## External load

In frozen soils under the effect of external loads, mechanical stresses developed by means of interrelated and successive processes are: elastic deformations (conventional-momentary and elastic aftereffect), plastic deformation (attenuating creep, viscous flow, unattenuating creep or progressive flow) and destruction (brittle, when the body loses its continuity, or plastic with loss of soil stability). In these processes the frozen soils show rheological properties, caused by their characteristic inner relations and by the presence of ice, which is an ideal fluid body. The specific composition and structure of frozen soils, sharply different from other solid bodies, essentially results, when under even small loads, in a process of deformation which may continue for a long period of time. The type of deformation depends on the load value. Under small permanent loads and with free lateral deformation (compression, extension or dislocation) the process develops at a declining rate, i.e. there is attenuating creep (Fig. 3.10a; curves <j7 — (78). With larger loads, when the stress in the frozen soil is above a certain limit called the limit of long-term strength (7lon or creep limit, the relative deformations develop at an increasing rate, i.e., there is unattenuating creep (Fig. 3.10a; curves a1 — a6).

Fig. 3.10. Creep curves for frozen soil: (a) group of curves for various applied loads ax > a, >... > a7 > a8; (b) ideal curve of creep (/-/// - creep stages); (c), (d) compilation of relaxation curves from curves of creep (1 - at s = constant; 2 - at s, = constant; 3 - long-term strength).

Fig. 3.10. Creep curves for frozen soil: (a) group of curves for various applied loads ax > a, >... > a7 > a8; (b) ideal curve of creep (/-/// - creep stages); (c), (d) compilation of relaxation curves from curves of creep (1 - at s = constant; 2 - at s, = constant; 3 - long-term strength).

A schematized curve of creep can be divided into several segments showing different stages of deformation (Fig. 3.10b). Segment 0,4 represents momentary deformation. Depending on the load value, this deformation can be elastic or elastic-plastic, and it entirely or partially disappears when the load is lifted. The segment AB shows deformation at a reducing rate, the unsettled creep stage (stage I). At this stage, the deformation of frozen soils after lifting of the load disappears only partially with time (plastic aftereffect), because the deformation includes structurally reversible and structurally irreversible and plastic deformations. The attenuating creep either continues until reaching a certain final value s depending on stress a, or develops continuously at a reducing rate. In this case the deformation does not stabilize, but grows without limit (the so-called secular creep). In both cases the deformation rate s approaches zero. During unattenuating deformation, when s = const (segment BC), the stage of stable or viscoplastic flow (stage II) occurs. This stage is entirely irreversible.

At high stresses, a progressive stage appears (segment CD), the stage of destruction (stage III). At this stage, the deformation rate grows and results in brittle or viscous failure of the frozen soil. Stage III should be divided into two sub-stages; during the first (segment CE) the plastic deformation still develops and failure is not reached, while the second substage (segment ED) leads to failure through intensive microfissuring and catastrophically rapid growth of deformations.

The periods and roles of individual stages of creep depend on the amount of load. The larger is the load, the shorter is the stage of settled flow and the sooner the progressive flow starts. The role and significance of different deformation stages also depend on soil properties. In frozen soils all three stages of creep develop, and the more plastic the ground and the higher its ice content, the greater is the significance of stages II and III. Unattenuating creep in ice appears practically at any stress; that is why the stages with constant and increasing rates are the most important. For example, the value of the critical displacing stress for ice, when its plastic flow starts, is not above 0.01 Mpa. The limit of the long-term strength of ice during displacement (at — 0.4°C) is not more than 0.02 MPa.

Relaxation of stresses is another, no less important manifestation of rheological processes in frozen soils. This means that to maintain a constant value (unchanging with time) of deformation, it is necessary to gradually (with time) reduce the load, i.e. to reduce stresses in the soil. In other words, the identical value of deformation of frozen soils (e = const) can be reached by applying different loads, but a longer time period is necessary, if the load is reduced. Consequently, the longer the influence of the load on the frozen soil sample (displacement, compression or distension), the less load is needed to reach failure.

The plotting of the stress relaxation curve a = f(r) is carried out with a group of creep curves e = f(r) (Fig.3.10c) at any fixed value of deformation, e = const (Fig. 3.10d; curves 1 and 2). By using the creep curves, a curve of long-term strength of frozen soil can be plotted (curve 3). For this purpose, the moments of time when progressive flow starts are marked on the curves of unattenuating creep (the beginning of stage III of deformation), and then curve a = f(r) is plotted from the obtained data. The initial ordinate of the curve corresponds to the momentary strength of frozen soil cmom. With a sufficiently long time period, when the change in curve o = f(r) can be practically disregarded, the stress value will correspond to the limited long-term resistance to failure of frozen soil (7lon. This value is an important characteristic, because at o > alon the creep will be unattenuating (which leads finally to destruction of the frozen soil), whereas at o < alon the creep deformations will be attenuating (in time). Consequently, (7lon is the highest stress at which progressive flow does not appear.

A study of behaviour of different bodies and materials under load shows that for elastic bodies this is sufficiently well described by the dependence between stress and deformation (Hooke's law). The deformation in viscous bodies, however, can grow with time under constant load, and the dependence 'stress-deformation' loses its significance. In this case, the 'stress-defor-

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