Revised Orowan Strengthening: Effective Interparticle Spacing and Strain Field Considerations
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INTRODUCTION
THE flow strength of engineering alloys by dispersed unshearable particles at the onset of yielding can be predicted by the Orowan equation (s = Gb/k) in its simplest form.[1] Nevertheless, there have been various refinements to k in order to incorporate the particle size effect, as well as the randomness of the obstacle distributions.[1–4] Several expressions were proposed based on various models and assumptions; most of these are best generically described by Eq. [1]. The various parameters used in this equation take into account the change of interparticle spacing with volume fraction and convert ideal particle distributions to random distributions. Moreover, they take into account the screw and edge effects during dislocation bow-out. Dp Dp G Xb Gb ln ¼C ½1 ln DryOrowan ¼ C L L=Dp Dp Xb Xb Table I shows the various parameters employed by several authors. Accordingly, depending on the expressions used for C and L, the yield strength predictions can be quite different. Moreover, none of the proposed expressions provide a completely satisfactory prediction of Orowan strengthening due to the presence of secondphase dispersoids. In particular, the proposed expressions J.B. FERGUSON and BEN SCHULTZ, Research Associates, HUGO LOPEZ, Professor, DANIEL KONGSHAUG, Research Assistant, and PRADEEP ROHATGI, State of Wisconsin and UWM Distinguished Professor, are with the Materials Department, CEAS, University of Wisconsin–Milwaukee, Milwaukee, WI 53211. Contact e-mail: [email protected] Manuscript submitted August 3, 2011. Article published online January 28, 2012 2110—VOLUME 43A, JUNE 2012
of an interparticle mean free path for dislocation motion are not accurate enough. Hence, in this work, an expression is developed to describe the effective interparticle spacing with improved precision. II.
MODEL
A. Mean Interparticle Free Path The interparticle mean free path for dislocation motion affects the distance dislocations can bow out before they are able to move freely or new dislocations or a new surface are irreversibly generated. In impenetrable particles, this mean free path distance is equivalent to the average dispersoid spacing distance minus the average dispersoid diameter. Table II shows the experimental data reported for Co-WC composites[5] as well as simulations for Orowan-type strengthening.[6] Figure 1 shows that the yield strength for the Co-WC system follows the Orowan relation described by Eq. [2], until L reaches about 1.7 lm (i.e., 1/L is slightly greater than 0.50 lm1). Beyond this point, the behavior deviates from Orowan strengthening. Rule of mixtures, as described in Eq. [3], may provide a better fit in this regime. Fang proposed a modified rule-of-mixtures equation[7] to describe the yield stress. However, it was developed for conventional Co-WC composites, where L is usually less than 2 lm and, therefore, does not fit the data of Table II well in the low volume fraction regime, where the material is more like a second-phase particle-strengthened alloy. ry ¼ r0 þ
C L
ðOrowan-type strengt
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