extended finite element modeling of shear band

J. Basic. Appl. Sci. Res., 3(10)443-455, 2013

© 2013, TextRoad Publication

ISSN 2090-4304 Journal of Basic and Applied

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Corresponding Author: Zarinfar Meysam, Civil Engineering Department, K.N.Toosi University of Technology. Tel: +989183521041; Fax: +982188779476; Email address: [email protected]

Extended Finite Element Modeling of Shear Band Localization of Fluid-Saturated Porous Media for Integral-Type Nonlocal Continuum

Mechanics

Zarinfar Meysam, Kalantary Farzin

Civil Engineering Department, K.N.Toosi University of Technology Received: May 31 2013 Accepted: June 29 2013

ABSTRACT Localization phenomena, as e.g. shear bands, is a narrow zone of local concentrations of plastic strains. Classical continuum theory does not include material length scale parameters and is inadequate when localization phenomena occur. Therefore, numerical results based on classical continuum theory strongly depend on the mesh size of the finite element discretization. The present study explored the application of an extended finite element model in the nonlocal continuum mechanics of fluid-saturated material for numerical simulation of shear band localization. In the present paper, a new relation was proposed to calculate the stress rate based on the integral-type nonlocal model. It was shown that the inclusion of fluid viscosity and nonlocal plasticity for saturated cases leads to a regularization of the shear band problem. The goal of the present paper was to propose a new numerical method for reduction of computational effort and mesh dependency. For nonlocal plasticity, element size had a critical effect on the solution. Sufficiently refined meshes were required for an accurate solution without mesh dependency. It was shown that an extended finite element method can be applied to the problem to decrease the required mesh density close to the localization band. Numerical simulations of biaxial test were used to investigate the effect of element size, permeability and boundary conditions on the localization problem. KEYWORDS: XFEM, Nonlocal plasticity, Shear band localization, Element size, Permeability, Boundary

conditions, Saturated porous media, Mesh dependency, Computational effort

1. INTRODUCTION Strain localization is of considerable interest because of its importance in the prediction of failure and

residual bearing capacity for many engineering structures. It is well known that strain localization is typically followed by a significant reduction in the load-carrying capacity of a structure as loading proceeds. Loret and Prevost [1], Sluys [2] and others have demonstrated that classical continuum theory is inadequate when localization phenomena occurs, since it leads to ill-posed problems in the numerical solution of the governing field equations. When localization or material softening occurs, the governing static equations lose ellipticity or the governing dynamic equations lose hyperbolicity [3, 4]. Once ellipticity for a static problem or hyperbohcity for a dynamic problem has been lost, the localization band width based on inelastic rate-independent constitutive equations strongly depends on the mesh size of the finite element discretization because, as the element mesh is refined, the shear band width decreases until a singular surface is obtained. This shortcoming arises because classical continuum theory does not have a material length scale parameter, so that when the mesh is refined, the plastic strain is localized in a narrower region.

Previous studies have attributed the localization of deformation to instability in the constitutive description. It has been applied to study strain localization in the context of one-phase rate-independent materials since the pioneering works of Hill [5] and Rice [6]. Stability analysis in quasi-static and dynamic states can be found in the works of Rudnicki [7] and Vardoulakis [8] in the context of elastoplasticity theory for fluid-saturated material. In the framework of single phase solids, several numerical techniques have been proposed for strain localization problems, such as rate dependent constitutive models [1, 2, 9], gradient plasticity models [2, 10, 11], consideration of micropolar degrees of freedom [12-14] and non-local integral models [15-17]. Computational results for multiphase materials have shown that mesh dependency is not as dramatic as it is in single-phase materials and shear band width remains dependent on mesh size [18]. The localization problem of multiphase geomaterials was studied numerically using rate-

