The MicroBooNE Collaboration ††thanks<EMAIL_ADDRESS>
# Search for an Excess of Electron Neutrino Interactions in MicroBooNE Using
Multiple Final State Topologies

P. Abratenko  Tufts University, Medford, MA, 02155, USA  R. An  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  J. Anthony  University
of Cambridge, Cambridge CB3 0HE, United Kingdom  L. Arellano  The University
of Manchester, Manchester M13 9PL, United Kingdom  J. Asaadi  University of
Texas, Arlington, TX, 76019, USA  A. Ashkenazi  Tel Aviv University, Tel Aviv,
Israel, 69978  S. Balasubramanian  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  B. Baller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  C. Barnes  University of Michigan,
Ann Arbor, MI, 48109, USA  G. Barr  University of Oxford, Oxford OX1 3RH,
United Kingdom  V. Basque  The University of Manchester, Manchester M13 9PL,
United Kingdom  L. Bathe-Peters  Harvard University, Cambridge, MA 02138, USA
O. Benevides Rodrigues  Syracuse University, Syracuse, NY, 13244, USA  S.
Berkman  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
A. Bhanderi  The University of Manchester, Manchester M13 9PL, United Kingdom
A. Bhat  Syracuse University, Syracuse, NY, 13244, USA  M. Bishai  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Blake  Lancaster
University, Lancaster LA1 4YW, United Kingdom  T. Bolton  Kansas State
University (KSU), Manhattan, KS, 66506, USA  J. Y. Book  Harvard University,
Cambridge, MA 02138, USA  L. Camilleri  Columbia University, New York, NY,
10027, USA  D. Caratelli  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  I. Caro Terrazas  Colorado State University, Fort
Collins, CO, 80523, USA  F. Cavanna  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  G. Cerati  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  Y. Chen  Universität Bern, Bern
CH-3012, Switzerland  D. Cianci  Columbia University, New York, NY, 10027, USA
G. H. Collin  Massachusetts Institute of Technology (MIT), Cambridge, MA,
02139, USA  J. M. Conrad  Massachusetts Institute of Technology (MIT),
Cambridge, MA, 02139, USA  M. Convery  SLAC National Accelerator Laboratory,
Menlo Park, CA, 94025, USA  L. Cooper-Troendle  Wright Laboratory, Department
of Physics, Yale University, New Haven, CT, 06520, USA  J. I. Crespo-Anadón
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid E-28040, Spain  M. Del Tutto  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  S. R. Dennis  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  P. Detje  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  A. Devitt  Lancaster University,
Lancaster LA1 4YW, United Kingdom  R. Diurba  University of Minnesota,
Minneapolis, MN, 55455, USA  R. Dorrill  Illinois Institute of Technology
(IIT), Chicago, IL 60616, USA  K. Duffy  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  S. Dytman  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  B. Eberly  University of Southern Maine, Portland,
ME, 04104, USA  A. Ereditato  Universität Bern, Bern CH-3012, Switzerland  L.
Escudero Sanchez  University of Cambridge, Cambridge CB3 0HE, United Kingdom
J. J. Evans  The University of Manchester, Manchester M13 9PL, United Kingdom
R. Fine  Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  G.
A. Fiorentini Aguirre  South Dakota School of Mines and Technology (SDSMT),
Rapid City, SD, 57701, USA  R. S. Fitzpatrick  University of Michigan, Ann
Arbor, MI, 48109, USA  B. T. Fleming  Wright Laboratory, Department of
Physics, Yale University, New Haven, CT, 06520, USA  N. Foppiani  Harvard
University, Cambridge, MA 02138, USA  D. Franco  Wright Laboratory, Department
of Physics, Yale University, New Haven, CT, 06520, USA  A. P. Furmanski
University of Minnesota, Minneapolis, MN, 55455, USA  D. Garcia-Gamez
Universidad de Granada, Granada E-18071, Spain  S. Gardiner  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  G. Ge  Columbia
University, New York, NY, 10027, USA  V. Genty  Columbia University, New York,
NY, 10027, USA  S. Gollapinni  University of Tennessee, Knoxville, TN, 37996,
USA  Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA  O.
Goodwin  The University of Manchester, Manchester M13 9PL, United Kingdom  E.
Gramellini  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  P. Green  The University of Manchester, Manchester M13 9PL, United
Kingdom  H. Greenlee  Fermi National Accelerator Laboratory (FNAL), Batavia,
IL 60510, USA  W. Gu  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  R. Guenette  Harvard University, Cambridge, MA 02138, USA  P. Guzowski
The University of Manchester, Manchester M13 9PL, United Kingdom  L. Hagaman
Wright Laboratory, Department of Physics, Yale University, New Haven, CT,
06520, USA  O. Hen  Massachusetts Institute of Technology (MIT), Cambridge,
MA, 02139, USA  C. Hilgenberg  University of Minnesota, Minneapolis, MN,
55455, USA  G. A. Horton-Smith  Kansas State University (KSU), Manhattan, KS,
66506, USA  A. Hourlier  Massachusetts Institute of Technology (MIT),
Cambridge, MA, 02139, USA  R. Itay  SLAC National Accelerator Laboratory,
Menlo Park, CA, 94025, USA  C. James  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  X. Ji  Brookhaven National Laboratory (BNL),
Upton, NY, 11973, USA  L. Jiang  Center for Neutrino Physics, Virginia Tech,
Blacksburg, VA, 24061, USA  J. H. Jo  Wright Laboratory, Department of
Physics, Yale University, New Haven, CT, 06520, USA  R. A. Johnson  University
of Cincinnati, Cincinnati, OH, 45221, USA  Y.-J. Jwa  Columbia University, New
York, NY, 10027, USA  D. Kaleko  Columbia University, New York, NY, 10027, USA
D. Kalra  Columbia University, New York, NY, 10027, USA  N. Kamp
Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA  N.
Kaneshige  University of California, Santa Barbara, CA, 93106, USA  G.
Karagiorgi  Columbia University, New York, NY, 10027, USA  W. Ketchum  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  M. Kirby
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T.
Kobilarcik  Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510,
USA  I. Kreslo  Universität Bern, Bern CH-3012, Switzerland  R. LaZur
Colorado State University, Fort Collins, CO, 80523, USA  I. Lepetic  Rutgers
University, Piscataway, NJ, 08854, USA  K. Li  Wright Laboratory, Department
of Physics, Yale University, New Haven, CT, 06520, USA  Y. Li  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  K. Lin  Los Alamos National
Laboratory (LANL), Los Alamos, NM, 87545, USA  A. Lister  Lancaster
University, Lancaster LA1 4YW, United Kingdom  B. R. Littlejohn  Illinois
Institute of Technology (IIT), Chicago, IL 60616, USA  W. C. Louis  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  X. Luo  University of
California, Santa Barbara, CA, 93106, USA  K. Manivannan  Syracuse University,
Syracuse, NY, 13244, USA  C. Mariani  Center for Neutrino Physics, Virginia
Tech, Blacksburg, VA, 24061, USA  D. Marsden  The University of Manchester,
Manchester M13 9PL, United Kingdom  J. Marshall  University of Warwick,
Coventry CV4 7AL, United Kingdom  D. A. Martinez Caicedo  South Dakota School
of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  K. Mason  Tufts
University, Medford, MA, 02155, USA  A. Mastbaum  Rutgers University,
Piscataway, NJ, 08854, USA  N. McConkey  The University of Manchester,
Manchester M13 9PL, United Kingdom  V. Meddage  Kansas State University (KSU),
Manhattan, KS, 66506, USA  T. Mettler  Universität Bern, Bern CH-3012,
Switzerland  K. Miller  University of Chicago, Chicago, IL, 60637, USA  J.
Mills  Tufts University, Medford, MA, 02155, USA  K. Mistry  The University of
Manchester, Manchester M13 9PL, United Kingdom  A. Mogan  University of
Tennessee, Knoxville, TN, 37996, USA  T. Mohayai  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. Moon  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  M. Mooney  Colorado State
University, Fort Collins, CO, 80523, USA  A. F. Moor  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  C. D. Moore  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  L. Mora Lepin  The University of
Manchester, Manchester M13 9PL, United Kingdom  J. Mousseau  University of
Michigan, Ann Arbor, MI, 48109, USA  M. Murphy  Center for Neutrino Physics,
Virginia Tech, Blacksburg, VA, 24061, USA  D. Naples  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  A. Navrer-Agasson  The University of
Manchester, Manchester M13 9PL, United Kingdom  M. Nebot-Guinot  University of
Edinburgh, Edinburgh EH9 3FD, United Kingdom  R. K. Neely  Kansas State
University (KSU), Manhattan, KS, 66506, USA  D. A. Newmark  Los Alamos
National Laboratory (LANL), Los Alamos, NM, 87545, USA  J. Nowak  Lancaster
University, Lancaster LA1 4YW, United Kingdom  M. Nunes  Syracuse University,
Syracuse, NY, 13244, USA  O. Palamara  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  V. Paolone  University of Pittsburgh,
Pittsburgh, PA, 15260, USA  A. Papadopoulou  Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA  V. Papavassiliou  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  S. F. Pate  New Mexico
State University (NMSU), Las Cruces, NM, 88003, USA  N. Patel  Lancaster
University, Lancaster LA1 4YW, United Kingdom  A. Paudel  Kansas State
University (KSU), Manhattan, KS, 66506, USA  Z. Pavlovic  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  E. Piasetzky  Tel Aviv
University, Tel Aviv, Israel, 69978  I. D. Ponce-Pinto  Wright Laboratory,
Department of Physics, Yale University, New Haven, CT, 06520, USA  S. Prince
Harvard University, Cambridge, MA 02138, USA  X. Qian  Brookhaven National
Laboratory (BNL), Upton, NY, 11973, USA  J. L. Raaf  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  V. Radeka  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  A. Rafique  Kansas State
University (KSU), Manhattan, KS, 66506, USA  M. Reggiani-Guzzo  The University
of Manchester, Manchester M13 9PL, United Kingdom  L. Ren  New Mexico State
University (NMSU), Las Cruces, NM, 88003, USA  L. C. J. Rice  University of
Pittsburgh, Pittsburgh, PA, 15260, USA  L. Rochester  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  J. Rodriguez Rondon  South
Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA  M.
Rosenberg  University of Pittsburgh, Pittsburgh, PA, 15260, USA  M. Ross-
Lonergan  Columbia University, New York, NY, 10027, USA  B. Russell  Wright
Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA
G. Scanavini  Wright Laboratory, Department of Physics, Yale University, New
Haven, CT, 06520, USA  D. W. Schmitz  University of Chicago, Chicago, IL,
60637, USA  A. Schukraft  Fermi National Accelerator Laboratory (FNAL),
Batavia, IL 60510, USA  W. Seligman  Columbia University, New York, NY, 10027,
USA  M. H. Shaevitz  Columbia University, New York, NY, 10027, USA  R.
Sharankova  Tufts University, Medford, MA, 02155, USA  J. Shi  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  J. Sinclair  Universität Bern,
Bern CH-3012, Switzerland  A. Smith  University of Cambridge, Cambridge CB3
0HE, United Kingdom  E. L. Snider  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  M. Soderberg  Syracuse University, Syracuse,
NY, 13244, USA  S. Söldner-Rembold  The University of Manchester, Manchester
M13 9PL, United Kingdom  S. R. Soleti  University of Oxford, Oxford OX1 3RH,
United Kingdom  Harvard University, Cambridge, MA 02138, USA  P. Spentzouris
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. Spitz
University of Michigan, Ann Arbor, MI, 48109, USA  M. Stancari  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  J. St. John  Fermi
National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  T. Strauss
Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  K.
Sutton  Columbia University, New York, NY, 10027, USA  S. Sword-Fehlberg  New
Mexico State University (NMSU), Las Cruces, NM, 88003, USA  A. M. Szelc
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  W. Tang
University of Tennessee, Knoxville, TN, 37996, USA  K. Terao  SLAC National
Accelerator Laboratory, Menlo Park, CA, 94025, USA  M. Thomson  University of
Cambridge, Cambridge CB3 0HE, United Kingdom  C. Thorpe  Lancaster University,
Lancaster LA1 4YW, United Kingdom  D. Totani  University of California, Santa
Barbara, CA, 93106, USA  M. Toups  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  Y.-T. Tsai  SLAC National Accelerator
Laboratory, Menlo Park, CA, 94025, USA  M. A. Uchida  University of Cambridge,
Cambridge CB3 0HE, United Kingdom  T. Usher  SLAC National Accelerator
Laboratory, Menlo Park, CA, 94025, USA  W. Van De Pontseele  University of
Oxford, Oxford OX1 3RH, United Kingdom  Harvard University, Cambridge, MA
02138, USA  B. Viren  Brookhaven National Laboratory (BNL), Upton, NY, 11973,
USA  M. Weber  Universität Bern, Bern CH-3012, Switzerland  H. Wei  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA  Z. Williams  University of
Texas, Arlington, TX, 76019, USA  S. Wolbers  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  T. Wongjirad  Tufts University,
Medford, MA, 02155, USA  M. Wospakrik  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  K. Wresilo  University of Cambridge, Cambridge
CB3 0HE, United Kingdom  N. Wright  Massachusetts Institute of Technology
(MIT), Cambridge, MA, 02139, USA  W. Wu  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  E. Yandel  University of California, Santa
Barbara, CA, 93106, USA  T. Yang  Fermi National Accelerator Laboratory
(FNAL), Batavia, IL 60510, USA  G. Yarbrough  University of Tennessee,
Knoxville, TN, 37996, USA  L. E. Yates  Massachusetts Institute of Technology
(MIT), Cambridge, MA, 02139, USA  H. W. Yu  Brookhaven National Laboratory
(BNL), Upton, NY, 11973, USA  G. P. Zeller  Fermi National Accelerator
Laboratory (FNAL), Batavia, IL 60510, USA  J. Zennamo  Fermi National
Accelerator Laboratory (FNAL), Batavia, IL 60510, USA  C. Zhang  Brookhaven
National Laboratory (BNL), Upton, NY, 11973, USA

