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= Paleostress Inversion =

It refers to the determination of paleostress history from a number of evidence found in existing rocks at the present, based on the principle that past tectonic stress should have left traces in the rocks. Such relationship has been discovered from field studies for years, that qualitative and quantitative analyses of deformation structures are reliable in understanding the distribution and evolution of paleostress fields across successive tectonic events. The deformation ranges from microscopic to regional scale, and from brittle to ductile behaviour, depending on the rheology of the rock, orientation and magnitude of the stress etc. Therefore, detailed observations in outcrops, as well as thin sections, with abundant deformation information is important in reconstructing the paleostress trajectories.

It is essential to note that there are always assumptions in the inversions, in order to simplify the sophisticated geological processes in nature and thus facilitate the application of models onto the conditions. Since the first introduction of the methods by Wallace and Bott, similar assumptions have been used throughout the decades. The stress field is assumed to be uniform for a considered rock mass containing faults and temporarily constant over the history of faulting in that region. Moreover, as the maximum shear stress resolved on the fault surface from the known stress field, the slip on each of the fault surface has the same direction and magnitude.

Grain boundary piezometer
Piezometer is the instrument used in the measurement of pressure (non-directional) or stress (directional). According to the paleostress inversion principle, a rock mass under stress should exhibit strain in both macroscopic and microscopic scale, while the latter happens at the grain boundaries. The grain boundary here refers to the interface between crystal grains at the magnitude below $$10^2 \mu m$$. The average grain size arising from dynamic recrystallization (DRX) is primarily dependent on flow stress. As the recrystallized grain size is easily measured under the optical microscope and believed to be stable during the post-deformation period, when compared to other stress indicators, it is often used as an indicator of paleostress in tectonically active regions such as crustal shear zones, orogenic belts and the upper mantle.





Dynamic recrystallization (DRX) is considered to be one of the key mechanisms of grain size reduction in shear zones (Tullis and Yund, 1985). DRX is defined as a nucleation-and-growth process because local grain boundary bulging (BLG) and subgrain rotation (SGR), which belong to mechanisms of nucleation, and grain boundary migration (GBM), which belongs to mechanisms of grain growth, all present in the deformation. Such evidence is observed in deformed quartz, a commonly used piezometer, from ductile shear zone by optical microscope and transmission electron microscope (TEM), that SGR and GBM occur sequentially. The nucleation process of DRX takes place when materials are strained to certain critical values, and occurs at the margins of existing grains. For BLG, nuclei grow at the expense of existing grains and form a ‘necklace’ structure. While for SGR, strain-induced polygonization results from the progressive misorientation, also called ‘recrystallization in situ’ without considerable grain growth. Microstructures indicative of such process are typical in minerals like quartz, olivine, calcite and halite that have gone through syn-tectonic deformation or manual high-temperature creep. Therefore, BLG and SGR are differentiated as discontinuous and continuous DRX respectively.

Static energy-balance model
The theoretical basis of grain size piezometry was first laid by Robert J. Twiss in late 1970s. By comparing free dislocation energy and grain boundary energy, he derived a static energy balance model applicable to subgrain size. The coefficient K was set to be temperature-dependent so that the grain size is solely dependent on stress. Such relation has been represented by an empirical equation between normalized value of grain size and flow stress, which is universal for various materials:

$$\frac{d}{b}=K\left (\frac{\sigma}{\mu}\right )^{-p}$$,

d is the average grain size; b is the length of the Burgers vector; K is a non-dimensional constant, which is typically in the order of 10; µ is the shear modulus; σ is the flow stress.

However, it did not account for the continuously changing nature of microstructures observed in DRX, so it is not applicable to recrystallized grain size.

Nucleation-and-growth models
Unlike the previous model, these models consider the sizes of individual grains vary over time and space, therefore, they derive an average grain size from a dynamic balance between nucleation-and-grain growth. The scaling relation of the grain size is as followed:

$$\hat{d}=a\left (\frac {\dot{R}}{I} \right )^{1/4}$$,

where d is the mode of logarithmic grain size, I is the nucleation rate per unit volume, and a is a scaling factor. Upon this basic theory, there are still plenty of arguments on the details, which are reflected in the assumptions of the models, so there are various modifications.

Derby–Ashby model

They considered BLG nucleation at grain boundary sites pinned by sub-boundaries in determining the nucleation rate (I). Thus this model describes the microstructures of discontinuous DRX (DDRX).

Shimizu model

Because of a contrasting assumption that SCR nucleation in continuous DRX (CDRX) should be considered for the nucleation rate, Shimizu has come up with another model, which has also been tested in laboratory:

$$\frac{d}{b}=B\left (\frac{\sigma}{\mu}\right )^{-p} \exp \left (- \frac{\Delta Q}{mRT}\right )$$.

Simultaneous operation of dislocation and diffusion creeps
Field boundary model

In the above model, surface energy is neglected when considering the driving force of GBM, yet it may be significant if grain size is reduced substantially via DRX. As the creep mechanism may switch from dislocation creep to diffusion creep when grains are sufficiently small, and the grain would become larger again. Hence De Bresser et al. calculated the boundary zone between the fields of dislocation creep and diffusion creep where recrystallized grain size tends to stabilize, as a supplement to the above model. It is important to note the assumption that the average grain size reduction always occurs due to DRX in the dislocation creep field, and grain grows in the diffusion creep field, does not agree with the nucleation-and-growth models mentioned before.

Common piezometers
Quartz is a common crustal mineral whose creep microstructures are sensitive indicators of deformation conditions in middle to lower crust. Before starting to infer flow stress magnitude, the mineral has to be calibrated carefully in laboratory. Quartz has been found to exhibit different piezometer relations during different recrystallization mechanisms, which are local grain boundary migration (dislocation creep), subgrain rotation (SGR) and the combination of these two, as well as at different grain size.

Other common minerals used for grain size piezometers are calcite and halite, which also demonstrate difference in piezometer relation for distinct recrystallization= mechanisms.

Examples
= Reference =