Finite element analysis of localization and micro–macro structure relation in granular materials. Part II- Implementation and simulations

Finite element analysis of localizationand micro–macro structure relation in granularmaterials. Part II: Implementation and simulations

H. Arslan1, S. Sture2

1 Exxon Mobil Upstream Research Company, Houston, TX, USA2 Department of Civil Engineering, University of Colorado – Boulder, Boulder, CO, USA

Received 5 May 2007; Accepted 26 September 2007; Published online 25 January 2008

� Springer-Verlag 2008

Summary. The formation of strain localization influences the stability and stiffness of the soil mass or

geosystem. The thickness of shear bands provides insight about overall strength and stiffness inside the granular

body, and the shear band angle gives information about the failure surface in a given soil or soil mass. Thus, it is

important to be able to predict when a shear band forms and how this zone of intense deformation is located and

oriented within the granular medium. A rational finite element analysis for capturing the formation and

development of shear bands has been performed and implemented by using a Cosserat continuum in finite

element simulations. An extension of plane strain Drucker–Prager elastoplasticity to Cosserat continua is

implemented in ABAQUS by using its User-defined ELement (UEL) option. The finite element formulation is

discussed in the companion paper. The length scale–size effect relation has been investigated to understand the

micro–macro structure relation. Several practical engineering problems are simulated in two dimensions by

using the finite element code ABAQUS together with analyst-supplied extensions. The effect of Cosserat

parameters on the finite element simulations has been simulated.

1 Introduction

Failure in soil masses generally takes place when shear stresses exceed the shear resistance, or when

excess deformations occur often due to the emergence of discontinuities or narrow bands of localized

deformations. This process is often associated with strain localization and formation of shear bands.

The occurrence of shear bands during the deformation process limits the strength of the soil. As soon

as the localization condition is satisfied at a point, failure is often initiated.

Geotechnical systems often involve analysis and design that deal with highly discrete, particulate

and different size ranges that in many ways resemble a discrete system rather than a continuum. In

fact, the geotechnical engineering literature often refers to the highly discrete nature of soils and yet

engineers mainly use continuum principles and definitions to describe their behavior. With increased

Correspondence: Haydar Arslan, Exxon Mobil Upstream Research Company, Houston, TX 77093, USA

e-mail: [email protected]

Acta Mech 197, 153–171 (2008)

DOI 10.1007/s00707-007-0514-0

Printed in The NetherlandsActa Mechanica

computational analysis capabilities it is now possible to use the relatively more complex Cosserat

continuum theory in practice, which accounts for additional and more complex stress variables that

may simulate soil behavior at the elemental level more realistically. Continuum vs. discrete has

always been an issue in geotechnical engineering. Cosserat continua in many ways consider the

nature of soils more accurately. Thus, it is worthwhile exploring this possibility.

There are several shortcomings of the traditional shear band theory (i.e., shear band analysis based

on local constitutive laws). Stress and strain are related to each other through constitutive relations

which are expected to contain all necessary information about the mechanical characteristics of

granular materials. Traditional continuum approaches do not relate the micro-characteristics of granular

assemblies to the macro-behavior of the continua. However, the particulate nature of soil materials is

directly responsible for their complex overall behavior [1]–[6]. However, the particulate nature of soil

materials is directly responsible for their complex overall behavior [1]–[6]–[16]–[17]–[18].

Non-local theories take a characteristic length into consideration associated with the physical size

effect in the micro-structure. Intuitively, this seems strongly linked to soil mechanics concepts. Thus,

non-local theories, particularly those based on Cosserat theory, remove many of the restrictions of

the traditional analysis. The nature of the constitutive response offers three key advantages over

other existing models. First, it provides the correct level of resolution to enable shear bands to be

captured and simulated in the analysis. This will assist in realistic load-displacement modeling.

Second, the non-local character obviates the mathematical difficulties of traditional analyses and

makes possible an investigation of the evolution of shear bands.

Valanis and Peters [7] showed that in the traditional theory the problem becomes ill-posed at the

onset of localization as the governing partial differential equations change from being elliptic to

hyperbolic. In contrast, those based on non-local constitutive models remain elliptic enabling an

analysis of the post-localization regime. Third, the constitutive law is expressed in terms of physical

properties of particles and their interactions (e.g., particle stiffness coefficients, coefficients of inter-

particle rolling friction and sliding friction) which have always been considered of fundamental

importance in soil behavior.

