Page 199 - Introduction to Continuum Mechanics

184 Stress

If dni, cbi2 t and dn^ can vary independently of one another, then Eq. (i) gives the familiar
w
condition for the determination of the triple («i,«2> 3) f° r tne stationary value of 7j

VI If I *- ftlf / . \Sttr *
But the dni,dn2 and dn$ can not vary independently. Indeed, taking the total derivative of
Eq. (4.6.6), i.e., nj+nl+wi = 1. we obtain

If we let

and

then by substituting Eqs. (iii) (iv) and (v) into Eq. (i), we see clearly that Eq. (i) is satisfied if
Eq. (4.6.6) is enforced. Thus, Eqs. (iii), (iv), (v) and (4.6.6) are four equations for the
W an
determination of the four unknown values of n^ «2» 3 d A which correspond to stationary
values of 7j. This is the Lagrange multiplier method and the parameter A is known as the
Lagrange multiplier (whose value is however of little interest).

Computing the partial derivatives from Eq. (4.6.4), Eqs. (iii), (iv), and (v) become

From Eqs. (vi), (vii), (viii) and (4.6.6), the following stationary points (ni^^s) can be obtained
(The procedure is straight forward, but the detail is somewhat tedious, we leave it as an
exercise.):