Failure in geomaterials: continuous and discrete analyses

Abstract

Various modes of failure in geomaterials have been observed in practice. Different criteria have been proposed to analyse these failures. In particular, Hill's Condition of Stability and diffuse modes of failure are considered in this paper in a dual framework: continuum mechanics and discrete mechanics. With the assumption of continuous media, experiments have shown that q constant loading paths (q characterizes the second stress invariant. For axisymmetric conditions, q is equal to: q=σ₁−σ₃, where σ₁ is the axial stress and σ₃ the lateral stress) can exhibit non-localized failure modes and are analyzed by the second order work criterion. With the assumption of discrete media, grain avalanches are considered, and spatial and temporal correlations between bursts of kinetic energy and peaks of negative values of second order work are demonstrated from discrete computations. It is concluded that the second order work criterion (under its dual form: continuous and discrete) can be a proper tool to analyse diffuse modes of failure in geomaterials.

Introduction

One of the most challenging problems in geomechanics is how to define, analyse and simulate failure. A general way to introduce the phenomenon of failure is to note that in elementary laboratory experiments, some loading paths lead to limit stress states. Along these paths, these stress states can not be exceeded: experimentally they constitute asymptotical limit states.

By restricting our study to elasto-plastic behaviors, the general expression of rate-independent constitutive equations is given by

$$\text{d}\text{σ}{}_{\mathrm{ij}}\text{=L}{}_{\mathrm{ijkl}}\text{d}\text{ε}{}_{\mathrm{kl}}\text{,}$$

where the elasto-plastic tangent operator $\text{L}\text{∼}$ depends on the previous stress–strain history (characterized by state variables and memory parameters $\text{h}\text{∼}$) and on the direction of $\text{d}\text{∼}\text{ε}$ (characterized by unit matrix $\text{v}\text{∼}\text{=}\text{d}\text{∼}\text{ε/∥}\text{d}\text{ε}\text{∼}\text{∥)}$. In the six-dimensional associated space the constitutive equation takes the form

$$\text{d}\text{σ}{}_{\mathrm{\alpha }}\text{=N}{}_{\mathrm{\alpha \beta }}\text{(v}{}_{\mathrm{\gamma }}\text{)}\text{d}\text{ε}{}_{\mathrm{\beta }}\text{(α,β=1…,6),}$$

$\text{v}{}_{\mathrm{\gamma }}\text{=}\text{d}\text{ε}{}_{\mathrm{\gamma }}\text{/∥}\text{d}\text{∼}\text{ε∥}\text{and}\text{∥}\text{d}\text{∼}\text{ε∥=}\text{d}\text{ε}{}_{\mathrm{\alpha }}\text{d}\text{ε}{}_{\mathrm{\alpha }}\text{.}$

From a mathematical point of view, limit stress points are characterized by

$$\text{d}\text{∼}\text{σ=}\text{0}\text{∼}\text{with}\text{∥}\text{d}\text{∼}\text{ε∥≠0.}$$

Condition (3) implies

$$\text{det}\text{N}\text{∼}{}_{\mathrm{<mtext>h</mtext><mtext>\sim </mtext>}}\text{(}\text{v}\text{∼}\text{)=0}\text{and}\text{N}\text{∼}{}_{\mathrm{<mtext>h</mtext><mtext>\sim </mtext>}}\text{(}\text{v}\text{∼}\text{)}\text{d}\text{∼}\text{ε=0.}$$

If the constitutive relation is incrementally linear, $\text{N}\text{∼}$ is independent of $\text{v}\text{∼}$. The first equation (4) represents precisely the plastic limit criterion because this is the equation of an hyper-surface in the six-dimensional stress space. The second equation represents the material flow rule, since this equation gives the direction of $\text{d}\text{∼}\text{ε}$ when the plastic limit surface has been reached.

