# Toughening mechanisms and damage propagation in Architected-Interfaces

Michelle L. S. Hedvard  Marcelo A. Dias<EMAIL_ADDRESS>Michal K.
Budzik<EMAIL_ADDRESS>Department of Mechanical and Production Engineering,
Aarhus University, 8000 Aarhus C, Denmark  Institute for Infrastructure &
Environment, School of Engineering, The University of Edinburgh, Edinburgh EH9
3FG Scotland, UK

###### Abstract

We investigate fracture toughness of architected interfaces and their ability
to maintain structural integrity and provide stable damage propagation
conditions beyond the failure load. We propose theoretical and numerical
frameworks to evaluate the fracture properties of architected interfaces
sandwiched between two (face) materials. The microscopic geometries of these
interfaces are chosen as 2D cells—pillar, tetrahedron, and hexagon—as well as
their 3D counterparts—namely, pillar array, octet truss, and Kelvin cell. Our
model, both numerical and analytical, exhibits a high level of accuracy in
predicting the compliance before failure and failure loads. Novel results are
obtained during the damage propagation regime, indicating fulfilment of the
so-called fail-safe design. Some of the cell geometries unfold during
fracture, thus increasing the failure load and ensuring stable and controlled
damage propagation conditions.

###### keywords:

Mechanical metamaterial , Architected materials , Interfaces , Fracture,
Adhesive, Bonding

##  1 Introduction

Recent advances in manufacturing have enabled fabrication of ever more complex
architected materials, which has brought large interest from scientific and
industrial communities to advance the future functional materials [1].
Although architected materials, also referred to as Mechanical Metamaterials
(MMs) [2], seem promising to innovate mechanical design, such as performance
weighted against the effective use of resources [3], their applications are
still limited and focused on compressive and impact loading [4, 5]. When it
comes to damage resistance, one important feature of such materials is a
multitude of failure modes depending on the choice of cell geometries and/or
direction of applied load. For instance, if the cell symmetry line is aligned
with the applied load during compressive loading, cell features localise
stresses leading to, for instance, buckling which subsequently create so-
called kink bands—such process is often similar to the one known from cellular
solids [6, 7, 8]. The failure is energetically expensive but often not
critical from the structural integrity perspective [9]. However, once failure
occurs under the tensile loading, symmetry considerations are a key concept: a
lattice structure with its local (_i.e._ cell) symmetry plane aligned with the
globally applied loading conditions lacks capability of redistributing the
stresses and the instantaneous excess of stored energy leads to spontaneous
failure. Such failure under tensile loading have been associated with
resistance against fracture [10] in the hopes that fracture mechanics can
enable better future designs. A number of studies focused on defining
effective stress intensity factors and successfully deduce effective fracture
properties of MMs [11, 10, 12]. One of the potential applications of MMs is
adhesive bonding [13], with the notion that for wide use of MMs, they will
need to be integrated into existing and future designs. In adhesive bonding,
or integrated scenario, MMs will be confined between two layers. We shall
refer to such use of MMs as Architected-Interfaces (ArchIs).

The verge of adhesive bonding is associated with aerospace engineering and up
until this point in the history of adhesive bonding, the majority of aerospace
applications adhesive layers are kept thin, ca. $100-300\,{\mu}$m. However,
the expansion of bonding to wind turbine blades, maritime structures or civil
engineering requires substantially different look at adhesive layer when
compared against those adopted in aerospace joints. Most of the time, in the
aforementioned applications, joints are designed to bond dissimilar materials
and manufacturing tolerances as well as on-site conditions are usually far
from controllable laboratory conditions characteristic for aerospace.
Moreover, physical, chemical, and mechanical requirements for bonding
materials are far different from that in aerospace. Here, the preference seems
to be given to a more flexible but tougher materials. It is noteworthy that
the micrometer bondline thicknesses are never used, instead they vary from
millimeters to centimeters. For instance, $20-30$ mm thick bondlines are used
for joining wind-turbine blades [14, 15]. Many challenges present themselves
in the context of bonding of large engineering structures. With added
flexibility and toughness, the volume of adhesive used is substantially
increased, while adhesives are usually not regarded as environmentally
friendly—although still new, green and recyclable compositions are being
developed. Therefore, it seems reasonable to claim, that what can be a
solution is ‘high volume’-to-‘low mass’ bondline layers. This is indeed the
very core of ArchIs working principle [13].

The mechanical properties of the bondlines are also affected by the thickness.
If the adhesive material is elastic-brittle, corner singularities will lead to
interface crack onset and premature failure. Therefore, ductile materials are
preferred for thick bondlines with advantage of higher fracture toughness and
increased damping capabilities [16, 17]. Such capabilities are within the
reach of ArchIs, where properties can be adjusted through geometrical design.
However, the full understanding of ArchIs confined between the joined
materials, affecting, distribution of cells as well as global and local
symmetries under loading is currently lacking. In addition, as adhesive joints
are multi-scale structures, ArchIs will add at least two further scales—one of
a cell size and the other from size of the most fundamental structural
building blocks, i.e. trusses, plates or shells. In that respect, modelling
and evaluation of metamaterial-inspired bondlines require development. In
recent work [13], a theoretical analysis to compute failure loads for simpler
2D lattice systems mimicking bondlines was proposed lead to establishing
relations between the geometry and the load at damage initiation. In another
development [10], the damage onset was investigated experimentally through
tomography technique. Some studies, such as those exploring numerical methods
for toughening in 2D lattice materials and investigating crack growth in
elastoplastic lattices for enhanced toughness through multi-phase
reinforcement strategies [18, 19], have focused on revealing the architecture
and material impact on fracture behaviour. However, still limited research has
been conducted on the damage propagation stage in architected materials,
especially for those of confined nature. Therefore, the focus in this work is
two fold. Primarily we wish to expand metamaterial-inspired bondlines concept
by introducing a numerical framework for studying these structures in
3-dimensions, with focus on post failure-onset behaviour. The second focus is
on witnessed fracture toughness increase during the crack growth that can be
harnessed as safe-fail design concepts. We shall focus on the the Double
Cantilever Beam (DCB) setup for the numerical experiments, where ArchIs are
used as beam-joining interfaces.

##  2 Theoretical analysis of architected interfaces

We study damage propagation in ArchIs in 2-dimensions (2D) and 3-dimensions
(3D). 2D interfaces are constructed from pillar, triangular, and hexagonal
cells, whereas in 3D, pillar array, octet truss, and Kelvin cells are
used—these geometries are shown in Fig. 1.

Figure 1: (a) Schematic representation of the DCB setup. (b) 3D cells, the
dimensions of the cells are $s\times s\times s$. (c) 2D cells, the dimensions
are given in the figure.

Pre-failure mechanical properties of each interface is derived from a
homogenised elastic modulus that is endowed from each architecture [20]. We
consider a DCB configuration, where an ArchI is sandwiched between two beams.
While beams are loaded perpendicularly to their plane, the interface
intermediates load transfer and it can be regarded as being subjected to mode
I loading conditions. We shall assume a simplification to this problem, so as
to ensure the validity of the Winkler foundation approximation [21].
Irrespective of the cell type, a single member of each geometry is identified
as a truss of cross-section $A^{*}=\pi r^{2}$, where $r$ is the truss radius,
and the Young’s modulus of the base material is given by $E^{*}$. If all cells
have a characteristic length $s$, we define the dimensionless aspect ratio
$\tilde{l}\equiv r/s$ between the truss’ smallest dimension and the cell size.

