# Tensile material instabilities in elastic beam lattices lead to a bounded
stability domain

G. Bordiga  Department of Civil, Environmental, and Mechanical Engineering,
University of Trento, Trento, Italy  D. Bigoni111Corresponding author: e-mail:
<EMAIL_ADDRESS>phone: +39 0461 282507. Department of Civil,
Environmental, and Mechanical Engineering, University of Trento, Trento, Italy
A. Piccolroaz  Department of Civil, Environmental, and Mechanical Engineering,
University of Trento, Trento, Italy

###### Abstract

Homogenization of the incremental response of grids made up of preloaded
elastic rods leads to homogeneous effective continua which may suffer
macroscopic instability, occurring at the same time in both the grid and the
effective continuum. This instability corresponds to the loss of ellipticity
in the effective material and the formation of localized responses as, for
instance, shear bands. Using lattice models of elastic rods, loss of
ellipticity has always been found to occur for stress states involving
compression of the rods, as usually these structural elements buckle only
under compression. In this way, the locus of material stability for the
effective solid is unbounded in tension, i.e. the material is always stable
for a tensile prestress. A rigorous application of homogenization theory is
proposed to show that the inclusion of sliders (constraints imposing axial and
rotational continuity, but allowing shear jumps) in the grid of rods leads to
loss of ellipticity in tension, so that the locus for material instability
becomes bounded. This result explains (i.) how to design elastic materials
subject to localization of deformation and shear banding for all radial stress
paths; (ii.) how for all these paths a material may fail by developing strain
localization and without involving cracking.

#### Keywords

Tensile buckling · Sliding interface · Material instability · Homogenization

##  1 Introduction

A design strategy leading to metamaterials capable of effectively filtering
and conditioning wave propagation is the composition of elastic structures via
periodic lattices [1, 2, 3, 4, 5, 6, 7, 8, 9]. In these structures, different
effects related to out-of-plane or in-plane deformations, presence of bending
moment or prestress have been explored [10, 11, 12, 13]. Still, many important
issues remain unknown, so that the present article addresses one of these,
namely, the possibility of defining structured materials capable of suffering
instabilities for all possible prestress states, including tensile.

The incremental response of a periodic grid of preloaded elastic rods can be
homogenized to obtain an effective, prestressed elastic solid, linearly
relating the increments of the first Piola-Kirchhoff stress and of the
displacement gradient, [14, 15, 16, 17]. Two types of instability may occur in
the grid, classified as ‘microscopic’ and ‘macroscopic’. Only the latter is
captured by the homogenized material response and corresponds to its loss of
ellipticity, which, in turn, coincides with the condition of strain
localization, and, as a special case, shear band formation [18, 19, 20, 21,
22].

Recently, two-dimensional grids of prestressed elastic rods, subject to in-
plane incremental normal and shear forces and bending moment, have been
advocated as materials that can be designed to exhibit instabilities inside
the elastic range as well as to display tunable effective properties [23].
However, loss of ellipticity in these materials has been so far shown to be
possible only when compression is involved, so that the locus of material
stability for the homogenized material is unbounded in tension. This
circumstance is a direct consequence of the fact that usually elastic rods
only buckle in compression. However, real materials exhibit localization of
strain for all stress states, including tension [24]. Therefore, it might be
superficially (and erroneously) concluded that it is impossible to design an
artificial material with a bounded stability domain, using a grid of elastic
rods. In contrast with this erroneous conclusion, it is shown in this article
that employing slider constraints (permitting only relative transverse sliding
between the connected ends of two rods) inside the rods forming the periodic
lattice may lead to a macroscopic buckling in tension, formally corresponding
to loss of ellipticity in the homogenized response for tensile stress states.

As a consequence, it is rigorously proven that homogenization leads to a
smooth and bounded domain where failure determined by strain localization and
shear bands is excluded. This failure occurs as soon as this domain is
touched, as it happens for every radial stress paths emanating from the
unloaded state.

Therefore, a new way is found to design architected materials with a stability
(or ‘failure’) domain that is bounded for all stress ‘directions’. These
materials, designed as grids of elastic beams endowed with slider constraints
as sketched in Fig. 1, will be shown to greatly extend their compliance under
stretching as a consequence of the occurrence of a localized shear deformation
band.

Figure 1:  A periodic two-dimensional grid of (axially and flexurally
deformable) elastic rods equipped with sliders leads to tensile and
compressive global bifurcations corresponding to loss of strong ellipticity in
the effective material. The rods are axially preloaded in tension or
compression from a stress-free configuration (left). The response to an
incremental shear displacement $\bm{u}(s)$ on the boundary of the grid leads
to shear forces and bending moments (shown in red) in the rods and it
corresponds in the effective material to an incremental shear (shown in
orange), so that the bending moments in the rods do not contribute to the mean
stress (right).

The methodology developed in this article allows for the control and design of
the bifurcation pattern triggered by both compressive and tensile stress in a
2D lattice material. These can be effectively leveraged for a wide range of
engineering applications, as recent studies show that suitably controlled
buckling instabilities can be harnessed to realize innovative devices. Some
instances of direct exploitation of a specific kind of ‘tensile’ buckling
(namely an instability triggered by local compression produced by a global
stretch) have been explored for the design of flexible 3D electronics [25, 26]
and bio-inspired active skins [27]. Therefore, the extension of the design
space to structures and materials exhibiting buckling under pure tension can
open unexplored routes to novel fabrication processes for periodic electro-
mechanical systems at different scales.

The results shown in the present article are directly connected to the
discovery of tensile buckling [28, 29] and are obtained through a rigorous
application of homogenization theory providing an energetic match between a
preloaded lattice (Section 2) and an effective elastic continuum (Section 3).
Examples of materials characterized by a bounded stability domain and their
characteristics are provided (Section 4), followed by the analysis of the ‘re-
stabilization’ occurring in the effective continuum, while the elastic grid is
subject to local instabilities (Section 5). Finally, concluding remarks are
drawn at the end of the paper (Section 6).

##  2 Incremental response of lattices incorporating sliding constraints

A two-dimensional periodic lattice of elastic rods, deformable in the plane
both axially and flexurally, is considered, in which all structural members
are axially prestressed from an unloaded reference configuration
$\mathcal{B}_{0}$. Each junction between the rods is assumed to be one of two
types: a fully welded junction or a sliding constraint. The former is standard
as it prescribes perfect continuity of displacements and rotations, while the
latter allows for a displacement jump along the sliding direction, here
assumed to be orthogonal to the rod’s axis, see Fig. 1.

The prestress is produced by tensile or compressive dead loading acting at
infinity, while body forces in the lattice are not considered for simplicity.
The preload is postulated not only to satisfy equilibrium, but also to
preserve periodicity and leave the structure free from flexure. The
incremental response is analyzed by considering arbitrary deformations, which
include development of bending moment and axial and shear forces.

The prestressed configuration $\mathcal{B}$ is periodic along two linearly
independent vectors $\\{\bm{a}_{1},\bm{a}_{2}\\}$, defining the direct basis
of the lattice, so that the structure can be constructed from a single unit
cell $\mathcal{C}$, assumed to be composed of $N_{\text{B}}$ nonlinear elastic
rods with Euler-Bernoulli incremental kinematics, as sketched in Fig. 1.

By considering in-plane flexural and axial incremental deformations, the
incremental displacement field of the $k$–th rod in a given unit cell is
defined by the vector field

$\bm{u}_{k}(s_{k})=\\{u_{k}(s_{k}),v_{k}(s_{k})\\}^{\mathsf{T}},\quad\forall
k\in\\{1,...,N_{\text{B}}\\}\,,$ (1)

where $s_{k}$ is the coordinate along the $k$–th rod, $u_{k}(s_{k})$ and
$v_{k}(s_{k})$ are the axial and transverse incremental displacements. The
incremental rotation of the rod’s cross-section $\theta_{k}(s_{k})$ is assumed
to satisfy the unshearability condition of the Euler-Bernoulli kinematics,
namely, $\theta_{k}(s_{k})=v_{k}^{\prime}(s_{k})$.

In order to formulate the problem of incremental equilibrium for the lattice,
the contributions to the second-order incremental energy are derived for the
single rod in Section 2.1 and for the slider constraint in Section 2.2. These
are then combined in Section 2.3 to obtain the unit cell equilibrium.

###  2.1 Analytic solution for the prestressed elastic rod

The incremental equilibrium equations for an elastic rod obeying Euler-
Bernoulli kinematics, prestressed with an axial load $P$ (assumed positive in
tension), and pre-stretched by $\lambda_{0}>0$, are the following

$\displaystyle A(\lambda_{0})\,u^{\prime\prime}(s)=0\,,$ (2a) $\displaystyle
B(\lambda_{0})\,v^{\prime\prime\prime\prime}(s)-P(\lambda_{0})\,v^{\prime\prime}(s)=0\,,$
(2b)

where $A(\lambda_{0})$ and $B(\lambda_{0})$ are the incremental axial and
bending stiffnesses, respectively, and $s\in(0,l)$ with $l$ being the current
length of the rod. It is worth noting that the current axial and bending
stiffnesses are, in general, functions of the current pre-stretch
$\lambda_{0}$, which in turn depends on the axial load $P$ (see for instance
[23]). In the following, the parameters $A(\lambda_{0})$ and $B(\lambda_{0})$
will simply be denoted as $A$ and $B$, and treated as independent quantities
for generality.

