J-integral
[1] The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov[2] and independently in 1968 by James R. Rice,[3] who showed that an energetic contour path integral (called J) was independent of the path around a crack.
Experimental methods were developed using the integral that allowed the measurement of critical fracture properties in sample sizes that are too small for Linear Elastic Fracture Mechanics (LEFM) to be valid.
[4] These experiments allow the determination of fracture toughness from the critical value of fracture energy JIc, which defines the point at which large-scale plastic yielding during propagation takes place under mode I loading.
[1][5] The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading.
[6] This is generally true, under quasistatic conditions, only for linear elastic materials.
For materials that experience small-scale yielding at the crack tip, J can be used to compute the energy release rate under special circumstances such as monotonic loading in mode III (antiplane shear).
The strain energy release rate can also be computed from J for pure power-law hardening plastic materials that undergo small-scale yielding at the crack tip.
Also, Rice showed that J is path-independent in plastic materials when there is no non-proportional loading.
Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence.
The two-dimensional J-integral was originally defined as[3] (see Figure 1 for an illustration) where W(x1,x2) is the strain energy density, x1,x2 are the coordinate directions, t = [σ]n is the surface traction vector, n is the normal to the curve Γ, [σ] is the Cauchy stress tensor, and u is the displacement vector.
The strain energy density is given by The J-integral around a crack tip is frequently expressed in a more general form[citation needed] (and in index notation) as where
If the faces of the crack do not have any surface tractions on them then the J-integral is also path independent.
Rice also showed that the value of the J-integral represents the energy release rate for planar crack growth.
The J-integral was developed because of the difficulties involved in computing the stress close to a crack in a nonlinear elastic or elastic-plastic material.
Hence, The J-integral may then be written as Now, for an elastic material the stress can be derived from the stored energy function
for a closed contour enclosing a simply connected region without any elastic inhomogeneity, such as voids and cracks.
For isotropic, perfectly brittle, linear elastic materials, the J-integral can be directly related to the fracture toughness if the crack extends straight ahead with respect to its original orientation.
) is For Mode III loading, the relation is Hutchinson, Rice and Rosengren [7][8] subsequently showed that J characterizes the singular stress and strain fields at the tip of a crack in nonlinear (power law hardening) elastic-plastic materials where the size of the plastic zone is small compared with the crack length.
Hutchinson used a material constitutive law of the form suggested by W. Ramberg and W. Osgood:[9] where σ is the stress in uniaxial tension, σy is a yield stress, ε is the strain, and εy = σy/E is the corresponding yield strain.
The model is parametrized by α, a dimensionless constant characteristic of the material, and n, the coefficient of work hardening.
If a far-field tensile stress σfar is applied to the body shown in the adjacent figure, the J-integral around the path Γ1 (chosen to be completely inside the elastic zone) is given by Since the total integral around the crack vanishes and the contributions along the surface of the crack are zero, we have If the path Γ2 is chosen such that it is inside the fully plastic domain, Hutchinson showed that where K is a stress amplitude, (r,θ) is a polar coordinate system with origin at the crack tip, s is a constant determined from an asymptotic expansion of the stress field around the crack, and I is a dimensionless integral.
The relation between the J-integrals around Γ1 and Γ2 leads to the constraint and an expression for K in terms of the far-field stress where β = 1 for plane stress and β = 1 − ν2 for plane strain (ν is the Poisson's ratio).
