Issue 33

V. Shlyannikov et alii, Frattura ed Integrità Strutturale, 33 (2015) 335-344; DOI: 10.3221/IGF-ESIS.33.37 340 In accordance with the approach of Hutchinson [11], the plastic stress intensity factor P K in pure Mode I (or pure Mode II) can be expressed directly in terms of the corresponding elastic stress intensity factor using Rice’s J -integral. That is       2 2 1 1 0 ; ' ' n n P K J I K E E      (6)             ; 1 1 2 1 2 0 1 1 2 0 2 1                         n n n n P I waY I K K           waY K 1 1     (7) where w K K 1 1  is normalized by a characteristic size of cracked body elastic stress intensity factor and ' E E  for plane stress and   2 ' 1 E E    for plane strain. In the above equations,  and n are the hardening parameters, wa   is the dimensionless crack length, w is characteristic size of specimen (for our case that is specimen diameter),  is the nominal stress, and  0 is the yield stress. The numerical constant   n I  is obtained from the singularity analysis by means of the conjugation solutions for the far and near fields. For small-scale yielding, i.e., when the plastic zone near the crack tip is quite small compared with the crack length, the amplitude of the singularity P K can be determined directly by application of the J -integral. In the framework of the present work, the conditions at the crack tip are described using the J -integral and an inherent parameter I n that must be determined for the tested hollow specimens as well as for the structural elements to enable the applicability of the results. To address the theoretical aspects relevant to the experiments, finite element analysis is used to determine the governing parameter I n of the asymptotic behavior of the stresses at the crack tip. In the classical first-term singular HRR-solution [11], the numerical parameter I n is a function of only the material strain hardening exponent n . Shlyannikov and Tumanov [5] reconsidered the HRR-solution for both plane strain and plane stress and supposed that under small-scale yielding, the expression for I n depends implicitly on the dimensionless crack length and the specimen configuration. In this section, we extend the analysis to the I n -integral behavior in an infinitely sized cracked body [11] to treat the test specimen’s specified geometries. The use of the Hutchinson’s theoretical definition for the In -factor directly adopted in the numerical finite element analyses leads to [5]       , , , ( , , , FEM FEM n p P I M n a w M n a w d          (8)       1 , , , cos sin 1 FEM FEM FEM FEM n FEM FEM FEM FEM r P e rr r r du du n M n a w u u n d d                                                1 cos . 1 FEM FEM FEM FEM rr r r u u n             In this case, the numerical integral of the crack tip field I n changes not only with the strain hardening exponent n but also with the relative crack length b/D and the relative crack depth a/D . More details to determine the I n factor for different test specimen configurations are given by Refs. [5-7]. The distribution of the elastic-plastic constraint parameters along the crack front in the direction from the free surface toward the mid-plane is plotted in Fig. 6 for the hollow specimens under different loading conditions. The left picture in Fig. 6 depicts the behavior of the I n -factor, whereas the right picture in Fig. 6 gives us the distribution of the stress triaxiality parameter h for the fixed curvilinear crack front position. The constraint parameters are plotted against the normalized coordinate R . In these plots R = 0.0 is the crack border (the specimen free surface) while R = 1.0 is the mid- plane of the specimen. It can be observed from these figures that both constraint parameters sufficiently changed along the crack front from the free surface toward the mid-plane as a function of loading conditions. Fig. 7 represents the distributions of the I n -factor for the pure tension and pure torsion as well as the combined loading conditions for different crack front position in hollow specimens. The last part of the numerical calculations of the present study is devoted to the determination of the plastic stress intensity factors in hollow samples. In Fig.8 shown the distributions of the elastic and plastic SIF's for the same tensile loading conditions along the same crack front. Fig. 8 gives a clear illustration of the necessity to take into account the