Critical comments to the Oliver–Pharr measurement technique of hardness and elastic modulus by instrumented indentations and refinement of its basic relations

Abstract

A critical analysis was made of the Oliver and Pharr method for determination of the hardness and elastic moduli of materials by instrumented indentations with the continuous recording of the P–hdiagrams (P is the force acting on the indenter, h is the approach of the indenter and a sample). Mistakes and insufficient justification were revealed in the basic theoretical relations of this method. In particular, this refers to an incorrect definition of the depth of the elastic contact h_c, which is the base of these relalations. New refined basic relations and formulas for the determination of hardness and elastic modulus are given, in which the above defects are eliminated and which are based only on the assumption of elastic unloading of the indenter according to the classic theories of the elastic contact. In addition to the foregoing, using the data of the P–h diagram measured in the arbitrary laboratory coordinate system, an improved method of the stable determination of the contact stiffness S = dP/dh at the P–h diagram position have been proposed in the commonly accepted theoretical coordinate system in which its basic classic model relations are recorded. These refinements have been derived without additional assumptions to hypotheses to the Oliver and Pharr method and additional experimental measurements.