In[]:=
Clear["Global`*"]​​(*ExactAnalyticalusingAiry(Real)StressPotentialFunctions)*)​​(*Case-1:End-LoadedCantileverin2DPlane-Stressb<<a*)​​endLoad=
P
end
;​​height=2*a;​​width=b;​​length=L;​​
σ
13
=0;​​
σ
23
=0;​​
σ
33
=0;​​
σ
31
=0;​​
σ
32
=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*
x
1
*
x
2
/(4*(height/2)*width))+(endLoad*
x
1
*(
x
2
^3)/(4*((height/2)^3)*width))​​​​(*StessesDefinedFromAiry'sFunction*)​​
σ
11
=D[ϕ,{
x
2
,2}]-Ω;​​
σ
22
=D[ϕ,{
x
1
,2}]-Ω;​​
σ
12
=D[ϕ,
x
1
,
x
2
];​​
σ
21
=D[ϕ,
x
1
,
x
2
];​​​​(*TheStressTensor*)​​stress={{
σ
11
,
σ
12
,
σ
13
},{
σ
21
,
σ
22
,
σ
23
},{
σ
31
,
σ
32
,
σ
33
}};​​​​(*TheConstitutiveEquations-Hooke'sLawForIsotropicSolid*)​​strain=((1+poissonR/modulusE))*stress-(poissonR/modulusE)*(
σ
11
+
σ
22
+
σ
33
)*IdentityMatrix[3];​​
ϵ
11
=strain[[1,1]]​​
ϵ
22
=strain[[2,2]]​​
ϵ
33
=strain[[3,3]];​​
ϵ
12
=strain[[1,2]]​​
ϵ
23
=strain[[2,3]];​​
ϵ
13
=strain[[1,3]];​​
ϵ
21
=strain[[2,1]];​​
ϵ
32
=strain[[3,2]];​​
ϵ
31
=strain[[3,1]];​​​​(*CheckForTheStrainCompatabilityIn2D*)​​​​(*TheStrain-DisplacementPDEquationsForLinearElasticity*)​​strainDisp11=D[u1[
x
1
,
x
2
],
x
1
]-
ϵ
11
​​strainDisp22=D[u2[
x
1
,
x
2
],
x
2
]-
ϵ
22
​​strainDisp12=1/2*(
∂
x
2
u1[
x
1
,
x
2
]+
∂
x
1
u2[
x
1
,
x
2
])-
ϵ
12
​​​​(*SolvingtheDifferentialEquationforDisplacements*)​​solDisp11=DSolve[strainDisp110,u1[
x
1
,
x
2
],{
x
1
,
x
2
}]​​solDisp22=DSolve[strainDisp220,u2[
x
1
,
x
2
],{
x
1
,
x
2
}]​​​​
Out[]=
-
3
P
end
x
1
x
2
4ab
+
P
end
x
1
3
x
2
4
3
a
b
Out[]=
-
3ν
P
end
x
1
x
2
2
3
a
b
+
31+
ν


P
end
x
1
x
2
2
3
a
b
Out[]=
-
3ν
P
end
x
1
x
2
2
3
a
b
Out[]=
1+
ν

-
3
P
end
4ab
+
3
P
end
2
x
2
4
3
a
b
Out[]=
3ν
P
end
x
1
x
2
2
3
a
b
-
31+
ν


P
end
x
1
x
2
2
3
a
b
+
(1,0)
u1
[
x
1
,
x
2
]
Out[]=
3ν
P
end
x
1
x
2
2
3
a
b
+
(0,1)
u2
[
x
1
,
x
2
]
Out[]=
-1+
ν

-
3
P
end
4ab
+
3
P
end
2
x
2
4
3
a
b
+
1
2

(0,1)
u1
[
x
1
,
x
2
]+
(1,0)
u2
[
x
1
,
x
2
]
DSolve
:To avoid possible ambiguity, the arguments in 
3ν
P
end
x
1
x
2
2
3
a
b
-
31+
ν


P
end
x
1
x
2
2
3
a
b
+
(1,0)
u1
[
x
1
,
x
2
]0 should literally match the independent variables.
Out[]=
DSolve
3ν
P
end
x
1
x
2
2
3
a
b
-
31+
ν


P
end
x
1
x
2
2
3
a
b
+
(1,0)
u1
[
x
1
,
x
2
]0,u1[
x
1
,
x
2
],{
x
1
,
x
2
}
DSolve
:To avoid possible ambiguity, the arguments in 
3ν
P
end
x
1
x
2
2
3
a
b
+
(0,1)
u2
[
x
1
,
x
2
]0 should literally match the independent variables.
Out[]=
DSolve
3ν
P
end
x
1
x
2
2
3
a
b
+
(0,1)
u2
[
x
1
,
x
2
]0,u2[
x
1
,
x
2
],{
x
1
,
x
2
}