An Asymptotic Approach to the Calculation of Wave Fields in a Layer with a Defect with a Small Characteristic Size

AbstractDimensional reduction strategy is an effective approach to derive reliable conceptual models to describe flow in fractured porous media. The fracture aperture is several orders of magnitude smaller than the characteristic size (e.g., the length of the fracture) of the physical problem. We identify the aperture to length ratio as the small parameter 𝜖 with the fracture permeability scaled as an exponent of 𝜖. We consider a non-Newtonian fluid described by the Carreau model type where the viscosity is dependent on the fluid velocity. Using formal asymptotic approach, we derive a catalogue of reduced models at the vanishing limit of 𝜖. Our derivation provides new models in a hybrid-dimensional setting as well as models which exhibit two-scale behaviour. Several numerical examples confirm the theoretical derivations of the upscaled models. Moreover, we have also studied the sensitivity of the upscaled models when a particular upscaled model is used beyond its range of validity to provide additional insight.

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Cosserat Modeling of Size Effects in Crystalline Solids

MRS Proceedings ◽

10.1557/proc-653-z8.2 ◽

2000 ◽

Vol 653 ◽

Author(s):

Samuel Forest

Keyword(s):

Stress State ◽

Size Effects ◽

Elastic Inclusion ◽

Characteristic Size ◽

Infinite Matrix ◽

Crystalline Solids ◽

Generalized Continua ◽

Absolute Size ◽

Kink Bands ◽

The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

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