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Meysam and Farzin, 2013

dependent constitutive models [19-21], gradient plasticity models [22, 23] and consideration of micropolar degrees of freedom [24-26]. The nonlocal concept was first extended into continuum mechanics for plasticity by Eringen [27, 28]. However, Eringen’s formulation cannot be used as a localization limiter because the averaging operator is applied to the total strain tensor which could cause spurious instabilities (zero energy modes)[29]. Pijaudier-Cabot and Bazant [30] proposed using nonlocal treatment only for those variables that control the strain softening process. This formulation has the advantage that the differential equations of motion or equilibrium remain the same as in classical theory, without extra terms in the form of integrals or higher-order derivatives [31]. This considerably simplifies numerical implementation. Bazant and Lin [31] proposed a nonlocal plasticity model without the problem of convergence at mesh refinement in the softening continuum. The nonlocal continuum has been used in a microplane model [32], damage plasticity model [33, 34] and hypoplasticity constitutive model [35]. Finite element implementation of nonlocal plasticity was presented by Stromberg and Ristinmaa [36], Brinkgreve [37], Marcher and Vermeer [38], Brunig et al. [39], Maier [40] and Tejchman [41]. Jirasek [42] presented an overview of the integral-type nonlocal model for damage and fracture, Bazant and Jirasek [43] did for plasticity and damage, and Jirasek and Rolshoven [44] did for plasticity. It seems that formulations based on nonlocal averaging, widely used in single phase materials, have not been fully explored in the context of the multiphase porous media. The XFEM method, including the partition of unity concept, was discussed by Melenk and Babuska [45]. This technique was applied to crack growth problems by Belytschko and Black [46] and has been used successfully in the modeling of crack propagation. Fries and Belytschko [47] provided an overview of XFEM applications. This numerical technique has been used to simulate shear bands and strain localization [48-53]. The advantage of XFEM is the independence of the localization band geometry on the mesh.

The goal of the present paper is to propose a new numerical method for simulation of shear band localization in the nonlocal continuum mechanics of fluid-saturated material without mesh dependency. No research was found using nonlocal integral models for numerical analysis of multiphase materials in the literature. However, simulation of strain localization in fluid-saturated material requires numerical models capable of capturing the coupling of the solid deformation with pore-fluid flow. The model for the two-phase system is based on the general framework of averaging theories [54-56]. The interested readers can refer to the papers of de Boer [57] for a review of the history of porous media theories.

In the present paper, a new relation is proposed to calculate the stress rate based on the integral-type nonlocal model. This relation depends on the local plastic strain rate and integral averaging of the local plastic strain rate. In the context of the nonlocal model, the mesh must be sufficiently refined near the high strain gradient region to resolve mesh dependency. Adaptive mesh refinement can be used to overcome this, however, adaptivity may produce stress oscillations cause by the poor ability of approximation functions to capture the non-smooth pattern of strain near the softening zone [48, 58]. The extended finite element method is an alternative that can be applied to decrease the required mesh density close to the shear band. This numerical strategy allows modeling of the mesh-independent approximation of the non-smooth solutions using enrichment functions which are close to the exact localization mode. In addition, the computational effort is reduced compared to remeshing methods. In summery, the scientific contributions of this paper are 1) a new integral-type nonlocal model is proposed 2) nonlocal theory is combined with XFEM for reduction of computational effort and mesh dependency 3) integral type nonlocal plasticity combined with XFEM is extended for numerical simulation of saturated porous media.

The remainder of the paper was organized as follows: In section 2 governing equations for deformable and saturated porous media are summarized, in section 3 XFEM is applied to the governing equations of porous media in an updated Lagrangian framework, followed by a generalized Newmark scheme used for the time domain discretization, in section 4 a new integral-type nonlocal model is proposed and the tangent stiffness matrix is derived for the mixed integral type nonlocal-XFEM formulation, Finally, in section 5 numerical simulations of biaxial test is used to demonstrate the efficiency of the mixed XFEM–integral type nonlocal model in shear band localization modeling without mesh dependency and to investigate the effect of element size, permeability and boundary conditions on the localization problem.

