Complex Stress Functions (Optional Section) (§ 2.7.0)

At this point, we know that the general solution is ψ which already satisfies the equilibrium requirements, but the exact form of the solution must also satisfy the Biharmonic Equation (derived from combining compatibility and the constitutive relations) and the boundary conditions. In 1939, Westergaard proposed a complex stress function for crack problems loaded by opening (Mode I). The stress function was later expanded by Sih, in 1966, to cover the in-plane shear mode of loading/failure (Mode II). It should be noted that in this document, Westergaard's stress function is given, not derived. According to Westergaard, ψ is a hierarchical function:

(55)

where Equations 56, 57, 58 and 59 hold true:

(56)

(57)

(58)

(59)

Using the Chaucy-Reimann differential rules for complex numbers, the Westergaard solution can be solved for the individual stresses:

σxx = δ2ψ δy2 = Re Z - y Im Z '

(60)

σyy = δ2ψδx2 = Re Z + y Im Z '

(61)

τxy = - δ2ψδxδy = - y Re Z '

(62)

It can be shown that the Westergaard solution does indeed satisfy the Biharmonic equation, so now compatibility requirements are satisfied (ψ already satisfied equilibrium requirements in the previous section).

Defined Boundary Conditions
	Near-Field	Far-Field
Mode I	τxy (-a < x < a and y = 0) = 0 σyy (-a < x < a and y = 0) = 0	τxy (y = ± ∞ and x = ± ∞) = 0 σyy (y = ± ∞) = σ0
Mode II	τxy (-a < x < a and y = 0) = 0 σyy (-a < x < a and y = 0) = 0	τxy (y = ± ∞ and x= ± ∞) = σ0 σyy (y = ± ∞) = 0

Lastly, based on the boundary conditions found in § 2.1.5 of this tutorial and summarized above, Westergaard further proposed that Z be a function of the following form:

where a is the crack length and ξ = r e^{i θ} = r cos θ + i sin θ . This equation, coupled with the trigonometric relations developed in the following section, can be used to derive the final form of the stresses ahead of a crack tip.

Derivation of Equation 63:

Above, we defined in Equation 61 that σ_yy can be defined as follows:

σ_yy = Re Z - y Im Z ' (64)

When using our boundary conditions, we find that y = 0, and thus:

σ_yy = Re Z (65)

Using our definition of Re Z, we can define σ_yy as:

And furthermore, we can obtain an equation for Z(z):

By moving the origin to crack tip, o, as illustrated in the figure to the right, we are able to put Equation 67 in terms of the complex variable, ξ:

When ξ < a,

becomes insignificant, so that:

where a is the crack length and ξ = r e^{i θ} = r cos θ + i sin θ . This equation, coupled with the trigonometric relations developed in the following section below, can be used to derive the final form of the stresses ahead of a crack tip.

Exercise 4: Knowing that

, find Z ' (ξ)