Eqs. (2) is a system of linear ODEs for the functions $u(s)$ and $v(s)$. As
the system is fully decoupled, the solution is easily obtained in the form

$u(s)=C_{1}^{u}+C_{2}^{u}\,s\,,\qquad
v(s)=C_{1}^{v}\,e^{-\beta\,s}+C_{2}^{v}\,e^{\beta\,s}+C_{3}^{v}\,s+C_{4}^{v}\,,$
(3)

where $\\{C_{1}^{u},C_{2}^{u},C_{1}^{v},...,C_{4}^{v}\\}$ are 6 arbitrary
complex constants and $\beta=\sqrt{P/B}$.

For a rod of length $l$, the following nomenclature can be introduced

$u(0)=u_{1}\,,\quad v(0)=v_{1}\,,\quad\theta(0)=\theta_{1}\,,\quad
u(l)=u_{2}\,,\quad v(l)=v_{2}\,,\quad\theta(l)=\theta_{2}\,,$ (4)

so that the vector
$\bm{q}=\\{u_{1},v_{1},\theta_{1},u_{2},v_{2},\theta_{2}\\}^{\mathsf{T}}$
collects the degrees of freedom of the rod expressed in terms of end
displacements. Solving the conditions (4) for the constants
$\\{C_{1}^{u},C_{2}^{u},C_{1}^{v},...,C_{4}^{v}\\}^{\mathsf{T}}$ allows the
solution (3) to be rewritten as

$\bm{u}(s)=\bm{N}(s;P)\,\bm{q}\,,$ (5)

which is now a linear function of the nodal displacements $\bm{q}$. The
$2{\times}6$ matrix $\bm{N}(s;P)$ acts as a matrix of prestress-dependent
‘shape functions’ and therefore the representation (5) can also be considered
as the definition of a ‘finite element’ endowed with shape functions built
from the exact solution. Moreover, these shape functions reduce to the
solution holding true in the absence of prestress, because in the limit

$\lim_{P\to
0}\bm{N}(s;P)=\begin{bmatrix}1-\frac{s}{l}&0&0&\frac{s}{l}&0&0\\\\[2.84526pt]
0&\frac{(l-s)^{2}(l+2s)}{l^{3}}&\frac{(l-s)^{2}s}{l^{2}}&0&\frac{(3l-2s)s^{2}}{l^{3}}&\frac{s^{2}(s-l)}{l^{2}}\end{bmatrix}\,,$

the usual shape functions (linear and Hermitian for axial and flexural
displacements, respectively) are retrieved.

By employing Eq. (5), the incremental stiffness matrix of a prestressed rod
can be computed, so that for the $k$-th rod the elastic strain energy at
second order is given by

$\mathcal{E}_{k}=\frac{1}{2}\int_{0}^{l_{k}}\left(A_{k}\,u^{\prime}_{k}(s_{k})^{2}+B_{k}\,v^{\prime\prime}_{k}(s_{k})^{2}\right)\,\text{d}s_{k}=\frac{1}{2}\,\bm{q}_{k}^{\mathsf{T}}\left(\int_{0}^{l_{k}}\bm{B}_{k}(s_{k};P_{k})^{\mathsf{T}}\bm{E}_{k}\,\bm{B}_{k}(s_{k};P_{k})\,\text{d}s_{k}\right)\bm{q}_{k}\,,$
(6)

where $\bm{E}_{k}$ is a matrix collecting the stiffness terms, while
$\bm{B}_{k}(s_{k};P)$ is the strain-displacement matrix, defined as follows

$\bm{E}_{k}=\begin{bmatrix}A_{k}&0\\\ 0&B_{k}\\\
\end{bmatrix}\,,\qquad\bm{B}_{k}(s_{k};P_{k})=\begin{bmatrix}\frac{\partial}{\partial
s_{k}}&0\\\ 0&\frac{\partial^{2}}{\partial s_{k}^{2}}\\\
\end{bmatrix}\bm{N}_{k}(s_{k};P_{k})\,.$

The ‘geometric’ contribution due to the presence of the axial prestress is now
included in the potential energy,

$\mathcal{V}_{k}^{g}=\frac{1}{2}P_{k}\int_{0}^{l_{k}}v^{\prime}_{k}(s_{k})^{2}\,\text{d}s_{k}=\frac{1}{2}\,\bm{q}_{k}^{\mathsf{T}}\left(P_{k}\int_{0}^{l_{k}}\bm{b}_{k}(s_{k};P_{k})^{\mathsf{T}}\bm{b}_{k}(s_{k};P_{k})\,\text{d}s_{k}\right)\bm{q}_{k}\,,$
(7)

where $\bm{b}_{k}(s_{k};P_{k})=\begin{bmatrix}0&\frac{\partial}{\partial
s_{k}}\end{bmatrix}\bm{N}_{k}(s_{k};P_{k})$ is a vector collecting the
derivatives of the shape functions describing the transverse displacement $v$.
A combination of Eqs. (6) and (7), yields the potential energy for the $k$-th
rod in the form

$\mathcal{V}_{k}=\mathcal{E}_{k}+\mathcal{V}_{k}^{g}\,.$ (8)

Note that, as the equilibrium equations for the rods have been linearized
around an axially preloaded configuration, the potential (8) represents the
incremental potential energy with respect to the current configuration.

From Eqs. (6), (7) and (8) the prestress-dependent stiffness matrix is defined
as

$\bm{K}_{k}(P_{k})=\int_{0}^{l_{k}}\bm{B}_{k}(s_{k};P_{k})^{\mathsf{T}}\bm{E}_{k}\,\bm{B}_{k}(s_{k};P_{k})\,\text{d}s_{k}+P_{k}\int_{0}^{l_{k}}\bm{b}_{k}(s_{k};P_{k})^{\mathsf{T}}\bm{b}_{k}(s_{k};P_{k})\,\text{d}s_{k},$

so that

$\bm{K}_{k}=\begin{bmatrix}\frac{A_{k}}{l_{k}}&0&0&-\frac{A_{k}}{l_{k}}&0&0\\\\[5.69054pt]
0&\frac{12B_{k}}{l_{k}^{3}}\varphi_{1}(p_{k})&\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&0&-\frac{12B_{k}}{l_{k}^{3}}\varphi_{1}(p_{k})&\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})\\\\[5.69054pt]
0&\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&\frac{4B_{k}}{l_{k}}\varphi_{3}(p_{k})&0&-\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&\frac{2B_{k}}{l_{k}}\varphi_{4}(p_{k})\\\\[5.69054pt]
-\frac{A_{k}}{l_{k}}&0&0&\frac{A_{k}}{l_{k}}&0&0\\\\[5.69054pt]
0&-\frac{12B_{k}}{l_{k}^{3}}\varphi_{1}(p_{k})&-\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&0&\frac{12B_{k}}{l_{k}^{3}}\varphi_{1}(p_{k})&-\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})\\\\[5.69054pt]
0&\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&\frac{2B_{k}}{l_{k}}\varphi_{4}(p_{k})&0&-\frac{6B_{k}}{l_{k}^{2}}\varphi_{2}(p_{k})&\frac{4B_{k}}{l_{k}}\varphi_{3}(p_{k})\end{bmatrix},$

where the $\varphi_{j}$ are functions of the non-dimensional measure of
prestress $p_{k}=P_{k}l_{k}^{2}/B_{k}$ given by

$\displaystyle\varphi_{1}(p_{k})$
$\displaystyle=\frac{p_{k}^{3/2}}{12\left(\sqrt{p_{k}}-2\tanh\left(\sqrt{p_{k}}/2\right)\right)}\,,$
$\displaystyle\varphi_{2}(p_{k})$
$\displaystyle=\frac{p_{k}}{6\sqrt{p_{k}}\coth\left(\sqrt{p_{k}}/2\right)-12}\,,$
$\displaystyle\varphi_{3}(p_{k})$
$\displaystyle=\frac{p_{k}\cosh\left(\sqrt{p_{k}}\right)-\sqrt{p_{k}}\sinh\left(\sqrt{p_{k}}\right)}{4\sqrt{p_{k}}\sinh\left(\sqrt{p_{k}}\right)-8\cosh\left(\sqrt{p_{k}}\right)+8}\,,$
$\displaystyle\varphi_{4}(p_{k})$
$\displaystyle=\frac{\sqrt{p_{k}}\left(\sinh\left(\sqrt{p_{k}}\right)-\sqrt{p_{k}}\right)}{\left(4\sqrt{p_{k}}\coth\left(\sqrt{p_{k}}/2\right)-8\right)\sinh^{2}\left(\sqrt{p_{k}}/2\right)}\,.$

Note that the tangent stiffness matrix $\bm{K}_{k}$ representative of the
$k-$rod in the prestressed lattice reduces, in the limit of vanishing
prestress (or unitary pre-stretch $\lambda_{0k}=1$), to the usual stiffness
matrix of an Euler-Bernoulli beam with Hermitian shape functions, so that

$\lim_{p\to 0}\varphi_{j}(p)=1,\quad\forall j\in\\{1,...,4\\}.$

###  2.2 Second-order energy contribution of the slider constraint

For the formulation of the incremental equilibrium of the lattice, the
contributions to the potential energy of the constraints between the rods need
to be introduced. As the constraints considered in this work are clamps and
sliders, the derivation of the incremental contribution of the latter is
addressed in the following, while the former simply impose continuity of
displacements and rotations.

Two rods subject to the same axial load $P$ are considered, connected through
a slider constraint, as sketched in Fig. 2.

Figure 2:  Stress-free (left), stretched (center), and incrementally deformed
(right) configurations of two rods connected to each other through a slider
constraint and subject to an axial load $P$. The slider imposes a constraint
on the displacement jump,
$\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\bm{n}_{\text{S}}=0$, and on
the rotation, $\llbracket\theta\rrbracket_{\text{S}}=0$.