2 Implementation of the model into the finite element code ABAQUS

The finite element implementation for the discretized equations shown in the companion paper [19]

was performed using the non-linear commercial finite element program ABAQUS [8]. This program

does not provide an element with material rotation; therefore a User Element Subroutine (UEL) is

needed to solve the system of the finite element equations within the micro-polar framework.

A 4-noded isoparametric element with four integration points was used. However, a selective

reduced integration technique was used to avoid any possible volumetric locking during the

softening regime. In this sense, full integration was used for all the state variables and only a reduced

integration technique was used for the volumetric strains.

The finite element program ABAQUS uses a Newton Raphson iteration technique to fulfill the

static equilibrium equations, and the load–displacement increments are updated using an implicit

integration scheme within the standard version of ABAQUS. The problem in hand is a mixed control

problem (load–displacement control) and all the internal state variables (such as stresses, plastic

work, void ratio, etc.) are updated within the UEL using the explicit forward Euler integration

scheme. Thereafter the ABAQUS post-processor is used to show the analysis results.

As shown in the previous section, the finite element governing equations were represented in their

weak form and this system of equations can be decoupled into stress and couple stress based

components,

154 H. Arslan and S. Sture

Kuu Kux

Kxu Kxx

� �_U_x

� �¼

_Fu

_Fx

� �: ð1Þ

An incremental form of the principles of virtual work, ignoring body forces and body couples, can be

written as:ZK

_rijd_eijdVþZK

_lijd _jijdV ¼Z

o1K

_td _uidAþZ

o2K

_mdxidA; ð2Þ

where the traction boundary conditions for stress and couple stresses are defined on different

portions of the boundary of the domain, i.e.,

_rijnj ¼ _ti on o1K; ð3Þ_lijnj ¼ _mi on o2K: ð4Þ

3 Solution technique for the governing equations

The constitutive relations used in this study are highly non-linear and so much caution is required

during the implementation. The loading mechanism in this study is applied as a mixed control type

of loading; initially the specimen is confined with hydrostatic pressure and the second step is the

deviatoric loading through strain-controlled loading. A total displacement is subincremented over a

certain period of time Tt. However, the problem in hands is a time independent problem since all the

relations used here are homogeneous in time.

The finite element program ABAQUS uses the well-known Newton–Raphson method to solve the

equilibrium equations in which the solution for highly non-linear equations will converge most of

time. The time increment used in ABAQUS can be a variable within minimum and maximum

values. ABAQUS will always choose the largest increment that will lead the solution to converge; in

other words it always tries to save in the computational cost and reach a solution. Since the finite

element solution is a numerical approximation some tolerance is used.

With a coupled system at the ith N–R iteration:

KiDuiþ1 ¼ Ri; ð5Þuiþ1 ¼ ui þ Duiþ1; ð6Þ

where Ri is the residual force vector. Then the convergence criterion will require that:

Duiþ1� tolerance; ð7ÞfDRi� tolerance: ð8Þ

The convergence criteria are to be satisfied inside ABAQUS and the user can simply control the

required tolerance.

4 Verification of Cosserat solutions with plane strain tests results

The Cosserat Finite Element solution is calibrated with the plane strain experiments performed by

Alshibli at University of Colorado-Boulder in 1995 [9]. In that study the effect of particle size and

confinement on the orientation and thickness of the shear band was investigated. The plane strain

numerical model is shown in Fig. 1. Instead of applying the displacement from the bottom of the

Finite element analysis of localization and micro–macro structure relation. Part II 155

specimen as in plane strain tests, the vertical displacement is applied to the top of the specimen to

avoid any numerical problems. The elastic material properties that are used in the Finite element

analysis are summarized in Table 1

As Figs. 1, 2, and 3 illustrate, the finite element results are in good agreement with the

experiments. The model captures both thickness and orientation of localization quite well. The

prediction of the stress–strain curves showed that the model fits the experimental results of granular

materials under plane strain.

