·Table of Contents
·Late Received Papers
ON THE NEW METHOD TO COMPUTE THE FATIGUE THRESHOLD - INFLUENCE OF THE MICROSTRUCTURE AND RESIDUAL STRESSES
K.NECIB ; M.A. Belouchrani ; A. BRITAH
Laboratoire Génie des Matériaux,
Bp 17, Bordj-El-Bahri - Algérie,
Tél. : (2132) 86.34.69 /
Fax : (2132) 86.32.04
The aim of this work is to compute a stress intensity factor corresponding to the shakedown state of the cracked structure, by the application of the shakedown theory using the finite element method and a non-linear mathematical optimization. We will show that these factors can be regarded as parameters being able to be assimilated to fatigue threshold taking into accounts the residual stresses that develop in the vicinity of the crack tip and that, if they are in compression, speculate the cause-list of shield against the crack growth. The shakedown load is evaluated for a cracked plate loaded in mode I using the F.E.M.
The first results show that these factors are crack length independent, what involves that they characterize a threshold of the stress intensity under of which, the structure is in a safe state. We have then computed shakedown stress intensity factor for some materials that we have compared to the corresponding fatigue thresholds given by some authors. The ascertain discard are owed essentially to residual stresses that develop in the neighbourhood of the crack tip, and prevent its propagation. We have observed also that there exists a linear relationship between the shakedown stress intensity factor and the diameter of the grain of the material for examples treated in this work.
We have concluded that the application of shakedown theory can be considered as an efficient approach for the estimation of the security of cracked elastic-perfectly-plastic structures and a new method to compute the fatigue threshold.
Materials used in industry are often ductile (metals and alloys). They support irreversible plastic deformations before to break, even when they contain manufacture defects or crack.
It does not exist material totally exempt of defect. The role of the inventor is to distinguish between these defects, these are harmful and that it are not, and are able to subsist in the system builds.
When a crack is detected in a system, the inventor makes an analysis, where it will have to explore the brutal rupture risk, the plastic ruin risk and the prediction of the evolution of the crack. The results of this analysis arouse many questions:
In order that, one makes call the fracture mechanics concepts, which have replaced the classical methods, because of the geometrical singularity fathered by the crack.
- the brutal rupture risk is excluded ?
- the considered crack does not growth ?
- the growth speed allows to guarantee a duration of life in fatigue well superior that the number of anticipated cycle ?
Nevertheless, one knows that the fracture mechanics can not to it alone insure that the crack is admissible. It attends only what happens in the vicinity of the crack tip (front), where the strong stress and deformation can make yield the material, bringing thus the propagation of the crack. Criteria of defect admissibility of the fracture mechanics are therefore by local gasoline, attached to events being able to happen at the crack tip. One knows also that even perfect, a structure can not support unlimited loads, it finishes by ruining.
To take into account these two types of ruin and to anticipate the stability and the security of the cracked structure, Belouchrani (1997) Belouchrani & Weichert (1999), have proposed a new approach by the application of the shakedown theory. They propose an extension of shakedown theory for cracked bodies using the crack analysis developed by Nguyen Quoc Son (1980) in the general framework of the GSM, the thermodynamics of irreversible processes and the theory of crack growth based on the global energy balance of the entire system proposed by Griffith (1921),(1924). The main idea is to split the total dissipation into two parts related to, respectively, plastic flow and crack growth. These phenomena have been, in general, excluded from shakedown analysis. This lack of investigation is explained by the fact that, according to classical shakedown theorems, the shakedown limit load of cracked body should be zero, because elastic stresses at the crack tip are singular.
On the other side, experiments show that there exist a certain limited value for load intensity below which the cracked body is in a safe state and the crack does not propagate.
A first heuristic attempt to bring together shakedown analysis and stable crack propagation can be found in Weichert (1989). Huang & Stein (1996), used an analytical shakedown method developed by the same authors (1994) for the investigation of the influence of a crack on the shakedown behaviour of a sheet under tension. Here, the crack-tip was represented by a notch according to the Neuber concept (1958). The aim of this work is to compute a stress intensity factor corresponding to the shakedown state of the cracked structure, by using the finite element method and a non-linear mathematical optimization. We will show that theses factors can be regarded as parameters being able to be assimilated to fatigue threshold.