By denoting the strain-energy density of the rods as $\psi$, the potential
energy can be written as

$\mathcal{V}=\int_{0}^{L_{1}}\psi\,\text{d}x_{1}+\int_{0}^{L_{2}}\psi\,\text{d}x_{2}-P\,u_{2}(L_{2})+P\,u_{1}(0)\,,$
(9)

where the local coordinates $x_{1}$ and $x_{2}$ as well as the rods’ length
are referred to the stress-free reference configuration. The strain-energy
density $\psi$ is dependent on the local stretch $\lambda$ and curvature
$\chi$ of the rod, which are defined as

$\lambda=\left(1+u^{\prime}(x)\right)\cos\theta(x)+v^{\prime}(x)\sin\theta(x)\,,\quad\chi=\theta^{\prime}(x)\,,$

with the rotation field $\theta$ satisfying the unshearability constraint
$\theta=\arctan\left[v^{\prime}/(1+u^{\prime})\right]$. The dependence of
$\psi$ on $\lambda$ and $\chi$ is assumed in the form
$\psi(\lambda,\chi)=\psi_{\lambda}(\lambda)+\psi_{\chi}(\chi)$.

The equilibrium configuration of the connected rods is provided by the
principle of virtual work, effective to the stationary condition

$\delta\mathcal{V}=\int_{0}^{L_{1}}\delta\psi\,\text{d}x_{1}+\int_{0}^{L_{2}}\delta\psi\,\text{d}x_{2}-P\,\delta
u_{2}(L_{2})+P\,\delta u_{1}(0)=0\,,$ (10)

whose solution is to be sought in the space of the displacement fields
satisfying the slider constraint between points $\text{S}^{-}$ and
$\text{S}^{+}$, which reads

$\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\underbrace{\left(\cos\theta_{\text{S}}\,\bm{e}^{\text{t}}+\sin\theta_{\text{S}}\,\bm{e}^{\text{n}}\right)}_{\bm{n}_{\text{S}}}=0\,,$
(11)

where $\llbracket\bm{u}\rrbracket_{\text{S}}$ denotes the displacement jump
across the slider,
$\llbracket\bm{u}\rrbracket_{\text{S}}=\bm{u}_{\text{S}^{+}}-\bm{u}_{\text{S}^{-}}$,
the unit vector $\bm{n}_{\text{S}}$ describes the orientation of slider in the
deformed configuration, and the rotation field is continuous,
$\theta_{\text{S}}=\theta_{\text{S}^{-}}=\theta_{\text{S}^{+}}$, as sketched
in Fig. 2.

Instead of dealing with Eq. (10) subject to the constraint (11), the
equilibrium can be equivalently formulated by means of the Lagrange multiplier
method, so that the potential energy (9) becomes

$\hat{\mathcal{V}}=\underbrace{\int_{0}^{L_{1}}\psi\,\text{d}x_{1}+\int_{0}^{L_{2}}\psi\,\text{d}x_{2}-P\,u_{2}(L_{2})+P\,u_{1}(0)}_{\mathcal{V}}+\underbrace{\mu\,\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\left(\cos\theta_{\text{S}}\,\bm{e}^{\text{t}}+\sin\theta_{\text{S}}\,\bm{e}^{\text{n}}\right)}_{\mathcal{S}},$
(12)

with $\mu$ being the Lagrange multiplier of the slider constraint. The
augmented potential (12) is now used to obtain the straight stretched
equilibrium configuration and the second-order potential governing the
incremental equilibrium.

The straight equilibrium configuration is easily obtained by solving the
vanishing condition of the first variation of functional (12), evaluated for a
displacement field of the form

$u_{0k}(x_{k})=u_{0k}(0)+(u_{0k}(L_{k})-u_{0k}(0))\frac{x_{k}}{L_{k}}\,,\qquad
v_{0k}(x_{k})=0\,\qquad\forall k\in\\{1,2\\}\,,$

so that stationarity of the augmented potential (12) reads as

$\begin{multlined}\delta\hat{\mathcal{V}}_{0}=\int_{0}^{L_{1}}\psi_{\lambda}^{\prime}(\lambda_{01})\,\delta
u_{1}^{\prime}\,\text{d}x_{1}+\int_{0}^{L_{2}}\psi_{\lambda}^{\prime}(\lambda_{02})\,\delta
u_{2}^{\prime}\,\text{d}x_{2}-P\,\delta u_{2}(L_{2})+P\,\delta u_{1}(0)+\\\
+\mu\left(\delta u_{2}(0)-\delta
u_{1}(L_{1})\right)+\delta\mu\left(u_{02}(0)-u_{01}(L_{1})\right)=0\,,\quad\forall\delta
u_{1},\delta
u_{2},\delta\mu\,,\end{multlined}\delta\hat{\mathcal{V}}_{0}=\int_{0}^{L_{1}}\psi_{\lambda}^{\prime}(\lambda_{01})\,\delta
u_{1}^{\prime}\,\text{d}x_{1}+\int_{0}^{L_{2}}\psi_{\lambda}^{\prime}(\lambda_{02})\,\delta
u_{2}^{\prime}\,\text{d}x_{2}-P\,\delta u_{2}(L_{2})+P\,\delta u_{1}(0)+\\\
+\mu\left(\delta u_{2}(0)-\delta
u_{1}(L_{1})\right)+\delta\mu\left(u_{02}(0)-u_{01}(L_{1})\right)=0\,,\quad\forall\delta
u_{1},\delta u_{2},\delta\mu\,,$ (13)

yielding the following equilibrium conditions

$\psi_{\lambda}^{\prime}(\lambda_{01})=\psi_{\lambda}^{\prime}(\lambda_{02})=\mu_{0}=P\,,\qquad
u_{02}(0)=u_{01}(L_{1})\,,$ (14)

where $\lambda_{01}$ and $\lambda_{02}$ denote the stretch values of the rods,
while $\mu=\mu_{0}$ denotes the value of the Lagrange multiplier at the
equilibrium. These conditions are derived from equation (13) by taking
advantage of homogeneity of the stretch in the two rods (so that
$\lambda_{01}$ and $\lambda_{02}$ are constants) and noting also that the
residual bending moment $\psi_{\chi}^{\prime}(0)$ vanishes on a straight
configuration.

Upon the straight stretched configuration defined by Eq. (14), the incremental
equilibrium is governed by the second-order expansion of the augmented
potential, Eq. (12). Letting $\Delta\bm{u}_{1}$, $\Delta\bm{u}_{2}$, and
$\Delta\mu$ be the increments with respect to the straight equilibrium
configuration, the expansion assumes the form

$\hat{\mathcal{V}}(\bm{u}_{01}+\Delta\bm{u}_{1},\bm{u}_{02}+\Delta\bm{u}_{2},\mu_{0}+\Delta\mu)\sim\underbrace{\delta\hat{\mathcal{V}}_{0}}_{=0}+\delta^{2}\mathcal{V}_{0}(\Delta\bm{u}_{1},\Delta\bm{u}_{2},\Delta\mu)+\delta^{2}\mathcal{S}_{0}(\Delta\bm{u}_{1},\Delta\bm{u}_{2},\Delta\mu),$
(15)

where the subscript $(\cdot)_{0}$ highlights reference to the straight
equilibrium configuration.

The second-order part of Eq. (15) is composed of two terms: the term
$\delta^{2}\mathcal{V}_{0}$ accounting for the incremental energy distributed
on the rods (as if the slider were absent) and the term
$\delta^{2}\mathcal{S}_{0}$ that accounts only for the contribution of the
slider constraint. The rods’ contribution $\delta^{2}\mathcal{V}_{0}$ assumes
the usual form, Eq. (8) (see [23] for details), while the slider contribution
$\delta^{2}\mathcal{S}_{0}$ has to be determined. Using the definition of
$\mathcal{S}$ from Eq. (12) and recalling the properties of the straight
equilibrium configuration, Eqs. (14), $\delta^{2}\mathcal{S}_{0}$ becomes

$\displaystyle\delta^{2}\mathcal{S}_{0}(\Delta\bm{u}_{1},\Delta\bm{u}_{2},\Delta\mu)$
$\displaystyle=\mu_{0}\,\llbracket\Delta\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\Delta\theta_{\text{S}}\bm{e}^{\text{n}}+\Delta\mu\,\llbracket\Delta\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\bm{e}^{\text{t}}$
(16) $\displaystyle=\mu_{0}(\Delta v_{2}(0)-\Delta
v_{1}(L_{1}))\Delta\theta_{\text{S}}+\Delta\mu(\Delta u_{2}(0)-\Delta
u_{1}(L_{1}))\,,$

where the incremental rotation of the slider $\Delta\theta_{\text{S}}$ can be
written in terms of the transverse displacement as

$\Delta\theta_{\text{S}}=\frac{1}{\lambda_{01}}\frac{\partial\Delta
v_{1}}{\partial x_{1}}(L_{1})=\frac{1}{\lambda_{02}}\frac{\partial\Delta
v_{2}}{\partial x_{2}}(0)\,,$

or also as

$\Delta\theta_{\text{S}}=\frac{\partial\Delta v_{1}}{\partial
s_{1}}(l_{1})=\frac{\partial\Delta v_{2}}{\partial s_{2}}(0)\,,$

obtained by simply updating the reference configuration from the stress-free,
described by the coordinate $x_{k}\in[0,L_{k}]$, to the current stretched
configuration function of $s_{k}=\lambda_{0k}x_{k}\in[0,l_{k}]$.