The distribution of equivalent plastic strain, micro-rotations and curvature of the micro-rotations

are shown in Figs. 4, 5, 6, and 7. The FE simulations and experimental results are summarized in

Table 2. Orientation and thickness of the shear band for different particle sizes are compared for two

different confining stresses. The numerical simulations confirm the experimental results. Thickness

and orientation of the shear band is dependent on the length scale for the finite element simulation as

they are dependent on particle sizes experimentally.

y

x Fig. 1. Plane strain experiment simula-

tion

Table 1. Material parameters for the granular material

Young’s modulus (E) (kPa) 72,000

Poisson’s ratio (v) 0.26

Length scale (mm) Changes

Cosserat shear modulus (Gc) 0.5 · G

Friction angle 40�Cohesion (kPa) 5.0


q

a) b)Fig. 2. Plane strain results a

experimental [9], b finite element

600

500

400

300

200

100

0 0 1 2 3

Confining pressure - 15.0kPa

Confining pressure - 100.0 kPa

Predicted Measured

4 5 6 7 8 Axial strain (%)

Dev

iato

ric

stre

ss (

kPa)

Fig. 3. Finite element-experiment

comparison of deviatoric stress–strain

behavior of granular materials under

plane strain condition

Fig. 4. FE-experimental comparison of

shear band thickness and inclination

angle for the medium–dense F-sand

under confining pressure of 15.0 kpa


5 Effect of boundary conditions

The behavior of the granular material is dependent on the boundary conditions. The following

example will illustrate the boundary conditions effect on the shear band formation. The finite

element study was performed by constraining both lateral movement and rotation at the top and

0.15

0.10

0.05

0.00 0.00 1.00 2.00 3.00 4.00 5.00

Distance along shear band (mm)

Equ

ivel

ant p

last

ic s

trai

n

d=0.29 mm

Fig. 5. Plastic strain profile to predict the

shear band thickness

0.0

–0.2

–0.4

–0.6

–0.8

–1.0 1.0 2.0 3.0 4.0


Mic

ro-r

otat

ion

(Rad

)

Fig. 6. Cosserat rotation profile along the

shear band

0.004

0.003

0.002

0.001

0

–0.001

–0.002

–0.003

–0.0040 1 2 3 4 5 6 7Distance along shear band (mm)

Cur

vatu

re (

%)

Fig. 7. Curvature of Cosserat rotations

across the shear band centerline


bottom boundaries. The finite element model result will be compared with Alshibli’s [9] and

Alshibli’s and Sture’s [10], [11] plane strain test results. The movement of the bottom of the

specimen was restrained in all directions during the finite element and experimental procedure.

Figures 8 and 9 illustrate that multiple shear bands develop in the specimen if the bottom boundary

is fixed. The finite element results show that shear band location and mode were found to be highly

influenced by the boundary conditions which are consistent with experimental observations.

The Cosserat continuum finite element plane strain results demonstrate that the following two

principal mechanisms of localized deformation may occur in granular materials under plane strain:

• A mechanism consisting of the formation of a single shear band, initiating in the hardening

regime and yielding a strong softening.

• More than one shear band can occur if the movement of the bottom boundary is restrained under

plane strain condition.

6 Mesh sensitivity study

The finite element analysis has been conducted for three different mesh sizes for the same material

properties (Figs. 10–15). Stress–strain and couple stress along the shear band were plotted for the

three different element sizes shown in Figs. 16 and 17.

The Cosserat model was used for a mesh sensitivity analysis of a plane strain test. Contour plot,

deformed mesh and vector plots of the velocity field for three different mesh sizes are shown in

Figs. 10–17. The velocity vector field shows the direction of the block-sliding mechanism. The

Table 2. Summary of the FE-experiment comparison

Soil type Confining

pressure (kPa)

d50 (mm) Shear band thickness (mm) Shear band inclination

Experimental FE Experimental FE

F-75 Ottawa 15 0.22 2.97 3.00 51.6 52.0

F-75 Ottawa 100 0.22 2.91 2.90 53.7 54.0

Coarse slica sand 15 1.60 17.33 17.50 51.4 52.0

Coarse slica sand 100 1.60 17.00 17.50 53.2 52.0

PEEQ(Ave. Crit.: 75%)

+5.264e-01+4.826e-01+4.388e-01+3.950e-01+3.512e-01+3.073e-01+2.635e-01+2.197e-01+1.759e-01+1.321e-01+8.828e-02+4.447e-02+6.555e-04

Fig. 8. Shear band simulations for the

restrained lateral and rotational

boundary


Cosserat finite element implementation nicely captures the shear band formation. As is well known,

the finite element calculations of a classical continuum approache are mesh sensitive and the width

of the predicted shear band collapses to the size of the element used in the finite element simulations.

