Error analysis in bending tests of short specimens and the new accurate formulas for elastic and shear moduli

The techniques to measure elastic modulus and shear modulus of a material in bending tests are studied and improved. Assuming specimen has a symmetry plane and a material is homogeneous and isotropic (ice, ceramics, alloys, and dispersion-filled composites with metallic, ceramic or polymer matrices and so on) we examine the manifold of features and peculiarities of bending tests, the factors influencing force – deflection curve and moduli values extracted from its linear part, we examine error sources and capabilities to minimize them and to extend the feasibility range of bending tests. We analyze the theoretic force-deflection curve obtained in three-point bending test in the symmetry plane of a specimen and its dependence on the ratio l of a specimen to the height of its cross-section (of arbitrary shape) taking into account the influence of shear stress and strain on a beam deflection (it is significant for beams with l < 10). The analytical estimate of the systematic error of the standard method for elastic modulus evaluation, which does not account for shear deformations, is obtained as a function of a beam span l and its cross-section shape and size. It is shown that the error depends on the Poisson ratio rather than on shear modulus. For materials with a positive Poisson’s ratio, the error is 11 – 15 % for beams with l = 5 and about 5 % for beams with l = 10. For materials with a negative Poisson’s ratio, the error is significantly lower. The new accurate explicit formulas are derived for simple calculation of elastic modulus, shear modulus, and Poisson’s ratio of the material based on two tests with different span l, eliminating the systematic error of the standard method. The formula for elastic modulus is independent of specific shape and size of a specimen’s cross-section and require only seven arithmetic operations. The applicability and high accuracy of the new formula for the effective longitudinal elastic modulus of fibrous and layered composites are demonstrated using bending test data of three structurally different composites with varying degrees of anisotropy (unidirectional carbon fiber-reinforced plastic, woven carbon fiber-reinforced plastic, and woven glass fiber-reinforced plastic with a fiber volume fraction of 61 – 63 %). The error didn’t exceed 2 – 4 % in any case.