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

Contents

1. The problem
2. Examples of fluid/heat/stress interactions

3. Similarities between the equations for displacement and velocity
4. Deduction of the associated stresses and strains
5. The "SIMPLE" algorithm for the computation of displacements

6. A simple example of flow-influenced stress
7. The computational problem
8. How PHOENICS solves it

9. Where further examples can be found
10. The future of the solid-stress option

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 TWO COMPUTER CODES are used, one for the flows and the other for the stresses; iterative interfacing between them is then employed, often with considerable awkwardness.

This interfacing is rendered unnecessary by the SOLID-STRESS OPTION of PHOENICS, which permits both processes, and the interaction between them, 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).

Preliminary illustration

To whet the appetite for what follows, results from a recent study will first be shown. They the calculation of fluid-flow, conjugate heat transfer and solid stress in the situation illustrated by the sketch below.

           ///////////////////////////////////////////
           --------------- adiabatic wall ------------
    cooling air-->>           duct                    ----->
           -------------                 -------------
           // steel ///|     cavity      |/// steel //
           -------------------------------------------
           ////////////// aluminium //////////////////
           -------------------------------------------

Velocity vectors above, displacement vectors below.

The solids are supposed to be fixed at the bottom left-hand corner /phoenics/d_polis/d_lecs/d_lvel/fil0 The temperature field which caused the thermal expansion.

The two upper blocks, which were heated, have a higher thermal- conductivity than the plate to which they are fixed.

                              Air



   Block 1                                          Block 2



                             Plate

/phoenics/d_polis/d_lecs/d_lvel/fil5 This scalar equation also yields the gap between walls, which is often needed. especially when surface-to-surface radiation is present. Its values are shown here.

/phoenics/d_polis/d_lecs/d_lvel/fil6 The x-direction strains, deduced from the x-direction displacements. The thermal-expansion coefficients of the plate and blocks differ

/phoenics/d_polis/d_lecs/d_lvel/fil1 The y-direction strains, deduced from the y-direction displacements.

/phoenics/d_polis/d_lecs/d_lvel/fil2 The x-direction stresses

/phoenics/d_polis/d_lecs/d_lvel/fil3 The y-direction stresses

/phoenics/d_polis/d_lecs/d_lvel/fil4

The interaction problem arises in the following (and similar) connexions:-

• air-cooled gas-turbine blades under transient conditions;

• "residual stresses" resulting from casting or welding;

• thermal stresses induced in liquid-sodium containers and immersed structures in fast-breeder nuclear reactors during emergency shut-down;

• manufacture of bricks, ceramics and other materials;

• flows induced by the contraction of "heat-shrinking" plastics;

• stresses in the pistons and cylinder blocks of diesel engines;

• the failure of steel-frame-supported buildings during fires.

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.

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.

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

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:

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

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.

The following sketch illustrates a combined mechanical-and-thermal- stress situation, in which the fluid flow plays a part. The task is to compute displacements, and thence strains and stresses, in the heated block.

     ------------------------------------------------------

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

• The independent variables are y & z. (the flow is steady; and the body is axi-symmetrical, so neither time nor circumferential coordinate x has an influence)

• The dependent variables are: v & w (velocities or displacements) P (pressure or dilatation) T (temperature)

• Derived variables: Te (linear thermal expansion) ey, ez (strains in y & z directions) strx, strz (stresses in y & z directions)

• Boundary conditions: as indicated on above sketch.

• The differential equations to be solved are:-

  • the Navier-Stokes equations for the fluid region;
  • the displacement equations (above) for the solid region;
  • the thermal-energy equation for both regions.

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.

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

At the present time (January 1997), only linear stress-strain relations are allowed for; but the Young's Modulus and the thermal- expansion coefficient can be temperature-dependent.

A library of input-files is supplied with the solid-stress option of the PHOENICS. These mainly concern rather elementary situations, several being of the simple kind which permit analytical solution, for comparison and confidence-building purposes.

Graphically-displayed results from one of these cases are displayed in the Applications Album supplied with PHOENICS.

Two pictures are shown here.

The first 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. /phoenics/d_polis/d_applic/d_design/stress_1 This picture shows the computed temperature distribution which caused (most of) the displacements. /phoenics/d_polis/d_applic/d_design/stress_2

The difference between the computational work of solving for the in-solids displacements and that of NOT doing so is rather small. Therefore, when the testing of the option has proceeded sufficiently far, it is to be expected that STRA=T will become the default option whenever immersed solids are present, and material-property data are available.

No BFC-grid capability has been developed at the present time, because it may well prove more economical to use a cartesian grid with the new X-cell procedure (see PHENC entry: X-Cell...), which allows the use of rectangular cells to be split along diagonals, with solid on one side and fluid on the other.

The next two pictures show, first, an element of the X-Cell grid, and, secondly, the result of an early use of X-Cell for simultaneous solution of solid stress and fluid flow.

/phoenics/d_polis/d_lecs/d_xcell/FIG51COL

Fluid flow on the left; displacements and temperatures on the right

/phoenics/d_polis/d_lecs/d_xcell/fig61

BFC grids are indeed often configured so as to avoid placing cells in (what have been) the uninteresting interiors of objects.

With the coming of the solid-stress option, the interiors are often just as interesting as the surrounding fluid-accessible spaces.

However, it may be mentioned that, if it is desired, the X-Cell and BFC techniques can be combined, as the next picture shows.

The picture illustrates also that the X-Cell grid may be confined to only a part of the integration domain.

This feature is still (January 1997) under development.

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/phoenics/d_polis/d_lecs/d_xcell/fig63