dx[u_]:=Cos[t]*D[u,r]-Sin[t]*D[u,t]/r
dy[u_]:=Sin[t]*D[u,r]+Cos[t]*D[u,t]/r
dxx[u_]:=dx[dx[u]]
dyy[u_]:=dy[dy[u]]
dxy[u_]:=dx[dy[u]]
dyx[u_]:=dy[dx[u]]
lap[u_]:=dxx[u]+dyy[u]
hes2[u_]:=dxx[u]^2+dyy[u]^2+2*dxy[u]^2

ne={-Cos[t],-Sin[t]}

dne[u_]:=dx[u]*ne[[1]]+dy[u]*ne[[2]]

Non commutatività di derivata cartesiana e derivata normale alla circonferenza

D[dx[t],t]

dx[D[t,t]]

D[dy[t],t]

v[r*Cos[t],r*Sin[t]]

S_ε prima riga

Se1[u_]:=-D[dxx[u]*ne[[1]]*dx[t]+dyy[u]*ne[[2]]*dy[t]+ dxy[u]*(ne[[1]]*dy[t]+ne[[2]]*dx[t]),t]

S_ε seconda riga

Se2[u_]:=-D[dxx[u]*Cos[t]*Sin[t]-dyy[u]*Sin[t]*Cos[t]- dxy[u]*Cos[2*t],t]/r

se1=Simplify[Se1[v[r*Cos[t],r*Sin[t]]]]

se2=Simplify[Se2[v[r*Cos[t],r*Sin[t]]]]

Simplify[se1-se2]

S_ε terza riga modificata

Se3[u_]:=dne[dne[dne[u]]]-dne[lap[u]]-(dxx[u]*dx[ne[[1]]]+dxy[u]*dx[ne[[2]]]+dxy[u]*dy[ne[[1]]]+dyy[u]*dy[ne[[2]]])-(D[dx[t],t]*(dxx[u]*ne[[1]]+dxy[u]*ne[[2]])+D[dy[t],t]*(dxy[u]*ne[[1]]+dyy[u]*ne[[2]]))

se3=Simplify[Se3[v[r*Cos[t],r*Sin[t]]]]

Simplify[se1-se3]

dne[ne]

Candidato W

u=Sqrt[α/(193*Pi)]*r^(3/2)*((Cos[t/2]-(7/3)*Cos[3*t/2])+Sqrt[21]*(Sin[t/2]-(5/3)*Sin[3*t/2]))

Candidato Φ

v=Sqrt[α/(193*Pi)]*r^(3/2)*(-(Cos[t/2]-(7/3)*Cos[3*t/2])+Sqrt[21]*(Sin[t/2]-(5/3)*Sin[3*t/2]))

Simplify[lap[lap[u]]]

Equipartizione dell'energia

Integrate[Integrate[Simplify[hes2[u]]*r,{r,0,R}],{t,-Pi,Pi}]

Condizione sul crack tip

Integrate[R*Simplify[hes2[u]*Cos[t]+2*	dx[u]*( D[dxx[u]*Cos[t]*dx[t]+dyy[u]*Sin[t]*dy[t]+ dxy[u]*(Cos[t]*dy[t]+Sin[t]*dx[t]),t]-dne[lap[u]])],{t,-Pi,Pi}]/.r->R

Condizione sul crack tip

Integrate[R*Simplify[hes2[u]*Cos[t]+2*dx[u]*( Se3[u]-dne[lap[u]])],{t,-Pi,Pi}]/.r->R

Condizione sul crack tip  con una funzione test ζ

Integrate[R*Simplify[hes2[u]*Cos[t]*ζ[r*Cos[t],r*Sin[t]]+2*dx[u]*( Se3[u]-dne[lap[u]])*ζ[r*Cos[t],r*Sin[t]]],{t,-Pi,Pi}]/.r->R

Limit[%,R->0]

k1=Integrate[R*Simplify[hes2[u]*Cos[t]],{t,-Pi,Pi}]/.r->R

k2=Integrate[R*Simplify[2*	dx[u]*( D[dxx[u]*Cos[t]*dx[t]+dyy[u]*Sin[t]*dy[t]+ dxy[u]*(Cos[t]*dy[t]+Sin[t]*dx[t]),t]-dne[lap[u]])],{t,-Pi,Pi}]/.r->R

dy[dy[u]]/.t->π

dy[dy[u]]/.t->-π

dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->π

dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->-π

W=u/.α->1

Φ=v/.α->1

ParametricPlot3D[{r*Cos[t],r*Sin[t],W,Hue[W]},{r,0,1},{t,-π,π},ViewPoint->{-0.683,- 1.036, 0.870}]

