Computing of stress intensity factor using J-integral method with F.E.A.

Tierean, Mircea ; Baltes, Liana

1. INTRODUCTION

Path-independent integrals have interesting applications in computational fracture mechanics. The reason is that stress-intensity factors are defined by the near tip fields of stress or displacement which are inaccurate in numerical computations. The J-integral discovered independently by Rice (1968) and Cherepanov (1968) originated from a conservation law given by Eshelby (1951) who introduced an integral having the meaning of a generalized force on an elastic singularity in a continuum lattice (Bui, 2006).

The fracture behaviour of linear-elastic materials can be easily analysed with the stress intensity factor (K). To calculate K through analytic methods, there are lots of typical calculus relations of elasticity theory available (Tierean, 1999), valid only in particular cases. The great majority of technical materials have linear-elastic behaviour only in special conditions (temperature, loading velocity, radiation) and the validity domain of analytic formulas is reduced. In these conditions it is necessary to use different methods to study the fracture resistance.

The J-integral approach has a greater validity area than the stress intensity factor method, being applicable to linear-elastic and elastic-plastic materials too. Selecting the favourable integration path, this method can be used for different pieces' geometry, because of path-independence on the integral domain.

2. THEORETICAL ASPECTS

For a correct analysis the finite element method is preferred. Many commercial programs for finite element analysis have fracture mechanics features, using the displacement method.

The stress intensity factor for the opening mode in P(r,9) point, in the crack neighbourhood (fig. 1), can be obtained with the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The J-integral is computed using the expression:

J = [[integral].sub.l]([W.sub.dy] - [bar.p][partial derivative][bar.D]/[partial derivative]x ds), (2)

defined on the contour l (fig. 2), where:

* W is the strain energy density;

* ds is the arc element along the l path;

* [bar.p] is the traction vector;

* [bar.D] is the strain vector on the l path, both of them related to the external perpendicular line [bar.n].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Because of J-integral's solution independence by the domain radius, "r" is chosen whatever the size. If r [right arrow] [infinity] this results into

J = [alpha]/E([K.sup.2.sub.I] + [K.sup.2.sub.II]) + (1 + v)/E [K.sup.2.sub.III], (3)

where

[alpha] = {1 for the plane stress state {1 - [v.sup.2] for the plane strain state. (4)

With this relation the stress intensity factor is determined when the J-integral value is known. Further, on a simple application, comparative results for stress intensity factor using analytical relations and the finite element method will be presented.

3. PLATE WITH TWO EDGE CRACKS EXAMPLE

One considers a square plate (2b=2000 mm), g=10 mm thickness, with two edge cracks (a=120 mm), loaded at traction ([sigma]=50 MPa) (fig. 3). The problem is solved with the analytic solution (5) (Keer & Freedman, 1973), valid for the infinite plate

[K.sub.I] = [sigma][square root of [pi]a 1.12 - 0.61 a/b + 0.13[(a/b).sup.3]/[square root of 1 - a/b] (5)

or with the relation (6) presented in (Gross & Srawley, 1972)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[K.sub.I] = 1.13 [sigma] [square root of [pi]a][[2b/[pi]a tg([pi]a/2b)].sup.1/2]. (6)

Because of the symmetry, only the left up quarter of the plate was used for finite element modelling. An alloyed steel with E=2.1 x [10.sup.11] Pa and v=0.28 was selected as material.

The influence of mesh shape (8-nodes [P8N] rectangular plane elements or 6-nodes [T6N] triangular plane elements) on the results was studied with CosmosM software. Although the path-independence of J-integral was proved by J.R. Rice, in the case of F.E.A. analysis, three different contours have been chosen to study the influence of approximation level to the obtained values. In fig. 4, the meshing using the 8-nodes rectangular plane elements, maximum size 20 mm is presented. The presence of the three integration contours is to be seen.

The analysis with Comsol Multiphysics was done using only triangular plane elements, maximum size 20 mm. In this case three rectangular integration contours were considered, each of them as sub domain, having the same size as in the first case (fig. 5). In fig. 6, von Mises stresses in the plate with two edge cracks are presented, analysed with Comsol Multiphysics. The heavy concentration of stress in the crack tip is to be seen.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The results of the analytic calculi as well as the simulations were presented in table 1. A good correlation between the values obtained with the large integration paths using CosmosM with rectangular plane elements and internal path using Comsol Multiphysics are to be observed.

4. CONCLUSION

The determination of stress intensity factor based on J-integral method was successfully used in the case of F.E.A. software that includes fracture mechanics facilities. With this method results near the analytic solutions were obtained, errors being smaller than 5%.

The unchallenged advantage of this method is the possibility to use it for elastic-plastic materials and different shapes, distinctively from standard geometries for which solutions through elasticity theory methods were determined. Because this integration is path-independent, it could be evaluated along with a path chosen to give the greatest computational advantage.

5. REFERENCES

Baltes, L.S. (2006). J-Integral method, a feasible method for stress intensity factor determination, In: 4th International Conference in Higher Education and Research in the 21st Century, CHER 2006, pp 79-81, ISBN-10:954-580-206-5

Bui, H.D. (2006). Fracture Mechanics. Inverse Problems and Solutions, Springer, ISBN-10 1-4020-4837-8, Dordrecht

Srawley, J.E. & Gross, B. (1972). Engineering Fracture Mechanics, Vol. 4, No. 3, pp 587-589, ISSN: 0013-7944

Keer, L.M. & Freedman, J.M. (1973). International Journal of Engineering Science, Vol 11, No. 12, pp 1265-1275, ISSN: 0020-7225

Tierean, M.H. (1999). Mecanica ruperii (Fracture Mechanics), Editura Lux Libris, ISBN 973-9240-98-4, Brasov

Tab. 1. Computed stress intensity factors.

 Stress intensity factor KI [MPa x [m.sup.1/2]]

Domain Analytic relation (5) Analytic relation (6)

Internal path -- --
Average path -- --
External path -- --

Total 34.27 34.90

 Stress intensity factor KI [MPa x [m.sup.1/2]]

 F.E.A. F.E.A. F.E.A.
Domain CosmosM [P8N] CosmosM [T6N] Comsol Multiphysics

Internal path 35.96 33.54 35.99
Average path 34.94 33.64 41.94
External path 34.81 33.63 43.53

Total -- -- --