a b s t r a c tThe shells of avian eggs are very brittle, but how brittle? Fracture toughness, K c is a standard measure used widely to characterise engineering materials. We devised a novel way to measure K c and applied it to commercial hens' eggs, obtaining a value of 0.3 MPa p m. This value is much lower than previous published values, which we argue are incorrect and misleading. We discuss how this exceptionally low toughness value (in comparison to that of other natural materials made from calcium carbonate) has been achieved by prevention of toughening mechanisms. Eggshell has an unusual combination of mechanical properties (low fracture toughness combined with high Young's modulus), making it ideally suited as a container for the developing chick, which must be stiff and rigid but also brittle enough to be broken when required. Further testing and analysis using the Theory of Critical Distances and Weibull probability theory allowed us to describe the effects of defects of various types: cracks, holes and notches, on the strength of whole eggs. These results are of commercial importance because many eggs break prematurely as a result of microscopic defects. Statement of SignificanceThis paper presents the first accurate measurements of the fracture toughness of eggshell. These results are important because eggshell is a brittle material which fails from microscopic defects, so knowledge of the fracture toughness is essential to understand its mechanical performance. The toughness value obtained is discussed in the context of other mechanical properties, of eggshell and other natural materials. This is useful for understanding how eggshell's stiffness and toughness make it ideally suited to its purpose, and the mechanisms by which toughness is achieved. The paper also contains analysis of the effect of defect type, including cracks, notches and holes, to provide a fuller picture of defect tolerance which will be useful in the egg producing industry.

It is shown by using self-consistent (so that action and reaction torques are fully accounted for) expressions for the viscous drag and white noise terms in the equations of motion of the fixed axis rotator version of the itinerant oscillator model that the characteristic (secular) equation of the system automatically factorizes. Hence all the correlation functions of the model (which consists of a “cage” of dipolar molecules surrounding a tagged molecule) may be given in closed form. In particular, the orientational correlation functions of the tagged molecule become the products of single particle ones namely those of a free Brownian rotator and a damped harmonic oscillator while the orientational correlation functions of the cage are simply those of the free Brownian rotator. The equations of motion of the system likewise decouple when the restrictions of small oscillations and of rotation about a fixed axis are removed. Thus, irrespective of the form of the interaction potential between the cage and the tagged molecule, the relaxation modes of the system separate into those of the tagged molecule and those of its surroundings which behave as an inertia corrected free Brownian rotator.

The double-transcendental nature of the correlation function for the complex polarizability of the itinerant oscillator model indicates that an infinite number of oscillatory decay modes exist thus giving rise to an infinite number of harmonic peaks in the complex polarizability. The approximate formula for the complex polarizability given by Coffey and Walsh (1997 J. Chem. Phys. 106 7625) does not include these decay modes as this formula is based on taking the first two terms of the Taylor series expansion of the correlation function in terms of singletranscendental functions. It is demonstrated by calculating numerically the Fourier transform of the correlation function that the approximate formula provides an accurate approximation to the complex polarizability since the contribution of the higher-order decay modes is negligible. In addition, a new dimensionless set of parameters for the description of the model which also reduces the number of parameters needed to fit the model to experimental spectra is proposed. The polarizability spectra calculated with the new parameter set from the model compare favourably with the experimental spectra in methyl chloride.

No abstract

Exact solutions obtained by Gross [J. Chem. Phys. 23, 1415 (1955)] and Sack [Proc. Phys. Soc. London, Sect. B 70, 402 (1957)] for the complex polarizability of assemblies of nonelectrically interacting rotators subjected to a variety of collisions and various approximations to that quantity, specifically the Rocard equation are reappraised in view of recent attempts to use a variety of forms of that equation for the interpretation of far infrared resonance absorption spectra. It is shown that for small values of the inertial parameter (heavy damping) the Rocard equation yields a really good approximation for the complex polarizability only for the small collision model considered by Gross and Sack. In the case of large inertial parameter values it is emphasized by means of plots of the complex polarizability that such an approximation always exhibits behavior characteristic of a sharply resonant system, i.e., a pronounced absorption peak well in excess of the Debye peak and a strongly negative real part, while the exact complex polarizability spectrum for the same parameter values merely displays inertia corrected Debye relaxation. Therefore, an explanation of the resonant term other than that based on a Rocard equation with a large inertial parameter must be sought as that equation strictly applies to inertia corrected Debye (heavily damped) relaxation only. The application of the itinerant oscillator model and the three variable Mori theory to the problem is discussed in view of this conclusion.