# Probing CPV mixing in the Higgs sector in VBF at 1 TeV ILC

N. Vukas̆inović<EMAIL_ADDRESS>I. Božović-Jelisavc̆ić  G.
Kac̆arević  I. Smiljanić  I. Vidaković  “VINC̆A” Institute of Nuclear Sciences
- National Institute of the Republic of Serbia, University of Belgrade, 11001
Belgrade, Serbia

###### Abstract

With the current precision of measurements by the ATLAS and CMS experiments,
it cannot be excluded that a SM-like Higgs boson is a CP violating mixture of
CP-even and CP-odd states. We explore this possibility here, assuming Higgs
boson production in ZZ-fusion, at 1 TeV ILC, with unpolarized beams. The full
simulation of SM background and fast simulation of the signal is performed,
simulating 8 ab-1 of data collected with the ILD detector. We demonstrate that
the CP mixing angle $\Psi_{\mathrm{CP}}$ between scalar and pseudoscalar
states can be measured with the statistical uncertainty of 3.8 mrad at 68% CL,
corresponding to 1.44 $\cdot 10^{-5}$ for the CP parameter $f_{\mathrm{CP}}$,
for the pure scalar state. This is the first result on sensitivity of an
$e^{+}e^{-}$ collider to measure $f_{\mathrm{CP}}$ in the Higgs production
vertex in vector boson fusion.

††preprint: APS/123-QED

##  I  Introduction

Since the experimentally established CP violation (CPV) in the quark sector is
not sufficient to explain the baryon asymmetry of the observable Universe,
exploring the possibility that CP is violated in the Higgs sector is an
important part of the physics program both at ongoing experiments and future
Higgs factories. Although a purely CP-odd state for the Higgs boson is already
excluded by the ATLAS and CMS experiments [1, 2], there is still a possibility
that CP symmetry is violated in Higgs interactions with bosons and fermions.

Experiments at the International Linear Collider (ILC) [3, 4] will be able to
explore a plethora of Higgs production and decay mechanisms to probe bosonic
and fermionic vertices for CPV. A tentative list of the processes of interest
at ILC is illustrated in Table 1 [5]. The CPV effect is typically weaker in
Higgs interactions with vector bosons ($HVV$) in comparison to those with
fermions ($Hff$) since the pseudoscalar state does not directly couple to the
Standard Model (SM) particles, and sensitivity targets to measure CPV effects
in these interactions are thus different. In order to provide a common
platform for interpretation of the CPV measurements in bosonic $HVV$ and
fermionic $Hff$ vertices, as well as to interpret projections for different
future experiments, a common framework is defined in [6], based on the CPV
parameter $f_{\mathrm{CP}}$. The parameter $f_{\mathrm{CP}}$ quantifies the
CP-odd contribution to a Higgs interaction [6] as:

$f_{CP}^{hX}=\frac{\Gamma_{h\rightarrow
X}^{CP^{\mathrm{odd}}}}{\Gamma_{h\rightarrow
X}^{CP^{\mathrm{odd}}}+\Gamma_{h\rightarrow X}^{CP^{\mathrm{even}}}}$ (1)

assuming here the Higgs decay to a final state $X$. In order to explain the
baryon asymmetry of the Universe assuming the 2HDM model of an extend Higgs
sector, a minimal 10% contribution of the CP-odd state is required [6]. This
sets the theoretical target for future colliders to measure $f_{\mathrm{CP}}$
with an absolute precision better than 10-2 (10-6) in $Hff$ ($HVV$) vertices.
The state-of-the-art projections on $f_{\mathrm{CP}}$ sensitivity at different
colliders to measure a pure scalar state with 68% CL are given in Table 2. The
projected precision of measurements in $HVV(V=Z,W)$ vertices at future
$e^{-}e^{+}$ colliders is based on [7] where the estimate is given for the
$HZZ$ production vertex in Higgstrahlung, at 250 GeV center-of-mass energy,
assuming 2.5 ab-1 of data. The study is performed at the generator level and
thus without realistic simulation of a detector response for signal and
background. Estimates at higher center-of-mass energies for $HVV$ production
vertices (marked with ‘*’ in Table 2) are obtained by scaling of the result at
250 GeV to the corresponding integrated luminosities indicated in Table 2.

Table 1:  Possible Higgs production and decay modes to probe CPV at various center-of-mass energies at ILC. mode | fermion couplings
---|---
decay |  H$\rightarrow\tau^{+}\tau^{-}$ |  250+ GeV
production |  $e^{-}e^{+}\rightarrow Ht\bar{t}$ |  500+ GeV
| boson couplings
production |  $e^{-}e^{+}\rightarrow HZ$ |  250+ GeV
decay |  H$\rightarrow ZZ$ |  250+ GeV
decay |  H$\rightarrow WW$ |  250+ GeV
production |  $e^{-}e^{+}\rightarrow He^{-}e^{+}$ |  1000+ GeV
Figure 1:  Feynman diagram of the signal process. Table 2:  Expected precision of the CP parameter $f_{CP}$ (68% CL) for measurements in $HXX$ ($X=f,V$) vertices for the pure scalar state at various colliders. The mark ‘$\checkmark$’ indicates feasibility of such a measurement. The entries marked with ‘*’ are obtained by scaling the precision to measure $f_{CP}$ in $HZZ$ interaction at 250 GeV to the corresponding integrated luminosities indicated in the table. This table is adapted from Table I of [6].  Collider |  $pp$ |  $pp$ |  $pp$ |  $e^{-}e^{+}$ |  $e^{-}e^{+}$ |  $e^{-}e^{+}$ |  $e^{-}e^{+}$ |  $e^{-}p$ |  $\gamma\gamma$ |  $\mu^{+}\mu^{-}$ |  $\mu^{+}\mu^{-}$ |  target
---|---|---|---|---|---|---|---|---|---|---|---|---
E(GeV) |  14,000 |  14,000 |  100,000 |  250 |  350 |  500 |  1,000 |  1,300 |  125 |  125 |  3,000 |  (theory)
$\mathcal{L}$ (fb-1) |  300 |  3,000 |  30,000 |  2500 |  3500 |  5000 |  10,000 |  1,000 |  250 |  20 |  1,000 |
$HZZ/HWW$ |  4.0 $\cdot 10^{-5}$ |  2.5 $\cdot 10^{-6}$ |  ✓ |  3.9 $\cdot{10^{-5}}$ |  2.9 $\cdot{10^{-5}}^{(*)}$ |  1.3 $\cdot{10^{-5}}^{(*)}$ |  3.0 $\cdot{10^{-6}}^{(*)}$ |  ✓ |  ✓ |  ✓ |  ✓ |  $<10^{-5}$
$\mathcal{L}$ (fb-1) |  300 |  3,000 |  30,000 |  250 |  350 |  500 |  1,000 |  1,000 |  250 |  20 |  1,000 |
$H\gamma\gamma$ |  |  0.50 |  ✓ |  |  |  |  |  |  0.06 |  |  |  $<10^{-2}$
$HZ\gamma$ |  |  $\sim$ 1 |  ✓ |  |  |  |  $\sim$ 1 |  |  |  |  |  $<10^{-2}$
$Hgg$ |  0.12 |  0.011 |  ✓ |  |  |  |  |  |  |  |  |  $<10^{-2}$
$Ht\bar{t}$ |  0.24 |  0.05 |  ✓ |  |  |  0.29 |  0.08 |  ✓ |  |  |  ✓ |  $<10^{-2}$
$H\tau\tau$ |  0.07 |  0.008 |  ✓ |  0.01 |  0.01 |  0.02 |  0.06 |  |  ✓ |  ✓ |  ✓ |  $<10^{-2}$
$H\mu\mu$ |  |  |  |  |  |  |  |  |  |  ✓ |  |  $<10^{-2}$

