Micromechanistic expressions of continuum microscale parameters for stable crack growth
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I.
INTRODUCTION
F O R high toughness materials stable crack growth can occur at J values several times Jm prior to instability.~ A measure of a material's resistance to stable crack extension is the rate at which J increases with increasing crack length. Such data are often expressed at the tearing modulus, TR, EdJ
T.
[1]
o-~ da
which is the dimensionless slope of a J vs a curve and where E is Young's modulus, 0-0 is the flow stress, and a is the crack length. 2 Rice and co-workers have developed a continuum model expressing TR in terms of microscale parameters l* and 6c, where 6< is a critical crack opening which must be maintained a characteristic distance l* behind the crack tip during stable crack growth. 3,4 According to their treatment, TR = ~ 2 da
O'o a l *
- - In
[2]
where c~ and /3 are constants which can be evaluated through finite element methods (e.g., /3 ~ 5.5 for u = 0.3) 5 and R is a distance which scales approximately with the radius of the plastic zone of a non-stationary crack. The assumed geometry here is analogous to that in Figure 1. While 6< can be measured directly, 6,7,8l* can be determined only by fitting the model to experimental results. 9 Our interests are both to test this continuum approach and to determine how and to what extent the tearing modulus is determined by inclusion spacing and parameters characterizing the fine scale microstructure. To provide a basis for such studies, expressions for the microscale parameters 6< and l* are proposed.
II.
of longitudinal and transverse void growth DL and D are given by Oc = (h/R,)
D r = (gv/R,) M = ( h / w ) = DL/ZDT
where h is the void depth, Rv is the transverse void ra Rx is the radius of the void nucleating particle and the width w = 2Rv. The quantity M is the surface m roughness. 10.H In using these parameters it must be ke mind that on every fracture surface there is a wide ran void sizes, possibly representing nucleation at a varie different types of void nuclei (e.g., in steels, sulfides dissolved carbides, precipitated carbides, oxides, and cates, as well as other possible inclusions). These parameters might be different for voids initiated by the ous populations of second phase particles. Therefore, void used in making these measurements ideally st be identified with a particle. Sectioning studies at st prior to fracture are sometimes necessary to make identification. Rice and Tracey have developed equations for the grt of an initially spherical void of radius Rt 12 modified by and Johnson for conditions just ahead of a crack tip. ~3qequations allow the calculation of three void radii, Rz and Ry. Ry is the radius of the void perpendicular tc plane of the crack; Rx and Rz are void radii in the plar crack (Figure 3). Rz is the radius parallel to the crack fi These equations are dRz
Rz
= 0.322 exp(1.50"/o'0) dep
FRACTOGRAPHIC PARAMETERS
Proposed expressions for I* and 6,. utilize parameters which can be obtained directly from the fracture surface. The three parameters (Figure 2) to be used include measures WARREN M. G
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