Ox

Eyy = 9y dw äz dv du Y*y = Tx + Vy dw dv du dw d z dx

(b) In polar coordinates with displacements ur, ug and uz along r, 0 and z respectively: these equations become:

with

1 duz dug

3z dr ue r

8.25. The strain equations of transformation

Using the experimental or theoretical procedures described in earlier sections it is possible to derive the values of the direct and shear stresses acting at a point on a body. These are normally obtained with reference to some convenient set of X, Y coordinates which, for example, may be parallel to the edges of the component considered. Sometimes, however, it may be more convenient to refer the values obtained to some other set of axes X'Y' at an angle 9 to the original axes.

In this case the two-dimensional versions of eqns. (8.73) and (8.74) apply equally well to the new axes (Fig. 8.31), du' dv' dv' du'

i.e. sX'X' = — £V'V< = — and yyv< =--1--

Fig. 8.31. Alternative coordinates to which strains may be referred.

Now, using the partial differentiation chain rule, du'

sin 9 cos 9

dy \dx 3 y/ sX'X' = Sjcx cos2 9 + Syy sin2 9 + yxy sin 9 cos 9

Or, in terms of the double angle 29, eX'x' = \(eXx + Byy) + \(exx - Syy) cos 2d + \ y^ sin 20

This is the same as eqn. (14.14) obtained in §14.10^ for the normal strain on any plane in terms of the coordinate strains. Indeed, the above represents an alternative proof for what are really similar requirements.

826. Compatibility

Equations (8.73) and (8.74) relate the six components of strain (three direct and three shear) to the equivalent displacements under a three-dimensional stress system. If, however, the situation arises where the six strain components are known, as they could well be following some theoretical or experimental strain analysis, then the above equations represent three in excess of that required for solution of the three unknown displacements (three unknowns require only three equations for solution). Thus, unless the solution obtained from any three equations satisfies the other three equations, then the values cannot be accepted as a valid solution. Certain specific relations must therefore be satisfied before a valid solution is obtained and these are termed the compatibility relations.

The problem can be considered physically as follows: consider a body divided into a large number of small cubic elements. When load is applied the elements deform and simple measurements of length and angle changes will yield the direct and shear strains in each element. These can be summated to produce the overall component strains if required. If, however, the deformed elements are separated and provided in their deformed shapes as a jigsaw puzzle, the puzzle can only be completed, i.e. the elements fully assembled without voids or discontinuities, if each element is correctly strained or deformed. The procedure used to check this condition then represents the compatibility equations. The compatibility relationships in terms of strain are derived as follows:

d2£xx d3u 3 y2 dxdy2

Therefore differentiating once with respect to x and once with respect to >•, d2Vxy 3 3M dxdy dxdy2 dx2dy aVxy _ Pe xx , dxdy dy2 dx2

dydz 3z2 dy2

These are three of the compatibility equations.

It can also be shownf that a further three compatibility relationships apply, namely

Exx=dx dv

Similarly, and t A.E.H. Love, Treatise on the Mathematical Theory of Elasticity, 4th edn., Dover Press, New York, 1944.

dzdx

dzdx

'dxdy

a dx

dYxy

tyzx dy

dy>7 dx

d dy

. dz

dyzx dy

dyyz dx _

a dz

3/xy dZ

+ 3/zx dy

The compatibility equations can also be written in terms of stress as follows: Consider the first of the strain compatibility relationships given in eqn. (8.41).

d2sx dy2

d ev yy dx2

d2Yxy dxdy

For plane strain conditions (and a similar derivation shows that the equation derived is equally appropriate for plane stress) we have:

0 0

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