The analysis presented below is the first result obtained for the Higgs
production in vector boson fusion (VBF), specifically in $ZZ$-fusion
$e^{-}e^{+}\rightarrow He^{-}e^{+}$, assuming 8 ab-1 of data collected with
the ILC operating at 1 TeV center-of-mass-energy with unpolarised beams. Since
$ZZ$-fusion is a t-channel process (Fig. 1), electrons and positrons in signal
events are peaked at small polar angles. The interplay between the production
cross-section and centrality of signal events makes 1 TeV an optimal energy
for CPV studies of the $HZZ$ vertex in VBF at an $e^{+}e^{-}$ collider.

The paper is organized as follows: Section II introduces event samples and
software tools, methodology of the measurement is discussed in Section III.1,
while statistical interpretation and discussion of the obtained results are
given in Section III.2.

##  II Event samples

In this analysis we consider Higgs boson production in $ZZ$-fusion (Fig. 1)
with the Higgs boson decaying to $b\bar{b}$ with a branching fraction of
$\sim$ 56% in the SM [8]. Exclusive reconstruction of the Higgs decays enables
us to avoid the high cross-section $e^{-}e^{+}\rightarrow e^{-}e^{+}\gamma$
background that would otherwise be present in an inclusive analysis. We have
generated 6 $\cdot 10^{5}$ signal events in Whizard 2.8.3 [9], using the Higgs
characterization model [10] within the UFO framework to allow contribution of
the CP-odd component to the 125 GeV Higgs mass eigenstate. In this model the
parametrization of CP mixing is entirely realized in terms of the mixing angle
between scalar and pseudoscalar states, allowing for a completely general
description of CP-mixed states [10]111In order to remove total cross section
dependence on the mixing angle, copuling of the CP-odd component in the
effective Lagrangian of [10] was set to $\kappa_{AZZ}\,=\,3.57$, for the CP-
even couplings $\kappa_{HZZ}=0$ and $\kappa_{H\partial Z}=1$..

Further interactions of signal with the detector are simulated assuming a
generic detector for ILC with the fast simulation DELPHES 3.4.2 (ILCgen cards)
[11]. For a smaller sample of signal events ($\sim$ 3500 events) the response
of the ILD detector [12] is fully simulated with detailed Geant4 [13]
simulation implemented in the Mokka toolkit [14]. These events are
reconstructed using realistic Marlin processors [15]. The Standard Model
background is also fully simulated and reconstructed with the ILD detector,
using the same simulation tools as for the signal. Backgrounds due to
beamstrahlung and hadron photo-production are overlaid onto the fully
simulated events in the digitization phase. Fragmentation and hadronization
are simulated in Pythia 6.4 [16] for all events. Particle identification is
based on the Particle Flow Approach implemented in the DDMarlinPandora
processor [17] of Marlin and, in a simplified version, in the fast simulation
DELPHES. The considered processes and their cross-sections are given in Table
3.

Table 3:  Cross-sections for signal and background processes with expected number of events in the full range of polar angles in 8 $\mathrm{ab^{-1}}$ of data, and number of reconstructed Monte Carlo events. Signal |  $\sigma(fb)$ |  Expected in 8 $\mathrm{ab^{-1}}$ |  Reconstructed MC events
---|---|---|---
$e^{-}e^{+}\rightarrow He^{-}e^{+}$; $H\rightarrow b\bar{b}$ |  13 |  104000 |  6 $\cdot 10^{5}$222DELPHES 3495888full sim.
Background††
$e^{-}e^{+}\rightarrow q\bar{q}e^{+}e^{-}$ |  2.4 $\cdot 10^{3}$ |  19 $\cdot 10^{6}$ |  2 $\cdot 10^{5}$
$e^{-}e^{+}\rightarrow qq$ |  3.6 $\cdot 10^{3}$ |  29 $\cdot 10^{6}$ |  4 $\cdot 10^{5}$
$e^{-}e^{+}\rightarrow q\bar{q}e\nu$ |  3 $\cdot 10^{3}$ |  24 $\cdot 10^{6}$ |  2.6 $\cdot 10^{6}$
$e^{-}e^{+}\rightarrow llll$ |  8 $\cdot 10^{3}$ |  64 $\cdot 10^{6}$ |  1.5 $\cdot 10^{6}$
$e^{-}e^{+}\rightarrow eeqqqq$ |  37 |  30 $\cdot 10^{4}$ |  1 $\cdot 10^{4}$
$e^{-}e^{+}\rightarrow e\nu_{e}qqqq$ |  51 |  4 $\cdot 10^{5}$ |  1 $\cdot 10^{6}$
$e^{-}e^{+}\rightarrow qq\nu_{e}ee\nu_{e}$ |  5.6 |  45 $\cdot 10^{3}$ |  5 $\cdot 10^{4}$

Since the signal signature is one electron-positron pair and two $b$-jets in
the final state, the event selection is based on identification of exactly one
isolated electron and one isolated positron per event while the remaining
Particle Flow Objects (PFOs) are clustered into two jets by the Durham
algorithm [18]. Electron and positron candidates are required each to have
energy above 60 GeV.

Electron isolation is based on different observables for events processed in
fast or full detector simulation. For signal events processed in DELPHES,
electrons are considered isolated if there is no additional particle with
transverse momentum greater than 0.5 GeV in a radius $R=$ 0.5 cone in
$\eta-\phi$ space around the electron candidate direction, and if the sum of
transverse momenta of all other particles within the cone is less than 12% of
the transverse momentum of the electron candidate. For fully simulated and
reconstructed events, electrons are selected if their transverse and
longitudinal impact parameters are less than 0.1 mm and 1 mm respectively,
ratio of depositions in electromagnetic versus hadronic calorimeter is larger
than 0.95, and the additional energy in a cone of size 0.1 rad around the
electron direction satisfies
$E_{\mathrm{c}one}^{\mathrm{2}}<40\,\mathrm{GeV}\cdot
E_{e^{\pm}}-20\,\mathrm{GeV^{2}}$.

