ECUACIONES DE LA ELASTICIDAD

INICIO

SetDirectory[NotebookDirectory[]]

<<StructuralMechanics`

Off[General::"spell1"]
Off[General::"spell"]

VECTOR DESPLAZAMIENTOS EN UN PUNTO

us=DisplacementComponents[u,Cartesian[x,y,z],Notation->Indicial]

%//MatrixForm

VECTOR DEFORMACIONES EN UN PUNTO

strs=StrainComponents[e, Cartesian[x,y,z],Notation-> Indicial]

%//MatrixForm

VECTOR TENSIONES EN UN PUNTO

stss=StressComponents[σ,Cartesian[x,y,z],Notation->Indicial]

%//MatrixForm

RELACIONES CINEMATICAS = DEFORMACIONES-DESPLAZAMIENTOS

sTd=StrainDisplacementRelations[e,u,Cartesian[x,y,z],Notation->Indicial];

%//TableForm

RELACIONES CONSTITUTIVAS = TENSIONES-DEFORMACIONES

TENSIONES-DEFORMACIONES

Lame Coefficients

cons=IsotropicStressStrainRelations[σ,e,{μ,λ},Cartesian[x,y,z],Notation->Indicial];

%//TableForm

Young's Modulus & Poisson's Ratio

conv=ConvertCoefficients[{λ, μ},{Em,ν},YoungsModulus->Em,LameCoefficients->{λ, μ},BulkModulus->k,PoissonsRatio->ν]

eqns=(cons/.conv);

%//TableForm

stresses=Simplify[{Solve[eqns,stss]}];

%//TableForm

DEFORMACIONES-TENSIONES

strains=Simplify[{Solve[eqns,strs]}];

%//TableForm

({{({{(Subscript[e, x y])->(((1+ν) (Subscript[σ, x y]))/Em)}, {(Subscript[e, x z])->(((1+ν) (Subscript[σ, x z]))/Em)}, {(Subscript[e, y z])->(((1+ν) (Subscript[σ, y z]))/Em)}, {(Subscript[e, x x])->(((Subscript[σ, x x])-ν ((Subscript[σ, y y])+(Subscript[σ, z z])))/Em)}, {(Subscript[e, y y])->((-ν (Subscript[σ, x x])+(Subscript[σ, y y])-ν (Subscript[σ, z z]))/Em)}, {(Subscript[e, z z])->((-ν (Subscript[σ, x x])-ν (Subscript[σ, y y])+(Subscript[σ, z z]))/Em)}})}})

PROBLEMAS DE TENSION PLANA Y DEFORMACION PLANA

TENSIONES PLANA

eqns

EcuacionesTensionPlana=eqns/.{(Subscript[σ, z z])->0}/.{(Subscript[σ, x z])->0}/.{(Subscript[σ, y z])->0}

TensionPlana=Solve[EcuacionesTensionPlana,{(Subscript[σ, x x]),(Subscript[σ, y y]),(Subscript[σ, x y]),(Subscript[e, z z]),(Subscript[e, x z]),(Subscript[e, y z])}]

TensionPlana[[1,1]][[2]]

(Subscript[e, x x])

FullForm[%]

D[TensionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,x]]]]

D[TensionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,x]]]]

MatrizTensionPlana=Simplify[{{D[TensionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[TensionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[TensionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,y]]]]},{D[TensionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[TensionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[TensionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[x,y]]]]},{D[TensionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[TensionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[TensionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[x,y]]]]}}];

%//MatrixForm

DEFORMACION PLANA

eqns

EcuacionesDeformacionPlana=eqns/.{(Subscript[e, z z])->0}/.{(Subscript[e, x z])->0}/.{(Subscript[e, y z])->0}

{(Subscript[σ, x x]),(Subscript[σ, y y]),(Subscript[σ, x y]),(Subscript[σ, z z]) ,(Subscript[σ, x z]),(Subscript[σ, y z])}

DeformacionPlana=Simplify[Solve[EcuacionesDeformacionPlana,{(Subscript[σ, x x]),(Subscript[σ, y y]),(Subscript[σ, x y]),(Subscript[σ, z z]),(Subscript[σ, x z]),(Subscript[σ, y z])}]]

Factor[-1+ν+2 (ν^2)]

DeformacionPlana[[1,1]][[2]]

Subscripted[e[HoldForm[x*x]]]

D[DeformacionPlana[[1,1]][[2]], Subscripted[e[HoldForm[x*x]]]]

MatrizDeformacionPlana=Simplify[{{D[DeformacionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[DeformacionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[DeformacionPlana[[1,1]][[2]],Subscript[e,HoldForm[Times[x,y]]]]},{D[DeformacionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[DeformacionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[DeformacionPlana[[1,2]][[2]],Subscript[e,HoldForm[Times[x,y]]]]},{D[DeformacionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[x,x]]]],D[DeformacionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[y,y]]]],D[DeformacionPlana[[1,3]][[2]],Subscript[e,HoldForm[Times[x,y]]]]}}];

%//MatrixForm

MORE ...

ShowPlane[{1,0,0}]

ShowPlane[{0,1,0}]

ShowPlane[{0,0,1}]

tns=VectorToTensor[strs]

MatrixForm[tns]

PrincipalStresses[{(Subscript[σ, xx]),(Subscript[σ,  yy]),(Subscript[σ,  zz]) ,0,0,0}]

PrincipalStressDirections[{(Subscript[σ, xx]),(Subscript[σ,  yy]),(Subscript[σ,  zz]) ,0,0,0}]

PrincipalStressDirectionsPlot[{1,1,1,.5,0,0}];

cube=UnitCubePlot[{5,4,7,0.01,0.03,2.91}];

dirs=PrincipalStressDirectionsPlot[{5,4,7,.01,.03,2.91}];

DisplacementComponents[u,Cartesian[x,y,z]]

DisplacementComponents[u,Cartesian[x,y,z],Notation->Indicial]

DisplacementComponents[u,Cartesian[x,y,z]]

IndicialForm[%]

StrainComponents[ϵ,Cartesian[x,y,z]]

StrainComponents[ϵ, Cartesian[x,y,z],Notation-> Indicial]

VectorToTensor[%]//MatrixForm

StressComponents[σ,Cartesian[x,y,z]]

StressComponents[σ, Cartesian[x,y,z],Notation->Indicial]

VectorToTensor[%]//MatrixForm

ForceComponents[f,Cartesian[x,y,z],Notation->Indicial]

ForceComponents[f,Cartesian[x,y,z]]

StrainDisplacementRelations[ϵ,u,Cartesian[x,y,z],Notation->Indicial]