Sponsor: Northrup-Grumman Aerospace Corp.
Investigators: Prof. A.R. Ingraffea and Dr. P. Wawrzynek.
Student Investigator: Changyu Hwang (Ph.D.).
Objective: To create the capability to calculate energy release rates and their higher order derivatives for a system of multiple interacting cracks
Publications: Hwang, C., Wawrzynek, P.A. and A.R. Ingraffea,"On the Virtual Crack Extension Method for Calculation of Rates of Energy Release Rate", Engineering Fracture Mechanics, Vol.59 No.4, pp521-541, 1998
OVERVIEW:
In some areas of fracture mechanics, higher order derivatives of energy release rate due to crack extension are required for prediction of the stability and arrest of a single crack, stability of multiple crack systems, and the prediction of fatigue crack growth rate. In multiple crack systems for example, cracks often interact. In this case, the variation of energy release rate at one crack tip due to the growth of any other crack must be calculated to determine the strength of the interaction. Another use of the higher order derivatives is for the size effect "law" which relates the nominal stress at maximum load (nominal strength) to the structure size. In the universal size effect model proposed by Bazant [1], the first and the second order derivatives of energy release rate are needed.
Hence, an important requirement of fracture mechanics analysis is to accurately evaluate the energy release rate and its higher order derivatives for a body containing multiple cracks subjected to arbitrary loadings, including crack-face loading, thermal loading and body forces. A new virtual crack extension method which provides the direct integral forms of energy release rate and its first order derivative was introduced by Lin and Abel [2]. This technique keeps all the advantages of the similar virtual crack extension techniques introduced by deLorenzi [3-4], Haber and Koh [5], and Barbero and Reddy [6] while adding the capability of calculating higher order derivatives of energy release rate.
This study further develops the analytical virtual crack extension method presented by Lin and Abel. It provides derivations for the verification of the following extension to the general case of multiple crack systems, extension to the axisymmetric case, inclusion of crack-face and thermal loading and evaluation of the second order derivative of energy release rate.
Several 2-D numerical examples with exact solutions or with solutions available in the literature are solved to demonstrate the accuracy of the current method. These examples include: a pressurized crack in an infinite plane for the crack-face loading case, a center cracked infinite plane subjected to a remote stress, to show the evaluation of the second order derivative of energy release rate, a circular crack subjected to symmetric point loads in an infinite 3-D space, to illustrate an axisymmetric case, and a system of multiple parallel edge cracks subjected to thermal loading in a semi-infinite plane.
It is also shown that the number of rings of elements surrounding the crack tip that are involved in the mesh perturbation due to the virtual crack extension has an effect on the solution accuracy for higher order derivatives of energy release rate. When more rings of surrounding elelements from the crack tip are used in the perturbation, more accurate solutions for the higher order derivatives of energy release rate are obtained. The maximum computed errors were about 0.2 % for energy release rate, 2-3 % for its first order derivative and 10 % for its second order derivative between the simulated solutions with the proposed method and the true infinite medium solutions for the mesh density used in the examples.
REFERENCES
1. Z.P. Bazant, Fract.ure Mechanics of Concrete strructures,
Proceedings FRAMCOS-2, (1995) 515-534.
2. S.C. Lin and J.F. Abel, Int. J. of Fracture 38 (1988)
217-235
3. H.G. deLorenzi, Int. J. of Fracture 19 (1982)
183-193
4. H.G. deLorenzi, Eng. Fract. Mechanics 21 (1985) 129-143.
5. R.B. Haber and H.M. Koh, Int. J. for Num. Methods in Eng. 21 (1985) 301-315.
6. E.J. Barbero and J.N. Reddy, Communications in applied num.
methods, 6 (1990) 507-518