A Multivariate Analysis (MVA) is used to further reduce the contribution from
several high cross-section background processes, in particular
$e^{-}e^{+}\rightarrow q\bar{q}e^{+}e^{-}$ with its signal-like signature. The
Boosted Decision Tree (BDT) classifier is employed, as implemented in the TMVA
toolkit [19]. Ten input observables are used: di-jet invariant mass
$m_{j\bar{j}}$, invariant mass $m_{e^{+}e^{-}}$ and transverse momentum
$p_{T}{{}_{ee}}$ of the final state $e^{+}e^{-}$ system, polar angle of the
di-jet system $\theta_{j\bar{j}}$, number of particle flow objects per event
$N_{\mathrm{PFO}}$, energies of final state $e^{+}$ and $e^{-}$, transverse
momenta of jets $p_{T}{{}_{j_{1}}}$ and $p_{T}{{}_{j_{2}}}$. Additional
requirements are applied to suppress background after the MVA application:
$m_{j\bar{j}}>110$ GeV, $p_{T}{{}_{j_{2}}}>160$ GeV and
$N_{\mathrm{PFO}_{1,2}}>10$, where $N_{\mathrm{PFO}_{1,2}}$ stands for the
number of particle flow objects in jets. The signal efficiency is obtained as
the ratio of the number of selected signal events and the number of signal
events with both $e^{+}$ and $e^{-}$ in the tracking region of polar angles
($|\cos\theta|<$ 0.98) and it is found to be 70%. Only two MC background
events remain after the selection, corresponding to around 240 background
events expected in 8 ab-1 of data. Selected signal and background are
collectively referred to as reconstructed data in further text, if not stated
otherwise.

##  III Method

In the Standard Model the Higgs boson is a CP-even state with the CP-
conserving couplings. In models with an extended Higgs sector, the 125 GeV
Higgs mass eigenstate ($h$) could be a mixture of CP-even ($H$) and CP-odd
($A$) states:

$h=H\cdot\cos\Psi_{\mathrm{CP}}+A\cdot\sin\Psi_{\mathrm{CP}},$ (2)

where $\Psi_{\mathrm{CP}}$ is the mixing angle violating CP symmetry in Higgs
interactions for non-zero values. There are several observables sensitive to
non-zero values of $\Psi_{\mathrm{CP}}$ [20], one of the most sensitive is the
angle $\Delta\Phi$ between scattering planes illustrated in Fig. 2.
$\Delta\Phi$ is the rotation angle of the positron plane with respect to the
electron plane in the Higgs reference frame, around the axis defined by the
momentum of the $Z$ boson emitted by the initial electron, following the
right-hand rule.

Figure 2:  Illustration of $\Delta\Phi$ angle between the scattering planes,
defined in the Higgs boson rest frame. Initial electron and positron are drawn
in red.

0.08(4.9,-2.05) ILD

Figure 3:  $\Delta\Phi$ distribution of $e^{-}e^{+}\rightarrow
q\bar{q}e^{+}e^{-}$ background after preselection of one isolated $e^{-}$ and
$e^{+}$ per event, with energies greater than 60 GeV.

$\Delta\Phi$ can be calculated as the angle between unit vectors
($\overrightarrow{n}_{1}$ and $\overrightarrow{n}_{2}$) orthogonal to electron
and positron scattering planes, respectively:

$\Delta\Phi=\mathrm{sgn}(\Delta\Phi)\cdot\arccos(\overrightarrow{n}_{1}\cdot\overrightarrow{n}_{2})$
(3)

where:

$\mathrm{sgn}(\Delta\Phi)=\frac{\overrightarrow{q}_{1}\cdot(\overrightarrow{n}_{1}\times\overrightarrow{n}_{2})}{|\overrightarrow{q}_{1}\cdot(\overrightarrow{n}_{1}\times\overrightarrow{n}_{2})|},$
(4)

and:

$\overrightarrow{n}_{1}=\frac{\overrightarrow{q}_{e^{-}_{i}}\times\overrightarrow{q}_{e^{-}_{f}}}{|\overrightarrow{q}_{e^{-}_{i}}\times\overrightarrow{q}_{e^{-}_{f}}|}\hskip
8.5359pt\mathrm{and}\hskip
8.5359pt\overrightarrow{n}_{2}=\frac{\overrightarrow{q}_{e^{+}_{i}}\times\overrightarrow{q}_{e^{+}_{f}}}{|\overrightarrow{q}_{e^{+}_{i}}\times\overrightarrow{q}_{e^{+}_{f}}|},$
(5)

$\overrightarrow{q}_{e^{-(+)}_{i(f)}}$ is the momentum of initial (final)
state electron (positron) and $\overrightarrow{q}_{1}$ is momentum of the $Z$
boson emitted by the initial electron.

The distribution of $\Delta\Phi$ for background is flat reflecting the fact
that background is CP insensitive, as illustrated for $q\bar{q}e^{+}e^{-}$
final state background remaining after selection of one isolated $e^{-}$ and
$e^{+}$ per event with energies greater than 60 GeV (Fig. 3). The event
selection described in the previous section does not bias the sensitive
observable. This is illustrated in Fig. 4.

Figure 4:  Signal selection efficiency w.r.t. the sensitive observable
$\Delta\Phi$. Figure 5:  $\Delta\Phi$ distribution for different
$\Psi_{\mathrm{CP}}$ values illustrating the shift of the $\Delta\Phi$ minimum
for non-zero values of $\Psi_{\mathrm{CP}}$. Figure 6:  $\Delta\Phi$
distributions for generated signal in the full physical range of polar angles,
reconstructed signal in DELPHES and reconstructed signal corrected for the
limited detector acceptance.  Figure 7:  Positions of the minima of
$\Delta\Phi$ distributions ($b/a$) over the true values of
$\Psi_{\mathrm{CP}}$.

