Comparison of the constraint parameters in elastic-plastic fracture mechanics

To describe the effect of crack-tip constraint on the stress field for finite cracked bodies, the following most widely used constraint parameters can be employed in elastic-plastic fracture mechanics, namely, local triaxiality parameter, constraint parameter Q, the second fracture mechanics parameter A using a three-term elastic-plastic asymptotic expansion. To establish the relationship between different constraint parameters, the crack-tip stress fields are employed. Relationship between the crack-tip constraint parameters A, A2, Q and p S of elastic-plastic fracture mechanics is investigated.


INTRODUCTION
t present time, it becomes obvious that classical one-parameter fracture mechanics based on traditional parameters, such as the stress intensity factor, the J-integral, etc., is not sufficient for describing the stress field in the vicinity of the crack tip for finite cracked bodies. It is necessary to take into account the crack-tip constraint effect, i.e. to employ models and criteria of two-parameter fracture mechanics. To describe the effect of crack-tip constraint in elastic-plastic two-parameter fracture mechanics, the following possible constraint parameters should be introduced into basic equations and criteria of fracture mechanics in the case of small scale yielding and large scale yielding conditions, namely, local triaxiality parameter [1], constraint parameter Q [2,3], the second fracture mechanics parameter A of a three-term elastic-plastic asymptotic expansion [4][5][6]. Some aspects of the relationship between most widely used crack-tip constraint parameters have been discussed [7,8]. Solutions of the parameter A for 2D test specimens are obtained by extensive finite element analysis [7]. Moreover, an engineering method of estimating the second parameter A in three-term elastic-plastic asymptotic expansion is developed to cover small-scale yielding to large-scale yielding loading conditions [9]. In particular, it was shown that the parameter A can be calculated through the elastic non-singular T-stress in the case of small-scale yielding. Some aspects of application of the J-A approach in elastic-plastic fracture mechanics as well as solutions of the parameter A for 3D test specimens are discussed in Refs. [8,10,11]. A The constraint parameter Q in the J-Q approach proposed by O'Dowd and Shih [2,3] has been widely used in engineering applications. This parameter can be estimated as the deviation of the crack-tip stress field from that based upon possible reference fields. Relationship between the parameter Q and the non-singular T-stress was also established. To quantify in-plane and out-of-plane constraints and their interaction, a unified constraint characterization parameter A p was proposed [12][13][14]. This parameter is based on the plastic area surrounded by the equivalent plastic strain isoline in the vicinity of the crack tip. A sole linear relation between the normalized fracture toughness in terms of the J-integral and √ was observed. The present paper concentrates on relationship between some crack-tip constraint parameters of elastic-plastic fracture mechanics.

TWO-PARAMETER J-A CONCEPT
The J-A asymptotic elastic-plastic crack-tip stress field and fracture criterion n the case of the elastic-plastic material deformed according to the Ramberg-Osgood power-law strain hardening curve the stress field in the vicinity of the mode I plane strain crack is described by the three-term asymptotic expansion [5,6] ( Fig. 1) Here, 0  σ 0 is the yield stress,  is the hardening coefficient, n is the hardening exponent (n > 1), 0 is the yield strain and E is Young's modulus, J is the energetic integral proposed by Cherepanov [15] and Rice [16], A is the second fracture parameter, ij  are stress components r  ,   and r  in the polar coordinate system r with origin at the crack tip, ( ) k ij  are dimensionless angular stress functions obtained from the solution of asymptotic problems of order (0), (1) and (2). Angular stress functions ( 0) ij  and (1) ij  are scaled so as maximal equivalent Mises stress is equal to unity, i.e. . Dimensionless radius  is defined by the following formula Comparing Eqn. (2) and the crack-tip stress field proposed by Hutchinson [17] and Rice and Rosengren [18] ((so-called the HRR field), it is easy to see that the first term of the asymptotic expansion (2) is exactly the HRR field. It can be seen that the HRR field does not describe correctly stresses in the region Fig. 1) that is significant for fracture process. Finite element solutions of elastic-plastic crack problems show that the J-A stress field is much closer to finite element results than the HRR field [8]. The three terms of expansion (2) are controlled by two parameters, namely, the J-integral and the parameter A. The parameter A is a measure of stress field deviation from the HRR field. Assuming that cleavage fracture of a specimen and a cracked structure occurs when stress fields near the crack tip are same it is possible to formulate two-parameter J-A fracture criterion. This assumption also justifies the use of A as a constraint parameter [8]. The two-parameter fracture I criterion implies comparison of computed J-integral for a cracked structure and the experimental fracture toughness JC corresponding to computed value of the constraint parameter A Figure 1: Comparison of the J-A stress field and the HRR field for edge cracked plate.
Determination of the dependency J C (A) can be accomplished by testing cracked specimens with different constraint at fracture load. Change of the constraint conditions is achieved by varying crack length in the specimen. To avoid numerous experiments it is desirable to develop computational approaches for predicting the fracture toughness as a function of the constraint parameter A. Estimating change of the fracture toughness J C with variation of the constraint parameter A can be based on an assumption that fracture under different constraint conditions corresponds to constant fracture probability P f [19].