###### Abstract

We present a measurement of $\nu_{e}$ interactions from the Fermilab Booster
Neutrino Beam using the MicroBooNE liquid argon time projection chamber to
address the nature of the excess of low energy interactions observed by the
MiniBooNE collaboration. Three independent $\nu_{e}$ searches are performed
across multiple single electron final states, including an exclusive search
for two-body scattering events with a single proton, a semi-inclusive search
for pionless events, and a fully inclusive search for events containing all
hadronic final states. With differing signal topologies, statistics,
backgrounds, reconstruction algorithms, and analysis approaches, the results
are found to be either consistent with or modestly lower than the nominal
$\nu_{e}$ rate expectations from the Booster Neutrino Beam and no excess of
$\nu_{e}$ events is observed.

††preprint: APS/123-QED

MicroBooNE is the first liquid argon time projection chamber (LArTPC) to
acquire high statistics samples of neutrino interactions on argon. Using this
unique data set, MicroBooNE has pioneered a large body of results on neutrino
interactions [1, 2, 3, 4, 5, 6, 7], astrophysical [8, 9] and beyond the
Standard Model physics [10, 11], neutrino event reconstruction [12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23], and detector properties [24, 25, 26, 27,
28, 29, 30, 31, 32, 33]. Here, we report the first measurement of electron
neutrinos produced in the Fermilab Booster Neutrino Beamline (BNB) using the
MicroBooNE detector. This multi-pronged search is aimed at investigating the
as-yet unexplained low energy excess of electromagnetic activity observed by
the MiniBooNE collaboration [34].

Over the past decade, there has been a rich and evolving landscape of
theoretical interpretations to explain the origin of the observed MiniBooNE
excess, including standard processes [35] as well as new physics involving
sterile neutrinos [36, 37], dark sector portals [38, 39, 40], heavy neutral
leptons [41, 42], non-standard Higgs physics [43, 44, 45, 46], new particles
produced in the beam [47, 48], and mixed models of sterile neutrino
oscillations and decay [49, 50]. A number of scenarios have also been ruled
out [51]. Because of the variety of theoretical explanations and their
possible signatures, MicroBooNE has developed three distinct $\nu_{e}$
searches targeting the MiniBooNE signal: an exclusive search for two-body
$\nu_{e}$ charged current quasi-elastic (CCQE) scattering, a semi-inclusive
search for pionless $\nu_{e}$ events, and an inclusive $\nu_{e}$ search
containing any hadronic final state. Additionally, a companion single-photon-
based search focused on radiative decays of the $\Delta$ resonance is reported
elsewhere [52]. This work capitalizes on the broad capabilities of a LArTPC to
perform high purity measurements of electron neutrinos across multiple signal
topologies and with significantly improved ability to distinguish whether an
electromagnetic shower is electron or photon-induced compared to Cherenkov-
based detectors such as MiniBooNE.

The advantage of this particular probe of the MiniBooNE signal is that
MicroBooNE is located in the same neutrino beamline and at roughly the same
location as MiniBooNE, but uses an imaging detector capable of mm-scale
spatial resolution and substantially lower energy detection thresholds for
many particle types. The MicroBooNE LArTPC [53, 54] itself contains 85 tons of
liquid argon and is sited 72.5 m upstream of the MiniBooNE detector hall at a
distance of 468.5 m from the BNB proton target.

The data used in this work are taken from an exposure of $7\times 10^{20}$
protons on target (POT) collected in neutrino mode, a $93.7\%$ $\nu_{\mu}$
($5.8\%$ $\overline{\nu}_{\mu}$) pure beam, from February 2016 to July 2018.
These results represent an initial probe into electron neutrino production in
the BNB using roughly half of the total data collected by MicroBooNE. Two
different data streams are used in this analysis: an on-beam data sample
triggered by BNB neutrino spills and an off-beam data sample taken during
periods when no beam was received. The off-beam data sample is used for a
direct data-based measurement of cosmic-induced backgrounds which are of
importance given MicroBooNE’s location near the surface.

$\nu_{e}$ Final State | Signal Constraints | Reconstruction Approach
---|---|---
$1e1p(0\pi)$ CCQE |  $\nu_{\mu}$ CCQE | Deep-Learning [55]
$1eN(\geq 1)p0\pi$, $1e0p0\pi$ |  $\nu_{\mu}$ CC | Pandora [56]
$1eX$ |  $\nu_{\mu}$ CC, $\nu_{\mu}$ CC $\pi^{0}$, $\nu_{\mu}$ NC $\pi^{0}$ | Wire-Cell [57]
Table 1: Summary of signal definitions, signal-constraining data sets, and
reconstruction approaches used for each of the three MicroBooNE $\nu_{e}$
searches. All samples require fully contained events with the exception of the
$1eX$ analysis which additionally uses both partially and fully contained
$\nu_{\mu}$ CC samples as constraints.

While probing different event topologies with distinct event reconstruction
methods, the three independent electron neutrino searches in MicroBooNE share
several aspects in common. To simulate neutrino interactions in argon, the
analyses rely on a Geant4-based [58] simulation of the neutrino beam [59], a
variation of the genie v3 event generator [60] specifically tuned to data that
reflects our best knowledge of neutrino scattering in the BNB energy range
[61], and a Geant4-based [58] detector simulation for particle propagation,
with the processing of the charge response of the TPC and modeling of
scintillation light implemented in the LArSoft framework [62]. The data-driven
detector simulation represents a significant upgrade from what has been
historically available to model LArTPCs and incorporates pioneering work
performed by MicroBooNE on wire signal processing [28, 29], noise removal
[31], electric field mapping [26, 25], and detector calibrations [27].

A common framework is additionally used to evaluate neutrino flux, neutrino
cross section, and detector systematics. The evaluation of neutrino flux
uncertainties is built on techniques developed by MiniBooNE [59]. A total of
more than 50 model parameters are varied within genie to assess uncertainties
related to simulating neutrino interactions and final state effects in argon
[61]. A full complement of LArTPC detector systematics are modeled, many with
a novel data-driven technique built on comparisons of wire response in data
and simulation [63]. In addition, beam-related $\nu_{e}$ event predictions and
uncertainties are further constrained through calculation of conditional means
and variances using data-driven measurements of neutrino charged current (CC)
and neutral current (NC) interactions in MicroBooNE, with constraint samples
tailored to each of the $\nu_{e}$ analysis approaches. Table 1 summarizes the
signal definitions, constraint samples, and reconstruction approaches for the
different analyses.

Common approaches are also used for assessing agreement between observed and
predicted $\nu_{e}$ samples and for hypothesis testing related to the presence
or absence of an anomalous $\nu_{e}$ rate. In each search, the neutrino energy
($E_{\nu}$) is based on reconstructed final state particle energies [55, 56,
57]. Binned reconstructed $E_{\nu}$ distributions for data and simulation are
then compared using a combined Neyman-Pearson test statistic, $\chi^{2}_{CNP}$
[64], which approximates well to the Poisson-likelihood test statistic for low
event numbers. Similar statistical studies are performed to test a model of
the $\nu_{e}$ event excess with an $E_{\nu}$ spectrum and normalization
representative of that observed in the MiniBooNE experiment.

A blind analysis scheme was adopted in which the signal region BNB $\nu_{e}$
data were only accessed after each analysis was completed. Details on each
$\nu_{e}$ analysis, including the optimization of purity and efficiency in
each approach, data-based sidebands and simulated data sets used to validate
the analyses, as well as data and Monte Carlo comparisons in a variety of
kinematics can be found in Refs. [55, 56, 57]. The next three sections
describe the strategy behind each of the $\nu_{e}$ searches and a summary of
their results.

##  I Two-Body $\nu_{e}$ CCQE Scattering ($1e1p$)

An exclusive selection of $\nu_{e}$ candidates satisfying two-body CCQE
kinematic constraints is performed in MicroBooNE with an analysis strategy
focused around the use of deep-learning techniques [55]. The CCQE process is
predicted to dominate at low energies, with 77% of $\nu_{e}$ events below 500
MeV (in true $E_{\nu}$) expected to interact via this channel in MicroBooNE. A
strength of this selection is its clean final state topology, which simplifies
event reconstruction and selection. The pure CCQE requirement also allows the
use of kinematic variables for signal selection, such as proton momentum,
transverse momentum, neutrino direction, momentum transfer, and Bjorken $x$.
For this analysis, only fully-contained candidate $\nu_{e}$ and $\nu_{\mu}$
interactions with one reconstructed final state lepton and proton are
selected. Final-state particle content and kinematics are reconstructed by
applying a combination of conventional and deep-learning tools to prepared
event images. Conventional tools are used to remove cosmic backgrounds [15],
identify candidate neutrino interaction vertices [65], and reconstruct final-
state particle candidates and their kinematics [65, 66], while convolutional
neural networks (CNNs) are used to differentiate track-like from shower-like
image pixels [67] and determine particle species contained in event images
[68]. After basic data quality selection criteria, Boosted Decision Tree (BDT)
ensembles exploit 23 (16) kinematic and topological variables, including those
described above, to collect a purified CCQE $\nu_{e}$ ($\nu_{\mu}$) event set.
The CNN designed for particle identification is then used to select final
candidates from this set.

