In the present paper a nonlocal plasticity model is described, intended to reproduce the mechanical behaviour of stiﬀ ﬁne-grained soils, including the objective simulation of strain localization; the phenomenon of accumulation of deformations in narrow zones in the form of shear bands or fractures. A number of analyses have been performed to assess the developed formulation. Relevant aspects have been addressed such as the thickness of the shear band, its orientation, and the onset of localization in a boundary value problem (BVP). Results provide useful insigths into relevant aspects of the numerical simulation of strain localization

Gens Solé, Antonio

Mánica, Miguel A.

Ruiz, Daniel F.

Vaunat, Jean

English

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Environment and GeosciencesA alysis of strain localization with a non-local plasticity modelXIV International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS XIVE. On˜ate, D.R.J. Owen, D. Peric & M. Chiumenti (Eds)ANALYSIS OF STRAIN LOCALIZATION WITH ANONLOCAL PLASTICITY MODELMIGUEL A. MA´NICA∗, ANTONIO GENS†, JEAN VAUNAT† ANDDANIEL F. RUIZ†∗†Department of Geotechnical Engineering and Geo-sciencesUniversitat Polite`cnica de Catalunya (UPC)Campus Nord, 08034 Barcelona, Spain∗e-mail: miguel.angel.manica@estudiant.upc.eduKey words: Strain localization, nonlocal plasticity, plane strain, stiff fine-grained soilsAbstract. In the present paper a nonlocal plasticity model is described, intended toreproduce the mechanical behaviour of stiff fine-grained soils, including the objectivesimulation of strain localization; the phenomenon of accumulation of deformations innarrow zones in the form of shear bands or fractures. A number of analyses have beenperformed to assess the developed formulation. Relevant aspects have been addressedsuch as the thickness of the shear band, its orientation, and the onset of localization in aboundary value problem (BVP). Results provide useful insigths into relevant aspects ofthe numerical simulation of strain localization.1 INTRODUCTIONIn conventional geotechnical engineering situations stiff fine-grained materials show aquasi-brittle behaviour under deviatoric loading [1]. The resulting strain field is generallynon homogeneous and deformations tend to localize into thin zones of intense shearing inthe form of fractures or slip surfaces. This phenomenon is known as strain localization,observed also in other geomaterials like concrete, rocks or dense sands. The numericalsimulation of such phenomenon under the framework of continuum mechanics involvesa number of difficulties, as standard formulations tend to deliver non-objective resultsdue to the loss of ellipticity of the governing equation at the onset of localization. Thisnon-objectivity is recognized by a strong dependency of results with the employed mesh; avanishing energy dissipation and localization into a zone of vanishing volume are obtainedby reducing the size of elements. Enriched continuum formulations must be employed,providing the material with an internal length scale, not present in the standard formula-tion, which tend to prevent the common pathologies arising from the simulation of strainlocalization.In this contribution, a nonlocal plasticity model is described, capable of simulatingstrain localization objectively, without resulting in mesh-dependent results. The model1606Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. Ruizis intended for the simulation of stiff fine-grained soils. A number of analyses have beenperformed to assess the developed formulation for the simulation of localized deformationpatterns. Relevant aspects have been addressed, such as the thickness of the shear band,its orientation, and the onset of localization in a BVP.2 MODEL FORMULATION2.1 Local modelThe local model refers to the employed standard elasto-plastic constitutive law, withoutthe enrichment provided by the nonlocal approach. It is based on the simple conceptualscheme depicted in Fig. 1, for the strength of stiff clays under shearing, where the followingcharacteristic pattern is modelled: after reaching the peak, a first rapid reduction ofstrength is identified, associated with the degradation and breakage of interparticle bonds;then, a more gentle strength reduction is observed, associated with the realignment of clayparticles tending towards the residual state, where no further strength reduction occurs.Experimental evidence supporting this conceptual scheme is summarized in [1], as well asa comprehensive review on the hydromechanical behaviour of these materials.PeakPost-ruptureResidualParticlere-orientationDisplacement 'c'pPeakFissure post-ruptureResidualPolishing /orientationp'f'r'StrainFigure 1: Conceptual scheme for the strength of stiff plastic clays [2]Inside the yield surface, the response is assumed linear elastic. The employed yield cri-terion is defined by a hyperbolic approximation of the Mohr-Coulomb envelope, expressedas,F = − (c∗ + p tanφ∗) +√J2f(θ)+ (c∗ + pt tanφ∗)2 (1)where c∗ is the apparent cohesion, φ∗ is the apparent friction angle, pt is the isotropictensile strength, p is the mean stress, J2 is the second invariant of the deviatoric stresstensor, θ is the Lode’s angle, and f(θ) is a function defining the shape in the