J-Integral Theory
The J-integral is a path independent integral, used for characterizing the severity of the loading at a crack tip. It can be used both in an elastic and an elastoplastic analysis. In the case of elasticity, the value of the J-integral can be shown to be identical to the strain energy release rate G, and also to be related to the stress intensity factor K.
In 2D, the J-integral is defined as
(3-216)
where Ws is the strain energy density, σ is a stress tensor, and m is the outward normal of the integration contour Γ. In Equation 3-216, the crack is assumed to extend in the positive x direction, and integration is made over , which can be any closed path around crack tip, see Figure 3-44. However, if there is no loading on the crack face, the contribution to J on Γface is zero, and it is sufficient to only perform the integration over Γ.
Figure 3-44: Circular J-integral around a notch in two dimensions.
For a general crack extension direction p, Equation 3-216 can be generalized to
In 3D, the path integral can be taken around an arbitrary point of the crack front, and Γtot can be any closed path around the point in the normal plane to the crack front. It then turns out that there is also an additional term, to be integrated over the area enclosed by Γtot. The expression is then further extended to
where t is the tangent to the crack front. In practice, this is also the normal to the plane of the integration contour.
In 2D axisymmetry, m and p are located in the rz-plane, and t is oriented in the azimuthal direction. The surface integral can then be simplified, so that
where er is the base vector of the global r direction.
Loads on Crack Faces
When a boundary load FA is applied on the crack face, it contributes with an additional term to the J-integral since the contour integrals on the crack face become nonzero. This additional contribution, Jface, is defined by
where Γface is schematically defined in Figure 3-44.
Computing Stress Intensity Factors From J-Integral
From the J-integral, it is possible to compute the stress intensity factor K for the three modes of fracture:
Computation of the stress intensity factors is done by utilizing the following decomposition of the J-integral
(3-217)
where Eeff is an effective Young’s modulus that accounts for the stress state at the crack front. Assuming that KII = βKI and KIII = γK/ (1 + ν), where β and γ are coefficients that account for the mode mixture, it follows from Equation 3-217 that the stress intensity factors are
(3-218)
The mode mixture coefficients are defined by
where ΔuI, ΔuII, and ΔuIII are the components of the displacement vector across the crack, Δu, defined in the coordinate system corresponding to the three modes of fracture.
In 2D, mode III is not relevant, and Equation 3-218 simplifies to
The definition of the effective Young’s modulus depends on the stress state at the crack tip
Hence, in 2D it depends on the 2D approximation used in the physics interface, while in 3D it is assumed that the stress state is approximately plane strain.