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Unlike conventional first-order continuum model of which material parameters can be identified via an inverse problem conducted at material point that exhibits homogeneous deformation, higher-order continua requires information from the derivative of the deformation gradient. This study concerns a integrated experimental-numerical procedure designed to identify material parameters for higher-order continuum models. Using a combination of micro-CT images and macroscopic stress-strain curves as the database, we construct a new finite element inverse problem which identifies the optimal value of material parameters that matches both the macroscopic constitutive responses and the meso-scale micropolar kinematics. Our results indicate that the optimal characteristic length predicted by the constrained optimization procedure is highly sensitive to the types and weights of constraints used to define the objective function of the inverse problems. This sensitivity may in return affects the resultant failure modes (localized vs. diffuse), and the couple stress responses. This result signals that using the mean grain diameter alone to calibrate the characteristic length may not be sufficient to yield reliable forward predictions.
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International Journal for Multiscale Computational Engineering, 14 (4): 389–413 (2016)
Kun Wang,1WaiChing Sun,1,Simon Salager,2SeonHong Na,1&
Ghonwa Khaddour2
1Department of Civil Engineering and Engineering Mechanics, Columbia University, 614 SW
Mudd, Mail Code: 4709, New York, New York 10027, USA
2UJF-Grenoble 1, Grenoble-INP, CNRS UMR 5521, 3SR Laboratory, 38041 Grenoble, France
Address all correspondence to: WaiChing Sun, E-mail:
Unlike a conventional first-order continuum model, the material parameters of which can be identified via an inverse
problem conducted at material point that exhibits homogeneous deformation, a higher-order continuum model requires
information from the derivative of the deformation gradient. This study concerns an integrated experimental-numerical
procedure designed to identify material parameters for higher-order continuum models. Using a combination of micro-
CT images and macroscopic stress–strain curves as the database, we construct a new finite element inverse problem
which identifies the optimal value of material parameters that matches both the macroscopic constitutive responses and
the meso-scale micropolar kinematics. Our results indicate that the optimal characteristic length predicted by the con-
strained optimization procedure is highly sensitive to the types and weights of constraints used to define the objective
function of the inverse problems. This sensitivity may in return affect the resultant failure modes (localized vs. dif-
fuse), and the coupled stress responses. This result signals that using the mean grain diameter alone to calibrate the
characteristic length may not be sufficient to yield reliable forward predictions.
KEY WORDS: micro-CT imaging, micro-polar plasticity, critical state, higher-order continuum, Hostun
Granular matters are some of the most commonly encountered materials in our daily lives. Ranging from natural
materials, such as sand, slit, gravel, fault gauges to the man-made materials, such as pills, sugar powder, and candy,
granular matters may be composed of grains with various shapes, sizes, and forms. The mechanical behavior of
granular assemblies are therefore not only depending on the material properties of the grains that form the assemblies,
but also the evolution of the grain contact topology or fabrics (Kuhn et al., 2015; Lenoir et al., 2010; Li and Dafalias,
2011; O’Sullivan, 2011; Satake, 1993; Subhash et al., 1991; Sun et al., 2013; Walker et al., 2016; Wang and Sun,
2015). In the past decades, theoretical and numerical studies have achieved great success to describe and predict the
granular materials that exhibit solid-like behavior when subjected to sufficient confining pressure. In those cases,
granular materials are often treated as first-order Boltzmann continua which possess no internal microstructures or
length scale. For instance, constitutive models that employ the critical state soil mechanics concept are now able
to replicate the monotonic and cyclic responses of sand of different relative densities, confining pressure and initial
states (Been and Jefferies, 1985; Borja and Sun, 2007; Dafalias and Manzari, 2004; Lade and Duncan, 1975; Manzari
1543–1649/16/$35.00 c
2016 by Begell House, Inc. 389
390 Wang et al.
and Dafalias, 1997; Pestana and Whittle, 1999; Schofield and Wroth, 1968; Sun, 2013; Wu et al., 1996). In these
macroscopic models, microstructural attributes, such as pore size distribution, grain shapes, and size distribution,
grain-scale heterogeneity are not explicitly taken into account at the grain scale. Instead, they are represented by a set
of internal variables that represents the loading history. The evolution of these internal variables are then governed
by phenomenological evolution laws that require material parameters calibrated from experiments. Nevertheless, the
success on replicating constitutive responses at a material point level often does not automatically lead to realistic
simulated responses. For instance, the classical first-order continuum model does not possess physical length scale and
this may cause finite element or finite difference models to exhibit spurious mesh dependence, unless a regularization
limiter is applied (Bazant et al., 1984; Belytschko et al., 1988; Fish et al., 2012; Lasry and Belytschko, 1988; Liu
et al., 2015; Na and Sun, 2016; Needleman, 1988; Sun and Mota, 2014). This regularization limiter can be introduced
via various means, ranging from incorporating rate dependence, inserting embedded strong discontinuity to adapting
nonlocal and gradient-based constitutive relations.
For instance, classical bifurcation analysis by Rudnicki and Rice (1975) and Issen and Rudnicki (2000) can be used
to predict the onset and orientiation of a deformation band. Beyond the bifurcation point, additional degree of freedom
that represent an embedded strong discontinuity can be introduced via assumed strain [e.g., Borja (2000)], extended
finite element [e.g., Song et al. (2006)] or localization or bifurcated elements [e.g., Belytschko et al. (1988); Ortiz
et al. (1987); Yang et al. (2005)]. Nevertheless, recent experimental and theoretical studies have both indicated that
granular materials may initially form multiple deformation bands at the bifurcation point, but the interaction among
deformation band may lead to some of the deformation band vanishing. Meanwhile, a dominated persistent shear band
may emerge among them in the post-bifurcation regime (Borja et al., 2013; Gajo et al., 2004; Rechenmacher, 2006).
This discovery indicates that the adaptive insertion of the enhanced mode(s) may lead to mesh bias unless there is a
coarsen mechanism in place to prevent the fixture of shear band localization (Lin et al., 2015).
Opposite to the enrichment approach, higher-order continuum theory is another attractive option to circumvent the
pathological mesh dependence exhibited in the Boltzmann continua. The history of generalized continuum theories
can be tracked back to the work of the Cosserat brothers who published a book titled “Th´
eorie des corps d´
in 1909” (Cosserat and Cosserat, 1909). As pointed out by Maugin and Metrikine (2010), there exist multiple inter-
pretations and generalizations of Cosserats’ work. Nevertheless, the necessary ingredients of the higher-order con-
tinuum theory are the incorporation of higher-order kinematics (micro-rotation, micro-stretch, micro-torsion, etc.)
and additional internal degree of freedom that breaks the symmetry of the Cauchy stress (Eringen, 1999; Maugin
and Metrikine, 2010; Mindlin, 1964). The micropolar continuum theory is a sub-class of the generalized continuum
theory in which the material point is associated with micro-structures exhibiting not only deformation but changing
orientations that are independent of the macroscopic deformation. These changes of orientations can be represented
by Euler’s angle, quaternions, and spinors, among other mathematical tools (Duhem, 1893). By incorporating the
additional rotational degrees of freedom at the microscopic level, micropolar effect of granular materials, especially
those associated with the onset of strain localization as observed by Desrues and Viggiani (2004); Hall et al. (2010),
can be replicated properly in numerical simulations without exhibiting the spurious mesh dependence.
Nevertheless, incorporating the micropolar effect to improve the accuracy and physical underpinning of the nu-
merical models also comes with a price. For instance, the identification of the the micropolar material parameters in
addition to the non-polar counterparts can be difficult for material testing procedures that are originally designed for
homogeneous specimen subjected to first-order boundary conditions. The assumption that the strain field developed
inside the specimen remains approximately homogeneous in convectional triaxial and shear apparatus is unavoidably
violated when high-order kinematics and couple stress are considered. These difficulties are partly responsible for the
limited applications of higher-order continuum theory in engineering practices.
Our objective is to overcome this technical barrier by introducing a simple experimental-numerical method de-
signed specifically for identifying material parameters for higher-order continua. In particular, we introduce the meso-
scale kinematics data from X-ray tomographic images obtained from drained triaxial compression tests to construct
a multiscale objective function. The tomographic images at selected axial strain levels are combined with the con-
ventional macroscopic stress–strain curve and the state path to constitute the measured data set. A new objective
function is then defined to simultaneously minimize the discrepancy between the simulated and measured macro-
scopic responses while enforcing the simulated grain-scale responses to be consistent with the kinematics information
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 391
provided by the tomographic images. Our results show that the incorporation of microstructural information from
the grain scale may profoundly change the predicted length scale and failure mode of the numerical specimen. More
importantly, the predicted coupled stress response is also highly sensitive to the way microstructural information is
incorporated in the inverse problem.