Generally speaking, the homogenised elastic modulus of each architecture can
be written as:

$\tilde{E}_{\mathrm{i}}=\frac{E_{\mathrm{i}}}{E^{*}}=C_{\mathrm{i}}\,\tilde{l}{\,}^{\alpha_{\mathrm{i}}},$
(1)

where $E_{\mathrm{i}}$ are effective Young’s moduli of specific architecture
and the index $\mathrm{i}$ is introduced in order to denote respective choices
of architecture, _i.e._ $\mathrm{i}\in\\{\mathrm{p},\mathrm{t},\mathrm{h}\\}$
in 2D for _pillar_ , _triangular_ and _hexagonal_ , and
$\mathrm{i}\in\\{\mathrm{p},\mathrm{o},\mathrm{k}\\}$ in 3D for _pillar_ ,
_octet_ and _Kelvin_. The powers $\alpha_{\mathrm{i}}$ are related to the
topology of the architecture, meaning that $\alpha_{\mathrm{i}}=2$ for
_stretching dominated_ cells and $\alpha_{\mathrm{i}}=4$ for _bending
dominated_ cells. Notice that a different result is obtained when assuming 2D
lattice material to be extended throughout the entire width. In this case, the
power-law is given by $\alpha_{\mathrm{i}}=1$ and $3$ for stretching and
bending-dominated structures, respectively [20]. In this work, we have chosen
to limit the geometry to the width of the interface by a strut diameter ($2r$)
instead, resulting in an increase of one for each power.

For pillar interfaces, both 2D and 3D, given that the trusses are initially
aligned with the externally applied loading, the forces are considered to be
transmitted entirely along the pillar’s longitudinal direction. The
corresponding cell stress,
$\sigma_{\mathrm{p}}=E_{\mathrm{p}}\epsilon_{\mathrm{p}}$, on the cross-
section area $A_{\mathrm{p}}=s^{2}$ must equate to that of the single member,
_i.e._
$\epsilon_{\mathrm{p}}A_{\mathrm{p}}E_{\mathrm{p}}=\epsilon^{*}A^{*}E^{*}$,
where the respective strains of the pillar unit and cell are
$\epsilon_{\mathrm{p}}$ and $\epsilon^{*}$. It is, therefore, found that
$C_{\mathrm{p}}=\pi$ and $\alpha_{\mathrm{p}}=2$. The triangular and octet
truss cells are stretching dominated under Maxwell counting principle [22],
for which it has been shown that the stiffness scales with the aspect ratio
with a power $\alpha_{\mathrm{i}}=2$ [20]. The normalised stiffness of the
triangular interface is given by $C_{\mathrm{t}}=\pi(3+\sqrt{2})/7$, whereas
for the octet truss $C_{\mathrm{o}}=2\sqrt{2}\pi$. This is derived from the
stiffness matrix method [23], which results in the effective interface
property in the vertical direction $E_{\mathrm{o}}=\sqrt{2}\pi
E^{*}r^{2}/l^{2}$, where $l=s/\sqrt{2}$ is the length of a single truss.
Hexagonal and Kelvin interfaces are both bending dominated [22], which makes
their stiffness scale in $\tilde{l}$ with a power $\alpha_{\mathrm{i}}=4$ [20,
24]. For such cases, deformations are assumed to only come from bending of the
incline trusses. The normalised stiffness of the hexagonal cell is, therefore,
given by eq. (1) with $C_{\mathrm{h}}=32\sqrt{3}\pi$, and the Kelvin cell
yields
$C_{\mathrm{k}}=1536\pi\cos{(\phi/2)}^{5}/[\sqrt{2}\sin{(\phi/2)}^{2}+\sin{(\phi/2)}]^{2}$,
where the angle $\phi=4\arctan{[(\sqrt{23}-2\sqrt{2})/5]}$ determines the
shape of the Kelvin cell.

###  2.1 Cells’ failure conditions

To incorporate the possible failure mechanisms in _stretching dominated_
architectures, we consider the material’s tensile strength,
$\sigma_{\mathrm{f}}$, and the critical buckling force, $F_{c}$. This leads to
the two strain based failure modes, in tension and compression:

$\displaystyle\epsilon_{\mathrm{i}}^{(t)}$
$\displaystyle=J_{\mathrm{i}}\frac{\sigma_{\mathrm{f}}}{E^{*}}$ (2a)
$\displaystyle\epsilon_{\mathrm{i}}^{(c)}$
$\displaystyle=-K_{\mathrm{i}}\pi^{2}\tilde{l}^{2}$ (2b)

where $J_{\mathrm{i}}$ and $K_{\mathrm{i}}$ are constants depending on the
cell geometry. For the pillar interface, the force is applied in the axial
direction of the truss leading to $J_{\mathrm{p}}=1$. Euler’s critical load
for the pillar structure fixed in both ends are given by
$F_{\mathrm{c}}=4\pi^{2}E^{*}I^{*}/h_{\mathrm{p}}^{2}$, where
$h_{\mathrm{p}}=ns$ is the height of the interface leading to
$K_{\mathrm{p}}=1/n^{2}$, and $I^{*}$ is the second moment of the pillar
cross-section area. For the triangular interface
$J_{\mathrm{t}}=\sqrt{2}\pi/C_{\mathrm{t}}$ and, from the buckling critical
loading, we obtain $K_{\mathrm{t}}=J_{\mathrm{t}}/2$. Similarly, for the octet
truss interface $J_{\mathrm{o}}=4\sqrt{2}\pi/C_{\mathrm{o}}$ and
$K_{\mathrm{o}}=J_{\mathrm{o}}/2$.

For the _bending dominated_ interfaces, the failure of the material is assumed
to occur when the bending stress of the inclined truss reaches the failure
stress of the material, under either tension or compression loading, leading
to the following bounds:

$\displaystyle\epsilon_{\mathrm{i}}^{(t)}$
$\displaystyle=J_{\mathrm{i}}\frac{\sigma_{\mathrm{f}}}{E^{*}}\tilde{l}^{-1}$
(3a) $\displaystyle\epsilon_{\mathrm{i}}^{(c)}$
$\displaystyle=-\epsilon_{\mathrm{i}}^{(t)}$ (3b)

where $J_{\mathrm{h}}=16\pi/(3C_{\mathrm{h}})$ and
$J_{\mathrm{k}}=\sin{(\phi/2)}/12[1-\sin{(\phi/2)}^{2}]$ for the hexagonal at
the octet truss interface respectively.

###  2.2 Analytical representation of DCB

We contrast results from two beam models, namely the Euler–Bernoulli beam
theory (EBT) and the limit case of zero curvature beam, or a rigid. The focus
is on the most important points needed for analysis, with additional details
of calculations provided in A. The ArchIs are represented by the Winkler type
elastic foundation, with homogenised materials properties of the cells (Fig.
1) given by eq. (1). The critical forces for failure of the interfaces are
determined, together with the compliance of the DCB. Three Boundary Value
Problems (BVP) are considered as outlined in Table 1. Here, $Q_{xz}(x)$ and
$M(x)$ are respectively the shear force and the moment in the beam, and $w(x)$
is the displacement of the beam in the $z$-direction.

