The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material.[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 elasticmaterials. 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.
The quantity J is not path-independent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. 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. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials.
Two-dimensional J-integral[edit]
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
![{\displaystyle W=\int _{0}^{[\varepsilon ]}[{\boldsymbol {\sigma }}]:d[{\boldsymbol {\varepsilon }}]~;~~[{\boldsymbol {\varepsilon }}]={\tfrac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8498e04defe4fad1820f6588c1718856baa64eb0)
The J-integral around a crack tip is frequently expressed in a more general form[citation needed] (and in index notation) as
where is the component of the J-integral for crack opening in the direction and is a small region around the crack tip. Using Green's theorem we can show that this integral is zero when the boundary is closed and encloses a region that contains no singularities and is simply connected. If the faces of the crack do not have any surface tractions on them then the J-integral is also path independent.
https://en.wikipedia.org/wiki/J-integral
Tuesday, August 10, 2021
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