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
156 H. Arslan and S. Sture
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
Finite element analysis of localization and micro–macro structure relation. Part II 157
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
Distance along shear band (mm)
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
158 H. Arslan and S. Sture
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
Finite element analysis of localization and micro–macro structure relation. Part II 159
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]
160 H. Arslan and S. Sture
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
Fig. 12a. Contour, b deformed mesh
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
Finite element analysis of localization and micro–macro structure relation. Part II 161
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
Fig. 14a. Contour, b deformed 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
162 H. Arslan and S. Sture
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
Distance along shear band (mm)
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
Finite element analysis of localization and micro–macro structure relation. Part II 163
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
164 H. Arslan and S. Sture
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
Finite element analysis of localization and micro–macro structure relation. Part II 165
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)
166 H. Arslan and S. Sture
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)
Fig. 26. Plane strain results for
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)
Fig. 25. Plane strain results for
specimen B (L/H = 10/20)
Finite element analysis of localization and micro–macro structure relation. Part II 167
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
168 H. Arslan and S. Sture
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
Finite element analysis of localization and micro–macro structure relation. Part II 169
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.
170 H. Arslan and S. Sture
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