# Complex Variable Formulation of Plane Elasticity Problems

4.1.1 Navier’s Equations for the Plane Elastic Problem

We take axes X, Хг lying in the plane of the structure, and axis x- perpendicular to it. Plane states always require а в — oy = 0, while 033 = 0 in generalized plane stress, and £33 = 0 in plane strain.

Restricting attention to the in-plane components of vectors and tensors (i. e., restricting indices to values I and 2), the equilibrium equation for negligible body forces arc reduced to state that the 2D divergence of the stress tensor must vanish;

‘ ‘ 0 ‘ (4-1.1)

where subscript j implies partial derivative with respect to the corresponding cartesian coordinate (i. e., fj = df/dxj). Repeated indices imply summation over г = 1,2.

The plane version of Hooke’s law may be reduced to (see, for example, Malvern 1969):

(Tij — X’ekk^ij + IGzij

where G is the shear modulus and A’ is an effective plane Lame constant. These elastic constants can be written as

q _ __ w gV

2(14- и) ’ Л "(1-і/*)

andCToo = Kjn/у/7гоо in (7.5.25) and (7.5.26) and take the limits for ao > oo. The first terms of Maclaurin’s series expansion of sec ж and ln{ 1 + x) are 1 + x2/2 + ■ ■ and x + ■ ■ ■, respectively.]

7.23 Consider a Dugdale model for the asymptotic limit of large crack in large body, (a) Use (7.5.70) to determine the function q{r]); (b) calculate Kin as a function of R; (c) calculate the crack tip opening displacement иіт as a function of Ft and (d) compare the results of parts (b) and (c) with those of parts (a) and (b) in the previous exercise.

7.25 Show that for the Dugdale model, Даоо = Я/3.

7.26 For a rectangular softening, determine the asymptotic values, RC and Дажс at peak load as a function of ich■