Schematic diagram used in the analysis of the hardness indentation. The five triangular portions of the material being tested slip relative to each other, thus performing work under the applied load. Figure adapted from M.F. Ashby and Dr. H. Jones, Engineering Materials 1: an introduction to their properties and applications, Butterworth/Heinemann.

Hardness tests often are used to quantify strength and are considered to be nondestructive in most applications because the indentations are small and do not adversely affect surface quality. In the case of steel, there is a common relationship between the Brinell hardness number (BHN) and the ultimate tensile strength (UTS) given in pounds force per square inch (psi), or MPa:


It often is assumed that this relationship is only valid for tempered martensitic structures. However, similar relationships can be shown for brass, aluminum and cast irons. A relationship between yield strength and hardness usually is not shown, and it may be instructive, and perhaps surprising, to show that these relationships have a theoretical basis.

Slip-line field theory usually is used to develop the hardness relationship. However, for the sake of simplicity, the problem is reduced to a two-dimensional example as shown in Fig. 1. Here, a force F is applied to an indenter of area A and the indenter is pushed into the metal a distance d. In this example, slip of the metal is restricted to the motion of five triangular portions near the indenter. The metal is assumed to be isotropic and that slip occurs at a shear stress assumed to be one-half the yield stress, sy. Figure 1 shows the displacement of each triangle as a function of the indenter displacement, d.

If the work performed by the indenter, Fd, is equated to the work performed by each triangle against the shear stress, the following equality is obtained, where each term represents the product of the number of sliding interfaces, the force and the displacement.

A more sophisticated analysis using slip-line field theory yields a value of 2.97sy. If the hardness is now measured in terms of the applied force and the area, the relationship between hardness and strength is complete.

Of all the various hardness measures, the Brinell test is perhaps ideal for these property relationships because the values are given in units of mass per area of indentation; i.e., kgmm^2. (In most cases, a 10-mm ball is used as the indenter and a mass of 3000 kg is applied.) It is important to note that most hardness values are quoted in terms of mass and that the mass must be multiplied by an appropriate acceleration to obtain the force in calculating stress.

Using the required conversion factors to obtain units of psi yields the following equation:

The downfall of this analysis is that metals work-harden during the indentation and, as a result, the hardness correlates much better with the ultimate tensile strength rather than the yield strength. If the metal did not work harden, the indenter in Fig. 1 would penetrate through the entire thickness of the specimen by displacing the triangles.

A compilation of ultimate tensile strength versus Brinell hardness number for selected metals based on handbook data.

A comparison of ultimate tensile strength with Brinell hardness is shown in Fig. 2 for several metals. Many of the copper-base alloys have a much higher UTS than predicted and the gray irons tend to have a much lower UTS than that predicted. In the case of gray iron, the graphite flakes serve as crack initiators in tension, whereas these flakes are under compression during the hardness test, and work hardening that occurs increases the hardness. Ductile irons also are limited in tensile strain, or elongation, but not as severely as gray irons because the graphite is nodular rather than flake. Resulfurized steel would behave in a manner similar to that of ductile vs. gray irons, when compared with plain carbon steel.