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STRESS AND STRAIN IN SOLIDS; their calculation by means of PHOENICS

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  1. The problem
    1. Its essential nature
    2. Practical occurrence
    3. The conventional solution
    4. A better solution
  2. A multi-physics example
    1. Stresses resulting from radiation, conduction and convection
    2. Vector and contour plots
  3. The mathematics of the method
    1. Similarities between the equations for displacement and velocity
    2. Deduction of the associated stresses and strains
    3. The "SIMPLE" algorithm for the computation of displacements
    4. More details of the equations
    5. The open-source Fortran coding
  4. A simple example of flow-influenced stress
    1. Description
    2. The computational problem
    3. How PHOENICS solves it
  5. References

1. The problem

(a) Its essential nature

It is frequently required to simulate fluid-flow and heat-transfer processes in and around solids which are, partly as a consequence of the flow, subject to thermal and mechanical stresses.

Often, indeed, it is the stresses which are of major concern, while the fluid and heat flows are of only secondary interest.

(b) Practical occurrence

Engineering examples of fluid/heat/stress interactions include:

(c) The conventional solution

It has been customary for two computer codes to be used for the solution of such problems, one for the fluid flow and the other for the stresses

Iterative interaction between the two codes is then employed, often with considerable awkwardness.

(d) A better solution

This interfacing is rendered unnecessary by the solid-stress option of PHOENICS, which permits fluid flow, heat flow and solid deformation, and the interactions between them, all to be simulated at the same time.

The method of doing so exploits the similarity between the equations governing velocity (in fluids) and those governing displacement (in solids).

2. A multi-physics example

(a) Description

In order to demonstrate what this method can achieve, the results from PHOENICS library case s400 will be shown. This concerns the calculation of fluid-flow, conjugate heat transfer and solid stress in the situation illustrated by the sketch below.

    cooling | air
          | V |/////// hot radiating wall ///////////|
          |   ----------------------------------------
          |                       duct              ----->  exit
          |-------------                 -------------
          |// steel ///|     cavity      |/// steel /|
          |////////////// aluminium /////////////////|

The task is to discover what stresses arise in the metal blocks as a result of the non-uniformities of temperature.

Further details are:

  1. The Reynolds number (based on the inflow velocity and horizontal duct width) is 1000; therefore the LVEL model is used for simulation of the turbulence.

  2. The radiative heat transfer is represented by the IMMERSOL model, which is economical and fairly accurate for such situations.

  3. Both LVEL and IMMERSOL make use of the distributions of distance from the wall (WDIS) and distance between walls (WGAP), both of which are calculated by solving a scalar equation for the LTLS variable.

  4. The temperature distributions in the metal blocks are the consequence of radiative heating and convective cooling at their exposed surfaces, and of heat conduction within them.

  5. The stresses within them result primarily from the differences in their thermal-expansion coefficients. namely:

(b) Vector and contour plots

3. The mathematics of the method

(a) Similarities between the equations for displacement and velocity

The similarities referred to in section 1 (d) are here decribed for only one cartesian direction; but they prevail for all three directions.

  1. The x-direction displacement, U, obeys the equation:
    [del**2]* U + [d/dx]* [ D*C1 - Te*C3 ] + Fx*C2 = 0

  2. When the viscosity is uniform and the Reynolds number is low, so that convection effects are negligible, the x-direction velocity, u, obeys the equation:

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



  1. The two equations are here set one below the other, so that they can be easily compared:

  2. The equations can thus be seen to become identical if:


    fx * c2 = Fx * C2

  3. The expressions for C1, C2 and C3 are:

  4. A solution procedure designed for computing velocities will therefore in fact compute the displacements if:

(b) Deduction of the associated stresses and strains

The strains (ie extensions ex, ey and 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}


(c) 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.

Its essential features are:

All that it is necessary to do, in order to modify this algorithm for solving for displacements simultaneously, is to treat the dilatation B as the mass source error and to ensure that p obeys the above linear relation to it.

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

  1. eliminating the convection terms (ie setting Re low); and
  2. making D linearly dependent on p and temperatureT.

The "staggered grid" used as the default in PHOENICS proves to be extremely convenient for solid-displacement analysis; for the velocities and displacements are stored at exactly the right places in relation to p.

4. A simple example of flow-influenced stress

(a) Description

The following sketch illustrates a combined mechanical-and-thermal-stress situation, in which the fluid flow plays a part. It is the focus of attention in a series of case to be found in the Input-File Library of the solid- stress option,


           load V V                          <---------   ^ y
                |-----------|                             |
                |///////////|                cold-air jet |
                |// heated /|                             |
                |// block //|                             |
                |///////////|                <------z-----O

The task is to compute displacements, and thence strains and stresses, in the heated block, subject to a variety of constraints as follows:

(b) The computational problem

(c) How PHOENICS solves the problem

PHOENICS solves the fluid-flow, displacement and thermal-energy equations simultaneously, being enabled to do so easily because:

  1. it does not need to solve for both displacements and velocities at the same place;
  2. the displacement equations are so similar to those for velocity that the same solution algorithm will do for both.

The user needs only to set STRA=T in the Q1 file, and to provide appropriate boundary conditions for displacement, etc; then PHOENICS will test, cell-by-cell, whether a solid or a fluid is present, and will solve whichever equation is appropriate there.

This is done whenever the solid-stress option is present.

Only linear stress-strain relations are allowed for; but the Young's Modulus and the thermal- expansion coefficient can be temperature-dependent.

Graphically-displayed results for this case are as follows:

Fig.18 shows simultaneously the displacement vectors and the velocity vectors; the second shows the temperature field which caused the thermal expansion.

The arrows represent velocity vectors in the upper region and displacement vectors in the block, and Fig.19 shows the computed temperature distribution which caused (most of) the displacements.

A library of input-files is supplied with the solid-stress option of the PHOENICS.

5. References

  1. The differential equations governing displacements, stresses and strains in elastic solids of non-uniform temperature can be found in numerous textbooks, for example:

    It has not been common to choose the displacements as the dependent variables in numerical-solution procedures. However, this has been done by:

    Their numerical procedure differ from that used here, which was that of

  2. The first use of the present method for solving the solid-displacements and fluid-velocity equations simultaneously appears to have been made by CHAM, late in 1990.

    Reports describing the early work include:

    From that time onwards, the solid-stress option was made available as a (little-advertised) option in successive issues of PHOENICS,

  3. Open-literature and conference publications have been few, but include: