Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load

In preceding studies, the type of cumulative probability distribution functions (cdf) of strength and of static lifetime of quasibrittle structures, including their tails, was mathematically derived from atomistic scale arguments based on nano-scale cracks propagating by many small, activation energy-controlled, random breaks of atomic bonds in the nanostructure. It was shown that a quasibrittle structure (of positive geometry) must be modeled by a finite (rather than infinite) weakest-link model, and that the cdf of structural strength as well as lifetime varies from nearly Gaussian to Weibullian as a function of structure size and shape. Excellent agreement with the observed distributions of structural strength and static lifetime was demonstrated. Based on the same theoretical framework, the present paper formulates the statistics of the residual structural strength, which is the strength after the structure has been subjected to sustained loading. A strength degradation equation is derived based on Evans' law for static crack growth during sustained loading. It is shown that the rate of strength degradation is not constant but continuously increasing. The cdf of residual strength of one RVE is shown to be closely approximated by a graft of Weibull and Gaussian (normal) distributions. In the left tail, the cdf is a three-parameter Weibull distribution consisting of the (n+1)th power of the residual strength, where n is the exponent of the Evans law and the threshold is a function of the applied load and load duration. The finiteness of the threshold, which is typically very small, is a new feature of quasibrittle residual strength statistics, contrasting with the previously established absence of a threshold for strength and lifetime. Its cause is that there is a non-zero probability that some specimens fail during the static preloading, and thus are excluded from the statistics of the overload. The predictions of the theory are validated by available test data on glass–epoxy composites and on borosilicate and soda-lime silicate glasses. The size effect on the cdf of residual strength is also determined. The size effect on the mean residual strength is found to be as strong as the size effect on the mean initial strength.     

Enhanced fracture energy and effects of temperature at spalling in ceramics: continuum damage modelling

Summary Enhanced fracture energy losses at spalling and the temperature dependence of the spalling strength of alumina ceramic bars are analysed on the basis of the experimental tests conducted both in room temperature and within the temperature range up to 1500°C at strain rates of some 500 s⁻¹. The experimental method and the measurements are first shortly outlined. The mechanical response of ceramic bars is modelled then as a heterogeneous distribution of brittle-elastic mesoelements undergoing continuum damage at the known strain history, corresponding to that registered in the experiments. The mesoelements are characterised by the values of initial damage randomly fluctuating within a given band-width superposed on a deterministic distribution, which corresponds to the fabrication conditions of the ceramic bars. The model has been tested in the evaluation of room-temperature experiments, its parameters: the average value of the initial damage, Young's modulus of the undamaged material and the energy absorption capacity in continuum damage are taken from the calibration fitting the experimental data. The registered energy losses at spalling, which exceed the static values of fracture energy by almost an order of magnitude, can be explained thus by the enhancement of the dissipation due to bulk damage, which is computed on the basis of the above parameters. The temperature change of the Young's modulus of the matrix material is taken as corresponding to the measured change of the uniaxial wave velocity in the bar, and corrected by the temperature change of the mass density. The analysis of the model shows that the drop in the spalling strength of the specimens with the increase of the temperature is phenomenologically related to the falling energy absorption capacity within the continuum damage mechanism. An explanation of this phenomenon is attempted, based on the grain-size-related mechanisms of the microfracture from pre-existing intergranular flaws distributed over the bulk of ceramics. Received 7 May 1999; accepted for publication 14 June 1999