Fatigue threshold and influence of the microstructure
The fatigue analysis consists in the study of crack propagation in a structure under an alternated loading. One defines the existence of a threshold KS below of which the crack growth is quasi - vanished. And, we can write :
Among parameters influencing the fatigue threshold, we can cite the microstructure of the material. Generally, KS increases with the grain size, this raise has been observed on ferritic steels by several authors (Radhakrishna  (1984), Taylor  (1984), Wasen et al.  (1988)). It has been suggested that the dependence between KS and x1/2 (x is the diameter of the grain) can have a linear equation [13-15] :
Where KS is in MPa.m1/2 and x in m, a1and b1 are the material constants.
|KI < K S no propagation
Formulation of the shakedown theorem for cracked body
It has been suggested , that a cracked body B, occupying a volume W with a surface G consisting of disjoint parts Gs and Gu , where statical and kinematical conditions are prescribed, respectively, shakes down with respect to given loading history, if a time-independent state of residual stress r 0 exists, such that for all times t > 0 :
With a supplementary condition imposed on the crack length. This condition controls its propagation and avoids the brutal propagation.
Here, F is the plastic yield surface assumed of Von-Mises type, convex by definition, syis the yield stress,ac the critical crack length for which unstable crack propagation occurs and alim is the largest admissible crack length by means of shakedown analysis.
In inequality (5), sc(t) is the time-dependent stress state for a purely elastic comparison problem, differing from the original problem only by the fact that the material reacts purely elastically with the same elastic moduli as for the elastic part of the material law in the original problem.
For the Chaboche & Lemaitre model  adopted for ductile fracture, alim is given by 
Where, a0 is the initial crack length, m and K are material constants characterizing the R-curve parameters and a is the shakedown safety factor. L is a positive definite and time-independent tensor of elastic moduli, with the usual symmetries Lijkl= Lklij=Ljikl=Lijlk .
For the application of the shakedown analysis to the cracked structures, the problem of stress singularity of the elastic stress field deserves special attent ion. In this case, no time-independent field of residual stresses r 0(x) satisfying Eq. (5) can be found and classical shakedown theory does not deliver comprehensive results, even for loads for which limits states exist. We bypass this problem, which is physically in contradiction with the experimentation, by assimilates the crack tip to a sharp notch, similar idea to the concept of the material block introduced by Neuber  (1968).
Fig1: Modified notch and crack|
So following the expression of stress distribution at the neighborhood of the notch root given by Creager  (1966), the equation (8) gives the expression of rf (fig. 1) :
With rf the effective notch root, e the length of the Neuber material block (assumed to be a material constant), and n as factor depending of the loading type. The factor n is equal to 2 in mode I and 0.5 in mode III (Neuber  (1968)). Radaj and Zhang  (1993) gives values of n for the mode II and the mixed mode I-II :
Following this concept the effective notch radius is equal to the original notch radius augmented of n time the dimension of the Neuber material block.
In the case of a sharp crack, the effective crack front radius, denoted rf, can be obtained by putting r=0 in the equation (8) :
Equation (9) indicates implicitly that the crack can be treated as a notch with tip radius rf (fig.1)
Physically, Neubers material block may be explained as being the sum of the minimum number of individual microscopic material particles (such as grains in polycrystalline metals). The properties of which may differ from each others, but in average they should have the property of the macroscopic material. In Huang & Stein  (1996) work, rf is put to be about ten time the size of a grain, for the mode I loading. Following this suggestion we write
where x is the grain diameter.
