Continuum Mechanics by P. Chadwick

Author:P. Chadwick [P. Chadwick]
Language: eng
Format: epub
Publisher: Dover Publications, Inc
Published: 2016-09-14T16:00:00+00:00

and equations (16) and (17)2 yield the following expressions for the normal stress and the shear stress norm:

(a) Suppose first that the principal stresses are distinct. Since the values of ni are subject to the constraint , the extremal values of σ(n) and τ(n) are found by solving the equations

and

in which μ and ν are Lagrange multipliers. On substituting from (A) and (B) and carrying out the differentiation we obtain the relations

and

It follows from (D) that two of the ni must be zero. Thus, from (A), the extremal values of the normal stress are the principal stresses and σ(n) = σi. when n = ±si, that is when n is aligned with the appropriate principal axis of stress.

From (E) we infer that at least one of the ni must be zero, since otherwise the quadratic equation σ2 – 2σσ(n) – 2ντ(n) = 0 would have three roots, namely σ1, σ2 and σ3. If n1 = 0, n2 ≠ 0, n3 ≠ 0, equations (C), (A) and (E) give

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