﻿ Sample Problems > Applications > Stress > anisotropic_stress

# anisotropic_stress

Navigation:  Sample Problems > Applications > Stress >

# anisotropic_stress   { 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 }

 C12 =  3.199e5    C13 = 23.836e5    C15 = -3.148e5    C21 = C12    C22 = 67.2e5    C23 =  3.199e5    C25 =  8.997e5    C31 = C13    C32 = C23    C33 = 49.22e5    C35 = -3.148e5    C51 = C15    C52 = C25    C53 = C35    C55 = 24.335e5 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