Shakedown stress intensity factor
We consider an elastic-plastic plate, subjected to variable mode I loading P(t). The values of P(t) vary arbitrarily with time t, but remains between a prescribed loads Pmin and Pmax. One then looks for the maximum value of the load factor a, such that the plate will shake down under the loads aP(t). This load factor will be called the shakedown load factor aSD and can be determined as solution of the following optimization problem :
With the shakedown load factor aSD computed for a cracked plate loaded in mode I, we will compute the stress intensity factor KSD, corresponding to the shakedown state by
The numerical assessment is made on the finite elements method, and on the mathematical programming procedure to maximize the load factor, with the condition that plastic criterion is violated in no point of the plate, and that the crack length remains inferior to its critical length. The resolution of the shakedown analysis problem under constraint, uses the procedure of the Augmented Lagrangien, proposed by Pierre & Lowe  (1975) and necessitates :
To this end, we use the finite element force method based on the principle of minimum complementary energy (Gallagher & Dhalla ). This approach uses stress functions for the construction of the complementary energy function and represents an algebraic dual to the finite element displacement method. This method has been used by Belyschko & Hodge  and Weichert & Gross-Weege , respectively, for the study of limit and shakedown analysis of two-dimensional structures.
- the solution of the problem of purely elastic comparison problem, corresponding to the same boundary conditions that the real problem,
- the construction of a time independent residual stress field.
Influence of the crack length on the shakedown stress intensity factor
The example considered in this work is a rectangular plate containing a lateral crack and subjected to a uniform traction P (fig. 2).
Fig 2: Plate with lateral crack.
Fig 3: Independence of KSD with a/W
We have computed the shakedown stress intensity factor for different crack lengths. Obtained results (fig. 3), show that KSD is independent of the crack length, and can therefore be regarded as a security parameter against the failure of cracked structures by plasticity and crack growth for the mode I of solicitation.
Comparison of the values of KSD to fatigue threshold KS of some materials
To validate the proposed approach, a comparison is made between the values of K SD computed in the case of a rectangular plate solicited in mode I, the fatigue threshold KS given by Wasen et al.  (1988) and the shakedown stress intensity factor Ksh given by Huang & Stein  (1996) for some materials. The characteristics of these materials are given in the Table 1.
||diameter of grain in
|| ssen MPa
|| suen MPa
|Table 1: Diameter of the grain and mechanical material data.|
According to the results given in the Table 2, we remark that the values of KSD agree with the values of Ksh given by Huang & Stein (1996). On the other hand, we notice a disparity with the fatigue threshold KS given by Wasen et al. (1988) for materials A and E, where the yield stress is weak for E and important for A. This is explained by the fact that for the material E, which possesses a weak yield stress, the failure of the structure is caused by the plasticity, what has given a value of KSD < KS. On the other hand, for the material A, as the yield stress is more important, we have an increase of residual stresses in compression at the crack tip, what increases the resistance to the crack growth and gives a value KSD > KS. However, the results indicate that indeed KSD can be considered as a fatigue threshold taking into account residual stresses.
|Table 2: Fatigue threshold and shakedown stress intensity factor|
Influence of the microstructure on the values of KSD
For the materials given in Table 1, we show in figure (4) the influence of the grain size on the obtained shakedown stress intensity factor KSD (Table 2) normalized by the corresponding yield stress, the results show that the ration fluctuates linearly with :
By comparing the equation (17) to (2) relative to the fatigue threshold, one notes that the yield stress influences the shakedown stress intensity factor KSD. This ca be explained by the fact that the shakedown stress intensity factor is computed in supporter counts the yielding of the material and the limit bound of the crack growth. More, knowing that the limit of plastic flow has a relationship with the diameter of the grain in the following form (Wasen et al.  (1988), Xu Dong Li  (1996)) :
ssincreases when the diameter of the grain decreases and as the residual stress intensity increases with a raise of the yield stress, we can say, with the equations (2) and (17) that the disparities ascertained between the values of KSD and KS are produced by the residual stresses.
Fig 4: Relationship between
We conclude that the shakedown stress intensity factors computed in the previous paragraph are just fatigue threshold taken into accounts residual stresses.
In this work, we have computed from shakedown load, a stress intensity factor corresponding to the shakedown state. The first results show that these factors are crack length independent, what involves that they characterize a threshold of the stress intensity under of which, the structure is in a state of security. We have then computed shakedown stress intensity factor for some materials that we have compared to the corresponding fatigue thresholds. The ascertain discard are owed essentially to residual stress that develop in the neighborhood of the crack tip, and prevent its propagation. It was also observed that there exists a linear functional relationship between the shakedown stress intensity factor and the diameter of the grain of the material for the examples treated in this work. The application of shakedown theory can be considered as an efficient approach for the estimation of the security of cracked elastic-perfectly-plastic structures and a new method to compute the fatigue threshold.
