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Crack Growth Resistance in Metallic Alloys: The Role of Isotropic Versus Kinematic Hardening

Emilio Martínez Pañeda's picture

We have always modelled crack propagation under monotonic/static loading in metals assuming isotropic hardening. However, we show that anisotropic/kinematic hardening effects play a significant role due to non-proportional straining with crack advance; the isotropic hardening idealization leads to steady state fracture toughness predictions that could be 50% lower. I hope that some of you find this work interesting.

Emilio Martínez-Pañeda, Norman A. Fleck. Crack Growth Resistance in Metallic Alloys: The Role of Isotropic Versus Kinematic Hardening. 

Journal of Applied Mechanics (2018) Vol. 85 (11), 111002 (6 pages)

The sensitivity of crack growth resistance to the choice of isotropic or kinematic hardening is investigated. Monotonic mode I crack advance under small scale yielding conditions is modeled via a cohesive zone formulation endowed with a traction–separation law. R-curves are computed for materials that exhibit linear or power law hardening. Kinematic hardening leads to an enhanced crack growth resistance relative to isotropic hardening. Moreover, kinematic hardening requires greater crack extension to achieve the steady-state. These differences are traced to the nonproportional loading of material elements near the crack tip as the crack advances. The sensitivity of the R-curve to the cohesive zone properties and to the level of material strain hardening is explored for both isotropic and kinematic hardening.

As usual, a post-print can be found at


kourousis's picture

Can you provide the values of c_k and gamma_k (10 parameters in total)? These are currently missing from the paper.

kourousis's picture

Can you also explain this? "The choice of n=10 brings the CAF model into alignment with (4) to within 0.04% for the range of true tensile strain 0<=e<=2.0"

In particular, why did you choose to capture the initial loading branch (tension) and not the subsequent unloading-reloading branches (compression-tension) of the hysteresis loops? 

Emilio Martínez Pañeda's picture

Thank you Dr Kourousis for your comments. 

Regarding your first question. We did not include the values of c_k and gamma_k because it is basically a curve-fitting. In fact, if you use a commercial finite element code like Abaqus what you provide is the uniaxial stress-strain curve (and the number of parameters, k=10), and Abaqus does the fitting for you. Thus, going from isotropic hardening to kinematic hardening just involves changing one line in the input file. Moreover, note that if you use other approaches, such as multilinear kinematic hardening (implemented in ANSYS since a few years ago and in Abaqus since 2017) you will get similar results in the crack growth problem. In any case, one can request the values of c_k and gamma_k, along with the fitting precision, so I have looked into my notes and the parameters used seem to be the following:

N=0.2 - c_1: 0.64945, gamma_1: 1.02E-04c_2: 1.8295, gamma_2: 8.25E-04c_3: 5.5242, gamma_3: 3.77E-03c_4: 17.798, gamma_4: 1.61E-02c_5: 60.82, gamma_5: 7.22E-02c_6: 195.53, gamma_6: 0.32148c_7: 652.78, gamma_7: 1.4174c_8: 2182.2, gamma_8: 6.4444c_9: 7391, gamma_9: 30.334c_10: 23610, gamma_10: 173.42.

N=0.1 - c_1: 2.77E-02, gamma_1: 0; c_2: 0.21698, gamma_2: 4.62E-04c_3: 0.89669, gamma_3: 2.99E-03c_4: 3.5643, gamma_4: 1.47E-02c_5: 15.397, gamma_5: 7.34E-02c_6: 62.707, gamma_6: 0.36018c_7: 263.63, gamma_7: 1.7468c_8: 1072.1, gamma_8: 8.6006c_9: 3986.5, gamma_9: 41.256c_10: 12163, gamma_10: 220.63.

Regarding your second question. This automated fitting to the tensile part of the uniaxial stress-strain curve shows that, for 10 c_k and gamma_k, the kinematic hardening model matches the isotropic hardening model up to a stain of 2 with the largest difference being of 0.04%. We are dealing with monotonic/static loading, where the translation of the yield surface with crack advance is always neglected. What we show is that if you model crack propagation with a kinematic hardening model that matches the isotropic case in uniaxial tension, then significant differences arise in the crack growth resistance curves. In other words, one should not neglect anisotropic/kinematic hardening effects in monotonic/static loading.

Thank you
​​​​Emilio Martínez Pañeda​

































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