To the best knowledge of the authors, this work is the first contribution to employ X-ray tomographic images
to identify material parameters via a multiscale inverse problem for granular materials. In addition, the proposed
method provides a new method to quantify the sensitivity of characteristic length predicted by material identification
procedure. The findings from the Hostun sand experiment indicate that a multiscale model capable of replicating
the macroscopic responses may nevertheless produce incorrect bifurcation modes and completely different micro-
scale responses. This result has important implications for material modeling, as it clearly shows that calibrating
macroscopic responses alone is insufficient to provide reliable forward prediction for multiscale models. In other
words, a seemingly good match between experimental and simulated responses at the macroscopic level may be a
consequence of high-quality grain-scale simulations or a product manufactured by excess tuning and manipulations
of meso-scale material parameters.
The rest of this paper is organized as follows. In Section 2, we first describe the experimental procedure used to
obtain the tomographic images. Then, the ingredients that constitute the inverse problem for a micropolar critical state
plasticity model are discussed in Section 3. The calibration of a single unit cell and the finite element inverse problem
are described in Sections 4 and 5, followed by a discussion of major findings. Finally, concluding remarks are given
in Section 6.
As for notations and symbols, bold-faced letters denote tensors; the symbol “·” denotes a single contraction of
adjacent indices of two tensors (e.g., a·b=aibior c·d=cij djk ); the symbol “:” denotes a double contraction of
adjacent indices of tensor of rank two or higher (e.g., C:ϵe=Cijkl ϵe
kl); the symbol “” denotes a juxtaposition
of two vectors (e.g., ab=aibj) or two symmetric second order tensors [e.g., (αβ) = αijβkl]. As for sign
conventions, we consider the direction of the tensile stress and dilative pressure as positive.
This section introduces the triaxial test performed for this research at 3SR Lab and details the experimental proce-
dures followed. The objective of performing the triaxial tests is to provide a multiscale benchmark database for model
calibration and material identification. The bemchmark data consist of the macroscopic stress–strain curve and the
grains position at different stress levels during the shearing, characterizing the grains-scale behavior of the granular
media. These data form the basis for a multiscale objective function to search the optimal material parameters to pre-
dict the most consistent mechanical behavior measured by a weighted norm that minimizes the discrepancy between
simulations and experimental observations at both grain- and specimen scales.
2.1 X-Ray CT
X-ray micro-CT is a non-destructive, 3D high-resolution imaging method that allows the internal structure of the
scanned objects to be investigated (And`
o et al., 2012; Hall et al., 2010; Lenoir et al., 2010; Sun et al., 2011). Micro-
CT technology yields images that map the variation of x-ray attenuation within the objects, based on the composition
of the object. Because sand and water have different x-ray attenuation coefficients, a significant contrast for the two
phases is observed in an x-ray transmission image, Fig. 1. A quantitative analysis that provides detailed information
on the arrangement and distribution of particles can be built from the processed 3D images.
The imaging process can be summarized as follows: the cell first is placed on a turntable stage whose rotation
can be accurately controlled. An x-ray source generates a continuous x-ray beam; the beam passes through the object
and casts an x-ray shadow onto the detector, see Fig. 2. The radiation that hits the detector is converted into an
electronic charge, that is subsequently passed to a computer to create a radiograph (i.e., digital image). A series
of images (radiographies) is acquired while rotating the object step by step through 360at a pre-defined angular
increment. These radiographies record projections that contain cumulative information on the position and density of
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392 Wang et al.
FIG. 1: Reconstructed x-ray image shows the clear contrast among grain, water, and air. The experiments in this study
are performed using x-ray setup of Laboratory 3SR at Grenoble University.
FIG. 2: X-ray CT scanner at Laboratory 3SR (left) and the whole arrangement of triaxial setup inside x-ray cabinet
the absorbing features within the specimen. The data obtained can be used to perform the numerical reconstruction of
the final 3D image (And`
o et al., 2012; Hall et al., 2010).
2.2 Triaxial Cell
The triaxial cell used in this work is designed to allow x-ray scanning and to fit to the mechanical loading system
available in 3SR Laboratory. The cell is made of PMMA, which is transparent to X-rays, and quite resistant to cell
pressure (confinement) applied in triaxial tests. Figure 3 shows the PMMA cell in front of X-ray source, with the
specimen installed at its place inside the cell. The loading system applies and measures the axial force compressing
the specimen from below, and the vertical displacement of the lower piston. The maximum axial force that can be
measured by the force meter, which is in contact with the bottom of the lower piston is 0.5 kN. The measurement of
the axial displacement is made by a linear variable differential transformer (LVDT) which is attached to a tie bar and
measures the vertical displacement of the loading head. These measurements allow plotting the relation between the
axial stress and axial shortening during the triaxial test. The speed range for the loading head is from 0.1 µm/min to
100 µm/min, which corresponds to a strain rate from 0.0005 % to 0.5% per minute (for a specimen of height 20 mm).
The speed range considered in this work for the three triaxial tests is 20 µm/min (strain rate is 0.1% for a specimen of
height 20 mm). The motor is driven remotely from a laptop used for data acquisition. The system of data acquisition
registers the force, the confinement pressure, and LVDT measurements which were calibrated at the beginning of the
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 393
FIG. 3: PMMA cell in front of x-ray source with the specimen installed at its place inside the cell
2.3 Material Used
The experimental program is conducted on Hostun sand (HN31). Hostun sand is used as a reference material in
different laboratories [cf. Amat (2008); Desrues and And`
o (2015); Sadek et al. (2007)]. Its chemical components
principally consist of silica (SiO2>98%). The grain shape is angular as shown in the scanning electron microscope
image (SEM) of a few grains of Hostun sand, taken from Flavigny et al. (1990), in Fig. 4(a). A particle size distribution
analysis for Hostun sand is shown in Fig. 4(b).
2.4 Specimen Preparation
The specimen prepared for the triaxial tests is a cylindrical specimen of 1 cm diameter and 2 cm height (slenderness
ratio 2). The technique used for specimen preparation in the three triaxial tests is water pluviation, and thus all the
FIG. 4: (a) SEM image of Hostun sand from Flavigny et al. (Reprinted with permission from Jacques Desrues,
Copyright 1990), (b) Grain size distribution of Hostun sand from Amat (2008)
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394 Wang et al.
pores in the specimen are completely filled with de-mineralized water in the initial state. This procedure ensures that
the specimen is fully saturated at the beginning of the mechanical tests.
2.5 Macroscopic Response
The axial stress is calculated by the axial force measured over the cross section of the specimen at the initial state.
The axial force comes from the raw measurement of force. The axial strain (in %) is obtained from the shortening
applied by axial compression of the specimen, with respect to its initial height. The specimen behaves as expected for
a dense granular material: there is a peak in the specimen’s axial stress response, followed by strain softening until a
plateau of residual stress is reached. The peak is reached at axial strain ϵa= 5.5%, and at axial stress σ= 620 kPa.
The residual stress is σ= 440 kPa. Figure 5 shows the cross section of the deformed specimen at different axial strain
level. While the initial deformation remains relatively homogeneous at low axial strain, the specimen subsequently
deforms into a barrel shape while large strain (ϵa= 21%) develops toward the end of the test.
2.6 Segmentation
Measurements of the spatial distribution of grains require a separation of the solid and water phases. The volume has
been “binarized” using thresholds, one between air and water peaks, and the second one between water and grain
peaks. This last one is chosen in such a way that volume of voxels identified as grain phase correspond to volume of
grains that has been estimated from the weight of the sample (in a dry state). The threshold between air and water
is chosen to correspond to the local minimum in the gray level distribution. Figure 6 shows the cross section of the
processed images of which the spatial distribution of the solid and water phases are idenified based on the threshold
FIG. 5: Micro-CT images of the cross section of the specimen at different axial strain level
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 395
FIG. 6: Cross sections of the “binarized” two-phase images at axial strain = 0, 8, and 21%.
2.7 Grains Position
Water phase is separated from the whole 3D volume using Visilog by setting the threshold to a gray value equal to 128.
In order to remove the noise inside the images, represented by water volumes of one voxel, a morphological process
of erode and dilate by one voxel is applied. Later, water clusters are labeled. Labeling process can be summarized as
giving each individual water cluster an Identification (Id). This Id is represented in the images by colors (i.e., each color
represents a specific water cluster). The process of labeling is very useful to be able to refer to any water cluster directly
by its Id number. These images where each water cluster has a specific Id and color can be considered as a mask applied
to the reconstructed image (i.e., the gray-scale information is always kept). At this point, more information about
water clusters is needed as the number of clusters, the center of mass, 3D volume, 3D area, ...etc. This information
is obtained using another tool of Visilog “analysis individual.” The measurements of clusters properties are saved in a
DAT file type associated primarily with data which can be read by a text editor.
This section presents the calibration procedure for the micropolar hypoplasticity model via the micro-CT images.
