Next: Discussion Up: GARNET ANVIL CELL (GAC) Previous: The Garnet Anvil Cell

Temperature and pressure distribution

As a consequence of the inner geometry of the cell, the temperature gradients over a sample sandwiched between the anvils remain negligibly small. This was checked by heating homogeneous discs of metals with a low melting point. This reveals no significant spatial dependency of melting onset under visual detection up to $330\, ^\circ$ C and atmospheric pressure (melt transitions of Sn, Bi and Pb at $231,9\, ^\circ$ C, $271,4\, ^\circ$ C and $327,5\, ^\circ$ C, respectively).

Under uniaxial loading, the stresses generated between the two Bridgman anvils obey the general relationship

$\begin{displaymath}\frac{d\sigma_r}{dr} + \frac{\sigma_r-\sigma_{\theta}}{r} + \frac{2f\sigma_z}{h} = 0 , \eqno(2) \end{displaymath}$

with the radial stress $\sigma_r$ , tangential stress $\sigma_{\theta}$ , and axial stress component $\sigma_z$ depending on radius r. h is the thickness of the sample disc and f is the coefficient of friction between the sapphire anvil and the sample surfaces [7]. Its solution for a cylinder with radius R under the external loading force F is given either by

$\displaystyle \sigma_r(r)$

$\displaystyle 2f\sigma_z (R-r) / 3h,$

for completely elastic deformations, or alternatively by

$\displaystyle \sigma_r(r)$

$\displaystyle \sigma_{\theta}(r) = \sigma_z(r) - \sigma_0 \cr$

for a cylinder deformed in the fully plastic regime, where $\sigma_0$ is the yield stress according to the Tresca yield criterion [7]. The radial pressure distribution inside the sample chamber is given by

$\begin{displaymath}p(r) = \frac{1}{3} \, [\sigma_r(r) + \sigma_{\theta}(r) + \sigma_z ] \eqno(5) \end{displaymath}$

and is either independent on radius r in the elastic limit according to eq. (3), or decreases exponentially with r in the flow region, in dependence of the friction coefficient f and the aspect ratio R/h, as shown in eq. (4).

It can be shown, that, for uniaxial loading, the central part of a sample between rigid cylindrical anvils remains at nearly hydrostatic conditions, separated from a surrounding ring showing plastic flow [8]. In this sense, the outer plastic ring of the sample acts as a gasket for its inner parts.

In addition to uniaxial loading, we are able to apply independent shear forces by rotating the lower anvil. The shear stresses produced in the sample disc are, under boundary conditions of no-slip, a linearly increasing function of rin the central (nearly elastic) region. The total shear stress experienced by the sample is therefore gradually increasing from zero at the centre and reaches its maximum near to yield stress at the periphery ring. The displacement field at the surface of the sample may be reconstructed either by image processing (subtraction of subsequent digitized images), or by means of strain markers distributed over the sample. In both cases, the accuracy is mainly limited by the spatial resolution of the digitizing device (video camera or scanner).

Next: Discussion Up: GARNET ANVIL CELL (GAC) Previous: The Garnet Anvil Cell

Michael Riedel
1999-01-27