The results of the $\nu_{e}$ CCQE analysis are shown in Fig. 1. This selection
is predicted to produce a 75% (75%) pure sample of $\nu_{e}$ ($\nu_{\mu}$)
CCQE events in the reconstructed $E_{\nu}$ range between 200 and 1200 MeV,
with an efficiency of 6.6% for all true $\nu_{e}$ CCQE interactions in the
LArTPC active volume. The average $E_{\nu}$ energy resolution for selected
$\nu_{e}$ events is estimated to be 16.5%, with negligible bias predicted
between true and reconstructed $E_{\nu}$. The $\nu_{e}$ CCQE prediction is
constrained using a high-statistics $\nu_{\mu}$ CCQE candidate data set, which
increases predicted $\nu_{e}$ counts by 6% and reduces systematic
uncertainties on the prediction by a factor of 2.0. Statistical uncertainties
dominate the final measurement. The post-constraint prediction yields
$29.0\,\pm\,5.2$ (stat.) $\pm\,1.9$ (syst.) events in the full $E_{\nu}$ range
given above (statistical errors follow the CNP formalism [64]), with most
events (89%) predicted to arise from CCQE and non-CCQE $\nu_{e}$ interactions
intrinsic to the beam. In the final selection, a total of 25 data candidates
are observed in this range. The $\chi^{2}_{\text{CNP}}$ test statistic
calculated between predicted and observed distributions is found to be 25.3
for the analysis in ten $E_{\nu}$ bins (where 6.9 units of
$\chi^{2}_{\text{CNP}}$ result from a single bin at 850 MeV), leading to a
$p$-value of 0.014. Below 500 MeV, the $\chi^{2}_{\text{CNP}}$ contribution is
7.9 for three $E_{\nu}$ bins, with data in two of the three bins falling
slightly below predicted values.

Figure 1: Reconstructed neutrino energy for $1e1p$ CCQE candidate events in
the deep-learning-based analysis. Backgrounds include contributions from
cosmics and $\nu_{\mu}$ interactions. The $\nu_{e}$ prediction constrained
using $\nu_{\mu}$ data is shown without (solid histogram) and with (red
dotted) a model of the MiniBooNE low energy excess included (further detail in
text). Systematic uncertainties on the constrained prediction are shown as a
hatched band.

##  II Pionless $\nu_{e}$ Scattering ($1eNp0\pi$, $1e0p0\pi$)

A higher statistics search for pionless $\nu_{e}$ interactions that includes
any number of protons in the final state uses the Pandora event reconstruction
package [22], which has been exercised over the years to produce a wide
variety of MicroBooNE physics measurements [2, 3, 4, 5, 6, 7, 10, 11]. The
Pandora pattern recognition software, which reconstructs and classifies LArTPC
events, is combined with specialized tools that further remove cosmic-ray
background events as well as identify the different particles produced in a
neutrino interaction [69] and reconstruct their energies [6]. This search
focuses on two exclusive channels with one electron and no pions in the final
state: one with at least one visible proton ($1eNp0\pi$, $N\geq 1$) and one
with no visible protons ($1e0p0\pi$). A strength of this selection is that the
two topologies combined exactly replicate the electron-like signal event
signature in MiniBooNE. This selection on fully contained events spanning
neutrino energies from 10 to 2390 MeV provides an efficiency of 15$\%$ (9$\%$)
with a purity of 80$\%$ (43$\%$) for $1eNp0\pi$ ($1e0p0\pi$) events. The
typical energy resolution is $2\%$ for protons, $3\%$ for muons, and
approximately $12\%$ for electrons, resulting in a predicted $E_{\nu}$
resolution of $15\%$ with $\sim 5\%$ bias. To constrain neutrino flux and
cross section uncertainties on the predicted intrinsic $\nu_{e}$ event rate,
this analysis uses a high-statistics, 77$\%$ pure $\nu_{\mu}$ CC inclusive
event sample [56] and makes use of the cosmic-ray tagger detector system in
MicroBooNE [54] to further reduce cosmic backgrounds. This constraint reduces
the systematic uncertainties in the $\nu_{e}$ selections by a factor of 1.7
and the result remains dominated by statistical uncertainties. This analysis
is also validated using MicroBooNE data from the NuMI beam [70] that provides
a large number of $\nu_{e}$-argon interactions at a similar energy range as
the BNB.

The results of the Pandora-based pionless $\nu_{e}$ analysis are shown in Fig.
2. For the $1eNp0\pi$ channel, $64$ $\nu_{e}$ data events are observed
compared to $86.8\,\pm\,8.8$ (stat) $\pm\,11.5$ (syst) events expected
(statistical errors follow the CNP formalism), in a reconstructed $E_{\nu}$
range between 10 and 2390 MeV. For the $1e0p0\pi$ channel, $34$ $\nu_{e}$ data
events are observed compared to $30.2\,\pm\,5.6$ (stat) $\pm\,4.3$ (syst)
events expected over that same energy range. The data are consistent with the
prediction: in the region $150~{}\text{MeV}\leq E_{\nu}\leq 1550~{}\text{MeV}$
where the final statistical tests are performed, the
$\chi^{2}_{\text{CNP}}/ndf$ (and associated $p$-values) relative to the
nominal prediction are 14.9/10 (0.194), 16.7.9/10 (0.116), and 31.56/20
(0.097) for the $1eNp0\pi$ channel, $1e0p0\pi$ channel, and both combined,
respectively. As with the $1e1p$ CCQE search results, the data for the
$1eNp0\pi$ channel fall slightly below prediction. For the $1e0p0\pi$ channel,
the observed event count below 500 MeV is above prediction, albeit in a region
with lower predicted $\nu_{e}$ purity.

Figure 2: Reconstructed neutrino energy for pionless $\nu_{e}$ candidate
events in the Pandora-based analysis: $1eNp0\pi$ (left) and $1e0p0\pi$
(right). Each of the plots extend from 10 to 2390 MeV with a 140 MeV bin
width. The unconstrained number of predicted events is shown broken down by
true interaction topology. The constrained predictions using $\nu_{\mu}$ data
are shown both with (red) and without (black) a model of the MiniBooNE low
energy excess included (further detail in text). Systematic uncertainties on
the constrained prediction are shown as a shaded band.

##  III Inclusive $\nu_{e}$ Scattering ($1eX$)

The highest statistics $\nu_{e}$ analysis in MicroBooNE searches inclusively
for all possible hadronic final states such as the type of analyses that will
be performed in the future wide-band Deep Underground Neutrino Experiment
(DUNE) which will have larger contributions from additional inelastic
scattering processes at higher energies. This analysis uses the Wire-Cell
reconstruction paradigm [71] which forms three-dimensional images of particle-
induced electron ionization tracks and showers via 1D wire position
tomography. The 3D images are then processed by clustering algorithms and
matched to light signals for cosmic rejection [72, 14, 13], before a deep
neural network [73] is used to determine the neutrino candidate vertex.
Finally, the events are characterized in terms of energy deposit, topology,
and kinematics, for eventual event building, classification (e.g. $\nu_{e}$
CC, $\nu_{\mu}$ CC, $\pi^{0}$, cosmic), and neutrino energy reconstruction.
The strengths of this approach are its high efficiency and high purity. After
all selections, the predicted efficiency for selecting inclusive $\nu_{e}$ CC
($\nu_{\mu}$ CC) events is 46% (68%) with a purity of 82% (92%) for
$0<E_{\nu}<2500$ MeV. For fully contained events, the predicted calorimetric-
based $E_{\nu}$ resolution is 10–15% (15–20%) for $\nu_{e}$ CC ($\nu_{\mu}$
CC) events with $\sim$7% (10%) bias. In addition to the $\nu_{\mu}$ CC data
samples, which include both fully and partially contained events in the
detector, CC and NC interactions with a reconstructed $\pi^{0}$ serve as
additional constraints for reducing systematic uncertainties and therefore
maximizing sensitivity. A high statistics sample of $\nu_{e}$ events from the
NuMI beam also serves to validate the analysis. The constraints reduce the
fractional uncertainty on the predicted number of fully-contained $\nu_{e}$ CC
events with reconstructed $E_{\nu}<600$ MeV by a factor of 3.5 relative to the
unconstrained prediction. After constraints, the largest systematic
uncertainties for fully-contained $\nu_{e}$ CC events are associated with
limited Monte Carlo statistics associated with this rare event search,
detector effects (mainly recombination and wire response), and neutrino cross
section modeling [61]. Compared to all systematic uncertainties, however, the
statistical uncertainty on the data remains dominant.

Figure 3:  Reconstructed neutrino energy for inclusive $\nu_{e}$ candidate
events in the Wire-Cell based analysis. The predicted event sample is
dominated by $\nu_{e}$ events intrinsic to the beam (green) while the other
background contributions are described in the legend. The constrained
predictions are shown both with (red) and without (black) a model of the
MiniBooNE low energy excess included (further detail in text). Systematic
uncertainties on the constrained prediction are shown as a shaded band.

Fig. 3 shows the results of this inclusive $\nu_{e}$ search. The post-
constraint $\nu_{e}$ CC inclusive analysis finds a modest deficit compared to
the prediction: 56 (338) data events in $E_{\nu}<600$ MeV ($0<E_{\nu}<2500$
MeV) with $69.6\pm 8.0~{}\mathrm{(stat.)}\pm 5.0~{}\mathrm{(syst.)}$
($384.9\pm 19.2~{}\mathrm{(stat.)}\pm 15.9~{}\mathrm{(syst.)}$) events
expected. Good agreement is found between the data and the expectation from
the BNB, with $\chi_{Pearson}^{2}/ndf=17.9/25$ and a corresponding $p$-value
of $0.848$, across all energies. Notably, agreement between the data and
expectation is also apparent when the Wire-Cell inclusive event sample is
studied in terms of its exclusive components, $1e0pX\pi$ and $1eNpX\pi$
($X\geq 0$), where these subsamples are further described in Ref. [57].

Signal-enhanced region comparison

| $1e1p$ |  |  |
---|---|---|---|---
| CCQE | $1eNp0\pi$ | $1e0p0\pi$ | $1eX$
$E_{\nu}$ (MeV) | 200-500 | 150-650 | 150-650 | 0-600
Predicted, | $8.8\pm 3.0$ | $30.4\pm 6.1$ | $19.0\pm 5.3$ |  $69.6\pm 9.4$
no eLEE |
Predicted, | $18.5\pm 4.4$ | $39.0\pm 6.8$ | $22.3\pm 5.7$ |  $104\pm 12$
w/ eLEE |
Observed | 6 | 21 | 27 | 56

Final fit results
$1e1p$ CCQE $1eNp0\pi$ $1e0p0\pi$ $1eX$ $E_{\nu}$ (MeV) 200-1200 150-1550
150-1550 0-2500  $p~{}(\chi_{x=0}^{2})$ $1.4\times 10^{-2}$ 0.18 0.13 0.85
$p~{}(\Delta\chi^{2}<\text{obs.})$, $1.6\times 10^{-4}$ $2.1\times 10^{-2}$
0.93 $9.0\times 10^{-5}$ w/ eLEE $x$ observed, $1\sigma$ [0.00,0.08]
[0.00,0.41] [1.91,8.10] [0.00,0.22]  $x$ observed, $2\sigma$ [0.00,0.38]
[0.00,1.06] [0.77,24.3] [0.00,0.51]  $x$ expected 0.98 1.44 4.64 0.56 upper
limit, $2\sigma$

Table 2: Left: Observed and predicted $\nu_{e}$ candidates in the signal-
enhanced neutrino energy range predefined by each analysis prior to
unblinding, in the absence ($x=0$) or presence ($x=1$) of a MiniBooNE-like
$\nu_{e}$ event excess. This energy range is a subset of the full fit range,
also chosen prior to unblinding. Predicted events include the alternate-
channel constraints of each analysis, and include statistical (following the
CNP formalism) and constrained systematic uncertainties. Right: Frequentist-
derived $p$-values of the data observations compared to the prediction
assuming no excess, $p~{}(\chi_{x=0}^{2})$, and under a simple hypothesis test
comparing an excess to no excess,
$p(\Delta\chi^{2}=\chi^{2}_{x=0}-\chi^{2}_{x=1}<\text{obs.})$, assuming the
eLEE model ($x=1)$. Also quoted are the 1$\sigma$ and 2$\sigma$ confidence
intervals for extracted signal strength $x$ over the full fit range and the
expected 2$\sigma$ upper endpoint of the interval on $x$ assuming no excess.

