J integral

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by Cherepanov and in 1968 by Jim Rice independently, who showed that an energetic contour path integral (called J) was independent of the path around a crack.

Later, experimental methods were developed, which allowed measurement of critical fracture properties using laboratory-scale specimens for materials in which sample sizes are too small and for which the assumptions of Linear Elastic Fracture Mechanics (LEFM) do not hold, and to infer a critical value of fracture energy $J_{\rm Ic}$. The quantity $J_{\rm Ic}$ defines the point at which large-scale plastic yielding during propagation takes place under mode one loading.

The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading. This is true, under quasistatic conditions, both for linear elastic materials and for materials that experience small-scale yielding at the crack tip.

Two-dimensional J-integral

Figure 1. Line J-integral around a notch in two dimensions.

The two-dimensional J-integral was originally defined as (see Figure 1 for an illustration)

$J := \int_\Gamma \left(W~dx_2 - \mathbf{t}\cdot\cfrac{\partial\mathbf{u}}{\partial x_1}~ds\right)$

where $W(x_1,x_2)$ is the strain energy density, $x_1, x_2$ are the coordinate directions, $\mathbf{t} = \mathbf{n}\cdot\boldsymbol{\sigma}$ is the surface traction vector, $\mathbf{n}$ is the normal to the curve $\Gamma$, $\sigma$ is the Cauchy stress tensor, and $\mathbf{u}$ is the displacement vector. The strain energy density is given by

$W = \int_0^\epsilon \boldsymbol{\sigma}:d\boldsymbol{\epsilon} ~;~~ \boldsymbol{\epsilon} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right] ~.$

The J-Integral around a crack tip is frequently expressed in a more general form (and in index notation) as

$J_i := \lim_{\epsilon\rightarrow 0} \int_{\Gamma_\epsilon} \left(W n_i - n_j\sigma_{jk}~\cfrac{\partial u_k}{\partial x_i}\right) d\Gamma$

where $J_i$ is the component of the J-integral for crack opening in the $x_i$ direction and $\epsilon$ is a small region around the crack tip. Using Green's theorem we can show that this integral is zero when the boundary $\Gamma$ 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.

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. Rice showed that if monotonic loading was assumed (without any plastic unloading) then the J-integral could be used to compute the energy release rate of plastic materials too.

Proof that the J-integral is zero over a closed path

To show the path independence of the J-integral, we first have to show that the value of $J$ is zero over a closed contour in a simply connected domain. Let us just consider the expression for $J_1$ which is $J_1 := \int_{\Gamma} \left(W n_1 - n_j\sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) d\Gamma$ We can write this as $J_1 = \int_{\Gamma} \left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right)n_j d\Gamma$ From Green's theorem (or the two-dimensional divergence theorem) we have $\int_{\Gamma} f_j~n_j~d\Gamma = \int_A \cfrac{\partial f_j}{\partial x_j}~dA$ Using this result we can express $J_1$ as $\begin{align} J_1 & = \int_{A} \cfrac{\partial}{\partial x_j}\left(W \delta_{1j} - \sigma_{jk}~\cfrac{\partial u_k}{\partial x_1}\right) dA \\ & = \int_A \left[\cfrac{\partial W}{\partial x_1} - \cfrac{\partial\sigma_{jk}}{\partial x_j}~\cfrac{\partial u_k}{\partial x_1} - \sigma_{jk}~\cfrac{\partial^2 u_k}{\partial x_1 \partial x_j}\right]~dA \end{align}$ where $A$ is the area enclosed by the contour $\Gamma$. Now, if there are no body forces present, equilibrium (conservation of linear momentum) requires that $\boldsymbol{\nabla}\cdot\boldsymbol{\sigma} = \mathbf{0} \qquad \implies \qquad \cfrac{\partial\sigma_{jk}}{\partial x_j} = 0 ~.$ Also, $\boldsymbol{\epsilon} = \tfrac{1}{2}\left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T\right] \qquad \implies \qquad \epsilon_{jk} = \tfrac{1}{2}\left(\cfrac{\partial u_k}{\partial x_j} + \cfrac{\partial u_j}{\partial x_k}\right) ~.$ Therefore, $\sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \tfrac{1}{2}\left(\sigma_{jk}\cfrac{\partial^2 u_k}{\partial x_1 \partial x_j} + \sigma_{jk}\cfrac{\partial^2 u_j}{\partial x_1 \partial x_k}\right)$ From the balance of angular momentum we have $\sigma_{jk} = \sigma_{kj}$. Hence, $\sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \sigma_{jk}\cfrac{\partial^2 u_j}{\partial x_1 \partial x_k}$ The J-integral may then be written as $J_1 = \int_A \left[\cfrac{\partial W}{\partial x_1} - \sigma_{jk}~\cfrac{\partial\epsilon_{jk}}{\partial x_1}\right]~dA$ Now, for an elastic material the stress can be derived from the stored energy function $W$ using $\sigma_{jk} = \cfrac{\partial W}{\partial\epsilon_{jk}}$ Then, using the chain rule of differentiation, $\sigma_{jk}~\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial\epsilon_{jk}}~\cfrac{\partial\epsilon_{jk}}{\partial x_1} = \cfrac{\partial W}{\partial x_1}$ Therefore we have $J_1 = 0$ for a closed contour enclosing a simply connected region without any stress singularities.

