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Bauville and Yamato (2021, https://doi.org/10.1029/2020gc009280) (G‐cubed) propose model‐based methods to convert metamorphic pressures to depths based on the claim that pressure data from global (ultra)high‐pressure([U]HP) rocks challenge the lithostatic assumption and support their model which invokes excessive overpressures. It is argued here that the opposite is true: Natural pressure data are fully consistent with the lithostatic assumption. They reflect selection of (U)HP rocks by accessibility and preservation. The data are however inconsistent with the model predictions of Yamato and Brun (2017, https://doi.org/10.1038/ngeo2852) and Bauville and Yamato (2021, https://doi.org/10.1029/2020gc009280). Furthermore, their model requires critical assumptions that are not justified by the principles of rock mechanics and unsupported by microstructures from (U)HP rocks. Key Points: Pressure data from (ultra)high pressure rocks are compatible with lithostatic assumption but are inconsistent with excessive overpressuresModel invoking excessive overpressures requires assumptions not justifiable and is unsupported by rock microstructuresTectonic fabrics cannot represent stress states and Mohr‐Coulomb plasticity is independent of stress rotation [ABSTRACT FROM AUTHOR]

Quartz c‐axis fabrics in natural mylonites can vary to such an extent that they apparently give opposite senses of shear in a single thin section. Many hypotheses have been invoked to explain this. Here, we couple our self‐consistent multiscale approach for flow partitioning with the viscoplastic self‐consistent model for c‐axis fabric simulation to investigate quartz c‐axis fabric development. Quartz aggregates are regarded as microscale Eshelby inhomogeneities embedded in a macroscale medium whose effective rheology is represented by a hypothetical homogeneous equivalent medium which is assumed to be rheologically isotropic or has a planar anisotropy. We reproduced the observed quartz c‐axis fabrics. We found that although the microscale flow fields are distinct from one another and from the macroscale flow, the microscale vorticity in every inhomogeneity has the same sense as the macroscale vorticity. This implies that one can use the average of the microscale vorticity axes determined through the crystallographic vorticity axis analysis to obtain the macroscale vorticity axis. However, quartz c‐axis fabrics cannot be used to determine the vorticity number where flow partitioning is significant. Key Points: Multiscale simulation of quartz c‐axis fabrics coupling Multi order power law approach and visco‐plastic self‐consistent modelingObserved natural and experimental quartz c‐axis fabric variations explained by flow partitioningNatural quartz c‐axis fabric variation reflected macroscale finite strain gradients without sense of vorticity reversal [ABSTRACT FROM AUTHOR]

An accurate flow law for dislocation creep of quartz aggregates is critical for the understanding of continental rheology, numerical modeling of lithospheric processes, and the interpretation of microstructures of natural quartz‐bearing mylonites. Despite many decades of research, considerable discrepancies still exist among quartz flow laws determined from different experiments. We demonstrate that the key to reconcile these discrepancies is to consider the pressure dependence of the activation enthalpy. From existing high‐quality creep experiments on quartz aggregates, we critically identified test runs having microstructures and stress‐strain curves indicating steady state regimes 2 and 3 dislocation creep and used the data to obtain a set of flow law parameters most consistent among existing experiments. Because estimates of strain rate and stress from natural mylonites are bound with large uncertainties, they cannot be used to construct a flow law better than one determined from well‐controlled creep experiments. A large number of geological studies and modern GPS observations suggest that crustal scale ductile shear zones likely flow at a strain rate range between 10−13 and 10−11 s−1. In this strain rate range, our flow law predicts flow stresses broadly consistent with stress estimates from many natural mylonites based on well‐calibrated piezometers for dynamically recrystallized quartz grains. Key Points: Pressure dependence of activation enthalpy is key to reconcile discrepancies among flow laws for quartz derived from different experimentsA consistent flow law for climb‐accommodated dislocation creep for quartzite is obtained from critically selected high‐quality experimental dataThe new flow law is discussed in the context of stress estimates from natural mylonites based on various piezometric methods [ABSTRACT FROM AUTHOR]