##  IV Fit Results

The three aforementioned analysis selections are not designed to be disjoint
to each other, and there is an overlap in the selected events. Of the 25
events selected in the $1e1p$ CCQE analysis, 16 are selected in either the
pionless or inclusive analysis. Of the 98 events selected across both pionless
analysis selections, 46 are selected in the inclusive analysis. All three
analyses observe $\nu_{e}$ candidate event rates in general agreement with or
below the predicted rates. Given the similar baseline and neutrino energies
sampled by MicroBooNE and MiniBooNE, this picture appears to disfavor an
interpretation of MiniBooNE’s observed electron-like excess signature as
arising purely from an anomalously high rate of charged current $\nu_{e}$
interactions. To more quantitatively address the comparison with the observed
MiniBooNE data excess, all three analyses have performed statistical tests
comparing datasets to a simple model of a MiniBooNE-like excess of $\nu_{e}$
interactions [74].

Using MiniBooNE simulation, a response matrix is constructed translating the
true incident $E_{\nu}$ to reconstructed $E_{\nu}$ under a quasi-elastic
assumption to the true incident $E_{\nu}$, accounting for detector response,
acceptance, resolutions, event reconstruction, and selection efficiencies.
Following a multi-dimensional unfolding procedure [75] on the MiniBooNE
observation [76] and using only the statistical uncertainties on the MiniBooNE
data and simulated events (to avoid any correlated uncertainties in flux and
interaction models with MicroBooNE), MicroBooNE extracts an energy-dependent
event rate of $\nu_{e}$ interactions. The resulting scaling template, found to
be robust against the number of unfolding iterations after an initial starting
point corresponding to the MiniBooNE prediction, is derived from the increase
in event rate relative to the MiniBooNE prediction and then applied to
simulated intrinsic $\nu_{e}$ events in MicroBooNE to form an “eLEE” signal
model, shown by the dashed lines in Figs. 1–3. This scaling template varies
only in true neutrino energy, thus the eLEE model otherwise assumes the same
kinematics and final-state topologies as MicroBooNE’s $\nu_{e}$ simulation –
additional kinematic information from the MiniBooNE excess result is not
considered. The range of the eLEE signal model is $200<$ true $E_{\nu}<800$
MeV, as the unfolding procedure does not consider data below 200 MeV in
neutrino energy and finds no significant excess at higher energies.

This simple model reproduces a median MiniBooNE electron-like excess to which
MicroBooNE’s results are compared, either by choosing a fixed excess
normalization, $x$, matching MiniBooNE ($x$=1), or by treating $x$ as a free
parameter to be extracted. While the MiniBooNE uncertainties are not directly
included in the eLEE model nor the statistical tests presented in this letter,
the reported significance of the excess from the MiniBooNE neutrino-mode data
[34], $4.69\sigma$, translates to a $1\sigma$ confidence interval on the eLEE
signal strength parameter of $1\pm 0.21$, illustrating how the MiniBooNE
excess would appear. More rigorous comparisons of consistency with MiniBooNE
in the future will need to consider correlated uncertainties in the neutrino
flux and cross section models, as well as additional kinematic measurements.

Figure 4: Ratio of observed to predicted $\nu_{e}$ candidate events – assuming
no eLEE – in each analysis’s signal-enhanced neutrino energy range (see Tab.
2, left) with the relative contributions shown from non-$\nu_{e}$ backgrounds
(light blue) and intrinsic $\nu_{e}$’s (green). Statistical errors are shown
on the observations (black), while systematic errors are shown around the
prediction (gray). The expected ratio assuming the MiniBooNE-like eLEE signal
model with its median signal strength is also shown (red).

Prior to unblinding of the data, each analysis defined a signal-enhanced low-
energy region and determined the predicted number of events with and without
the excess model in that region. Those predictions and the observed number of
events are shown in Table 2 (left) and as ratios relative to the $x=0$
prediction in Fig. 4.

Each analysis performs two statistical analyses to test the signal hypothesis.
First, a simple hypothesis test uses
$\Delta\chi^{2}_{\text{CNP}}=\chi^{2}_{x=0}-\chi^{2}_{x=1}$ as a test
statistic, comparing the observations to a frequentist
$\Delta\chi^{2}_{\text{CNP}}$ distribution derived from model simulations
assuming $x=0$ and $x=1$. The $p$-values corresponding to
$\Delta\chi^{2}_{\text{CNP}}$ being less than the observed value assuming an
eLEE (no eLEE) signal is $1.6\times 10^{-4}$ ($0.02$), $0.021$ ($0.29$),
$0.93$ ($0.98$), and $9.0\times 10^{-5}$($0.33$) in the $1e1p$ CCQE,
$1eNp0\pi$, $1e0p0\pi$, and $1eX$ selections, respectively. Each selection
shows a strong preference for the absence of an electron-like MiniBooNE
signal, with the exception of the $1e0p0\pi$ selection, driven by a data
excess in the lowest energy bins, which also contain the highest contributions
from non-$\nu_{e}$ backgrounds.

Second, each analysis performs a nested hypothesis test where the eLEE signal
strength $x$ is varied, with a lower bound constraint at $x=0$. Each analysis
independently finds a best-fit signal strength, $x_{\text{min}}$, by
minimizing $\chi^{2}_{\text{CNP}}$. Following this, a test statistic defined
as
$\Delta\chi^{2}(x)=\chi^{2}_{\text{CNP}}(x)-\chi^{2}_{\text{CNP}}(x_{\text{min}})$
can be constructed for varying hypothetical signal strengths. A Feldman-
Cousins method [77] is used to construct confidence intervals around the best-
fit signal strength, which are shown in Table 2 (right) and Fig. 5. Consistent
with the observed deficit of events at low reconstructed energies, the $1e1p$
CCQE, $1eNp0\pi$, and $1eX$ selections each find a best fit signal strength of
$x=0$, corresponding to the absence of an observed event excess, with
$2\sigma$ upper bounds at $x<0.38$, $<1.06$, and $<0.51$, respectively. The
expected $2\sigma$ upper bounds for these selections, assuming no signal, are
shown in Table 2. Consistent with the fact that in most analyses the observed
number of events is less than the predicted number in the low energy regions,
the measured upper endpoints of the $2\sigma$ interval are lower than
expected. The best-fit signal strength for the $1e0p0\pi$ selection is
$x=4.0$, but with a wide confidence interval due to the low sensitivity of
this channel. The best-fit signal strength for the $1eNp0\pi$ and $1e0p0\pi$
channels combined is $x=0.36$, with $x<1.86$ at the $2\sigma$ confidence level
(and an expected upper bound where there is no signal at $x<1.37$), with more
details in [56].

Figure 5: Result of best-fit eLEE signal strength ($x$) in each analysis
(black), along with the $1$ and $2\sigma$ confidence intervals (solid and
dashed lines, respectively). The expected $2\sigma$ upper bound for each
analysis, assuming no eLEE signal, is also shown (red). Signal strength values
approximated from the MiniBooNE statistical and systematic errors (at
$1\sigma$) are shown for comparison (blue). Note that the vertical scale is
presented as linear from $x=0$ to $x=2$, while in logarithmic scale beyond
that.

##  V Conclusions

The MicroBooNE experiment has performed a set of inclusive and exclusive
searches for $\nu_{e}$ CC events using $7\times 10^{20}$ POT of Fermilab BNB
neutrino-mode data, about half of the collected dataset, with each analysis
considering a hypothesis for the nature of the MiniBooNE low-energy excess.
This work and Ref. [52] represent the first detailed study of this excess,
noting that future MicroBooNE and SBN [78] measurements will continue to
scrutinize the MiniBooNE results. The independent MicroBooNE search approaches
have been led by distinct groups with each using a different fully automated
event reconstruction software and common data-blindness scheme. All results
reported here are unchanged since data unblinding.

Afforded by the capabilities of the LArTPC technology to image various
leptonic and hadronic final states, the searches all feature excellent signal
identification and background rejection. In addition, the analyses use data-
driven $\nu_{e}$ estimates constrained by high-statistics samples of $\pi^{0}$
and $\nu_{\mu}$ CC events. The expected event rate is dominated by intrinsic
$\nu_{e}$ CC events originating from the beamline, rather than background
events involving photons. Despite the near-surface location, cosmic rays
represent a sub-dominant and usually negligible contribution to the
backgrounds.

No excess of low-energy $\nu_{e}$ candidates is observed, and the mutually
compatible, statistics-limited measurements are either consistent with or
modestly lower than the predictions for all $\nu_{e}$ event classes, including
inclusive and exclusive hadronic final-states, and across all energies. With
the exception of the $1e0p0\pi$ selection which is the least sensitive to a
simple model of the MiniBooNE low-energy excess, MicroBooNE rejects the
hypothesis that $\nu_{e}$ CC interactions are fully responsible for that
excess ($x=1$) at $>$97% CL for both exclusive ($1e1p$ CCQE, $1eNp0\pi$) and
inclusive ($1eX$) event classes. Additionally, MicroBooNE disfavors generic
$\nu_{e}$ interactions as the primary contributor to the excess, with a
1$\sigma$ (2$\sigma$) upper limit on the inclusive $\nu_{e}$ CC contribution
to the excess of 22% (51%). While the MiniBooNE excess remains unexplained,
our sensitive measurements are so far inconsistent with a $\nu_{e}$
interpretation of the excess.

###### Acknowledgements.

##  VI Acknowledgements

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); and The European
Union’s Horizon 2020 Marie Sklodowska-Curie Actions. Additional support for
the laser calibration system and cosmic-ray tagger was provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses,
without whom this work would not have been possible.

