In[]:=
Clear["Global`*"]​​(*ExactAnalyticalusingAiry(Real)StressPotentialFunctions)*)​​(*Case-1:End-LoadedCantileverin2DPlane-Stressb<<a*)​​endLoad=
P
end
;​​height=2*a;​​width=b;​​length=L;​​σ13=0;​​σ23=0;​​σ33=0;​​σ31=0;​​σ32=0;​​(*MaterialConstantsApplicabletoLinearElastic,Isotropic*)​​poissonR=ν;​​modulusE=;​​λ=(ν*)/((1+ν)*(1-2*ν));​​μ=/(2*(1+ν));​​​​(*BodyForce*)​​Ω=0;​​(*Airy'sFunction*)​​ϕ=phi;​​phi=-(3*endLoad*x1*x2/(4*(height/2)*width))+(endLoad*x1*(x2^3)/(4*((height/2)^3)*width))​​​​(*StessesDefinedFromAiry'sFunction*)​​σ11=D[ϕ,{x2,2}]-Ω;​​σ22=D[ϕ,{x1,2}]-Ω;​​σ12=D[ϕ,x1,x2];​​σ21=D[ϕ,x1,x2];​​​​(*TheStressTensor*)​​stress={{σ11,σ12,σ13},{σ21,σ22,σ23},{σ31,σ32,σ33}};​​​​(*TheConstitutiveEquations-Hooke'sLawForIsotropicSolid*)​​strain=((1+poissonR/modulusE))*stress-(poissonR/modulusE)*(σ11+σ22+σ33)*IdentityMatrix[3];​​ϵ11=strain[[1,1]]​​ϵ22=strain[[2,2]]​​ϵ33=strain[[3,3]];​​ϵ12=strain[[1,2]]​​ϵ23=strain[[2,3]];​​ϵ13=strain[[1,3]];​​ϵ21=strain[[2,1]];​​ϵ32=strain[[3,2]];​​ϵ31=strain[[3,1]];​​​​(*CheckForTheStrainCompatabilityIn2D*)​​​​(*TheStrain-DisplacementPDEquationsForLinearElasticity*)​​strainDisp11=D[u1[x1,x2],x1]-ϵ11​​strainDisp22=D[u2[x1,x2],x2]-ϵ22​​strainDisp12=1/2*(
∂
x
2
u1[x1,x2]+
∂
x
1
u2[x1,x2])-ϵ12​​​​(*SolvingtheDifferentialEquationforDisplacements*)​​solDisp11=DSolve[strainDisp110,u1[x1,x2],{x1,x2}]​​solDisp22=DSolve[strainDisp220,u2[x1,x2],{x1,x2}]​​​​
Out[]=
-
3x1x2
P
end
4ab
+
x1
3
x2
P
end
4
3
a
b
Out[]=
-
3x1x2ν
P
end
2
3
a
b
+
3x1x21+
ν


P
end
2
3
a
b
Out[]=
-
3x1x2ν
P
end
2
3
a
b
Out[]=
1+
ν

-
3
P
end
4ab
+
3
2
x2
P
end
4
3
a
b
Out[]=
3x1x2ν
P
end
2
3
a
b
-
3x1x21+
ν


P
end
2
3
a
b
+
(1,0)
u1
[x1,x2]
Out[]=
3x1x2ν
P
end
2
3
a
b
+
(0,1)
u2
[x1,x2]
Out[]=
-1+
ν

-
3
P
end
4ab
+
3
2
x2
P
end
4
3
a
b
Out[]=
u1[x1,x2]
3
2
x1
x2
P
end
4
3
a
b
+

1
[x2]
Out[]=
u2[x1,x2]-
3x1
2
x2
ν
P
end
4
3
a
b
+

1
[x1]