J integral

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The J-integral represents a way to calculate work (energy) per unit fracture surface area in a material.^[1] The theoretical concept of J-integral was developed in 1967 by Cherepanov ^[2] and in 1968 by Jim Rice^[3] 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 $J I c$ . The quantity $J 1 c$ defines the point at which large-scale plastic yielding during propagation takes place under mode one loading.^[1] Physically the J-integral is related to the area under a curve of load versus load point displacement.^[4].

1 Two-dimensional J-Integral
2 J-Integral and Fracture Toughness
3 See also
4 References
5 External links

[edit] Two-dimensional J-Integral

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

The two-dimensional J-integral was originally defined as^[3] (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 traction vector, $\mathbf{n}$ is the normal to the curve $Γ$ , $σ$ 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 $ε$ is a small reqion around the crack tip. Using Green's theorem we can show that this integral is zero when the boundary $Γ$ is closed and encloses a region that contains no singularities and is simply connected. If the faces of the crack do not have any 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 patch 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 $Γ$ . 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 $σ j k = σ k j$ . 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 a 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 $Γ = Γ 1 + Γ + + Γ 2 + Γ -$ . 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.

[edit] J-Integral and Fracture Toughness

The J-integral can be described as follows ^[1]

$J=\oint_{C} \frac {F}{A}\frac{du}{dl_0}=\int_{}^{}\sigma d\varepsilon\,$

where

F is the force applied at the crack tip
A is the area of the crack tip
$\frac{du}{dl_0}$ is the change in energy per unit length
$σ$ is the stress
$d\varepsilon$ is the change in the strain caused by the stress

Fracture toughness is then calculated from the following equation ^[1]

$J_{1c} = K_{1c}^2(\frac{1-v^2}{E})$

where

$K 1 c$ is the fracture toughness in mode one loading
v is the Poisson's ratio
E is the Young's Modulus of the material

[edit] See also

[edit] References

^ ^a ^b ^c ^d Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials", [1]
^ G. P. Cherepanov, The propagation of cracks in a continuous medium, Journal of Applied Mathematics and Mechanics, 31(3), 1967, pp. 503-512.
^ ^a ^b 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.
^ Meyers and Chawla (1999): "Mechanical Behavior of Materials," 445-448.

[edit] 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)