2. Governing equations for deformable and saturated porous media

Macroscopic balance equations for a saturated porous medium can be derived by the application of averaging theory to the microscopic balance equations of the constituents at microscopic level and introducing simplifying assumptions [59]. Isothermal conditions and compressible constituents were

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assumed for the present study. Convective acceleration terms, fluid density variation in space, and high frequency phenomena were neglected; such assumptions are common in soil mechanics [60]. The macroscopic balance equations in an updated Lagrangian framework consist of a linear momentum balance equation for saturated porous media, linear momentum balance equation (Darcy’s law) for water species and mass balance.

0div buσ ρρ (1)

)grad( bupw ff ρρk (2)

0divtr wεp αQ1* (3)

Where pIσσ α is the total stress, σ refers to the modified effective stress (Bishop’s stress), I is the unit second-order tensor, p and u denote fluid pressure and displacement of solid skeleton, w is the average water velocity relative to the solid phase, k stands for the dynamic permeability, ρ and ρf are the mixture density and the fluid density respectively and the parameter

1

fs* knknαQ ,with ks, kf

which are the bulk moduli of the solid and fluid phases respectively and n is the porosity. For an isotropic material the Biot’s constant is sT kkα where kT is the bulk moduli of the medium. In order to solve the governing equations, using the appropriate initial and boundary conditions is necessary. The Dirichlet boundary conditions are uu on Γ=Γu and pp on Γ=Γp. The Neumann boundary conditions are

tσnt on Γ=Γt and nwwn on Γ= Γw. 3. Extended FEM Formulation of deformable saturated porous media The spatial discretization involving the variables u and p is achieved by suitable shape functions.

Therefore, the finite element polynomial displacement and pressure fields are enriched with regular Heaviside function which models the high displacement gradient in the localization band. The approximation of displacement (u) and pressure (p) fields can be expressed in the following form:

unod

unod

unod

i

unod n

1iti

uEnri

n

1iti

ui

n

1itiφφ

ui

n

1iti

uit, NNψψRNN )()()()()()()()()()()( [ aua]uu xxxxxxxx (4)

unod

pnod

pnod

i

pnod n

1iti

pEnri

n

1iti

pi

n

1itiφφ

pi

n

1iti

pit, NNψψRNN )()()()()()()()()()()( ][ bpbpp xxxxxxxx (5)

where unodn and p

nodn denote the number of element nodes, uiN and p

iN the standard finite element

shape function associated with node i, iu and ip the standard nodal displacement and pressure, ia and ib the additional degrees of freedom (DOF) associated with node i. In the above relation, ψ(x) denotes the appropriate enrichment function and φ(x) is the distance from the strain localization interface. R(x) is the ramp function which resolves difficulties in blending elements [47].

enrn

iiNR )()( xx (6)

where nenr is the number of enriched nodes of the element. This numerical technique improves the description of the displacement field inside the localized zone by adding the special enrichment functions which can model high gradient of displacement field. These functions and their gradients must be similar to the profile of displacement and strain fields. In this study, the hyperbolic tangent is employed to describe the corresponding profiles, as shown in Fig 1. This function is defined as:

βφ2 tanhψφ(x) x (7) where β is the parameter which controls width of the shear band. By selecting several values of β, one

can construct a series of enrichment functions describing the displacement profile near the strain localization interface.

By substituting equation (4) in to strain rate definition, strain matrix can be defined as:

unodn

i(t)

i(t)Enrii

1i a

uBBε xx

)()( (8)

in which

445


1

u Enri

2

u Enri

2

u Enri

1

u Enri

i(x)Enr

1

ui

2

ui

2

ui

1

ui

i

xN

x

N

xN

0

0 x

N

xN

x

N

xN

0

0 x

N

)()(

)(

)(

)()(

)(

)(

)(

xx

x

x

xx

x

x

x BB (9)

