Elastic Modulus/Strength

Elastic modulus or Young’s modulus (E with units of GPa) describes the response of a linear elastic material to an applied mechanical load [16]. Elastic modulus relates the applied load to the resulting strain as expressed by Hooke’s law (7).

Подпись: (7)s = E e

where s is the applied stress (GPa) and e is the strain (no units).

Under an applied load, deformation of the solid requires that the atoms be moved closer together (compressive load) or farther apart (tensile load). As such, dimen­sional changes are related to the strength of the bonds among the atoms [8]. When the component ions of a material have high bond strengths, the material typically displays high elastic modulus and low coefficient of thermal expansion. For example, SiC has a high bond strength giving sintered a-SiC a coefficient of thermal expansion of 4.02 ppm K-1 and a modulus of 410 GPa [56]. Conversely, NaCl has a low bond strength resulting in a coefficient of thermal expansion of 11.0 ppm K-1 and a modulus of 44 GPa [16, 57]. Modulus can be measured using either acoustic methods (ASTM E 1876 Standard Test Method of Dynamics Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation of Vibration) or by directly measuring displacement as a function of an applied load using a deflectometer.

Although modulus is a fundamental material property related to the strength of the chemical bonds among atoms, the measured strength (s with units of MPa) is affected by specimen characteristics, the testing environment, the type of test performed, and other factors. The theoretical tensile strength can be estimated as the stress required to break the chemical bonds among the atoms of a solid [57]. However, brittle materi­als fail at applied stresses two or more orders of magnitude below the theoretical strength values due to stress concentration around physical features of the solids such as pores, defects, grain boundaries, and edges. The Griffith criterion is often used to relate the fundamental material properties such as modulus to observed strength using specimen characteristics such as flaw size [1], although the predictions are qualitative at best. The Griffith criterion can be used to understand the statistical nature of frac­ture of brittle materials if the distribution of flaw sizes within a given specimen is considered [16]. Strength can be measured in many different manners ranging from compression (ASTM C1424 Standard Test Method for Monotonic Compressive Strength of Advanced Ceramics at Ambient Temperatures) to tension (ASTM C1273 Standard Test Method for Tensile Strength of Monolithic Advanced Ceramics at Ambient Temperatures). The strength of advanced ceramics is most often measured using relatively small specimens in three – or four-point bending, which determines the so-called flexural strength (ASTM C1161 Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature). Because traditional refrac­tory materials have grain sizes that approach or exceed the size of the specimens used for testing of advanced ceramics, strengths must be determined using alternate methods (ASTM C133 Standard Test Methods for Cold Crushing Strength and Modulus of Rupture of Refractories) that accommodate course grain specimens.