(c) Similarities between the equations for displacement and velocity

The similarity is here shown for only one direction, but prevails for all three directions.

(1) the x-direction DISPLACEMENT, U, obeys, for linear strain:


  [del**2]* U  +  [d/dx]* [ D*C1  -  Te*C3 ]  +  Fx*C2  =  0

  where:
  Te = temperature * linear thermal expansion coefficient
  D  = [d/dx]* U + [d/dy]* V + (d/dz]* W (ie the dilatation)
  Fx = external force per unit volume in x-direction
  V & W = displacements in y and z directions
  C1, C2 & C3 are functions of Young's modulus & Poisson's
                                                    ratio

(2) the x-direction VELOCITY, u, obeys, at low Reynolds number:


  [del**2]* u - [d/dx]* [ p*c1 ] + fx*c2 = 0 ,

  where  
  p  = pressure,
  fx = external force per unit volume in x-direction,
  c1 = c2 = 1 / viscosity.

Notes:

(1) The equations are similar if:


           p*c1  = D*C1  -  Te*C3
  ie:      D     = [p*c1  +  Te*C3 ] / C1
  and:     fx*c2 = Fx*C2

(2) The expressions for C1, C2 and C3 are:


  C1 = 1//(1  - 2*PR)
  C2 = 2*(1 + PR) / YM        where PR   = Poisson's Ratio
  C3 = 2 *(1 + PR)/(1 - 2*PR)       YM   = Young's Modulus

(3) In Te, the local temperature is measured above that of the unstressed solid in the zero-displacement condition.

(4) The linear relation between D ( ie [d/dx]* U + ...) , p and Te can be effected by inclusion of a pressure- and temperature- dependent "mass-source" term.

(d) Deduction of the associated stresses and strains

The strains (ie extensions ex, ey end ez) are obtained from differentiation of the computed displacements.

Thus: ex = [d/dx]* u, ey = [d/dx]* v, ez = [d/dx]* w .

Then the corresponding normal stresses, sx, sy, sz, and shear stresses tauxy, tauyz, tauzx, are obtained from the strains by way of equations such as:


  sx    = {YM / (1 - PR**2)} * {ex + PR*ey}
  tauxy = {YM / (1 - PR**2)} * {0.5 * (1 - PR)*gamxy}
          where: gamxy = [d/dy]*u - [d/dx]*v

The relevant computer coding in PHOENICS is to be found in the open- source Fortran sub-routine GXSTRA.F .

(e) The "SIMPLE" algorithm for the computation of displacements

PHOENICS employs (a variant of) the "SIMPLE" algorithm of Patankar & Spalding (1972) for computing velocities from pressures, under a mass-conservation constraint.

In this algorithm, p is computed from D above, (with u, not U), the f(p,D) function being linear in simple circumstances.

Therefore a CFD code based on SIMPLE can be made to solve the displacement equations by:

eliminating the convection terms (ie setting Re low); and
making D linearly dependent on p and temperature T .

The modular structure of PHOENICS has made this rather easy to do.

The "staggered grid" used as the default in PHOENICS proves to be extremely convenient for solid-displacement analysis; the u, v and (in 3D) w are stored at exactly the right places in relation to P.