To obtain the weak form of the governing equations, Galerkin’s procedure is used. The test functions δ u(x,t) and δp(x,t) which have the same form as u and p are respectively multiplied by equations (1) and (3) and integrated over the domain Ω. Using the Divergence theorem leads to the following equations as:

0dΩΩ

(1)T Fb

pQσB

a

uM

(10)

0ˆˆ

(2)Fb

pS

b

pH

au

Q

(11)

in which

ΩT

ΩT

ΩT

ΩT

dΩ dΩ

dΩ dΩ

uuuu

uuuu

NNNN

NNNNM

EnrEnrEnr

Enr

ρρ

ρρ (12)

Ω Ω

Ω Ω

dΩ dΩ

dΩ dΩpEnr

TEnr

pTEnr

pEnr

TpT

αα

αα

mNBmNB

mNBmNBQ (13)

dΓdΩ

dΓdΩ

t

t

ΓT

ΩT

ΓT

ΩT

tNbN

tNbNF

uu

uu(1)

EnrEnr ρ

ρ (14)

Ω Ω

Ω Ω

dΩ dΩ

dΩ dΩ ˆ

EnrTTp

EnrTTp

Enr

EnrTTpTTp

αα

αα

BmNBmN

BmNBmNQ (15)

Ω Ω

Ω Ω

dΩ dΩ

dΩ dΩ

pEnr

TpEnr

pTpEnr

prEn

TppTp

NkNNkN

NkNNkNH (16)

Ω ΩTT

Ω ΩTT

ΩdQ1 Ωd

Q1

ΩdQ1 Ωd

Q1

ˆpEnr

pEnr

ppEnr

pEnr

ppp

NNNN

NNNNS (17)

w

w

ΓT

ΩT

ΓT

ΩT

dΓdΩ

dΓdΩ

qNbkN

qNbkNF(2)

pEnrf

pEnr

pf

p

ρ

ρ (18)

In above equations, m is the vector defined as mT=[1 1 0 ]. The governing equations (10) and (11) are discretized in the time domain by means of the Newmark’s scheme. The generalized Newmark GN22 method is employed for the displacement field (u,a) and GN11 method for the pressure field (p,b) as :

446


2t2

2ttt

2t2

2ttt

tt

tt

t1

tt

t1

tt

tt

tt

tt

tt

tt

tt

ttt

ttt

tt

tt

tt

tt

ΔtΔβ21

Δt21

Δt

ΔtΔβ21Δt

21Δt

ΔβΔt

ΔβΔt

Δ

Δ

ΔΔβΔt

ΔΔβΔt

Δ

Δ

aaaa

uuuu

a

u

aΔaa

uΔuu

a

u

aa

uu

a

u

bbb

ppp

b

p

bb

pp

b

p

t

t

t

t

tt

tt

(19)

21 β and β,β are the Newmark parameters. For unconditional stability of the numerical procedure, it is

required that 5.β β and .5β 12 . It must be noted that equation 19 is obtained at the known time t i.e. tp , tp , tu , tu and tu are the known values of pressure and displacement field. Substituting relation (19)

into the space-discrete equations (10) and (11), the following nonlinear equation can be achieved.

2

1

1

22

ΔtβˆˆΔtβ

ΔtβΔtβ .5

Ψ-

Ψ-

b

p

a

u

HS Q

Q- KM

t

t

t

t

(20)

In above equation, Ψ1 and Ψ2 denotes the vector of known values at time t and K is the tangential stiffness matrix.

tt

tt

t

t

t

t

tt

tt

2t

t

tt

tt

tt

tt2

1tt

tt

2ttt

2ttt

t

t1

ˆΔt

Δtˆ

5.

5.

