I am working on the validation of CTOD.

In the evaluation of CTOD using the
J-integral approach as applied in ASTM E1290 and E1820, the plastic J
component is defined as (Npl*Apl)/(Bn*Bo)
Npl=dimensionless constant(plasticity)
Apl=area under load vs. displacement plot
Bn=ligamen length (W-a)
Bo=section thickness

On E1820 there are different functions provided for Npl if Apl is
evaluated differently (CMOD or load-line displacement).

Do anybody know what are the relation of the Npl to Apl and how they interact
for the J-integral evaluation?

Submitted to the Journal of Applied Mechanics on 2/1/2010.

The path-dependence of the J-integral is investigated numerically, via the finite element method, for a range of loadings, Poisson's ratios, and hardening exponents within the context of J2-flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This feature allows for a dense finite element mesh within the plastic zone and accurate infinite boundary conditions. Features of the crack tip field that have been computed previously by others, including the existence of an elastic sector in Mode I loading, are confirmed. The somewhat unexpected result is that J for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for Mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for Mode II. The applications of T or S stresses, whether applied proportionally with the K-field or prior to K, have only a modest effect on the path-dependence.

Hello everyone,

Here is the problem I have: I'm modeling the geometry of a simple straight edge crack in a 2D elastic medium using Abaqus. I assume plain stress conditions. The crack makes an angle with the horizontal, is small enough to be considered as embedded in an infinite domain (ratio crack length/size of domain < 1/10) and I apply a vertical compressive load on top of my domain. I fixed one point in displacement at the bottom of it and the whole bottom edge is constrained not to move vertically.

Abaqus allows the computation of the J-integral and the stress intensity factors KI, KII by use of the Contour integral keyword. Results in tension are pretty accurate, however I seem to get problems when it comes to compressive loads. According to many authors ([Maji91], [Lauterbach98], [Rao03], [Zhu05], [Albrecht06], etc.), this problem gives non-zero shear mode SIF (KII=σ∞√(πa).cosα.sinα), which is understandable, but zero mode I SIF, which I explain as the closed character of the crack under such conditions. Abaqus gives rather good values for KII and their path-independence is correct, but also gives very negative values for KI, of the same order as the values of KII. The J-integral is calculated with Abaqus according to the following formulation (as defined by Rice, for instance):

J=lim_{Γ→0}∫_{Γ}[Wδ1j-σijui,1]njdΓ

This integral is then transformed into a surface integral (using the well-known ramp function q), and an interaction integral method can be used to separate modes I and II stress intensity factors, using the fact that in the framework of linear elastic mechanics in an homogeneous medium, we have:

J=1/E'.(KI2+KII2)

Of course the compressive character of the fields is not taken into account in the definition of the J-integral presented above, which is valid in case the crack is opened. I have been trying to reconstruct the analytical field around the crack tip in the conditions of my problem (that is a slanted crack subjected to a remote compressive vertical load in an infinite domain), but did not quite manage to do so using simple terms. My questions are:

1. Do you think there would be any way to "extract" the compressive part of the stress field and substract it from the whole field to get the singular field around the crack tip -- then use this field to compute the SIF?
2. Does the negative character of mode I stress intensity factor physically mean anything? I personally would not think it does, since plugging those negative values of KI into the singular displacement field fomulation around a crack tip (in √r) would impose an interpenetration of the crack lips.
3. Is there any closed-form formulation for the displacement/stress fields for such a problem?

Thank you in advance for any remarks/suggestions you may have regarding my issue.

Kind regards,

Julien Jonvaux

Ph.D student at the University of Illinois at Chicago