###  III.1  $\Delta\Phi$ and CP mixing angle $\Psi_{\mathrm{CP}}$

Differently from $Hf\bar{f}$ vertices where the dependence of $\Delta\Phi$ on
$\Psi_{\mathrm{CP}}$ can be derived from the differential cross-section for
$H\rightarrow f\bar{f}$ decay [21], CP violating contributions in bosonic
$HVV$ vertices occur at the loop level and there is no simple analytical
dependence of the sensitive observable $\Delta\Phi$ on the CP mixing angle
$\Psi_{\mathrm{CP}}$. The dependence is therefore to be empirically
determined, in this case by correlating the position of the minimum of
$\Delta\Phi$ distribution to the true value of the mixing angle
$\Psi_{\mathrm{CP}}$ used in event generation. As can be seen from Fig. 5, the
position of the minimum of $\Delta\Phi$ shifts to larger values for positive
values of $\Psi_{\mathrm{CP}}$ (and similarly to the left for negative values
of $\Psi_{\mathrm{CP}}$). Before determining the position of the minimum,
reconstructed data has to be corrected for effects of detector acceptance, in
order to retrieve the information on the CP state of the Higgs boson in the
full physical range of polar angles. Fig. 6 illustrates the $\Delta\Phi$
distribution from generated signal in the full physical range, reconstructed
signal with the fast simulation and the signal corrected for the detector
acceptance to compensate for the limited acceptance in polar angles of the
central detector tracking system. The acceptance function is obtained as the
ratio of the generated $\Delta\Phi$ distributions for signal in the central
tracking region ($|\cos\theta|<$ 0.98) and in the full range of polar angles.

Figure 8:  Fit of $\Delta\Phi$ distribution of the selected and corrected
reconstructed data with the function $f(\Delta\Phi)$ from Eq. 6, in order to
obtain the position of the minimum ($b/a$).  Figure 9:  $\Psi_{\mathrm{CP}}$
values ($\Psi_{\mathrm{exp}}$) determined from Eq. 7 versus
$\Psi_{\mathrm{CP}}$ true values ($\Psi_{\mathrm{true}}$). The red line is
illustrating the ideal dependence $\Psi_{\mathrm{exp}}=\Psi_{\mathrm{true}}$.

The minimum of $\Delta\Phi$ distribution from the reconstructed data can be
determined by a local fit with the function $f(\Delta\Phi)$:

$f(\Delta\Phi)=A+B\cdot\cos(a\cdot\Delta\Phi-b)$ (6)

where $A,B,a$ and $b$ are free parameters. From the principle of the first
derivative, the ratio $b/a$ determines the minimum of $\Delta\Phi$
distribution.

(a)

(b)

Figure 10:  (a) Statistical dissipation of measured $\Psi_{\mathrm{CP}}$
values ($\Psi_{\mathrm{exp}}$) w.r.t. the true ones ($\Psi_{\mathrm{true}}$).
(b) Pull distribution for 2000 pseudo-experiments.

For $\Psi_{\mathrm{CP}}$ values up to 200 mrad, the variable
$(b/a)/\Psi_{\mathrm{CP}}$ is to a good approximation a linear function of
true values of $\Psi_{\mathrm{CP}}$, as shown in Fig. 7, with coefficients $k$
and $m$ determining a slope and a constant term, respectively. Knowing the
parameters $k$ and $m$ from simulation, $\Psi_{\mathrm{CP}}$ values can be
determined by solving the quadratic equation:

$k\cdot\Psi^{2}_{\mathrm{CP}}+m\cdot\Psi_{\mathrm{CP}}-(b/a)=0$ (7)

where the minimum $b/a$ is measured from experimental or in this case from the
reconstructed pseudo-data. The fit of reconstructed data corrected for the
detector acceptance is illustrated in Fig. 8. The Fig. 9 illustrates that
$\Psi_{\mathrm{CP}}$ values extracted this way are in agreement with the true
ones within the statistical uncertainties. Statistical uncertainties are
derived from the uncertainties of the fit parameters $a$ and $b$ from Eq. 6
and from the uncertainties of parameters $k$ and $m$ from Eq. 7. The method is
applicable to measure CP mixing angles up to approximately 200 mrad above
which the $\chi^{2}$ fit with the function $f(\Delta\Phi)$ (Eq. 6)
significantly deteriorates.

###  III.2  Statistical uncertainty and interpretation of the measurement

From the fit to the single pseudo-experiment assuming 8 ab-1 of data
illustrated in Fig. 8, one determines $\Psi_{\mathrm{CP}}$ = (2.4 $\pm$ 4.0)
mrad by solving Eq. 7. In order to estimate the statistical dispersion of
results of repeated $\Psi_{\mathrm{CP}}$ measurements, we performed 2000
pseudo-experiments each with 8 ab-1 of data. The dispersion of the results
assuming a pure scalar state is found to be 3.8 mrad at 68% CL, as illustrated
in Fig. 10 (a). Dispersion of errors from repeated pseudoexperiments is 0.4
mrad. The pull distribution shown in Fig. 10 (b) illustrates that the estimate
of the statistical uncertainty on $\Psi_{\mathrm{CP}}$ is reasonable. Allowing
parameters $k$ and $m$ (from Eq. 7) to vary within their uncertainties, we
have estimated a systematic uncertainty from modeling to be significantly less
than 1 mrad.

To interpret the obtained precision of measurement of the mixing angle in
terms of sensitivity to the CP-odd amplitude $f_{\mathrm{CP}}^{HZZ}$,
following [6] we assume that $f_{\mathrm{CP}}^{HZZ}$ will vary from zero as
$\sin^{2}(\Delta(\Psi_{\mathrm{CP}}))$ for the pure scalar state, where
$\Delta(\Psi_{\mathrm{CP}})$ is the absolute statistical uncertainty of the
$\Psi_{\mathrm{CP}}$ measurement. The statistical uncertainty of 3.8 mrad of
the $\Psi_{\mathrm{CP}}$ determination translates into $f_{\mathrm{CP}}^{HZZ}$
sensitivity of 1.44 $\cdot 10^{-5}$ at 68% CL. The comparable results can be
obtained if polarized samples of signal and background are considered, as the
ILC operation foresees 80% (20%) polarization for electron (positron) beams at
1 TeV center-of-mass energy.

##  IV Conclusion

This analysis brings the first result of the CP mixing angle measurement in
$HVV$ interactions where the Higgs boson is produced in vector boson fusion.
We assume $Hee$ production in $ZZ-$ fusion with the reconstruction of
exclusive Higgs decays to $b\bar{b}$ with the fast detector simulation.
Standard Model background is fully simulated assuming the ILD detector
response to 8 ab-1 of data collected at 1 TeV center-of-mass energy with
unpolarised beams. This measurement relies on the model-independent hypothesis
that the 125 GeV Higgs mass eigenstate could be a mixture of CP-even and CP-
odd states with the mixing angle $\Psi_{\mathrm{CP}}$. From the shape of
distribution of the CP sensitive angle between the scattering planes, the
mixing angle $\Psi_{\mathrm{CP}}$ can be extracted with a statistical
uncertainty of 3.8 mrad at 68% CL, for the pure scalar state. This translates
to the sensitivity of the CP parameter $f_{\mathrm{CP}}^{HZZ}$ of 1.44 $\cdot
10^{-5}$.