Fb

pS

bb

ppH

aa

uuQΨ

Fbb

ppQ

aaa

uuuK

a

uMΨ

(21)

Ω Ω

Ω ΩΩ

dΩ dΩ

dΩ dΩ

,(

dΩ

EnrTEnr

TEnr

EnrTT

d

d

BDBBDB

BDBBDB

a)u

σBK

T

(22)

where D is the appropriate constitutive matrix. The non-linear coupled equation system is linearised in a standard way thus yielding the linear algebraic equation system (20) which can be solved using an appropriate approach, such as the Newton-Raphson procedure. 4. The mixed XFEM - integral type nonlocal formulation Nonlocal continuum theory uses integral averaging of the variable around its neighborhood, instead of a local definition. For example, nonlocal averaging of plastic strain tensor ( p

ijε ) at location x may be defined

by the local plastic strain tensor ( pijε ).

Vpij

pij dVε,αε ξξxx)( (23)

where V denotes volume of the body, ),(α ξx is suitable weighting function. In this paper Gaussian distribution function is used as a weighting function.

V2

2

dVe

e,α 2

2

xζ

xξ

ξx (24)

in which γ is a scalar which is related to material length scale parameter. For numerical finite element computation, the integral definition (23) can be approximated by summation relation. As a conclusion, nonlocal plastic strain tensor can be written as follow:

k

kpijk

pij ε,α)(ε xxxx (25)

where k denotes gauss points which are closer to point x than 2γ. The value of ),(α ξx for gauss points with greater distance than2γ is negligible.

447


In this paper, equation (26) is proposed to calculate the effective stress vector rate from the local and nonlocal plastic strain vector rate.

pepee εDεDεDσ mm )(1 (26)

where De is linear elastic matrix and m is a constant parameter. Using equation (26), the stress rate at a certain point is related to the plastic strain rate in its neighborhood. If the local plastic strain distribution in a body is smooth, nonlocal plastic strain is approximately equal to local plastic strain; thus, it can be concluded that the proposed formulation is approximately identical to the local formulation. When the distribution of local plastic strain becomes non-smooth, the difference between nonlocal plastic strain and local definition increases and differences between the two formulations are revealed. The formulation presented by Bazant and Lin [31] is a special case of equation (26) where m is considered to be 1; therefore, different patterns for the load-displacement curve can be obtained by changing m. This allows simulation of a wider range of materials. Since the proposed stress-strain relation uses a combination of local and nonlocal plastic strains, this formulation acts as a localization limiter [44]. The value of mE/H in which E denotes the elastic modulus should remain constant during loading to prevent numerical locking. Thus, the model describes the complete loss of material resistance without artificial locking effects.

Since the dissipative energy function is defined in local space, as in classic continuum mechanics, the associated flow rule can be used [44]. As a result, local plastic strain rate is obtained by equation (27).

σεp

Fλ (27)

where λ denotes the plastic multiplier and F the yield function which can be defined as: 0κ)f(κ)(F σσ, in which κ is the hardening-softening parameter. Applying the consistency

condition to the yield function, λ can be calculated as:

ε

σDσDσ

FFHFλ eT

eT (28)

where H denotes the plastic tangential modulus and is equal to pκκF ε . By substituting equations (28) in to (27), vector form of plastic strain rate is obtained as a function of total strain vector rate.

εεσDσ

Dσσεe

e

FFHFFT

Tp (29)

By using equations (25), (26) and (29), stress rate at gauss point k can be defined by:

ll

lklkkkkk εxxΛDεΛIDσ ,αm m)(1 ee (30)

Where I is identity matrix and l denotes gauss points which are closer to gauss point k than 2γ. The tangent stiffness matrix (equation (22)) for mixed XFEM-integral type nonlocal plasticity for a specified element can be modified using numerical integration and Eq. (30).

k l lll

llllk

kk

kk

WJ

WJ

WJ

)Enr(xlkek

T)Enr(x)(xk

ek

T)Enr(x

)Enr(xkek

T)(x)(xk

ek

T)(x

)Enr(xkek

T)Enr(x)(xk

ek

T)Enr(x

)Enr(xkek

T)(x)(xk

ek

T)(x

)Enr(xkek

T)Enr(x)(xk

ek

T)Enr(x

)Enr(xkek

T)(x)(xk

ek

T)(x

lklk

lklk

kkkk

kkkk

kkkk

kkkk

,α,α

,α,α m-...