As the three different mesh solutions illustrate, Cosserat implementation represents a good model for

the finite element simulations, and the results are almost mesh-independent. Deviatoric stress and

couple stress distributions for the three different meshes sizes are shown in Figs. 16 and 17. As can

be seen, the stresses are not dependent on the mesh sizes. It can be said that the Cosserat effect

remedies the mesh dependence with respect to shear band thickness-orientation and stress–strain

behavior of granular materials.

Fig. 10a. Contour, b deformed mesh

plot of equivalent plastic strain for fine

mesh

Fig. 9. Experimental simulation

of shear band [9]


7 Length scale study

The importance of the length scale will be investigated next by solving plane strain problems.

Different length scales will be used for a fixed mesh size, and the effect will be assessed relative to

stress–strain behavior, plastic strain magnitude, and the thickness of the shear band will be simulated


plot of equivalent plastic strain for

medium mesh

PE, Max. In-Plane PrincipalPE, Min. In-Plane PrincipalPE, Out-of-Plane Principal

ABA QUS/STANDARD Version 6.5-1 Thu Aug 10 01:21:27:Fig. 13. Vector plot of the velocity

field for medium mesh

PE, Max. In-Plane PrincipalPE, Min. In-Plane PrincipalPE, Out-of-Plane Principal

Fig. 11. Vector plot of the velocity

field for fine mesh


for four different length scales. The possible magnitude of the length scale will be investigated for a

more realistic finite element simulation. The plane strain problem has been solved for different

length scales. First, the effect of the length scale on the plastic strain magnitude and thickness of the

shear band will be investigated. Then the importance of the length scale on the stress–strain

behavior, peak stress and post-peak behavior will be simulated for different length scales.

Yoshida et al. [12] observed that large strain gradients are present within a shear zone. This is also

seen from Fig. 18. Notice, however, that the use of a larger length scale enables the diffusion of the

PE, Max. In-Plane PrincipalPE, Min. In-Plane PrincipalPE, Out-of -Plane Principal

ABA QUS/STANDARD Version 6.5-1 Thu Aug 10 01:21:27:Fig. 15. Vector plot of the velocity

field for coarse mesh


plot of equivalent plastic strain for

coarse mesh

600

500

400

300

200

100

0 0 2 4 6 8 10

Axial strain

Coarser Mesh Medium Mesh Finer Mesh

Dev

iato

ric

stre

ss (

KPa

)

Fig. 16. Comparison of deviator stress–

strain behavior of three different mesh

sizes


concentration of plastic strains and results in a consistent width of the shear band. The maximum

value of Cosserat rotation becomes too small due to relatively high values of the length scales. Thus,

the gradient of the micro-rotation can be negligible inside the shear band for large length scales. This

yields to much more consistent plastic strain inside the shear band for larger granular materials.

Figure 19 shows the resulting shear banding for four different values of the internal length scales.

A relatively thin band is observed for the small internal length and a much thicker band results for

the larger internal length. Past researchers have observed the thickness of the shear band within

granular materials to vary between 5 and 20 times the mean grain diameter d50. As Table 3 and

Fig. 20 illustrate, this observation is correct for the relatively small length scales. If the length scales

increase dramatically to 25.00 mm, the thickness of the shear band becomes 1.50 times as large as

the length scales. The length scale study shows that the shear band thickness may be approximately

equal to the length scale. Thus, the range of the shear band thickness should be defined as to vary

from 1.5 to 20 times the mean grain diameter.

The effect of the length scale on the stress–strain curve with a fixed mesh size is illustrated in

Fig. 21. The dependency of the maximum predicted force on the gradient term is illustrated in

Fig. 22. The figure shows the variation of the maximum deviator stress as a function of the length

scale. It can be seen that both the maximum value of the stress and the magnitude of the softening are

dependent on the length scale.

It appears that using a very small length scale leads to more softening of the post-peak deviatoric

stress in granular materials. The peak stresses are given in Table 4 for different length scales.