Proof that the J-integral is path-independent

Figure 2. Integration paths around a notch in two dimensions. Consider the contour $\Gamma = \Gamma_1 + \Gamma^{+} + \Gamma_2 + \Gamma^{-}$. Since this contour is closed and encloses a simply connected region, the J-integral around the contour is zero, i.e. $J = J_{(1)} + J^{+} - J_{(2)} - J^{-} = 0$ assuming that counterclockwise integrals around the crack tip have positive sign. Now, since the crack surfaces are parallel to the $x_2$ axis, the normal component $n_1 = 0$ on these surfaces. Also, since the crack surfaces are traction free, $t_k = 0$. Therefore, $J^{+} = J^{-} = \int_{\Gamma} \left(W n_1 - t_k~\cfrac{\partial u_k}{\partial x_1}\right) d\Gamma = 0$ Therefore, $J_{(1)} = J_{(2)}$ and the J-integral is path independent.

J-integral and fracture toughness

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.

For plane strain, under Mode I loading conditions, this relation is

$J_{\rm Ic} = G_{\rm Ic} = K_{\rm Ic}^2 \left(\frac{1-\nu^2}{E}\right)$

where $G_{\rm Ic}$ is the critical strain energy release rate, $K_{\rm Ic}$ is the fracture toughness in Mode I loading, $\nu$ is the Poisson's ratio, and E is the Young's modulus of the material.

For Mode II loading, the relation between the J-integral and the mode II fracture toughness ($K_{\rm IIc}$) is

$J_{\rm IIc} = G_{\rm IIc} = K_{\rm IIc}^2 \left[\frac{1-\nu^2}{E}\right]$

For Mode III loading, the relation is

$J_{\rm IIIc} = G_{\rm IIIc} = K_{\rm IIIc}^2 \left(\frac{1+\nu}{E}\right)$

See also

• Fracture toughness
• Toughness
• Fracture mechanics
• Stress intensity factor

External links

• J. R. Rice, "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks", Journal of Applied Mechanics, 35, 1968, pp. 379-386.
• Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [2]
• Fracture Mechanics Notes by Prof. Alan Zehnder (from Cornell University)
• Nonlinear Fracture Mechanics Notes by Prof. John Hutchinson (from Harvard University)
• Notes on Fracture of Thin Films and Multilayers by Prof. John Hutchinson (from Harvard University)
• Mixed mode cracking in layered materials by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
• Fracture Mechanics by Prof. Piet Schreurs (from TU Eindhoven, Netherlands)
• Introduction to Fracture Mechanics by Dr. C. H. Wang (DSTO - Australia)
• Fracture mechanics course notes by Prof. Rui Huang (from Univ. of Texas at Austin)