- Belouchrani M. A., "Contribution to the shakedown analysis of inelastic cracked structures", Ph. D. Thesis, University of Lille, (1997).
- Belouchrani M. A. and Weichert D., "An extension of the static shakedown theorem to inelastic cracked structures", International Journal of Mechanical Sciences, 41, 163-177, (1999).
- Nguyen Q. S., "Méthodes énergétiques en mécanique de la rupture", Journal de Mécanique, vol.19, N° 2, (1980).
- Griffith A. A., "The phenomena of rupture and flow in solids", Philosophical Transactions of the Royal Society of London A221, 163-198, (1921).
- Griffith A. A., "The theory of rupture", Proceedings of First International Congress of Applied Mechanics, Delft, 55-63, (1924).
- Weichert D., "Failure assessment of structures using refined material laws", In: Advances in constitutive laws for Engineering Materials, Proc. ICCLM 1989, Ed.: Jinghong F. & Murakami S., Pergamon Press, 665-670, (1989).
- Huang Y. and Stein E., "Shakedown of a cracked body consisting of kinematic hardening material.", Engineering Fracture Mechanics, 54, 1, 107-112, (1996)
- Huang Y. and Stein E.; "An analytical method for shakedown problems with linear kinematic hardening materials", International Journal solids and Structures, 31, 18, 2433-2444, (1994).
- Neuber H., Kerbspannungslehre, Springer-verlag, Berlin, (1958).
- Irwin G. R., "Relation of stresses near a crack to the crack extension force", International Congress in Applied Mechanics, pp. 245, Brussels, (1956).
- Barthelemy B., "Notions pratiques de la mécanique de la rupture", Eyrolles, Paris, (1980).
- Bui H. D., "Mécanique de la rupture fragile", Masson, Paris, (1978).
- Radhakrishna V. M., In C. J. BEEVERS (Ed.), Fatigue '84, Proc. 2nd International Conference On Fatigue and Fatigue Thresholds, Birmingham, September 3-7, 1984, Engineering and Materials Advisory Services, Warley, 371, (1984).
- Taylor D., In : C. J. BEEVERS (Ed.), Fatigue '84, Proc. 2nd International Conference On Fatigue and Fatigue Thresholds, Birmingham, September 3-7, 1984, Engineering and Materials Advisory Services, Warley, 479, (1984).
- Wasen J., Hamberg K. and Karlsson B., "The influence of Grain Size and Fracture Surface Geometry on the Near-threshold Fatigue Crack Growth in Ferritic Steels", Materials Science and Engineering A, 102, 217-226, (1988).
- Lemaitre J. and Chaboche J. L.," Mécanique des matériaux solides", Dunod, Paris, (1985).
- Neuber H., "Über die Berücksichtigug der Spannungskonzentration bei Festigkeitsberechnungen", Konstruktion 20 (7), 245-251, (1968).
- Creager M., Master Thesis, Lehigh University, (1966).
- Radaj D. and Zhang S., "On the relation between notch stress and crack stress intensity in plane shear and mixed mode loading", Eng. Frac. Mech., vol. 44, N°5, 691-704, (1993).
- Pierre D. A. and Lowe M. J., "Mathematical programming via Augmented Lagrangians", London : Addision-Wesley, (1975).
- Gallagher R. H. and Dhalla A. K., "Direct flexibility finite element elasto-plastic analysis", Englewood Cliffs, New Jersey, USA, (1975).
- Belytschko T. and Hodge P. G. "Plane Stress limit Analysis by Finites elements", Proc. ASCE, Journal Engineering Mechanics Division, vol. 96, EM 6, 931-944, (1970).
- Gross-Weege J. and Weichert D., "Elastic-plastic shells under variable mechanical and thermal loads", International Journal of Mechanical Sciences, 34, 863-880, (1992).
- Xu-Dong Li and Edwards L., "Theoretical Modeling of Fatigue Threshold for Aluminum Alloys", Engineering Fracture Mechanics, 54, 35-48, (1996).