The macroscopic constitutive model and the parameter identification procedure conducted via the open source toolkit
Dakota [cf. Adams et al. (2009)] are briefly described. We will first treat the entire specimen as a single unit cell and
we will use the gradient based approach to obtain the material parameters. The open-source optimization code Dakota
is employed. Then we make use of the micro-CT images to construct the geometry of the Hostun sand sample at the
beginning of the triaxial compression loading. The initial distribution of void ratio is also inferred from micro-CT
information. They serve as the starting point of the model simulations. We calibrate the material parameters by two
sets of benchmark data. One set consists of the macroscopic stress ratio and volumetric strain vs. the axial strain
curves. The other set contains the geometry and void ratio distribution when the axial stress reaches the peak and at
the residual stage.
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396 Wang et al.
3.1 Micropolar Finite Element Model
The inverse problem introduced in this paper is applicable to a wide spectrum of constitutive laws that considers
high-order kinematics. In this study, we extend the previous two-dimensional implementation of the micropolar hy-
poplasticity model in Lin and Wu (2015), Lin et al. (2015) to a 3D formulation and use the 3D model to calibrate
with experimental data. This model can be considered as an extension of the non-polar hypoplasticity model in Wu
et al. (1996) and is suitable to characterize the Houstun sand specimen obtained for this study. For completeness, we
include a brief outline of the 3D finite element formulation of the micropolar hypoplasticity model. Readers interested
at the details of the micropolar hypoplasticity constitutive model may refer to Wu et al. (1996), Lin and Wu (2015),
Lin et al. (2015).
The non-symmetric Cauchy stress sand the coupled stress µare given in nonlinear path-dependent rate form in
terms of the current state of the stresses, the strain rate ˙e, the curvature rate ˙
κ, and the void ratio e. The micropolar
constitutive equations writes
s=C1tr(s)˙e+C2tr(˙e)s+C3ψs+C4fd|| ˙e||2+l2|| ˙
κ+C2tr(˙e)µ+C3ψµ+ 2C4fd|| ˙e||2+l2|| ˙
µare the Jaumann rates of sand µ,sis the deviatoric part of s, and
fd= (1 a)eemin
ecemin +a(3)
is a linear scalar function that accounts for the effect of critical state on the constitutive behavior. The material pa-
rameters to be determined are C1,C2,C3,C4,a, the critical void ratio ec, the minimum void ratio emin, and the
characteristic length l, which is the only additional parameter in micropolar framework compared to classical con-
tinuum hypoplasticity model. Note that this micropolar hypoplasticity model is valid under the assumption that ˙
µare all skew-symmetric. An explicit numerical integration scheme (Forward-Euler) is adopted to integrate the
highly nonlinear constitutive equations Eq. (1), i.e.,
κ,et0)∆t, µt1µt0+˙
κ,et0)∆t. (4)
The finite element implementation for 2D small strain problems was previously described in great detail (Lin
et al., 2015). However, the 3D finite element counterpart has not yet been established. Since the tomographic images
are three-dimensional and the bifurcation modes can be either symmetric or non-symmetric, an extension to three-
dimensional space is necessary for comparing experimental data and simulated responses at the meso-scale level.
As a result, the micropolar hypoplasticity model is implemented with a 3D micropolar finite element model. Con-
sequently, the degrees of freedom for each node consist of three translational DOFs and three rotational DOFs, i.e.,
u= [u1u2u3ϕ1ϕ2ϕ3]T. Adopting the Voigt notation, the vectors that store the components of the generalized
strain and stress tensors read
{s}= [s11 s22 s33 s12 s21 s23 s32 s13 s31 µ21 µ32 µ31]T,
{e}= [e11 e22 e33 e12 e21 e23 e32 e13 e31 κ21 κ32 κ31]T.(5)
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 397
The element strain-displacement matrix Beis modified accordingly and takes the form:
Be= [B1,B2, . . . Bnen ],Ba=
Na,10 0 0 0 0
0Na,20 0 0 0
0 0 Na,30 0 0
Na,20 0 0 0 Na
0Na,10 0 0 Na
0Na,30Na0 0
0 0 Na,2Na0 0
Na,30 0 0 Na0
0 0 Na,10Na0
0 0 0 0 Na,10
0 0 0 0 0 Na,2
0 0 0 0 0 Na,1
where nen is the number of element nodes, Nais the shape function of element node a,Na,1,Na,2,Na,3are the
derivatives of the shape function Nawith respect to x1,x2,x3directions in a Cartesian coordinate system. Consider
a body Bsubjected to traction tand body force bpartitioned into element Be. The matrix form of the balance of linear
and angular momentum therefore reads
FINT =Anel
e=1 Be
BeT·sdV ;FEXT =Anel
e=1 Be
NeT·bdV +Be
where Ais the assembly operator [cf. Hughes (2012)] and Ne= [N1,N2,N3, ...Nnen ].
3.2 Optimization Algorithms for Material Parameter Identification
Material parameter identification of soil samples can be done via solving inverse problems. Previous works, such as
Herle and Gudehus (1999), use multiple simple experiments, such as angle of repose, minimum and maximum index
density, shear test, and oedometric compression test to identify material parameters in an almost one-by-one fashion
for a non-polar hypolasticity model parameter for granular materials. Meanwhile, Ehlers and Scholz (2007) introduce
a two-step staggered procedure for micropolar constitutive model for granular materials in which the identification
algorithm first seeks the standard non-polar elasto-plastic material parameters from a series of cyclic triaxial tests that
keeps the specimen in homogeneous states, then searches for the optimal values for the micropolar counterpart with
another set of biaxial compression tests. In both approaches, the material identification procedure requires multiple
mechanical tests on multiple specimens for the same materials. Nevertheless, due to the particulate nature of the
granular materials, it is almost impossible to reconstruct specimens that have identical microstructural attributes.
In this work, we propose a different approach that only requires one mechanical test to identify all material pa-
rameters. Instead of using multiple tests to constitute sufficient constraints to identify all material parameters, the new
approach uses tomographic imaging techniques to track the spatial variability of the void ratio and deformation and
uses this kinematics information as the additional constraints for the objective function. One key assumption we made
here is that the spatial relative density or porosity variation is the dominating factorfor the post-bifurcation responses,
but the material parameter variation is negligible. This approximated approach has been found to be successful in
yielding consistent macroscopic constitutive responses and bifurcation modes for sand specimen that evolves with
complex bifurcation modes. In particular, Borja et al. (2013) found that by taking account only the relative density
Volume 14, Issue 4, 2016
398 Wang et al.
variation, one may correctly predict the onset of multiple shear bands at the bifurcation points, the sequential interac-
tion mechanisms of these shear bands, and the persistent shear band that lasts till the end of the loading program.
The calibration of the micropolar hypoplasticity model is completed by finding the optimal values of the material
parameters that minimize the discrepancy between the simulated results and the experimental data via iterations.
How “optimal” the values of the material parameters are is determined by the specific objective function used for the
inverse problem. If a global minimum of the objective function exists, then an optimization algorithm may start from
differential initial guess but find the same set of optimal material parameters. In this work, we use a gradient-based
method to find the optimal values of the material parameters. As a result, multiple boundary value problems are run
and the results and their corresponding gradient with respect to the parametric space are computed. The objective
function we used takes the following form:
f(x) =
wiri[si(x), di]2,(9)
where xis the vector of all material parameters, Nis the number of available experimental data for calibration, diis
the ith data point, wiis its weight, and si(x)is the corresponding result obtained from model simulation using the
parameter set x.riis the residual between the simulation data si(x)and the experimental data di. In this work, we
normalize the residuals by defining them as the relative errors, i.e.,
ri[si(x), di] = si(x)di
since the laboratory data we employed for parameter calibration include the stress ratio, volumetric strain, and void
ratio, which are not in the same order.
In this work, we employ the Dakota analysis toolkit which provides flexible interface between simulation codes
and robust optimization algorithms (Adams et al., 2009). Among the available methods we choose the gradient-based
least-squares algorithm (NL2SOL) which minimizes f(x)using its gradient:
f(x) = J(x)r(x), Jij =∂ri
and approximates the Hessian matrix with Gauss–Newton method:
2f(x) = J(x)2.(12)
The parameter calibration procedure requires an input file specifying the chosen optimization algorithm, the pa-
rameters to be determined, their initial values, upper and lower limits, and the experimental data set to be compared.
At the beginning of each iteration, Dakota passes the current guess of the material parameters to the finite element
simulation code developed for this study. The focus of this research is not the development of state-of-the-art opti-
mization algorithm to identify material parameters. Interested readers may refer to, for instance, Mahnken and Stein
(1996), Cooreman et al. (2007), Adams et al. (2009), Hu and Fish (2015) for details.
