Encyclopaedia Index
### (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.