###### Acknowledgements.

This research is institutionally funded by the Ministry of Science and
Technological Development of the Republic of Serbia and by the Science Fund of
the Republic of Serbia through the Grant No. 7699827, IDEAS HIGHTONE-P.
Authors would also like to acknowledge our colleagues from ILC IDT Working
Group 3 and the ILD Detector Concept Group for useful discussions, and
especially to Prof. Aleksander Filip Żarnecki for sharing ideas leading to a
better understanding of the sensitive observable behavior w.r.t. the CP mixing
angle. We are also grateful to Dr. Daniel Jeans for careful reading of the
text.

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# The key to the enhanced performance of slab-like topologically interlocked
structures with non-planar blocks

Ioannis Koureas  Mohit Pundir  Shai Feldfogel  David S. Kammer
<EMAIL_ADDRESS>
###### Abstract

Topologically interlocked structures are assemblies of interlocking blocks
that hold together solely through contact. Such structures have been shown to
exhibit high strength, energy dissipation, and crack arrest properties. Recent
studies on topologically interlocked structures have shown that both the peak
strength and work-to-failure saturate with increasing friction coefficient.
However, this saturated structural response is only achievable with
nonphysically high values of the friction coefficient. For beam-like
topologically interlocked structures, non-planar blocks provide an alternate
approach to reach similar structural response with friction properties of
commonly used materials. It remains unknown whether non-planar blocks have
similar effects for slab-like assemblies, and what the achievable structural
properties are. Here, we consider slab-like topologically interlocked
structures and show, using numerical simulations, that non-planar blocks with
wave-like surfaces allow for saturated response capacity of the structure with
a realistic friction coefficient. We further demonstrate that non-planar
morphologies cause a non-linear scaling of the work-to-failure with peak
strength and result in significant improvements of the work-to-failure and
ultimate deflection – values that cannot be attained with planar-faced blocks.
Finally, we show that the key morphology parameter responsible for the
enhanced performance of non-planar blocks with wave-like surfaces is the local
angle of inclination at the hinging points of the loaded block. These findings
shed new light on topologically interlocked structures with non-planar blocks,
allowing for a better understanding of their strengths and energy absorption.

###### keywords:

Frictional Contact , Wave-faced Blocks , Topologically Interlocked Structures
, Architected Surfaces

[inst1]organization=Institute for Building Materials, addressline=ETH Zurich,
city=Zurich, country=Switzerland

##  1 Introduction

Topologically interlocked structures (TIS) are assemblies of interlocking
blocks that rely solely on contact along the interfaces to achieve structural
integrity [1, 2]. Slab-like TIS panels (also referred to as plate-like [3, 4,
5]), a commonly used type of TIS (as shown in Fig. 1a), exhibit high strength,
energy dissipation [6], crack arrest abilities [7, 8, 9], and work-to-failure
(the area under the load-deflection curve, elsewhere referred to as toughness
[9, 10] or loading energy [11, 12]).

High work-to-failure $U$ results from the combination of a high peak load
$F_{\mathrm{max}}$ and a high ultimate deflection $\delta$. Regarding peak
load, it has been shown in [11, 13, 14] that, in TIS with planar-faced blocks
(henceforth referred to as planar blocks), the peak load saturates for high
friction coefficients and cannot exceed a well-defined upper bound, see Fig.
1b. Regarding ultimate deflection, the ‘envelope’ saturated response is
commonly capped at relatively small normalized deflection compared to the
theoretical upper bound [15]. For instance, the achieved deflection for a
slab-like TIS with 5-by-5 internal blocks is $\delta/h\approx 1.3$ (see Fig.
1b), which is $\approx 2.3$ times smaller than the theoretical upper bound
$\delta/h=3$. The limited ability to achieve a high deflection is a direct
consequence of the fact that these TIS systems fail in a brittle-like manner
as soon as the peak load is reached (see sharp load drop for saturated
response in Fig. 1b), which results in a linear scaling between the work-to-
failure and the peak load (see Fig. 1c). As a result, the work-to-failure
cannot exceed the value associated to the saturated response (see theoretical
envelope in Fig. 1b), which points to a large untapped potential for increased
work-to-failure (see dark grey region in Fig. 1b).

In fact, the untapped potential for work-to-failure is practically even larger
because reaching the theoretical envelope requires materials with
unrealistically high friction coefficients $\mu$. With realistic $\mu$ in
common TIS (i.e., $\mu=0.23$ as used in [9]), the maximally achievable peak
load $F_{\mathrm{max}}$ is considerably lower (indicated by the black dot on
the black curve in Fig. 1b obtained with numerical simulations), and since
there is a linear dependence between $F_{\mathrm{max}}$ and $U$, the
realistically achievable work-to-failure $U$ is also much lower (see arrows in
Fig. 1c).

Possible avenues for increasing the work-to-failure in slab-like TIS made from
common materials consist of 1) increasing the peak load to approach the
theoretical envelope, and 2) increasing the ultimate deflection to go beyond
this theoretical envelope into the untapped potential. As we will show here,
this is achievable through the use of blocks with non-planar surface
morphologies (henceforth referred to as non-planar blocks). Such non-planar
morphologies include osteomorphic (double curvature) blocks [16, 17, 18, 19,
20, 21, 22], double-face osteomorphic blocks [4, 23, 24, 25], and hierarchical
morphologies [26].

Figure 1: (a) Schematic illustration of slab-like TIS with planar-faced blocks
along with a single planar block and its cross-sectional area showing the
inclination angle $\theta$. (b) Normalized load-deflection curve
$F_{y}/(E\cdot h^{4}/L^{2})$ vs $\delta/h$, where $E$ is Young’s modulus, $h$
the height and $L$ the side length of the panel. The light red curve
corresponds to the saturated response of the examined TIS with planar-faced
blocks achieved with an (unrealistic) friction coefficient of $\mu=7.0$. The
black curve shows the load-deflection numerically obtained with $\mu=0.23$, a
value commonly used for TIS structures in experiments [9]. The light red
shaded region corresponds to the established theoretical envelope and the grey
shaded region shows the untapped potential based on [15]. The black points
correspond to the peak loads of each curve. (c) Normalized peak load versus
work-to-failure. The linear relationship between the two measures shows that
for planar blocks, saturation in one measure leads to saturation in the other,
achieved with very high values of friction coefficients.