0.30

0.25

0.20

0.15

0.10

0.05

5.00 10.00


I=0.1 mm I=1.0 mm

I=10.0 mm I=25.0 mm

15.00 20.00 25.00 30.00

Equ

ival

ent p

last

ic s

trai

n

Fig. 18. Length scale–plastic strain

relations

0.015

0.001

0.005

0 0

–0.0005

–0.001

–0.0015 Distance along shear band (m)

Finer Mesh

Coarse Mesh

Medium Mesh

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Cou

ple

stre

ss (

kPa)

Fig. 17. Comparison of deviator couple

stress of three different mesh sizes


35.00

30.00

25.00

20.00

15.00

10.00

5.00

0.00 0.00 5.00

Length scale (mm)10.00 15.00 20.00 25.00 30.00

Shea

r ba

nd th

ickn

ess

(mm

)

Fig. 19. Length scale–shear band

thickness relations

Table 3. Length scale–shear band thickness; length scale–normalized shear band thickness relations

Length scale (mm) Predicted shear band

thickness (t) (mm)

t/l

0.10 1.75 17.5

1.00 13.50 13.50

10.00 24.00 2.40

25.00 38.00 1.52

20.0 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

0.00 10.00 Length scale (mm)

5.00 15.00 20.00 25.00 30.00

Nor

mal

ized

she

ar b

and

thic

knes

s (t

/I)

Fig. 20. Length scale–normalized shear

band thickness relations

600

500

300

200

100

0 0 1 2 3 4 5 6 7 8

Axial strain (%)

I=25.00 mm

I=1.00 mm I=10.00 mm

I=0.10 mm

Dev

iato

nic

stre

ss (

kPa)

400

Fig. 21. Effects of length scales on the

stress–strain behavior of granular mate-

rials


It is also important to note that the thickness of the shear zone is affected significantly by the

increase of the length scale. However, the length scale also has a great effect on the overall stress–

strain behavior of granular materials. The rotational stiffness increases as the length scale increases

and leads to a stiffer response. If the value of the length scale is quite small, a clear mechanism of

failure is recognized by considering the areas where Cosserat rotation has reached significant values.

The response of the granular material in the failure zone is highly dependent on the displacements

and rotations of grains in the neighborhood of the point in consideration.

Standard finite element analysis using a Cauchy–Boltzmann continuum encounters significant

numerical difficulties and diverges due to the emergence of many negative eigenvalues in the tangent

stiffness matrix. However, with a micro-polar continuum the incremental boundary value problem

remains well posed and the analysis can be continued until a collapse load analysis is attained.

Nevertheless, this analysis clearly demonstrates the stabilizing effect of the micro-polar approach in

the numerical simulation and the significance of the length scale on the analysis of the behavior of

granular materials. The enrichment eliminates computational difficulties and provides to observe the

effect of the micro-structure on the behavior of granular materials.

8 Length scale–size effect study

The next example compares the effect of length scales for three different sizes as shown in Fig. 23.

The length scale–size effect study aims to observe the effect of the length scale on different

specimen sizes. The aspect ratio (length/height) of the specimen has been fixed but the dimensions

have been changed for three different specimen sizes (Fig. 23). The plane strain problem will be

solved for three different specimen sizes as follows:

A ¼ L=H ¼ 1=2; B ¼ L=H ¼ 10=20; C ¼ L=H ¼ 100=200;

where L is the width of the specimen and H is the height of the specimen.

600.00

590.00

580.00

560.00

570.00

550.00

540.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00

Length scale (mm)

Peak

dev

iato

r st

ress

(kP

a)

Fig. 22. Effect of length scales on the

peak stress level

Table 4. Length scale–peak stress relations for plane strain loading conditions

Length scale (mm) Peak stress (kPa)

0.10 546.20

1.00 568.30

10.00 593.40

25.00 595.60


The material properties of the specimen are the same as of the previous examples in Section 7.2.

The plane strain simulations have been carried out with the fixed moderate mesh size and with

100 kPa of confinement pressure. Three issues related to the length scale will be addressed. First:

comparing the length scale effect for different specimen sizes. Second: to simulate the length scale–

size effect on the peak stress and post-peak behavior of granular materials. Third: to compare the

axial strain level at peak stresses for different length scales and different sizes.

As Fig. 24 illustrates, the specimen A reaches the peak stress at approximately 2.4% of the axial

strain level for all three length scales. The peak stresses are increasing from 498.0 to 565.0 kPa with

the increasing length scale from 0.1 to 10 mm. The post-peak behavior is more ductile for larger

length scales than for smaller length scales.

Figure 25 illustrates that the strain level for peak stress is 2.00% for the specimen B for all three

length scales. The peak stresses are 496.0, 515.0, and 538 kPa for length scales of 0.1, 1.00, and

10.00 mm, respectively. There is more softening at the post-peak behavior for smaller length scales.

