Modeling for Creep Crack Growth

Focusing on the tensile deformation instability of a (noncreeping) ligament with cross-sectional area A and uniformly applied stress (on average) a, and the apphed force P = a A. The change in force is given by  

At the maximum load, the change in load is zero Le., dP = 0) such that, for a time-independent, power-hardening material one that does not creep)  

In other words, the strain at the onset of tensile deformation instability (maximum load point) is equal to the strain-hardening exponent. For a time-dependent, power-hardening material (i.e, one that creeps), on the other hand, deformation is enhanced by creep, such that  

According to Landes and Wei [2], the connection between the steady-state creep rate and the crack-driving force (characterized by K) is derived through the use stress-strain results from elastic-plastic analysis by Hutchinson [9] and Rice and Rosengren [10], According to these models, crack-tip stress and strains in the loading direction (y-direction) are given by Eqn. (6.7). 

Differentiating strain (Eqn. (6.7b)) with respect to time, the strain rate at the position (r, 0) ahead of the crack tip becomes  

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In structural ceramic composites, the principal effect considered was one of crack-face closure tractions, or cohesive forces, brought about, for instance, by bridging fibers. A rigorous evaluation of the crack tip fields where the crack faces are not traction free has not yet been attempted. However, an approximate approach for the small-scale creep case is to assume that the crack tip fields are not functionally altered by crack-face tractions, with the effect of the traction being only to introduce a zone of crack tip shielding. This allows for the development of preliminary models for creep crack growth which is inclusive of the role of crack bridging. These preliminary models predict that,... 

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