Page 177 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 157 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 157

where A* is the area enclosed by Γ*. By invoking the definition of strain energy density given by
Equation (3.20), we can evaluate the first term in square brackets in Equation (A3.14):

∂w = ∂w ∂ε ij = σ ∂ε ij
∂x ∂ε ij ∂ x ij ∂ x (A3.15)

Note that Equation (A3.15) applies only when w exhibits the properties of an elastic potential.
Applying the strain-displacement relationship (for small strains) to Equation (A3.15) gives

∂ 
∂w = σ   ∂  ∂u i  + ∂  u j  
1

∂x 2 ij   ∂x   ∂x j   ∂  x i  
x
∂ 
∂  u (A3.16)
∂ 
= σ ij ∂x j   x i 
∂ 
since s = s . Invoking the equilibrium condition:
ij
ji
∂σ ij
∂x j = 0 (A3.17)

leads to

∂  u ∂  ∂  u ∂ 
i
σ ij x ∂ j   x i  = ∂ x ∂  j   σ ij x ∂   (A3.18)

which is identical to the second term in square brackets in Equation (A3.14). Thus the integrand
in Equation (A3.14) vanishes and J = 0 for any closed contour.
Consider now two arbitrary contours Γ and Γ around a crack tip, as illustrated in Figure A3.3.
2
1
If Γ and Γ are connected by segments along the crack face (Γ and Γ ), a closed contour is formed.
3
1
2
4
The total J along the closed contour is equal to the sum of contributions from each segment:
J J = J + J + J + = 0 (A3.19)
1 2 3 4

FIGURE A3.3 Two arbitrary contours Γ 1 and Γ 2 around the tip of a crack. When these contours are connected
by Γ 3 and Γ 4 , a closed contour is formed, and the total J = 0.