A Statistical Theory of the Damage of Materials

Abstract

A statistical theory of disperse damage of materials gained under loading is proposed. It is based on the idea of distribution of potential damage spots within a specimen similar to the distribution of the strength values found by testing a set of identical specimens. A relation between damage risk and the probability of damage of a single specimen is assumed. It conforms to the relation between risk and probability of strength distribution of a set of identical specimens, according to Weibull’s statistical theory of material strength. The damage risk just like the damage probability are assumed to be functions of loading and time. Damage is modeled regarding two mechanisms—thermo-fluctuation damage based on the kinetic theory of material strength, and damage depending on a parameter called loading degree (percentage), which depends on load and time. The first mechanism adopts two relations regarding energy barrier reduction due to loading. Equations of damage advance under short-term loading (with a constant rate of load increase) and under long-term (constant) loading are derived. Theoretical relations of damage development are compared with experimental evidence gained via specific tests on short glass fibre reinforced polyoximethylene. The damage itself consists of the accumulation of additional internal surfaces within the entire material volume as measured by small angle scattering X-ray refractometry. It is shown that the mechanism of damage as a function of loading degree where time participates implicitly, gains advantage over the kinetic mechanism where time is explicitly present. The cumulative functions derived for damage accumulation can be used to assess not only damage, but also for statistical analysis of the strength and other quantities.