The finite element code then runs the simulations using the material parameters provided by Dakota and returns
the simulation results back to Dakota to compute the discrepancy measured by the objective function, and the cor-
responding gradient. Based on these inputs, Dakota then determines the next iteration of material parameters and
repeats the gradient estimation procedure and the discrepancy is again measured after another run of the simulation
code (Adams et al., 2009). This procedure repeats until the discrepancy is below a pre-defined tolerance or when the
number of iterations reaches its maximum.
The Dakota calibration scheme is first applied to calibrate a unit cell represented by a tri-linear 8-node quadrilateral
element representing the entire granular sample. This test is conducted under the additional assumptions that (1)
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 399
the specimen is composed of an effective medium of homogeneous properties inferred from the real heterogeneous
specimen and (2) the deformation remains homogeneous. With these additional assumptions, the inverse problem
is simplified such that only macroscopic responses from the experiment are utilized in the objective function [as
demonstrated previously in Liu et al. (2016)]. The target data for curve-fitting are the stress ratio s (σ1/σ3)lab and
volumetric strain elab
valong the triaxial compression test. As a result, the objective function reads
f(x) =
vj(x), elab
where Nσ1/σ3and Nevare the number of stress ratio data and volumetric strain data, respectively. Both data have 30
data points and the weights are equal to 1. The subscript idenotes the ith data point. The unit cell and the boundary
conditions are presented in Fig. 7. The initial void ratio and the minimum void ratio are set to be 0.6, which is the
average value of the sample. Note that there is no micropolar effect in this single element example, thus the material
length ldoes not affect the macroscopic response. It is set to be equal to the element size 10 mm. The parameters to
be identified via Dakota are C1,C2,C3,C4,ec, and a.
The Dakota calibration procedure takes in total 92 evaluations, of which 72 evaluations are performed for deter-
mination of the gradient of the 6 material parameters, while the remaining 20 evaluations are making guesses based
on the gradients. To demonstrate the convergence of the material parameters, the trial material parameters and the cor-
responding values of the objective function are presented in Table 1. The macroscopic responses obtained by the trial
material parameters compared with the experimental data are shown in Fig. 8. Despite the large discrepancy between
the initial guess and the laboratory results, f(x)decreases rapidly: it is reduced by about 95.7% after 50 evaluations.
This example demonstrates the robustness of the NL2SOL scheme in Dakota for nonlinear models and least-squares
problems for which the residuals do not tend to vanish.
FIG. 7: Domain and boundary condition for single unit cell calibration
TABLE 1: Evolution of the material parameters during the Dakota calibration procedure
C1C2C3C4eca f(x)
Initial guess 33.33 104.61 336.44 105.90 0.800 0.900 143.77
Evaluation 20 31.63 62.44 484.71 107.15 0.650 0.905 97.60
Evaluation 50 40.60 1054.35 1565.66 138.86 0.651 0.923 6.18
Calibration result 63.14 1831.49 2563.92 237.52 0.732 0.848 2.85
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400 Wang et al.
FIG. 8: Macroscopic stress ratio (left) and volumetric strain (right) responses from selected value sets of material
parameters in unit cell calibration
To analyze how the spatial variability of porosity (or relative density) affects the macroscopic responses, we recon-
struct a detailed 3D numerical specimen with the exact geometry and porosity distribution of the laboratory specimen
using the data extracted from micro-CT image analysis. The process of converting the micro-CT experimental data
into numerical specimen is illustrated in Fig. 9. After obtaining the images from the X-ray CT scan, the position and
effective diameter of each grain are recorded. Three micro-CT images taken at initial (0% axial strain), peak (6%
axial strain) and residual (15% axial strain) stages are used. The boundary particles are identified and thus the outer
boundary of the 3D specimen can be extrapolated from the position of these particles. Following this step, the domain
of the specimen is discertized by finite element and the void ratio of each finite element is calculated by established
the total solid volume using the positions and effective diameters of the particles, as shown in Fig. 9.
All micro-CT based finite element simulations are performed on the numerical specimen with identified initial
geometry and initial void ratio distribution. Here we adopt the hypothesis that the dominating factor that governs the
transition from compressive to dilatant behavior of granular materials is the relative density or porosity, the same
simplification used in Borja et al. (2013). As a result, the material parameters C1,C2,C3,C4,ec,a, and lare assumed
to be homogeneous within the specimen, while the spatial variation of the void ratio inferred from micro-CT image
of the initial configuration is incorporated to study the effect of the spatial variation of voidratio. Note that the results
of the inverse problem depend on how the boundary conditions are applied in model simulations. In this study, these
conditions are defined based on the experimental setup with assumptions and simplifications. For example, because
of the complexity of the interaction between the Hostun sand sample and the loading pistons of the triaxial cell, the
top and bottom surfaces of the specimen are not fully constrained, neither in terms of the transnational nor rotational
degrees of freedom (as shown in Fig. 4). In particular, we observed that the loading plates placed on the top of the
specimen has slid. In this study, the authors assume that the nodal displacements on the top surface of the specimen
are totally constrained, while the bottom surface is compressed under a constant strain rate in the Z direction and all
nodes at the bottom boundary have the same vertical displacement in the XY plane in order that the surface area does
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 401
FIG. 9: Construction of the numerical sample and void ratio distribution from micro-CT experimental data at the
initial stage (0% axial strain) of the triaxial compression test
not change. This constraint is applied via the Lagrange multipliers. Meanwhile, the rotations on both surfaces are
prohibited. The lateral surface of the FEM model is under constant confining pressure of 100 kPa and is free to rotate,
neglecting the effect of the rubber membrane in the testing apparatus.
The Dakota calibration procedure of the material parameters is carried out for two inverse problems. In the first
problem (Case A), only the macroscopic responses serve as the constraints for material parameters. The second prob-
lem (Case B) takes into account, in additional to the macroscopic constitutive responses, the local void ratio developed
at the peak and residual stages, thus adopting information from micro-CT images as additional target. The weight of
the void ratio data in Case B is intentionally set larger than the macroscopic data such that the microstructural evolu-
tion can be compatible. These two extreme cases are studied to separate the influence of either macroscopic behavior
or meso-scale behavior on the parameter calibration. In the third numerical experiments, we introduce a new mul-
tiscale objective function that takes account of both macroscopic and meso-data in a more balanced way. We then
reuse the calibrated material parameter sets from Case A and Case B as the initial guesses for the restarted material
parameter identification procedure to study the sensitivity of the calibration procedure.
5.1 Case A: Results from Macroscopic Objective Function
The objective function for Case A is the same as Eq. (13), which only consists of macroscopic stress ratio and volu-
metric strain data. Both data types contain 30 data points, thus their weights are identical. Unlike elasto-plastic models
for granular materials, the micropolar hypoplastic constitive model does not separate elastic and plastic parameters.
Thus the material parameters are calibrated simultaneously, not in a stepwise manner (Ehlers and Scholz, 2007). The
initial guess of the material parameters and the calibrated results in Case A is presented in the Table 2. The macro-
scopic responses obtained by the parameter sets from initial guess, the 20th evaluation, the 50th evaluation, and the
final calibration result are compared in Fig. 10. The evolution of the curves shows that the iterations converge to the
final solution that minimizes the objective function. However, the local void ratio distribution does not converge to the
actual experiment data. This is shown in Fig. 11 where the experimental data of void ratio map in cross-section YZ
and the relative error map [defined in Eq. (10)] computed from simulations are presented. Since the sample geometry
and void ratio distribution are not included in the objective function of Case A, the calibration procedure does not
correct the void ratio discrepancy with the initial guess and leads to a numerical solution that a dominant shear band
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402 Wang et al.
TABLE 2: Calibration of material parameters of entire sample using Case A: only macroscopic responses; Case
A1: equal weights of stress ratio, volumetric strain and local void ratio data, starts from results of Case A; Case B:
macroscopic responses and local void ratio distributions; Case B1: equal weights of stress ratio, volumetric strain and
local void ratio data, starts from results of Case B
Number of C1C2C3C4eca l
Initial guess 68.00 767.60 2742.70 257.50 0.650 0.980 0.200 mm
Case A 74 70.57 832.40 2524.10 261.70 0.637 0.960 0.468 mm
Case A1 30 69.67 1372.67 2075.62 251.90 0.641 0.976 0.364 mm
Case B 117 67.21 920.08 2312.79 259.45 0.636 0.971 0.977 mm
Case B1 30 67.91 1229.29 2244.82 262.41 0.6358 0.970 0.979 mm
FIG. 10: Stress ratio and volumetric strain responses of full sample simulation during the calibration procedure using
only macroscopic responses (Case A)
is formed inside the sample, while the actual specimen developed a “barrel” shape that exhibit diffusive bands (Ikeda
et al., 2003).