Having determined the second-order contribution of the slider, Eq. (16), the
symbol $\Delta$, introduced to denote incremental quantities, will be omitted
in order to ease the notation. Thus, all quantities are assumed in the
following to be incremental quantities unless stated otherwise. Accordingly,
the contribution of the slider, Eq. (16), is simply denoted by $\mathcal{S}$
and written as

$\mathcal{S}\left(\llbracket\bm{u}\rrbracket_{\text{S}},\mu\right)=P\,\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\theta_{\text{S}}\bm{e}^{\text{n}}+\mu\,\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\bm{e}^{\text{t}}\,,$
(17)

where the equality $\mu_{0}=P$ provided by Eq. (14) has been used. The
stationary of Eq. (17) with respect to the Lagrange multiplier $\mu$ yields
the condition
$\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\bm{e}^{\text{t}}=0$,
expressing continuity of the axial displacement across the slider. Hence, Eq.
(17) is finally simplified as

$\mathcal{S}\left(\llbracket\bm{u}\rrbracket_{\text{S}}\right)=P\,\llbracket\bm{u}\rrbracket_{\text{S}}\bm{\cdot}\theta_{\text{S}}\bm{e}^{\text{n}}\,.$
(18)

The following subsection combines the incremental energy of the rods, Eq. (8),
with that pertaining to the sliders, Eq. (18), to construct the incremental
equilibrium of the entire lattice structure.

###  2.3 Incremental equilibrium for the unit cell

The equations governing the incremental equilibrium are formulated for a unit
cell of the lattice with respect to the current preloaded configuration. The
incremental potential energy $\mathcal{V}(\bm{q})$ of a unit cell can be
evaluated by summing the contribution of each rod, Eq. (8), as well as of each
slider, Eq. (18), so that

$\mathcal{V}(\bm{q})=\sum_{k=1}^{N_{\text{B}}}\mathcal{V}_{k}(\bm{C}_{k}\bm{q})+\sum_{i=1}^{N_{\text{S}}}\mathcal{S}_{i}(\llbracket\bm{u}\rrbracket_{S_{i}})\,,$
(19)

where $N_{\text{S}}$ is the number of sliders in the unit cell, $\bm{q}$ is
the vector collecting the degrees of freedom of the unit cell,
$\mathcal{S}_{i}$ denotes the contribution of $i$-th slider, and $\bm{C}_{k}$
is the connectivity matrix of the $k$-th rod, such that
$\bm{q}_{k}=\bm{C}_{k}\bm{q}$, which imposes the appropriate constraints at
the junctions between the rods (continuity of all the fields for welded
junctions and continuity of rotation and axial displacement for sliders).

The current configuration of the unit cell is subject to external generalized
incremental forces $\bm{f}$ (including bending moments) transmitted by the
rest of the lattice from which the cell is thought to be ideally ‘excised’.
Hence, the incremental equilibrium for a single unit cell can be stated
through the principle of virtual work as

$\delta\mathcal{V}(\bm{q},\delta\bm{q})=\bm{f}\bm{\cdot}\delta\bm{q}\,,\quad\forall\delta\bm{q}\,,$
(20)

where $\delta\bm{q}$ is the virtual counterpart of $\bm{q}$. Note that, due to
the assumption of absence of body forces, the external virtual work
$\bm{f}\bm{\cdot}\delta\bm{q}$ only involves forces applied on the unit cell
boundary, as the only non-vanishing external forces acting on a unit cell are
those transmitted by the neighboring cells.

The incremental equilibrium equations are therefore obtained from the
variational statement, Eq. (20), yielding

$\bm{K}(\bm{P})\,\bm{q}=\bm{f}\,,$ (21)

where

$\bm{K}(\bm{P})=\frac{\partial^{2}\mathcal{V}(\bm{q})}{\partial\bm{q}\,\partial\bm{q}}\,,$
(22)

is the symmetric (as derived from a scalar potential) stiffness matrix of the
unit cell, function of the vector $\bm{P}=\\{P_{1},...,P_{N_{\text{B}}}\\}$,
which collects the axial prestress of the rods. The dimension of the system
(21) is $3N_{j}$ where $N_{j}$ is the number of nodes in the unit cell.

It is worth pointing out that the contribution $\mathcal{V}_{k}$ from each
rod, appearing in the potential energy, Eq. (19), is positive definite
whenever the preload is tensile ($P_{k}>0$), while the slider contribution
$\mathcal{S}_{k}$ is indefinite even for a tensile preload. This property
implies that in the absence of slider constraints bifurcations would be
excluded for tensile preload state. Therefore, the presence of the sliders
allows $\mathcal{V}$ to vanish for a non-trivial deformation even when
$P_{k}>0\,,\,\forall k\in\\{1,...,N_{\text{B}}\\}$ and thus tensile
instabilities of the lattice (and its effective continuum) become possible.

##  3 The effective prestressed elastic continuum

As the preloaded configuration of the lattice is assumed to be spatially
periodic, the homogenized incremental response of an effective prestressed
elastic solid can be defined by computing the average strain-energy density,
associated to an incremental displacement field (defined for the $j$-th node
by the displacement and rotation components, respectively,
$\bm{q}_{u}^{(j)}=\\{u^{(j)},v^{(j)}\\}^{\mathsf{T}}$ and
$\bm{q}_{\theta}^{(j)}=\\{\theta^{(j)}\\}$) which obeys the Cauchy-Born
hypothesis [30, 31, 32]. The latter, for a single unit cell, prescribes that
the displacement of the lattice’s nodes be decomposed into the sum of an
affine incremental deformation (ruled by a constant second-order tensor
$\bm{L}$) and a periodic field (defined by a displacement
$\tilde{\bm{q}}_{u}^{(j)}$ and a rotational $\tilde{\bm{q}}_{\theta}^{(j)}$
component) as

$\bm{q}_{u}^{(j)}=\tilde{\bm{q}}_{u}^{(j)}+\bm{L}\,\bm{x}_{j}\,,\qquad\bm{q}_{\theta}^{(j)}=\tilde{\bm{q}}_{\theta}^{(j)}\,,\qquad\forall
j\in\\{1,...,N_{j}\\}\,,$ (23)

where $\bm{x}_{j}$ is the position of the $j$-th node.

The periodic term $\tilde{\bm{q}}$ satisfies
$\tilde{\bm{q}}^{(p)}=\tilde{\bm{q}}^{(q)}$ for all $\\{p,q\\}$ such that
$\bm{x}_{q}-\bm{x}_{p}=n_{1}\bm{a}_{1}+n_{2}\bm{a}_{2}$ (with
$n_{j}\in\\{0,1\\}$). This term can be expressed as a function of its
independent components through a partition of the degrees of freedom, to be
made in accordance with the location of the nodes present inside the unit
cell. Specifically, by denoting with $\tilde{\bm{q}}^{i}$ the degrees of
freedom located inside the unit cell, with $\tilde{\bm{q}}^{l}$,
$\tilde{\bm{q}}^{r}$, $\tilde{\bm{q}}^{b}$, $\tilde{\bm{q}}^{t}$ those on the
left, right, lower, and upper edge respectively, and with
$\tilde{\bm{q}}^{lb}$, $\tilde{\bm{q}}^{rb}$, $\tilde{\bm{q}}^{lt}$,
$\tilde{\bm{q}}^{rt}$ those located at the four corners, the periodic field
can be expressed as

$\tilde{\bm{q}}=\begin{Bmatrix}\tilde{\bm{q}}^{i}\\\ \tilde{\bm{q}}^{l}\\\
\tilde{\bm{q}}^{b}\\\ \tilde{\bm{q}}^{lb}\\\ \tilde{\bm{q}}^{r}\\\
\tilde{\bm{q}}^{t}\\\ \tilde{\bm{q}}^{rb}\\\ \tilde{\bm{q}}^{lt}\\\
\tilde{\bm{q}}^{rt}\end{Bmatrix}=\begin{bmatrix}\bm{I}&\bm{0}&\bm{0}&\bm{0}\\\
\bm{0}&\bm{I}&\bm{0}&\bm{0}\\\ \bm{0}&\bm{0}&\bm{I}&\bm{0}\\\
\bm{0}&\bm{0}&\bm{0}&\bm{I}\\\ \bm{0}&\bm{I}&\bm{0}&\bm{0}\\\
\bm{0}&\bm{0}&\bm{I}&\bm{0}\\\ \bm{0}&\bm{0}&\bm{0}&\bm{I}\\\
\bm{0}&\bm{0}&\bm{0}&\bm{I}\\\
\bm{0}&\bm{0}&\bm{0}&\bm{I}\end{bmatrix}\begin{Bmatrix}\tilde{\bm{q}}^{i}\\\
\tilde{\bm{q}}^{l}\\\ \tilde{\bm{q}}^{b}\\\
\tilde{\bm{q}}^{lb}\end{Bmatrix}\,,$ (24a) which may succinctly be rewritten
as $\tilde{\bm{q}}=\bm{Z}_{0}\,\tilde{\bm{q}}^{*}\,,$ (24b)

where $\bm{Z}_{0}$ and $\tilde{\bm{q}}^{*}$ are defined according to Eq.
(24a). The same partitioning is also used for the vectors $\bm{q}$ and
$\bm{f}$. Note that the periodicity conditions (24) represent the long-
wavelength limit of the Floquet-Bloch conditions used in wave propagation
problems [33, 23].

In order to enforce the Cauchy-Born conditions into the equations of
incremental equilibrium (21), it is convenient to rewrite Eq. (23) as

$\bm{q}(\tilde{\bm{q}}^{*},\bm{L})=\bm{Z}_{0}\,\tilde{\bm{q}}^{*}+\hat{\bm{q}}(\bm{L})\,,$
(25)

where the affine part of the deformation $\hat{\bm{q}}(\bm{L})$ is a vector-
valued function linear in $\bm{L}$ and such that

$\hat{\bm{q}}(\bm{L})_{u}^{(j)}=\bm{L}\,\bm{x}_{j}\,,\qquad\hat{\bm{q}}(\bm{L})_{\theta}^{(j)}=\bm{0}\,,\qquad\forall
j\in\\{1,...,N_{j}\\}\,,$

where the same notation introduced with Eq. (23) has been used so that the
subscript $u$ (subscript $\theta$) denotes displacement (rotation) components.