The reason non-planar blocks may outperform planar ones in terms of
practically achievable work-to-failure is that they enable a the saturated
response with realistic $\mu$, as has been shown for beam-like TIS in [12]. If
this is also true for slab-like TIS (which are the most common TIS application
and the one at the focus of the present study), this would mean an immediate
major improvement in the practically achievable work-to-failure by extending
the range of accessible values on the linear relation shown in Fig. 1c. In
addition, if non-planar blocks allowed for less brittle failure, this would
correspond to a different scaling between the peak load and the work-to-
failure, which, in turn, could lead to further increases in the work-to-
failure. Therefore, the degree to which non-planar blocks can help access the
untapped work-to-failure potential of slab-like TIS depends on the following
questions: Is it possible to reach the saturated response obtained of planar
blocks with high $\mu$ by using non-planar morphologies and realistic $\mu$?
Is it possible to reach higher ultimate deflections with non-planar blocks
than with planar ones? What do such potentially larger deflections mean in
terms of achievable work-to-failure and the latter’s scaling with the peak
load?

While the many studies in the TIS literature that considered non-planar blocks
[4, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], provide valuable data and
insight, to our knowledge, they did not address the aforementioned questions
in the context of slab-like TIS. The objective of the present study is,
therefore, to address these questions with a focus on wave-like surface
morphologies, and thereby shedding new light on TIS with non-planar blocks,
and pointing to ways in which their hitherto untapped work-to-failure
potential can be exploited.

Figure 2: Schematic illustration of (a) a 5-by-5 slab-like topologically
interlocked structure with non-planar blocks and structural length
$L=50~{}\mathrm{mm}$. The immovable peripheral blocks (indicated by grey)
support the structure. (b) A single non-planar block with number of
oscillations $n=3$, amplitude $\mathcal{A}=0.25~{}\mathrm{mm}$, height
$h=3.18~{}\mathrm{mm}$ and average $l=8.33~{}\mathrm{mm}$. (c) Schematic of a
quarter of the structure used in the numerical analyses. The rigid indenter is
pushed down on the central block by a prescribed displacement $\delta$. (d)
Cross-sectional representation of a block’s non-planar surface with
$\mathcal{A}=0.25~{}\mathrm{mm}$, $n=3$, wavelength $\lambda$, and an average
inclination $\overline{\theta}=0^{\circ}$ (left) and
$\overline{\theta}=5^{\circ}$ (right).

##  2 Problem Statement

###  2.1 Physical problem

In this study, we consider the 5-by-5 slab-like TIS depicted in Fig. 2a. Each
block has length $l=8.33~{}\textrm{mm}$ and height $h=3.18~{}\textrm{mm}$ (see
Fig. 2b). The immovable peripheral blocks are halves and quarters (in the four
corners) and are used to support the structure (see Fig. 2c). We consider an
isotropic and linear elastic material with Young’s modulus
$E=18.7~{}\mathrm{GPa}$, and Poisson’s ratio $\nu=0.2$. A friction coefficient
value of $\mu=0.23$ is assumed at the blocks’ interface. This configuration
was chosen because it allows us to compare the effects of non-planar
interfaces with those of planar interfaces [9, 13] with high friction
coefficients [13], as available in the literature.

###  2.2 Planar-faced blocks

Slab-like TIS with planar-faced blocks as depicted in Fig. 1a have been widely
used in the literature [9, 13, 15, 28, 29, 30, 31]. Here, we use them as a
reference for comparison with non-planar block morphologies.

###  2.3 Blocks with non-planar morphologies

Following [12, 26, 32, 33, 34, 35, 36, 37], we consider sinusoidal wave-like
patterns at the interface of the TIS (see Fig. 2) as prototypes of general
non-planar surface morphologies. They are defined by wavelength $\lambda$,
number of oscillations $n$, and amplitude $\mathcal{A}$ (see Fig. 2d) with
average inclination angle $\overline{\theta}=0^{\circ}$ (i.e., simple
rectangular block). To understand the effects of each of these variables and
to find the governing non-dimensional parameters of non-planar morphologies,
we consider different combinations of $n=2,3,4$ and $\mathcal{A}=0.025$,
$0.05$, $0.1$, $0.15$, $0.2$, $0.25$, $0.3~{}\mathrm{mm}$.

###  2.4 Numerical model

The 5-by-5 assembly is loaded by applying displacement $\delta$ on a rigid
spherical indenter. The peripheral blocks are fully fixed (in all their
nodes). Using the structures’ symmetry about the $x$\- and $z$-axes, we model
only a quarter of the TIS (see Fig. 2c). The sharp edges of the blocks are
filleted to reduce stress concentrations. We note that our model does not
account for block fracture. Therefore, we define the failure of a structure
when the loaded central block loses contact with all its neighbors.

The presented model is solved using the Finite Element software Abaqus [38].
We use a static non-linear solver that accounts for geometrical non-
linearities and uses a penalty-based surface-to-surface frictional contact
algorithm for handling contact and frictional constraints at interfaces. For
describing the frictional forces along an interface and the stick-slip
transition, we apply Coulomb’s friction law ($T=\mu N$, where $T$ and $N$ are
tangential and normal interface tractions, respectively). The blocks are
discretized using first-order tetrahedral elements. The element size was
determined through a mesh convergence analysis in which the effects of further
mesh refinement on the mechanical response were found to be negligible.

##  3 Results and Discussion

###  3.1 Load-deflection response for planar-faced blocks

The load-deflection curves ($F_{y}-\delta$) with planar blocks, are shown in
Fig. 3a. Similarly to the Level-Set-Discrete-Element-Method (LS-DEM) [13], our
FEM model captures the saturation to the peak load as observed for
unrealistically high $\mu$ (i.e., $\mu\geq 5.0$). In addition, for lower
values of $\mu$ (i.e., $\mu=0.6$) the blocks initially stick leading to a peak
load and then slip of the central block occurs causing a sharp load drop as
shown in Fig. 3a. When $\mu$ is decreased even further to experimentally
equivalent values [9], (i.e., $\mu=0.23$) the peak load is kept to minimum and
failure occurs due to sliding mechanism.

###  3.2 The effects of non-planar surface morphologies on load-deflection
response

Turning to TIS with non-planar blocks, we consider the $F_{y}-\delta$ response
with the sinusoidal patterns depicted in Fig. 3b. Similarly to planar-faced
block structures, we observe a saturation to $F_{\mathrm{max}}$. However,
contrary to planar-faced blocks, where unrealistic $\mu$ are needed to reach
saturation, with high values of $\mathcal{A}$, the peak load saturates at the
relatively low value of $\mu=0.23$. This means that a theoretical upper bound
capacity of slab-like TIS can be obtained through non-planar block
morphologies.

In terms of ultimate deflection and the associated work-to-failure, both are
visibly larger with non-planar blocks compared with planar ones. Specifically,
for cases with $\mathcal{A}\geq 0.05$ mm, the ultimate deflection approaches
$\delta\approx 2.3h$, considerably larger than that with planar-faced blocks
($\delta\approx 1.3h$ as shown in Fig. 1b). These findings suggest that non-
planar morphologies are inherently superior to planar-faced blocks in terms of
(structural) ductility and energy absorption/dissipation.