Numerical estimation of the J-integral and parameter A
The equivalent domain integral method (EDI) [20,21] is employed for computing the J-integral. Determination of the constraint parameter A is based on stress calculation in the vicinity of the crack tip by means of the finite element method. If stresses are known at points ( ) , i i   then the value of the parameter A at ith point is found from the following where FEM  is stress calculated by the finite element method. Solution of Eqn. (5) produces different A values at different points due to deviation of actual stress field from the three term asymptotic expansion. Better estimate of A for the set of points is obtained by minimizing sum of squares of deviations of the J-A asymptotic field from the finite element results. Application of the least squares method leads to a cubic equation for the parameter A [6]   , ,

RELATIONSHIP BETWEEN THE PARAMETER A AND OTHER ELASTIC-PLASTIC CONSTRAINT PARAMETERS
he most widely used constraint parameters mentioned in the introduction of this paper can be employed in engineering applications. To establish the relationship between these parameters, the crack-tip stress fields can be very useful. Relationship between the constraint parameters A, A 2 and Q was discussed in Refs. [7,8]. A constraint characterization parameter p A [12][13][14] is also analyzed in connection with the constraint parameter A.

Relationship between the parameters A and A 2
The well-known J-A 2 approach for two-dimensional body with a mode I crack under plane strain conditions is based on the following expression of the three-term asymptotic expansion for stress in the vicinity of the crack tip in elastic-plastic material [22] Here, A1=     , s , 2 s s t s t s     . Therefore, the following relationship between the constraint parameters A and A 2 in the three-term asymptotic expansions of elastic-plastic crack-tip stress field can be written Thus, J-A and J-A 2 approaches are mathematically equivalent. The difference of these approaches is in distance scaling of r according to Eqs. (2) and (7). Comparison of parameters A and A2 is shown in Fig. 2. Dependencies of both parameters are plotted for 3PB specimen with cracks of same aspect ratios a/W. In contrast to the parameter A, the parameter A2 does not have its small-scale yielding value. It can be seen that the parameter A 2 tends to infinite value with load decrease. It is less convenient for constraint interpretation and engineering application.

Relationship between the parameters A and Q
The constraint parameter Q in the J-Q approach [2,3] is defined by the deviation of the crack-tip stress field under consideration from a reference stress field. Possible reference fields can be the HRR field or field corresponding to smallscale yielding conditions The following relationship between the parameters A and Q can be obtained from Eq. (2) and Eq. (9) Fig. 3 presents dependencies of A and Q as functions of the applied load for cracks of different depth in 3PB specimen. Constraint parameters A and Q demonstrate similar behavior and can be used in engineering applications. But, the Q parameter can be considered as a qualitative measure of crack-tip constraint, because small errors in the crack-tip stress T obtained by finite element method lead to the significant error in the value of Q [23]. The J and A parameters allow describing the elastic-plastic stress field near the crack tip with sufficient accuracy.

Relationship between the parameters A and p
is introduced into consideration as the unified constraint parameter to characterize inplane and out-of-plane crack-tip constraint [12][13][14].
The parameter p A is denoted as p S . The modified constraint parameter p S should be less dependent on the J-integral.
In this case, the separation of the constraint parameter and the J-integral improves the quality of a fracture criterion that is based on these parameters. Behavior of the parameter p S as a function of A/ASSY is investigated using the J-A elastic-plastic asymptotic field. The area surrounded by the equivalent plastic strain contour with specified value of strain p  is estimated with the following algorithm [8]. Stress p  corresponding to specified plastic strain p  can be represented as The rectangular area in the vicinity of the crack tip is divided into square cells. Equivalent stress is computed at each cell by means of the J-A elastic-plastic asymptotic field. If this stress is larger than p  , the cell area is added to PEEQ A .
Computation of the plastic area is performed in coordinates 0 / x J  and 0 / y J  , so the resulting area is PEEQ A . . Largest plastic zone is related to A max (shown in Fig. 4). It can be seen that there is a unique relationship between the constraint parameters p S and A in a wide range of specified values of strain p  .

CONCLUSIONS
elationship between the crack-tip constraint parameters A, A2, Q and p S of elastic-plastic fracture mechanics is based on analysis of the stress fields in the vicinity of the crack tip. The following conclusions can be drawn. It was demonstrated that J-A and J-A 2 approaches are mathematically equivalent. The difference of these approaches is in distance scaling r. But, in contrast to the parameter A, the parameter A2 does not have its value under small scale yielding conditions. Constraint parameters A and Q show similar behavior. At the same time, it should be noted that small errors in the crack-tip stress obtained by finite element method lead to the significant error in the value of