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Comparing improved crack tip plastic zone estimates considering corrections based on T-stresses and on complete stress fields

Last modified: 2011-02-25

#### Abstract

Ductile cracked structures yield locally under load to form a plastic zone (pz) around their crack

tips. As the crack behavior strongly depends on this pz size, and as most cracked structures design routines

depend on it, its precise estimation is a problem of major practical importance. The first classical pz estimates

are based only on the stress intensity factor (SIF), but their precision is limited to very low nominal stresses.

Improved estimates have been proposed considering the T-stress, obtained from the Williams series zero order

term. However, neither SIF, nor SIF+T-stress based estimates can reproduce linear elastic (LE) stress fields that

satisfy all boundary conditions in cracked components. In particular, the nominal stresses far from the crack tip,

which have a major influence on the predicted pz size and shape. To prove this point, this paper first presents

the complete LE stress field solution for the Griffith plate, using three different methods to arrive in the same

analytical solution: the first is based on its Westergaard stress function, the second starts from the equivalent

Inglis plate (considering its elliptical notch root as equal to about half the crack tip opening displacement), and

the third is based on the complete Williams series. Next it introduces the equilibrium corrections necessary to

compensate for the stress limitation inside the plastic zone. Then it compares pz estimates generated from this

complete solution with classical SIF and SIF+T-stress pz estimates for various nominal stress to yield strength

ratios, demonstrating the importance of using correct stress fields to evaluate such pz, particularly for the

relatively high ratios used in high-performance structures. Finally, it speculates that for more complex

structures, where the component geometry and type of loading may also significantly influence pz sizes and

shapes, the plastic zones can be better estimated by a similar approach.

tips. As the crack behavior strongly depends on this pz size, and as most cracked structures design routines

depend on it, its precise estimation is a problem of major practical importance. The first classical pz estimates

are based only on the stress intensity factor (SIF), but their precision is limited to very low nominal stresses.

Improved estimates have been proposed considering the T-stress, obtained from the Williams series zero order

term. However, neither SIF, nor SIF+T-stress based estimates can reproduce linear elastic (LE) stress fields that

satisfy all boundary conditions in cracked components. In particular, the nominal stresses far from the crack tip,

which have a major influence on the predicted pz size and shape. To prove this point, this paper first presents

the complete LE stress field solution for the Griffith plate, using three different methods to arrive in the same

analytical solution: the first is based on its Westergaard stress function, the second starts from the equivalent

Inglis plate (considering its elliptical notch root as equal to about half the crack tip opening displacement), and

the third is based on the complete Williams series. Next it introduces the equilibrium corrections necessary to

compensate for the stress limitation inside the plastic zone. Then it compares pz estimates generated from this

complete solution with classical SIF and SIF+T-stress pz estimates for various nominal stress to yield strength

ratios, demonstrating the importance of using correct stress fields to evaluate such pz, particularly for the

relatively high ratios used in high-performance structures. Finally, it speculates that for more complex

structures, where the component geometry and type of loading may also significantly influence pz sizes and

shapes, the plastic zones can be better estimated by a similar approach.

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