As Fig. 26 illustrates, the specimen reaches the peak stress at 1.80% axial strain level. There is not

much difference at the peak stresses for different length scales. The peak stress is 485 kPa for the

length scale equal to 0.10 and it is 525 kpa for the length scale equal to 10.00. Specimen C shows

more softening than specimens A and B, whose specimen sizes are smaller than C.

Figures 24, 25, and 26 illustrate that the length scale has a similar effect for different sizes of the

specimens. If the length scale is increasing, the peak stresses are increasing also. However, the

difference between the peak stresses is not the same for different sizes. If the specimen size

increases, the effect of the length scale on the peak stress is decreasing. It is observed that a smaller

specimen gives more ductile behavior and the ductility is increasing with an increase of the length

scale. The size of the specimen affects the strain level at peak stress. If the specimen size is

Fig. 23. Plane strain simulation for

three different specimen sizes

600

500

400

300

200

100

0 0 1 2 3

I=0.10 I=1.00 I=10.00

4 5 6 7 8

Axial strain (%)

Dev

iato

ric

stre

ss (

kPa)

Fig. 24. Plane strain results for

specimen A (L/H = 1/2)


increasing, the specimen reaches the peak stress at a lower strain level. This is because of the change

of stiffness of the material for a different specimen size. This parametric study illustrates that the

length scale and the size of the specimens have substantial influence on the behavior of granular

materials.

9 Influences of Cosserat parameters on the elastic finite element solutions

The Cosserat continuum has been widely used for post-peak behavior. However, the influences of

Cosserat parameters on the Elastic solution have not been considered in any detail. Cosserat elastic

constants (based on isotropic theory) were used to predict the stress–strain behavior of a soft layer

(Fig. 27), and the results are compared with the classical isotropic elastic solution. The material

properties of the model are given in Table 5.

Shear stress and micro-rotation of the soft layer have been shown in Figs. 28 and 29. Shear stress

and micro-rotations are localized at the corner of a soft layer. As can be seen, micro-rotations are

observed in the localized zone.

The effect of the Cosserat shear modulus on the elastic solution will be illustrated in Figs. 30 and 31.

As Figs. 30 and 31 illustrate, the stiffness of the soft layer is decreasing with the decrease of Cosserat

shear modulus values. The elastic response of the soft layer is highly dependent on the values of the

Cosserat shear modulus in Micro-polar elastic finite element simulation.

600

500

400

300

200

100

0 0 1 2 3 4 5

I=0.10 I=1.00 I=10.00

6 7 8Axial strain (%)

Dev

iato

ric

stre

ss (

kPa)


L/H = 100/200

600

500

400

300

200

100

0 0 1 2 3 4 5 6 7 8

Axial strain (%)

I=0.10 I=1.00 I=10.00 D

evia

tori

c st

ress

(kP

a)


specimen B (L/H = 10/20)


10 Summary and discussions

The finite element solution of the classical continuum is mesh sensitive. To solve the mesh

sensitivity, Cosserat continuum theory is implemented into the Drucker–Prager criterion and

subsequently used in finite element analysis. The finite element analyses were carried out using the

commercial non-linear finite element code ABAQUS. The Cosserat element with the additional

degrees of freedom was implemented in ABAQUS using the UEL interface. This formulation was

used in order to analyze selected geotechnical problems or configurations. The finite element results

were verified with plane-strain experimental results.

Constitutive relations have been derived for a micro-polar continuum using a thermomechanical

approach. The thermomechanical approach guarantees that the resulting micro-polar models are

consistent with the laws of thermodynamics. The first constraint of the thermodynamics laws leads to

Plane of symmetry

Platen/Soft layer

Platen/Soft layer

2L

Rigid

Rigid

Soft layer

CL

2h

Fig. 27. Soft layer model illustration

Table 5. Material parameters for soft layer

Young’s modulus (E) (kPa) 6 · 106

Poisson’s ratio (v) 0.20

Length scale (mm) 1.0

Cosserat shear modulus (Gc) 0.5 · G

2

3 1ODB: arslan_ben.odb ABAQUS/STANDARD Version 6.5-1 sat Jun 24 12:21:47 Mountain DaylightStep : Step-1Increment 1: Step Time = 1.000Primary Var : 5, 512Deformed Var: U Information Scale Factor: +1.000c+00

8, 812SVEG, (fraction = -1.0)(Ave. Crit.: 759)

+1.656e+04+1.300e+04+1.204e+04+8.280e+03+5.520e+03+2.760e+03+1.465e-03-1.760e+03-5.520e+03-8.280e+03-1.104e+04-1.380e+04-1.656e+04

Fig. 28. Shear stress distribution of soft layer


the conservation of momentum for a micro-polar continuum. This results in a restriction on the form

of the free energy, specifically of the micro-strain.

