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A global model is presented to account for the specific rheology of a two-phase material. Examples of observations are taken from a crystallizing magma and these are applied to a partially molten rock, plastic deformation and soil liquefaction. The general behaviour of the viscosity is drawn as a function of the strain rate and the amount of solid phase. It constitutes a 3D diagram developing a cubic surface. The cubic equation is justified by thermodynamic considerations. It results from the mixing of a Newtonian (n = 1) and a power law (n = 3) type of deformation. The diagram shows two types of rheological response. At high strain rate values, the viscosity contrast between the two phases is the lowest. The resulting en masse behaviour is observed during tectonic activity. It manifests itself by homogeneous transport of magma during emplacement and fabric development. An equivalent medium, with average viscosity is a good proxy. Conversely, at low strain rate values, the viscosity contrast between the two phases is the highest. The two end members behave according to their respective rheology. In between, a transitional state develops, in which instability occurs depending on the strain rate and stress conditions. In the 3D diagram it appears as a cusp shape. Rheology presents continuous jumps between the liquid-like and the solid-like rheology. They result in strain localization or phase segregation. The latter preferentially develops during magma crystallization. Deformation under a constant amount of each phase is also possible, resulting in pressure dissolution-like processes. A bifurcation in the solution plane of the equation of viscous motion causes instability. It is comparable with strain softening. A similar situation should develop when mixing Newtonian and power law rheology, for example during diffusion and dislocation creep, or water-saturated sediment deformation. Owing to the continual jumps between the two types of rheology, hysteresis or memory effect may develop. Rapid cyclic deformation may drive strain to extreme straining. The effect of simple shear seems much more effective than pure shear (compaction) to separate the weak phase from its strong matrix. The development of instabilities and continuous jumps from one rheology to the other lead to discontinuous motion of the weak phase. In a molten region, it corresponds to discontinuous bursts of magma that are extracted.

Rheology of a two-phase material with applications to partially molten rocks, plastic deformation and saturated soils