of 12
European
Journal of Mechanics / A Solids 103 (2024) 105156
Available
online 10 October 2023
0997-7538/© 2023 The Author(s).
Published by Elsevier Masson SAS. This is an open access article under the CC BY license
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European Journal of Mechanics / A Solids
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Failure of topologically interlocked structures — a Level-Set-DEM approach
Shai Feldfogel
a
, Konstantinos Karapiperis
b
, Jose Andrade
c
, David S. Kammer
a
,
a
Institute for Building Materials, ETH Zurich, Switzerland
b
Department of Mechanical and Process Engineering, ETH Zurich, Switzerland
c
Department of Mechanical and Civil Engineering, Caltech, Pasadena, CA, USA
A R T I C L E I N F O
Keywords:
Topologically interlocked structures
Level-set
Discrete-element model
A B S T R A C T
Topological Interlocking Structures (TIS) are assemblies of interlocking building blocks that hold together solely
through contact and friction at the block interfaces and thus do not require any connective elements. This
salient feature makes them highly energy-absorbent, resistant to crack propagation, geometrically versatile,
and reusable. It also gives rise to failure mechanisms that, differently from ordinary structures, are governed
by multiple contact interactions between blocks and frictional slip at their interfaces. Commonly-used modeling
tools for structural analysis struggle to capture and quantify these unusual failure mechanisms. Here, we
propose a different approach that is well-suited for modeling the complex failure of slab-like TIS. It is based on
the Level-Set-Discrete-Element-Method, which was originally developed for granular mechanics applications.
After introducing the basic assumptions and theoretical concepts underlying our model, we show that it
accurately captures the slip-governed failure of slab-like TIS panels as observed in the literature, that it
can closely estimate the force–displacement curves, and that it is can be used to explore important features
governing the structural mechanics of TIS. The theoretical foundation, together with the results of this study,
provide a proof-of-concept for our new approach and point to its potential to improve our ability to model
and understand the behavior of interlocked structural forms.
1. Introduction
Topological Interlocking Structures (TIS) are assemblies of inter-
locking building blocks that hold together solely through contact and
friction at the block interfaces and thus do not require any connective
elements, see
Fig. 1
left. This defining feature sets them apart from
ordinary structural forms and it is responsible for their unique behavior
and advantageous properties (
Dyskin et al.
, 2005; Molotnikov et al.
,
2007; Carlesso et al.
, 2012, 2013; Dyskin et al.
, 2019, 2012). In spite of
their attractive properties, TIS’ promising potential is yet to translate to
large-scale prevalence, likely because our ability to predict their failure
– a prerequisite for designing them safely – is far from fully developed.
Developing predictive capabilities for the behavior and failure of
TIS is challenging because TIS blocks are not connected by any me-
chanical means (e.g., adhesives or bolts) and the structural integrity
therefore relies on transmission of forces through the interfaces. These
interfacial forces are difficult to quantify and predict because the
interfacial contact conditions that govern them are: (a) geometrically
irregular and dynamically changing by nature (
Djumas et al.
, 2017); (b)
highly dependent on local slip failures (
Djumas et al.
, 2017; Mirkhalaf
et al. , 2019; Koureas et al.
, 2022); (c) coupled with all other inter-
faces through the global response; and (d) sensitive to unavoidable
Corresponding author.
E-mail address:
dkammer@ethz.ch
(D.S. Kammer).
geometrical imperfections (
Mirkhalaf et al.
, 2019
; Barthelat and Zhu
,
2011).
As shown ahead, commonly-used models struggle to capture and
quantify the slip-governed failure of TIS, pointing to the potential ben-
efit of alternative modeling approaches. The main aim of this study is
to establish a proof-of-concept for a new computational approach, one
based on applying the Level-Set-Discrete-Element-Method (LS-DEM),
originally developed for granular applications, to structural analysis of
TIS, see
Fig. 1
.
The most commonly-used tool to model the behavior and failure of
TIS is the Finite Element Method (FEM), see
Williams and Siegmund
(2021), Short and Siegmund
(2019), Djumas et al.
(2017), Mirkhalaf
et al.
(2019), Schaare et al.
(2008), Dalaq and Barthelat
(2020) and
Dalaq and Barthelat
(2019). This is a natural choice due to FEM’s
ability to handle arbitrarily shaped solids and to accurately resolve their
stress and deformation fields. In cases where the response was entirely
governed by a stick regime and the specimens were not loaded up to
failure, FEM obtained a very good agreement with experimental and
analytical results (
Schaare et al.
