A substantial fraction of the mysteries associated with crack extension might be eliminated if the description of fracture experiments could include some reasonable estimate of the stress conditions near the leading edge of a crack particularly at points of onset of rapid fracture and at points of fracture arrest. It is pointed out that for somewhat brittle tensile fractures in situations such that a generalized plane-stress or a plane-strain analysis is appropriate, the influence of the test configuration, loads, and crack length upon the stresses near an end of the crack may be expressed in terms of two parameters. One of these is an adjustable uniform stress parallel to the direction of a crack extension. It is shown that the other parameter, called the stress-intensity factor, is proportional to the square root of the force tending to cause crack extension. Both factors have a clear interpretation and field of usefulness in investigations of brittle-fracture mechanics.
Cracks are present in all structures, they can exist in the basic defect form in the material or may be induced during construction, these cracks are the main reason for the most failures that occur in structures and parts of machines in service, subjected to static or dynamic forces 1. The purpose of fracture mechanics is to study and predict the cracks initiation and propagation in solids. The start of the study of brittle materials rupture began in 1920, with the work of Griffith, before reappearing in the 1950s and 1960s, when the discipline took off really with the new works of Irwin and Rice. As for the study of the rupture of ductile materials, it only begins at the late of 1960s and through the 1970s, with the fundamental works of Rice and Tracey 2 and Gurson 3. Linear elastic mechanics is interested in the rupture of brittle materials. It is widely used by engineers because it allows the use of global energy criteria such as stress intensity factors FIC 4. Griffith 5 was interested in the problem of rupture in an elastic cracked medium from an energetic view point. He thus highlighted a variable called later the rate of energy restitution characterizing the fracture, and whose critical value is a characteristic of material 6. The first theoretical developments in the analysis of stress and strain fields nearly to a crack in elasticity. These studies, in particular by Irwin 7, allowed to define the FIC, characterizing the state of stress of the region in which the rupture occurs. The development of the finite element method made it possible to study numerical the mechanic of rupture, thus proposing more precisely solutions to more complex problems. Then appeared a multitude of methods allowing to calculate the stress intensity factors 8. Among these methods the method of the principle of superposition, extrapolation of displacements and the collocation method borders 9.
In this work, finite element method was used to determine the normalized stress intensity factors for different configurations. For this, a 2-D numerical analysis with elastic behavior was undertaken in pure I mode. This simulation was carried out using a numerical calculation code. On the basis of the numerical results obtained from the different models treated, there is a good correlation between the nodal displacement extrapolation method (DEM) and the energy method based on the Rice integral (J) to evaluate the normalized stress intensity factors and this for different crack lengths. For each configuration, the increase in the crack size causes an amplification of normalized intensity stresses fators.
G.R. Irwin., "Estimates of stress intensily and rivet force for a crack arrested by arivited stiffener. Discussion based on 'Analysis of stress and strains near the end of a crack traversing a plate," Journal of Applied Mechanics, vol. 24, pp. 361-364, 1957.
Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic. 
But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics is small scale yielding, the condition that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures.
In the mid-1960s James R. Rice (then at Brown University) and G. P. Cherepanov independently developed a new toughness measure to describe the case where there is sufficient crack-tip deformation that the part no longer obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear elastic (or monotonic deformation theory plastic) deformation ahead of the crack tip, is designated the J-integral. This analysis is limited to situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. It also demands that the assumed non-linear elastic behavior of the material is a reasonable approximation in shape and magnitude to the real material's load response. The elastic-plastic failure parameter is designated JIc and is conventionally converted to KIc using the equation below. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior.
The elastic-plastic multi-scale finite element method is formulated and applied to the analysis of thefracture test on compact tension specimen with 304 stainless steel. The calculation was carried out onABAQUS. The elastic-plastic stress field near the crack tip, and the plastic zone are first determined insufficient detail. The numerical results show that elastic-plastic stress singularity exists at the cracktip. Then, the critical updated Mises stress intensity factors of compact tension specimen and middlecracktension specimen with 304 stainless steel are determined by using the fracture test results of thespecimens. It is found that the magnitudes of these two critical updated Mises stress intensity factorsare related to the specimen and differ greatly, they cannot be regarded as a characterization of theinherent fracture toughness of 304 stainless steel. This study shows that: the elastic-plastic multi-scalefinite element method can offer the numerical results of the mechanical parameters of the singular stressfield near the crack tip accurately enough, it provides an analytical basis for the development of fracturecriteria of elastic-plastic fracture mechanics; combined with the fracture experiments of the crackedspecimens, the elastic-plastic multi-scale finite element method provides an effective numerical analysisfor the experimental verification of elastic-plastic fracture criterion.
Early in the 1970s and 1980s, Rice et al.  first devoted to the development of elasticplasticfinite element method, for the study of elastic-plastic fracture mechanics [13-17].Thereafter, the finite element analysis of elastic-plastic deformation near the crack tip has beenthe focus of research [18-24]. Based on the elastic-plastic finite element analyses of compacttension specimen with ABAQUS, the CTOD estimation obtains improvements of nearly 25%over the existing formula . However, in order to evaluate the reliability of the existingelastic-plastic fracture criteria more accurately, the finite element analysis technique for theelastic-plastic field near the crack tip has to be further improved. There are two issues thatneed special attention: 1. As approaching the crack tip, the accuracy of finite element analysismust be maintained; 2. Because the elastic-plastic deformation near the crack tip will reach avery large extent, it is necessary to mesh the crack tip finite elements appropriately to avoidthe calculation interruption due to the excessive shape distortion of the crack tip elements.
The present paper focused on the application of the elasticplasticmulti-scale finite element method in Elastic-Plastic FractureMechanics (EPFM). The Elastic-Plastic Multi-Scale Finite ElementMethod (EPMSFEM) complemented with incremental theory ofplasticity is formulated and applied to the analysis of the fracturetest on compact tension specimen with 304 stainless steel (304-C(T)). The experimental curve of engineering stress vs engineeringstrain of 304 stainless steel is fitted by Ramberg-Osgood equation. Based on the Ramberg Osgood equation, the numericalrelationship between true stress and logarithmic plastic strain isobtained. The multi-scale finite element mesh near the crack tip of304-C(T) is optimized. So that, even if the plastic deformation ofcrack tip elements is up to a large extent, the whole calculation canbe completed in one loading step without re-meshing. On this basis,the elastic-plastic stress field near the crack tip, and the plastic zoneare first determined in sufficient detail. From the numerical results,it is seen that the crack tip field of 304-C(T) shows the characteristicsof elastic-plastic stress singularity, even if the crack tip is in largescaleyield. So, the updated Mises stress intensity factor can becalculated with the output data. Then, the critical updated Misesstress intensity factor of compact tension specimen and middlecracktension specimen of 304 stainless steel are determined byusing the fracture test results of the two kinds of specimens .Because the magnitudes of these two critical updated Mises stressintensity factors are related to the specimen and differ greatly, theycannot be regarded as a characterization of the inherent fracturetoughness of 304 stainless steel. The research in this paper showsthat: the elastic-plastic multi-scale finite element method can offerthe numerical results of the mechanical parameters of the singularstress field near the crack tip accurately enough, it provides ananalytical basis for the development of fracture criteria of elasticplasticfracture mechanics; combined with the fracture experimentsof the cracked specimens, the elastic-plastic multi-scale finiteelement method provides an effective numerical analysis for theexperimental verification of elastic-plastic fracture criterion. 2b1af7f3a8