Effects of Boundary Conditions and Anisotropy on Elastically Bent Silicon

In four-point bending, the rollers that are used for load application impose additional
constraints on the specimen that affect the anticlastic specimen curvature and cause the specimen displacement and stress profiles to deviate from the pure beam bending case. In this study, x-ray microdiffraction is used to map both the principal and anticlastic curvatures of elastically bent, rectangular (100)-type Si strips possessing width:thickness ratios of 40:1. We quantify the amount of roller constraint and show that the region over which the anticlastic specimen curvature is affected away from the roller is approximately five times the roller diameter. Consequently, for bending tests used to determine Poisson’s ratio, if a region on the sample that is free from roller effects is not chosen, measurement errors as high as 46% can occur. Furthermore, we show that, due to the anisotropy of single crystal Si, this roller-constraining effect depends on crystallographic orientation and is more pronounced when the principal bending axis lies along the <100> direction as compared with the <110> direction. Introduction
Flexural techniques, such as four-point loading, are frequently employed for accurately
loading brittle materials in a variety of mechanical tests.1-3 Depending on the sample geometry and dimensions, either the elementary beam solution or a plate solution is typically used to describe the displacement and stress profiles of a sample loaded in this configuration.4,5 These solutions, however, do not consider the constraints imposed by the bending jig rollers used for load application. If the rollers affect either the longitudinal or transverse curvatures, errors in the displacements or stresses predicted by an analytical solution can occur. In this paper, we compare the measured curvatures of specimens loaded in four-point bending with beam solution predictions.