For incrementally piecewise linear constitutive relations, where $\text{N}\text{∼}$ depends in a discontinuous manner from $\text{v}\text{∼}$ (inside “tensorial zones”, see Darve and Labanieh [1]), det $\text{N}\text{∼}{}_{\mathrm{<mtext>h</mtext><mtext>\sim </mtext>}}\text{(}\text{v}\text{∼}\text{)=0}$ is still the equation of an hyper-surface in the stress space, possibly involving several plastic mechanisms. Besides, $\text{N}\text{∼}{}_{\mathrm{<mtext>h</mtext><mtext>\sim </mtext>}}\text{(}\text{v}\text{∼}\text{)}\text{d}\text{∼}\text{ε=0}$ is a singular generalized flow rule with vertices that are locally represented by pyramids [2].

This classical elasto-plastic view of failure has given rise to the wide domain of the “limit analysis”, where the entire body is assumed to have reached the plastic limit condition. However, in practice, various modes of failure are encountered: localized modes in shear bands [3], [4], [5], in compaction bands or in dilation bands, diffuse modes due to geometric instabilities (buckling, etc.) or with a chaotic displacement field. For non-associated materials (essentially because of the non-symmetry of the elasto-plastic matrix $\text{N}\text{∼}$), the elasto-plastic theory shows effectively that, by considering proper bifurcation criteria, these localized or diffuse modes of failure can precede the plastic limit condition [6].

In relation to these various modes of failure, different bifurcation criteria exist in the literature. With respect to shear band formation by plastic strain localizations, Rice's criterion [3] based on the description of such an incipient shear band of normal n corresponds to vanishing values of the so-called “acoustic tensor”:

$$\text{det}\text{(}{}^{\mathrm{t}}\text{n}\text{L}\text{∼}\text{n)=0.}$$

For non-associated materials, Eq. (5) can be satisfied before plasticity criterion (4). This has been verified experimentally for dense sand [7]. Strain localization corresponds to a bifurcation of the strain mode from a diffuse one to a strictly discontinuous one. This kind of bifurcation can be called “discontinuous bifurcation”.

“Continuous bifurcations” [8] as opposed to the previous one, correspond also to a failure mode, but without strain localization. Such failure mode is called “diffuse failure” and this is the response path which is subjected to a bifurcation with a loss of constitutive uniqueness at the bifurcation point. According to the control mode (stress, strain or mixed control) of the loading path, different response paths are possible from the bifurcation point. For certain control modes, the loading is no more “controllable” in Nova's sense [9].

These continuous bifurcations and the related diffuse modes of failure can be detected by Hill's condition of stability [10] which corresponds, for unstable states, to vanishing values of second order work, i.e., vanishing values of the determinant of the symmetric part of the matrix $\text{N}\text{∼}$ for incrementally linear constitutive relations. Section 2 of this paper is devoted to recalling briefly Lyapunov's definition of stability [11] and Hill's condition, then to applying them to the well known case of failure of undrained loose sands. The generalization to axisymmetric paths allows the numerical and analytical determination of the “unstable” domain (i.e. the failure domain) and, inside this domain, the “unstable” stress directions which form cones.

In Section 3, this analysis is applied to axisymmetric g-constant loading paths, since some experiments (see Chu [12] for example) have shown failure modes without localization along these paths in the case of loose sands. 2 Material instabilities, 3 illustrate the application of Hill's condition in the framework of continuum mechanics. It is also possible to consider a discrete form of second order work [14] and to apply it to boundary value problems described by discrete mechanics. In this perspective the best known examples of typical failures are the so-called “grain avalanches”. This phenomenon has been extensively studied specially by physicists of granular media (see Hermann [15] for example). It has been also considered as a paradigm of the concept of “self-organized criticality” [16]. In Section 4 of this paper, grain avalanches are analyzed in the strict framework of discrete mechanics. The, more or less local, failures is characterized by bursts of kinetic energy (computed by a discrete element method). Strong spatial and temporal correlations between these bursts and the peaks of negative values of second order work are demonstrated. These results represent yet another illustration of the relationship between certain failure modes and second order work, which could constitute a novel indicator of failure.