Note that, since the lattice is subject to a non-vanishing prestress state,
the macroscopic incremental deformation gradient defined by $\bm{L}$ must be
an arbitrary second-order tensor, not constrained to be symmetric (as it
happens in the absence of prestress [20, 31, 32]). As explained in the next
section, this lack of symmetry is essential for the correct evaluation of the
incremental fourth-order tensor defining the effective continuum,
‘macroscopically equivalent’ to the lattice.

###  3.1 Incremental constitutive tensor for the effective continuum

Before introducing the homogenization technique, it is important to recall
that, as shown in Section 2, the equilibrium equations for the lattice are
(i) obtained in the context of a linearized theory, and  (ii) referred to a
prestressed reference configuration.  Therefore, the effective continuum, for
the moment unknown, has to be formulated in the context of the incremental
theory of nonlinear elasticity by means of a relative Lagrangian description
as introduced by Hill [34], see also [24]. As a consequence, the response of
the effective material is defined by an incremental constitutive law in the
form

$\dot{\bm{S}}=\mathbb{C}[\bm{L}]\,,$ (26)

relating the increment of the first Piola-Kirchhoff stress $\dot{\bm{S}}$ to
the gradient of incremental displacement $\bm{L}$, through the elasticity
tensor $\mathbb{C}$. The most general form for the constitutive tensor
$\mathbb{C}$ is

$\mathbb{C}=\mathbb{E}+\bm{I}\boxtimes\bm{T}\qquad\mbox{in
components}\qquad\mathbb{C}_{ijkl}=\mathbb{E}_{ijkl}+\delta_{ik}T_{jl}\,,$
(27)

where $\delta_{ik}$ is the Kronecker delta, $\bm{T}$ is the Cauchy stress,
defining the prestress, and $\mathbb{E}$ is a fourth-order elastic tensor,
endowed with all usual (left and right minor and major) symmetries

$\mathbb{E}_{ijkl}=\mathbb{E}_{jikl}=\mathbb{E}_{ijlk}=\mathbb{E}_{klij}\,,$
(28)

so that $\mathbb{C}$ lacks the minor symmetries but possesses the major
symmetry. The symmetries of $\mathbb{C}$ explain the reason why the full
incremental deformation gradient $\bm{L}$, and not only its symmetric part,
appears in the Cauchy-Born hypothesis, Eq. (23), of the lattice. Moreover, Eq.
(27) shows that $\bm{L}$ can be restricted to be symmetric only in the absence
of prestress, $\bm{T}=\bm{0}$.

The incremental strain-energy density for the prestressed continuum is
referred to the prestressed configuration. It can be expressed in terms of a
second-order expansion with respect to the incremental deformation gradient
$\bm{L}$ as follows

$\mathcal{W}(\bm{L})=\underbrace{\bm{T}\bm{\cdot}\bm{L}}_{\mathcal{W}_{1}(\bm{L})}+\underbrace{\mathbb{C}[\bm{L}]\bm{\cdot}\bm{L}/2}_{\mathcal{W}_{2}(\bm{L})}\,,$
(29)

where the first-order increment $\mathcal{W}_{1}(\bm{L})$ accounts for the
work expended by the current prestress state $\bm{T}$ (due to the relative
Lagrangian description the first Piola-Kirchhoff stress coincides with the
Cauchy stress), while the second-order term $\mathcal{W}_{2}(\bm{L})$ is the
strain-energy density associated with the incremental first Piola-Kirchhoff
stress given by Eq. (26).

It is also worth noting that a calculation of the second gradient of the
incremental energy density, Eq. (29), with respect to $\bm{L}$ yields the
constitutive fourth-order tensor $\mathbb{C}$ relating the stress increment to
the incremental displacement gradient. Taking the first gradient provides,
when evaluated at $\bm{L}=\bm{0}$, the prestress $\bm{T}$. The latter property
will be used to dissect the effect of prestress in the homogenized response of
the lattice.

###  3.2 First and second-order matching of the incremental strain-energy
density

The homogenization of the lattice response is based on the equivalence between
the average incremental strain-energy associated to a macroscopic incremental
displacement gradient applied to the lattice and the incremental strain-energy
density of the effective elastic material subject to the same deformation. In
the classical homogenization theory, this condition is known as macro-
homogeneity condition, or Hill-Mandel theorem [35, 36, 20, 31], which provides
the link between the microscopic and macroscopic scales.

In the following, the macro-homogeneity condition is enforced to obtain the
incremental energy density (29) that matches the effective behavior of the
prestressed lattice at first- $\mathcal{W}_{1}(\bm{L})$ and at second-
$\mathcal{W}_{2}(\bm{L})$ order. Thus, the homogenization scheme is based on
the following steps:

  1. (i)

An incremental deformation gradient $\bm{L}$ is considered, so that the
incremental energy density for the unknown effective continuum is defined by
Eq. (29);

  2. (ii)

following the Cauchy-Born hypothesis, Eq. (25), the incremental displacement
field for the lattice is prescribed by the given tensor $\bm{L}$ and the
periodic vector $\tilde{\bm{q}}^{*}$ necessary to enforce the equilibrium of
the lattice;

  3. (iii)

with the solution of the lattice in terms of $\bm{L}$ (the periodic vector
$\tilde{\bm{q}}^{*}$ becomes in solution a function of $\bm{L}$) the
incremental energy density is calculated for the lattice;

  4. (iv)

the two incremental energy densities in the continuum and in the lattice are
matched, so to obtain the components of the incremental elastic tensor
defining the effective solid.

#### Determination of the periodic displacement field for the lattice.

By substituting condition (25) into Eqs. (21) and pre-multiplying by
$\bm{Z}_{0}^{\mathsf{T}}$, the incremental equilibrium becomes

$\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0}\,\tilde{\bm{q}}^{*}+\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\hat{\bm{q}}(\bm{L})=\bm{Z}_{0}^{\mathsf{T}}\bm{f}\,,$
(30)

where the right-hand side can be written more explicitly using the
partitioning introduced by Eq. (24a) as

$\bm{Z}_{0}^{\mathsf{T}}\bm{f}=\begin{Bmatrix}\bm{f}^{i}\\\
\bm{f}^{l}+\bm{f}^{r}\\\ \bm{f}^{b}+\bm{f}^{t}\\\
\bm{f}^{lb}+\bm{f}^{rb}+\bm{f}^{lt}+\bm{f}^{rt}\end{Bmatrix}\,.$

The fact that the only non-vanishing forces are assumed to be the internal
forces transmitted at the unit cell boundary by the neighboring cells implies
$\bm{f}^{i}=\bm{0}$. Moreover, as the displacement field satisfying the
Cauchy-Born hypothesis generates internal forces in the infinite lattice that
are periodic along the direct basis $\\{\bm{a}_{1},\bm{a}_{2}\\}$, any single
unit cell is subject to external boundary forces that are anti-periodic.
Consequently, $\bm{f}^{l}=-\bm{f}^{r}$, $\bm{f}^{b}=-\bm{f}^{t}$ and
$\bm{f}^{lb}=-\bm{f}^{rb}-\bm{f}^{lt}-\bm{f}^{rt}$, so that the term
$\bm{Z}_{0}^{\mathsf{T}}\bm{f}$ vanishes and Eq. (30) becomes

$\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0}\,\tilde{\bm{q}}^{*}=-\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\hat{\bm{q}}(\bm{L})\,.$
(31)

The solution of the linear system (31) provides the incremental periodic
displacement field $\tilde{\bm{q}}^{*}$ internal to the lattice for every
given $\bm{L}$. As a consequence of the linearity of $\hat{\bm{q}}(\bm{L})$,
the solution $\tilde{\bm{q}}^{*}(\bm{L})$ is, in turn, a linear function of
$\bm{L}$.

A few considerations have to be made about the solvability of the system (31).
In fact, it is easy to show that the matrix
$\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0}$ is always singular,
regardless of the specific lattice structure under consideration. This is
proved by considering a vector $\tilde{\bm{q}}^{*}=\bm{t}$ defining a pure
rigid-body translation and observing that
$\bm{K}(\bm{P})\,\bm{Z}_{0}\,\bm{t}=\bm{0}$, which, in turn, implies that the
dimension of the nullspace of
$\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0}$ is at least 2, as two
linearly independent rigid-body translations exist for a 2D lattice. Any other
deformation mode, possibly contained in
$\ker(\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0})$, is therefore a
zero-energy mode, called ‘floppy mode’ [37, 38]. These modes are excluded in
the following analysis to ensure solvability of system (31), so that
$\ker(\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0})$ contains only two
rigid-body translations. Floppy modes can be always recovered in limits of
vanishing stiffness and can eliminated or introduced playing with prestress
[39, 40].

Having excluded floppy modes and observing that the right-hand side of Eq.
(31) is orthogonal to
$\ker(\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0})$,

$\bm{t}\bm{\cdot}\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\hat{\bm{q}}(\bm{L})=0,$

for all rigid-body translations $\bm{t}$, the solution
$\tilde{\bm{q}}^{*}(\bm{L})$ can now be determined.

#### Match of the second-order incremental strain-energy density and
determination of the incremental constitutive tensor.

