On Brinell and Boussinesq indentation of creeping solids (Q1322763)

In order to examine coldworking, annealing, ductility, carbon content, alloying in heat treated steel, the Brinell and Vickers tests are the most commonly employed. These tests are performed by impressing hard indentors, in the shapes of balls or cones, into a specimen of the material under examination, and determining the resultant contact region and the load necessary for obtaining a prescribed penetration of the indentor. If the indentor is rigid and the material is elastic, the relationships between load and penetration are determined by two classical solutions of the theory of elasticity, known, respectively, as Hertz's and Boussinesq's solutions. But the test must often be done upon polymers or metals at elevated temperatures, in which creep is by far dominant with respect to elasticity. For a large range of materials at different temperatures and stress levels, the most satisfactory constitutive equation describing the creep was proposed by Norton (1929) and has the form \({d\varepsilon\over dt}= \left({\sigma\over \sigma_c}\right)^m\), where \(\varepsilon\), \(\sigma\), \(t\) denote strain, stress, and natural time, and where \(m\), \(\sigma_c\) are characteristic parameters. Norton's formula holds for one-dimensional strain-stress states, but its generalization to three- axial states is easy. Once Norton's equation is introduced, it is possible to formulate the contact problem of a rigid indentor with a creeping half-plane by writing the balance equations and the appropriate mixed boundary conditions. In contrast to the elastic case, the problem is now nonlinear and time- depending. A natural simplification is that of considering only axisymmetric punches, whose shapes, in polar coordinates, have the equation \(f(r)= D(r/D)^p\), when \(D\) is the curvature of the indentor and \(p\geq 1\). A further restriction is that of studying slow motions, so that the inertia terms can be neglected and the times occurs as a parameter in the equations. In the present paper the problem is treated numerically by using the finite element method.