Table 1: BVP investigated in this study. BVP | Case A | Case B | Case C
---|---|---|---
Beam model | EBT | EBT | Rigid beam
BC |  | $Q_{xz}(x=0)=F$
---
$M(x=0)=0$
$w^{\prime}(x=L)=0$
$w(x=L)=0$
| $Q_{xz}(x=0)=F$
---
$M(x=0)=0$
$Q_{xz}(x=L)=0$
$M(x=L)=0$
| $Q_{xz}(x=0)=F$
---
$M(x=0)=0$

In Case A, an infinitely-long interface is assumed, i.e. $L\rightarrow\infty$,
and $L\gg a_{0}$, where $L$ is the total length of the beam and $a_{0}$ is the
initial crack length, such that there is neither rotation nor deflection of
the beam far away from the point $x=0$, where the force $F$ is applied. In
Case B, the length of the interface section is assumed finite and of length
$L$. Hence, neither force nor moment can be transmitted at $x=L$. In Case C,
the beam is behaving as infinitely rigid in both shear and bending, which is
adequate for many practical cases including soft and/or short interface
regions or crack/damage approaching the terminal beam edge, _i.e._
$L-a\rightarrow 0$, where $a$ is the current crack length. Notice that,
practically, the three BVP are likely to be mapped onto one another during the
crack growth.

The beams shown in Fig. 1 (a) are of rectangular cross section, with the
second moment of area given by $I=bh^{3}/12$, where $b$ is the width and $h$
is the thickness of the beam. The material properties of these beams is given
by their Young’s modulus, which is here denoted by $E$. The total thickness of
the interface is given by $h_{\mathrm{i}}$. The elastic Winkler foundation
should be regarded as distributed linear elastic springs with the stiffness
$k_{i}=2E_{\mathrm{i}}b/h_{\mathrm{i}}$, given a distributed load on the beam
of $q(x)=k_{i}w(x)$ for $x>a$ [21]. Assuming the Euler-Bernoulli beam
kinematics, the governing equation is given by [21]:

$\diff[4]{w}{x}=-H(x-a)\frac{2E_{\mathrm{i}}b}{h_{\mathrm{i}}EI}w(x),$ (4)

where $H(x-a)$ is the Heaviside step function, which determines the position
of the crack front. By defining
$\lambda_{\mathrm{i}}\equiv\left({2h_{\mathrm{i}}EI}/{E_{\mathrm{i}}b}\right)^{1/4}$,
the governing equation can be written as

$\diff[4]{w}{x}+\frac{4}{\lambda_{\mathrm{i}}^{4}}wH(x-a)=0,$ (5)

where $\lambda_{\mathrm{i}}$ is a characteristic wavelength that determines
the location for changes in load direction along the interface, from being
loaded under tension to compression. Rewriting $\lambda_{\mathrm{i}}$ using
the normalised parameters $\tilde{h}_{\mathrm{i}}\equiv h_{\mathrm{i}}/h$,
$\tilde{E}_{i}\equiv E_{\mathrm{i}}/E^{*}$ and $\bar{E}\equiv E/E^{*}$ gives:

$\lambda_{\mathrm{i}}=\lambda_{0}\left(\frac{\tilde{h}_{i}}{\tilde{E_{\mathrm{i}}}/\bar{E}}\right)^{1/4},$
(6)

where $\lambda_{0}\equiv h/6^{1/4}$ is the characteristic wavelength for a DCB
when the interface is of zero thickness [25]. Eq. (5) is solved separately for
$x\leq a$ and $x>a$. The two solutions are then connected using continuity of
the point of intersection $x=a$ (details provided in A).

On the other hand, for the rigid beam case, we may consider that the
deflection is represented by a linear equation. This deflection is found from
minimisation of the potential energy, $\Pi=U-W$. Since the beam is assumed to
be rigid, the strain energy will be stored only in the ArchIs, represented by
the Winkler foundation, thus leading to
$U=1/2\int_{a}^{L}[2E_{\mathrm{i}}bw(x)/h_{\mathrm{i}}]dx$. The work done by
the external forces is $W=F\delta$. Further details can be found in A.

###  2.3 Critical Force

The critical force, defined as the force at which failure of the interface
begins, is determined by considering the critical strain of the interface,
given eqs. (2) for each choice of geometry. The maximum deflection of the
interface, and the related strain, are positioned at $\tilde{x}=\tilde{a}$,
where $\tilde{x}\equiv x/\lambda$ and $\tilde{a}\equiv a/\lambda$ are the
normalised position and the crack length, respectively. Likewise,
$\tilde{L}\equiv L/\lambda$ is the normalised length of the DCB, and the
normalised deflection is given by $\tilde{w}=w/h_{\mathrm{i}}$. Thus, the
normalised maximum deflection is found to be given by

$\displaystyle\tilde{w}_{\mathrm{max}}=\frac{\tilde{F}}{2}\times\begin{cases}(1+\tilde{a})&\text{Case
A}\\\
\frac{e^{4\tilde{L}}(\tilde{a}+1)+e^{4\tilde{a}}(\tilde{a}-1)-2e^{2\tilde{L}+2\tilde{a}}[\tilde{a}\cos{(2\tilde{L}-2\tilde{a})}+\sin{(2\tilde{L}-2\tilde{a})]}}{e^{4\tilde{L}}+e^{4\tilde{a}}+2e^{2\tilde{L}+2\tilde{a}}[\cos(2\tilde{L}-2\tilde{a})-2]}&\text{Case
B}\\\
\frac{2\tilde{L}^{2}-\tilde{a}^{2}-\tilde{L}\tilde{a}}{(\tilde{L}-\tilde{a})^{3}}&\text{Case
C},\end{cases}$ (7)

where the applied normalised force $\tilde{F}$ is defined as [13]:

$\tilde{F}\equiv\left(\frac{\lambda_{\mathrm{i}}^{3}}{EIh_{\mathrm{i}}}\right)F.$
(8)

Such definition of $\tilde{F}$ is used later to normalise the obtained
numerically results. Depending on the choice of BVP, the minimum deflection of
the interface region $(\tilde{x}>\tilde{a})$ can be found either within the
interface region or at the end of the DCB specimen, $\tilde{x}=\tilde{L}$. For
the Case A, the loading lead to a compression zone within the interface region
for which the beam deflection is negative with the minimum deflection found by
solving the equation ${d\tilde{w}}/{d\tilde{x}}=0$. Assuming a finite length
of the interface section, as in Case B, the minimum deflection is localised
either inside the interface region or at $\tilde{x}=\tilde{L}$, depending on
the geometry and stiffness of the beam. In this situation, the normalised
minimum deflection $\tilde{w}_{\mathrm{min}}$ is determined numerically.

The normalised compliance, using minimum deflection, is defined as
$\tilde{c}_{\mathrm{min}}\equiv\tilde{w}_{\mathrm{min}}/\tilde{F}$. This will
later be used to determine the critical force. Considering Case C, the minimum
deflection will be localised at $\tilde{x}=\tilde{L}$. Therefore, its value
can be found analytically:

$\tilde{w}_{\mathrm{min}}=-\frac{\tilde{F}}{2}\times\begin{cases}\sqrt{\frac{1}{2}+\tilde{a}(1+\tilde{a})}e^{\arctan(1+2\tilde{a})-\pi}&\text{Case
A}\\\
\frac{\tilde{L}^{2}-2\tilde{a}^{2}+\tilde{L}\tilde{a}}{(\tilde{L}-\tilde{a})^{3}}&\text{Case
C.}\end{cases}$ (9)

From the expressions derived for the maximum, eq. (7), and minimum, eq. (9),
deflections together with the critical strain in both tension and compression,
eqs. (2a)–(3b), the critical force is found from
$\epsilon_{\mathrm{i}}^{(t)}=2w_{\mathrm{max}}$ and
$\epsilon_{\mathrm{i}}^{(c)}=2w_{\mathrm{min}}$. Therefore, the critical force
under tensile loading of the interface is given by