The solution of the linear system (31) allows the incremental displacement,
Eq. (25), to be expressed only in terms of the macroscopic displacement
gradient $\bm{L}$ as $\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})$. Therefore,
the second-order incremental strain-energy stored in a single unit cell of the
lattice undergoing a macroscopic strain can be evaluated as follows

$\mathcal{E}(\bm{L})=\frac{1}{2}\,\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})\bm{\cdot}\bm{K}(\bm{P})\,\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})\,,$
(32)

which is a quadratic form in $\bm{L}$, because
$\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})$ is linear in $\bm{L}$. By equating
the second-order strain-energy density of the continuum
$\mathcal{W}_{2}(\bm{L})=\mathbb{C}[\bm{L}]\bm{\cdot}\bm{L}/2$ to the average
energy of the lattice (32), the following equivalence condition is obtained

$\underbrace{\frac{1}{2}\,\mathbb{C}[\bm{L}]\bm{\cdot}\bm{L}}_{\text{Continuum}}=\underbrace{\frac{1}{\lvert\mathcal{C}\lvert}\,\mathcal{E}(\bm{L})}_{\text{Lattice}}\,,$
(33)

where $\lvert\mathcal{C}\lvert$ is the area of the unit cell.

Finally, a calculation of the second gradient of Eq. (33) with respect to
$\bm{L}$ yields the incremental constitutive tensor for the effective Cauchy
material, in the form

$\mathbb{C}=\frac{1}{\lvert\mathcal{C}\lvert}\,\frac{\partial^{2}\,\mathcal{E}(\bm{L})}{\partial\bm{L}\,\partial\bm{L}}=\frac{1}{2\lvert\mathcal{C}\lvert}\,\frac{\partial^{2}}{\partial\bm{L}\,\partial\bm{L}}\Big{[}\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})\bm{\cdot}\bm{K}(\bm{P})\,\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})\Big{]}\,,$
(34)

which becomes now an explicit function of the prestress state, as well as of
all the mechanical parameters defining the lattice.

#### Match of the first-order incremental strain-energy density and
homogenization of the prestress state.

So far, the incremental constitutive tensor $\mathbb{C}$ of a continuum
‘equivalent’ to a prestressed elastic lattice, Eq. (34), has been obtained
through homogenization. It is important now to ‘dissect’ from $\mathbb{C}$ the
effect of the prestress $\bm{T}$ and, as a consequence, to obtain tensor
$\mathbb{E}$.

It will be shown below that the current prestress state $\bm{T}$ of the
homogenized material can directly be linked to the preload state
$\bm{P}=\\{P_{1},...,P_{N_{\text{B}}}\\}$ of the lattice. In fact, by
observing that equation (33) represents the second-order incremental strain
energy, equal to
$\mathcal{W}_{2}(\bm{L})=\dot{\bm{S}}(\bm{L})\bm{\cdot}\bm{L}/2$, an
equivalence analogous to that expressed by equation (33) can be obtained
considering the first-order increment of the strain energy,
$\mathcal{W}_{1}(\bm{L})=\bm{T}\bm{\cdot}\bm{L}$. Thus, the first-order term
can be identified as the average work done by the prestress state $\bm{P}$
during the lattice deformation $\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})$
induced by $\bm{L}$. Accordingly, the following equivalence can be stated

$\underbrace{\bm{T}\bm{\cdot}\bm{L}}_{\text{Continuum}}=\underbrace{\frac{1}{\lvert\mathcal{C}\rvert}\,\bm{f}_{\bm{P}}\bm{\cdot}\bm{q}(\tilde{\bm{q}}^{*}(\bm{L}),\bm{L})}_{\text{Lattice}}\,,$
(35)

where vector $\bm{f}_{\bm{P}}$ collects the forces that emerge at the nodes of
the unit cell and are in equilibrium with the axial preload $\bm{P}$, in the
current configuration assumed as reference. As a consequence, the forces
$\bm{f}_{\bm{P}}$ are independent of $\bm{L}$ and linear in $\bm{P}$.

Equation (35) requires that the work done by axial loads $\bm{f}_{\bm{P}}$ for
nodal displacements $\bm{q}$ associated to a skew-symmetric velocity gradient
$\bm{L}=\bm{W}$ be zero, namely

$\bm{f}_{\bm{P}}\bm{\cdot}\bm{q}(\tilde{\bm{q}}^{*}(\bm{W}),\bm{W})=0\,.$ (36)

This statement is a direct consequence of the principle of virtual work for
rigid body incremental motions, because
$\bm{q}(\tilde{\bm{q}}^{*}(\bm{W}),\bm{W})$ represents an incremental rotation
of the lattice and $\bm{f}_{\bm{P}}$ satisfies equilibrium. Hence, taking into
account the property (36), the homogenized prestress $\bm{T}$ can be obtained
as the gradient of the equivalence condition (35) with respect to the
symmetric part of $\bm{L}$, denoted as $\bm{D}$,

$\bm{T}=\frac{1}{\lvert\mathcal{C}\rvert}\,\frac{\partial}{\partial\bm{D}}\Big{[}\bm{f}_{\bm{P}}\bm{\cdot}\bm{q}(\tilde{\bm{q}}^{*}(\bm{D}),\bm{D})\Big{]}\,.$
(37)

##  4 Tensile material instabilities in a preloaded lattice

The analytical framework developed in Sections 2, and 3 is now applied to a
particular lattice structure in order to showcase a concrete example of a
material displaying static instabilities that are triggered by both
compressive and tensile presstress states.

(a) Unit cell

(b) Periodic lattice

Figure 3:  The rectangular lattice of preloaded rods endowed with sliding
constraints used to realize a material capable of losing ellipticity under
tensile and compressive loadings. Linear springs are introduced to stiffen the
sliders, thus preventing floppy modes at vanishing preload.

The lattice under analysis is the rectangular grid illustrated in Fig. 3 which
is preloaded along the two principal directions and endowed with sliding
constraints. The sliders are stiffened by linear springs to prevent trivial
floppy modes that would otherwise be present when the preload vanishes. The
unit cell of the periodic structure, Fig. 3(a), is chosen to provide the
minimum number of rods and thus simplify the computations involved in the
stability analysis.

The lattice configuration is parametrized by the following dimensionless
ratios

$p_{i}=\frac{P_{i}\,l_{i}^{2}}{B_{i}}\,,\quad\lambda_{i}=l_{i}\sqrt{\frac{A_{i}}{B_{i}}}\,,\quad\kappa_{i}=\frac{k_{i}\,l_{i}^{3}}{B_{i}}\,,\quad\xi=l_{2}/l_{1}\,,\quad\chi=A_{2}/A_{1}\,,\quad\forall
i\in\\{1,2\\}\,,$ (38)

where the index $i$ identifies the horizontal and vertical rods according to
Fig. 3(a), while $k_{i}$ denotes the sliding stiffness of the sliders provided
by the linear springs.

Buckling of structures similar to those considered here and embedding sliding
constraints has been considered in [41], under the hypotheses that the
structure has a finite-size and is subject to equibiaxial loading. In [41] the
rods are assumed as axially extensible, but rigid under bending, while both
deformations are taken into account in the present work. Moreover, an
effective continuum material for such structures has not been given in [41].

The general homogenization method developed in Section 3, applied to the
lattice material represented in Fig. 3, allows the identification of the
effective incremental constitutive tensor $\mathbb{C}$ and prestress tensor
$\bm{T}$ (which may be conveniently made dimensionless through multiplication
by $l_{1}/A_{1}$) as explicit functions of all parameters (38). The resulting
expression of the constitutive tensor $\mathbb{C}$ is quite lengthy and is
deferred to Appendix A, while the homogenized prestress tensor $\bm{T}$ can be
compactly expressed as

$\bm{T}=\frac{P_{1}}{l_{2}}\,\bm{e}_{1}\otimes\bm{e}_{1}+\frac{P_{2}}{l_{1}}\,\bm{e}_{2}\otimes\bm{e}_{2}\,.$

###  4.1 Positive definiteness, strong ellipticity, and lattice stability

A comprehensive stability analysis of the orthotropic lattice under study
requires the determination of the threshold for the applied preload which
triggers one or multiple non-trivial static bifurcations [23].

With regards to the stability of the effective medium, the positive
definiteness (PD) of the incremental constitutive operator $\mathbb{C}$
ensures the uniqueness of the incremental boundary value problem subject to
arbitrary boundary conditions [24]. The effective material is defined PD if

$\mathbb{C}[\bm{L}]\bm{\cdot}\bm{L}>0\,,\qquad\forall\bm{L}\neq\bm{0}\,,$ (39)

which, due to the major symmetry of $\mathbb{C}$, is equivalent to the
positiveness of all the eigenvalues of $\mathbb{C}$. When PD is lost at a
given loading threshold, condition (39) does not hold true and therefore zero-
energy modes (or floppy modes) arise. Thus if the effective material is PD,
the lattice can be considered ‘macroscopically stable’ against arbitrary
disturbances. Otherwise, some macroscopic deformation exist, which is
associated to a zero energy expenditure.

Strong ellipticity (SE) characterizes the stability of the effective medium
with respect to perturbations that vanish on the boundary of an arbitrary
small sphere (corresponding to the so-called ‘van Hove problem’ [24]) and is
defined as the positive definiteness of the acoustic tensor
$\bm{A}^{(\mathbb{C})}(\bm{n})$ associated to the incremental fourth-order
tensor $\mathbb{C}$

$\bm{g}\bm{\cdot}\bm{A}^{(\mathbb{C})}(\bm{n})\,\bm{g}>0\qquad\forall\bm{n}\neq\bm{0}\quad\forall\bm{g}\neq\bm{0}\,,$
(40)

where
$\bm{A}^{(\mathbb{C})}(\bm{n})\,\bm{g}=\mathbb{C}[\bm{g}\otimes\bm{n}]\,\bm{n}$.