, 2008; Short and Siegmund
, 2019). In
the context of beam-like assemblies with few blocks, FEM was also able
to correctly capture the experimentally-observed slip-governed failure
https://doi.org/10.1016/j.euromechsol.2023.105156
Received 4 October 2022; Received in revised form 15 September 2023; Accepted 6 October 2023
European
Journal of Mechanics / A Solids 103 (2024) 105156
2
S. Feldfogel et al.
Fig. 1.
Illustration of presented concept: Based on the similarities between TIS and granular media on the one hand, and LS-DEM unique ability to model the latter’s mechanics
on the other, we apply LS-DEM to model the complex failure of TIS, which common structural analysis tools struggle to capture.
mechanism and match well the global load displacement curves (
Dalaq
and Barthelat
, 2020
). However, as stated in
Dalaq and Barthelat
(2020),
computational-cost issues arise when modeling TIS with more than a
few blocks using FEM, and this becomes problematic in the context of
slab-like TIS, which typically comprise dozens of blocks.
In spite of the computational challenges that the slip-governed
behavior of slab-like TIS poses, FEM has been able to correctly cap-
ture and quantify experimentally-observed slip-governed failure of a
dynamically-loaded TIS panel made of tetrahedral blocks (
Feng et al.
,
2015). This required calibrating two parameters — the contact stiff-
ness
and the friction coefficient
. Reasonable agreement was also
obtained in
Rezaee Javan et al.
(2017) in terms of the load–deflection
curves of dynamically-loaded TIS panels, but the failure mechanism
predicted by FEM were not shown. The most computationally challeng-
ing context is, arguably, the quasi-static regime. This is expressed, for
example, by large over-prediction of the peak load (
Mirkhalaf et al.
,
2019) and by divergence of the analyses from the experimental results
close to failure (
Djumas et al.
, 2017).
The Discrete Element Method (DEM) was originally designed to
model dynamically-evolving contact and friction interactions between
multiple spherical grains (
Cundall and Strack
, 1979). It relies on an ex-
plicit dynamic framework, a rigid-body assumption, elementary block
shapes (mostly spherical, and generally convex), and a penalty-enforced
contact between the blocks, properties that make it a natural frame-
work to addressing the intricate behavior and failure of TIS.
DEM’s potential for TIS is, curiously, supported by
Schaare et al.
(2008), where excellent agreement with experimental results was ob-
tained using extremely coarse FEM meshes with only 8 elements per
block (three orders of magnitude less than in
Djumas et al.
(2017)).
This suggests that a coarse representation of block deformation, one
that is also possible in DEM as will be explained ahead, may suffice to
capture the essential features in TIS’ structural response.
DEM was used by
Brugger et al.
(2008, 2009) to model centrally
loaded slab-like TIS with cube shaped blocks, but this approach has not
been further explored. The limiting element in DEM as a general mod-
eling approach to TIS is the lack of geometrical generality necessary to
fully address the variety of TI blocks and their complex contacts.
Recently, a geometrically versatile DEM variant called Level-Set-
DEM (LS-DEM) (
Kawamoto et al.
, 2016) was developed. LS-DEM is able
to represent arbitrary block geometries and resolve the complex contact
kinematics that arise between them through a node-based discretization
of block boundary. This makes LS-DEM a potentially attractive ap-
proach for TIS, see
Fig. 1
-right and
Karapiperis et al.
(2022). Recently,
LS-DEM’s original contact formulation has been adapted, enabling us to
use it for structural analysis (
Feldfogel et al.
, 2022). However, LS-DEM
ability to realistically capture and predict the behavior and failure of
TIS as observed in experiments – a necessary validation test for a model
– has not yet been established.
Summarizing, efficient and reliable computational tools are indis-
pensable to modeling the slip-governed failure of TIS. Yet, capturing
this complex phenomenon still poses major modeling and compu-
tational challenges to FEM and DEM alike. The main objectives of
this manuscript are to present the concepts underlying LS-DEM as a
computational model for TIS and to show that it is a viable and useful
modeling alternative
2. Methodology
2.1. Assumptions
The modeling assumptions underlying our LS-DEM model involve
global considerations, the blocks, and the interfaces. Globally, the
structural response is defined by the 3D rigid body motions of the
blocks, which are governed by Newton’s generalized laws of motion.
