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The MultiOrder PowerLaw Approach (MOPLA)

MOPLA stands for MultiOrder PowerLaw Approach (Jiang and Bentley, 2012). It is a multiscale model, realized by embedding ellipsoidal elements within ellipsoid elements – hence the Multi-Order. The primary (first-order) elements make up the deforming continuum and higher-order elements define fabrics. Fabric-defining elements follow the deformation in their embedding host elements. All rheological elements, regardless of orders, are power-law viscous – hence the PowerLaw.

MOPLA was developed to address the multiscale deformation in Earth’s rheologically heterogeneous lithosphere. It includes theoretical formulation (mainly in Jiang 2014, 2016) and numerical (MathCad and Matlab) implementations.

The backbone theory of MOPLA is the solutions of an Eshelby inclusion problem in a general powerlaw viscous material. The original Eshelby inclusion problem deals with an ellipsoid elastic inclusion in an infinite uniform elastic matrix subjected to homogeneous deformation at infinity. Eshelby (1957, 1959) gives the formal exact solutions which express the mechanical fields inside the inclusion (called the interior fields) and around the inclusion in the matrix (called the exterior fields) in terms of the far-field stress and strain fields. The formal solutions are now referred to as the “partitioning equations” or “interaction equations”. Although Eshelby’s original solutions were based on isotropic linear elastic materials, it has been developed to include general anisotropic elastic and general (isotropic and anisotropic) Newtonian materials (Mura, 1987; Bilby and Kolbuszewski, 1977, Bilby et al. 1975; Jiang 2013). Although there are no exact solutions for the Eshelby inclusion problem in nonlinear materials, the formal solutions (interaction equations) obtained from linear rheologies can be applied to nonlinear rheological materials as good approximate solutions with a linearization approach. We refer to these formal solutions based on the linearization approach as the Generalized Eshelby Solutions (GES) below.

Earth’s lithosphere is a heterogeneous material which at any scale of observation is made of many Rheologically Distinct Elements (RDEs). To apply GES to the deformation and associated fabric development in Earth’s lithosphere, MOPLA makes the following assumptions: First, a RDE is treated as an Eshelby ellipsoid inclusion. This assumption means that the results from the application of the Eshelby formal solutions should be regarded as the averaged fields in the RDE. Second, the heterogeneous lithosphere surrounding any given RDE is replaced by a hypothetical Homogeneous-Equivalent Medium (HEM), the rheology of which is obtained from the rheologies of all RDEs contained in a Representative Volume Element (RVE) centered at the RDE. The procedure to get the HEM rheology from the rheologies of the constituent RDEs is called homogenization. Homogenization equations are also formulated in terms of GES (Jiang 2014, 2016). The volume of the RVE must be large enough in order to be representative of the local bulk material and to satisfy the condition of separation of scales so that the RDE is much smaller than the RVE. This separation of scales justify the use of the inclusion solution which is based on an inclusion within an infinite medium. However, the RVE should not be so large that the macroscale variation of the large-scale deformation is not captured. Homogenization equations and interaction equations must be solved simultaneously in a self-consistent way. Once self-consistent solutions are obtained, local mechanical fields in any given RDE are used as boundary conditions to calculate the deformation and fabric development inside that RDE. The algorithm thus moves to a high-order element.

Both the homogenization equations and the interaction equations are derived from the Eshelby inclusion formalism and are solved simultaneously in a self-consistent way (details in Jiang 2014).

References:

Bilby, B.A., Eshelby, J.D., and Kundu, A.K., 1975, The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity: Tectonophysics, v. 28, p. 265-274.

Bilby, B.A., Kolbuszewski, M.L., 1977. The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow. Proceedings of the Royal Society of London, A355, 335-353.

Eshelby, J.D. 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems: Proceedings of the Royal Society of London, v. A241, p. 376-396.

Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proceedings of the Royal society of London, A 252, 561-569.

Jiang, D., 2013. The motion of deformable ellipsoids in power-law viscous materials: Formulation and numerical implementation of a micromechanical approach applicable to flow partitioning and heterogeneous deformation in Earth’s lithosphere. Journal of Structural Geology 50, 22-34.

Jiang, D. 2014. Structural geology meets micromechanics: A self-consistent model for the multiscale deformation and fabric development in Earth's ductile lithosphere. Journal of Structural Geology, 68, 247-272. doi:10.1016/j.jsg.2014.05.020.

Jiang, D. 2016. Viscous inclusions in anisotropic materials: Theoretical development and perspective applications. Tectonophysics 693, 116-142. doi:10.1016/j.tecto.2016.10.012

Jiang, D. and Bentley, C., 2012. A micromechanical approach for simulating multiscale fabrics in large-scale high-strain zones: Theory and application. Journal of Geophysical Research 117: B12201, doi:12210.11029/12012JB009327.

Lebensohn, R.A., and Tomé, C.N., 1993, A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys: Acta Metallurgica et Materialia, 41, 2611-2624.

Molinari, A., Canova, G.R., and Ahzi, S., 1987, A self consistent approach of the large deformation polycrystal viscoplasticity. Acata Mettalurigica, v. 35, 2983-2994.

Mura, T., 1987. Micromechanics of defects in solids. Martinus Nijhoff Publishers, Dordrecht/Boston/Lancaster, 587 pp.