$\tilde{F}_{c}^{(t)}=\epsilon_{\mathrm{i}}^{(t)}\times\begin{cases}\frac{1}{1+\tilde{a}}&\text{Case
A}\\\
\frac{e^{4\tilde{L}}+e^{4\tilde{a}}+2e^{2\tilde{L}+2\tilde{a}}[\cos(2\tilde{L}-2\tilde{a})-2]}{e^{4\tilde{L}}(\tilde{a}+1)+e^{4\tilde{a}}(\tilde{a}-1)-2e^{2\tilde{L}+2\tilde{a}}[\tilde{a}\cos{(2\tilde{L}-2\tilde{a})}+\sin{(2\tilde{L}-2\tilde{a})]}}&\text{Case
B}\\\
\frac{(\tilde{L}-\tilde{a})^{3}}{2\tilde{L}^{2}-\tilde{a}^{2}-\tilde{L}\tilde{a}}&\text{Case
C},\end{cases}$ (10)

whereas the critical force for failure of the interface region under
compression loading of the interface is

$\tilde{F}_{c}^{(c)}=-\epsilon_{\mathrm{i}}^{(c)}\begin{cases}\frac{e^{\pi-\arctan(1+2\tilde{a})}}{\sqrt{\frac{1}{2}+\tilde{a}(1+\tilde{a})}}&\text{Case
A}\\\ -\frac{1}{2\tilde{c}_{\mathrm{min}}}&\text{Case B}\\\
\frac{(\tilde{L}-\tilde{a})^{3}}{-\tilde{L}^{2}+2\tilde{a}^{2}-\tilde{L}\tilde{a}}&\text{Case
C.}\end{cases}$ (11)

where $\lambda_{\mathrm{i}}$ is calculated from eq. (6).

###  2.4 Pre-failure compliance

The compliance for the DCB is defined by $c\equiv\delta/F$, where $F$ is the
applied force and $\delta$ is the displacement of the left end on the bean
shown in Fig. 1. The normalised compliance is found by using the normalised
force, $\tilde{F}$, and displacement, $\tilde{\delta}=\delta/h_{\mathrm{i}}$
and, hence, $\tilde{c}\equiv\tilde{\delta}/\tilde{F}=cEI/\lambda^{3}$. Then,
considering e.g. Case A:

$\tilde{c}=\frac{1}{6}\left[1+2(1+\tilde{a})^{3}\right].$ (12)

##  3 Numerical model

###  3.1 Numerical implementation

The main scope of the present work is numerical modelling of damage of the
interface with metamaterial cells presented in Fig. 1, and thus complementing
findings of [13, 26] with non-existing damage growth predictions. For that,
the finite element method (FEM) is used. The adherent thicknesses
$h=h_{\mathrm{i}}$ is used with the interface height set to
$h_{\mathrm{i}}=$20\text{\,}\mathrm{m}\mathrm{m}$$ and the initial crack
length $a_{0}=10h$. The pre-processing, solving and post-processing were
carried out in COMSOL Multiphysics (v5.8) for both the 2D and 3D geometries.
For the 2D case, the Euler-Bernoulli beam elements are used for modelling both
the adherents and the cell trusses. For the 3D case, the Euler-Bernoulli beam
elements are used only for the cell trusses, while the adherents are modelled
via the shell elements. A linear-elastic material model is used for the
adherents assuming the Young’s modulus
$E=$100\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ and Poisson’s ratio
$\nu=0.3$—the values corresponding roughly to titanium and its alloys or
carbon fibre composites. The initial modulus of the material of the interface
$E^{*}=$5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, while the Poisson’s ratio
is taken as $\nu^{*}=0.3$—roughly properties of structural polymers. Note,
that the absolute values are here provided only for reproducibility reason and
out of numerical implantation necessity. Our theoretical framework [13]
removes such limitation allowing comparison of any geometries and materials as
long as kinematic assumptions are obeyed.

###  3.2 Damage model

Modelling material failure without impeding progressing damage zone is
numerically challenging and computationally expansive. Such costs are even
higher in the case of architected materials with complex cells geometries and
multiple trusses per cell. Practically, once one of the cell trusses fails its
stiffness can be considered as being zero. However, this provokes numerical
instability arising from singularities in the numerical solution. To avoid
such situation, in the current model the stiffness of the material is reduced
$k=10^{6}$ times upon failure, thus, making the material sufficiently ‘soft’.
Note, that constant $k$ will depend on the number of beam elements/trusses $n$
used within the model. In the present case $n$ varies for different cells but
always fulfil $n\ll k$, and, thus, the accumulated stiffness effect of failed
trusses is negligible. As a conservative “rule-of-thumb”, we suggest keeping
the ratio $k/n\geq 100$. The material model for the interface material
becomes:

$E(\epsilon)=\begin{cases}E^{*}/k,&\text{if
}\epsilon<\epsilon_{\mathrm{f}}^{(c)},\\\ E^{*}{\,\,\,\,\,\,},&\text{if
}\epsilon_{\mathrm{f}}^{(c)}\leq\epsilon\leq\epsilon_{\mathrm{f}}^{(t)},\\\
E^{*}/k,&\text{if }\epsilon>\epsilon_{\mathrm{f}}^{(t)}\end{cases}$ (13)

resembling damage model formulations. Here, $\epsilon$ is the axial strain of
the beam elements, $\epsilon_{\mathrm{f}}^{(t)}$ and
$\epsilon_{\mathrm{f}}^{(c)}$ are the failure strain under tension and
compression, respectively. For stretching dominated cells, the failure under
tension loading occurs when the ultimate tensile stress $\sigma_{\mathrm{f}}$
is reached, $\epsilon_{\mathrm{f}}^{(t)}={\sigma_{\mathrm{f}}}/{E^{*}}$. Since
the truss is considered slender, the failure under compression loading is
expected when the buckling load $F_{\mathrm{f}}=({E^{*}\pi^{3}r^{4}})/{s^{2}}$
is reached, _i.e._ $\epsilon_{\mathrm{f}}^{(c)}={F_{\mathrm{f}}}/({E^{*}\pi
r^{2}})=-\pi\tilde{l}^{2}$. For the bending dominated cells, the truss fails
due to material failure under both tension and compression, hence
$\epsilon_{\mathrm{f}}^{(t)}=-\epsilon_{\mathrm{f}}^{(c)}={\sigma_{\mathrm{f}}}/{E^{*}}$
including stresses due to bending.

###  3.3 Boundary conditions

All the simulations are performed under assumption of centre line symmetry.
For the 2D case, following Fig. 1, the symmetry around the $xy$-plane is
considered, while for the 3D case both the $xy$\- and $xz$\- symmetry planes
are used. When using the symmetry plane, some of the beams are cut into half.
These are modelled as semicircles with the second moment of inertia with
respect to the axes $x^{\prime}$ and $y^{\prime}$, given by
$I_{x^{\prime}}=I_{y^{\prime}}=\frac{\pi r^{4}}{8}$, where $r$ is the radius
of the truss.

The displacement $\delta$ is applied on the left side of the beam following
Fig. 1. The calculation is then carried by increasing $\delta$ and includes
geometrical non-linearity. Two non-linear solvers are used _Constant Newton_
(with a damping factor of 1 which was found adequate for majority of cases) or
_Automatic Newton_ (computationally more expensive as the the damping factor
is reevaluated at each step. In the present case it varied in the range of
$0.1-1$) depending on the solution convergence.

