J-integral methods

The fracture toughness of a semi-crystalline polymer (PP) filled with mineral submicron and micron scale particles is investigated according to the J-integral method determination of the crack initiation energy (J ), and the crack propagation resistance dJ/d(Aa). Ultrafine mineral... 

The w can be obtained if 1/t ratio is large enough to ensure plane-stress condition in the ligament area and it is proved to be a material constant for a given sheet thickness [Mai and Cotterell, 1986 Mai et al., 1987 Mai and Powell, 1991]. With a reduction of 17t ratio, plastic constraint increases and the plane-stress/plane-strain fracture transition may occur at a certain 1/t ratio. Theoretical analysis shows that the specihc essential work of fracture method is equivalent to the J-integral method for all three fracture modes [Mai and Powell, 1991 Mai, 1993]. 

The J integral method is more widely used especially in cases of strong plasticization of the material during its rupture. This method substantially follows the K factor approach with the difference that the parameter to be evaluated is, now, a special integral operator, called the J integral (Rice, 1968). 

Lu ML, Chiou KC, Chang FC. Fracture toughness characterization of a PC/ABS blend under different strain rates by various J-integral methods. Polym Eng Sci 1996 September 36(18) 2289-95. 

In order to obtain crack-tip quantities such as the strain energy release rate g, the complex stress intensity factor K, and the mode-mixity xp, the following procedure may be adopted first, the strain energy release rate Q is directly computed via a contour integral evaluation - the J-integral method, or the VCCT second, the modulus of K, can be computed from Equation (10) and third, the crack surface displacements may be substituted in Equation (8) and with the knowledge of e, the parameter a is computed. Finally, the stress intensity factors may be expressed as ... 

As part of the post-processing phase, ABAQUS provides output of the mode-specific strain energy release rates Q, g and which are computed internally. These results are used to compute the total strain energy release rate g with the aid of Equation (16) which is then compared with the corresponding calculation based on the J-integral method. The displacement jumps in the wake of the crack-tip, and 6x,m-n2. are used to compute Ki, K,, and... 

Ip based on [3] and compared with corresponding computations from the J-integral method. 

The total mixed-mode strain energy release rate, obtained via the J-integral method is shown Figure 5. These results are reported as a fimction of the specimen width, w, and different normalized crack lengths, a/f. Additionally, the steady-solution for the material and geometry parameters of the current model, obtained via the analytical formulation in [3] is also included in Figure 5. The steady-state solution, Qss, is given by Equation (20) [3]. 

Figure 5. The crack-tip total strain energy release rate, computed via the J-integral method.

Figure 7. The crack-tip total energy release rate, g, computed via the VCCT and the J-integral method for a/f = 0.30. For comparison, the steady-state solution obtained from [3] is also included.

Figure 7, the results pertaining to the strain energy release rate, g, comiwted with the J-integral method and the steady-state solution [3] are also superimposed. It is clear that the results are in reasonable agreement with each other. 

Figure 8. Crack-tip quantities, F, /f, X/, and Kn, obtained via the VCCT and J-integral method for the normalized crack length, a/( = 0.30. The results are self-consistent...

---