# Mechanical Spectroscopy

When using the method of mechanical spectroscopy one observes the strain of a solid sample while an external mechanical stress acts to it. The strain behavior gives hints about microscopic, structural processes in the sample. Especially the dynamic of lattice defects (e.g. grain boundaries, dislocations, guest atoms) or phase transitions can cause some typical anelastic effects. So mechanical spectroscopy is a good method to determine the parameters of these microscopic processes or to verify a model.

## Macroscopic Behavior

Consider a mechanical stress acting on a solid sample constantly in time. One observes an instantaneous elastic strain fullfilling Hooks law for stresses which are small enough.

Additionally one may observe a time dependent additional strain. This can reach a maxium value (**elastic aftereffect**, **anelastic addional strain**, see figure) or increase continuously in time up to breaking (creeping). In the absence of the external stress the sample constricts instantaneously by the value of the instantaneous elastic strain. The anelastic strain vanishes with a delay in time: the process is full revesible. Creeping is not reversible in this sense.

## Microscopic Explanation

The reasons for the elastic aftereffect are phase transitions or diffusion of lattice defects. This is easy understandable when using the example of diffusing interstitiell guest atoms: Consider guest atoms occupying some interstitiell sites in an unstrained solid. Among other things the occupation propabilities of the different sites depend on temperature and guest atoms concentration (for calculation of the propabilities one needs the configuration entropy and the (elastic) energy).

If the lattice gets strained the distances between the host atoms and whith it the energies of the interstitiell sites alter. Vividly speaking now some sites provide more space than others. Hence the guest atoms prefer to occupy these sites which provide more space (lower energy). So the lattice is additionally strained one can observe a macroscopic elastic aftereffect.

The reorientation of the guest atoms and with it the appearance of the additional strain do not happen instantaneously because the atoms must overcome an energy barrier in order to change the site. This energy they get of their thermal energy. So the mean jumping rate depends on the barrier energy (so called activation energy) and the temperature.

## Measure Methods

With **static measurements** (stress constant in time (Fig. 1)) one can receive the mean jump rate and the activation energy as well as verify models of the jump path.

If the mean jump time is too small for proper static measurements one can execute **dynamic measurement**s using stress oscillating harmonicly in time. The time delay of the jumps generates a phase shift between stress and strain which is measurable macroscopically as attenuation (**internal friction**). The results of measurements of this friction with different frequencies and temperatures are the mean jump time and the activation energy as well as a check of the jump model.

For the realisation of the measurements one usually uses samples in the shape of a reed. These get fixed in a clamp at one end and are inflected or excited to inflection or torsion vibrations. As an alternative one can fix the sample at a few points which must be the nodes of the vibration.