ParametricPlot3D[{r*Cos[t],r*Sin[t],Φ,Hue[Φ]},{r,0,1},{t,-π,π},ViewPoint->{-0.683,- 1.036, 0.870}]

v=r^p*ψ[t]

Simplify[lap[lap[v]]]

Condizione sul crack per le omogenee

ψ[t_]:=d1+d2*t+d3*Cos[2*t]+d4*Sin[2*t]

(* p=2 *)

e1=ψ''[t]+2*ψ[t]/.t->Pi

e2=ψ''[t]+2*ψ[t]/.t->-Pi

e3=ψ'''[t]+4*ψ'[t]/.t->Pi

e4=ψ'''[t]+4*ψ'[t]/.t->-Pi

Solve[{e1==0,e2==0,e3==0,e4==0},{d1,d2,d3,d4}]

(* p=0 *)

e1=ψ''[t]/.t->Pi

e2=ψ''[t]/.t->-Pi

e3=ψ'''[t]+2*ψ'[t]/.t->Pi

e4=ψ'''[t]+2*ψ'[t]/.t->-Pi

Solve[{e1==0,e2==0,e3==0,e4==0},{d1,d2,d3,d4}]

(* p=1 *)

ψ[t_]:=d1*Cos[t]+d2*Sin[t]+d3*t*Cos[t]+d4*t*Sin[t]

e1=ψ''[t]+ψ[t]/.t->Pi

e2=ψ''[t]+ψ[t]/.t->-Pi

e3=ψ'''[t]+ψ'[t]/.t->Pi

e4=ψ'''[t]+ψ'[t]/.t->-Pi

Solve[{e1==0,e2==0,e3==0,e4==0},{d1,d2,d3,d4}]

Condizione sul crack per le omogenee p =3/2

ψ[t_]:=c1*Cos[3/2*t]+c2*Sin[3/2*t]+c3*Cos[t/2]+c4*Sin[t/2]

u=r^(3/2)*ψ[t]

e1=dy[dy[u]]/.t->π

e2=dy[dy[u]]/.t->-π

e3=dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->π

e4=dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->-π

Solve[{e1==0,e2==0,e3==0,e4==0},{c1,c2,c3,c4}]

Condizione sul crack per le omogenee p generico semiintero

ψ[t_]:=c1*Cos[(h+3/2)*t]+c2*Sin[(h+3/2)*t]+c3*Cos[(h-1/2)*t]+c4*Sin[(h-1/2)*t]

u=r^(h+3/2)*ψ[t]

e1=Simplify[dy[dy[u]]/.t->π,h∈Integers]

e2=Simplify[dy[dy[u]]/.t->-π,h∈Integers]

e3=Simplify[dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->π,h∈Integers]

e4=Simplify[dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->-π,h∈Integers]

Solve[{e1==0,e2==0,e3==0,e4==0},{c1,c2,c3,c4}]

Condizione sul crack per le omogenee p generico reale

u=r^p*(c1*Cos[p*t]+c2*Sin[p*t]+c3*Cos[(p-2)*t]+c4*Sin[(p-2)*t])

e1=Simplify[dy[dy[u]]/.t->π]

e2=Simplify[dy[dy[u]]/.t->-π]

e3=Simplify[dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->π]

e4=Simplify[dy[dy[dy[u]]]+2*dx[dx[dy[u]]]/.t->-π]

MatrixForm[{{-(p-1)*p*Cos[p*π]*r^(p-2),-(p-1)*p*Sin[p*π]*r^(p-2),-(p-1)*(p-4)*Cos[p*π]*r^(p-2),-(p-1)*(p-4)*Sin[p*π]*r^(p-2)},{-(p-1)*p*Cos[p*π]*r^(p-2),(p-1)*p*Sin[p*π]*r^(p-2),-(p-1)*(p-4)*Cos[p*π]*r^(p-2),(p-1)*(p-4)*Sin[p*π]*r^(p-2)},{(p-1)*(p-2)*p*Sin[p*π]*r^(p-3),-(p-1)*(p-2)*p*Cos[p*π]*r^(p-3),(p-1)*(p-2)*(p+2)*Sin[p*π]*r^(p-3),-(p-1)*(p-2)*(p+2)*Cos[p*π]*r^(p-3)},{-(p-1)*(p-2)*p*Sin[p*π]*r^(p-3),-(p-1)*(p-2)*p*Cos[p*π]*r^(p-3),-(p-1)*(p-2)*(p+2)*Sin[p*π]*r^(p-3),-(p-1)*(p-2)*(p+2)*Cos[p*π]*r^(p-3)}}]

Simplify[Det[%]]