##  4 Results and Discussion

###  4.1 Compliance and beam kinematics prior to failure

To verify the homogenisation steps of the proposed model, Fig. 2 gathers the
elastic loading compliance data for all the investigated cells and compares
the numerical data with the analytical result of eq. (12). All the data
collected here are for the DCB specimen with long interface sections (only
Case A, Table 1, is considered).

Figure 2: The initial normalized compliance for the DCB using the different
interfaces geometries. Linear scale of the axis. The markers show the FEM
numerical results and the solid-line show the analytical result from eq. (12).

Fig. 2 shows a very good agreement between the analytical and the numerical
model is obtained. However, some deviation can be expected for the case of
extremely short cracks and bending dominated Kelvin and hexagonal cells. For
short cracks the beam bending is very limited. However, the exerted rotation
breaks the initial symmetry of the cells. For the bending cells such situation
leads to conversion from bending to axial stress, and as a result lower
compliance/higher stiffness results from the FEM. Such local/topological
effects are not mirrored by the homogenised model and the Winkler formulation.
The homogenisation assumes that the symmetry of the cell is maintained, thus,
all the trusses undergoes only bending loading mapped to effective foundation
stiffness—this is not the case one the symmetry is broken. In addition, within
Winkler foundation formulation the stress is not transferred between the
cells—the broken symmetry implies additional interaction beyond the limit of
the model used. Similar effect on compliance is not found for all the cells
experiencing tensile stress state.

In Fig. 3 deflection along the beam, (a) and (c), and fracture forces, (b) and
(d), for two different aspect ratios $\tilde{l}$ and for the different BVP
formulations investigated from Table 1 are presented.

Figure 3: Deflection of the beam along the beam (a) and (c) for two different
aspect ratios $\tilde{l}$ and BVPs (b) and (d) Respective fracture forces. The
pillar interface is here used for the FEM, where for $\tilde{l}=0.1$ the
interface is failing under tensile loading when $\tilde{l}=0.01$, the
interface is failing under compression lading.

In Fig. 3 (a) and (c) the kinematic behaviour of the beam, and beam–interface,
is depicted outlining the effect of finite length of the interface region.
Namely, given the known bending stiffness of the beam, the analytical problem
may fall into different BVP. More specifically, Case A is valid for
sufficiently long interface regions, ensuring the applicability of far field
assumptions. If the interface region is of the order of characteristic length
scale of the problem $\lambda_{p}$ BVP is changed. On the other hand, if the
interface region is short, or sufficiently soft (high $\lambda_{p}$ value),
the deflection is correctly captured only by BVP Case B, and, finally, C. It
is important to notice that, for the given aspect ratio and material choice,
the analytical predictions remain in excellent agreement with the numerical
model. This is also the case for very small aspect ratios of the lattice
truss, Fig. 3 (c). Figs. 3 (b) and (d) shows how important it is to formulate
the BVP appropriately. Given reduction of the interface region length
$\tilde{L}-\tilde{a}$, the critical force undergoes a smooth transition
between different formulations, with Case B capturing such effects. The
importance stems also from the fact that, during damage propagation, the BVP
will indeed change. By using the far-field formulation, Case A can lead to a
significant overestimation of fracture onset forces and non-conservative
design. Finally, the Case C formulation, rigid body motion of the beam, offers
high accuracy for small values of $\tilde{L}-\tilde{a}$. This is useful
information enabling design of experiments targeting respective BVP for which
the data reduction scheme is straightforward.

###  4.2 Failure onset and damage propagation

From this point forward we focus our attention to Case A, where an infinitely-
long interface is assumed ($L\rightarrow\infty$) and $L\gg a_{0}$ in a DCB.
This lies in the desire to focus on the critical region around the crack tip.
By assuming an infinitely long interface, complexities associated with the
beam’s length are neglected. The condition $L\gg a_{0}$ ensures that far from
the crack, the beam behaves as if there were no crack present, simplifying the
overall analysis. We consider the microstructure to be perfect, ensuring that
the symmetry conditions initially adopted remain valid during the fracture
process. This approach is chosen to isolate and fully understand the
fundamental mechanics that govern crack propagation, highlighting the
influence of geometry, material properties, and applied forces in the
immediate vicinity of the tip of the crack. Let us now look at the process
with the most stable propagation of damage, which is achieved with the pillar
interface, as seen in Fig. 4. In Fig. 4 (a) the force-displacement curves for
this case, with pillars of different aspect ratios—for both 2D and 3D
geometries, where the BVP Case A is considered, are shown.

Figure 4: Failure of the pillar interface using both 2D and 3D models. (a)
Force-displacement plot. (b) Zoom force-displacement plot showing failure of
low aspect ratios interfaces. (c) Interface the two different aspects ratios
at stages (1) and (2) corresponding to the force-displacement curves. Green
colour denotes damaged trusses. The colour scale bar gives the normalised
stress (by $\sigma_{\mathrm{f}}$) from compression (blue) to tension (red).

For all the models a constant distance between the pillars was maintained as
well as interface thickness. The 3D models are considered for $b=s$ and
$b=2s$, giving one and two rows of pillars along the interface respectively.
The critical force is marked by either upwards triangle or downwards triangle,
each case indicating whether the failure is due to tension or compression,
respectively. A very good agreement between the 2D and 3D models is evident
for all the different aspect ratios $\tilde{l}$ depicted. For the pillar
interfaces the force-displacement curve can be interpreted via two stages: (i)
initial, linear, loading stage corresponding to the elastic response of
material that upon reaching the critical force bifurcates to (ii) the non-
linear, damage propagation, stage, governed by a power-law given by
$F/b\propto\delta^{-0.5}$. Fig. 4 (b) provides further details of propagation
stage driven by compressive failure. In Fig. 4 (c), the green colour is used
to denote damaged pillars, while the colour gradient from red to blue denotes
the tension-compression stresses within the interface region. For the cases
with higher aspect ratio a stick-slip softening is observed during the damage
propagation. It is noteworthy to remark that, upon externally applied loading,
the failure features of ArchIs at the cell level do not necessarily related to
fracture mechanics in the standard sense— _i.e._ the individual trusses can
fail in multiple ways corresponding to very different failure criteria.
However, once integrated into the structure and constrained by other
materials, the entire interface can be regarded as a “crack” growth plan. An
otherwise continuous fracture process, for constant fracture energy, that is
expected for homogeneous materials, is instead replaced by an iterative pillar
failure process, with the onset and arrest forces oscillating around the
mentioned fracture scaling relation given by $F/b\propto\delta^{-0.5}$. The
magnitude and wavelength of this oscillation are directly related to the
geometry of the cell. In the pillar case to $\tilde{l}$. On the other hand,
for lower aspect ratios, instead of the growth we observe sudden snap-down
phenomena. In these cases, we see a large section of the interface region fail
due to the buckling phenomena.

###  4.3 Toughening mechanisms of complex interfaces

We consider the damage propagation for the remaining, topologically more
complex interfaces: triangular, octet truss, hexagonal and Kelvin. As before,
we focus on the same BVP given by case A from Table 1. In Figs. 5 to 8 the
force-displacement curves for the mentioned interface types (with an aspect
ratio $\tilde{l}=0.1$) are depicted. For better comparison, the force has been
normalised the following way: $\tilde{F}\equiv Fc_{0}/h_{\mathrm{i}}$, where
$c_{0}$ is the initial compliance.

