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Multiphysics Role in Instabilities in Geomaterials: a Review

Tomasz HUECKEL

Duke University, Durham, USA

Multiphysical nature is characterized by complex phenomena involved in instabilities, strain localization and bifurcation in landslides, borehole instability in nuclear waste disposal and drying cracking, as seen in situ and in the laboratory. The multiphysics includes the effect of heat generated during precursor creep in the development of landslides, as well as the effect of geochemical reactions, the effect of heat on inducing possible failure through pressurization of pore water, the effect of evaporation-induced suction and air entry during drying and subsequent cracking of soils. The phenomena illustrated with specific natural or engineered events are interpreted as scenarios of processes that are either simultaneous or sequential, and that are coupled or result from an accumulation of dissipative processes. As pointed out by Terzaghi (1950), the causes of the instabilities are often long-term phenomena, rather than single events, such as major rainfalls, which are contributing factors. The need for a proper description of these long-term phenomena and their coupling with variable mechanical properties of soil and rock is emphasized.

1.1. Introduction

The engineering practice in all branches of geomechanics is now at an interesting stage of development, when the customary tools of evaluation of the margin of safety, such as "admissible stress" and "factors of safety", are felt to lead to an oversimplification of what we are capable of saying about a sample, soil/rock mass or structure. This is mainly because of the developed computational capabilities of contemporary engineering, as well as experimentally supported modeling capabilities, including coupled fields, through which soil and rock behavior is mathematically described. In the statement above I have adapted the words of Giulio Maier, with which he opens the foreword to a fascinating book by Davide Bigoni (2012) on bifurcation and material instabilities. While the book refers to bifurcations in a larger class of materials than just geomaterials, the above pronouncement catches the situation exactly: we can predict much more in detail than we could a few years ago: the stress field evolution, together with strain and/or damage progress along a process of loading following multiple scenarios of coupling with temperature; concentration of ions; salts or reaction progress field. It potentially includes patterns related to failure/instability and their precursors. However, how this information could/should be utilized to quantify "the distance from failure" or "factor of safety" (FOS) often remains an open question.

The purpose of this chapter is to provide an overview of a series of phenomena in geomechanics which qualify as instabilities/failures of various kinds. The use of this less-than-strictly defined term is intentional, as we want to encompass the widest possible class of phenomena for which the criteria are not necessarily within a single type of definition, but which correspond in loose terms to Lyapunov's definition: an unlimited response to a limited solicitation. Solicitation refers to a trigger of any sort: mechanical, hydraulic, thermal or chemical. We shall start with classical phenomena associated with purely mechanical loading induced instabilities and their criteria and implications to expand into an array of non-classical multiphysics instability phenomena. Current observations and understanding of geomechanical processes indicate a critical role of non-mechanical variables, whereas the conceptual base is lagging behind. Material (local) and field (global) instabilities based on the actual instability events leading to failure are both discussed.

1.2. General remarks

As we started with a promise to be wide open and inclusive, we have to issue several warnings in order to try to wave off an inevitable confusion that the subject brings, despite the appearance of a strictly rigorous approach.

To start with, in a geoengineering/geophysics context, instability or, better, loss of stability may mean instability of a material per se (at a point), instability of a soil/rock mass, or speaking mathematically of a boundary value problem. In other terms, we speak of a local or global stability. A local loss of stability at at least a single point of the continuum is considered a necessary (but far from sufficient) condition for global instability. Similarly, a local instability in a volume around a tunnel opening may be critical for a highway authority, but of no relevance to the stability of the mountain in which it is built. On the other end of the spectrum, local fault instability may induce global slope instability, or trigger an earthquake. It depends on the geometrical constraints that the considered boundary value problem implies.

Local stability is usually tested in a materials laboratory on a uniform specimen, or in a mathematical model, for a single material point. Global stability can rarely be tested on a large scale, but attempts have been made to monitor known landslide sites or earthquake source sites, or to conduct large liquefaction experiments.

Physically, instability may mean many things depending on the type of material and on the geometrical scale of consideration. In the plainest case, a macroscopically homogeneous material element in a laboratory under sufficiently low stress deforms in a homogeneous manner when a uniform traction is applied at its boundary. However, for unspecified physical reasons, at a certain stress level it responds with an unconstrained strain to, for example, a small stress perturbation. Often, the homogeneous strain is associated with a diffuse dilatancy (increased volume). This is a classical representation of instability. The key point is the homogeneity of the response maintained during the unstable phase.

Alternatively, we perceive as critical a loss of uniqueness of response, which means that a repetition of what is theoretically the same experiment would yield a different, still homogeneous, response. An additional option is to treat as unstable a response in which the increase in internal energy over a virtual displacement is less than the work of the external forces. Each of the above critical conditions, in principle, leads to a different criterion, both locally and globally.

In addition, when in a homogeneous sample the straining becomes more advanced, this strain homogeneity may be spontaneously lost. It is inevitable that, due to the actual irregularities in the distribution of minerals with a different stiffness, interface properties, pre-existence of structural defects, etc., high concentrations of stress and/or strain occur. Such concentrations may result in various forms of local damage, like mineral decohesion, micro-fissures, nucleation of pores, and intermineral or intergrain slips, depending on the structure, mineralogy and history of the material. The interaction of such local singularities gives rise to a variety of failure mechanisms. Despite their differences, their common feature is the possibility of microdefects developing into large-scale mechanisms of failure, such as slip-surface, fault or compaction band, leading to landslides, sinkholes, faulting, subsidence, etc. Such mechanisms are often characterized as macroscopic strain localization, when certain features of continuity abruptly cease to persist, like intergranular or intermineral phase contact. Commonly, more than one mechanism occur simultaneously, as in Figure 1.1, which was obtained for a sample of sand, and in which two separate slip surfaces and a substantial volume of diffuse dilatancy are observed after a triaxial compression test.

A mathematical representation of the physics above is equally complex. Often, Lyapunov-type instability (defined as an unconstrained response to a limited perturbation) is implied to result from the solution of a system of differential equations describing the nonlinear material behavior. The instability consists of bifurcation of the solution of the system of equations, which clearly implies that the solution is not unique.

Figure 1.1. Multiple shear bands and a volume increase due to diffuse dilatancy

The criteria for exclusion of instability may be local (or for uniformly deformed systems) or global for a piece of continuum. There are several different criteria, expressing very similar, but not identical conditions.

The first type of criteria refers to certain characteristics of material energy or work required to deform in a unique manner (meaning one uniform solution) under the conditions of equilibrium, for a specific stress-strain relationship and geometrical strain-displacement relationship. The second type allows for strain localization, with a particular geometrical form of deformed configuration that satisfies the conditions of equilibrium that allow certain forms of discontinuity, that is, it admits a certain type of discontinuity (shear or compaction bands). Global criteria consist of the same requirements but integrated over the entire volume of the considered body. The very definition of "bifurcation point" varies from author to author, although in any specific situation there is usually no doubt about what should be called a bifurcation point. The above types of instabilities and their implications have been studied in the context of geomaterials for nearly 50 years, and an example of their implications and their relationships...