## References

  * Abratenko  _et al._ [a] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.14023 [hep-ex] .
  * Abratenko  _et al._ [2021a] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 104, 052002 (2021a), arXiv:2101.04228 [hep-ex] .
  * Abratenko  _et al._ [2020a] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 102, 112013 (2020a), arXiv:2010.02390 [hep-ex] .
  * Abratenko  _et al._ [2020b] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 125, 201803 (2020b), arXiv:2006.00108 [hep-ex] .
  * Abratenko  _et al._ [2019] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 123, 131801 (2019), arXiv:1905.09694 [hep-ex] .
  * Adams  _et al._ [2019a] C. Adams _et al._ (MicroBooNE), Phys. Rev. D 99, 091102 (2019a), arXiv:1811.02700 [hep-ex] .
  * Adams  _et al._ [2019b] C. Adams _et al._ (MicroBooNE), Eur. Phys. J. C 79, 248 (2019b), arXiv:1805.06887 [hep-ex] .
  * Abratenko  _et al._ [2021b] P. Abratenko _et al._ (MicroBooNE), JINST 16, P04004 (2021b), arXiv:2012.14324 [physics.ins-det] .
  * Abratenko  _et al._ [2021c] P. Abratenko _et al._ (MicroBooNE), JINST 16, P02008 (2021c), arXiv:2008.13761 [physics.ins-det] .
  * Abratenko  _et al._ [2021d] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 127, 151803 (2021d), arXiv:2106.00568 [hep-ex] .
  * Abratenko  _et al._ [2020c] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 101, 052001 (2020c), arXiv:1911.10545 [hep-ex] .
  * Abratenko  _et al._ [b] P. Abratenko _et al._ (MicroBooNE), arXiv:2109.02460 [physics.ins-det] .
  * Abratenko  _et al._ [2021e] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Applied 15, 064071 (2021e), arXiv:2101.05076 [physics.ins-det] .
  * Abratenko  _et al._ [c] P. Abratenko _et al._ (MicroBooNE), arXiv:2012.07928 [hep-ex] .
  * Abratenko  _et al._ [2021f] P. Abratenko _et al._ (MicroBooNE), JINST 16, P06043 (2021f), arXiv:2011.01375 [physics.ins-det] .
  * Abratenko  _et al._ [2021g] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 103, 052012 (2021g), arXiv:2012.08513 [physics.ins-det] .
  * Abratenko  _et al._ [2021h] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 103, 092003 (2021h), arXiv:2010.08653 [hep-ex] .
  * Abratenko  _et al._ [2021i] P. Abratenko _et al._ (MicroBooNE), JINST 16, P02017 (2021i), arXiv:2002.09375 [physics.ins-det] .
  * Adams  _et al._ [2020a] C. Adams _et al._ (MicroBooNE), JINST 15, P02007 (2020a), arXiv:1910.02166 [hep-ex] .
  * [20] C. Adams _et al._ (MicroBooNE), arXiv:1812.05679 [physics.ins-det] .
  * Adams  _et al._ [2019c] C. Adams _et al._ (MicroBooNE), Phys. Rev. D 99, 092001 (2019c), arXiv:1808.07269 [hep-ex] .
  * Acciarri  _et al._ [2018] R. Acciarri _et al._ (MicroBooNE), Eur. Phys. J. C 78, 82 (2018), arXiv:1708.03135 [hep-ex] .
  * Acciarri  _et al._ [2017a] R. Acciarri _et al._ (MicroBooNE), JINST 12, P03011 (2017a), arXiv:1611.05531 [physics.ins-det] .
  * Abratenko  _et al._ [d] P. Abratenko _et al._ (MicroBooNE), arXiv:2104.06551 [physics.ins-det] .
  * Abratenko  _et al._ [2020d] P. Abratenko _et al._ (MicroBooNE), JINST 15, P12037 (2020d), arXiv:2008.09765 [physics.ins-det] .
  * Adams  _et al._ [2020b] C. Adams _et al._ (MicroBooNE), JINST 15, P07010 (2020b), arXiv:1910.01430 [physics.ins-det] .
  * Adams  _et al._ [2020c] C. Adams _et al._ (MicroBooNE), JINST 15, P03022 (2020c), arXiv:1907.11736 [physics.ins-det] .
  * Adams  _et al._ [2018a] C. Adams _et al._ (MicroBooNE), JINST 13, P07006 (2018a), arXiv:1802.08709 [physics.ins-det] .
  * Adams  _et al._ [2018b] C. Adams _et al._ (MicroBooNE), JINST 13, P07007 (2018b), arXiv:1804.02583 [physics.ins-det] .
  * Acciarri  _et al._ [2017b] R. Acciarri _et al._ (MicroBooNE), JINST 12, P12030 (2017b), arXiv:1707.09903 [hep-ex] .
  * Acciarri  _et al._ [2017c] R. Acciarri _et al._ (MicroBooNE), JINST 12, P08003 (2017c), arXiv:1705.07341 [physics.ins-det] .
  * Acciarri  _et al._ [2017d] R. Acciarri _et al._ (MicroBooNE), JINST 12, P09014 (2017d), arXiv:1704.02927 [physics.ins-det] .
  * Abratenko  _et al._ [2017] P. Abratenko _et al._ (MicroBooNE), JINST 12, P10010 (2017), arXiv:1703.06187 [physics.ins-det] .
  * Aguilar-Arevalo  _et al._ [2021] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 103, 052002 (2021), arXiv:2006.16883 [hep-ex] .
  * Giunti  _et al._ [2020] C. Giunti, A. Ioannisian, and G. Ranucci, JHEP 11, 146 (2020), [Erratum: JHEP 02, 078 (2021)], arXiv:1912.01524 [hep-ph] .
  * [36] K. N. Abazajian _et al._ , arXiv:1204.5379 [hep-ph] .
  * Bai  _et al._ [2016] Y. Bai, R. Lu, S. Lu, J. Salvado, and B. A. Stefanek, Phys. Rev. D 93, 073004 (2016), arXiv:1512.05357 [hep-ph] .
  * Bertuzzo  _et al._ [2018] E. Bertuzzo, S. Jana, P. A. N. Machado, and R. Zukanovich Funchal, Phys. Rev. Lett. 121, 241801 (2018), arXiv:1807.09877 [hep-ph] .
  * Abdullahi  _et al._ [2021] A. Abdullahi, M. Hostert, and S. Pascoli, Phys. Lett. B 820, 136531 (2021), arXiv:2007.11813 [hep-ph] .
  * Alvarez-Ruso and Saul-Sala [2017] L. Alvarez-Ruso and E. Saul-Sala, in  _Prospects in Neutrino Physics_ (2017) arXiv:1705.00353 [hep-ph] .
  * Ballett  _et al._ [2019] P. Ballett, S. Pascoli, and M. Ross-Lonergan, Phys. Rev. D 99, 071701 (2019), arXiv:1808.02915 [hep-ph] .
  * Gninenko [2011] S. N. Gninenko, Phys. Rev. D 83, 093010 (2011), arXiv:1101.4004 [hep-ex] .
  * Dutta  _et al._ [2020] B. Dutta, S. Ghosh, and T. Li, Phys. Rev. D 102, 055017 (2020), arXiv:2006.01319 [hep-ph] .
  * Asaadi  _et al._ [2018] J. Asaadi, E. Church, R. Guenette, B. J. P. Jones, and A. M. Szelc, Phys. Rev. D 97, 075021 (2018), arXiv:1712.08019 [hep-ph] .
  * Abdallah  _et al._ [2021] W. Abdallah, R. Gandhi, and S. Roy, Phys. Rev. D 104, 055028 (2021), arXiv:2010.06159 [hep-ph] .
  * Abdallah  _et al._ [2020] W. Abdallah, R. Gandhi, and S. Roy, JHEP 12, 188 (2020), arXiv:2006.01948 [hep-ph] .
  * Chang  _et al._ [2021] C.-H. V. Chang, C.-R. Chen, S.-Y. Ho, and S.-Y. Tseng, Phys. Rev. D 104, 015030 (2021), arXiv:2102.05012 [hep-ph] .
  * Brdar  _et al._ [2021] V. Brdar, O. Fischer, and A. Y. Smirnov, Phys. Rev. D 103, 075008 (2021), arXiv:2007.14411 [hep-ph] .
  * [49] S. Vergani, N. W. Kamp, A. Diaz, C. A. Argüelles, J. M. Conrad, M. H. Shaevitz, and M. A. Uchida, arXiv:2105.06470 [hep-ph] .
  * Fischer  _et al._ [2020] O. Fischer, A. Hernández-Cabezudo, and T. Schwetz, Phys. Rev. D 101, 075045 (2020), arXiv:1909.09561 [hep-ph] .
  * Jordan  _et al._ [2019] J. R. Jordan, Y. Kahn, G. Krnjaic, M. Moschella, and J. Spitz, Phys. Rev. Lett. 122, 081801 (2019), arXiv:1810.07185 [hep-ph] .
  * Abratenko  _et al._ [e] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.00409 [hep-ex] .
  * Acciarri  _et al._ [2017e] R. Acciarri _et al._ (MicroBooNE), JINST 12, P02017 (2017e), arXiv:1612.05824 [physics.ins-det] .
  * Adams  _et al._ [2019d] C. Adams _et al._ (MicroBooNE), JINST 14, P04004 (2019d), arXiv:1901.02862 [physics.ins-det] .
  * Abratenko  _et al._ [f] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.14080 [hep-ex] .
  * Abratenko  _et al._ [g] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.14065 [hep-ex] .
  * Abratenko  _et al._ [h] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.13978 [hep-ex] .
  * Agostinelli  _et al._ [2003] S. Agostinelli _et al._ (GEANT4), Nucl. Instrum. Meth. A 506, 250 (2003).
  * Aguilar-Arevalo  _et al._ [2009] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 79, 072002 (2009), arXiv:0806.1449 [hep-ex] .
  * Andreopoulos  _et al._ [2010] C. Andreopoulos _et al._ , Nucl. Instrum. Meth. A 614, 87 (2010), arXiv:0905.2517 [hep-ph] .
  * Abratenko  _et al._ [i] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.14028 [hep-ex] .
  * [62] E. Church _et al._ (LArSoft), https://larsoft.org, arXiv:1311.6774 [physics.ins-det] .
  * Abratenko  _et al._ [2021j] P. Abratenko _et al._ (MicroBooNE) (2021) arXiv:2111.03556 [hep-ex] .
  * Ji  _et al._ [2020] X. Ji, W. Gu, X. Qian, H. Wei, and C. Zhang, Nucl. Instrum. Meth. A 961, 163677 (2020), arXiv:1903.07185 [physics.data-an] .
  * Abratenko  _et al._ [2021k] P. Abratenko _et al._ (MicroBooNE), JINST 16, P02017 (2021k), arXiv:2002.09375 [physics.ins-det] .
  * Abratenko  _et al._ [j] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.11874 [hep-ex] .
  * Abratenko  _et al._ [2021l] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 103, 052012 (2021l), arXiv:2012.08513 [physics.ins-det] .
  * Abratenko  _et al._ [2021m] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 103, 092003 (2021m), arXiv:2010.08653 [hep-ex] .
  * Abratenko  _et al._ [k] P. Abratenko _et al._ (MicroBooNE), arXiv:2109.02460 [physics.ins-det] .
  * Adamson  _et al._ [2016] P. Adamson _et al._ , Nucl. Instrum. Meth. A 806, 279 (2016), arXiv:1507.06690 [physics.acc-ph] .
  * Qian  _et al._ [2018] X. Qian, C. Zhang, B. Viren, and M. Diwan, JINST 13, P05032 (2018), arXiv:1803.04850 [physics.ins-det] .
  * Abratenko  _et al._ [2021n] P. Abratenko _et al._ (MicroBooNE), JINST 16, P06043 (2021n), arXiv:2011.01375 [physics.ins-det] .
  * Abratenko  _et al._ [l] P. Abratenko _et al._ (MicroBooNE), arXiv:2110.13961 [hep-ex] .
  * Abratenko  _et al._ [m] P. Abratenko _et al._ (MicroBooNE), https://microboone.fnal.gov/wp-content/uploads/MICROBOONE-NOTE-1043-PUB.pdf.
  * D’Agostini [1995] G. D’Agostini, Nucl. Instrum. Meth. A 362, 487 (1995).
  * Aguilar-Arevalo  _et al._ [2018] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 121, 221801 (2018), arXiv:1805.12028 [hep-ex] .
  * Feldman and Cousins [1998] G. J. Feldman and R. D. Cousins, Phys. Rev. D 57, 3873 (1998), arXiv:physics/9711021 .
  * [78] M. Antonello _et al._ (MicroBooNE, LAr1-ND, ICARUS-WA104), arXiv:1503.01520 [physics.ins-det] .


# On exact relations for the calculation of effective properties of composites

Biswajit Banerjee<EMAIL_ADDRESS>and Daniel O. Adams
Dept. of Mechanical Engineering, University of Utah, Salt Lake City, USA

(June 2002)

## Abstract

Numerous exact relations exist that relate the effective elastic properties of
composites to the elastic properties of their components. These relations can
not only be used to determine the properties of certain composites, but also
provide checks on the accuracy on numerical techniques for the calculation of
effective properties. In this work, some exact relations are discussed and
estimates from finite element calculations, the generalized method of cells
and the recursive cell method are compared with estimates from the exact
relations. Comparisons with effective properties predicted using exact
relations show that the best estimates are obtained from the finite element
calculations while the moduli are overestimated by the recursive cell method
and underestimated by the generalized method of cells. However, not all exact
relations can be used to make such a distinction.

##  1 Introduction

Exact relations for the effective elastic properties of two-component
composites can be classified into three types. The first type consists of
relations that have been determined from the similarity of the two-dimensional
stress and strain fields for certain types of materials. These exact relations
are called duality relations [1]. The second type of exact relations, called
translation-based relations, state that if a constant quantity is added to the
elastic moduli of the component materials then the effective elastic moduli
are also “translated” by the same amount. Microstructure independent exact
relations, valid for special combinations of the elastic properties of the
components, form the third category [2]. The known exact relations are
directly applicable only to a limited range of properties of the components.
Therefore the utility of these relations lies not only in determining the
effective elastic properties of a small range of composites but also in
evaluating the accuracy of numerical and analytical methods of computing
effective properties. In this work, predictions from exact relations are
compared with estimates from finite element calculations, the generalized
method of cells (GMC) [3], and the recursive cell method (RCM) [4]. The goal
is to assess the effectiveness of these relations in evaluating the accuracy
of the three numerical methods, especially with regard to high modulus
contrast materials such as polymer bonded explosives.

