Aspects of Grain Size Strengthening in Polycrystals
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Aspects of Grain Size Strengthening in Polycrystals Dilip Chandrasekaran and Kjell Pettersson Dept. of Materials Science and Engineering, Royal Institute of Technology S-100 44 Stockholm, Sweden ABSTRACT The strengthening effect of grain boundaries is well established and observed experimentally as the Hall-Petch relationship. In this paper different mechanisms proposed in the literature to explain the observed Hall-Petch effect are reviewed critically. The fundamental implications of the different approaches are discussed with reference to experimental data for two different classes of materials; -Materials with locked dislocations, i.e. with a sharp yield point behaviour. -Materials without locked dislocations, i.e. with a smooth yielding behaviour. It is shown that a simple model (Bergström) can be used to understand the grain size strengthening in the latter class of materials while more work is needed to quantitatively understand the behaviour of materials showing a sharp yield point. INTRODUCTION The strengthening in polycrystals due to grain boundaries has been experimentally well established ever since Hall [1] proposed his relation between the grain size and the yield stress. The Hall-Petch relation (as shown below) is found to be experimentally valid for a range of different alloy systems.
σ = σ0 + k ⋅d
− 12
(1)
where σ is the (upper or lower) yield stress or the flow stress, σ0 the contribution from other strengthening mechanisms, d the grain size and k is a constant, often known as the Hall-Petch constant. In order to explain the experimental observations of the Hall-Petch effect, several different types of mechanisms have been proposed in the literature. In this paper different models explaining the grain size strengthening effect are critically evaluated and their ability to predict the experimental information is reviewed. The focus is on the fundamental mechanisms behind the models and their implications are discussed in reference to two different classes of materials. MODELS FOR GRAIN SIZE STRENGTHENING Pile-up Models One of the earliest attempts to explain the Hall-Petch behaviour was the pile-up model by Hall [1], with subsequent modifications by Petch and Cottrell [2]. The basic idea here, is that dislocations are assumed to pile-up against a grain boundary, thereby causing a stress concentration. When the stress concentration equals a critical stress, assumed to activate new dislocation sources, yielding starts in the next grain, thereby allowing for plastic deformation to proceed from grain to grain. This model leads finally to the following expression for the tensile shear stress:
BB2.8.1
τ s = τ 0 + k1
τ c µb − ⋅d π
1 2
(2)
This relation is the same as eq (1). The content under the root can be identified with k in eq (1), d is the grain-size and k1 is a constant with the numerical value depending on the nature of the pile-up and the assumption coupling the length of the pile-up with the grain diameter. There are several attractive features with this theory, it gives an explanation
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