Figure 5: (a) Force-displacement curve triangular interface (blue) where
$\tilde{l}=0.1$. The curve is divided into three sections; an elastic zone
(EL), a toughening zone (T) and a zone for the crack growth (CG). The curve
for the pillar interface (p) is included for comparison. (b) Stresses and
damage inside the triangular interface. Green colour denotes damaged trusses.
The colour scale bar gives the normalised stress (by $\sigma_{\mathrm{f}}$)
from compression (blue) to tension (red).

Starting with Fig. 5, the stresses and damage stages of triangular interface
are shown. For comparison, a corresponding $\tilde{l}$ curve for pillar
interface is shown (denoted with letter ‘p’). As it was for the pillar
interface, Fig. 4, the force increase linearly until failure of the first
truss. Then, a bridging zone connecting the bottom and top plates is created,
similar to the well-known from composite materials phenomenon. This allows the
force to increase beyond the initial failure force, thus resulting in a
toughening effect. The force continues to increase until a sudden drop,
marking failure of some of the created bridging links. Notice that at this
stage, for the triangular cells, the failure force is almost doubled, compared
to the first truss failure loading, and that the stored energy would have
quadrupled. The damage begins to propagate with characteristic stress pattern
within the ArchI, visible in the second and third inset from the top of Fig. 5
(b). By comparing the triangular case with the pillar interface, the
$F\propto\delta^{-0.5}$ law is not followed. This leads us to conclude that
the process is not self-similar and it can be split into the following
physical phenomena: (i) linear loading below the initial failure load (denoted
as EL); (ii) unfolding of topologically preferable links bridging the
face/beam materials, thus resulting in significant toughening effects (T);
(iii) failure of some of the bridges and creation of quasi self-similar damage
zone composed of both the remaining bridges and leftover interface core; (iv)
and finally crack growth (CG) of such a pattern. As these are novel
observations, some additional comments should be added. Within the loaded
portion of the interface, and especially at the very edge of the interface,
the loading direction is skewed locally in relation to initial symmetry planes
of the cell. Different features and damage patterns are to be expected
depending on the side of the cell, _i.e._ one side is free from direct
interactions with other cells and thus more compliant and capable to higher
extensions, while the other is constrained by such interactions. This effect
is captured in Fig. 5 (b) through the inset zooming the edge of the interface.
Even if the main stress field, and its characteristic pattern, propagated by a
distance of few interface thicknesses, the edge link is still carrying the
load causing deviation from the fracture power-law. The significant effect is
stemming from the fact that this link is unfolding and approaching pure
tension conditions.

Similar behaviour is observed for the octet interface, as shown in Fig. 6.

Figure 6: (a) Force-displacement curve of the octet interface (orange) where
$\tilde{l}=0.1$. (b) Stresses and damage inside the octet interface. Green
colour denotes damaged trusses. The color scale bar gives the normalised
stress (by $\sigma_{\mathrm{f}}$) from compression (blue) to tension (red).

Here, the interface also experiences a toughening effect, while a bridging
zone is created and, qualitatively, the same features of the load response
curve could be recognised. However, by contrast with the 2D triangular
interface, it is noticed that the CG phase presents a stable and self-similar
process with the characteristic power low. The pillar cell interface is left
for comparison and it is denoted by the letter p.

Figs. 7 and 8 show the load response of the hexagonal and Kelvin interfaces,
respectively.

Figure 7: (a) Force-displacement curve of the hexagonal interface where
$\tilde{l}=0.1$. The curve is divided into two sections; an elastic zone (EL)
and a toughening zone (T+G). For comparison a pillar interface case is also
plotted and denoted with p. (b) Stresses and damage of hexagonal interface at
three different stages corresponding to (a). Green colour denotes damaged
trusses. The colour scale bar gives the normalised axial stress (by
$\sigma_{\mathrm{f}}$) from compression (blue) to tension (red).

As for the previous interfaces, both tension and compression zones could be
found along the interface. As previously, loading the DCB with the hexagonal
interface, Fig. 7, the force is initially increased in linear way until the
point where the first truss fails. After this point, a bridging zone is
created while the displacement $\tilde{\delta}$ is increased, which causes the
creation of a toughening effect. However, neither steep increase nor sudden
drop of the force are observed. Focusing on the remaining trusses in the
bridging zone, it is shown that these trusses are creating a pattern, for
which 6 trusses can unfold to a pillar-like structure shown at the zoomed plot
in the top of Fig. 7 (b), where for clarity only axial stresses are plotted.
After that, the self-similar damage propagation process begins (CG). When the
Kelvin interface undergoes the damage process, as in Fig. 8, the first trusses
to fail are not at the very front of the interface, but in the second column,
where topology is different and the trusses are further constrained by their
neighbours.

Figure 8: (a) Force-displacement curve of the Kelvin interface (purple) where
$\tilde{l}=0.1$. The curve is divided into three sections; an elastic zone
(EL), a toughening zone (T) and a zone for the crack growth (CG). The curve
for the remaining curves are included for comparison. (b) Stresses and damage
of Kelvin cell interface. Green colour denotes damaged trusses. The colour
scale bar gives the normalised axial stress (by $\sigma_{\mathrm{f}}$) from
compression (blue) to tension (red).

Shortly after that, a large section of the interface fails at once which leads
to snap-down in the force-displacement curve. This is followed by a damage
propagation, which aligns with the fracture scaling law and it is similar to
the one observed for e.g. the pillar interface. Therefore, this leads to the
consideration that the cells in their unloaded configuration have well defined
finite in number symmetries, which are related to, or coincides with, the
externally applied loading directions. However, due to fracture loading, these
symmetries are broken, as the number of local degrees of freedom are different
at the “crack tip” (or better described as “damage front”), compared to other
locations within the lattice [27, 28]. It is observed that certain topologies,
those for which the arc length of the trusses between the joined surfaces
exceeds the interface thickness, are likely and can be designed to unfold or
rotate during damage propagation. This resembles the fracture phenomena in
homogeneous thermoplastic materials where, within the damage zone, the
macromolecular chains re-conform upon loading activating significant energy
dissipation.

###  4.4 Failure maps

For each of the interface types considered here, we provide a failure map
depicting the relationship between the normalised failure force
$\tilde{F}_{c}$ and the aspect ratio $\tilde{l}$. These, which establish
design spaces for ArchIs, enable further verification of results obtained
through the analytical Eqs. (10) and (11). As previously, we focus on Case A
BVP. For the pillar interface the definition of the critical force for the
numerical studies is straightforward, as the force-displacement curve, Fig. 4
is similar to what is known from a homogenised interface, with the difference
of the stick-slip behaviour. The critical force is defined as the maximum
force coinciding for pillar interfaces with the force under which failure of
the first truss is recorded. The failure map for the pillar interface is
presented in Fig. 9, which shows a good agreement between the analytical and
numerical results.

Figure 9: Failure map for the pillar interfaces. The region to the left (blue)
shows compressive failure, whereas the region to the right (red) shows tensile
failure.

For the remaining interfaces, triangular, octet, hexagonal and Kelvin, the
damage propagation is more complex, making it more ambiguous as for which
force should be chosen as the critical force. Several different failure
criteria have been considered, such as force corresponding to the failure of
the first truss, the maximum force, and the force at which the damage
propagation begins. All of these failure mechanisms are then collected for the
failure maps for each failure mechanism.