octahedralplane. Eq. (1) describes a curved envelope at low mean stresses, with a limited tensilestrength, as generally occurs in stiff clayey materials. For high mean stresses, the criteriontends to converge to a linear Mohr-Coulomb envelope, with φ = φ∗ and c = c∗.Isotropic non-linear hardening/softening was considered in such a way that the maincharacteristics of the conceptual scheme for the strength of stiff clays (Fig. 1) were incor-2607Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. Ruizporated. Hardening/softening is driven by the evolution of the strength parameters withplastic strains, characterized by a scalar state variable defined as,peq = (p : p)1/2 (2)where p is the plastic strain tensor. Strength parameters vary in a piecewise manner asshown in Fig. 2.Figure 2: Hardening/softening rulesA non-associated flow rule is adopted. Rather than deriving a specific function for theplastic potential, the flow rule us directly obtained in the following way,∂G∂σ= ω∂F∂p∂p∂σ+∂F∂J2∂J2∂σ+∂F∂θ∂θ∂σ(3)whereG is the plastic potential and ω is a constant that controls the volumetric componentof plastic deformations.Other important features exhibited by these materials are not included here, such asstiffness and strength anisotropy, consolidation processes (i.e. hydromechanical coupling),creep, or yielding under volumetric loading. The main purpose of the present work is theobjective simulation of localization under deviatoric loading, and therefore the incorpo-ration of these features within the present approach will be addressed in future work.2.2 Nonlocal extensionA nonlocal model is one where the behaviour at a material point (or at a Gauss pointin a finite element simulation) depends not only on its state, but also on the state ofneighbouring points within a certain region. This is accomplished by replacing a givenvariable by its nonlocal counterpart. If f(x) is some local field within a body of volumeV , the nonlocal field can be expressed as,f¯(x) =∫Vw(x, ξ)f(ξ)dξ (4)where w(x, ξ) is a weight function controlling the importance of neighbouring points asa function of its position (ξ), relative to the position of the actual point (x). Typically,3608Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. Ruizonly the distance between them is considered, and a Gaussian function is employed asthe weighting function, where the width of the bell-shaped curve implicitly introduces alength scale to the continuum formulation.Different nonlocal models are obtained depending on which variable (or variables) isconsidered nonlocal (see [3] for a compenhensive review). Here, we applied the approachgiven by [4], for the enrichment of the local model described in Section 2.1. The variableassumed nonlocal is the state variable controlling softening (Eq. 2), but using the alter-native weight function depicted in Fig. 3. Its main characteristic is that the influence ofthe actual point is removed, and the highest weight is located at a distance of 0.707ls,where ls controls the width of the symmetrical curves, acting as the internal length scale.00.10.20.30.40.50.60.7w(||   -   ||)x||   -   ||x Gaussian functionGalavi & Schweiger [4]3 2 1 0 1 2 3 44 x ξ ls x  ξosξ   el x2w Figure 3: Representation of the weight function from [4]In the implementation of this approach, the nonlocal state variable is computed onlyfor points in the softening regime, and considering neighbouring points at distances lowerthan 2ls [4].3 STRAIN LOCALIZATION ANALYSESA number of two-dimensional analyses were performed to assess the developed con-stitutive model for the simulation of localized deformation patters, corresponding to adrained biaxial plane strain test under displacement control. The specimen is 6 cm widthand 10 cm height and fixed horizontal displacement were considered at the top and bot-tom ends, in order to develop a non-homogeneous stress/strain field and favour the onsetof localization.Fig. 4a shows the contours of shear strain (s = (1 − 3)/2) of three analyses withdifferent meshes and ls = 1.0 cm. Because of the rough boundary conditions, stressesconcentrate at the four corners of the model, allowing the simultaneous formation of twoX-shaped shear bands. In the three analyses, the same localization pattern and the samewidth of shear bands were obtained regardless the number of elements. In addition, analmost unique force-displacement curve was obtained (Fig. 4b), demonstrating the abilityof the approach to deliver objective results.4609Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. RuizB01 (343 elements) B02 (789 elements) B03 (1303 elements)0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4b)a)0.0 1.0 2.0 3.0 4.0 5.0 6.0Vertical displacement [mm]0.05.010.015.020.025.030.035.0Deviator load [kN/m]B03 (1303 elements)B02 (789 elements)B01 (343 elements)  = 1.0 cmslFigure 4: (a) Contours of shear strain and (b) force-displacement curves for different meshesThe effect of ls on the configuration of shear bands is shown in Fig. 5a. As ls isdecreased, the interaction zone (considered neighbouring Gauss points) is also decreased,and plastic deformations tend to localize in a narrower zone. The width of the numericalshear bands is roughly equal to ls, as already obtained by [4]. However, a thinner shearband renders a lower energy