Accordingly, the total number of degrees of freedom equals the number
European
Journal of Mechanics / A Solids 103 (2024) 105156
3
S. Feldfogel et al.
Fig. 2.
Contact modeling — (a) the continuum-based approach; and (b) LS-DEM’s discretized nodal forces (penetrations are grossly exaggerated for illustrative purposes).
of blocks times six (the number of rigid body degrees of freedom), and
it is smaller by three orders of magnitude compared to FEM models
reported in the literature, see, e.g. Djumas et al.
(2017
). The energy
dissipation mechanisms comprise sliding friction, restitution losses, and
global damping.
The blocks are assumed to be unbreakable rigid bodies; their mass
corresponds to their true material density (no mass scaling); and the
forces acting on them comprise gravity, contact and friction interface
forces by adjacent blocks, support reactions by Dirichlet boundaries,
and damping forces.
The interfaces are assumed to be adhesion-less, so only normal com-
pressive forces and tangential friction forces are considered; contact
is enforced in a linear-penalty sense. A regularized Coulomb law is
adopted, whereby the shear traction is given as:
=
||
||
min(
||
||
,휇
||
||
)
(1)
where
denotes a shear penalty parameter with units of traction per
unit displacement, analogously to
,
is the friction coefficient, with
no distinction being made between static and kinetic friction, and
is
the normal traction at the contact region.
2.2. Mathematicalformulation
The mathematical formulation of LS-DEM has been detailed else-
where (Kawamoto et al.,
2016;
Feldfogel et al.,
2022) and it is not
repeated here in full for the sake of brevity. Nevertheless, the adapted
contact formulation introduced in
Feldfogel et al.
(2022) and adopted
here is briefly described for completeness.
As illustrated in
Fig. 2(a), we adopt a continuum-based contact
approach wherein contacting block surfaces are thought of as elastic
foundations, exerting equal and opposite normal compressive tractions
proportional to the penetrations
푗,푖
at each contact point. Accord-
ingly, the penetration stiffness
has dimensions of traction per unit
penetration and it is analogous to the elastic foundation modulus.
In LS-DEM, the block surfaces are discretized by seeding nodes
across them, as schematically shown on block i in
Fig. 2(b). Accord-
ingly, the continuous contact tractions in
Fig. 2(a) are represented by
discrete nodal forces, shown as red arrows in
Fig. 2(b). The nodal force
푛,푎
at contact node
reads
1
:
푛,푎
=
푗,푖
̂
푗,푖
(2)
1
To avoid redundant symbols,
from Feldfogel et al.
(2022) has been
denoted here by
, with the understanding that its dimension is still traction
per unit displacement and not force per unit displacement as in the original
LS-DEM formulation.
where the subscript
represent the
’th contact node and where
is the nodal tributary area. Note that the method allows for robust
treatment of corners, as encountered at the edges of blocks. This is due
to the computation of normals (gradients of the level set function) by
means of trilinear interpolation from the nearest points in the level set
grid. The interested reader is referred to
Kawamoto et al.
(2016) for
more details.
2.3. Modelingdeformabilitywithrigidblocks
Under the rigid body assumption used in LS-DEM, it is not possible
to directly account for the in-plane deformability of the blocks, which
governs the stiffness and capacity of TIS. Instead, we account for
this deformability indirectly through the block penetrations and the
commensurate penetration stiffness
, as explained next.
2
Beginning with the simple case of two-blocks of total length
under in-plane compressive traction
depicted in
Fig. 3(a), the total
shortening is the sum of elastic shortenings of the blocks
=
,
see Fig. 3(b). In models where contact between deformable blocks is
imposed in a penalty sense (one of the two most popular approaches
in FEM, alongside the more accurate Lagrange multiplier methods),
the total shortening depicted in
Fig. 3(c) is the sum of the elastic
deformations and the interface penetration thus
퐹퐸푀
=
+
. In
our LS-DEM model, the total shortening depicted in Fig. 3(d) equals
the interface penetration
퐿푆퐷퐸푀
=
. By equating
퐿푆퐷퐸푀
to
and solving for
we obtain
=
푑푒푓
=
, a value that yields
the same elastic shortening and therefore the same effective in-plane
deformability. In this simplified approach,
has a dual function of
penetration stiffness and a correlate of the elastic modulus.
