Page 109 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 89 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 89

The governing equations of plane elasticity are given below for rectangular Cartesian coordi-
nates. Section A2.1.2 lists the same relationships in terms of polar coordinates.
A2.1.1 Cartesian Coordinates

Strain-displacement relationships:

u ∂ u ∂ 1  u ∂ u ∂ 
y
ε xx x ∂ x ε = yy ∂ y y ε = xy = 2   y ∂ x + x ∂   (A2.1)

where
x and y = horizontal and vertical coordinates
ε , ε , etc. = strain components
yy
xx
u and u = displacement components
y
x
Stress-strain relationships:
1. Plane strain

E
σ xx = + ν (1 − ν)(1 2 ) [(1 − νε + ) xx νε ] (A2.2a)
yy
E
σ yy = + ν (1 − ν)(1 2 ) [(1 − νε + ) yy νε ] (A2.2b)
xx
E
τ xy µ = ε xy = 2 ε xy (A2.2c)
1 + ν
σ zz ν σ= xx σ + ( yy ) (A2.2d)
ε zz ε = xz ε = yz τ = xz τ = yz = 0 (A2.2e)

where
σ and τ = normal and shear stress components
E = Young’s modulus
µ = shear modulus
υ = Poisson’s ratio

2. Plane stress
E
σ = [ ε xx ν + ε yy ] (A2.3a)
xx
1 − ν 2
E
σ yy = [ ε yy ν + ε xx ] (A2.3b)
1 − ν 2
E
τ xy µ = ε xy = 2 ε xy (A2.3c)
1 + ν
− ν
ε = ( ε xx ε + yy ) (A2.3d)
zz
1 − ν
σ zz ε = xz ε = yz τ = xz τ = yz = 0 (A2.3e)