Trigging Instabilities in Geomaterials within the Finite Element …people.duke.edu/~laursen/melosh09/paper5.pdf · 2012. 12. 18. · features of the geomaterial. In the case of localiza-tion,

Trigging Instabilities in Geomaterials within the Finite Element Method Framework 1. INTRODUCTION

Instability manifests itself through various defor-mation modes in geomechanics. For example, a natural slope may undergo large movements as a result of deformations either localizing into shear bands or developing in a diffuse manner through-out the entire mass. In both cases, the problem underlies a material instability phenomenon that originates in the small scale due to microstructural features of the geomaterial. In the case of localiza-tion, one finds distinctive forms of concentrated deformations such as shear bands, compaction bands and dilation bands [1-2]. These bands lead to an unstable response that is associated to a bi-furcation phenomenon at the material point level. That type of instability usually takes place near limit failure conditions. Paradoxically, another type of instability may occur well within the plas-tic limit surface [3]. For instance, when loose sand is sheared in undrained conditions, it may collapse at stress levels far from the plastic limit surface. In this form of instability, normally coined as diffuse instability, a rather generalized failure takes place. The implication is that stress states deemed to be safe with respect to a limit condition or localiza-tion can be still vulnerable to another type of in-stability such as of the diffuse type.

One of the main objectives of this study is to establish plausible models to numerically capture both localized and diffuse instabilities and explore the non-uniqueness associated to the underlying field equations in a boundary value problem. As anticipated, non-uniqueness can be triggered by introducing small perturbations in either the con-straints of the boundary value problem or the ini-tial conditions. It is shown in plane strain condi-tions that these small perturbations impact di-rectly on the position of the shear band that is formed, and hence on the overall response of the structure. Next, diffuse type of instability is exam-ined under both axisymmetric boundary and load-ing conditions. One aspect that is investigated is the role of the initial void ratio distribution in triggering instability. Both homogeneous and het-erogeneous void ratio distributions (uniform and Gaussian types) are considered. A density-stress-fabric dependent elasto-plastic model [4] is used to describe the constitutive behaviour of the material. All simulations are carried within the finite ele-ment framework.

2. DENSITY-STRESS-FABRIC DEPENDENT ELASTOPLASTIC MODEL

The ability of a constitutive model to describe soil behaviour within a high degree of fidelity is crucial in any failure analysis. A constitutive model for geomaterials must be able to capture essential be-havioural features such as non-linearity, irreversi-bility and dependencies of the mechanical behav-iour on stress level, density and fabric. The model herein used has many of the attributes described above. It was developed in [4] to describe the me-chanical behaviour of granular materials, especially sands. The core framework of this model, referred to as WG-model, is recalled here.

The WG-model is a two-surface elastoplastic model that accounts for both deviatoric and com-paction behaviours of granular materials. It is founded on Rowe’s stress-dilatancy theory [5] for predicting volumetric changes under deviatoric loading. Rowe’s original theory has been further enriched to address density, stress level and fabric dependencies as well as cyclic loading regime con-ditions. The two surfaces of the WG-model are the shear-yield surface f(s) and the cap-yield surface f(c), as shown in Figure 1.

1!

2!

3!

(a) (b)

p0

q

c.s.l.

c.s.l.

TCM1

1"M

#p

$ %sf$ %cf

cp

Figure 1: Generalized WG-model: (a) Trace of yield surfaces in the meridional plane. (b) Three-dimensional view in principal stress space. The acronym c.s.l. stands for critical state line (or surface).

The first yield surface treats deviatoric loading

governed by dilatancy whereas the second surface accounts solely for isotropic loading producing plastic volumetric compressive strains. A non-associated flow rule derived from the enriched stress-dilatancy theory is used to calculate the in-crement of plastic shear deformations. The update of both shear- and cap-yield surfaces is controlled

2 by two distinct hardening-softening laws. For the shear-yield surface, the mobilized friction angle acts as plastic hardening variable, whereas for the cap-yield surface, the pre-consolidation pressure, pc, governs plastic hardening. Table 1 summarizes the equations used in the model.