In the correction step (Case A1), we modify the objective function used as previous calibrated material parameters
set as an initial guess. This modified objective function incorporates additional terms to constrain the material param-
eter set such that the numerical specimen also exhibits the same peak and residual shear strength due to the last two
terms in (14), which reads
f2(x) = 1
wjrjevj(x), elab
k(x), epeaklab
m(x), eresiduallab
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 403
(a) Void ratio distribution from Micro-CT images
(b) Distribution of residuals [defined in Eq. (10)] at selected steps of the Dakota calibration in Case A
FIG. 11: Relative error of local void ratio distribution between full sample simulation and micro-CT data (shown in
cross section in plane YZ) during Dakota calibration procedure using Case A: only macroscopic responses
where eiare void ratio in element i,Nelement is the number of elements in the FEM model. The contributions of stress
ratio, volumetric strain, local void ratio data are balanced by the number of data points of each type. The weights of
each data type equal to 1.
5.2 Case B: Results from Multiscale Objective Function
The objective function for Case B takes the form
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404 Wang et al.
f2(x) =
rievi(x), elab
i(x), epeaklab
i(x), eresiduallab
In this function, the residuals of all available experimental data point are treated equally in the objective function.
Since the number of elements is much larger than the number of stress ratio and volumetric strain data, the objective
function Eq. (15) has much larger weight on micro-CT data than the macroscopic responses. The calibration procedure
in Case B employs the same initial guess of the material parameters previously used in Case A. The calibrated results
are also shown in Table 2 to show comparisons. Compared to other hypoplastic material parameters, the material
length parameter l, which accounts for the micropolar effect in the model, varies more significantly when experimental
data of local void ratio are included in the least square problem. The macroscopic responses obtained by the parameters
sets from initial guess, the 20th evaluation, the 50th evaluation, and the final calibration result are compared in Fig. 12.
Although the volumetric strain response approaches the experiment data along the iterations, the macroscopic stress
ratio response deviates from the experimental response in the sense that the peak stress and residual stress do not
coincide and the softening phenomenon is not apparent. As for the meso-scale data shown in Fig. 13, the calibrated
parameters lead to a deformed configuration much closer predication to actual specimen geometry than that of the
Case A results shown in Fig. 11. In particular, the Case B calibrated simulation correctly predicts the increased of
porosity at the middle of specimen, which is consistent to the observation in the laboratory. The Case A calibrated
simulation leads to porosity increase highly concentrated in a single persistent anti-symmetric shear band that was not
triggered in the actual experiment. This observation indicates the necessity of including micro-structural information
in material parameter identification procedures. Note that the relative errors of void ratio near the top and bottom
surfaces, unlike the central areas, are not significantly reduced during the Dakota calibrations. This is because the
boundary conditions in model simulation does not perfectly represent the experimental setup.
The objective function Eq. (14) is again adopted to perform the correcrtion step from the calibrated results
(Case B1). The correction is made by using the equilibrium weights of different types of experimental data. The
FIG. 12: Stress ratio and volumetric strain responses of full sample simulation during the calibration procedure using
macroscopic responses and local void ratio distribution (Case B)
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 405
Distribution of residuals at selected steps of the Dakota calibration in Case B
FIG. 13: Relative error of local void ratio distribution between full sample simulation and micro-CT data (shown in
cross section in plane YZ) during Dakota calibration procedure using Case B: macroscopic responses and local void
ratio distribution
results are recapitulated in Table 2 and the macroscopic responses in different cases are compared in Fig. 14. Recall
that in the correction step Case A1, the prediction of peak stress has been improved. In this case, the discrepancy
between the model response and experimental data in Case B1 has not been improved or even changed significantly
in the correction step, which indicates that both objective functions Eqs. (15) and (14) havesimilar local minimizers.
5.3 Discussion
Two calibration strategies are employed in this study. The resultant material parameters and calibrated simulations
are analyzed. In the first strategy, we find material parameter that allows the finite element simulations to replicate
the macroscopic responses as close as possible, but neglect all meso-scale information provided by the micro-CT
images. This optimized material parameter set (optimized in terms of macroscopic responses only), are then used as
the initial guess of the next inverse problem. Following this predictor step, another inverse problem is defined by a
new multiscale objective function that takes account of both the macroscopic data and local void ratio properties used
for calibration. This approach mimics the idea in Ehlers and Scholz (2007) for determining material parameters for
micropolar constitutive laws. The major departure here is the usage of micro-CT image and the elimination of the need
to use multiple experimental tests to generate constraints for the objective function. To analyze the importance of the
initial guess and whether a global optimal value for the material parameter set exists, we employ another alternative
strategy in which the meso-scale information is used right at the predictor step. Then, the same multiscale objective
function used in the corrector step [i.e., Eq. (14)] is used to balance the weights of global and local data.
5.3.1 Comparisons of Results
At the first look, the approach that starts with calibrating macroscopic parameter seems to be better in terms of replicat-
ing compatible shear stress history as shown in Fig. 14(a), even though both calibrated finite element simulations yield
Volume 14, Issue 4, 2016
406 Wang et al.
FIG. 14: Stress ratio and volumetric strain responses of full sample simulation during the calibration procedure using
Case A: only macroscopic responses; Case A1: equal weights of stress ratio, volumetric strain, and local void ratio
data, starts from results of Case A; Case B: macroscopic responses and local void ratio distribution; Case B1: equal
weights of stress ratio, volumetric strain, and local void ratio data, starts from results of Case B
similar volumetric responses. In particular, the second approach is unable to capture the macroscopic peak and resid-
ual shear stresses at the predictor step, and again fails to make any significant improvement in capturing the peak and
residual shear strength after switching to the multiscale objective function, as shown in Fig. 14. As a result, evidence
provided in the macroscopic responses seems to favor the staggered approach similar to the one proposed in Ehlers
and Scholz (2007) in which the calibration process begins with an inverse problem that first curve-fit macroscopic
behaviors, followed by a correction step that uses multiscale objective function to enforce consistency of kinematics.
However, a closer look at the deformed configuration and the meso-scale responses may lead to an opposite
conclusion. In particular, we find that the weight modification approach that begins with calibrating meso-scale in-
formation from micro-CT images actually yields the experimentally observed bifurcation mode, while the macro-
then-microscopic approach does not. In Case A, l= 0.468 mm, which is close to the mean particle diameter d50 =
0.338 mm, a persistent shear band has developed in the specimen. This persistent shear band is non-symmetric and has
not been observed in the micro-CT images captured during the drained triaxial compression test. In Case B, however,
l= 0.977 mm, which is 2.5 times larger than the mean particle diameter and the barrel-shaped deformed specimen
develops a barrel deformed configuration followed by the development of an X-shaped shear localization zone. These
kinematic features are consistent with what is observed in the physical experiment, even though the Case B simulation
does not replicate the shear stress responses as close as the Case A simulation does.
In other words, the set of material parameters that leads to the best replica of macroscopic responses observed in
laboratory does not necessarily yield the correct bifurcation mode. As a result, the reasonable strategy is to design
an objective function that acts as a compromise between matching macroscopic responses and maintaining consistent
kinematics at meso-scale level. Furthermore, the notable difference in the macroscopic and microscopic responses
predicted by the staggered approach and weight modification approach indicates that the calibration exercise is highly
path-dependent and multiple local minima are likely to exist, thus making it difficult to find the global minimum point
for the multiscale objective function in the parametric space.
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Identifying Micropolar Material Parameters via Micro-CT Images 407
5.3.2 Source of Error
The discrepancy between the experimental data and simulation results may also be attributed to the idealized bound-
ary condition applied on numerical specimen. In particular, the specimen-loading-plate interaction between the sample
and the loading pistons as well as the elastic membrane are not accurately and explicitly modeled (Albert and Rud-
nicki, 2001). Moreover, the assumption that material parameters are homogeneous throughout the specimen may also
oversimplify the spatial variability of the physical specimens. Finally, the micropolar finite element model is formu-
lated using the Jaumann rate of the Cauchy stress. The Jaumann rate is an objective rate which is suitable for materials
in the geometrical nonlinear regime with small strain and large rotation. However, some previous works, such as
Molenkamp (1986), have pointed out that the Jaumann rate should be avoided for problems with large deviatoric
strain. These limitations will be considered in future studies.
5.3.3 Length Scale, Higher-Order Kinematics and Bifurcation Modes
The higher-order quantities, namely the curvature κand the coupled stress µ, can be computed from the micropolar
model simulation, while this information is not available from micro-CT images. In this study, two types of shear bands
are encountered with different material parameters in the finite element simulations even though both simulations
began with the same initial geometry and void ratio distribution. In Case A and B, the calibrated material length l
varies significantly, compared to other material parameters.
