1. Critical Plane Approach in Stage I and Stage II of Fatigue Under Multiaxial Loading A. KAROLCZUK E. MACHA Opole University of Technology, Department of Mechanics and Machine Design, POLAND XIII INTERNATIONAL COLLOQUIUM, “MECHANICAL FATIGUE OF METALS” September 25-28, 2006, Ternopil, Ukraine

2. 2 Plan of the presentation Introduction A brief review of some multiaxial fatigue failure criteria based on the critical plane approach - The Findley Criterion - The Matake Criterion - The Maximum Normal Stress Criterion on the Critical Plane Damage Degree Accumulation and Fatigue Life Calculation Fatigue tests Stress and strain computations Evaluation and discussion Conclusions

3. 3 Introduction The critical plane approach (Stanfield 1935) This approach is based upon the experimental observation that fatigue cracks initiate and grow on certain material planes. Assumption: Only stress or/and strain components acting on the critical plane are responsible for the material fatigue failure. The critical plane criteria define different functions that combine the shear and normal stress or/and strain components on a plane into one equivalent parameter.

4. 4 Introduction Problems: - orientation of the critical plane - orientation of the fracture plane - It is commonly accepted that depends on the test conditions (loading level, temperature, material type, state of stress, ect.) material generally forms one of the two types of cracks - shear cracks or tensile cracks. - The shear cracks are formed on the maximum shear stress plane and Forsyth called this process as Stage I. - The tensile cracks are formed in Stage II that is predominated by the maximum normal stress component. Either under multiaxial and uniaxial fatigue tests the cracks may initiate and propagate on different planes – contradictory to the one critical plane orientation.

5. 5 Multiaxial fatigue failure criteria The Findley Criterion The critical plane orientation coincides with the plane orientation where the maximum value of this linear combination occurs. where  af, m  are the fatigue limit and the exponent of the S-N curve for fully reversed ( R=-1 ) torsion loading, respectively; N f is the considered number of cycles to failure; N  is the number of cycles corresponding to the fatigue limit  af for fully reversed torsion loading.

6. 6 Multiaxial fatigue failure criteria The Findley criterion adapted to random loading: The equivalent shear stress history  eq (t) at observation time T is then used as the cyclic counting variable.

7. 7 The Matake Criterion Multiaxial fatigue failure criteria The critical plane orientation coincides with the maximum shear stress amplitude The Matake criterion adapted to random loading:

8. 8 The Maximum Normal Stress Criterion on the Critical Plane (max{  n }) Multiaxial fatigue failure criteria where m  is the exponent of the S-N curve for fully reversed (R=-1) push-pull loading; N  is the number of cycles corresponding to the fatigue limit  af. For random loading the equivalent stress history is as follows

9. 9 Damage Degree Accumulation For the variable-amplitude loading, the linear damage accumulation hypotheses of Sorensen-Kogayev is used. where damage degree is computed by the general equation as follows where F is the generalised fatigue damage parameter (for the Findley criterion: F=  ), n (i) is the number of cycles assigned into the i- th stress level, a is a coefficient allowing to include amplitudes below F af in the damage accumulation, is a computed number of cycles to failure (S-N curve) for the i-th stress level NfNf FaFa S-N curve

10. 10 Damage Degree Accumulation For Serensen-Kogayev p is calculated according to the following equations: where f (i) is the frequency of the i-th stress/strain level, is the maximum amplitude of the generalised fatigue damage parameter (F= , , or  ) Accumulated damage degree D at observation time T is used to estimate the fatigue life according to the following expression

