Failure of topologically interlocked structures - a Level-Set-DEM approach Shai Feldfogel1 Konstantinos Karapiperis2 Jose Andrade3

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Failure of topologically interlocked structures -

a Level-Set-DEM approach

Shai Feldfogel1, Konstantinos Karapiperis2, Jose Andrade3,

and David S. Kammer1

1Institute for Building Materials, ETH Zurich, Switzerland

2Department of Mechanical and Process Engineering, ETH

Zurich, Switzerland

3Department of Mechanical and Civil Engineering, Caltech,

Pasadena, California, USA

October 26, 2022

Abstract

Topological Interlocking Structures (TIS) are assemblies of inter-

locking building blocks that hold together solely through contact and

friction at the blocks’ interfaces, and thus do not require any con-

nective elements. This salient feature makes them highly energy-

absorbing, resistant to crack propagation, geometrically versatile, and

re-usable. It also gives rise to failure mechanisms that, diﬀerently from

ordinary structures, are governed by multiple contact interactions be-

tween blocks and frictional slip at their interfaces. Commonly-used

modeling tools for structural analysis severely struggle to capture and

quantify these unusual failure mechanisms. Here, we propose a diﬀer-

ent approach that is well suited to model the complex failure of TIS. It

arXiv:2210.14155v1 [cond-mat.soft] 5 Oct 2022

is based on the Level-Set-Discrete-Element-Method, originally devel-

oped for granular mechanics applications. After introducing the basic

assumptions and theoretical concepts underlying our model, we show

that it well-captures experimentally observed slip-governed failure in

TIS slabs and that it estimates the force-displacement curves better

than presently available modeling tools. 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 to understand the behavior of interlocked structural forms.

1 Introduction

Topological Interlocking Structures (TIS) are assemblies of interlocking build-

ing blocks that hold together solely through contact and friction at the blocks’

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. These

include high energy absorption, high resistance to crack propagation, large

tolerance to missing blocks and to geometrical imperfections, geometrical ver-

satility, re-usability, and more [1–6]. In spite of these useful properties, TIS’s

promising potential is yet to translate to large-scale prevalence. One likely

reason is that our ability to predict their failure is far from fully developed.

As a result, the ability to design them safely, a prerequisite for widespread

application, is limited.

Developing predictive capabilities for the behavior and failure of TIS is

challenging because the blocks of TIS are, in general, not connected by any

mechanical means (e.g., adhesives or bolts). This means that the structural

integrity depends entirely on the transmission of forces through the interfaces,

and, these interfacial forces are diﬃcult to quantify and predict because the

interfacial contact conditions that govern them are: (a) geometrically irreg-

ular and dynamically changing by nature [7]; (b) highly dependent on local

slip failures [7–9]; (c) coupled with all other interfaces through the global

Load

Displacement

LS-DEM well-captures

granular mechanics response

Topologically Interlocked Structures (TIS)

Load

Individual

initial configuration

Complex contact and

friction governed

mechanical response

Structural analysis tools struggle to

capture TIS' highly nonlinear failure

Apply LS-DEM to

capture TIS' complex

structural mechanics

Granular media

Our concept

assembly

blocks grains

/arrangement

Figure 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.

response; and (d) sensitive to unavoidable geometrical imperfections [8, 10].

As shown and discussed ahead, commonly-used models struggle to cap-

ture and quantify the slip-governed failure of TIS, underscoring a need for

alternative approaches. The main objective of this study is to establish a

proof-of-concept for a new computational approach that is better equipped

to model the failure of TIS than presently available tools. The basic idea

underlying our approach is to apply the Level-Set-Discrete-Element-Method

(LS-DEM), originally developed for granular applications, to structural anal-

ysis 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 [7, 8, 11–15]. 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[12, 13]. In the context of beam-like assemblies with few

blocks, FEM was also able to correctly capture the experimentally observed

slip-governed failure mechanism and match well the global load displacement

curves [14]. However, as stated in [14]: ”we have limited the study to 5 blocks

(N = 5) because of expensive computational costs with larger N”. FEM’s dif-

ﬁculty with handling more than a few blocks becomes a major obstacle in the

context of the most common TIS application - slabs, which typically comprise

dozens of blocks. FEM’s diﬃculty with TIS slabs is expressed, for example,

by over-prediction of the peak load by an order of magnitude in [8], and by

divergence of the analyses from the experimental results close to failure and

an inability to capture the experimentally observed load drops [7]. In general,

FEM struggles to capture the experimentally observed slip-governed failure

in TIS slabs and to follow the corresponding load-displacement response up to

failure. Since these capabilities are important to properly model TIS failure,

alternatives to FEM are warranted.

The Discrete Element Method (DEM) was originally designed to model

dynamically-evolving contact and friction interactions between multiple spher-

ical grains [16]. As such, it is a natural starting-point framework for a model

that could better address the intricate behavior and failure of TIS better.

The ability of DEM to handle multiple dynamic contacts is due to an ex-

plicit dynamic framework, a rigid-body assumption, elementary block shapes

(mostly spherical, and generally convex), and a penalty-enforced contact be-

tween the blocks. The potential of a DEM-based approach for TIS is sup-

ported by the 3D FEM analysis in [13]. There, excellent agreement with

experimental results was obtained using extremely coarse meshes of only 8

elements per block (three orders of magnitude less than in [7]). This suggests

that a coarse representation of block deformation, one that is possible even

under the seemingly contradictory rigid body assumption as will be explained

ahead, may be suﬃcient to capture the essential features in the behavior of

TIS. However, diﬀerently from a coarse meshed FEM approach, which would

lack the spatial resolution of the contact kinematics necessary to capture

stick-slip transition and the slip regime, these pose no special diﬃculties for

a DEM-based model. DEM was used by Brugger et al. [17, 18] to model

centrally loaded TIS slabs with cube shaped blocks, but this approach has

not been further explored. The essential limitation of ordinary DEM as a

general modeling approach to TIS is the inability to fully address the geomet-

rical variety of TI blocks and their complex contacts. Recent DEM variants

were developed that can handle non-convex polygonal blocks in the context

of granular ﬂow [19]. However, these variants lack the ability to represent the

geometry of curved faced blocks, such as the popular osteomorphic blocks,

[1, 7, 20–22], and to resolve the conforming contact interactions between such

blocks. As such, they are not fully equipped to address the full range of TIS,

and therefore cannot provide a general modeling solution.

Recently, a DEM variant called Level-Set-DEM (LS-DEM) [23] was de-

veloped speciﬁcally to overcome the shape limitations of ordinary DEM. LS-

DEM is able to represent arbitrary block geometries and resolve the complex

contact kinematics that arise between them through a node-based discretiza-

tion of block boundary. This shape versatility, together with the ability to

handle non-convex shapes and the aforementioned DEM advantages, makes

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Failureoftopologicallyinterlockedstructures-aLevel-Set-DEMapproachShaiFeldfogel1,KonstantinosKarapiperis2,JoseAndrade3,andDavidS.Kammer11InstituteforBuildingMaterials,ETHZurich,Switzerland2DepartmentofMechanicalandProcessEngineering,ETHZurich,Switzerland3DepartmentofMechanicalandCivilEngineering,Cal...

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Failure of topologically interlocked structures - a Level-Set-DEM approach Shai Feldfogel1 Konstantinos Karapiperis2 Jose Andrade3

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