1 The small and large deformations of geomaterials

From the continuum mechanics point of view, a number of geomaterials are both

  1. damageable elastic solids in which highly localized features emerge as a result of failure and

  2. materials experiencing high, permanent strains that dissipate stresses.

In this sense, modelling their deformation lies between a solid mechanics (small deformations) and a fluid dynamics (large deformations) problem.

One important example is the Earth's crust, in which brittle fracturing and Coulomb stress redistribution are known to take place and for which scaling properties have been recognized for years (Kagan & Knopoff 1980; Turcotte 1992 and others). Along active faults, co-seismic fracturing activates aseismic creep, leading to deformations that can be larger than those associated with the fracturing itself (Cakir et al. 2012) and to slip rates that decrease progressively over years to decades due to various healing processes (Gratier et al. 2014). Creep relaxes a significant amount of elastic strain, retarding stress accumulation along some portions of faults and concentrating stresses on other locked portions. Hence this dissipative process should be included in earthquakes models (Cakir et al. 2012; Gratier et al. 2014). Another example is sea ice, which deforms rapidly under the action of the wind and ocean drags, in the brittle regime, and for which scaling properties have also been recently recognized (Marsan et al. 2004 and many others). In this case, much larger deformations occur once faults, or ice "leads" (see Fig. 1a, A), are formed and divide the ice cover into ice plates called "floes" (Fig. 1a, B), as these plates move relative to each other with much reduced mechanical resistance. In sea ice models, these large deformations must be accounted for as they set the overall drift and long-term evolution of the ice pack.

In such contexts, the challenge of the continuum modelling approach lies in the representation of the discontinuities that arise within a material due to fracturing processes using continuous variables and grid-cell averaged quantities. On the numerical point of view, another challenge arises as the methods employed must allow resolving the extreme gradients associated with these discontinuities while limiting the diffusivity associated with advective processes. Here, we present a simple continuum mechanical framework called Maxwell-Elasto-Brittle (Maxwell-EB), built in the view of allowing a transition between the small deformations associated with the fracturing and the larger, permanent, post-fracture deformations, while having the capability of damage mechanics models to reproduce the observed space and time scaling properties of the deformation of brittle geomaterials. The theoretical and numerical development of this new rheological model will be discussed of in two different geophysical contexts:

  1. modelling the drift and deformation of Arctic sea ice at regional scales and

  2. representing the pre-eruption deformation of a volcanic edifice.

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