The problem of sliding at a nonplanar grain boundary is considered in detail. The stress field, and sliding displacement and velocity can be calculated at a boundary with a shape which is periodic in the sliding direction (a wavy or stepped grain boundary): a) when deformation within the crystals which meet at the boundary is purely elastic, b) when diffusional flow of matter from point to point on the boundary is permitted. The results give solutions to the following problems. 1) How much sliding occurs in a polycrystal when neither diffusive flow nor dislocation motion is possible? 2) What is the sliding rate at a wavy or stepped grain boundary when diffusional flow of matter occurs? 3) What is the rate of diffusional creep in a polycrystal in which grain boundaries slide? 4) How is this creep rate affected by grain shape, and grain boundary migration? 5) How does an array of discrete particles influence the sliding rate at a grain boundary and the diffusional creep rate of a polycrystal? The results are compared with published solutions to some of these problems.

On grain boundary sliding and diffusional creep

We discuss the deformation of a curved interface between solid phases, assuming small strains in the bulk phases and neglecting accretion at the interfaces. Such assumptions are relevant to the deformation of solid microstructures when atomic diffusion and the formation of defects such as dislocations are negligible. We base our theory on a constitutive equation giving the (excess) free energy ψ of the interface when the interfacial limits of the displacement fields in the abutting phases as well as the limits of the displacement gradients are known. Using general considerations of frame invariance, we show that ψ can depend on these quantities at most through: firstly the normal and tangential components of the jump in displacement at the interface (stretch and slip), secondly the average of the projected strain in the tangent plane (average tangential strain), thirdly the tangential component of the jump in the projected displacement gradient at the interface (relative tangential strain and rel...

A general theory of curved deformable interfaces in solids at equilibrium

Publisher Summary This chapter discusses the elastic constants of crystals. The elastic properties of solid matter hold interest for both technology and basic research. In the first field applied elasticity is an important discipline for those fundamental considerations of engineering design which are usually included under the designation “strength of materials.” The treatment of structural materials requires semiempirical methods, because their compositions are complex and prior treatment has a pronounced effect. On the other hand, basic research into elastic properties is usually concerned with work on specimens in the simplest state which can be obtained reproducibly—for example, annealed single crystals. The complete tabulation of elastic constants for such specimens is valuable, not only for itself, but also because the data can be correlated with other physical measurements and thereby provide possible insight into the nature of the atomic forces in solid matter. The research aspect of elastic studies is of primary interest for this chapter. The case of the isotropic medium is important both for the chronological development of the subject of elasticity and also for its applicability to polycrystals and glasses.

The Elastic Constants of Crystals

Porous structures having a continuous solid phase with isolated pores were prepared by the addition of different amounts of crushed naphthalene to an alumina casting slip. Samples of from 5 to 50% porosity were fired together for comparable grain development, eliminating structural variables except porosity. Effects of porosity and temperature on strength, elastic modulus, modulus of rigidity, and coefficient of thermal expansion were investigated. Effects of porosity on thermal stress resistance and tor-sional creep properties were studied at constant temperature.

Effect of Porosity on Physical Properties of Sintered Alumina

CXVII. A theory of work-hardening of metal crystals

The mechanical behavior of grain boundaries in metals has been a subject of constant controversy. The present research is designed to examine thoroughly the mechanical behavior of grain boundaries in metals in a quantitative manner. A simple torsional apparatus has been devised for measuring four types of anelastic effects at very low stress levels, namely: internal friction at low frequencies; variation of dynamic rigidity with temperature; creep under constant stress; and stress relaxation at constant strain. All four types of anelastic effects have been studied in 99.991 percent polycrystalline aluminum as well as in single crystal aluminum; these effects are practically absent in single crystal aluminum. The four types of anelastic effects observed in polycrystalline aluminum are completely recoverable and are linear with respect to the applied stress and prior strain. They satisfy the interrelations derived by Zener from Boltzmann's superposition principle within experimental error. These are consistent with the viewpoint that the grain boundaries behave in a viscous manner. The maximum amount of shear stress relaxation in polycrystalline aluminum determined by the four types of anelastic measurements is about 33 percent. This is in good agreement with the theoretical value of 36 percent calculated by assuming the grain boundary to be viscous. The heat of activation associated with the viscous slip along the grain boundaries has been found to be 34,500 calories per mole. The coefficient of viscosity of the grain boundaries in aluminum estimated using this heat of activation, is consistent with that of molten aluminum at the same temperature. Similar anelastic effects have been also observed in polycrystalline magnesium, indicating that the viscous behavior is common to all metals.

Experimental Evidence of the Viscous Behavior of Grain Boundaries in Metals

In order to elucidate further the concept of relaxation of shear stress across grain boundaries in metals, the temperature dependence of internal friction and rigidity modulus of 99.991 percent aluminum have been measured as a function of frequency of torsional vibration and as a function of grain size of the specimen. It has been found that for the same specimen, an increase of frequency of vibration shifts the internal friction curve and the rigidity relaxation curve (${Q}^{\ensuremath{-}1}$ and $\frac{G}{{G}_{U}}$ versus temperature) to higher temperatures; and when the frequency of vibration is kept constant, a change in grain size of the specimen has the same effect as a change of the frequency of vibration The observed internal friction and rigidity relaxation can be expressed as functions of the parameter $(\mathrm{G}.\mathrm{S}.)\ifmmode\times\else\texttimes\fi{}f\ifmmode\times\else\texttimes\fi{}\mathrm{exp}(\frac{H}{\mathrm{RT}})$, where (G.S.) is the grain size or average grain diameter of the specimen, $f$ is the frequency of vibration, and $H$ is the heat of activation. It is shown that all these observed phenomena are necessary manifestations of the stress relaxation across grain boundaries arising from the viscous behavior of the grain boundaries in metals, which behavior has been demonstrated by previous anelastic-effect measurements.

T'ing-Sui Kê

Papers

Experimental Evidence of the Viscous Behavior of Grain Boundaries in Metals

Stress Relaxation across Grain Boundaries in Metals