axisymmetric_stress

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{  AXISYMMETRIC_STRESS.PDE  

 

  This example shows the application of FlexPDE to problems in

  axi-symmetric stress.

 

  The equations of Stress/Strain arise from the balance of forces in a

  material medium, expressed in cylindrical geometry as

       dr(r*Sr)/r - St/r  + dz(Trz) + Fr = 0

       dr(r*Trz)/r + dz(Sz) + Fz = 0

 

  where Sr, St and Sz are the stresses in the r- theta- and z- directions,

  Trz is the shear stress, and Fr and Fz are the body forces in the

  r- and z- directions.

 

  The deformation of the material is described by the displacements,

  U and V, from which the strains are defined as

       er = dr(U)

       et = U/r

       ez = dz(V)

       grz = dz(U) + dr(V).

 

  The quantities U,V,er,et,ez,grz,Sr,St,Sz and Trz are related through the

  constitutive relations of the material,

       Sr  =  C11*er + C12*et + C13*ez - b*Temp

       St  =  C12*er + C22*et + C23*ez - b*Temp

       Sz  =  C13*er + C23*et + C33*ez - b*Temp

       Trz =  C44*grz

 

  In isotropic solids we can write the constitutive relations as

       C11 = C22 = C33 = G*(1-nu)/(1-2*nu)     = C1

       C12 = C13 = C23 = G*nu/(1-2*nu)         = C2

       b = alpha*G*(1+nu)/(1-2*nu)

       C44 = G/2

 

  where G = E/(1+nu) is the Modulus of Rigidity

        E is Young's Modulus

        nu is Poisson's Ratio

  and   alpha is the thermal expansion coefficient.

 

  from which

       Sr  =  C1*er + C2*(et + ez) - b*Temp

       St  =  C1*et + C2*(er + ez) - b*Temp

       Sz  =  C1*ez + C2*(er + et) - b*Temp

       Trz =  C44*grz

 

  Combining all these relations, we get the displacement equations:

       dr(r*Sr)/r - St/r + dz(Trz) + Fr = 0

       dr(r*Trz)/r + dz(Sz) + Fz = 0

 

  These can be written as

       div(P) = St/r - Fr

       div(Q) = -Fz

 

  where P = [Sr,Trz]

  and   Q = [Trz,Sz]

 

  The natural (or "load") boundary condition for the U-equation defines the

  outward surface-normal component of P, while the natural boundary condition

  for the V-equation defines the surface-normal component of Q. Thus, the

  natural boundary conditions for the U- and V- equations together define

  the surface load vector.

 

  On a free boundary, both of these vectors are zero, so a free boundary

  is simply specified by

       load(U) = 0

       load(V) = 0.

 

  The problem analyzed here is a steel doughnut of rectangular cross-section,

  supported on the inner surface and loaded downward on the outer surface.

 

}

title "Doughnut in Axial Shear"

coordinates

   ycylinder('R','Z')

variables

   U           { declare U and V to be the system variables }

   V

definitions

   nu = 0.3           { define Poisson's Ratio }

   E  = 20             { Young's Modulus x 10^-11 }

   alpha = 0           { define the thermal expansion coefficient }

   G = E/(1+nu)

   C1 = G*(1-nu)/(1-2*nu)     { define the constitutive relations }

   C2 = G*nu/(1-2*nu)

   b = alpha*G*(1+nu)/(1-2*nu)

   Fr = 0             { define the body forces }

   Fz = 0

   Temp = 0           { define the temperature }

   Sr  =  C1*dr(U) + C2*(U/r + dz(V)) - b*Temp

   St  =  C1*U/r + C2*(dr(U) + dz(V)) - b*Temp

   Sz  =  C1*dz(V) + C2*(dr(U) + U/r) - b*Temp

   Trz =  G*(dz(U) + dr(V))/2

   r1 = 2             { define the inner and outer radii of a doughnut }

   r2 = 5

   q21 = r2/r1

   L = 1.0             { define the height of the doughnut }

initial values

   U = 0

   V = 0

equations               { define the axi-symmetric displacement equations }

   U:  dr(r*Sr)/r - St/r + dz(Trz) + Fr = 0

   V:  dr(r*Trz)/r + dz(Sz) + Fz = 0

boundaries

  region 1

    start(r1,0)

    load(U) =  0         { define a free boundary along bottom }

    load(V) =  0

    line to (r2,0)

    value(U) = 0         { constrain R-displacement on right }

    load(V) = -E/100     { apply a downward shear load }

    line to (r2,L)

    load(U) =  0         { define a free boundary along top }

    load(V) =  0

    line to (r1,L)

    value(U) = 0         { constrain all displacement on inner wall }

    value(V) = 0

    line to close

monitors

  grid(r+U,z+V)           { show deformed grid as solution progresses }

plots                       { hardcopy at to close: }

  grid(r+U,z+V)           { show final deformed grid }

  contour(U) as "X-Displacement"         { show displacement field }

  contour(V) as "Y-Displacement"         { show displacement field }

  vector(U,V) as "Displacement"           { show displacement field }

  contour(Trz) as "Shear Stress"

  surface(Sr) as "Radial Stress"

end