Five exact relations are explored in this work. The first is a duality-based
identity for the effective shear modulus that is valid for phase-
interchangeable materials [5]. The second is a set of duality relations that
are valid for materials that are rigid with respect to shear [1]. Two
translation-based relations are explored next - the CLM theorem [6] and a
relation for symmetric composites with equal bulk modulus [5]. The
microstructure independent Hill’s relation [7] is explored last.

##  2 Phase interchange identity

A symmetric composite is one that is invariant with respect to interchange of
the components. A checkerboard, as shown in Figure 1, is an example of a
symmetric composite.

Figure 1: Representative volume element for a checkerboard composite.

The phase interchange identity [5] for the effective shear modulus of a
symmetric two-dimensional two-component isotropic composite is a duality-based
exact relation that states that

$G_{\text{eff}}=\sqrt{G_{1}G_{2}}$ (1)

where $G_{1}$, $G_{2}$ are the shear moduli of the two components and
$G_{\text{eff}}$ is the effective shear modulus.

The phase interchange identity is valid only for isotropic composites. In a
finite-sized representative volume element (RVE) for a checkerboard composite
the shear modulus is not the same all directions and hence isotropy is not
achieved. The two-dimensional stress-strain relation for such a RVE with
“square symmetry” can be written as

$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{lll}K+\mu^{1}&K-\mu^{1}&0\\\
K-\mu^{1}&K+\mu^{1}&0\\\
0&0&\mu^{2}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ 2\left<\epsilon_{12}\right>_{\text{V}}\end{array}\right]$ (2)

where $\sigma_{11}$, $\sigma_{22}$, $\sigma_{12}$ are the stresses;
$\epsilon_{11}$, $\epsilon_{22}$, $\epsilon_{12}$ are the strains; $K$ is the
two-dimensional bulk modulus, $\mu^{1}$ is the shear modulus when shear is
applied along the diagonals of the RVE, and $\mu^{2}$ is the shear modulus for
shear along the edges of the RVE.

The numerical verification of the phase interchange identity, therefore,
requires that the components of the composite be chosen so that the difference
between $\mu^{1}$ and $\mu^{2}$ for the composite is minimal. This implies
that the components should have a weak modulus contrast.

Numerical estimates of the effective elastic properties of the checkerboard
composite shown in Figure 1 were obtained using finite elements (FEM), the
recursive cell method (RCM) and the generalized method of cells (GMC).
Following the requirement of low modulus contrast, both components were
assigned a Young’s modulus of 15,300 MPa. The Poisson’s ratio of the first
component was fixed at 0.32 while that of the second component was varied from
0.1 to 0.49. The FEM calculations were performed using a mesh of
256$\times$256 four-noded square elements. The RCM calculations used a grid of
64$\times$64 subcells with blocks of 2$\times$2 subcells and each subcell was
modeled using one nine-noded element. The GMC calculations used 64$\times$64
square subcells to discretize the RVE.

Figure 2 shows a comparison of the exact effective shear modulus for the
checkerboard composite with estimates of $\mu^{1}$ and $\mu^{2}$ from the
three numerical approaches.

Figure 2: Validation of FEM, RCM and GMC using the phase interchange identity
for a checkerboard composite.

The results show that all the three methods perform well (the maximum error is
0.1%) in predicting the effective shear modulus when the modulus contrast is
small, i.e., when the composite is nearly isotropic. It can also be observed
that the values of $\mu^{1}$ and $\mu^{2}$ are within 1% of each other for the
chosen component moduli.

###  2.1 Range of applicability

The question that arises at this point is whether the three numerical
approaches can predict the phase interchange identity for larger modulus
contrasts. Numerical calculations have been performed on the checkerboard
microstructure to explore this issue. The first component of the checkerboard
was assigned a Young’s modulus of 15,300 MPa and a Poisson’s ratio of 0.32.
For the second component, the Poisson’s ratio was fixed at 0.49 and the
Young’s modulus was varied from 0.7 MPa to 7000 MPa.

Figure 3 shows plots of the effective $\mu^{1}$ and $\mu^{2}$ versus shear
modulus contrast for a checkerboard RVE.

Figure 3: Variation of effective shear moduli with modulus contrast for a
checkerboard composite.

The plots confirm that when the modulus contrast between the components of the
checkerboard exceeds 2, the material can no longer be considered isotropic
since the values of $\mu^{1}$ and $\mu^{2}$ are considerably different from
each other. However, the values of $\mu^{1}$ predicted by FEM are quite close
to the effective shear modulus $G_{\text{eff}}$ predicted by the phase
interchange identity. This result suggests that the simulation of a diagonal
shear may not be necessary to predict the effective shear modulus of an
isotropic composite when the finite element approach is used. It also implies
that the phase interchange identity can be used for a much larger range of
modulus contrasts. The effective shear moduli predicted by GMC are
considerably lower than that from the exact relation while the values from RCM
are consistently higher. The RCM estimates worsen with increasing modulus
contrast. If only the value of $\mu^{1}$ is examined, the phase interchange
identity indicates that the FEM approach is much more accurate than the GMC
and RCM approaches. However, it is difficult to choose between GMC and RCM for
high modulus contrasts composites. While the exact value of $\mu^{1}$ is 10
times the value predicted by GMC, the corresponding RCM estimate is 10 times
the exact value. These results confirm the findings of detailed numerical
studies on high modulus contrast, high volume fraction polymer bonded
explosives [8, 9, 4].

###  2.2 Convergence of FEM calculations

The checkerboard material provides an extreme case to test the convergence of
the FEM solution because the corner singularities lead to high stresses that
can only be resolved with refined meshes. Figure 4 shows the convergence of
the effective $\mu^{1}$ and $\mu^{2}$ with increasing mesh refinement for a
checkerboard with a shear modulus contrast of about 25,000.

Figure 4: Convergence of effective moduli predicted by FEM with increase in
mesh refinement for a checkerboard composite with a shear modulus contrast of
25,000.

The effective $\mu^{1}$ converges to a steady value when 128$\times$128
elements are used to discretize the RVE. The shear modulus $\mu^{2}$ reaches a
steady value when 256$\times$256 elements are used. This is why the finite
element calculations in this work were performed using 256$\times$256 elements
or more.

RCM uses a finite element approach to homogenize blocks of subcells. When
blocks of 2$\times$2 subcells are used, some of these blocks can resemble
checkerboards - especially at the first level of recursion for a two-component
composite. The finite element convergence result suggests that RCM may
overestimate the effective shear moduli by a factor of two if a block of four
subcells is simulated using only four finite elements.

##  3 Materials rigid in shear

The stress-strain response of two-dimensional materials that are rigid with
respect to shear can be represented by

$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{ccc}S_{11}&S_{12}&0\\\
S_{12}&S_{22}&0\\\
0&0&0\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]$ (3)

where $\sigma_{11}$, $\sigma_{22}$, $\sigma_{12}$ are the stresses;
$\epsilon_{11}$, $\epsilon_{22}$ and $\epsilon_{12}$ are the strains, and
$S_{ij}$ are the components of the compliance matrix.

Two duality-based exact relations that are valid for two-component composites
composed of such materials are [1]:

Relation RS1

If $S_{11}S_{22}-(S_{12})^{2}=\Delta$ for each phase (where $\Delta$ is a
constant), then the effective compliance tensor also satisfies the same
relationship, i.e.,
$S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=\Delta_{\text{eff}}$.
This relation is true for all microstructures.

Relation RS2

If the compliance tensors of the two phases are of the form
$\mathbf{S}_{1}=\alpha_{1}\mathbf{A}$ and
$\mathbf{S}_{2}=\alpha_{2}\mathbf{A}$ where $\mathbf{A}$ is a constant matrix,
then the effective compliance tensor of a checkerboard of the two phases
satisfies the relation
$\det{\mathbf{S}_{\text{eff}}}=S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=\alpha_{1}\alpha_{2}(A_{11}A_{22}-(A_{12})^{2})$.

###  3.1 Relation RS1

Figure 5 shows a square array of disks occupying an area fraction of 0.7.

Figure 5: RVE for a square array of disks.

Numerical experiments have been performed on this array of disks to check if
Relation RS1 can be reproduced by finite element analyses, GMC and RCM. The
$\mathbf{S}$ matrices that have been used for the disks (superscript $1$) and
the matrix (superscript $2$), and the corresponding values of $\Delta$ are
shown below. These matrices have been chosen so that the value of $\Delta$ is
constant.

$\mathbf{S}_{1}=\left[\begin{array}[]{ccc}1000&-300&0\\\ -300&1000&0\\\
0&0&0.001\end{array}\right],~{}\Delta=9.1\times 10^{5},$

and

$\mathbf{S}_{2}=\left[\begin{array}[]{ccc}1094.3&-536.21&0\\\
-536.21&1094.3&0\\\ 0&0&0.001\end{array}\right],~{}\Delta=9.1\times 10^{5}.$

The shear modulus for both materials is $1000$ (arbitrary units) - around
$10^{6}$ times the Young’s modulus. Higher values of shear modulus have been
tested and found not to affect the effective stiffness matrix significantly.

Table 1 shows the values of $S^{\text{eff}}_{11}$, $S^{\text{eff}}_{12}$ and
$\Delta_{\text{eff}}$ calculated using finite elements (350$\times$350
elements), GMC (64$\times$64 subcells) and RCM (256$\times$256 subcells).

Table 1: Two-dimensional effective compliance matrix for a square array of disks. | $S^{\text{eff}}_{11}$ | $S^{\text{eff}}_{12}$ |  $\Delta_{\text{eff}}$ $(\times 10^{5})$ |  $\Delta_{\text{eff}}$/$\Delta$
---|---|---|---|---
FEM | 850.35 | -536.32 | 4.35 | 0.48
RCM | 847.25 | -538.17 | 4.28 | 0.47
GMC | 871.75 | -517.14 | 4.93 | 0.54

The ratio of the calculated $\Delta_{\text{eff}}$ to the original $\Delta$ are
also shown in the table. The modulus contrast between the two components of
the composite is small, so the calculated effective properties are expected to
be accurate (based on the results on the phase interchange identity for shear
moduli). However, the results in Table 1 show that all the three numerical
methods predict values of $\Delta_{\text{eff}}$ that are around half the
original $\Delta$. These results imply that all three methods (FEM, GMC and
RCM) overestimate the effective normal stiffness of the array of disks.
Relation RS1 for materials rigid in shear may therefore be a very sensitive
test of the accuracy of numerical methods even though the modulus contrast
that can be used is small.

###  3.2 Relation RS2

The second duality relation for materials that are rigid in shear requires
(Relation RS2) is valid for the checkerboard geometry shown in Figure 1. The
following values of the elastic properties have been used to test the accuracy
of FEM, RCM and GMC in predicting this relation.

$\displaystyle\mathbf{S}_{1}$
$\displaystyle=100\left[\begin{array}[]{cc}10&-3\\\
-3&10\end{array}\right]~{};~{}\mathbf{S}_{2}=1000\left[\begin{array}[]{cc}10&-3\\\
-3&10\end{array}\right]$ $\displaystyle\alpha_{1}$
$\displaystyle=100~{};~{}\alpha_{2}=1000$ $\displaystyle\mathbf{A}$
$\displaystyle=\left[\begin{array}[]{cc}10&-3\\\ -3&10\end{array}\right]$

The duality relation requires that the effective compliance matrix of the
checkerboard composite should be such that

$\det(\mathbf{S}_{\text{eff}})=S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=9.10\times
10^{6}~{}.$

The FEM calculations were performed using 350$\times$350 four-noded elements,
the RCM calculations used 64$\times$64 subcells (blocks of 2$\times$2
subcells) and the GMC calculations used 64$\times$64 subcells too. The results
from these three methods are tabulated in Table 2.

