A theoretical study of the Brinell hardness test

Abstract

Brinell tests have long been the preferred method of assaying the hardness of metals during forming operations. The general significance of the test has been codified in empirical laws, especially those of Meyer, O’Neill and Tabor. On the other hand, the indentation of elastoplastic media by a ball has never been thoroughly analysed in the context of modern mechanics of continua; this is the objective here. The actual boundary-value problem is non-steady but can be made steady in terms of reduced variables when the material response is suitably modelled. Namely, the strain should be infinitesimal and expressible as the tensor gradient of a potential function of the stress deviator; the function must be homogeneous of degree n +1 (≥ 2), but is otherwise arbitrary. Meyer’s law is then derivable rigorously ahead of a detailed solution. Moreover the predicted index is (2 n +1)/ n , substantiating O’Neill’s rule for materials whose strain under uniaxial tension varies as some n th power of the stress. It is predicted also that the piling-up or sinking-in around the indenter is correlated with n in the manner observed. These immediate implications of the model amount to a priori evidence of its overall ability to simulate elastoplastic response of the kind induced in Brinell tests. Evidence a posteriori was supplied by finite element computations for a standard potential whose level surfaces are of Mises type. Mixed nine-node quadrilateral elements were adopted; these are known to promote optimal convergence and are well suited to handling incompressibility. A carefully graded mesh provided about 24000 degrees of freedom. Computations were performed for n ═ 1, 2, 4 and 10, covering the practical range. The results include (i) distributions of the contact pressure and the radial and circumferential in-surface stresses; (ii) profiles of the deformed surface; and (iii) contours of representative strain in the main body of material. Excellent agreement was obtained with Tabor’s experimental findings that the representative strain at the contact perimeter is γ ═ 0.4 a / D for any n ( a is the contact radius and D the ball diameter), while the average pressure is 2.8 times the flow stress at strain γ in a tension test. Finally, strain paths at fixed stations were sampled to check the simulation of elastoplastic response locally as well as overall.