The total pressure a rock element is subjected to in Earth's lithosphere undergoing deformation may deviate from the lithostatic pressure. Despite many decades of research, the significance of this pressure deviation is still debated. Here, we apply the micromechanics approach based on the generalized Eshelby inclusion solutions for anisotropic power-law viscous materials to investigate the pressure deviation in rheologically heterogeneous rocks. We regard a rheologically distinct element (RDE) as a microscale heterogeneous inclusion which is embedded in and interacts with the heterogeneous macroscale medium. The latter is represented by a homogeneous-equivalent medium (HEM) with its effective rheology obtained self-consistently from the rheological properties of all constituent elements making up the ambient macroscale material embedding the RDE. Partitioning equations from the generalized Eshelby solutions and developed numerical implementations allow the pressure deviations inside and around the RDE to be calculated with quasi-analytical accuracy. We prove formally for limiting cases and demonstrate numerically that the maximum pressure deviation in and around any RDE is on the same order as the deviatoric stresses in the ambient medium or in the element for general power-law viscous materials, isotropic or anisotropic. The pressure deviation fields related to a RDE in an anisotropic HEM are lower than the pressure deviation fields related to the same RDE in an isotropic HEM. Pressure deviations due to an initial pressure anomaly are insignificant considering the viscoelastic interaction between an RDE and the HEM. Our results suggest that pressure deviations in Earth's lithosphere are insignificant for metamorphic processes if the differential stress in Earth's lithosphere is on a few hundred MPa level. Higher pressure deviations require correspondingly greater differential stresses in the lithosphere. They may be generated by short-term strong elastic interactions. [ABSTRACT FROM AUTHOR]

Theories and numerical solutions for a viscous ellipsoid in an infinite anisotropic viscous medium subjected to far-field homogeneous deformation lie at the heart of self-consistent homogenization models and multiscale simulations of texture and fabric development in Earth's lithosphere. There is considerable literature on ellipsoid inclusions, focused on anisotropic elastic materials, published in multi-disciplinary fields. To make this body of work more accessible as well as to advance viscous inclusion studies, an effort is made here to summarize recent advances and to further develop formally more explicit and, where possible, analytic solutions for incompressible viscous materials. The point-force concept and equivalent inclusion method of Eshelby are used together with the Green function approach. This leads to generalized equations for ellipsoid inclusion behaviors in anisotropic materials. In the particular case of isotropic materials, the new mathematical development here enables the use of existing methods for elastic materials to get solutions for corresponding viscous inclusion problems efficiently and accurately. A 2D formulation is also presented for elliptic cylinders in plane-straining flows of anisotropic materials, using the same Green function method that is adopted for 3D inclusions. The 2D formulation is benchmarked with existing analytic solutions. A reconnaissance investigation to compare the behaviors of 2D elliptic inclusions and triaxial ellipsoids in a matrix of planar anisotropy undergoing far-field plane-straining flows suggests that conclusions drawn from 2D cannot be applied to 3D in anisotropic cases. The application of the viscous inclusion theory to the rheologically heterogeneous and non-linear lithosphere is discussed. By regarding a heterogeneous element as an ellipsoidal inclusion embedded in a hypothetical homogeneous equivalent matrix whose effective rheology is obtained through micromechanical homogenization and assuming that the conditions of scale separation are satisfied, the viscous inclusion theory forms the backbone of multiscale models for natural deformation and the accompanying texture and fabric development. [ABSTRACT FROM AUTHOR]

To address the multiscale deformation and fabric development in Earth's ductile lithosphere, micromechanics-based self-consistent homogenization is commonly used to obtain macroscale rheological properties from properties of constituent elements. The homogenization is heavily based on the solution of an Eshelby viscous inclusion in a linear viscous medium and the extension of the solution to nonlinear viscous materials. The homogenization requires repeated numerical evaluation of Eshelby tensors for constituent elements and becomes ever more computationally challenging as the elements are deformed to more elongate or flattened shapes. In this paper, we develop an optimal scheme for evaluating Eshelby tensors, using a combination of a product Gaussian quadrature and the Lebedev quadrature. We first establish, through numerical experiments, an empirical relationship between the inclusion shape and the computational time it takes to evaluate its Eshelby tensors. We then use the relationship to develop an optimal scheme for selecting the most efficient quadrature to obtain the Eshelby tensors. The optimal scheme is applicable to general homogenizations. In this paper, it is implemented in a MATLAB package for investigating the evolution of solitary rigid or deformable inclusions and the development of shape preferred orientations in multi-inclusion systems during deformation. The MATLAB package, upgrading an earlier effort written in MathCad, can be downloaded online. [ABSTRACT FROM AUTHOR]

