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(7) and (10) are of similar form and will be identical provided that

Three values of 9 may be obtained to satisfy (12), i.e. 0,0+ 120° and 0 + 240°

Then, from (9), three corresponding values of z are obtained, namely

(6) then yields appropriate values of y and hence the required values of x via (2).

8.14. Stress invariants; Eigen values and Eigen vectors

Consider the special case of the "stress at a point" tetrahedron Fig. 8.3 where plane ABC is a principal plane subjected to a principal stress ap and, by definition, zero shear stress. The normal stress is thus coincident with the resultant stress and both equal to ap. If the direction cosines of op (and hence of the principal plane) are Ip, mp, np then:

Pxn — Op • I p Pyn = Op ■ mp Pzn — Op • n p i.e. substituting in eqns. (8.13), (8.14) and (8.15) we have:

Considering eqn. (8.43) as a set of three homogeneous linear equations in unknowns Ip, mp and np, the direction cosines of the principal plane, one possible solution, viz. lp = mp — np — 0, can be dismissed since I2 + m2 + n2 = 1 must always be maintained. The only other solution which gives real values for the direction cosines is that obtained by equating the determinant of the R.H.S. to zero:

Evaluating the determinant yields the so-called "characteristic equation"

oi - («XX + <Tyy + <*zz) rf, + [(°xxOyy + °yy°a + °7zaxx) ~ (<^y + <4 + Op

— Wxxayya7z + 2axyayzazx — (axxa^z + OyyO^ + er^^)] = 0

Thus, for any given set of cartesian stress components in three dimensions a solution of this cubic equation is required before principal stress value can be determined; a graphical solution is not possible.

### Eigen values

The solutions for the principal stresses cs\, oi and <73 from the characteristic equation are known as the Eigen values whilst the associated direction cosines Ip, mp and np are termed the Eigen vectors.

One procedure for solution of the cubic characteristic equation is given in §8.10.

### 8.15. Stress invariants

If, for the same applied stress system, the stress components had been given relative to some other set of cartesian co-ordinates x', y' and z', the above equation would still apply (with x' replacing x, y' replacing y and z replacing z) and would still produce the same principal stress values. It follows, therefore, that whatever axis system is chosen the coefficients of the various terms of the characteristics equation must have the same values, i.e. they are "non-varying quantities" or "invariant".

The equation can thus be re-written in the form:

with

/2 = (< +*£+<£) ' !3 = °Xl ' Oyy ■ Olz + 2(7;

the three quantities I\, 12 and /3 being termed the stress invariants.

If the reference axes selected are the principal stress axes in the system then all shear components reduce to zero and the equations (8.46) reduce to:

The first and second invariants are particularly important in development of the theory of plasticity since it is assumed that:

(a) 11 has no influence on initial yielding

(b) I2 — constant can be taken as an important criterion of yielding.

For biaxial stress conditions, i.e. 03 =0, the third stress invariant vanishes and the others reduce to

Now from eqn. (13.11)^ the principal stresses in a two-dimensional stress system are given by:

Now from eqn. (13.11)^ the principal stresses in a two-dimensional stress system are given by:

which is the general solution of the following quadratic equation:

The graphical solution of this equation is as follows (see Fig. 8.21):

Fig. 8.21. Graphical determination of principal stresses in a two-dimensional stress system from known stress invariant / values (solution for positive ¡2 value)

Fig. 8.21. Graphical determination of principal stresses in a two-dimensional stress system from known stress invariant / values (solution for positive ¡2 value)

(i) On a horizontal (direct stress) axis mark off a length OA = i-

(ii) Draw semi-circle on OA as diameter.

(iii) With centre O draw arc OB, radius JT2, to cut the semi-circle at B.

(iv) With centre A and radius AB draw semi-circle to cut stress axis at o\ and a2 the required principal stress values.

N.B. If /2 is negative (see §8.46), algebraically *JT2 > and the line OB cannot cut the semi-circle on OA as diameter and no solution can be obtained. In this case an alternative construction is required - see Fig. 8.22.

(ii) Erect perpendicular at O of length OC = */—I2.

(iii) With centre A and radius AC draw a circle to cut OA (produced as necessary) at o\ and (72 the required principal stress values.

Returning to a three-dimensional principal stress system a further interesting graphical relationship is obtained from the 3D Mohr circle construction - see Fig. 8.23.*

The three stress invariants are given in Fig. 8.23 in terms of the tangents to the three circles from the origin 0 as:

12 = o, o2+ o2o3 + o3o, = OB j + OB\ + OB\ /3 = cr, (72(73 = OB1 X OB2 x OB3

### 8.16. Reduced stresses

An alternative form of the cubic characteristic equation is obtained if a "hydrostatic stress" of 11/3 is substracted from the original stress system to produce "reduced stresses" cr' = (7-A/3.

Thus, replacing op by (o' +1\/3) in eqn. (8.45) we have:

Ji = + 3/2] J3 = ^[2/^+9/,/2 + 27/3] * M.G. Derrington and W. Johnson, The Defect of Mohr's Circle for Three-Dimensional Stress States.

The terms J J 2 and J 3 are termed the invariants of reduced stress and, again, have special significance in the consideration of yielding of metals and associated plastic theory.

It will be shown in §8.20 that the hydrostatic stress component does not affect the yield of metals and hydrostatic stress = +02+03)= 5/1

It therefore follows that first stress invariant I\ also has no significance on yielding and since the principal stress system can be written, as above, in terms of reduced stresses a' — (o — 1/3 /1) it also follows that it must be the reduced stress components which influence yielding.

(N.B.: "Reduced stresses" are synonymous with the deviatoric stresses introduced in §8.20.) Other useful relationships which can be derived from the above eqi ations are:

(«7, - a2f + (<r2 - <r3)2 + to - <*\ )2 = 6J2 (8.52)

and (a^-ayyf+ (oyy-azzf+ (a7Z-axxf+ e(x2xy + x2yz + r1zx] = 2/2 + 6/2 (8.53)

The left-hand sides of both equations are thus, in themselves, invarimt and are useful in further considerations of strain energy, yielding and failure.

For example, the shear strain energy theory of elastic failure uses tl e criterion:

(oi - a2)2 + (a2 — 03)2 + (0-3 - o\ )2 = 2er2 = const int which, from eqn. (8.52), can be simply re-written as

N.B.: It should be remembered that eqns. (8.52) and (8.53) are merely different ways of presenting the same information since:

8.17. Strain invariants

It has been shown in §14.10"!" that the basic transformation equations for stress and strain have identical form provided that e is used in place of a and y/2 in place of x. The equations derived above for the stress invariants will therefore apply equally for strain conditions provided that the same rules are followed.

8.18. Alternative procedure for determination of principal stresses (eigen values)

An alternative solution to the characteristic cubic equation expressed in stress invariant format, viz. eqn. (8.45), is as follows: Given the basic equation:

a3p-ha2p-I2ap-h=0 (8.45)bis the stress invariants may be calculated from:

11 = &xx + Oyy + &zz h = -(Oxx°yy + OyyOu. + Ozz(Txx) + T2y + X2yz + t^ (8.46)bis h = oxxayyalz + 2xxyxyzxzx - cxxr2z - OyyX^ - <Jzzxxy and the required principal stresses obtained from^:

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