In assemblies with
+1
blocks, see Fig. 3(e), the total shortening is
again
=
, Fig. 3(f), but
퐿푆퐷퐸푀
=
=
, where
is the
number of interfaces across which penetrations occur, Fig. 3(g). Equat-
ing
and
퐿푆퐷퐸푀
and solving for
yields the general closed-form
expression:
=
푑푒푓
=
(3)
2
It is tacitly assumed in the following derivation that the in-plane action
is the most dominant factor governing the response of TIS panels and that the
effects of shear deformations can be neglected as secondary. This approach
follows the thrust-line model (Krause et al.,
2012;
Khandelwal et al.,
2013;
Short and Siegmund,
2019) which also neglects the effects of shear and
considers the global in-plane action as the only load transfer mechanism.
Also, by virtue of Saint-Venant’s principle, we (similarly to the thrust-line
model) neglect the effect of stress concentrations near contact regions on the
all-important global in-plane deformability.
European
Journal of Mechanics / A Solids 103 (2024) 105156
4
S. Feldfogel et al.
Fig. 3.
Methodology — the in-plane deformability is accounted for in LS-DEM through interfacial penetrations: (a–d) a two-block case. (e–g) the multi-block case.
Unlike the penetration stiffness in common FEM/DEM/LS-DEM ap-
plications,
in Eq. (3) is an explicit correlate of
which requires
no calibration. As such, it is considered to be a structural property
involving the total number of blocks, the total length, and assuming
a common
to all the blocks. Cases involving blocks made of different
materials or blocks with significantly different dimensions may require
reconsidering
as a block-wise property, and are therefore beyond the
present scope.
2.4. Limitations
The three main limitations of our model are that (a) it is only
applicable to blocks made of relatively rigid materials for which
is
sufficiently large and the penetrations sufficiently small; soft materials
with very small
and
may induce too large penetrations that
could overly distort the actual (penetration-less) kinematics; (b) it
does not account for material non-linearity, specifically fracture, which
sometimes plays a role in TIS’ failure; and (c) it does not resolve the
bulk stresses.
3
3. Set-upandnumericalmodel
All the numerical examples in this manuscript consider centrally-
loaded square TIS panels studied in
Mirkhalaf et al.
(2019). No ex-
periments were done in this study. This experimental benchmark was
chosen because (a) centrally loaded slab-like TIS are the most common
TIS studied in the literature; (b) it contains detailed experimental
information and ample data for comparison and validation; and (c) the
polyhedral blocks used in
Mirkhalaf et al.
(2019) have planar faces
and therefore they interact across matching planes. Such interfaces
represent the simplest form of conforming contacts, as distinct from
the non-conforming contact typically modeled with discrete element
methods. As such, they are a natural starting point for a future investi-
gation of more complex cases of conforming contacts that characterize
TIS with curved-face (e.g., osteomorphic) blocks (Dyskin et al.,
2003;
Djumas et al.,
2017,
2016;
Estrin et al.,
2021).
Fig. 4(a) shows the truncated polyhedral block used in
Mirkhalaf
et al.
(2019) and its xz and yz cross-sections. The bottom face of the
3
Nevertheless, bulk stresses can be estimated at post-processing by solving
the continuum problem of blocks loaded by the contact surface tractions which
our model provides. This can be done using any continuum model, e.g., FEM.
blocks is a square with side length
, and the angle of inclination of its
sloping lateral faces is
. Fig. 4(b) shows the basic 5-block cell formed
by surrounding a block by four similar ones rotated with respect to it by
90
about the
axis. Fig. 4(c) shows an entire panel with the contour
of a basic cell around the central block marked in black. The panels’
dimensions are 50
×
50
×
3.18 mm, and they consist of boundary blocks
along the edges and internal blocks. The boundary blocks are either
halves or quarters (in the four corners) of the internal blocks in a way
that the assembled panel’s convex hull is a straight parallel-piped.
Panels with identical overall dimensions but with three block sizes
– medium, large, and small – are considered. The medium-block panel,
depicted in Fig. 4(c), has 5
×
5 internal blocks with
= 8
.
33 mm
, and it
is referred to as the 5
×
5 panel. The large- and small-block panels are
referred to, respectively, as the 3
×
3 and 7
×
7 panels, see Fig. 4(d).
The panels in Mirkhalaf et al. (2019) were confined by a stiff
peripheral frame that held the boundary blocks in place without pre-
compression. They were quasi-statically loaded by a pin indenter that
pushed the central block in the negative z direction at a rate of
0.01 mm/sec. The force
exerted by the indenter on the panel and
the corresponding indenter displacement
are indicated by a yellow
arrow in the -z direction in Fig. 4(c).
