This paper is concerned with a dense, randomly packed, granular material that consists of identical spheres or disks with elastic, frictional interactions, that is first isotropically compressed and subsequently loaded along an arbitrary stress path. An analytical relationship between the overall stress and strain increments is determined for the pre-failure regime. The purpose of the modelling is to understand how this relation depends upon the features of the packing and the particle interactions. From the outset it is recognised that the packing and interactive properties for these materials may vary substantially from grain to grain and the heterogeneity introduced in this manner is fully accounted for. Moment equilibrium equations are solved for each particle and force equilibrium equations are solved for each neighbourhood. Then, the heterogeneity of the aggregate is taken into account by introducing means and fluctuations in the description of the local deformations and the measures of the particles and interactions. The general development is illustrated with an example in two dimensions in which the packing and contact interactions are approximated by angular distributions and the heterogeneity is introduced by variations in these. For an isotropic medium with constant contact stiffnesses the theory provides predictions that compare well with results obtained from numerical simulations.

We present a quantitative treatment of the macroscopic behaviour of a sheared granitic magma using Biot's theory [Biot, 1941] and a shear‐dilatancy sensitive material model for the solids. The calculations are relevant to a magma in the solidosity range 0.55–0.7. The resulting excess pore pressure distribution is a function of the position and the time. Results are presented in graphical form. A scaling emerges that enables the results to be presented in non‐dimensional form. A sensitivity study is carried out of the parameters describing the rheology of the solid matrix (including permeability features). The model enables the estimate of pressure and flow rates, thus opening a way of understanding the features in the granite crystal mush that are caused by upflowing magma. At high strain rates (10−10S−1) flow rates due to shear far exceed melt movement due to buoyancy effects.

S U M M A R YWe present a model for flow and seepage in a deforming, shear-dilatant sensitive porous layer that enables estimates of the excess pore fluid pressures and flow rates in both the melt and solid phase to be captured simultaneously as a function of stress rate. Calculations are relevant to crystallizing magma in the solidosity range 0.5-0.8 (50-20 per cent melt), corresponding to a dense region within the solidification front of a crystallizing magma chamber. Composition is expressed only through the viscosity of the fluid phase, making the model generally applicable to a wide range of magma types. A natural scaling emerges that allows results to be presented in non-dimensional form. We show that all length-scales can be expressed as fractions of the layer height H, timescales as fractions of H 2 (nβ θ + 1)/(θ k) and pressures as fractions of Rċ 0 H 2 /(θ k). Taking as an example the permeability k in the mush of the order of magnitude 10 −15 m 2 Pa −1 s −1 , a layer thickness of tens of metres and a mush strength (θ ) in the range 10 8 -10 12 Pa, an estimate of the consolidation time for near-incompressible fluids is of the order of 10 5 -10 9 s. Using mush permeability as a proxy, we show that the greatest maximum excess pore pressures develop consistently in rhyolitic (high-viscosity) magmas at high rates of shear (ċ 0 > 10 −1 Pa s −1 ), implying that during deformation, the mechanical behaviour of basaltic and rhyolitic magmas will differ. Transport parameters of the granular framework including tortuosity and the ratio of grain size to layer thickness (a/H) will also exert a strong effect on the mechanical behaviour of the layer at a given rate of strain. For dilatant materials under shear, flow of melt into the granular layer is implied. Reduction in excess pore pressure sucks melt into the solidification front at a velocity proportional to the strain rate. For tectonic rates (generally 10 −14 s −1 ), melt upwelling (or downwelling, if the layer is on the floor of the chamber) is of the order of cm yr −1 . At higher rates of loading comparable with emplacement of some magmatic intrusions (∼10 −10 s −1 ), melt velocities may exceed effects due to instabilities resulting from local changes in density and composition. Such a flow carries particulates with it, and we speculate that these may become trapped in the granular layer depending on their sizes. If on further solidification the segregated grain size distribution of the particulates is frozen in the granular layer, structure formation including layering and grading may result. Finally, as the process settles down to a steady state, the pressure does not continue to decrease. We find no evidence for critical rheological thresholds, and the process is stable until so much shear has been applied that the granular medium fails, but there is no hydraulic failure.

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