PSI - Issue 81

Jesús Toribio et al. / Procedia Structural Integrity 81 (2026) 54–57

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3. Relationship between microstructure and strength: On the validity of the Hall-Petch equation The final aim of this paper is to establish the relationship, typical of material science approaches, between micro structure and macro scopic mechanical behaviour. In particular, the relationship between the pearlite interlamellar spacing and the conventional yield strength is the matter of interest. Fig. 2 shows both the pearlite interlamellar spacing Si and the conventional yield strength  02 plotted against the degree of cold drawing D i /D 0 (where Di is the wire diameter for any drawing degree and D 0 the initial diameter before cold drawing). The inverse relationship between the two variables in the y-axis indicates that the pearlite interlamellar spacing (decreasing with cold drawing) influences clearly the improvement of yield strength (increasing with cold drawing, final aim of manufacturing). However, a relationship of the Hall-Petch type cannot be fitted in this case, neither with the classical exponent in the form S i – 1/2 , as suggested by Embury and Fisher (1966), Langford (1977), Porter et al. (1978), Hyzak and Bernstein (1976), Dollar et al. (1988) and Alexander and Bernstein (1989), nor with other exponents, e.g. S i – 1 (Alexander and Bernstein, 1989). The inability of the Hall-Petch relationship to fit the experimental data in this case is probably due to the fact that cold drawing produces not only an increasing closeness of the pearlitic packing with decrease of pearlite interlamellar spacing, but also an evident microstructural orientation in the material, as reported by Toribio and Ovejero (1997, 1998a, 1998b, 1998c), and it confirms the recommendation of using the Hall-Petch equation with caution (Dieter, 1988) due to its evident uncertainties that produce a sort of ambiguity in the relationship between microstructure and strength in eutectoid pearlitic steels, mainly in the case of heavily cold drawn pearlitic steel wires with markedly oriented pearlitic microstructure.

Fig. 2. Relationship between microstructure and strength in the steels, in a plot representing the interlamellar spacing S i and the conventional yield strength  02 as a function of the degree of cold drawing. 4. Discussion: Embury-Fisher vs. Hall-Petch equation in the case of cold-drawn pearlitic steels As explained in previous sections of the paper, the validity of the Hall-Petch equation becomes a controversial topic if one considers the ability of such equation to describe the better mechanical performance ( increase of yield strength ) with the decrease of the size of a characteristic microstructural unit: grain size, particle size, or, in the case of pearlitic steels, colony size or interlamellar spacing, the latter being usually considered as the key microstructural length representing the free distance for a dislocation to move before being blocked by the cementite barrier. The phenomenon of blocking of free dislocation movement in the ferrite phase by the hardest cementite phase acting as a barrier provokes an increase of material strength. Although in randomly oriented pearlite (e.g., in a rail steel or a hot rolled pearlitic steel bar) the Hall-Petch equation can be valid to fit the strength versus material characteristic size, in the case of oriented pearlite (e.g., in a heavily cold drawn steel ) the relation between microstructure (represented by a characteristic length such as the pearlite interlamellar spacing) and strength does not fit a Hall-Petch equation. Therefore, although the Hall-Petch equation seems to be effective to describe the relationship between microstructure and strength in randomly oriented pearlitic microstructures, it does not properly work for drawn pearlite, i.e., for the case of oriented pearlitic microstructures). However, an Embury-Fisher equation (Embury and Fisher, 1966) can be proposed to describe the relationship between microstructure and strength in cold-drawn pearlitic steels (Fig. 3). The basic assumption of the approach is that the projection of the average interlamellar spacing over the transverse section of the wire is proportional to the wire diameter, so that the modified Hall-Petch equation fits very well the experimental results regarding both  Y and  max , the modification consisting on using a new microstructural length, namely the modified (or corrected) average pearlitic interlamellar spacing s 0 /cos  (  being the angle between the ferrite/cementite lamellae and the wire axis or cold drawing direction) instead of the conventional microstructural length, namely the average interlamellar spacing s 0 .