Fig. 4(e) shows an experimental
curve, with the main global response parameters indicated in red.
Turning to the LS-DEM model, the blocks were positioned and ori-
ented in the initial undeformed configuration of the panel as illustrated
in Fig. 4(c), and the boundary conditions were affected by fixing the
boundary blocks. Next, the assembly was subjected to gravity until
it reached a relaxed state, i.e., until the kinetic energy lowered to
effectively zero. The relaxed positions and rotations of the blocks were
then taken as the initial conditions for the main loading phase — the
indentation. For the indentation loading, the 2.5 mm spherical tip of the
indenter was prescribed a constant velocity in the negative z direction,
see Fig. 4(c,d). To expedite the analyses, the loading speed was taken
as high as possible, but always low enough to avoid inertial effects.
The loading rate values ranged between 3–6 mm/s. The density of the
alumina-silicate blocks was taken equal to
2
.
5
10
−6
kg
mm
3
, and a friction
coefficient
=
0.23 was used, in accordance with the data in
Mirkhalaf
et al.
(2019).
Numerical tests detailed in
Feldfogel et al.
(2022) were carried out
to determine the refinement of the surface discretization and of the
level-set geometrical representation of the blocks necessary for numer-
ical convergence of the results. The converged surface discretization
and Level-set parameters was found to be 0.06 mm and 0.025 mm,
and these values were used for all the analyses in this manuscript.
Additional computational details are given in
Appendix.
European
Journal of Mechanics / A Solids 103 (2024) 105156
5
S. Feldfogel et al.
Fig. 4.
Configuration — (a) a typical internal block with
= 10
, its two cross-sections, and its LS-DEM surface discretization; (b) a basic five-block interlocked cell; (c) The full
5
×
5 panel and its boundary conditions; (d) the 3
×
3 and 7
×
7 panels; (e) a typical load displacement curve; and (f) a typical strip to determining
푑푒푓
from Eq.
(3) .
4. Validation
We validate the LS-DEM model in the context of the FEM simula-
tions and experimental results reported in
Mirkhalaf et al.
(2019). For
this, we consider the three cases analyzed in
Mirkhalaf et al.
(2019)
with FEM, namely the 3
×
3, 5
×
5, and 7
×
7 panels with
=
2
.
5
. Specifically, we compare the load–deflection response as obtained
in FEM and LS-DEM, and examine the latter’s ability to capture the
experimentally-observed evolution of the failure mechanism, and the
internal force chains.
Starting with the ultimate deflection, it is somewhat overestimated
by both FEM and the LS-DEM simulations, see
Fig. 5
(a). LS-DEM is,
however, generally closer to the experimental benchmark, especially
for the 3
×
3 panel. Continuing with the peak-load estimates, both
LS-DEM and FEM considerably over-estimated the peak load in all
three cases, see the specific over-estimation (error) factors in
Table 1
.
To allow comparison of the shape of the load–deflection curves, the
FEM and LS-DEM loads were normalized by the over-estimation factors
in Table 1
so that they matched the experimental peak-load. The
normalized LS-DEM curves in all three cases are a bit closer to the
experimental reference compared with the normalized FEM curves.
These results show that, in spite of using a fraction of the degrees of
freedom, LS-DEM is comparable with FEM in terms of estimating the
global response parameters.
Turning to LS-DEM’s ability to capture the failure mechanisms,
Fig. 5
(b) shows that, for all three assemblies, the model captures the
failure mechanism observed in the experiments (
Mirkhalaf et al.
, 2019),
namely one where the central block gradually slips out of the assembly.
In terms of the load transfer mechanism,
Fig. 5
(c) shows an arch-
like internal forces chains in accordance with the thrust-line analogy
for slab-like TIS (
Khandelwal et al.
, 2012, 2013; Short and Siegmund
,
2019). These results support the physical modeling concepts described
in Section
2 and the LS-DEM model in general.
Notwithstanding the ability of LS-DEM to capture the
experimentally-observed failure mechanisms and the internal force
chains, the ability to estimate the peak load without parameter cal-
ibration is far from predictive, see
Table 1
. As a first step towards
better predictive capabilities, we focus in the next section on physical
aspects that cause peak-load over-estimates and on heuristic strategies
to account for them and thereby to get closer estimates of the structural
response.
Table 1
Factors of error of adapted LS-DEM and FEM relative to
Mirkhalaf et al.