Table 1: Main equations in WG-model for loading in-volving the shear-yield surface

Yield sur-face

(s)f : q M p"& ' ; m

TCm

6 sinM :

3 sin

"&' "

TC2

M : M(1 ) (1 )t"

(&

) ( ' '(

Potential function

(s)g : q M p*

& ' ; m f

m f

sin sinM :

1 sin sin*" ' "

&' " "

fnp

Ff csp

cs0

esin : sin

e

+ ,- ) . /0 /0" & "/0 /0 /- ) . 1 2

Hardening law

mnp

m cspcs0

esin sin

e

'+ ,. /0 /0" & "/0 /0 /- ) . 1 2

Other evo-lution law

csn

cs cs0 cs0

pe : e exp h

p

3 4+ ,5 6/0 /& ' 05 6/0 //05 61 27 8

where p: mean effective stress; q: deviatoric stress; t = sin39 (9: Lode’s angle); "m, "f, "cs: friction angles mo-bilized, at failure, and at critical state, respectively; e, ecs: void ratio at current and at critical states, respec-tively; .p: deviatoric plastic strain; ecs0, !, -F, -0, nf, nm, ncs and hcs are material parameters; p0 = 1 kPa.

3. LOCALIZED AND DIFFUSE INSTABILITIES

Stability in its simplest form evokes the idea that a small perturbation input or load results into a small-bounded output or response [6]. In this re-gard, the well-known phenomenon of plastic failure can be viewed as a problem of instability whereby a small or even null stress input leads to large de-formations. Thus, according to Lyapunov’s con-cept, a material would exhibit unstable behaviour whenever its stress state reaches a certain limit state defined by a failure criterion such as the Mohr-Coulomb. However, experimental tests have shown that a material may be unstable even before any failure criterion is violated [3, 7]. The next sub-sections define briefly two such types of insta-bility in geomechanics, namely, localized instabil-ity and diffuse instability. 3.1 Localized Instability The localized stress-strain response of geomaterials essentially results from their underlying micro- and macro-mechanical properties. Similarly, boundary

conditions, imposed stress path, relative density and fabric are among the most important factors that would trigger localization. A great amount of experimental and numerical effort has been ex-pended in an attempt to understand and weigh those factors [1, 8]. Mostly, from a theoretical standpoint, localized instability is viewed as a bi-furcation of the underlying continuum field equa-tions. In other words, an alternate mode of non-homogeneous deformation is possible apart from the homogeneous one due to loss of uniqueness of the governing incremental equilibrium equations. The non-homogeneous mode involves a surface with a kinematic discontinuity upon which interfa-cial slip occurs. By imposing continuity of trac-tions across the discontinuity surface and the same constitutive relationship throughout the body, Rudnicki & Rice’s criterion [1] for strain localiza-tion emerges as:

det( : : ) 0&n D n (1)

where D is the tangent constitutive matrix in the sense that d : d& D! " , and n is the normal to the discontinuity surface or shear band.

3.2 Diffuse Instability In contrast to localized deformations, diffuse in-stability is mathematically described using the en-ergy-based Hill’s stability criterion which is de-fined by the sign of the second-order work [9]. Whenever the second-order work given by the product of incremental stress d! and strain d" becomes negative at a material point during a loading increment, instability springs. If the loss of positiveness of the second-order work becomes pervasive within the structure, collapse will even-tually occur. Locally, the second-order work crite-rion is given by:

2W : d : d& ! " (2)

This notion of second-order work which in fact underlies a bifurcation problem can be extended to highlight the directional character of the loading-response behaviour of a material and its relation-ship to instability. For instance, the domain of stress-strain states for which at least one loading-response direction exists such that W2 becomes non-positive gives rise to a so-called bifurcation domain. This concept, though, has to be brought hand-in-hand with that of controllability. Control-lability, according to Nova [10], highlights the fact that instability is also a function of the loading programme: force, displacement, or mixed test control. To illustrate the raised point, in force con-trolled tests, failure will not occur before the plas-tic limit condition is reached.

3 4. FEM STUDY OF LOCALIZED AND DIFFUSE

INSTABILITY

In the following sub-sections, the occurrence of various localized and diffuse instability modes is numerically studied in a boundary value setting via finite element computations. The density-stress-fabric dependent elasto-plastic model de-scribed in Section 2 of this paper was implemented into the commercial finite element code Abaqus [11] through the user material sub-routine facility called UMAT. As such, full advantage of the ca-pabilities of Abaqus can be taken in terms of searching algorithm for locating limit or bifurca-tion points. However, for ensuring robustness in the plasticity calculations, an implicit stress return and consistent tangent operator algorithm together with a spectral decomposition of the stress tensor were developed and implemented in UMAT.