Our purpose here is to demonstrate the application of the constitutive relations derived in the

companion paper. Toward this goal, we have adopted several assumptions to simplify the analysis.

We allow no variation in the horizontal direction and the material response is a function of the

vertical position within the body. The normal stress on the boundaries is held constant, and there are

no body forces or moments.

160000

140000

120000 100000

80000

40000

20000 0

0 0.02 0.04 0.06 0.08 0.1 0.12 Average compressive strain

Classical Cosserat I=0.0Cosserat I=0.4Cosserat I=1.0A

vera

ge c

ompr

essi

ve s

tres

s

60000

Fig. 30. Stress–strain behavior com-

parison of classical and Cosserat

isotropic elastic solution (Gc = 0.1 G)

140000

120000

100000

80000

60000

40000

20000

0 0 0.02 0.04 0.06 0.08 0.1

Average compressive strain

Classical Cosserat I=0.0Cosserat I=0.4Cosserat I=1.0

Ave

rage

com

pres

sive

str

ess

0.12 Fig. 31. Stress–strain behavior com-

parison of classical and Cosserat

isotropic elastic solution (Gc = 0.1 G)

U6 (Cosserat Rotation) +5. 402e−02 +4. 502e−02 +3. 602e−02 +2. 701e−02 +1.801e−02 +9. 004e−03 +3. 725e−09 -9.004e−03 -1.001e−02 -2.701e−02 -3.602e−02 -4.501e−02 -5.402e−02

2

3 1ODB: arslan_ben.odb ABAQUS/STANDARD Version 6.5-1 sat Jun 24 12:21:47 Mountain Daylight Step : Step-1 Increment 1: Step Time = 1.000 Primary Var : U6 (Cosserat Rotation) Deformed Var: U information Scale Factor: +1.000c+00

Fig. 29. Micro-rotation of soft year


Alsaleh et al. [13, 14] and Alshibli et al. [15] implemented the enhanced Lade’s model that

accounts for the couple stress, Cosserat rotation and Intrinsic Length scale. These papers are the

latest and very systematic studies in this area. However, the authors did not mention so much about

the effect of Cosserat parameters on the formulations and Finite Element simulations of Cosserat

Elastoplastic analysis. They did not discuss the effects of Cosserat shear modulus, length scale–size

effect relations, etc., on the finite element analysis. The finite element simulations of this paper

illustrate that the increase of the Cosserat shear modulus gives stiffer response for Finite Element

simulation in a micro-polar continuum. The length scale and size of laboratory specimens have

substantial influence on the observed and measured behavior of granular materials. Smaller

specimens give more ductile behavior and the ductility is increasing with an increase of the length

scale. If the specimen size is increasing, the specimen reaches the peak stress at a lower strain level

because of stiffer pre-peak behavior. Softening is steeper in large size specimens. This was also

observed in experiments.

11 Conclusions

The purpose of this thesis was to evaluate the classical and micro-continuum approaches in elastic-

elastoplastic failure analysis. Similarities and differences of classical and micro-continuum

approaches are compared with kinematics and constitutive formulations. The major findings of

this research are:

• The shear band thickness increases as the mean particle size increases, and will decrease as the

confining pressure increases; while the inclination angle increases with increasing confining

pressure. This was also observed in experimental studies. However, if the shear band width is

normalized with the grain size diameter (t/d50), the normalized thickness is found to be as small

as 1.50 times the d50 for large diameter particles. This is in contrast to the traditional analysis

that shear band thickness is observed between 5 and 20 times of d50.

• The use of a larger length scale enables the diffusion of the concentration of plastic strains and

results in a consistent width of the shear band.

• Maximum Cosserat rotation is observed at the center of the shear band. Couple stresses and the

rotation curvature are nearly zero outside the shear band and they switch their direction at the

center line of the shear band. The magnitude of Cosserat rotation is decreasing with the increase

of the length scale. This seems physically and intuitively correct.

• The lack of an internal length scale in classical continuum models means that the size of the

localized zone may not be determined due to mesh sensitivity. However, micro-polar continua

enable the size of the shear zone to be predicted along with micro-structural properties.