(2019)
experiments in the geometrically perfect case.
Response parameter
Assembly Overestimation factor
index
FEM (
Mirkhalaf
et al. , 2019
)
LS-DEM with
푝푒푟
Peak load [N]
3
×
3
14–15
7.1
5
×
5
3.7
7
×
7
3.4
Loading energy [N mm]
3
×
3
9–13
6.2
5
×
5
5.6
7
×
7
6.0
5. Twoavenuestowardsimprovedpredictivemodeling
5.1. Accountingforinitialgapsbetweentheblocks
The over-estimation of the peak load by the FEM simulations was
attributed in
Mirkhalaf et al.
(2019) to the presence of ‘‘small gaps
between the blocks resulting from statistical variations in the shape
of blocks, an effect which has been previously found to significantly
affect the mechanical performance in similar materials (
Barthelat and
Zhu, 2011)’’. Following this reasoning, with which we concur, we
next present and test a simplified modeling approach to account in an
approximate way for the effect of gaps.
5.1.1. Asimplifiedapproachtoaccountingfortheglobaleffectsofgaps
From a structural mechanics perspective, the presence of gaps re-
duces TIS’s global stiffness and carrying capacity because they reduce
the in-plane stiffness. Under in-plane compression, the gaps reduce
without exerting tractions, leading to total deformation that is always
larger than when there are no gaps. Based on the fact that gap reduction
and block deformation contribute to the global deformation in-series,
we heuristically consider the penetration stiffness in the geometrically
imperfect case
푖푚푝
to be a resultant spring of two springs in series
푑푒푓
(which represents the blocks’ deformability, see Eq.
(3)) and
푔푎푝푠
(which represents the contribution of gap closure to the in-plane
deformability) thus:
=
푖푚푝
=
푑푒푓
푔푎푝푠
푑푒푓
+
푔푎푝푠
(4)
In general, the magnitude and distribution of initial gaps is not
known a-priori and so there is no close form expression for
푔푎푝푠
as the
European
Journal of Mechanics / A Solids 103 (2024) 105156
6
S. Feldfogel et al.
Fig. 5.
The performance of the LS-DEM model compared with the experimental and numerical results reported in
Mirkhalaf et al.
(2019) for the 3
×
3, 5
×
5, and 7
×
7 panels
with
= 2
.
5
. (a) Load–deflection curves for as obtained from the experiments in
Mirkhalaf et al.
(2019), the FEM simulations in
Mirkhalaf et al.
(2019), and the present LS-DEM
model. The numerical curves from the FEM and LS-DEM simulations are normalized by the peak experimental load with the normalization (error) factors listed in
Table 1
. (b)
Deformation mode snapshots obtained from the LS-DEM model. (c) The evolution of resultant contact forces for the 5
×
5 panel obtained from the LS-DEM model. The forces
directions are represented by the inclination of the cylinders axis and their magnitude is represented by the cylinders width.
one for
푑푒푓
. Therefore, when the effect of gaps is taken into account
using Eq.
(4), the
푔푎푝푠
component of
푖푚푝
requires calibration.
5.1.2. Calibrationofthegapparameter
푔푎푝푠
We calibrated
푔푎푝푠
in the context of the
= 5
5
×
5 panel
from
Mirkhalaf et al.
(2019), keeping
푑푒푓
as before at 2.25 GPa/mm.
Comparing the experimental and LS-DEM load–deflection curves in
Fig. 6
, the
푔푎푝푠
=
0.49 GPa/mm and
푔푎푝푠
=
0.65 GPa/mm curves
envelope the experimental one, with the former value closer in terms of
peak load, loading energy, and ultimate displacement. Comparing the
failure mechanism, the
푔푎푝푠
=
0.65 GPa/mm case correctly captures
the localized failure observed in the
Mirkhalaf et al.
(2019) experi-
ments, whereas the
푔푎푝푠
=
0.49 GPa/mm case does not. Based on this,
we chose the
푔푎푝푠
=
0.65 GPa/mm case as the best fit
푔푎푝푠
. As an
additional check on the validity of the choice of
푔푎푝푠
=
0.65 GPa/mm,
we plotted the internal force chains obtained with it and found that,
similarly to what we found for the closed-form
used in Section
4,
they are similar to those obtained in FEM simulations (
Khandelwal
et al. , 2012) and are inline with the thrust-line model (
Khandelwal
et al. , 2012, 2013; Short and Siegmund
, 2019).