Fig. 10 shows the failure maps for the two stretching dominated interfaces,
triangular and octet. While the analytical and numerical results deviates
quantitatively, it should the noted that the power low for the different
failure mechanisms of the numerical results and the analytical formulation is
in good agreement. The criterion chosen for the critical force obtained
through FEM is not affecting the results qualitatively.

Figure 10: Failure maps for, (a) triangular, and, (b) octet cell interfaces.
The region to the left (blue) shows compressive failure, whereas the region to
the right (red) shows tensile failure.

The failure maps in Fig. 11 correspond to the results for the bending-
dominated interfaces—both hexagonal and Kelvin cells. As all the loads are
leading to bending of individual trusses, these remain under tensile loading
within the entire range of $\tilde{\lambda}$. For the bending-dominated cells,
the critical strain-damage criterion is the same in both tension and
compression. However, the displacement along the interface is much bigger
within the tension zone than the compression leading to the trusses failing
only due to tension.

Figure 11: Failure maps for, (a) hexagonal, and, (b) Kelvin cell interfaces.
The region to the left (blue) shows compressive failure, whereas the region to
the right (red) shows tensile failure.

##  5 Summary and Conclusions

The behaviour of ArchIs during fracture loading was investigated, yielding
valuable insights into their ability to maintain structural integrity and
enable stable damage propagation beyond the failure load. We employed tailored
theoretical and numerical frameworks designed for assessing interfaces between
two face materials, considering diverse microscopic geometries in both 2D
(pillar, tetrahedron, hexagon) and 3D (pillar array, octet truss, Kelvin
cell). Both frameworks showed excellent agreement in predicting structural
compliance and critical failure loads. Figures 4–8 showcase fracture
performance in various architectural designs, emphasising crack propagation
and elucidating toughening phenomena. However, our primary findings relate to
the post-failure-initiation stage. Figures 9–11 delve into damage initiation,
highlighting the influence of topology, whether stretching or bending
dominated. Despite our study’s advancements, a notable limitation still exists
in the absence of a robust fracture mechanics framework for accurately
quantifying damage resistance parameters in architected materials, challenging
traditional metrics like crack stress fields and the self-similarity of the
fracture process zone.

Different types of cell geometries, prone to unstable failure under tensile
loading, lead to stable damage propagation during fracture loading, showcasing
the potential for realising a fail-safe design. This behaviour is attributed
to the breakage of cell symmetry and the emergence of new topological states
near the damaged front. It was observed that crack propagation unfolds in
distinct stages. Initially, under linear loading below the initial failure
load, the material undergoes stress (EL). As the process advances,
topologically preferable links emerge, bridging the face/beam materials and
introducing substantial toughening effects (T). Subsequent to this, the
failure of certain bridges occurs, giving rise to a quasi self-similar damage
zone composed of both the remaining bridges and the leftover interface core.
Finally, the crack undergoes growth, following the established pattern. This
sequential progression sheds light on the dynamic evolution of material
structure and the intricate mechanisms underlying crack propagation and
toughening effects. Stick-slip damage propagation is recorded, followed by
step-wise propagation of these specific patterns.

The distinguishing feature of ArchIs lies in the concurrent presence of both
tensile and compressive stress zones along the architected region, setting
them apart from conventional architected materials. This simultaneous
occurrence of tension and compression within the structure is a direct
consequence of the unique design and arrangement of confinement. Unlike
traditional architected materials where stress zones may be predominantly
given by a $K$-field, ArchIs exhibit, in our case, a balanced distribution of
tensile and compressive forces throughout their architected regions. This
design characteristic leads to the development of specific load profiles
closely tied to the cell type comprising the ArchIs. The interplay between
tension and compression within the material not only influences its overall
mechanical response but also contributes to the distinct behaviour observed
during fracture loading, making ArchIs a compelling subject for understanding
and harnessing the mechanical properties of MMs in various applications.
Although this work does not exhaust all possible damage scenarios, as it
studies only a few types of unit cells with perfect microstructures—an
idealisation unlikely due to manufacturing flaws—we hope it further elucidates
the potential to exploit the damage phenomena in architected materials and to
incorporate characteristic toughening mechanisms within the selected MMs.
Further exploration of ArchIs in this context is deemed necessary, offering an
attractive solution for the ever-increasing applications of adhesive bonding
to engineering solutions.

## CRediT authorship contribution statement

M. L. S. Hedvard, M. A. Dias, and M. K. Budzik performed the research,
analysed the methodologies and results, and wrote the paper. M. A. Dias and M.
K. Budzik designed the research and acquired funds.

## Acknowledgements

M. K. Budzik would like to thank the Velux Foundations for support under the
Villum Experiment program (VIL50302). M. A. Dias would like to thank UKRI for
support under the EPSRC Open Fellowship (EP/W019450/1).

##  Appendix A Complementary calculations performed using Euler–Bernoulli Beam
Theory

###  A.1 Cases A and B

For $x\leq a$ the Heaviside step function $H(x-a)=0$, thus eq. (5) can be
reduced to:

$\diff[4]{w}{x}=0.$ (14)

By use of the BC from Table 1, the shear force
$Q_{xz}(x)=EIw^{\prime\prime\prime}(x)=F$ and moment
$M(x)=w^{\prime\prime}(x)=Fx$ the solution to eq. (14) is:

$w_{x\leq a}(x)=\frac{F}{6EI}x^{3}+A_{1}x+A_{2}$ (15)

where $A_{1}$ and $A_{2}$ are arbitrary constants.

When $x>a$ the Heaviside step function $H(x-a)=1$, thereby the Winkler
foundation is taken into account in eq. (5). Since the equation is homogeneous
the complete solution is given by the complementary solution. The
complimentary solution for an equation of this form is given by [21]:

$w_{x>a}(x)=A_{3}e^{-x/\lambda}\cos{\frac{x}{\lambda}}+A_{4}e^{-x/\lambda}\sin{\frac{x}{\lambda}}+A_{5}e^{x/\lambda}\cos{\frac{x}{\lambda}}+A_{6}e^{x/\lambda}\sin{\frac{x}{\lambda}}.$
(16)

The constants $A_{2},A_{3},...,A_{6}$ in eqs. (15) and (16) are determined
using BC for case A and B from Table 1 and continuity conditions where
$w_{x\leq a}(a)=w_{x>a}(a)$, $w^{\prime}_{x\leq a}(a)=w^{\prime}_{x>a}(a)$,
$w^{\prime\prime}_{x\leq a}(a)=w^{\prime\prime}_{x>a}(a)$, and
$w^{\prime\prime\prime}_{x\leq a}(a)=w^{\prime\prime\prime}_{x>a}(a)$. The two
solutions for $w(x)$ is normalised by the relation $\tilde{w}\equiv
w/h_{\mathrm{i}}$ and using the relations for the normalised force
$\tilde{F}_{\mathrm{i}}\equiv F\lambda^{3}/(EIh_{\mathrm{i}})$, the normalised
position $\tilde{x}\equiv x/\lambda$, the normalised crack length
$\tilde{a}\equiv a/\lambda$, and the normalised length of the beam
$\tilde{L}\equiv L/\lambda$. The particular solution for case A and B are
given in Table 2.

