Deformation/strain and stress

The necessary vocabulary, fundamentals and definitions for ‘deformation’, ‘strain’ and ‘stress’ are provided. Types of deformation, incremental or progressive, pure shear and simple shear, deformation regimes, flow lines and vorticity number, shortening, extension and strain measurements are explained. The concept of stress acting on a surface, through its normal and shear components is presented, along with their graphical representation using the Mohr diagram. In the elastic domain that characterizes very small strains, the relationship between stress and strain is discussed through the elastic constants among which the shear modulus and Poisson’s coefficient are notable. Finally, the stress–strain relationships for the ductile (plastic) and viscous behaviours, characteristic of large deformations, are discussed. These form the basis of understanding the rheology of the Earth, and hence Tectonics.

New continuity-based velocity interpolation scheme for staggered grids

<p>In the marker-in-cell method combined with staggered finite differences, Lagrangian markers carrying information on material properties are advected with the velocity field interpolated from the staggered Eulerian velocity grid. With existing schemes, velocity interpolation from the grid points to markers violates (to some extent) mass conservation requirement that causes excess convergence/divergence of markers and opening marker gaps after significant amount of advection. This effect is especially well visible in case of diagonal simple shear deformation along planes that are oriented at 45 degrees to the grid and marker circulation through grid corners.</p><p>Here, we present a new second order velocity interpolation scheme that guarantees exact interpolation of normal strain rate components from pressure nodes (i.e. from the locations where these components are defined by solving of the mass conservation equation). This new interpolation scheme is thus applicable to both compressible and incompressible flow and is trivially expendable to 3D and to non-regular staggered grids.</p><p>Performed tests show that, compared to other velocity interpolation approaches, the new scheme has superior performance in preserving continuity of the marker field during the long-term advection including the diagonal simple shear deformation and marker circulation through grid corners. We showcase a performance-oriented implementation of the new scheme using Julia language's shared memory parallelisation features. The Julia implementation of the new advection schemes further augments the ParallelStencil.jl  related application collection with advection routines.</p>

New sights in crenulation geometry developed in anisotropic materials undergoing simple shear deformation

<p>Deformation of foliated rocks commonly leads to crenulation or micro-folding, with the development of cleavage domains and microlithons. We here consider the effect of mechanical anisotropy due to a crystallographic preferred orientation (CPO) that defines the foliation, for example by of alignment of micas. Mechanical anisotropy enhances shear localisation (Ran, et al., 2018; de Riese et al., 2019), resulting in low-strain domains (microlithons) and high-strain shear bands or cleavage domains. We investigate the crenulation patterns that result from moderate strain simple shear deformation, varying the initial orientation of the mechanical anisotropy relative to the shear plane.  </p><p>We use the Viscoplastic Full-Field Transform (VPFFT) crystal plasticity code coupled with the modelling platform ELLE (http://www.elle.ws; Llorens et al., 2017) to simulate the deformation of anisotropic single-phase material with an initial given CPO in dextral simple shear in low to medium strain. Deformation is assumed to be accommodated by glide along the basal, prismatic and pyramidal slip systems of a hexagonal model mineral. An approximately transverse anisotropy is achieved by assigning a small critical resolved shear stress to the basal plane. An initially point-maximum CPO at variable angles to the shear plane defines the initial straight foliation at different angles to the shear plane, limiting ourselves to orientations in which the foliation is in the stretching field. The resulting crenulation geometries strongly depend on the orientation of the foliation and we observe four types of localisation behaviour: (1) synthetic shear bands, (2) antithetic shear bands, (3) initial formation of antithetic shear bands and subsequent development of synthetic shear bands, and (4) distributed, approximately shear-margin parallel strain localisation, but no distinct shear bands.</p><p>The numerical simulations not only show the evolving strain-rate field, but also the predicted finite strain pattern of existing visible foliations. We show the results for layers parallel to the foliation, but also cases where the visible layering is at an angle to the mechanical anisotropy (e.g. in case of distinct sedimentary layers and a cleavage that controls the mechanical anisotropy). A wide range of crenulation types form as a function of the initial orientation of the visible layering and mechanical anisotropy (comparable to C, C' and C'' shear bands and compressional crenulation cleavage). Most importantly, some of may be highly misleading and may easily be interpreted as indicating the opposite sense of shear.</p><p>Reference</p><p>de Riese, T., et al. (2019). Shear localisation in anisotropic, non-linear viscous materials that develop a CPO: A numerical study. Journal of Structural Geology, 124, 81-90. DOI: 10.1016/j.jsg.2019.03.006</p><p>Llorens, M.-G., et al. (2017). Dynamic recrystallisation during deformation of polycrystalline ice: insights from numerical simulations. Philosophical Transactions of the Royal Society A, Special Issue on Microdynamics of Ice, 375: 20150346. DOI: 10.1098/rsta.2015.0346.</p><p>Ran, H., et al. (2018). Time for anisotropy: The significance of mechanical anisotropy for the development of deformation structures. Journal of Structural Geology, 125, 41-47. DOI: 10.1016/j.jsg.2018.04.019</p>

Explicit Representation of SO(3) Rotation Tensor for Deformable Bodies

Computing the rotation tensor is vital in the analysis of deformable bodies. This paper describes an explicit expression for the SO(3) rotation tensor R of the deformation gradient F, and successfully establishes an intrinsic relation between the exponential mapping Q = exp A and the deformation F. As an application, Truesdell's simple shear deformation is revisited.