Abstract. We develop a mathematical model of frost heave in colloidal soils. The theory accounts for heave and consolidation, while not requiring a frozen fringe assumption. Two solidification regimes occur: a compaction regime in which the soil consolidates to accommodate the ice lenses, and a heave regime during which liquid is sucked into the consolidated soil from an external reservoir, and the added volume causes the soil to heave. The ice fraction is found to vary inversely with the freezing velocity V while the rate of heave is independent of V , consistent with field and laboratory observations. Key words. frost heave, colloids, Lie-Backlund transformation AMS subject classifications. 86A40, 86A991. Introduction. Frost heave is a phenomenon in which the ground surface deforms and heaves under the action of freezing and thawing. The process leads to peculiar geological features (patterned ground) in cold regions of Earth as well as to challenging engineering problems [48]. In the early twentieth century it was discovered that frost heave is caused by the formation of repetitive layers of segregated ice in the soil, called ice lenses [43,10]. The most natural explanation for heave -expansion of ice upon freezing -has surprisingly little effect. In fact Taber [43] and Zhu et al. [50] showed that frost heave occurs in soils saturated with benzene and argon, materials which contract upon freezing. The heave occurs because the water which forms the ice lenses is drawn from other regions of the soil, and the frozen part comes to contain an excess mass of ice. The amount of heave is highly correlated with the volume of ice lenses formed [43,48].Mathematical models of frost heave tend to adopt the frozen fringe assumption [26,27,13,21,39]. A partially frozen region of soil is assumed to exist ahead of (i.e. at temperatures warmer than) the warmest ice lens. A combination of factors allow the partially frozen soil to segregate, revealing a new lens of pure ice [26,39]. Remarkably, frost heave experiments on highly compressible laboratory soils such as colloidal silica found that there is no pore ice near the ice lenses [6,7,45,46]. These results confirm early observations of Beskow that in clays and fine silts the soil between ice lenses is soft and unfrozen [5,28], but have yet to be explained theoretically.In the present work we develop a continuum theory of frost heave that does not depend on the frozen fringe assumption. The model bears some similarity to mushy layer models of alloy solidification [18,49], in that we treat the region containing segregated ice lenses as a continuum, rather than attempt to track individual ice crystals. In section 2 we discuss the phase diagram of a colloidal soil composed of silica microspheres, distinguishing between segregational freezing (ice lenses) and interstitial freezing (pore ice). The frost heave model is presented in section 3, and in section 4 we present results and comparison with experiment. Some analytical solutions for the frost heave model are given in the Appe...