4.1 Plane Strain Conditions The initial boundary value problem (BVP) illus-trated in Figure 2 is examined. It represents a dis-placement controlled biaxial test on a hypothetical 10 x 22 x 1 cm3 sand specimen isotropically com-pressed to 100 kPa and submitted to a 3 cm verti-cal displacement on its top surface. Deformations are only allowed to occur along components y and z. The strain component in the direction x is con-strained to zero so as to reproduce plane strain conditions. Material parameters and initial condi-tions are identical for all elements; therefore the specimen is perfectly homogeneous. A total of 220 three-dimensional C3D8 elements with eight nodes and full integration were used in these simulations.

x

z

y

100 kPa

100 kPa

Constraints:

3D element8 nodes8 Gauss points

3 cm

10 cm

22 c

m

xu 0 (front & back)&

zu 0 (bottom)&yu 0 (along x-axis)&

†

Position varies†

Element: C3D8

Figure 2. Geometry, mesh, boundary conditions and element type used in the finite element simulations.

In view of evaluating the effects of boundary

conditions on the loss of homogeneity in deforma-tions under the presence of a perfectly homogene-

ous field during loading history, the following sub-tle displacement constraints that block the rigid-body lateral motion were examined as shown in Figure 3. Among all configurations, the case (g) with both top and bottom middle nodes restrained laterally serves to further illustrate the effect of small perturbation on the final deformation pat-tern. As a matter of comparison, the case (h) in Figure 3 illustrates the fundamental solution which refers to the homogeneous deformation mode. This mode was obtained by rerunning the whole bound-ary value problem as a single element problem so that shear band formation would be overcome.

y

z

x

p.

0.100.160.210.270.320.380.43

(a) (b) (c) (g)

(d) (e) (f) (h)

Figure 3. Plastic deviatoric strain: (a)-(g) Deformed meshes showing different localized deformation re-sponses for the distinct initial boundary conditions. (h) Theoretical homogeneous response with no shear band.

Figure 3 also shows the numerical results in

terms of deformed configuration and values of plastic deviatoric strains for a homogeneous and initially dense specimen (initial void ratio, 0.60). Although homogeneous conditions are imposed, localized deformation in the form of an inclined shear band appears in all cases (a)-(g). From a theoretical viewpoint, the above numerical results clearly confirm that, under the presence of mate-rial instability, there is a loss of uniqueness in the solution of the underlying field equations govern-ing the structural response of the biaxial specimen. This loss of uniqueness allows a shear band to emerge with its position being dictated by the boundary constraint imposed.

Figure 4a shows the effective stress path fol-lowed by elements on the top boundary for all cases (a)-(h). Given the various positions and con-figurations of the shear band obtained in the finite element computations, the compelling consequence on the load versus displacement curves is that

4 various post peak branches are obtained with slightly different bifurcation points, as shown in Figure 4b. Note that in the pre-peak regime, specimens respond identically for all cases despite of small subtleties in the boundary constraint. It is worth noting that the above numerical results seem to point to the fact that small imperfections give rise to distinct post-peak response as soon as localization is implicated. The practical signifi-cance of this is that no two samples tested in the laboratory can result into the same response curve as a consequence of initial imperfections and boundary effects. The open question here is whether there exists a unique solution for the post peak behaviour irrespective of the nature of the initial imperfections. In other numerical experi-ments under plane-strain conditions (not presented here) we show, though, that material perturbation in which void ratio is assumed to scatter around a specific mean value in space is a more natural method whereby the bifurcated solution is more objectively found irrespective of the imperfections in boundary conditions.

0

300

600

900

0 200 400 600

p (kPa)

q (k

Pa)

Top-LTop-CTop-RBottom-LBottom-CBottom-R

Bottom-Top-CNo Shear Band

(a)

(b)

0

300

600

900

0 1 2 3vertical displ. (cm)

q (k

Pa)

650

750

850

2.14 2.3 2.46

c.s.l.

Figure 4. Average response for elements located on the top boundary.

4.2 Axisymmetric Conditions In this section we essentially examine the same prototype problem as in sub-section 4.1, except that the geometry and loading conditions are now axisymmetric just like in a so-called triaxial test in soil mechanics. The specimen’s size is the same as previously, i.e. 22 cm tall and 10 cm of diameter.

The initial imposed constraints are: uz = 0 on all nodes of the bottom surface and ux = uy = 0 on one node of the bottom surface located at the axis of geometrical symmetry of the problem. A total of 550 three-dimensional C3D8 elements with eight nodes and full integration were used.

In order to erase the influence of initial imper-fections as explored in the previous sub-section, material perturbation is introduced through a small fluctuation in material properties in the form of a random distribution of initial void ratio throughout the specimen. To specify a certain void ratio variation throughout the specimens, both a uniform and a Gaussian (normal) stochastic void ratio distribution are used as illustrated in Figure 5. In the uniform distribution case, the void ratio varies between 0.55 and 0.65, corresponding to a dense sand. By contrast, in the Gaussian distribu-tion, the void ratio is made to spread about a mean value of 0.60 with a standard deviation of 0.025. In either case, the distribution inside the specimen is random and non-spatially correlated.

