1,3}]g( i , j )= ( {{1, 0, 0},{0, r2, 0},{0, 0, r2 Sin[theta]2}} )Gamma( 1 2 2 )= -rGamma( 1 3 3 )= -r Sin[theta]2Gamma( 2 1 2 )= 1/rGamma( 2 2 1 )= 1/rGamma( 2 3 3 )= -Cos[theta] Sin[theta]Gamma( 3 1 3 )= 1/rGamma( 3 2 3 )= Cot[theta]Gamma( 3 3 1 )= 1/rGamma( 3 3 2 )= Cot[theta](2 u1)/r+u2 Cot[theta]+Dt[u1,r]+Dt[u2,theta]+Dt[u3,phi](*Physical components*)u1p=ur Sqrt[g[[1,1]]]u2p= utheta Sqrt[g[[2,2]]]//Simplifyu3p=uphi Sqrt[g[[3,3]]]//Simplifyurr2 utheta

uphi r2Sintheta2U=2 Ur/r + Utheta Cot(theta) + r Sin(theta) D(Uphi)/D(phi)+ rD(Utheta)/D(theta)+D(Ur)/D(r)........Solution