...m...

...II

II

BxxΛDBBxxΛDB

BxxΛDBBxxΛDB

BΛDBBΛDB

BΛDBBΛDB

BΛDBBΛDB

BΛDBBΛDB K

(31)

where J is determinant of Jacobian matrix and Wk denotes weight coefficient of the gaussian quadrature. If the second and third terms in equation (31) are ignored, the formulation becomes equivalent to local theory. The DOF of one element may become related to the DOF of a non-neighboring element because

448


of the third term of equation (31). In conclusion, the nonzero components of the tangent stiffness matrix increase. In this situation, the tangent stiffness matrix is not symmetrical because α(xk, xl) ≠ α(xl, xk).

5. Numerical simulation

In order to demonstrate a part of the wide range of problems that can be solved by the present approach, we have illustrated the effective performance of XFEM technique using the integral type nonlocal continuum in strain localization analysis. A computer program is developed to investigate the computational aspects of the enriched finite element model in a higher order continuum model. The finite element mesh employed in all simulations is eight-noded rectangular plane strain elements with nine integration points. The analysis starts with the standard FE model with no enrichment functions. The enrichment function is then implemented into the standard shape functions by tracing the evolution of shear band zone. The parameter β, which is defined the width of enrichment zone, is set to γ in all analyzes due to the fact that the shear band thickness is about γ. All elements closer to the strain localization interface than β are enriched by the hyperbolic tangent function.

By applying enrichment functions to the formulation, additional DOF must be assigned to the nodes. At the beginning of the analysis, the additional DOF are inactive. For this reason a criterion must be defined for the activation of additional DOF. This study used the simple numerical algorithm proposed by Khoei and Karimi [53] that uses the maximum gradient of displacement to locate strain localization interface.

For integration purposes, a decomposition of the elements into sub-elements that align with the interface is standard in the XFEM [47]. In the case of a rectangular element, the elements located on the interface were partitioned using triangular sub-elements (Fig. 2a) and 24 Gauss quadrature points were used for the elements cut by the shear band interface. For standard FE elements, a set of 3×3 Gauss points were used for numerical integration. If an interface surface was added during the evolution of the shear band zone (Fig. 2b), the number of Gauss quadrature points for an element may differ before and after each increment. In this case, the value of the stresses can be determined at the standard FE Gauss points of an element. To obtain these values at the Gauss quadrature points of sub-triangles, a simple interpolation on a support domain for each triangular Gauss point consisting of the three nearest standard FE Gauss points was used. The required stress value can be determined using a simple interpolation.

'

-1

-1

1

1

a b

Fig.1 The smoothed hyperbolic tangent function and its derivative

Fig.2 Modeling of localization interface in XFEM; (a) The sub-triangles associated with elements cut by the interface at step n, (b) the sub-triangles obtained by partitioning procedure at step n + 1

5.1. Comparison of the nonlocal and mixed XFEM-nonlocal model

In this example, the performance of the nonlocal formulation and mixed nonlocal-XFEM formulation were compared for fluid-saturated material. The geometry and boundary condition of biaxial test are shown in Fig. 3. The specimen is kept fixed horizontally and vertically in the middle of the bottom in order to preserve the stability of the specimen against sliding. The other nodes at the bottom are fixed vertically. Axial compression is applied to the specimen by vertical velocity (.1mm/s) of the top nodes. A Drucker-Prager yield criterion with associative flow rule and isotropic linear softening is used for the solid skeleton. The material parameters used during the computations are Young's modulus E=30 MPa, Poisson's ratio υ= 0.3, solid grain density ρs =2000 kg/m3, water density ρw=998.2 kg/m3, apparent cohesion c0=0.5 MPa, hardening modulus H = -1 MPa, angle of internal friction φ= 10°, initial porosity n= 0:20, solid grain bulk

449


modulus Ks= 6.78 GPa, water bulk modulus Kw= 0.20 GPa, permeability k= 5E-8 m/s. The shaded area of the specimen is taken as the weak zone, as shown in Fig 3. The initial apparent cohesion of this zone is assumed be equal to 60 percent of the apparent cohesion in other parts of the specimen. The parameter γ is set to 1cm and m is set to 1.5 to obtain the shear band thickness equal to 2.5 cm.