0

void ratio

200

400

600

800Fr

eque

ncy

0.50

0.52

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

UniformGaussian

Figure 5. Clustered frequency plots of void ratio for Uniform and Gaussian distributions for dense sand.

Figure 6 shows the deformed shapes of the cy-

lindrical sand specimens after applying 3 cm of vertical displacement for three cases: (a) initial homogeneous condition (e0 = 0.60), (b) initial uni-form and (c) initial Gaussian void ratio distribu-tions. As expected, the unperturbed case (a) gives the fundamental homogeneous response. However, the other two cases give a diffuse deformation mode. Note that depending on the distribution type there is no clear sight of strain localization. Either bulging at the top or at the bottom parts of the specimen is obtained.

At last, Figure 7 depicts the evolution of the second-order work throughout the middle section of the sample for the case of Gaussian distribution. It is noticeable how the zone of negative second-order work coincides with the bulged region of the specimen where diffuse failure manifests more in-tensely after pseudo-time t = 0.866. At the plastic limit condition, for any strain localization that would occur, the negative second-order work crite-

5 rion is violated. This brings about the fact that the localization condition is contained within the bifurcation domain derived from the second-order work criterion.

e

0.590.610.620.640.660.670.69

x y

z

(a) (b) (c) Figure 6. Undeformed-deformed configurations for: (a) Homogeneous, (b) Uniform, and (c) Gaussian void ratio distributions for dense sand

t = 0.226

t = 0.591 t = 0.666 t = 0.866 t = 1.000

t = 0.456 t = 0.507t = 0.351

W > 02 W : 02 Figure 7. Temporal evolution of second-order work for dense sand specimen with Gaussian void ratio distribu-tion

5. CONCLUSIONS

The question of failure in its various forms such as diffuse and localized deformation have been dis-cussed within the context of bifurcation and loss of uniqueness in the presence of material instability. These notions are illustrated through finite ele-ment computations of initial boundary value prob-lems. It is shown that various post-peak responses can be computed as a function of small fluctua-tions in the initial boundary conditions. Interest-ingly, it is possible to capture the bifurcation of a homogeneous deformation field into a heterogene-ous one with shear localization without the need to introduce any artificial non-homogeneity in the

calculations. This is due to enriched features such as strong dilatancy with pyknotropy, barotropy and anisotropy level sensitivities of the elastoplas-tic constitutive model that are sufficient to capture bifurcated failure modes. Under axisymmetric stress-strain conditions, it was demonstrated that, according to Hill’s stability criterion, diffuse insta-bility prevails over shear localization along certain loading directions in drained testing conditions. Finally, it was shown that a correlation exists be-tween the spatial distribution of second-order work and the zone of diffuse deformations in the sample.

REFERENCES

[1.] Rudnicki, J.W. & Rice, J.R. 1975. Conditions for the localization of deformation in pressure sensitive dilatant material. J. Mech. Phys. Solids 23: 371-394.

[2.] Vardoulakis, I. 1980. Shear band inclination and shear modulus of sand in biaxial tests. Intl. J. Num. Anal. Methods in Geomechanics 4: 103-119.

[3.] Darve, F. 1994. Stability and uniqueness in geoma-terials constitutive modeling. In Chambon, Des-rues, Vardoulakis (eds), Localisation and Bifurca-tion Theory for Soils and Rocks: 73-88, Rotterdam: Balkema.

[4.] Wan, R.G. & Guo, P.J. 2004. Stress dilatancy and fabric dependencies on sand behavior. J. Eng. Mech. 130(6): 635-645.

[5.] Rowe, P.W. 1962. The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Royal Soc. London. Series A, Math. Phys. Sciences, 269(1339): 500-527.

[6.] Lyapunov, A.M. 1907. Problème général de la sta-bilité des mouvements. Annals of the Faculty of Sciences in Toulouse, 9: 203-274.

[7.] Lade, P.V. 2002. Instability, shear banding, and failure in granular materials. Intl. J. Solids Struc-tures 39: 3337–3357.

[8.] Borja, R.I. 2002. Bifurcation of elastoplastic solids to shear band mode at finite strain. Comp. Me-thods Appl. Mech. Engrg. 191: 5287-5314.

[9.] Hill, R. 1958. A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids, 6: 236-249.

[10.] Nova R. 1994. Controllability of the incremental response of soil specimens subjected to arbitrary loading programs. J. Mech. Behavior of Materials, 5(2): 193-201.

[11.] Abaqus, 2006. ABAQUS/CAE: User’s Manual. Version 6.6, ABAQUS Inc, USA.