Breaking four-point and three-point bending tests

Since their initial standardizations in the 1930s and 1950s, the so-called four-point and three-point bending tests on unnotched beams have been embraced by practitioners as two popular methods to indirectly measure the tensile strength of concrete, ceramics, and other materials with a large compressive strength relative to their tensile strength. This is because of the ease that the tests afford in both the preparation of the specimen (a beam of rectangular cross section) and the application of the loads (simple supports pressing on the specimen). Yet, this practical advantage has to be tempered by the fact that the observations from both of these tests — being indirect experiments in the sense that they involve not uniform uniaxial tension but non-uniform triaxial stress states throughout the specimen — have to be appropriately interpreted to be useful. By making use of the phase-field fracture theory initiated by Kumar et al. (2018a), which has been recently established as a complete theory of fracture capable of accurately describing the nucleation and propagation of cracks in elastic brittle materials under arbitrary quasistatic loading conditions, the main objective of this paper is to carry out a thorough 3D quantitative analysis of when and where fracture nucleates and propagates in four-point and three-point bending tests and thereby establish how to appropriately interpret their results. The focus is on the fundamental case of materials that can be considered homogeneous, isotropic, linear elastic brittle at the length scale of the beam. As a corollary, the analysis provides an explanation for why four-point bending tests typically yield smaller flexural strengths than three-point bending tests, a source of constant headaches for practitioners who have been left to wonder which test — if any — would be more appropriate for their purposes. Results on the impact of strength stochasticity and the presence of a boundary layer are also presented and discussed. The final section of this paper presents simple formulas for deducing the uniaxial tensile strength of a material directly from the flexural strength measured in each of these tests.