Page 110 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 90 Thursday, April 14, 2005 6:28 PM

90 Fracture Mechanics: Fundamentals and Applications

Equilibrium equations:

∂σ xx + ∂τ xy
∂x ∂y = 0 (A2.4a)

∂ σ yy ∂ τ xy
∂y + ∂x = 0 (A2.4b)
Compatibility equation:

∇ 2 xx + (σ yy ) = σ 0 (A2.5)

where
∂ 2 ∂ 2
∇= +
2
∂x 2 ∂y 2
Airy stress function: For a two-dimensional continuous elastic medium, there exists a function
Φ(x, y) from which the stresses can be derived:
∂ Φ
2
σ = y ∂ 2 (A2.6a)
xx
∂ Φ
2
σ = x ∂ 2 (A2.6b)
yy
∂ Φ
2
τ =− ∂∂ (A2.6c)
xy
xy
where Φ is the Airy stress function. The equilibrium and compatibility equations are automatically
satisfied if Φ has the following property:

∂ Φ 4 ∂ Φ 4 + ∂ Φ 4
∂x 4 + 2 ∂∂y x 2 2 ∂y 4 = 0

or

∇∇ Φ = 0 (A2.7)
2
2
A2.1.2 Polar Coordinates
Strain-displacement relationships:

u ∂
ε = r ∂ r (A2.8a)
rr
u 1 ∂u θ
ε θθ = r r + r ∂ θ (A2.8b)

ε = 1  1 u ∂ r + u ∂ θ − u   (A2.8c)
θ

θ r
2  r ∂ θ r ∂ r 
where u r and u θ are the radial and tangential displacement components, respectively.