In order to show the capability of the non local extension with XFEM, the numerical results obtained with three different meshes are presented. The meshes consist of 120, 340 and 480 elements. The results of the mixed XFEM-nonlocal and nonlocal analysis are shown in Fig 4-6. Fig. 4 presents the deformed configuration for vertical compression 2.4cm. Fig. 5 and Fig. 6 present the nonlocal effective plastic strain and pressure contours for both formulations. As seen, the results for coarse mesh are different than for other mesh results in the nonlocal formulation. In this formulation, thickness and inclination of the localization band depend on the mesh size. In the proposed approach, good agreement can be observed between the three mesh sizes; therefore, the XFEM can be applied to the problem to decrease the required mesh density close to the localization band. The resulting effective plastic strains and pressure distribution at 2.4cm are drawn along the vertical cross-sections at the center in Fig. 7-9. Without XFEM, these profiles would greatly change for the three meshes considered. These figures confirm that the mixed nonlocal XFEM technique gives a good prediction of localization, even for the coarse mesh. As seen in Fig. 6 and 9, water pressure concentration with lower values can be observed at the centre of the shear bands due to the plastic dilatant behavior of the sand. a

34cm

imperviousboundary

10cm

cons

tant

wat

erpr

essu

reimperviousboundary

cons

tant

wat

erpr

essu

re

Fig.3 The biaxial specimen; the geometry and boundary conditions a b c d e f

Fig.4 The deformed mesh from; a,b,c) nonlocal model and d,e,f) mixed XFEM-nonlocal model; a,d) coarse mesh b,e) medium mesh c,f) fine mesh

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a b c d e

0.60.50.40.30.20.1

f

Fig.5 The effective plastic strain contour from; a,b,c) nonlocal model and d,e,f) mixed XFEM-nonlocal

model; a,d) coarse mesh b,e)medium mesh c,f) fine mesh a b c d e

45040035030025020015010050

f

Fig.6 The distribution of pressure from; a,b,c) nonlocal model and d,e,f) mixed XFEM-nonlocal model;

a,d) coarse mesh b,e) medium mesh c,f) fine mesh

120elements340elements480elements

0 5 10 15 20 25 30 350

0.2

0.4

0.6

height(cm)

_

120elements340elements480elements

0 5 10 15 20 25 30 350

100

200

300

400

500

600

height(cm)

p(KPa)

Fig.7 The effective plastic strain for different meshes along a vertical cross section through the centre from mixed XFEM-nonlocal model

Fig.8 The distribution of pressure for different meshes along a vertical cross section through the centre from mixed XFEM-nonlocal model

5.2. Permeability and boundary condition effect on the localization

In this example, the same cross-section and material properties as for the previous example were considered, but without a weak zone in the specimen. The nodes on the bottom edge were kept horizontally

451


fixed. In this case, a symmetric localization band was developed in the lower part of the specimen. In order to study the effect of permeability, three different permeabilities are considered. In Fig 10, the specimen reaction versus displacement is shown. Fig. 11 and Fig. 12 depict the nonlocal effective plastic strain and pressure contours for different value of permeability at the same load 59KN. Localization band thickness and maximum pressure value increased and the maximum effective plastic strain value decreased as permeability decreased. As seen, lower water pressure values at the center of the shear bands is specific for k=5E-11. After the peak, load decreased slowly with axial displacement for k=5E-11, in contrast to k=5E-5. This indicates that the presence of the fluid pressure delayed shear band formation.