Abstract: We test the hypothesis that small ductile shear zones are developed from initial rheologically weak domains. We regard weak domains as ellipsoidal inhomogeneities and apply Eshelby's formalism extended for power-law viscous materials to investigate the kinematics and finite strain evolution of the partitioned flow fields in weak domains. We show that, under an imposed bulk flow field, weak domains, regardless of their initial shapes and orientations, deform into zone-like features at relatively small bulk strains and the deformation paths inside weak domains have all characteristics expected in ductile shear zones. We apply our model to the Cap de Creus area Spain, where abundant small ductile shear zones exist. To relate the fabrics inside these shear zones with the regional deformation, we take a multi-scale approach. We assume that the area contained many weak domains which were randomly shaped and oriented initially and were deformed into shear zones eventually in response to the regional flow. We constrain the regional flow field by the fabric patterns across the area and compute numerically the partitioned flow fields in individual weak domains. The latter are related to fabrics inside shear zones. Our model reproduces first-order features of field-observed fabrics. Although the deformation path of each shear zone was close to simple shearing, the deformation of the whole belt was not. Our approach also resolves the strain compatibility problem for a finite-sized shear zone embedded in a far less deformed country rock. [Copyright &y& Elsevier]

Abstract: Earth’s lithosphere is heterogeneous in rheology on a wide range of observation scales. When subjected to a tectonic deformation, the incurred flow field can vary significantly from one rheologically distinct element to another and the flow field in an individual element is generally different from the bulk averaged flow field. Kinematic and mechanical models for high-strain zones provide the relations between prescribed tectonic boundary conditions and the resulting bulk flow field. They do not determine how structures and fabrics observed on local and small scales form. To bridge the scale gap between the bulk flow field and minor structures, Eshelby’s formalism extended for general power-law viscous materials is shown to be a powerful means. This paper first gives a complete presentation of Eshelby’s formalism, from the classic elastic inclusion problem, to Newtonian viscous materials, and to the most general case of a power-law viscous inhomogeneity embedded in a general power-law viscous medium. The formulation is then implemented numerically. The implications and potential applications of the approach are discussed. It is concluded that the general Eshelby formalism together with the self-consistent method is a powerful and physically sound means to tackle large plastic deformation of Earth’s lithosphere. [Copyright &y& Elsevier]

Abstract: Rocks are heterogeneous materials. Deformation structures and fabrics in the ductile lithosphere can result from the history of the three dimensional flow field, or the progressive deformation of some rheologically distinct elements embedded in the 3D flow field, or a combination of both. The flow and progressive deformation of a rheologically distinct element is generally non-steady even if the hosting 3D flow field is steady. Vorticity decomposition is the key to following the progressive history of the deformation of specific elements. A general theoretical background to the problem is presented. Flow and progressive deformation of surface (and layer-like) elements in 3D homogeneous flows are investigated specifically. Analytical procedures to calculate the sectional vorticity number and the sectional dilatancy number on a general section in a 3D flow field are introduced. The theoretical analysis is applied to various examples of sectional flow embedded in different types of 3D homogeneous and steady flow fields. Implications of this investigation for kinematic analysis of natural rock deformation are discussed. All types of material line histories are expected in sectional flows of layer-like elements. Shortened boudins can be formed in a single progressive deformation. In sequences of rocks with a well-developed transposition foliation common in large scale high-strain zones, each layer unit may have undergone a unique deformation history during its rotation to the final, shear zone boundary parallel orientation. In kinematic analysis of such areas, it is important to ascribe the correct kinematic significance to structural associations and fabrics. [Copyright &y& Elsevier]