In both cases, the distribution of the norm of the couple stress tensor |µ|=µ2
21 +µ2
32 +µ2
31 and the norm of the
curvature tensor |κ|=κ2
21 +κ2
32 +κ2
31 are shown in Figs. 15 and 16, respectively. The coupled stress and curvature
localize in the transition zone between the shear localization region and homogeneously deformed region, suggesting
that the micropolar effect becomes very important when high-gradient deformation occurs in granular materials. This
observation is consistent with numerical experiments conducted with discrete element method (Ehlers et al., 2003;
Oda and Iwashita, 2000; Wang and Sun, 2016a,b).
The evolution of shear bands simulated in Case A and Case B is illustrated in Fig. 17 using color map of |µ|, as
well as in Fig. 18 using color map of |κ|. At the beginning of the triaxial loading, two-axes symmetric localization
patterns emerge for both simulations (Ikeda et al., 2003). After the peak stress of Case A (at 3% axial strain), the
pattern bifurcates to an asymmetry pattern: one shear band becomes stronger than the other. Upon further loading, the
dominant shear band becomes persistent and the other weak band gradually dies out. As for Case B, the two bands
compete with each other along the deformation but neither predominates. The initial pattern bifurcates to bilateral
symmetry so that the symmetry with respect to the horizontal axis is broken. The diffuse mode is preserved to the end
(a) Case A (b) Case B
FIG. 15: Norm of the coupled stress tensor |µ|=µ2
21 +µ2
32 +µ2
31 for results of inverse problems A and B at the
residual stage
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408 Wang et al.
(a) Case A (b) Case B
FIG. 16: Norm of the curvature tensor |κ|=κ2
21 +κ2
32 +κ2
31 for results of inverse problems A and B at residual
FIG. 17: Evolution of shear localization in cross section YZ of calibraion results of Case A and Case B, illustrated in
the norm of coupled stress
of the loading. This analysis shows that, even though the spatial variability of the material parameters is neglected,
the numerical values of the material parameters, particularly the material length l, still impose strong effects on the
failure mode of numerical specimen.
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Identifying Micropolar Material Parameters via Micro-CT Images 409
FIG. 18: Evolution of shear localization in cross section YZ of calibraion results of Case A and Case B, illustrated in
the norm of curvature tensor
5.3.4 Critical State
The anisotropy density function fdof Eq. (3) is an indicator of the critical state, a condition at which particulate
materials keep deforming in shear at constant void ratio and stress (Casagrande, 1936; Roscoe et al., 1958). In the
micropolar hypoplasticity adopted in this study, the local void ratio approaches the critical void ratio ec,fdapproaches
1. The distribution of fdof the two numerical specimens captured at residual stage of Cases A and B are presented in
Fig. 19. Since the different calibrated material parameters are used in Case A and B, the failure modes as well as the
locations where the Hostun sand first reaches the critical state are also different. In Case A, the elements residing in
the persistent shear band are close to the critical state, while the elements outside the zone are not close to it. In Case
B, the pattern of fdalso coincides with the diffuse failure mode presented in Fig. 17, showing that the unit cells inside
the shear band are also closer to critical state than the host matrix, but the difference between the numerical value
of fdwithin the host matrix and inside the shear band are less significant than Case A. Both findings are consistent
with Tejchman and Niemunis (2006); Wang and Sun (2016b) where the strain localization triggered by the material
bifurcation in dense assemblies tends to have the void ratio approaching its critical value locally, but the specimen
itself does not necessarily reach critical state globally. Comparing this observation with the coupled stress norm shown
in Fig. 17, we observed that the shear band is not only much closer to the critical state, but also has significantly lower
coupled stress magnitude. This result is consistency with the norm of the curvature tensor shown in Fig. 18 where
the specimen has significant amount of micro-rotation at the boundary of the shear band and the host matrix but the
micro-polar kinematics is not significant inside the shear band and in the host matrix.
In this work, we incorporate information obtained from both macroscopic measurement and meso-scale kinematics to
analyze the sensitivity of the predicted characteristic length and mechanical responses of a 3D micropolar hypoplastic-
ity finite element model. To the best knowledge of the authors, this is the first contribution that incorporates micro-CT
Volume 14, Issue 4, 2016
410 Wang et al.
FIG. 19: Distribution of fdfor results of inverse problems A and B at the residual stage. fd= 1 indicates that the
material is in critical state.
images and multiscale objective function into the material parameter identification procedure for micropolar plasticity
for granular materials.
The results show that the incorporation of meso-scale information may significantly change the predicted length
scale obtained from the inverse problems and leads to different bifurcation modes in the finite element simulations.
Even though similar macroscopic responses are observed from simulations conducted with single-scale and multi-
scale objective functions, the macroscopic responses in the former case may yield meso-scale responses that are not
consistent with those of the real specimen. As a result, the apparently good curve-fitting of macroscopic response
is not a good indicator of forward prediction capacity. This result has important implications for the validations of
grain-scale simulation tools (such as discrete element, lattice-beam, and lattice spring models) in which macroscopic
stress–strain responses are often the only experimental data available for benchmark and validations. The numerical
results, particularly the difference of the failure modes obtained from different objective functions, indicate that using
macroscopic stress–strain curve alone to evaluate or validate grain-scale simulations is neither productive nor reliable.
Although the higher-order quantities, the curvature κand the coupled stress µ, are not available from micro-CT
images, they are computed from the micropolar simulations with the material parameters optimized for different ob-
jective functions. Depending on which set of material parameters is employed, the numerical specimen may either
develop a persistent shear band, which is not observed in the experiment, or a diffuse failure mode, which is observed
from micro-CT images. Comparisons between the simulation results with micro-CT images suggest that a staggered
predictor–corrector procedure that first employs the macroscopic objective function to curve-fit the macroscopic re-
sponses, then use the multiscale objective function to enforce kinematic constraints seem to yield a more compatible
macroscopic constitutive response with the experimental counterpart, butthe material parameters that lead to the best
curve-fitting macroscopic responses also lead to an incorrect bifurcation mode. This finding is alerting, as the material
parameters that lead to the wrong bifurcation mode in the backward calibration exercise is also likely to generate
even more unrealistic forward prediction. The apparently good match in the macroscopic curve can be misleading and
generate a false sense of confidence for the numerical model. This is a noteworthy concern, as there is an alerting
trend in which the forward-predictive capacity of grain-scale models are often incorrectly measured by how well they
curve-fit the macroscopic stress–strain curve, rather than how well they are able to generate compatible and consistent
mechanical behaviors across length scales. The issue associated with this calibration approach is not apparent when
calibration is conducted at the unit cell level in which only homogeneous deformation is considered. However, when
macroscopic stress–strain curve is used to calibrate meso-scale or grain-scale models, the dimensions of the paramet-
ric space can be larger than the number of constraints provided by the macroscopic responses. The insufficiency of
constraints then makes it possible to generate simulations that apparently match the macroscopic calibration with a
International Journal for Multiscale Computational Engineering
Identifying Micropolar Material Parameters via Micro-CT Images 411
completely inconsistent microstructure. By incorporating microscopic information from micro-CT images to calibrate
material parameters, this research provides important evidence to suggest that constraining micro-mechanical model
to match macroscopic responses is not sufficient. Nor is it a meaningful way to measure the quality of numerical
predictions. These lessons are important for the calibration and validation of high-order and multiscale finite element
This research is supported by the Earth Materials and Processes program at the US Army Research Office under
grant contract W911NF-14-1-0658 and W911NF-15-1-0581, the Mechanics of Material program at National Science
Foundation under grant contracts CMMI-1462760 and EAR-1516300, and Provost’s Grants Program for Junior Fac-
ulty who Contribute to the Diversity Goals of the University at Columbia University. These supports are gratefully
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... Macroscopic constitutive models designed to capture historydependent constitutive responses can be categorized into multiple families, such as hypoplasticity, elastoplasticity, and generalized plasticity [74][75][76][77]. For example, hypoplasticity models often do not distinguish the reversible and irreversible strain [8,78,79]. Unlike the classical elastoplasticity models where the plastic flow is normal to the stress gradient of the plastic potential and the evolution of it is governed by a set of hardening rules [80][81][82][83], hypoplasticity models do not employ a yield function to characterize the initial yielding. Instead, the relationship between the strain rate and the stress rate is captured by a set of evolution laws originated from a combination of phenomenological observations and physics constraints. ...
... These descriptors, such as volume fraction of void, dislocation density, degradation function, slip system orientation, and shape factor are often incorporated as state variables in a system of ordinary differential equations that leads to the constitutive responses at a material point. Classical examples include the family of Gurson models in which volume fraction of void is related to ductile fracture [144][145][146][147][148], critical state plasticity in which porosity and over-consolidation ratio dictates the plastic dilatancy and hardening law [79,82,[149][150][151][152], and crystal plasticity where activation of slip system leads to plastic deformation [153][154][155]. In those cases, a specific subset of descriptors is often incorporated manually such that the most crucial deformation mechanisms for the stress-strain relationships are described mathematically. ...
