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{ ANISOTROPIC_STRESS.PDE

 

 This example shows the application of FlexPDE to an extremely complex

 problem in anisotropic thermo-elasticity.  The equations of thermal

 diffusion and plane strain are solved simultaneously to give the

 thermally-induced stress and deformation in a laser application.

 

               -- Submitted by Steve Sutton

                  Lawrence Livermore National Laboratory

}

title "ANISOTROPIC THERMAL STRESS"

select

   errlim = 1e-4       { more accuracy to resolve stresses }

variables

   Tp(5)               { Temperature }

   up(1e-6)           { X-displacement }

   vp(1e-6)           { Y-displacement }

definitions

   Qs                 { The heat source, to be defined }

   Q0 = 3.16

   ro = 0.2           { Heat source radius }

   W = 2               { slab size constants }

   L = 0.5

   mag = 5000

   kp11 = 0.0135       { anisotropic conductivities }

   kp33 = 0.0135

   kp13 = 0.0016

   C11 = 49.22e5       { anisotropic elastic constants }

 

   ayy = 34.49e-6     { anisotropic expansion coefficients }

   axx = 34.49e-6

   azz = 25.00e-6

   axy = 9.5e-6

   h = 1.0

   Tb = 0.

   Q = Q0*(exp(-2*(x^2+y^2)/ro^2)) { Gaussian heat distribution }

              { some auxilliary definitions }

   qx = -kp33*dx(Tp) - kp13*dy(Tp)     { heat flux }

   qy = -kp13*dx(Tp) - kp11*dy(Tp)

                                      { expansion stress coefficients }

   apxx = C31*ayy + C32*azz + C33*axx + C35*axy

   apyy = C11*ayy + C12*azz + C13*axx + C15*axy

   apzz = C21*ayy + C22*azz + C23*axx + C25*axy

   apxy = C51*ayy + C52*azz + C53*axx + C55*axy

   exx = dx(up)                       { strain }

   eyy = dy(vp)

   exy = 0.5*(dy(up)+dx(vp))

                                      { stress }

   sxx = C31*eyy + C33*exx + 2*C35*exy - apxx*Tp

   syy = C11*eyy + C13*exx + 2*C15*exy - apyy*Tp

   szz = C21*eyy + C23*exx + 2*C25*exy - apzz*Tp

   sxy = C51*eyy + C53*exx + 2*C55*exy - apxy*Tp

initial values

   Tp = 5.

   up = 0

   vp = 0

equations

   Tp: dx(qx) + dy(qy) = Qs

   Up: dx(sxx) + dy(sxy) = 0.

   Vp: dx(sxy) + dy(syy) = 0.

constraints                             { prevent rigid-body motion: }

   integral(up) = 0                   { cancel X-motion }

   integral(vp) = 0                   { cancel Y-motion }

   integral(dx(vp) - dy(up)) = 0       { cancel rotation }

boundaries

region 1

   Qs = Q

  start(-0.5*W,-0.5*L)

      natural(up) = 0.               { zero normal stress on all faces }

      natural(vp) = 0.

      natural(Tp) = h*(Tp-Tb)         { convective cooling on bottom boundary }

  line to (0.5*W,-0.5*L)

      natural(Tp) = 0.               { no heat flux across end }

  line to (0.5*W,0.5*L)

      natural(Tp) = h*(Tp-Tb)         { convective cooling on top boundary }

  line to (-0.5*W,0.5*L)

      natural(Tp) = 0.               { no heat flux across end }

  line to close

monitors

  grid (x+mag*up,y+mag*vp)

  contour(Tp) as "Temperature"

plots

  grid (x+mag*up,y+mag*vp)

  contour(Tp) as "Temperature" on grid (x+mag*up,y+mag*vp)

  contour(Tp) as "Temperature" zoom(-.2,-.2,0.4,0.4) on grid (x+mag*up,y+mag*vp)

  contour(up) as "x-displacement" on grid (x+mag*up,y+mag*vp)

  contour(vp) as "y-displacement" on grid (x+mag*up,y+mag*vp)

  vector(up,vp) as "Displacement vector plot" on grid (x+mag*up,y+mag*vp)

  contour(sxx) as "x-normal stress" on grid (x+mag*up,y+mag*vp)

  contour(syy) as "y-normal stress" on grid (x+mag*up,y+mag*vp)

  contour(sxy) as "shear stress" on grid (x+mag*up,y+mag*vp)

  elevation(Tp) from (0,-0.5*L) to (0,0.5*L) as "Temperature"

  elevation(sxx) from (0,-0.5*L) to (0,0.5*L) as "x-normal stress"

  elevation(syy) from (0,-0.5*L) to (0,0.5*L) as "y-normal stress"

  surface(Tp) as "Temperature"

 

end