On the other hand, the bifurcation of an incrementally loaded periodic lattice
can be analyzed with a Floquet-Bloch technique as shown in [15, 23]. This
analysis shows that both local and global bifurcation modes can occur and only
the latter correspond to failure of ellipticity for the effective material
evaluated from homogenization [18, 20, 22].

###  4.2 A bounded stability domain

Stability domains represent an effective tool to characterize the regions of
the prestress space where a material is stable. Their boundaries define the
thresholds of instability.

Stability domains for the orthotropic lattice shown in Fig. 3 are computed in
the 2D dimensionless prestress space $\\{p_{1},p_{2}\\}$ for several values of
the dimensionless sliding stiffnesses $\kappa_{1}$ and $\kappa_{2}$ with the
purpose of investigating both cubic and orthotropic configurations, including
the limiting cases $\kappa_{1}\to\infty$ and $\kappa_{2}\to\infty$.
Physically, these limits correspond to sliders with infinite sliding
stiffness, thus realizing perfect ‘welding’ conditions. Several configurations
for ‘fully welded’ grids have been explored in [23] where it has been shown
that the stability domain of these materials is unbounded for tensile
preloads. However, the results reported in this section demonstrate that the
introduction of sliding constraints strongly alters the structure of the
stability domain.

For the computation of the stability domains, the slenderness values are set
equal to $\lambda_{1}=\lambda_{2}=20$, while the aspect ratio and the area
ratio are chosen as $\xi=\chi=1$.

Results are reported in Figs. 4–6. Each figure contains six plots
corresponding to different values of sliding stiffness, and in each plot three
kinds of stability domains are illustrated. The shaded regions in dark blue
and bounded by a dashed line represent preload states where the effective
material is PD, while the light blue regions bounded by a solid line define
the domain where the effective material is SE. Note that as a consequence of
Eqs. (39) and (40) PD regions are always contained inside SE regions. The
third domain is the one enclosed by the colored spots, which define the region
where the lattice is stable, so that both long and short-wavelength
bifurcations are excluded.

It can be observed in all of the three Figs. 4–6 that the boundary for lattice
stability (colored spots) coincides with the boundary of SE for the effective
material and encloses the origin. This implies that the critical bifurcation
occurring in the lattice is macroscopic, namely, a long-wavelength bifurcation
corresponding to the formation of a shear band.

The red and green spots denote the presence of a horizontal and vertical shear
band, respectively, while the insets depict the corresponding critical dyad
$\bm{n}_{\text{cr}}\otimes\bm{g}_{\text{cr}}$ responsible for the loss of SE
along the direction $\bm{n}_{\text{cr}}$ and with critical polarization
$\bm{g}_{\text{cr}}$. Note also that the diamond-shaped spots (two for each
domain) denote limit points characterized by the simultaneous occurrence of a
vertical and a horizontal shear band.

Figure 4:  Strong ellipticity (SE), positive definiteness (PD), and lattice
stability domains for the cubic grid with $\lambda_{1}=\lambda_{2}=20$,
$\xi=\chi=1$, and six values of sliding stiffness
$\kappa_{1}=\kappa_{2}=1,2,5,10,20,100$. The arrows sketched in the insets
represent the critical direction $\bm{n}_{\text{cr}}$ and the associated mode
$\bm{g}_{\text{cr}}$ responsible for the loss of strong ellipticity.

Figure 5:  Strong ellipticity (SE), positive definiteness (PD), and lattice
stability domains for the orthotropic grid with $\lambda_{1}=\lambda_{2}=20$,
$\xi=\chi=1$, and six values of sliding stiffness
$\kappa_{1}=\kappa_{2}/3=1,2,5,10,20,100$. Compared to the cubic case reported
in Fig. 4, the orthotropy induced by the different sliding stiffnesses
increases the size of the stability domain along the direction of the stiffest
sliders (vertical).  Figure 6:  As for Fig. 5, but for values of sliding
stiffness $\kappa_{1}=1,2,5,10,20,100$ and $\kappa_{2}=100$, representing the
case of a strongly orthotropic grid with an almost unbounded stability domain
for vertical tensile loading.

Fig. 4 shows stability domains for different slider stiffness
$\kappa_{1}=\kappa_{2}=\\{1,2,5,10,100\\}$ of a lattice with cubic symmetry.
Figs. 5 and 6 refer to $\kappa_{1}\neq\kappa_{2}$ and therefore refer to
orthotropy. In the case of cubic symmetry, Fig. 4, the equibiaxial loading
path $p_{1}=p_{2}$ becomes a symmetry axis for the stability regions, while
this symmetry is broken when the two sliders have a different stiffness,
$\kappa_{1}\neq\kappa_{2}$, Figs. 5, 6.

Remarkably, the stability domain of the lattice is bounded both in compression
and in tension, and tends to become unbounded in tension when the stiffness of
the sliders increases, so that in the limit $\kappa_{1}\to\infty$ and
$\kappa_{2}\to\infty$ the case of a ‘welded connection’ is recovered for which
there is no bifurcation in tension. In addition, three non-trivial features of
the stability domains can be highlighted:

  1. (i)

The SE and PD limit points coincide for uniaxial tension loading along the
horizontal ($p_{2}=0$) and vertical direction ($p_{1}=0$). This is clearly
visible in Figs. 4–4 and Figs. 5–5.

  2. (ii)

The transition from a single horizontal (occurring for $p_{2}<p_{1}$) to a
single vertical (occurring for $p_{2}>p_{1}$) shear band is marked by the
equibiaxial loading path $p_{1}=p_{2}$ regardless of the values of sliding
stiffness $\kappa_{1}$ and $\kappa_{2}$. This can be observed in Figs. 4–6 by
noting that the two limit points, leading to the formation of two shear bands
and marked with diamond-shaped spots in the figures, lie on the equibiaxial
line. Note also that these points are corner points of the otherwise smooth
stability boundary.

  3. (iii)

In Figs. 4–6, the region corresponding to uniqueness with respect to global
bifurcations in the grid of rods has a ‘leaf-shaped’ shape and is bounded by a
closed line marked with green and red spots, denoting occurrence of
macroscopic bifurcations. Outside this region, local bifurcations occur in the
lattice, so that uniqueness is possible only inside the ‘leaf-shaped’
boundary.

In the same figures, the regions corresponding to uniqueness for the grid of
rods mark the first SE boundary that is encountered in a radial (increasing)
stress path by the effective material. Thus, it is clear that loss of
uniqueness at the ‘leaf-shaped’ boundary coincides with the critical loss of
SE in the effective material, which correctly captures instability of the
elastic grid.

However, differently from the lattice, the effective material evidences zones
outside the boundary of global instability, where SE and even PD are recovered
(not shown in Figs. 4–6, but investigated in Section 5). In these zones the
response of the homogenized material returns to be stable, but this stability
does not reflect the behavior of the grid of rods, which is subject to local
instabilities, so that the homogenization scheme does not work properly. This
sort of ‘re-stabilization’ for the effective solid, which recovers SE and even
PD after having lost both in a radial path of increasing prestress, is
analyzed in Section 5.

###  4.3 Zero-energy modes at loss of PD of the effective material

The stability analysis presented in Section 4.2 demonstrates that the bounded
stability domain of the lattice material endowed with sliders is correctly
captured by the SE domain of the effective medium, which allows the prediction
of shear band formation.

An investigation on the deformation of the lattice and the effective material
is presented in Figs. 7 and 8, both referring to the following parameters of
the grid

$\lambda_{1}=\lambda_{2}=20\,,\qquad\xi=\chi=1\,,\qquad\kappa_{1}=\kappa_{2}=10\,,$

corresponding to one of the cases analyzed in Fig. 4.

In both figures, the grid and the effective material are prestressed up to a
point denoted with a red triangle in the prestress space (inset of the figure)
and then subject to an incremental deformation defined by the tensor $\bm{L}$.
The applied incremental deformation $\bm{L}$ is chosen to be the macroscopic
zero-energy mode corresponding to the given radial loading path. In turn, the
macroscopic zero-energy modes are obtained by evaluating, on the PD boundary,
the eigenvectors $\bm{L}_{\text{cr}}$ leading to failure of the PD condition
(39), or second-order work
$\mathbb{C}[\bm{L}_{\text{cr}}]\bm{\cdot}\bm{L}_{\text{cr}}=0$. Then, by
solving Eq. (31) and using Eq. (25), the actual displacement field of the
lattice can be determined and visualized. Fig. 7 refers to a uniaxial tensile
prestress state upon which an incremental simple shear
$\bm{L}=\bm{e}_{1}\otimes\bm{e}_{2}$ is applied. Fig. 8 refers to a biaxial
stress path inclined at $30^{\circ}$ with respect to the horizontal axis and
an incremental deformation
$\bm{L}=0.866025\,\bm{e}_{1}\otimes\bm{e}_{2}+0.5\,\bm{e}_{2}\otimes\bm{e}_{1}$.

It is worth noting that the zero-energy mode for uniaxial tension (Fig. 7(b))
highlights the peculiar interplay, occurring at the instability threshold,
between the local deformation of the unit cell and the macroscopic
deformation. In fact the bifurcation mode shows the sliders within each cell
opening vertically while enabling an overall horizontal macroscopic shearing.
Note also that under uniaxial tension, loss of positive definiteness PD for
the homogenized incremental constitutive operator coincides with loss of
strong ellipticity SE.

Similarly to Fig. 1, Figs. 7 and 8 report both the incrementally deformed grid
and the corresponding incrementally deformed effective material (shown in
orange). From both figures the following conclusions can be drawn:

  * •

The bending moments applied at the ends of the rods correspond to a null mean
stress and therefore do not provide any effect on the boundary of the
effective material. In fact, this effect is a higher-order contribution within
the homogenization scheme adopted here, so that it could be highlighted only
in a higher-order description employing Cosserat or Mindlin continua for the
effective material.