If nodes on the top edge are kept horizontally fixed instead of nodes on the bottom edge, symmetric localization band was observed in the upper part of the specimen (Fig 13). This helps us to explanation for reasons of the different pattern of shear zone.

In Fig 14 and 15, the specimen reaction versus displacement, the nonlocal effective plastic strain and pressure contours are shown for different values of parameter m. As seen, the maximum pressure value and effective plastic increased as m increased. Moreover, up to the peak, all curves coincide. Obviously, the nonlocal term first influenced the results when localization occurred. Thereafter, the amount of softening depended on the value of m. Higher values of m resulted in less softening and residual reaction force increased with increasing of m. Thus, different load-displacement curves can be obtained by changing m.

delta=1.8delta=2cmdelta=2.2cmdelta=2.4cm

0 5 10 15 20 25 30 350

100

200

300

400

500

600

height(cm)

p(KPa)

k=5E-8k=5E-11k=5E-5

0 0.5 1 1.5 2 2.5 3 3.50

20

40

60

80

100

(KN/m)

(cm)

P

Fig.9 Development of pressure along a vertical

cross section through the centre from mixed XFEM-nonlocal model

Fig.10 The variation of the specimen reaction with prescribed velocity for different value of

permeability from mixed XFEM-nonlocal model a b

0.140.120.10.080.060.04

c

a b

45040035030025020015010050

c

Fig.11 The effective plastic strain contour from

mixed XFEM-nonlocal model; a) k=5E-8 b) k=5E-11 c)k=5E-5

Fig.12 The distribution of pressure contours from mixed XFEM-nonlocal model; a) k=5E-8 b) k=5E-

11 c)k=5E-5

452


0.140.120.10.080.060.040.02

a

45040035030025020015010050

b

m=1.5m=2.25m=3

0 0.5 1 1.5 2 2.5 3 3.50

20

40

60

80

100

(KN/m)

(cm)

P

Fig.13 a) The effective plastic strain contour b) The

distribution of pressure contours Fig.14 The variation of the specimen reaction with

prescribed velocity for different value of m parameter

a

0.10.090.080.070.060.050.040.030.02

b c

1000800600400200

d

Fig.15 a,b) The effective plastic strain contour; c,d) The distribution of pressure for a,c) m=2.25 b,d)m=3

at the axial displacement 2cm 6. Conclusion

In the present study, an enriched finite element technique was employed based on an integral-type nonlocal theory for simulation of strain localization for fluid saturated porous media. Governing equations for deformable and saturated porous media were summarized in section 2. In section 3, XFEM was applied to the governing equations of porous media in an updated Lagrangian framework, followed by a generalized Newmark scheme used for the time domain discretization. Approximation of the displacement field in the localization band was improved by incorporating a set of special enrichments. The governing equations were regularized using nonlocal averaging of the plastic strain tensor, which included the material length scale parameter. The nonlocal model of Bazant and Lin [31] was extended in section 4. The tangent stiffness matrix was derived for the mixed integral type nonlocal-XFEM formulation. Standard FE analysis was first employed with no enrichment functions to perform the numerical simulation. The enrichment functions were then incorporated into the standard shape functions after the evolution of the localization band. In section 5, the efficiency of the proposed nonlocal model was demonstrated using simulated biaxial specimens. It was shown that the nonlocal model preserved the well-posedness of the governing equations in the post-localization regime and prevented pathological mesh sensitivity of the numerical results if the size of the element was smaller than γ. The mixed XFEM-nonlocal model guaranteed mesh independence even if the size of the elements was larger than γ. In the other words, coarser mesh can be used for XFEM combined with a nonlocal model than when using only a nonlocal model. The computational effort required for the mixed XFEM-nonlocal model was less than for the nonlocal formulation because coarser mesh can be used in simulation. For the mixed XFEM-nonlocal model, the width and inclination of the shear band was obtained independently of element size. Finally, the effect of permeability, boundary conditions and parameter m on the effective plastic strain, pressure contour and load-displacement curve were investigated.

453


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