Table 2: Effective compliance matrix for a checkerboard composite with components rigid in shear. | $S^{\text{eff}}_{11}$ | $S^{\text{eff}}_{12}$ | $\det(\mathbf{S}_{\text{eff}})(\times 10^{6})$ | $\det(\mathbf{S}^{\text{eff}})/\alpha_{1}\alpha_{2}\det{\mathbf{A}}$
---|---|---|---|---
FEM | 3282 | -2004 | 6.75 | 0.74
RCM | 1655 | -7090 | 2.23 | 0.24
GMC | 5007 | -2146 | 2.05 | 2.25

The finite element calculations lead to quite an accurate effective compliance
matrix and the deviation from the exact result is only around 25%. The GMC
calculations overestimate the compliance matrix and the determinant of the
compliance matrix is around 2.3 times higher than the exact result. On the
other hand, the RCM calculations predict a compliance matrix that has a
determinant that is only around 20

##  4 The CLM theorem

The Cherkaev, Lurie and Milton (CLM) theorem is a well known “translation”
based exact relation for two-component planar composites [6]. For a two-
dimensional two-component isotropic composite, this theorem can be stated as
follows.

Let the isotropic bulk moduli of the components be $K_{1}$ and $K_{2}$. Let
the shear moduli of the two components be $G_{1}$ and $G_{2}$. The effective
bulk and shear modulus of a two-dimensional composite made of these two
components are $K_{\text{eff}}$ and $G_{\text{eff}}$ respectively. Let us now
create two new materials that are “translated” from the original component
materials by a constant amount $\lambda$. That is, let the bulk and shear
moduli of the translated component materials be given by

$\displaystyle
1/K^{T}_{1}=1/K_{1}-\lambda~{};~{}1/K^{T}_{2}=1/K_{2}-\lambda~{};$
$\displaystyle
1/G^{T}_{1}=1/G_{1}+\lambda~{};~{}1/G^{T}_{2}=1/G_{2}+\lambda~{}.$

The CLM theorem states that the effective bulk and shear moduli of a two-
dimensional composite of the two translated materials, having the same
microstructure as the original composite, are given by

$1/K^{T}_{\text{eff}}=1/K_{\text{eff}}-\lambda~{};~{}1/G^{T}_{\text{eff}}=1/G_{\text{eff}}+\lambda.$
(6)

The requirement of isotropy can be satisfied approximately by choosing
component material properties that are close to each other. Since our goal is
to determine how well GMC and RCM perform for high modulus contrast, choosing
materials with small modulus contrast is not adequate. Another alternative is
to choose a RVE that represents a hexagonal packing of disks. However, such an
RVE is necessarily rectangular and cannot be modeled using RCM in its current
form. It should be noted that RCM can easily be modified to deal with elements
that are not square and hence to model rectangular regions.

Another problem in the application of the CLM theorem is that the value of
$\lambda$ has to be small if the difference between the original and the
translated moduli is large and vice versa. If the value of $\lambda$ is small,
floating point errors can accumulate and exceed the value of $\lambda$. On the
other hand, if $\lambda$ is large, the original and the translated moduli are
very close to each other and the difference between the two can be lost
because of errors in precision. Hence, the numbers have to be chosen carefully
keeping in mind the limits on the value of the Poisson’s ratio.

The translation relation has been tested on the square array of disks
occupying a volume fraction of 0.70 from Figure 5. This RVE exhibits square
symmetry, i.e., the shear moduli $\mu^{1}$ and $\mu^{2}$ shown in equation (2)
are not equal. A unique value of the effective shear modulus cannot be
calculated for this RVE. Instead, he value of the effective translated shear
modulus is calculated from equation (6) by first setting $G_{\text{eff}}$
equal to $\mu^{1}$ and then to $\mu^{2}$. These “exact” values are compared
with the $\mu^{1}$ and $\mu^{2}$ values predicted using finite element
analyses, GMC and RCM.

The original set of elastic moduli for the RVE is chosen to reflect the
elastic moduli of the constituents of polymer bonded explosives. These moduli
are then translated by a constant $\lambda=0.001$. The original and the
translated constituent two-dimensional moduli are shown in Table 3 (phase ’p’
represents the particles and phase ’b’ represents the binder). It can be
observed that the translation process creates quite a large change in the bulk
modulus of the particles.

Table 3: Original and translated two-dimensional constituent moduli for checking the CLM condition. | $K_{p}$ | $G_{p}$ | $K_{b}$ | $G_{b}$ | $K_{p}/K_{b}$ | $G_{p}/G_{b}$
---|---|---|---|---|---|---
| $(\times 10^{2})$ | $(\times 10^{2})$ |  |  | $(\times 10^{2})$ | $(\times 10^{2})$
Original | 9.60 | 4.80 | 10.07 | 0.20 | 0.95 | 23.8
Translated | 240.0 | 3.24 | 10.17 | 0.20 | 23.5 | 16.1

Table 4 shows the effective bulk and shear moduli of the original and the
translated material calculated using finite elements (350$\times$350
elements), GMC (64$\times$64 subcells) and RCM (256$\times$256 subcells). The
values of $\lambda_{\text{err}}$ shown in the table have been calculated using
the equation

$\displaystyle\lambda_{\text{err}}$ $\displaystyle=(\lambda/0.001-1)\times
100,$
$\displaystyle\lambda=1/K_{\text{eff}}-1/K^{T}_{\text{eff}}=1/\mu^{i(T)}_{\text{eff}}-1/\mu^{i}_{\text{eff}}.$

Table 4: Comparison of effective moduli for the original and the translated composites. | $K_{\text{eff}}$ | $\mu^{1}_{\text{eff}}$ | $\mu^{2}_{\text{eff}}$
---|---|---|---
| Orig. | Trans. |  $\lambda_{\text{err}}$(%) | Orig. | Trans. |  $\lambda_{\text{err}}$(%) | Orig. | Trans. |  $\lambda_{\text{err}}$(%)
FEM | 36.4 | 37.8 | -0.8 | 10.1 | 10 | -3.1 | 0.9 | 0.9 | 22
RCM | 42.5 | 44.5 | 6.1 | 29.8 | 29 | -6.9 | 1.3 | 1.3 | -292
GMC | 34.0 | 35.1 | -1.3 | 3.8 | 3.8 | 5.3 | 0.7 | 0.7 | 30

Even though the modulus contrast between the two components of the composite
is high, the effective properties predicted by FEM, GMC and RCM are close to
each other in magnitude. The effective moduli of the translated composite are
also quite close to that of the original composite as predicted by the CLM
condition. The interesting fact is that all the three methods satisfy the CLM
condition and the error is small (as seen by the values of
$\lambda_{\text{err}}$. Of the three methods, FEM and GMC produce the least
error while RCM produces the most error.

##  5 Composites with equal bulk modulus

The translation procedure can also be used to generate an exact solution for
the effective shear modulus of two-dimensional symmetric two-component
composites with both components having the same bulk modulus [5]. This
relation is

$\displaystyle K_{\text{eff}}$ $\displaystyle=~{}K~{}=~{}K_{1}~{}=~{}K_{2}$
$\displaystyle G_{\text{eff}}$
$\displaystyle=\frac{K}{-1+\sqrt{\left(1+\frac{K}{G_{1}}\right).\left(1+\frac{K}{G_{2}}\right)}}$
(7)

This relation has been tested on the checkerboard model shown in Figure 1
using the component material properties given in Table 5. The exact effective
properties for the composite, calculated using equation (7), are also given in
the table. The values of the effective moduli calculated using finite elements
(FEM), GMC and RCM are also shown in Table 5.

Table 5: Component properties, exact effective properties and numerically
computed effective properties for two-component symmetric composite with equal
component bulk moduli.

| $E$ | $\nu$ | $K$ | $G$
---|---|---|---|---
| $(\times 10^{2})$ |  | $(\times 10^{3})$ | $(\times 10^{2})$
Component 1 | 25.00 | 0.25 | 2.0 | 10.0
Component 2 | 1.19 | 0.49 | 2.0 | 0.4
Composite | 5.12 | 0.46 | 2.0 | 1.76

| $K_{\text{eff}}$ | $\mu^{1}_{\text{eff}}$ | Diff. | $\mu^{2}_{\text{eff}}$ | Diff.
---|---|---|---|---|---
| $(\times 10^{2})$ | $(\times 10^{2})$ | % | $(\times 10^{2})$ | %
FEM | 20 | 1.29 | -26.8 | 2.54 | 44.4
GMC | 20 | 0.77 | -56.3 | 0.77 | -56.3
RCM | 20 | 2.96 | 68.0 | 4.41 | 150.9

These results show that the effective two-dimensional bulk modulus is
calculated correctly by all the three methods. However, the shear moduli
calculated for the checkerboard microstructure are quite different from the
exact result. This exact result also shows that the FEM calculations are the
most accurate, followed by GMC and then RCM. The values of
$\mu^{1}_{\text{eff}}$ are also found to most closely approximate the value of
$G_{\text{eff}}$.

##  6 Hill’s equation

Hill’s equation [7] is an exact relation that is independent of
microstructure. This equation is valid for composites composed of isotropic
components that have the same shear modulus. For a two-dimensional two-
component composite, this equation can be written as

$\frac{1}{K_{\text{eff}}+G}=\frac{f_{p}}{K_{p}+G}+\frac{f_{b}}{K_{b}+G}$ (8)

where $f$ represents a volume fraction, $K$ represents a bulk modulus, and $G$
represents a shear modulus. The subscript ’$p$’ represents a particle
property, ’$b$’ represents a binder property, and ’eff’ represents the
effective property of the composite.

This relationship is verified using the RVE containing an array of disks
occupying 70% of the volume that is shown in Figure 5. Table 6 shows the
properties of the two components used to compare the predictions of finite
elements, GMC and RCM with the exact value of bulk modulus predicted by Hill’s
equation. It should be noted that the materials chosen are not quite
representative of polymer bonded explosive materials.

Table 6: Phase properties used for testing Hill’s equation and the exact effective moduli of the composite. | Vol. | $E$ | $\nu$ | $G$ | $K$
---|---|---|---|---|---
| Frac. | $(\times 10^{3})$ |  | $(\times 10^{3})$ | $(\times 10^{3})$
Disks | 0.7 | 3.00 | 0.25 | 1.20 | 2.40
Binder | 0.3 | 3.58 | 0.49 | 1.20 | 60.00
Composite | 1.0 | 3.22 | 0.34 | 1.20 | 3.82

Since the modulus contrast is small, the square array of disks is expected to
exhibit nearly isotropic behavior. Therefore, the predictions of finite
elements, GMC and RCM are expected to be close to the exact values of the
effective properties of the composite. The numerically calculated values of
the effective two-dimensional bulk and shear moduli of the composite are shown
in Table 7. The percentage difference of the effective bulk modulus from the
exact value is also shown in the table.

Table 7: Numerically computed effective properties for a square array of disks with equal component shear moduli. | $K_{\text{eff}}$ | % Diff. | $\mu^{1}_{\text{eff}}$ | $\mu^{2}_{\text{eff}}$
---|---|---|---|---
| ($\times 10^{3}$) |  | ($\times 10^{3}$) | ($\times 10^{3}$)
FEM | 3.98 | 4.4 | 1.20 | 1.20
RCM | 3.92 | 2.7 | 1.20 | 1.20
GMC | 3.66 | -4.2 | 1.20 | 1.20

The effective shear moduli predicted by all the three methods are exact. In
case of the effective bulk moduli, the RCM predictions are the most accurate
followed by GMC and the finite element based calculations. The finite element
based calculations overestimate the effective two-dimensional bulk modulus by
around 4.4% while GMC underestimates the bulk modulus by around 4.2%. Since
the error of estimation of all the three methods is small, it is suggested
that all three methods are accurate for low contrasts in the shear modulus.
However, Hill’s equation does not appear to be suitable for determining the
best numerical method of the three.