We note that the results in Fig. 3 are theoretical since our model does not
account for material non-linearity, damage, or fracture. It is likely that in
practice, these effects will lead to reduced peak loads and work-to-failure,
even for small interlocking angles.

Figure 3: Normalized load $F_{y}/(E\cdot h^{4}/L^{2})$ plotted against the
normalized prescribed displacement $\delta/h$ for (a) structures with planar-
faced blocks obtained with FEM model (solid curves) and those obtained in
Feldfogel et al. [13] using LS-DEM (dashed curves). (b) Structures with wave-
faced blocks and average inclination angle $\overline{\theta}=0^{\circ}$,
oscillation $n=3$, and different amplitudes $\mathcal{A}$. (c) The normalized
peak load $F_{\mathrm{max}}/(E\cdot h^{4}/L^{2})$ for each curve (for all $n$
and $\mathcal{A}$) is plotted against the normalized work-to-failure
$U/(E\cdot h^{4}/L)$ for TIS with planar-faced and wave-faced blocks. The
dark-shaded cluster denotes the area where $F_{\mathrm{max}}$ saturates for
wave-faced blocks. For the complete study of $F_{y}-\delta$ curves for slab-
like TIS with wave-faced blocks examined (i.e., $n=2$ and $n=4$), see A. (d,
e) Stick-slip mechanism for structures with $n=3$, $\mathcal{A}=0.15$ mm, and
$\mathcal{A}=0.025$ mm.

###  3.3 Non-linear scaling of load and work-to-failure for TIS with non-
planar blocks

Differently from the case of planar-faced blocks [9, 13, 31] where the work-
to-failure $U$ scales linearly with the peak load, see Fig. 3c, with non-
planar blocks, it scales sub-linearly. This is due to the latter’s ability to
sustain $F_{\mathrm{max}}$ (or values close to it) for greater deflections
(see Fig. 3b) compared to planar-faced blocks. In this case, $U$ continues to
increase even after $F_{\mathrm{max}}$ reaches saturation (see shaded cluster
in Fig. 3c). The sub-linear scaling with wave-faced blocks is favorable
compared to the linear scaling with planar blocks as it promotes increased
work-to-failure. We refer to this ability as pseudo-ductility since it does
not involve actual ductile deformation of the bulk material. We attribute it
to local enhanced interlocking, that is, the improvement of interlocking
between blocks through local non-planar morphologies111Based on this
definition, planar-faced blocks, which lack such local features, have a
smaller degree of overall interlocking..

To put this result into context with regards to TIS with planar-faced blocks,
we highlight two cases. First, we focus on TIS with the same $\mu$ (i.e.,
0.23) but with different block types, i.e. TIS with planar blocks (highlighted
by a red square marker in Fig. 3c) with TIS with non-planar blocks
(highlighted by shaded cluster in Fig. 3c). TIS with non-planar blocks not
only increase significantly the peak load (i.e. more than four times), as
could have been expected from beam-like observations [12], but also the work-
to-failure is increased by more than a factor of $19$ compared to TIS planar
blocks. This clearly demonstrates the capacity of TIS with non-planar surface
morphologies to significantly improve the work-to-failure – and this while
simultaneously increasing the peak load. In addition, if the same peak load
needs to be achieved with planar blocks, one would need a friction coefficient
as high as 1.0 (highlighted by a green square marker in Fig. 3c) and that
would still cause a decrease in work-to-failure by more than a factor $2$.
Hence, these results suggest that when a material is given for a particular
TIS application, introducing non-planar block surface morphology may improve
TIS performance both in terms of peak load and work-to-failure.

While TIS with non-planar blocks have been explored previously, they do not
exhibit the same improved performance we have identified. This stems from
specific surface characteristics, particularly the way the local angle of the
surface changes. If we consider osteomorphic blocks as an example, the
curvature of the interface promotes smooth sliding along the surface,
resulting in sustained work-to-failure, but lower peak loads. In contrast, a
wavy pattern, as suggested here with multiple oscillations, introduces
curvature that restricts neighboring blocks from sliding extensively across
the whole interface. This localized sliding on the curved surface allows us to
achieve both high peak loads and work-to-failure and therefore to access the
untapped potential.

###  3.4 The importance of the local angles of inclination at hinging contact
regions

The design of non-planar surface pattern creates enhanced interlocking due to
the high local inclination of the interface. While both unrealistically high
$\mu$ and non-planar surface patterns can lead to $F_{\mathrm{max}}$
saturation, high $\mu$ alone cannot provide pseudo-ductile behavior and
enhanced work-to-failure. This means that, regardless of how high the friction
coefficient is, it cannot provide the high levels of work-to-failure provided
by high local interlocking angles.

The effect of local enhanced interlocking is significant compared to
increasing the angle of inclination ($\theta$) in a planar-faced block because
the angle of inclination is limited (design constraint). On the contrary, non-
planar blocks with wave-like surfaces can locally introduce very high
inclinations, which favor this behavior of promoted work-to-failure while
maintaining blocks with attainable shapes. This raises a question regarding
the underlying characteristic of the non-planar interfaces that causes the
non-linear scaling response.

To quantify the effects of each of the parameters governing non-planar block
morphologies, we consider $U$ first as a function of $\lambda/4$ where
$\lambda$ is the wavelength of the surface pattern Fig. 4a, and next as a
function of waviness amplitude $\mathcal{A}$ in Fig. 4b. While $\lambda/4$
does not seem to directly influence $U$ as indicated by the large scatter of
the data points, there appears to be some correlation with $\mathcal{A}$.
Since, we expect the local interlocking to be the main factor, we consider $U$
as a function of $\mathcal{A}/(\lambda/4)$, which represents the average slope
at the top of the interface (see Fig. 4e) as a measure of the degree of
interlocking, and observe that the data points collapse and saturate neatly
(see Fig. 4c), as they also do for $F_{\mathrm{max}}$ in Fig. 4d. This
characteristic promotes large rotations and constraints sliding to a specific
local area of the overall interface (see Fig. 3d). This means that (a) the
local angle at the top of the interface of the loaded block (where it is in
contact with its neighbors, see $\theta^{\mathrm{top}}$ in Fig. 4e), not the
wavelength or the amplitude independently, is the governing non-dimensional
parameter defining the effects of non-planar blocks on the structural
response; (b) the local angle is the key to the enhanced work-to-failure and
ultimate deflection obtainable with non-planar interfaces; and (c) there is a
limit to $\theta^{\mathrm{top}}$ beyond which its further increase does not
affect the structural performance.

We note that high local angles are likely to be associated with high surface
tractions and bulk stresses, which can lead to material damage and fracture.
This means that the practical applicability of our findings might require
technical solutions to address these phenomena.