... Directly predicting the plastic flow from phenomenological observations between the evolution of hand-picked microstructural descriptors and the resultant macroscopic plastic flow is not a new idea. In fact, this treatment is commonly used in generalized plasticity models (Pastor et al., 1990;Dafalias and Manzari, 2004;Wang et al., 2016;Sun, 2013;Liu et al., 2016a). The key innovation here is the replacement of the hand-picked descriptors with a generic encoded feature vector approach that can efficiently reduce the dimensions of the topological information without losing the information that might otherwise compromise the accuracy of predictions. ...
The history-dependent behaviors of classical plasticity models are often driven by internal variables evolved according to phenomenological laws. The difficulty to interpret how these internal variables represent a history of deformation, the lack of direct measurement of these internal variables for calibration and validation, and the weak physical underpinning of those phenomenological laws have long been criticized as barriers to creating realistic models. In this work, geometric machine learning on graph data (e.g. finite element solutions) is used as a means to establish a connection between nonlinear dimensional reduction techniques and plasticity models. Geometric learning-based encoding on graphs allows the embedding of rich time-history data onto a low-dimensional Euclidean space such that the evolution of plastic deformation can be predicted in the embedded feature space. A corresponding decoder can then convert these low-dimensional internal variables back into a weighted graph such that the dominating topological features of plastic deformation can be observed and analyzed.
... Furthermore, bypass-35 ing the use of constitutive laws also eliminates the calibration process. For materials that exhibit complex 36 physics and/or do not possess symmetry, it becomes necessary to introduce more material parameters in 37 order to predict the material responses with sufficient precision [Ehlers and Scholz, 2007, Wang et al., 2016, 38 Liu et al., 2016. On-the-fly multiscale calculations, such as hierarchical upscaling methods (e.g., FEM 2 ) 39 Sun, 2016, Matouš et al., 2017] or concurrent multiscale domain coupling methods [Hughes 40 et al., 1998, Sun and Mota, 2014, Sun et al., 2017, Badia et al., 2008 may eliminate the need to compose con-41 stitutive laws that captures the responses of the effective medium through computational homogenization. ...
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This article introduces a manifold embedding data-driven paradigm to solve small-and finite-strain elasticity problems without a conventional constitutive law. This formulation follows the classical 6 data-driven paradigm by seeking the solution that obeys the balance of linear momentum and compatibility conditions, while remaining consistent with the material data through minimizing a distance measure. Our key point of departure is the introduction of a global manifold embedding as a means to learn the geometrical trend of the constitutive data mathematically represented by a smooth manifold. By training an invertible neural network to embed the data of an underlying constitutive manifold onto a Euclidean space, we reformulate the local distance-minimization problem that requires a computationally intensive combinatorial search to identify the optimal data points closest to the conservation law with a cost-efficient projection step. Meanwhile, numerical experiments performed on path-independent elastic materials of different material symmetries suggest that the geometrical inductive bias learned by the neural network is helpful to ensure more consistent predictions when dealing with data sets of limited sizes or those with missing data.
... Nevertheless, one key aspect that is often overlooked in the phase field fracture modeling is that materials with internal structures that enables size effects on damage and fracture may likely exhibit size effect in the elastic regimes. Examples of these materials include concrete, composite, particulate materials as well as some metamaterials [361][362][363][364][365]. Since these materials exhibit large internal length scales compared to the length scale of damage or fracture, suitable size effect must be carefully incorporated in both the elastic and pathdependent regimes to capture the size-dependence properly. ...
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Global challenges associated with extreme climate events and increasing energy demand require significant advances in our understanding and predictive capability of coupled multi- physical processes across spatial and temporal scales. While classical approaches based on the mixture theory may shed light on the macroscopic poromechanics simulations, accurate forward predictions of the complex behavior of phase-changing geomaterials cannot be made without understanding the underlying coupling mechanisms among constituents at the microstructural scale. To precisely predict the multi-physical behaviors originated by smaller scales, fundamental understandings of the micromechanical interactions among phase constituents are crucial. Hence, this dissertation discusses mathematical and computational frameworks designed to capture coupled thermo-hydro-mechanical-fracture processes in phase-changing porous media that incorporate necessary microscopic details. To achieve this goal, this dissertation aims to introduce a practical way to investigate how phase transition and evolving microstructural attributes at small scales affect the applicability of meso- or macroscopic finite element simulations, by leveraging the phase field method to represent the regularized interfaces of phase constituents. Firstly, a multi-phase-field microporomechanics model is presented to model the growth and thaw of ice lenses. In specific, we extend the field theory for ice lens that is not restricted to one-dimensional space. The key idea is to represent the state of the pore fluid and the evolution of freezing-induced fracture via two distinct phase field variables coupled with balance laws, which leads to an immersed approach where both the homogeneous freezing and ice lensing are distinctively captured. Secondly, a thermo-hydro-mechanical theory for geological media with thermally non-equilibrated constituents is presented, where we develop an operator-split framework that updates the temperature of each constituent in an asynchronous manner. Here, the existence of an effective medium is hypothesized, in which the constituents exhibit different temperatures while heat exchange among the phases is captured via Newton’s law of cooling. Thirdly, an immersed phase field model is introduced to predict fluid flow in fracturing vuggy porous media, where crack growth may connect previously isolated voids and form flow conduits. In this approach, we present a framework where the phase field is not only used as a damage parameter for the solid skeleton but also as an indicator of the pore space, which enables us to analyze how crack growth in vuggy porous matrix affects the flow mechanism differently compared to the homogenized effective medium while bypassing the needs of partitioning the domain and tracking the moving interface. Finally, we present a new phase field fracture theory for higher-order continuum that can capture physically justified size effects for both the path-independent elastic responses and the path-dependent fracture. Specifically, we adopt quasi-quadratic degradation function and linear local dissipation function such that the physical size dependence are insensitive to the fictitious length scale for the regularized interface, which addresses the numerical needs to employ sufficiently large phase field length scale parameter without comprising the correct physical size effect.
... Numerous combinations of microstructures and constitutive laws are widely applied to match constitutive responses during the calibration process, which makes use of just a few parameters, owing to data scarcity. Friedman (1997) refers to these arbitraries as the "curse of dimensionality", in which calibrated DEM models are unable to provide accurate prediction [126][127][128]. Hanaor et al. (2013) developed specialized particle geometries for 3D printing by combining three different shape descriptors: fractal surface overlay, directed polyhedral aggregation, and contour rotation interpolation [129]. ...
Full-text available
The study of soil–structure interface behavior contributes to the fundamental understanding of engineering performance and foundation design optimization. Previous research studies the effect of soil characteristics and surface roughness property on the soil–material interface mechanism via interface shear test. The reviews utilizing past established laboratory studies and more recent tests based on state-of-the-art technologies reveal that surface roughness significantly affects interface shear performances in the studies of soil–structure interactions, especially in peak shear strength development. A preliminary but original investigative study by the authors was also carried out using a sophisticated portable surface roughness gauge to define the material surface roughness properties in order to study the interface behavior parametrically. Additionally, using the authors’ own original research findings as a proof-of-concept innovation, particle image velocimetry (PIV) technology is applied using a digital single-lens reflex (DSLR) camera to capture sequential images of particle interactions in a custom-built transparent shear box, which validate the well-established four-stage soil shearing model. The authors also envisaged that machine learning, e.g., artificial neural network (ANN) and Bayesian inference method, amongst others, as well as numerical modeling, e.g., discrete element method (DEM), have the potential to also promote research advances on interface shear mechanisms, which will assist in developing a greater understanding in the complex study of soil–structure interactions.
... While this conventional workflow has led to numerous improvements in modeling, the more sophisticated models are often inherently harder to tune due to the expansion of parametric space. This expansion does not only make it less feasible to determine the optimal mathematical expressions through a manner trial-and-error effort (even after the causality of the yielding and hardening is known [9]), but also require a more complicated inverse problems to identify the material parameters [10,11,12]. ...
Full-text available
Conventionally, neural network constitutive laws for path-dependent elasto-plastic solids are trained via supervised learning performed on recurrent neural networks, with the time history of strain as input and the stress as input. However, training a neural network to replicate path-dependent constitutive responses require significantly more amount of data due to path dependence. This demand on diverse and abundance of accurate data, as well as the lack of interpretability to guide the data generation process, could become major roadblocks for engineering applications. In this work, we attempt to simplify these training processes and improve the interpretability of the trained models by breaking down the training of material models into multiple supervised machine learning programs for elasticity, initial yielding, and hardening laws that can be conducted sequentially. To predict pressure-sensitivity and rate dependence of the plastic responses, we reformulate the Hamliton-Jacobi equation such that the yield function is parametrized in product space spanned by the principle stress, the accumulated plastic strain, and time. To test the versatility of the neural network meta-modeling framework, we conduct multiple numerical experiments where neural networks are trained and validated against (1) data generated from known benchmark models, (2) data obtained from physical experiments, and (3) data inferred from homogenizing sub-scale direct numerical simulations of microstructures. The neural network model is also incorporated into an offline FFT-FEM model to improve the efficiency of the multiscale calculations.
