Page 111 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 91 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 91

Stress-strain relationships: The stress-strain relationships in polar coordinates can be obtained
by substituting r and θ for x and y in Equation (A2.2) and Equation (A2.3). For example,
the radial stress is given by

E
σ = + ν (1 − ν)(1 2 ) [(1 − νε + ) rr νε ] (A2.9a)
rr
θθ
for plane strain, and
E
σ = [ ε ν + ε ] (A2.9b)
rr
1 − ν 2 rr θθ
for plane stress.

Equilibrium equations:
∂σ + 1 ∂τ θ rr + σ rr −σ θ r θ
∂ r r ∂θ r = 0 (A2.10a)
1 ∂σ + ∂τ r θθ + 2τ rθ θ =
r ∂ r ∂θ r 0 (A2.10b)
Compatibility equation:

∇ 2 rr + (σ θθ ) =σ 0 (A2.11)

where

∂ 2 1 ∂ 1 ∂ 2
∇+ + +
2
∂R 2 r ∂r r 2 ∂θ 2
Airy stress function

∇∇ Φ = 0 (A2.12)
2
2
where Φ = Φ(r, θ) and
σ = r 1 2 ∂ Φ 2 θ 2 + 1 r ∂ ∂Φ ∂ r (A2.13a)
rr

∂ Φ
2
σ θθ = ∂r 2 (A2.13b)

∂Φ
1
τ θ r =− 1 ∂ Φ 2 θ rr ∂∂ + r ∂ θ (A2.13c)
2
A2.2 CRACK GROWTH INSTABILITY ANALYSIS
Figure 2.12 schematically illustrates the general case of a cracked structure with finite system
compliance C . The structure is held at a fixed remote displacement ∆ given by
M
T
∆= ∆ + CP (A2.14)
T
M