Читать книгу Instabilities Modeling in Geomechanics - Jean Sulem - Страница 14

To start with we will limit ourselves to considering exclusively incipient elastoplastic straining at small strains. At the moment, we will not deal with materials that exhibit strong viscous effects.

In what follows, we will outline the basic formulae of elasto-plasticity (equation [1.1]) at small strains to introduce basic concepts that will be referred to (Roscoe and Burland 1968). The given set of equations implies the existence of a yield limit, f = f(p′,q,r,pc′) in the effective stress invariant space, which means yielding if f = 0, and means an elastic unloading state if f < 0. The case of f > 0 is deemed as statically inadmissible. The core of the multiphysical aspects of soil/sediment/rock behavior is embedded in the last of the equations, which describes the evolution of the principal mechanical property of the material, which is the size of the yield locus pc′ = pc′ ₍εpl, μ₍ΔT,ξ,c...₎₎with changes in a series of the environmental (non-mechanical) variables (see, for example Hueckel 1992, 2002).

[1.1]

That implies that variables such as temperature, chemical mass removal (accretion) and ion concentration of pore fluid essentially affect the geomaterial strength, apparent preconsolidation pressure and elasto-plastic stiffness.

As is customary in inelasticity theories, the deformation can only be uniquely determined for an incremental stress, rather than total stress, as in elasticity. Therefore, the plastic strain rate is defined as

[1.2]

where Pij is the gradient of the plastic potential, or a resultant direction tensor of all inelastic strain components of various origins, including the irreversible change in elasticity due to plastic straining or damage (Hueckel 1975, 1976; Maier and Hueckel 1979; Bigoni and Hueckel 1991a, 1991b; Bigoni 2012). Most of the physical evidence suggests that Pij ≠ Qij, where is the yield locus gradient, but for a number of reasons it is often seen as convenient to ignore the difference.

Non-associativity, i.e. when Pij ≠ Qij (for instance via an elasto-plastic coupling), or any other forms of irreversible (not necessarily mechanical) straining leading to a non-symmetrical stress–strain incremental relationship, are notorious for inducing a premature loss of stability and/or strain localization.

Among several criteria for (local) material stability, the most commonly accepted is an energetic one: stating the positiveness of an increment of internal energy, or in other terms, of second-order work, in the sense of Hill (1958) or Drucker (1964). Following Drucker (1964), the continuum considered will be said to be stable in a given equilibrium configuration, if and only if an external agency imposing a kinematically admissible (compatible) infinitesimal geometric disturbance, by preserving equilibrium, performs non-negative (second-order) work regardless of what the disturbance is, i.e.

[1.3]

Let us consider the set of constitutive equations [1.1] and express the incremental effective stress as a function of an incremental strain (Maier and Hueckel 1979) and properties of the material behavior: elasticity tensor Eijkl , plastic hardening modulus H (positive during plastic strain hardening and negative during strain softening) and its critical value Hc, which may depend on stress and plastic strain , where Qij and Pij are, respectively, yield locus gradient and a second-order tensor defining the mode of incremental irreversible strain. We arrive at the following stress–strain incremental relationship characterized by elastoplastic stiffness tensor Dijkl, as follows:

[1.4]

Employing the material properties introduced above in a local condition of stability [1.2], we conclude that the criterion is articulated through the hardening modulus H, which, if it is larger than the first value or lower than the second one (Maier and Hueckel 1979; Bigoni and Hueckel 1991a),

(1.5)

ensures stability.

For the associative flow rules (Pij = Qij), the hardening modulus at loss of stability is H = H₁ = 0, while for materials with so-called subcritical softening, the hardening modulus at the point of re-gaining stability is H = H₂ = Hc. Hence, the stability range coincides with that of hardening and the postcritical range, while the softening range is all unstable (Figure 5.2). The most important departure from this rule is for all kinds of non-associative flow rules, for which it can be seen that H1 > 0 (Bigoni and Hueckel 1991a, 1991b). Notably, an earlier, particular version of such conditions for a non-associated flow rule for a Mohr–Coulomb material was provided by Mróz (1963).

A separate issue is the uniqueness of material response. It appears that in certain situations, a certain type of incremental solicitation is not admissible, in the sense that the response is not unique, which means that two possible or infinite possible responses can be expected. The classical result is that in the softening range, i.e. H≤0, dσij such that dσij Qij ≥ 0 is statically not admissible, as it violates the flow rules of equation [1.3]. In addition, dσij such that dσijQij <0 is admissible, but generates two possible types of response: an elastic unloading and an elastoplastic softening. In other terms, the range of H ≤ 0 is not stress controllable. Analogous ranges of static admissibility may be established for dεij , where a range of the hardening modulus where H ≤ Hc is not strain controllable. Consequently, the range 0 ≤ H ≤ Hc is strain controllable. The issues of controllability are relevant in experimental studies of material behavior.

Buscarnera and Nova (2011) have generalized the question of controllability to experiments with a mixed stress–strain control of loading programs. These are relevant, for instance, in undrained tests in which (under the assumption of incompressibility of solids and water) volumetric strain rate is imposed as zero.

An alternative form of instability is the one with strain localization. Originally identified in the seminal paper by Rudnicki and Rice (1975), the condition is less restrictive than [1.5]; nevertheless it does admit strain localization of a particular form of strain tensor in inequality [1.2], which is a product of a unit vector ni normal to the planar band and a vector gj defining the jump in the velocity derivative. This condition is equivalent to positive definiteness of all possible acoustic tensors ni Dijkl nl. A special case of the strain localization into a planar band takes place when the differential equations describing the behavior of the material in equilibrium cease to be elliptic. Two modes of strain rate discontinuity are possible depending on directions of vectors ni and gi, which are normal modes of split and compaction (Castellanza et al. 2009): when the two vectors are coaxial, and a shear band mode, when they are not.

The criterium for localization in a particular direction specified by versor ni orthogonal to the discontinuity shear band is given in the form of a hardening modulus

[1.6]

with G and v being the elastic shear modulus and Poisson coefficient. To determine a critical modulus at a given point, constrained maximization of equation [1.2] needs to be performed over all possible directions of ni.

A global condition of stability is guaranteed in the medium for the entire boundary value problem if

[1.7]

which is again stated as non-negativeness of the second-order work of tractions integrated over the whole boundary (plus that of body forces, over volume). Through a standard consideration, the condition is brought to the requirement of positiveness of the second-order work integrated over the entire volume of the body, under the assumption that both the constitutive laws and the geometric conditions are fulfilled. That condition clearly is much less restrictive than the local condition [1.5], which requires the same, but at every point of the body. Thus, [1.7] admits large areas of a negative second-order work, as long that there are areas that can outweigh the negative areas.

One other way to guarantee the stability of the solution of the boundary value problem in geomechanics is to ensure that the solution is unique, or in other terms, that there are no alternative solutions for the one obtained to snap through to. The solution uniqueness is guaranteed only under sufficient conditions of local or overall nature, the overall condition being much less stringent than the above sufficient conditions for stability (Hueckel and Maier 1977).

In geoengineering practice analyses, the assessment of stability is often made on the basis of the finite element result through detection of failure as a loss of global equilibrium seen as a lack of convergence of the solution identified by the lack of convergence within a certain iteration number (Griffiths and Lane 1999; Zienkiewicz et al. 2005). Alternatively, loss of stability, for instance of a slope, is identified as an onset of a kinematically admissible “sliding” mechanism through monitoring of the selected nodes as the solution evolves to detect a sudden increase in displacements (Hicks and Spencer 2010).