Initialization

<<Notation`
Off[Notation`Symbolize::bsymbexs];
Symbolize[(NotationTemplateTag[(Subscript[P, 0])])];
Symbolize[(NotationTemplateTag[(Subscript[(OverHat[P]), 0])])];
Symbolize[(NotationTemplateTag[(OverHat[μ])])];
Symbolize[(NotationTemplateTag[(OverHat[Π])])];
Symbolize[(NotationTemplateTag[(OverHat[s])])];
Symbolize[(NotationTemplateTag[(OverHat[L])])];
Symbolize[(NotationTemplateTag[(Subscript[L, 0])])];
Symbolize[(NotationTemplateTag[(Subscript[σ, f])])];
Symbolize[(NotationTemplateTag[(Sec^2)])];
(Sec^2)[x_]=((Sec[x])^2);
<<HaneeshPackages`
Styles={Red,Blue, Green,Cyan, Magenta,Orange, Black};

ClearAll[ComputeEnergy]
ComputeEnergy[(Subscript[(OverHat[P]), 0]):_,(OverHat[μ]):_,(OverHat[L]):_,ψ_]:=Module[{πval},
πval=(1/2)NIntegrate[(((1+(Subscript[(OverHat[P]), 0])Cos[ψ[s]]))^2)((ψ'[s])^2)+((Subscript[(OverHat[P]), 0])^2)(OverHat[μ])  ((Sin[ψ[s]])^2)-2 (Subscript[(OverHat[P]), 0])(OverHat[μ])(Cos[ψ[s]]-1),{s,0,(OverHat[L])}];
Return[πval]]

(*
(Subscript[L, 0]), (Subscript[P, 0]), ψ'[0]
10,-0.3->1.6;
10,-0.4->2.1;
10,-0.6->3.9;
10,-0.61->4.1;
10,-0.62->4.14;
10,-0.63->4.3;
10,-0.64->4.44;
10,-0.7->5.57;
10,-0.8,8.944;


5,-0.62,4.2;
5,-0.7,5.54;
5,-0.8,8.9;
*)
ℒ= 1;
𝓂=1000;
(Subscript[(OverHat[𝒫]), 0])=-0.5;
ClearAll[sols,ψsol]
Block[{(OverHat[μ])=𝓂,(Subscript[(OverHat[P]), 0])=(Subscript[(OverHat[𝒫]), 0]),(OverHat[L])=ℒ},
sols = Map[First[NDSolve[{D[(1+((Subscript[(OverHat[P]), 0])/(OverHat[μ]))Cos[ψ[s]])ψ'[s],s]==(Subscript[(OverHat[P]), 0])Sin[ψ[s]],ψ[0]==0,ψ'[(OverHat[L])]==0},ψ,s, Method->"BoundaryValues"->{"Shooting", "StartingInitialConditions"->{ψ[0] == 0, ψ'[0] == #}}]]&,
(*{1.414912848726396}*)
Range[1.3,1.414912848726396,0.01]
$Aborted
(*P=-0.5, μ=1000,{1.414912848726396},10, 1.488*)
];
{Plot[Evaluate[ψ[s]/.sols],{s,0,(OverHat[L])},PlotStyle->Styles,PlotRange->All],Plot[Evaluate[ψ'[s]/.sols],{s,0,(OverHat[L])},PlotStyle->Styles,PlotRange->All],Evaluate[ψ'[0]/.sols]
}

]

Deformed shapes

plt={};
For[i=1,i≤ Length[sols],i++,
(*Print[i];*)
Block[{(OverHat[L])=ℒ,usol,wsol, ψsol},
(*ClearAll[ψsol,usol, wsol];*)
ψsol[s_]=ψ[s]/.sols[[i,1]];
usol[s_]=u[s]/.NDSolve[{u'[s]==Cos[ψsol[s]]-1,u[0]==0},u,{s,0,(OverHat[L])}][[1,1]];
wsol[s_]=w[s]/.NDSolve[{w'[s]==Sin[ψsol[s]],w[0]==0},w,{s,0,(OverHat[L])}][[1,1]];
Print[wsol[(OverHat[L])]];
AppendTo[plt,ParametricPlot[{s+usol[s],wsol[s]},{s,0,(OverHat[L])},PlotStyle->Styles[[i]]]]
]
]
Show[plt,PlotRange->All]

Energy

πVec={};
Block[{(OverHat[μ])=𝓂,(Subscript[(OverHat[P]), 0])=(Subscript[(OverHat[𝒫]), 0]),(OverHat[L])=ℒ},
For[i=1,i≤ Length[sols],i++,
(*Print[i];*)
ClearAll[ψsol];
ψsol[s_]=Evaluate[ψ[s]/.sols[[i,1]]];
AppendTo[πVec,ComputeEnergy[(Subscript[(OverHat[P]), 0]),(OverHat[μ]),(OverHat[L]),ψsol ]]
];

]
Echo[πVec,"Energy values:"]

Old code

(*(Subscript[L, 0])=5
ClearAll[sol,ψsol]
Block[{μ=1,(Subscript[P, 0])=-0.20,L=(Subscript[L, 0])},
sol=NDSolve[{D[(1+(Subscript[P, 0])Cos[ψ[s]])ψ'[s],s]==μ (Subscript[P, 0])Sin[ψ[s]],ψ[0]==π/8,ψ'[L]==0},ψ,{s,0,L}];
ψsol[s_]=Evaluate[ψ[s]/.sol[[1]]];
{Plot[{ψsol[s]/π},{s,0,L}],
Plot[{ψsol'[s],0},{s,0,L},PlotRange->All]}
]
u[s_]:=NIntegrate[Cos[ψsol[ξ]]-1,{ξ,0,s}]
w[s_]:=NIntegrate[Sin[ψsol[ξ]],{ξ,0,s}]
*)

Results

Experimental parameters

Block[{ℓ,\[FormalCapitalR], (Subscript[σ, f]),(OverHat[μ]),ν,\[FormalCapitalE],\[FormalCapitalG],\[FormalCapitalI],\[FormalCapitalA],(Subscript[(OverHat[P]), 0])},
ℓ=Quantity[.5,"milli meters"];
\[FormalCapitalR]=Quantity[25.5,"micro meters"];
(Subscript[σ, f])=Quantity[1.4,"GPa"];
\[FormalCapitalE]=Quantity[28,"GPa"];
ν=0.3;
\[FormalCapitalG]=(\[FormalCapitalE]/(2(1+ν)));
\[FormalCapitalA]=π (\[FormalCapitalR]^2);
\[FormalCapitalI]=((π (\[FormalCapitalR]^4))/4);
(OverHat[μ])=((\[FormalCapitalG] \[FormalCapitalA] (ℓ^2))/(\[FormalCapitalE] \[FormalCapitalI]))]