Table 2: Solutions for the normalised beam deflections $\tilde{w}$ for the
four different cases described in Table 1. Case A
---
$\begin{aligned}
\tilde{w}(\tilde{x})=\frac{\tilde{F}_{\mathrm{i}}}{2}\times\begin{cases}&\text{for
}\tilde{x}\leq\tilde{a}\\\
&\frac{1}{3}[1+\tilde{x}^{3}+2(1+\tilde{a})^{3}-3\tilde{x}(1+\tilde{a})^{2}]\\\
\quad\\\ &\text{for }\tilde{x}>\tilde{a}\\\
&e^{\tilde{a}-\tilde{x}}[(1+\tilde{a})\cos(\tilde{a}-\tilde{x})+\tilde{a}\sin(\tilde{a}-\tilde{x})]\\\
\end{cases}\end{aligned}$
Case B
$\begin{aligned}
\tilde{w}(\tilde{x})=\frac{\tilde{F}}{2}\times\begin{cases}\begin{aligned}
&\text{for }\tilde{x}\leq\tilde{a}\\\
&\frac{1}{e^{4\tilde{L}}+e^{4\tilde{a}}+e^{2\tilde{L}+2\tilde{a}}[2\cos(2\tilde{L}-2\tilde{a})-4]}\\\
&\begin{aligned}
\times\Big{\\{}&\frac{1}{3}e^{4\tilde{L}}[1+\tilde{x}^{3}+2(1+\tilde{a})^{3}-3\tilde{x}(1+a)^{2}]\\\
&+e^{4\tilde{a}}[-1+\tilde{x}^{3}-2(1-\tilde{a})^{3}-3\tilde{x}(1-a)^{2}]\\\
&+2e^{2\tilde{L}+2\tilde{a}}[-2\tilde{x}^{3}+6\tilde{a}^{2}\tilde{x}-4\tilde{a}^{3}+12(\tilde{a}^{2}-\tilde{a}\tilde{x}-\frac{1}{2})\sin{(2\tilde{L}-\tilde{a})}\\\
&+(\tilde{x}^{3}+3\tilde{x}(1-\tilde{a}^{2})-6\tilde{a}+2\tilde{a}^{3})\cos{(2\tilde{L}-2\tilde{a})}]\Big{\\}}\end{aligned}\end{aligned}\\\
\quad\\\ \begin{aligned} &\text{for }\tilde{x}>\tilde{a}\\\
&\frac{B_{1}e^{\tilde{x}}\sin{\tilde{x}}-B_{2}e^{\tilde{x}}\cos{\tilde{x}}+B_{3}e^{\tilde{x}}\cos{\tilde{x}}-B_{4}e^{\tilde{x}}\sin{\tilde{x}}}{e^{4\tilde{L}}+e^{4\tilde{a}}+e^{2\tilde{L}+2\tilde{a}}[2\cos(2\tilde{L}-2\tilde{a})-4]}\\\
\end{aligned}\end{cases}\end{aligned}$
where
$\begin{aligned}
B_{1}=&e^{2\tilde{L}+\tilde{a}}[(\tilde{a}+1)\cos{(2\tilde{L}-\tilde{a})}-\tilde{a}\sin{(2\tilde{L}-\tilde{a})}-(2\tilde{a}+1)\cos{\tilde{a}}+\sin{\tilde{a}}]\\\
&+e^{3\tilde{a}}[(\tilde{a}-1)\sin{\tilde{a}}+\tilde{a}\cos{\tilde{a}}]\\\
B_{2}=&e^{2\tilde{L}+\tilde{a}}[\tilde{a}\cos{(2\tilde{L}-\tilde{a})}+(\tilde{a}+1)\sin{(2\tilde{L}-\tilde{a})}-\cos{\tilde{a}}-(2\tilde{a}+1)\sin{\tilde{a}}]\\\
&+e^{3\tilde{a}}[(1-\tilde{a})\cos{\tilde{a}}+\tilde{a}\sin{\tilde{a}}]\\\
B_{3}=&e^{2\tilde{L}+3\tilde{a}}[-\tilde{a}\cos{(2\tilde{L}-\tilde{a})}+(\tilde{a}-1)\sin{(2\tilde{L}-\tilde{a})}-\cos{\tilde{a}}+(1-2\tilde{a})\sin{\tilde{a}}]\\\
&+e^{4\tilde{L}+\tilde{a}}[(\tilde{a}+1)\cos{\tilde{a}}+\tilde{a}\sin{\tilde{a}}]\\\
B4=&e^{2\tilde{L}+3\tilde{a}}[(\tilde{a}-1)\cos{(2\tilde{L}-\tilde{a})}+\tilde{a}\sin{(2\tilde{L}-\tilde{a})}+(1-2\tilde{a})\cos{\tilde{a}}+\sin{\tilde{a}}]\\\
&+e^{4\tilde{L}+\tilde{a}}[\tilde{a}\cos{\tilde{a}}-(\tilde{a}+1)\sin{\tilde{a}}]\end{aligned}$
Case C
$\begin{aligned}
\tilde{w}(x)=\frac{\tilde{F}}{2}\frac{[-3(\tilde{L}+\tilde{a})\tilde{x}+2(\tilde{L}^{2}+\tilde{a}^{2}+\tilde{L}\tilde{a})]}{(\tilde{L}-\tilde{a})^{3}}\end{aligned}$

###  A.2 Rigid Beam - Case C

When the interface section becomes sufficiently short, relative to the other
dimensions and mechanical properties of the DCB, the beam will start to behave
as a rigid beam. When this happens, the beam can be described with a linear
equation:

$w(x)=C_{1}x+C_{2}$ (17)

where $C_{1}$ and $C_{2}$ are constants that depends on the geometry and are
here found using minimisation of the potential energy. The potential energy is
given by:

$\Pi=U-W$ (18)

where $U$ is the strain energy and $W$ is the work done by the external
loading.

The work done by the external forces $F$ is given by:

$W=Fw(x=0)=FC_{2}$ (19)

Since the beam is assumed to be rigid, the strain energy refers only to the
energy stored in the interface. As the interface is assumed to be a linear
elastic material, the strain energy can be expressed as:

$U=\frac{1}{2}\int\sigma_{ij}\epsilon_{ij}dV\\\ $ (20)

where $V$ refers to the volume of the interface. Within the Winkler foundation
assumptions, the stress and strain component are found solely along the
$z$-direction, whereas all the other components are zero. Thus,
$\sigma_{ij}\epsilon_{ij}=\sigma_{zz}\epsilon_{zz}$. Using Hooke’s law the
strain energy can be found as:

$\displaystyle
U=\frac{1}{2}\int_{a}^{L}E_{\mathrm{i}}\epsilon_{zz}^{2}dV=\frac{1}{2}\int_{a}^{L}E_{\mathrm{i}}w(x)\frac{2}{h_{\mathrm{i}}}bdx$
(21)

be aware that the $i$ in $E_{\mathrm{i}}$ here refers to the interface type.

Minimisation of the potential energy is used to determine the constants
$C_{1}$ and $C_{2}$, by setting:

$\frac{\partial\Pi}{\partial C_{1}}=\frac{\partial\Pi}{\partial C_{2}}=0.$
(22)

This gives:

$\displaystyle C_{1}=-\frac{3Fh_{\mathrm{i}}(L+a)}{E_{\mathrm{i}}b(L-a)^{3}}$
(23) $\displaystyle
C_{2}=\frac{2Fh_{\mathrm{i}}(L^{2}+a^{2}+La)}{E_{\mathrm{i}}b(L-a)^{3}}.$

Then:

$w(x)=\frac{Fh_{\mathrm{i}}}{E_{\mathrm{i}}b(L-a)^{3}}[-3(L+a)x+2(L^{2}+a^{2}+La)].$
(24)

For comparison of Cases A, B and C, eq. (24) is rewritten by using the
normalised parameters found when solving cases A and B. The normalised
solution for Case C is given in Table 2.

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