• Length scale and size of laboratory specimens have substantial influence on observed and

measured behavior of granular materials. A smaller specimen gives more ductile behavior and

the ductility is increasing with the increase of the length scale. If the specimen size is increasing,

the specimen reaches the peak stress at a lower strain level because of stiffer pre-peak behavior.

Softening is steeper in large size specimens. This was also observed in experiments.

Acknowledgements

The authors would like to thank Prof. Kaspar Willam for his guidance and help during the study. The authors

gratefully acknowledge the financial support provided by NASA under Contract No. NCC8-242.


References

[1] Oda, M., Kazama, H., Konishi, J.: Effects of induced anisotropy on the development of shear bands in

granular materials. Mech. Mat. 28, 103–111 (1998)

[2] Aifantis, E.: The physics of plastic deformation. Int. J. Plast. 3, 211–248 (1987)

[3] Arslan, H., Baykal, G.: Analyzing the crushing of granular materials by sound analysis technique.

J. Test. Eval., 34(6), 464–470 (2006)

[4] Oda, M., Kazama, H.: Microstructure of shear bands and its relation to the mechanisms of dilatancy

and failure of dense granular soils. Geotechnique 48(4), 465–481 (1998)

[5] Tordesillas, A., Walsh, S.D.C.: Incorporating rolling resistance and contact anisotropy in microme-

chanical models of granular media. Powder Tech. 124, 106–111 (2002)

[6] Alshibli, K.A., Alsaleh, M.I., Voyiadjis, G.Z.: Modeling strain localization in granular materials using

micropolar theory: numerical implementation and verification. Int. J. Numer. Anal. Meth. Geomech.

30(15), 1525–1544 (2006)

[7] Valanis, K.C., Peters, J.F.: Ill-posedness of the initial and boundary value problems in non-associative

plasticity. Acta Mech. 114, 1–25 (1996)

[8] ABAQUS Version 65 (2004), Manuals, Hibbitt, Karlsson and Sorensen Inc.

[9] Alshibli, KA.: Localized deformations in granular materials. Ph.D. dissertation, University of

Colorado at Boulder, CO (1995)

[10] Alshibli, K.A., Sture, S.: Sand shear band thickness measurements by digital imaging techniques.

J. Comput. Civil Engng. 13(2), 103–109 (1999)

[11] Alshibli, K.A., Sture, S.: Shear band formation in plane strain experiments of sand. J. Geotech.

Geoenviron. Engng. (ASCE) 126(6), 495–503 (2000)

[12] Yoshida, T, Tatsuoka, F., Siddiquee, M.S.A., Kamegai, Y.: Shear banding in sands observed in plane

strain compression. In: Chambon, R., et al. (eds.) Localization and Bifurcation Theory for Soils and

Rocks, pp. 165–179. Balkema, Amsterdam (1995)

[13] Alsaleh, M.I., Voyiadjis, G.Z., Alshibli, K.A.: Modeling strain localization in granular materials using

micropolar theory: mathematical formulations. Int. J. Numer. Anal. Meth. Geomech. 30(15), 1501–

1524 (2006)

[14] Alsaleh, M.I., Alshibli, K.A., Voyiadjis, G.Z.: Influence of micro-material heterogeneity on strain

localization in granular materials. ASCE: Int. J. Geomech. 6(4), 248–259 (2006)

[15] Alshibli, K.A., Alsaleh, M.I., Voyiadjis, G.Z.: Modeling strain localization in granular materials using

micropolar theory: numerica limplementation and verification. Int. J. Numer. Anal. Meth. Geomech.

30(15), 1525–1544 (2006)

[16] Arslan, H., Sture, S., Willam, K.J.: Analytical and geometrical representation of localization in

granular materials. Acta Mech. 194, 159–173 (2007)

[17] Arslan, H., Sture, S.: Finite element simulation of localization in granular materials by micropolar

continuum approach. Comput. Geotech., DOI: 10.1016/j.compgeo.2007.10.006 (2007)

[18] Arslan, H., Sture, S.: Evaluation of a physical length scale for granular materials. Comput. Mater. Sci.,

DOI: 10.1016/j.commatsci.2007.08.0 (2007)

[19] Arslan, H., Sture, S.: Finite element analysis of localization and micro–macro structure relation in

granular materials. Part I: Formulation. Acta Mech. 197, 135–152 (2008)


Documents

Finite element analysis of localization and micro–macro structure relation in granular materials. Part II- Implementation and simulations