Introduction
This is article 2 in my series on fracture mechanics of polymers. The focus of this article is how to extend basic Linear Elastic Fracture Mechanics (LEFM) to non-linear elastic materials. This modification is the foundation of Elastic-Plastic Fracture Mechanics (EPFM) and the J-integral. In this article I will derive the J-integral from scratch. It is pretty cool stuff.
J-Integral Theory
- LEFM is only valid if limited viscoplastic deformation occurs in a small region surrounding the crack tip.
- Elastic-plastic fracture mechanics can be used for materials that exhibit time-independent non-linear elastic behavior.
- The theory was derived by J.R. Rice (1968).
- Personal note: I took a graduate class from Prof. Rice at Harvard during my time at MIT. It was an excellent experience. The derivation that I will reproduce here is based on Rice’s original paper that has > 11,000 citations. That is pretty impressive! Also, did you know that Jim Rice received his Ph.D. at age 24. Hard to beat…
J.R. Rice
To start the derivation of the J-integral I will consider a 2D body with a crack. The potential energy for this body can be written:
\(\displaystyle\Pi(a) = \int_A W dA – \int_{\Gamma_T} T_i du_i ds\).
In this equation: W=strain energy density, \(T_i\) = surface tractions, \(u_i\) = displacement in direction i, \(\Gamma\) = path around the crack, a = crack length.
Next, take the derivative of the potential energy with respect to the crack length:
\( \displaystyle\frac{d\Pi}{da} = \int_A \frac{dW}{da}dA – \int_{\Gamma} T_i \frac{du_i}{da} ds.\)
Note that we can replace \(\Gamma_T\) with \(\Gamma\) since \(du_i/da=0\) on \(\Gamma_u\). So far, we have used a coordinate system located at the center of the edge crack. Now introduce a new coordinate system X1-X2 located at the tip of the crack. The relation between the two coordinate systems is: \(X_i = x_i – a \delta_{i1}\). The chain rule then gives:
\( \displaystyle\frac{d}{da}= \frac{\partial}{\partial a} – \frac{\partial}{\partial x_1} \)
The derivative of the potential energy with respect to crack length can therefore be written:
\( \displaystyle\frac{d\Pi}{da} = \int_A \left( \frac{\partial W}{\partial a} – \frac{\partial W}{\partial x_1} \right) dA – \int_{\Gamma} T_i \left(\frac{\partial u_i}{\partial a} – \frac{\partial u_i}{\partial x_1} \right)ds.\)
Utilize the following ‘trick’ (this is where the restriction to a non-linear elastic material response comes in):
\(\displaystyle \frac{\partial W}{\partial a} = \frac{\partial W}{\partial \varepsilon_{ij}} \frac{\partial \varepsilon_{ij}}{\partial a} = \sigma_{ij} \frac{\partial \varepsilon_{ij}}{\partial a}, \)
and therefore:
\(\displaystyle \int_A \frac{\partial W}{\partial a} dA = \int_A \sigma_{ij} \frac{\partial \varepsilon_{ij}}{da}dA = {\text{[Princ. Virtual Work]}} = \int_{\Gamma} T_i \frac{\partial u_i}{\partial a} ds.\)
Hence:
\(\displaystyle\frac{d\Pi}{da} = -\int_A \frac{\partial W}{\partial x_1} dA + \int_{\Gamma} T_i \frac{\partial u_i}{\partial x_1} ds. \)
Finally, we can convert the area integral to a path integral using the divergence theorem (recall: \(\int_A B_{i,i} dA = \int_{\Gamma} B_i n_i ds\)), where \(n_i\) is the outward normal to \(\Gamma\), i.e. \(n_1 ds = dx_2\). This gives the J-integral in “classical form”:
\( J \displaystyle\equiv \left( W n_1 – T_i \frac{\partial u_i}{\partial x_1}\right) ds.\)
- The J-integral is equal to the energy release rate (the change in potential energy due to crack growth).
- The J-integral can also be used for non-linear elastic materials (and elastic-plastic materials if no unloading).
- The J-integral is a global energy-based criterion.
- The J-integral can also be thought of as the energy flow into the crack tip.
- It is intuitively clear that the J-integral is path independent, and it is easy to prove that mathematically. I will leave that as an exercise.
Alternative J-Integral Formulation 1
\( J = \displaystyle\int_{\Gamma_0} \mathbf{C} \mathbf{n} \cdot \mathbf{e}_c ds\)
where \(\mathbf{C}=W \mathbf{I} – \mathbf{F}^T \mathbf{S}^{PK1}\) is the Eshelby tensor, \(\Gamma_0\) is the path in the reference configuration, n is the normal to the path, \(\mathbf{e}_c\) is the crack growth direction, and ds is an increment along the path. Note the following units: [C] = N/m2, [J] = J/m2 = N/m.
Alternative J-Integral Formulation 2
\(J = -\int_{A_0} \mathbf{C} : \nabla_X \mathbf{q} dA\)
where \(A_0\) is the area around the crack tip in the reference configuration, \(\nabla_X\) is the Lagrangian gradient, q is a virtual crack extension vector with length 1. This equation is particularly suitable for FE analysis.
Stress Intensity Factor
Hutchinson, Rice, and Rosengren (HRR) derived the stress and strain field at a crack for the case when the material is non-linear elastic following the Ramberg-Osgood material model:
\(\varepsilon = \displaystyle\frac{\sigma}{E} + K \left( \frac{\sigma}{E} \right)^n\).
In this equation, K is the hardening modulus, and n is the hardening coefficient. Under these conditions the stress field is given by:
\( \displaystyle\sigma_{ij} = k_1 (\theta,n) \cdot \left( \frac{J}{r} \right)^{1/(1+n)}. \)
Notes:
- For real materials the stress and strain do not go to infinity at the crack tip due to crack tip blunting.
- Similar in concept to LEFM, the J-integral is useful for quantifying the magnitude of the stress field if there is an area surrounding the crack that is described by the HRR fields.
- The J-integral is better suited than LEFM for predicting failure of polymers.
- Personal notes: I also took a graduate class from Prof. Hutchinson at Harvard. He has a very keen sense of how to solve mechanics of materials problems. Also Prof. Hurchinson received his Ph.D. at age 24.
J.W. Hutchinson