11. 11 Fatigue tests Fatigue tests were performed on the round full cross-section specimen made of 18G2A (A 765-94 ASTM) steel under constant- and variable-amplitude combined bending and torsion moments (bending: M b (t), torsion: M t (t)) NoδM b,max M radNm-[o][o][o][o] Constant-amplitude loading 108.0; 10.0; 10.30.6818.1 17.1  19.0 206.4; 7.4; 8.2; 9.80.9621.9 20.0  23.8 305.3; 6.2; 7.2;1.4426.5 23.8  29.2 4π/28.9; 9.2; 9.6; 10.30.6812.3 9.1  15.5 5π/28.30.988.4 7.3  9.5 6π/26.4; 7.21.4210.2 6.4  13.9 Variable-amplitude loading 7-18.4 (torsion)∞43.6/86.3 42.2  45.0/82.3  90.2 8-16.3 (bending)01.5 0.8  2.2

12. 12 Fatigue tests (i)one general crack orientation at the macroscale was observed under all the investigated constant-amplitude loadings (  =  zx,max /  zz,max  0.71) and under the variable-amplitude bending; (i)(i) (ii) (ii)two crack orientations were observed for the specimens subjected to variable-amplitude torsion. The first orientation with a crack length of around 0.3-0.6 mm is parallel to the specimen axis. The other orientation comes from branching of the primary crack, and these branching directions are inclined to the specimen axis by around 45 o Two fatigue crack behaviours were noticed:

13. 13 Stress and strain computations Stress and strain histories in an arbitrary point ( x, y ) of the specimen cross- section were computed from bending and torsion moments M b (t), M t (t) considering the plastic strains. The Chu multi-surface plasticity model of material behaviour with kinematic hardening was applied to determine the strain-stress relation and the influence of loading history on the strain state at each point of the specimen cross-section. It is assumed that the material in some regions of the specimen cross- section could be in the elastic–plastic state. However, it was also assumed, according to hypothesis that the beam cross-section remains flat, that the strains change along the specimen cross-section in a linear way.

14. 14 For every increment of bending  M b (t) and torsion  M t (t) moments the following quasi-static equilibrium equations were solved by the trust-region method Stress and strain computations Quarter of specimen cross-section and mesh

15. 15 Exemplary stress courses

16. 16 Evaluation and discussion Evaluation concerns: Orientation of the critical plane versus fracture plane orientation Fatigue lives Analysis was performed with the use of local approach, hot spot: y=R, x=0

17. 17 Evaluation and discussion Orientation of the critical plane versus fracture plane orientation The following error parameters were used to compare the experimental macroscopic fracture plane orientation with the critical plane orientation

18. 18 Evaluation and discussion Fatigue lives The following errors parameters were applied for the fatigue life verification: The mean error parameter E m reflects the general results conformity. The standard deviation error parameter E std is the superior parameter since it reflects the scatter of the results and therefore gives us the information about the failure criterion ability to correlate the different kind of multiaxial stress states and the equivalent damage parameter.

19. 19 Evaluation and discussion Comparison between the experimental fatigue lives T exp and the calculated fatigue lives T cal : (a) the maximum normal stress criterion, (a)(b) (b) the maximum shear stress criterion (the Findley criterion for k = 0)

20. 20 Evaluation and discussion (c)(d) Comparison between the experimental fatigue lives T exp and the calculated fatigue lives T cal : (c) the Matake criterion, k=0.4, It was assumed that selection of fatigue criterion could be made by the maximum damage degree computed by two simple criteria, i.e. the maximum normal stress criterion and the maximum shear stress criterion. For each specimen, the damage degree on the critical plane is computed by these two criteria (max{  n,  ns }) and than the fatigue life T cal is determined by the highest damage degree. (d) the max{  n,  ns } criterion

21. 21 Evaluation and discussion

22. 22 Conclusions The following general conclusion appears: 1.The critical plane and the fracture plane notions must be separated. The critical plane is simply a plane that used in the fatigue life assessment. The fracture plane at the microscale/macroscale is a plane where material cohesion is lost. 2.Depending on loading levels, state of stress etc. the critical plane and fracture plane orientations may or not coincide. We postulate that the critical plane approach may be successfully used in the fatigue life estimation under different test conditions but the proposed damage parameter should be equivalent to the uniaxial one not only in term of the total fatigue life but also in term of the macroscopic fracture plane behaviour.