Figure 4: Normalized work-to-failure, plotted against (a) $\lambda/4$ where
$\lambda$ is the wavelength of the surface pattern, (b) the amplitude
$\mathcal{A}$ and (c) the top inclination $\mathcal{A}/(\lambda/4)$. (d) The
normalized peak load for each curve is plotted against the top inclination
$\mathcal{A}/(\lambda/4)$. The dashed lines in (c) and (d) have been added as
a visual aid to demonstrate the saturation of the work-to-failure and peak
load, respectively. (e) Schematic illustration of the top inclination angle
between the central and a neighboring block.

###  3.5 Improving stability for low-amplitude configurations

Next, we consider the influence of the average inclination of the interface
$\overline{\theta}$, see Fig. 5a, on the structural response of TIS with wave-
faced blocks. Specifically, we focus on how $\overline{\theta}>0^{\circ}$ can
contribute to ameliorating the load-drop instabilities associated with small
amplitudes for $\overline{\theta}=0^{\circ}$ observed in Fig. 3b for
$\mathcal{A}=0.025$ mm. Specifically, we observe that the $F_{y}-\delta$
response undergoes a series of fluctuations characterized by multiple local
maxima where each local maximum is successively smaller in magnitude compared
to the previous one. This behavior is due to the ‘pushing-downward’ of the
central block from the top interface crest to the next one through a sliding
mechanism (see Fig. 3e). Reconsidering the cases with $\mathcal{A}=0.025$ mm
(see Fig. 6), but this time with $\overline{\theta}=5^{\circ}$, we see that,
for all the examined number of oscillations $n$, with
$\overline{\theta}=5^{\circ}$ the load-drops are smaller, the ultimate
deflections increase by roughly $25\%$, and, as a result, the work-to-failure
roughly doubles Fig. 5b. This speaks to the great contribution of the average
angle of inclination to the stability of TIS with non-planar blocks.

To gain a deeper understanding of how the average inclination contributes to
the stability of the structure, we consider its effect on the internal force
transmission by comparing the distribution of the minimum principal stress
with $\overline{\theta}=0^{\circ}$ and $\overline{\theta}=5^{\circ}$ (Fig. 5c)
and the contact forces of the loaded block (Fig. 5d). With
$\overline{\theta}=0^{\circ}$, the stress and contact force distribution are
symmetrical, meaning that the load is transferred equally in the $yz$ and $xy$
planes. In contrast, with $\overline{\theta}=5^{\circ}$, they are asymmetric,
with more load being transmitted in the $yz$ plane compared with the $xy$
plane as reflected by the larger compressive stresses indicated by blue. This
bias indicates that larger $\overline{\theta}$ make one direction (in this
case $yz$) stiffer than the perpendicular one, and it suggests that having a
stiffer load path contributes to the overall resilience of the structure.

Figure 5: (a) Schematic representation of a wave-faced block with average
inclination angle $\overline{\theta}=5^{\circ}$ as it results from the
combination of a planar-faced block with inclination angle $\theta=5^{\circ}$
and a wave-faced block with average inclination angle
$\overline{\theta}=0^{\circ}$. (b) Normalized load $F_{y}/(E\cdot h^{4}/L)$
against the normalized prescribed displacement $\delta/h$ for structures with
different oscillations $n$, and for amplitude $\mathcal{A}=0.025\mathrm{mm}$.
The dashed curves correspond to an average inclination angle
$\overline{\theta}=0^{\circ}$ and the solid curves to average inclination
angle $\overline{\theta}=5^{\circ}$. (c) Snapshots of minimum principal stress
distribution normalized with Young’s modulus $E$ of the structure at specific
$\delta/h$ (left) for $n=4$ and $\overline{\theta}=0^{\circ}$ and (right) for
$n=4$ and $\overline{\theta}=5^{\circ}$. (d) Snapshots of the stick-slip
behavior at the interfaces of the central block for structures (left) with
$\mathcal{A}=0.025$, $n=4$, and $\overline{\theta}=0^{\circ}$ and (right)
$\mathcal{A}=0.025$, $n=4$, and $\overline{\theta}=5^{\circ}$).

##  4 Conclusion

This study examined how, through the use of blocks with non-planar surface
morphologies, the hitherto untapped work-to-failure potential of slab-like
topologically interlocked structures (TIS) can be better exploited. For this
purpose, we considered blocks with wave-like sinusoidal surface morphologies
defined by the number of oscillations $n$, the amplitude $\mathcal{A}$, and
the average interface inclination angle $\overline{\theta}$. The effects of
these parameters on the mechanical response of the structure were investigated
and compared to TIS with planar-faced blocks, shedding new light on their
influence in achieving improved structural performance. The main conclusions
of our study are:

  * 1.

Unlike TIS with planar blocks, in slab-like TIS with non-planar blocks, the
structural response parameters saturate with a relatively low and realistic
value of the friction coefficient.

  * 2.

With non-planar blocks, the failure mode is more ductile compared to planar
ones. This allows to develop much larger work-to-failure, and results in a
sub-linear scaling of the work-to-failure as a function of the peak load.

  * 3.

The key non-dimensional parameter governing non-planar blocks with wave-like
surface morphologies, leading to the improved work-to-failure and ultimate
deflection, is the local angle of inclination at the hinging contact regions
of the structure.

In summary, we have shown that introducing local enhanced interlocking to the
structure, promotes the system’s stability, work-to-failure, and ultimate
deflection. This study can be further expanded by considering multi-layer TIS
[39] with architected surface patterns and structures with curved geometries
(i.e., arches) [40, 41]. Exploring such configurations could help further
improve structural stability, allowing us to obtain the peak load and ultimate
deflection simultaneously.

## Acknowledgements

DSK and MP acknowledge support from the Swiss National Science Foundation
under the SNSF starting grant (TMSGI2_211655).

##  Appendix A Load-deflection for all amplitudes and oscillations examined

Figure 6: Normalized load-deflection curves for structures with (a) 2, (b) 3
and (c) 4 oscillations $n$, and different amplitudes $\mathcal{A}$. The dashed
horizontal line indicates the saturation to the peak load $F_{\mathrm{max}}$.

Fig. 6 shows that the $F_{y}-\delta$ response saturates to $F_{\mathrm{max}}$
for all $n$, while further increase of $\mathcal{A}$ promotes pseudo-ductile
behavior causing increase in ultimate deflection and work-to-failure. After
reaching the saturated response for $F_{\mathrm{max}}$, we have observed
significant deformation in the elements, which impacts the accuracy of the
$F_{y}-\delta$ curves (see curves that surpass the saturation level in Fig. 6b
and c). This occurrence arises from significantly reducing the number of
contact points within the mesh at the interface between the central block and
its neighboring ones. The reduced contact points result from high block
rotations, resulting in numerical artefacts and recorded load value
inaccuracies.

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