##  7 Summary and conclusions

Predictions from the phase interchange identity for the shear modulus are
closely approximated by the finite element approach (FEM), the recursive cell
method (RCM) and the generalized method of cells (GMC) for checkerboard
composites with low modulus contrast. However, for higher modulus contrasts
the FEM approximations of shear moduli are the most accurate. The RCM
predictions overestimate the shear modulus while GMC underestimates the shear
modulus. The exact relations for materials that are rigid in shear show that
all three numerical techniques are inaccurate. The exact relation for this
class of materials that is applicable to checkerboard materials shows that the
FEM calculations are the most accurate while both RCM and GMC perform poorly
in comparison. Though the predictions of the CLM theorem are quite accurately
predicted by all three numerical methods for high modulus contrast composites,
the FEM results show the least error between the original and the translated
effective properties while the RCM results show the largest error. The exact
relation for isotropic composites with components that have the same bulk
moduli also shows that the FEM predictions are the most accurate though they
are somewhat higher than the exact values. However, no such distinction
between the three methods can be made using Hill’s equation. These results
reflect previous studies for high modulus contrast, high volume fraction
polymer bonded explosives using FEM, RCM and RCM and show that exact relations
can be used to determine the accuracy of numerical methods without performing
detailed numerical studies.

## Acknowledgements

This research was supported by the University of Utah Center for the
Simulation of Accidental Fires and Explosions (C-SAFE), funded by the
Department of Energy, Lawrence Livermore National Laboratory, under
subcontract B341493.

## Appendix

The components of the two-dimensional stiffness matrix can be computed from
two-dimensional plane strain finite element analyses. However, the components
of the two-dimensional compliance matrix cannot be directly determined from
two-dimensional plane strain finite element analyses. The reasons for these
are discussed in this appendix. The approach taken to approximate the two-
dimensional compliance matrix is also discussed.

###  A.1 Two-Dimensional Stiffness Matrix

The stress-strain relation for an anisotropic linear elastic material is given
by

$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\ \sigma_{33}\\\
\sigma_{23}\\\ \sigma_{31}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{cccccc}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\
C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\
C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\
C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\
C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\
C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{33}\\\ \epsilon_{23}\\\ \epsilon_{31}\\\
\epsilon_{12}\end{array}\right].$ (9)

For the plane strain assumption, we have,

$\epsilon_{33}~{}=~{}\epsilon_{23}~{}=~{}\epsilon_{31}~{}=~{}0.$ (10)

Therefore, the stress-strain relation can be reduced to

$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{ccc}C_{11}&C_{12}&C_{16}\\\
C_{12}&C_{22}&C_{26}\\\
C_{16}&C_{26}&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{12}\end{array}\right].$ (11)

The six terms in the apparent two-dimensional stiffness matrix reduce to four
is the material is orthotropic, i.e.,

$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{ccc}C_{11}&C_{12}&0\\\
C_{12}&C_{22}&0\\\
0&0&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{12}\end{array}\right].$ (12)

The three constants $C_{11}$, $C_{12}$ and $C_{22}$ can be determined by the
application of normal displacements in the ’1’ and ’2’ directions
respectively. The constant $C_{66}$ can be determined using shear displacement
boundary conditions in a finite element analysis. Hence, it can be seen that
the stiffness matrix can be calculated directly from two-dimensional plane
strain based finite element analyses. This is not true for the compliance
matrix.

###  A.2 Two-Dimensional Compliance Matrix

The strain-stress relation for an anisotropic linear elastic material can be
written as

$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ \epsilon_{33}\\\
\epsilon_{23}\\\ \epsilon_{31}\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccccc}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\\
S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\\
S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\\
S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\\
S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{23}\\\ \sigma_{31}\\\
\sigma_{12}\end{array}\right].$ (13)

The relationship between the stiffness matrix and the compliance matrix is

$\left[\begin{array}[]{cccccc}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\\
S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\\
S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\\
S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\\
S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{array}\right]=\left[\begin{array}[]{cccccc}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\
C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\
C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\
C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\
C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\
C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{array}\right]^{-1}$ (14)

or,

$\mathbf{S}=\mathbf{C}^{-1}.$ (15)

It is obvious from the above equation that the apparent two-dimensional
compliance matrix is not equal to the inverse of the apparent two-dimensional
stiffness matrix, i.e.,

$\left[\begin{array}[]{ccc}S_{11}&S_{12}&S_{16}\\\ S_{12}&S_{22}&S_{26}\\\
S_{16}&S_{26}&S_{66}\end{array}\right]\neq\left[\begin{array}[]{ccc}C_{11}&C_{12}&C_{16}\\\
C_{12}&C_{22}&C_{26}\\\ C_{16}&C_{16}&C_{66}\end{array}\right]^{-1}.$ (16)

Hence, we cannot determine the two-dimensional compliance matrix if we only
know the two-dimensional stiffness matrix.

Let us again examine the effect of the plane-strain assumption on the stress-
strain relation. We then have

$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&S_{13}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{26}\\\ S_{13}&S_{23}&S_{33}&S_{36}\\\
S_{16}&S_{26}&S_{36}&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (17)

For orthotropic materials, this relation simplifies to

$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&S_{13}&0\\\
S_{12}&S_{22}&S_{23}&0\\\ S_{13}&S_{23}&S_{33}&0\\\
0&0&0&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (18)

This equation shows that we need to know the stress $\sigma_{33}$ to determine
the terms of the compliance matrix and hence three-dimensional analyses are
necessary. If we assume plane stress, we can determine the terms of the matrix
$\mathbf{S}$ directly. However, the apparent two-dimensional compliance matrix
for plane stress is not equal to that for plane strain and hence we cannot
apply this method for our purposes. This is why the plane strain compliance
matrix cannot be determined using two-dimensional finite element analyses
only.

###  A.3 Approximation of Compliance Matrix

The two-dimensional compliance matrix can be determined approximately for
materials with square symmetry by assuming that $S_{13}$, $S_{23}$ and
$S_{33}$ are known. Let,

$S_{13}=S_{23}=-\frac{\nu_{3}}{E_{3}}\\\ S_{33}=\frac{1}{E_{3}}$ (19)

where, $\nu_{3}$ is the Poisson’s ratio in the out-of-plane direction and
$E_{3}$ is the Young’s ratio in that direction. Then, for a material with
square symmetry,

$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&-\frac{\nu_{3}}{E_{3}}&0\\\
S_{12}&S_{11}&-\frac{\nu_{3}}{E_{3}}&0\\\
-\frac{\nu_{3}}{E_{3}}&-\frac{\nu_{3}}{E_{3}}&\frac{1}{E_{3}}&0\\\
0&0&0&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (20)

Inverting the relation, we have,

$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\ \sigma_{33}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{cccc}C_{11}&C_{12}&C_{13}&0\\\
C_{12}&C_{11}&C_{23}&0\\\ C_{13}&C_{23}&C_{33}&0\\\
0&0&0&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ 0\\\ \epsilon_{12}\end{array}\right].$ (21) where,
$\displaystyle
C_{11}=\frac{E_{3}S_{11}-\nu_{3}^{2}}{E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}-E_{3}S_{12}^{2}+2\nu^{2}S_{12}}~{},$
$\displaystyle
C_{12}=\frac{-E_{3}S_{12}+\nu_{3}^{2}}{E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}-E_{3}S_{12}^{2}+2\nu^{2}S_{12}}.$

Note that it is not necessary to know $C_{13}$, $C_{23}$ and $C_{33}$ to
determine $S_{11}$ and $S_{12}$.

We can write the above relations between $C_{11},C_{12}$ and $S_{11},S_{12}$
in the form

$\displaystyle
E_{3}S_{11}^{2}-\left(\frac{E_{3}}{C_{11}}+2\nu_{3}^{2}\right)S_{11}-\left(E_{3}S_{12}^{2}-2\nu_{3}^{2}S_{12}-\frac{\nu_{3}^{2}}{C_{11}}\right)$
$\displaystyle=0,$ (22) $\displaystyle
E_{3}S_{12}^{2}-\left(\frac{E_{3}}{C_{12}}+2\nu_{3}^{2}\right)S_{12}-\left(E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}+\frac{\nu_{3}^{2}}{C_{12}}\right)$
$\displaystyle=0.$ (23)

In simplified form,

$\displaystyle A_{1}S_{11}^{2}+B_{1}S_{11}+C_{1}$ $\displaystyle=0,$ (24)
$\displaystyle A_{2}S_{12}^{2}+B_{2}S_{12}+C_{2}$ $\displaystyle=0.$ (25)

We can solve these quadratic equations to get expressions for $S_{11}$ and
$S_{12}$ as

$\displaystyle S_{11}$ $\displaystyle=\frac{-B+\sqrt{B^{2}-4AC}}{2A},$ (26)
$\displaystyle S_{12}$ $\displaystyle=\frac{-B-\sqrt{B^{2}-4AC}}{2A}.$ (27)

Knowing $C_{11}$, $C_{12}$, $E_{3}$ and $\nu_{3}$ these two equations can be
solved iteratively to determine $S_{11}$ and $S_{12}$. The values of $C_{11}$
and $C_{12}$ can be determined using the procedure outlined at the beginning
of this section. It remains to be discussed how $E_{3}$ and $\nu_{3}$ are to
be determined.

###  A.4 Determination of $E_{3}$ and $\nu_{3}$

Two methods can be used to determine the values of $E_{3}$ and $\nu_{3}$ for
our calculations. The first method is to assume that the rule of mixtures is
accurate enough to determine the effective properties in the ’3’ direction.
Thus, if the volume fraction of the first component is $f_{1}$ and that of the
second component is $f_{2}$, we have,

$\displaystyle E_{3}$ $\displaystyle=f_{1}E_{1}+f_{2}E_{2},$ (28)
$\displaystyle\nu_{3}$ $\displaystyle=f_{1}\nu_{1}+f_{2}\nu_{2}.$ (29)

where $E_{i}$ and $\nu_{i}$ are the Young’s modulus and the Poisson’s ratio of
the $i$th component.

The other option is to use the values of $S_{13}$, $S_{23}$ and $S_{33}$
obtained from GMC since these are also quite accurate for the out of plane
direction. Thus, we have,

$\displaystyle E_{3}$ $\displaystyle=\frac{1}{S_{33}^{\text{GMC}}},$ (30)
$\displaystyle\nu_{3}$ $\displaystyle=-S_{13}^{\text{GMC}}E_{3}.$ (31)

This is the procedure we have use to determine the effective compliance
matrices discussed in this paper.

## References

  * [1] J. Helsing, G. W. Milton, and A. B. Movchan.  Duality relations, correspondences, and numerical results for planar elastic composites.  J. Mech. Phys. Solids, 45(4):565–590, 1997.
  * [2] G. W. Milton.  Composites : a myriad of microstructure independent relations.  In T. Tatsumi, E. Watanabe, and T. Kambe, editors, Theoretical and Applied Mechanics (Proc. XIX International Congress of Theoretical and Applied Mechanics, Kyoto, 1996), pages 443–459. Elsevier, Amsterdam, 1997.
  * [3] J. Aboudi.  Micromechanical analysis of composites by the method of cells - update.  Appl. Mech. Rev, 49(10):S83–S91, 1996.
  * [4] B. Banerjee and D. O. Adams.  On predicting the effective elastic properties of polymer bonded explosives using the recursive cell method.  2002\.
  * [5] G. W. Milton.  Theory of Composites.  Cambridge University Press, New York, 2002.
  * [6] A. V. Cherkaev, K. A. Lurie, and G. W. Milton.  Invariant properties of the stress in plane elasticity and equivalence classes of composites.  Proc. R. Soc. Lond. A, 438(1904):519–529, 1992.
  * [7] R. Hill.  Theory of mechanical properties of fibre-strengthened materials: I. elastic behaviour.  J. Mech. Phys. Solids, 12:199–212, 1964.
  * [8] B. Banerjee and D. O. Adams.  Effective elastic moduli of polymer bonded explosives from finite element simulations.  2002\.
  * [9] B. Banerjee and D. O. Adams.  Application of the generalized method of cells to polymer bonded explosives.  2002\.