dissipation, and therefore a more brittle response (Fig. 5b).In the simulation of a given material, the constitutive softening rate can be adjusted tomatch the desired global force-displacement response for a given ls; a technique known assoftening scaling [4].C01 (l  = 0.8 cm) C02 (l  = 0.6 cm) C03 (l  = 0.4 cm)0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7s s sa)0.0 1.0 2.0 3.0 4.0 5.0 6.0Vertical displacement [mm]0.05.010.015.020.025.030.035.0Deviator load [kN/m]C02 (   = 0.6 cm)C01 (   = 0.8 cm)B03 (   = 1.0 cm)C03 (   = 0.4 cm)1303 elementsssssllllb)Figure 5: (a) Contours of shear strain and (b) force-displacement curves for different length scaleparametersThe onset of localization was identified by the evolution of the second derivative ofthe shear strain with respect to time, averaged from all Gauss points (¯′′s ), i.e. some sortof global shear strain acceleration. Fig. 6a shows the evolution of this variable during5610Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. Ruizone of the analysis. A distinct jump is clearly identified, suggested here as the onset oflocalization of the BVP. This point does not necessarily occurs at the global peak strength,and in this case occurs slightly before it. Its actual location seems to be controlled bythe amount of points entering to the softening regime before the peak strength, which inturn is mainly the result of the considered boundary conditions.Regarding the orientation of shear bands, this problem has been historically boundedby two limits: the upper bound given by Coulomb’s theory (θC = 45◦ + φ/2), and thelower bound given by Roscoe’s solution [5] (θR = 45◦ + ψ/2) (φ and ψ are the mobilizedfriction and dilation angles). The theoretical orientation given by these solutions hasbeen computed in one of the analyses, at a point within the shear band, and the resultshave been compared to the actual inclination obtained from the simulation (Fig. 6b).Coulomb’s orientation seems to overestimate the obtained shear band inclination, whichappears to coincide with Roscoe’s solution at the onset of localization.0.0 1.0 2.0 3.0 4.0 5.0 6.0Vertical displacement [mm]20.030.040.050.060.070.0Shear band orientation [º]Peak strengthOnset of localizationObtained shearband orientation45º+   /245º+   /2E030.0 1.0 2.0 3.0 4.0 5.0 6.0Vertical displacement [mm]-0.6-0.4-0.20.00.20.40.60.81.01.2       [1/day  ]Peak strengthOnset of localizationE032 s''b)a)Figure 6: (a) Onset of localization and (b) theoretical and obtained shear band orientationsSince the amount of dilation at the onset of localization seems to control the orientationof the shear band, a given BVP should render a different orientation if the flow rule (Eq. 3)of the constitutive law is modified. The latter is shown in Fig. 7, where the same analysiswas performed using different values of ω, controlling the amount of volumetric plasticstrains (vertical displacements were normalized with the corresponding value at the onsetof localization). As ω is reduced, a smaller ψ is attained at the onset of localization,rendering a gentler inclination of shear bands. Nevertheless, the obtained orientationscan again be explained in terms of Roscoe’s solution at the onset of localization.4 CONCLUSIONSA nonlocal plasticity model has been described, aimed to reproduce the strength char-acteristics of stiff fine-grained soils, including the objective simulation of strain localiza-tion. The selected results presented here, provide useful insights into relevant aspectsof the numerical simulation of strain localization, and demonstrate the capability of theapproach to simulate localized deformation patterns.6611Miguel A. Ma´nica, Antonio Gens, Jean Vaunat and Daniel F. Ruiz0.0 0.5 1.0 1.5 2.0Normalized vertical displacement [-]20.030.040.050.060.0Shear band orientation [º]Obtained shearband orientationG02 (     = 0.6)G01 (     = 0.8)G03 (     = 0.4)G04 (     = 0.2)G05 (     = 0.0)45º+   /2Onset of localizationFigure 7: Theoretical and obtained shear band orientations for different flow rulesREFERENCES[1] A. Gens, “On the hydromechanical behaviour of argillaceous hard soils-weak rocks(Keynote Lecture),” in Geotechnics of Hard Soils Weak Rocks, Proceedings of the15th European Conference on Soil Mechanics and Geotechnical Engineering, vol. 4,(London), pp. 71–118, Thomas Telford, 2011.[2] R. J. Jardine, A. Gens, D. W. Hight, and M. R. Coop, “Developments in understandingsoil behaviour,” in Advances in geotechnical engineering: The Skempton conference(R. J. Jardine, D. M. Potts, and K. G. Higgins, eds.), pp. 103–206, 2004.[3] Z. P. Bazˇant and M. Jira´sek, “Nonlocal Integral Formulations of Plasticity and Dam-age: Survey of Progress,” Journal of Engineering Mechanics, vol. 128, no. 11, pp. 1119–1149, 2002.[4] V. Galavi and H. F. Schweiger, “Nonlocal Multilaminate Model for Strain SofteningAnalysis,” International Journal of Geomechanics, vol. 10, no. 1, pp. 30–44, 2010.[5] K. H. Roscoe, “The influence of strains in soil mechanics,” Ge´otechnique, vol. 20, no. 2,pp. 129–170, 1970.7612

Analysis of strain localization with a nonlocal plasticity model

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Analysis of strain localization with a nonlocal plasticity model

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