Full-text available
Elastoplasticity models often introduce a scalar-valued yield function to implicitly represent the boundary between elastic and plastic material states. This paper introduces a new alternative where the yield envelope is represented by a manifold of which the topology and the geometry are learned from a set of data points in a parametric space (e.g. principal stress space, pi-plane). Here, deep geometric learning enables us to reconstruct a highly complex yield envelope by breaking it down into multiple coordinate charts. The global atlas that consists of these coordinate charts in return allows us to represent the yield surface via multiple overlapping patches, each with a specific local parametrization. This setup provides several advantages over the classical implicit function representation approach. For instance, the availability of coordinate charts enables us to introduce an alternative stress integration algorithm where the trial stress may project directly on a local patch and hence circumvent the issues related to non-smoothness and the lack of convexity of yield surfaces. Meanwhile, the local parametric approach also enables us to predict hardening/softening locally in the parametric space, even without complete knowledge of the yield surface. Comparisons between the classical yield function approach on the non-smooth plasticity and anisotropic cam-clay plasticity model are provided to demonstrate the capacity of the models for highly precise yield surface and the feasibility of the implementation of the learned model in the local stress integration algorithm.
This work looks into the passive failure of tunnel head in strain-softening soils modeled using a nonlocal constitutive model. The Drucker–Prager yield criterion and plastic potential are used, and an over-nonlocal approach is utilized to regularize the ill-posed boundary value problems emerging from material softening and strain localization inelasticity. The nonlocal variable is incorporated through a Helmholtz-type partial different equation (PDE), instead of nonlocal integration. A nonlocal implicit gradient enhancement is proposed, and its numerical implementation is discussed. The proposed methodology is verified to produce mesh-independent strain-softening responses accompanied by localized deformation. The passive failure analysis of tunnel head is conducted to study key parameters influencing plastic strain accumulations and ground surface deformation. It is revealed that the strain-softening mechanism can promote local concentrated passive deformation and substantial ground surface movement near the tunnel head. Increasing the friction angle increases face supporting force, but only affects the inelastic deformation in a minimal way. The increase in the dilation angle widens the extent of horizontal soil deformation and vertical ground deformation, while a larger cover depth suppresses the passive failure response of shallow tunneling.
Full-text available
This paper presents a combined experimental-modeling effort to interpret the coupled thermo-hydro-mechanical behaviors of the freezing soil, where an unconfined, fully saturated clay is frozen due to a temperature gradient. By leveraging the rich experimental data from the microCT images and the measurements taken during the freezing process, we examine not only how the growth of ice induces volumetric changes of the soil in the fully saturated specimen but also how the presence and propagation of the freezing fringe front may evolve the anisotropy of the effective media of the soil-ice mixture that cannot be otherwise captured phenomenologically in the isotropic saturation-dependent critical state models for plasticity. The resultant model is not only helpful for providing a qualitative description of how freezing affects the volumetric responses of the clayey material, but also provide a mean to generate more precise predictions for the heaving due to the freezing of the ground.
Full-text available
This paper introduces an explicit material point method designed specifically for simulating the micropolar continuum dynamics in the finite deformation and finite microrotation regime. The material point method enables us to simulate large deformation problems while circumventing the potential mesh distortion without remeshing. To eliminate rotational motion damping and loss of angular momentum during the projection, we introduce the mapping for microinertia and angular momentum between particles and grids through the affine particle-in-cell approach. The microrotation and the curvature at each particle are updated through zero-order forward integration of the microgyration and its spatial gradient. We show that the microinertia and the angular momentum are conserved during the projections between particles and grids in our formulation. We verify the formulation and implementation by comparing with the analytical dispersion relation of micropolar waves under the small strain and small microrotation, as well as the analytical soliton solution for solids undergoing large deformation and large microrotation. We also demonstrate the capacity of the proposed computational framework to handle a wide spectrum of simulations that exhibit size effects in the geometrical nonlinear regime through three representative numerical examples, i.e., a cantilever beam torsion problem, a fragment-impact penetration problem, and a micropolar fluid discharging problem.
Conference Paper
Full-text available
A micropolar discrete-continuum coupling model is proposed to link the collectively particulate mechanical simulations at high-order representative elementary volume to field-scale boundary value problems. By incorporating high-order kinematics to the homoge-nization procedure, contact moment and force exerted on grain contacts are homogenized into a non-symmetric Cauchy stress and higher-order couple stress. These stress measures in return become the constitutive updates for the macroscopic finite element model for micropolar continua. Unlike the non-lcoal weighted averaging models in which the intrinsic length scale must be a prior knowledge to compute the nonlocal damage or strain measures, the proposed model introduces the physical length scale directly through the higher-order kinematics. As a result, there is no need to tune or adjust the intrinsic length scale. Furthermore, since consti-tutive updates are provided directly from micro-structures, there is also no need to calibrate any high-order material parameters that are difficult to infer from experiments. These salient features are demonstrated by numerical examples. The classical result from Mindlin is used as a benchmark to verify the proposed model.
Full-text available
The present manuscript addresses the issue of computational complexity of optimizing nonlinear composite materials and structures at multiple scales. Several solutions are detailed to meet the enormous computational challenge of optimizing nonlinear structures at multiple scales including: (i) enhanced sampling procedure that provides superior performance of the well-known ant colony optimization algorithm, (ii) a mapping-based meshing of a representative volume element that unlike unstructured meshing permits sensitivity analysis on coarse meshes, and (iii) a multilevel optimization procedure that takes advantage of possible weak coupling of certain scales. We demonstrate the proposed optimization procedure on elastic and inelastic laminated plates involving three scales.
This paper proposes new expressions of the graph-theoretical analysis of granular assemblies. New mechanical quantities of void, i.e. void force, void displacement etc, are defined and relations between quantities of contact are given using loop matrix. Discrete Schaefer operators are introduced and the energy expression of a granular assembly is explained using new quantities and operators.
The objective of this research is to use grain-scale numerical simulations to analyze the evolution of stress anisotropy exhibited in wetted granular matter. Multiphysical particulate simulations of unsaturated granular materials were conducted to analyze how the interactions of contact force chains and liquid bridges affect macroscopic responses under various suction pressure and loading histories. To study how the formation and rupture of liquid bridges affect the mechanical responses of wetted granular materials, a series of suction-controlled triaxial tests were conducted with two grain assemblies, one composed of large particles of similar sizes, the other composed of a mixture of large particles with significant amount of fines. The results indicate that capillary stresses are anisotropic in both sets of specimens, and that stress anisotropy is more significant in granular assemblies filled with fine particles. A generalized tensorial Bishop's coefficient is introduced to analyze the connections between microstructural attributes and macroscopic responses. Numerical simulations presented in this paper indicate that the principal values and directions of this Bishop's coefficient tensor are path dependent.
Force chains within a granular material are the major load carriers responsible for the strength of a material. The authors study the spatial distribution of force chains across length scales in a three-dimensional (3D) assembly of spherical particles subject to simple shear. Specifically, the authors network sets of force chains by basing connectivity on the distance between chain particles. The authors quantify the connectivity of networked force chains at different length scales using ideas employed in landscape ecology. The authors find a geometric transition in force chain connectivity and observe a change in the character of this transition with respect to force chains restricted to the shear band. The spatial distribution of force chains in the shear band exhibits two dominant length scales: force chains are separated by lateral support, on average, by one particle diameter in one direction and two particle diameters in a second direction. The existence of two dominant length scales in the lateral support around those force chains within shear bands suggests improvements to predictive constitutive models by explicitly incorporating such variation in, for example, in-plane and normal to shear directions.
The thermo-hydro-mechanical (THM) coupling effects on the dynamic wave propagation and strain localization in a fully saturated softening porous medium are analyzed. The characteristic polynomial corresponding to the governing equations of the THM system is derived, and the stability analysis is conducted to determine the necessary conditions for stability for both non-isothermal and adiabatic cases. The result from the dispersion analysis based on the Abel-Ruffini theorem reveals that the roots of the characteristic polynomial for the thermo-hydro-mechanics problem can not be expressed algebraically. Meanwhile, the dispersion analysis on the adiabatic case leads to a new analytical expression of the internal length scale. Our limit analysis on the phase velocity for the non-isothermal case indicates that the internal length scale for the non-isothermal THM system may vanish at the short wavelength limit. This result leads to the conclusion that the rate-dependence introduced by multiphysical coupling may not regularize the THM governing equations when softening occurs. Numerical experiments are used to verify the results from the stability and dispersion analyses.