  * •

At loss of PD the effective material admits incremental deformations
corresponding to zero second-order energy. For these deformations (addressed
in the figures), an element of the effective material is subject to an
incremental strain with null surface tractions. In this case, the grid of rods
deforms only under incremental bending moments, which, although contributing
to the local equilibrium of the lattice, do not appear on the effective
continuum.

(a)

(b)

Figure 7:  An incremental simple shear deformation,
$\bm{L}=\bm{e}_{1}\otimes\bm{e}_{2}$, is superimposed upon a uniaxial
prestress state. The response of the unit cell, together with the prestress
position in the stability domain, is shown on the left, while the incremental
deformation of the grid, with superimposed the incremental deformation of the
effective continuum, is shown on the right. (7(a)) The prestress state is far
from the PD and SE boundaries, $\bm{p}\approx\\{3.32955,0\\}$. (7(b)) The
prestress state belongs to both the PD and SE boundaries (coinciding for
uniaxial stress), $\bm{p}_{\text{cr}}\approx\\{6.65910,0\\}$; in this case,
the reported incremental deformation is a zero-energy mode for the effective
material.

(a)

(b)

Figure 8: As for Fig. 7, except that the prestress states are tensile and
biaxial, located on a line inclined at $30^{\circ}$ with respect to
$p_{1}$-axis and that the superimposed deformation is an incremental
deformation
$\bm{L}=0.866025\,\bm{e}_{1}\otimes\bm{e}_{2}+0.5\,\bm{e}_{2}\otimes\bm{e}_{1}$.
(8(a)) The prestress is well inside the SE domain,
$\bm{p}\approx\\{3.53705,2.04212\\}$; (8(b)) The prestress state is on the PD
boundary, $\bm{p}_{\text{cr}}\approx\\{7.07411,4.08424\\}$, now inside the SE
boundary, and the reported incremental deformation represents the
corresponding zero-energy mode for the effective material.

##  5 Re-stabilization of the effective continuum induced by lattice periodic
microinstabilities

The occurrence of the ‘re-stabilization’ mentioned in Section 4.2 is
highlighted in Fig. 9. Here zones of PD (shaded dark gray) and SE (shaded
light gray) are shown that extend beyond the leaf-shaped zone corresponding to
the critical (in other words, the first encountered in a radial stress path)
loss of SE (shaded blue) for the effective material.

This occurrence can be explained on the basis of the homogenization scheme
described in Section 3. One of the key points of this procedure is the
solution of Eq. (31), so that to obtain the periodic displacement vector
$\tilde{\bm{q}}^{*}$ caused by a uniform deformation gradient $\bm{L}$ applied
to the nodes of the grid. The solvabilty of Eq. (31) relies on the absence of
so-called floppy modes, i.e. zero-energy modes, besides rigid translations, as
was already observed in Sec. 3.2. However, compressive/tensile prestress may
induce in the lattice periodic zero-energy modes, associated to buckling
shapes of a single isolated elastic link. The tensile and the first two
compressive buckling loads and shapes for a single link, both hinged and
clamped, are shown in Fig. 9(b).

When one of these lattice periodic microbifurcations is attained, the
coefficient matrix $\bm{Z}_{0}^{\mathsf{T}}\bm{K}(\bm{P})\,\bm{Z}_{0}$ in Eq.
(31) admits a non-trivial zero-energy mode (floppy mode), so that the
displacement vector $\tilde{\bm{q}}^{*}$ tends to infinity. As a result, the
incremental strain energy (32) becomes unbounded. Furthermore, continuing in a
radial stress path beyond the singularity, the effective material may recover
SE, condition (40), or even PD, condition (39). However, the grid of rods is
subject to local instabilities characterized by a periodic bifurcation mode,
as illustrated in Fig. 9. In these conditions the homogenization framework
does not capture the real behavior of the grid of rods, which remains unstable
after the first global bifurcation corresponding to loss of SE in the
effective material.

Fig. 9(b) reports the local modes of bifurcation occurring in all the rods, as
they were isolated from each other, at different re-stabilization points
(indicated with the letters (9(c))–(9(f)) in panel (9(a)) and referring to
corresponding panels denoted with the same letters). The square zones
highlighted in orange in the figure represent the element of the homogenized
continuum, which is left undeformed by the deformation mode corresponding to
the local bifurcation mode shown to occur in the lattice. This incremental
deformation in the grid is therefore ‘invisible’ to the effective continuum.

(a) Stability domains beyond the first bifurcation

(b) Buckling modes of a single elastic rod

(c) Equibiaxial tension: $p_{1}=p_{2}\approx 6.659$

(d) Equibiaxial tension: $p_{1}=p_{2}\approx 9.703$

(e) Equibiaxial compression: $p_{1}=p_{2}\approx-14.65$

(f) Equibiaxial compression: $p_{1}=p_{2}\approx-33.91$

Figure 9:  Re-stabilization of the effective material outside the domain of
initial SE. The latter is reached when the material is subject to a radial
increase in stress, but beyond this threshold, the material recovers SE and
even PD. In these re-stabilization regions, homogenization does not work, as
the grid of elastic rods is always subject to local bifurcations and is
therefore unstable. (a) Stability domains beyond the first bifurcation. (b)
Buckling modes of a single elastic rod endowed with a slider and a stiffening
spring: the tensile and the first two compressive bifurcation modes are shown
for both hinged and clamped configurations. Lattice microbifurcations inducing
re-stabilization of the effective material: (c) equibiaxial tension:
$p_{1}=p_{2}\approx 6.659$, (d) equibiaxial tension: $p_{1}=p_{2}\approx
9.703$, (e) equibiaxial compression: $p_{1}=p_{2}\approx-14.65$ and (f)
equibiaxial compression: $p_{1}=p_{2}\approx-33.91$.

##  6 Concluding remarks

Harnessing periodic lattices of elastic rods to design architected materials
may lead to the erroneous conclusion that the latter are characterized by an
ellipticity domain unbounded in tension. As a consequence, it may be believed
that these materials cannot fail under an ellipticity loss when these stress
states prevail. How these materials can be created to achieve a bounded
stability domain has been shown in the present article through the use of
sliders, namely, constraints allowing only relative sliding between two
connected pieces of rod. This result shows that homogenization leads to
bounded stability domains. Moreover, our results open new possibilities for
the realization of artificial materials with tunable properties and exhibiting
strain localization within the elastic regime and for all possible directions
in the stress space.

## Acknowledgements

G.B. and A.P. gratefully acknowledges the funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No 955944-REFRACTURE2. D.B. acknowledges financial
support from ERC-ADG-2021-101052956-BEYOND.

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##  Appendix A Full expression for the effective constitutive tensor

The complete analytic expression for the effective constitutive tensor of the
lattice analyzed in Section 4 is here reported in dimensionless form
$\mathbb{C}=\frac{A}{l}\,\bar{\mathbb{C}}$ (null components are omitted).

[style=,breakdepth=20]

$\bar{\mathbb{C}}_{1111}=\xi^{-1}\,,$ $\bar{\mathbb{C}}_{2222}=\chi\,,$
$\bar{\mathbb{C}}_{1212}=p_{2}^{3/2}\chi\left(-2\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{3/2}\sqrt{p_{2}}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\\\
+\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{5/2}\left(\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\right)+\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}-1\right)p_{1}^{2}\sqrt{p_{2}}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\\\
-2\kappa_{1}\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}+1\right)\sqrt{p_{1}}\sqrt{p_{2}}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+4\kappa_{1}\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}-1\right)\sqrt{p_{2}}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}+\\\
-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)-2\kappa_{1}\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}-1\right)p_{1}\left(\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\right)\right)/\left(\lambda_{2}^{2}\,D\right)\,,$
$\bar{\mathbb{C}}_{1221}=\bar{\mathbb{C}}_{2112}=2p_{1}p_{2}\chi\left(\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{3/2}-2\kappa_{1}\left(e^{\sqrt{p_{1}}}-1\right)\right)\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)/D\,,$
$\bar{\mathbb{C}}_{2121}=\left(-2\kappa_{1}\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{3/2}p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\\\
+\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{7/2}\left(-2\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}+\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{2}+4\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)+-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\sqrt{p_{2}}\right)+\\\
+\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}-1\right)p_{1}^{3}p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)-2\kappa_{1}\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}-1\right)p_{1}^{2}\left(-2\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}+\\\
+\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{2}+4\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\sqrt{p_{2}}\right)\right)/\left(\lambda_{1}^{2}\xi\,D\right)\,,$

where the coefficient $D$ is given by

$D=-2\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{3/2}p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}+1\right)p_{1}^{5/2}\left(-2\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}+\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{2}+4\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\sqrt{p_{2}}\right)+\\\
+\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}-1\right)p_{1}^{2}p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+4\kappa_{1}\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}-1\right)p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\\\
-2\kappa_{1}\lambda_{1}^{2}\xi\left(e^{\sqrt{p_{1}}}+1\right)\sqrt{p_{1}}p_{2}\chi\left(\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}-2\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)\right)+\\\
-2\kappa_{1}\lambda_{2}^{2}\left(e^{\sqrt{p_{1}}}-1\right)p_{1}\left(-2\left(e^{\sqrt{p_{2}}}+1\right)p_{2}^{3/2}+\left(e^{\sqrt{p_{2}}}-1\right)p_{2}^{2}+4\kappa_{2}\left(e^{\sqrt{p_{2}}}-1\right)-2\kappa_{2}\left(e^{\sqrt{p_{2}}}+1\right)\sqrt{p_{2}}\right)\,.$