# Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Michael Czajkowski  School of Physics, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA  D. Zeb Rocklin  School of Physics, Georgia
Institute of Technology, Atlanta, Georgia 30332, USA

###### Abstract

Mechanical metamaterials designed around a zero-energy pathway of deformation,
known as a mechanism, have repeatedly challenged the conventional picture of
elasticity.However, the complex spatial deformations these structures are able
to support beyond the uniform mechanism remain largely uncharted. Here we
present a unified theoretical framework, showing that the presence of any
uniform mechanism in a two-dimensional structure fundamentally changes its
elastic response by admitting a family of non-uniform zero-energy
deformations. Our formalism reveals a mathematical duality between these
stress-free strains, which we term “sheared analytic modes” and the supported
spatial profiles of stress. These modes undergo a transition from bulk
periodic response to evanescent surface response as the Poisson’s ratio $\nu$
of the mechanism is tuned through an exceptional point at $\nu=0$. We suggest
a first application of these unusual response properties as a switchable
signal amplifier and filter for use in mechanical circuitry and computation.

Classical elasticity is a field theory describing a structure’s deformation
from a single zero-energy shape. In contrast, a growing number of strategies
and methods to program special energy-free pathways of deformation directly
into designer materials Kadic  _et al._ (2012); Bertoldi  _et al._ (2017); Li
_et al._ (2019); Dieleman  _et al._ (2019); Liu  _et al._ (2021); Meloni  _et
al._ (2021) generate structures with continuous manifolds of (nearly) zero-
energy shapes. Such pathways, known as mechanisms, fundamentally challenge the
classical picture of elasticity by setting common elastic constants to
anomalous zero or negative values Nicolaou and Motter (2012); Kadic  _et al._
(2012); Bückmann  _et al._ (2014); Kane and Lubensky (2014); Paulose  _et al._
(2015a, b); Dudek  _et al._ (2016); Wu  _et al._ (2016); Rocklin  _et al._
(2017); Bertoldi  _et al._ (2017); Fruchart  _et al._ (2020); Bossart  _et
al._ (2021). Still further structures have been able to generate previously
forbidden (odd) elastic constants Scheibner  _et al._ (2020), gyroscopic
elastic behavior Nash  _et al._ (2015), multistable Jin  _et al._ (2020) and
hierarchical structures Gatt  _et al._ (2015); Coulais  _et al._ (2018);
Michel and Yunker (2019) as well as frustrated structures lacking a zero-
energy elastic reference state Kang  _et al._ (2014); Armon  _et al._ (2014);
and yet the generic consequences of a single mechanism on elastic response
have still remained largely unexplored.

Figure 1:  A characteristic class of planar mechanisms. (a) The parallelogram-
based mechanism designs we investigate are generated by setting a design angle
$\psi$, and includes the canonical “Rotating Squares” pattern at the point
$\psi=0$. (b) The mechanism itself is traversed via rotating each rigid
parallelogram (dark grey) opposite to its neighbors (light grey), which alters
the macroscopic strain as reflected in the lattice vectors (pink arrows).
Light and dark grey block coloration is purely for ease of viewing. (c) Such a
mechanism strain may be used to generate an orthonormal basis for strain which
divides any arbitrary strain into mechanism and nonmechanism components. (d)
The strains generated by varying the counter-rotation $\theta$ for a variety
of different $\psi$ (different lines) capture an arbitrary variety of
Poisson’s ratios.

A recent series of investigations has revealed that a purely dilational
mechanism fundamentally changes the response of a continuum material by
introducing an associated space of conformal soft modes Sun  _et al._ (2012);
Czajkowski  _et al._ (2022); Zheng  _et al._ (2021). In the continuum limit of
a perfect mechanism, these modes cost zero energy, and it was suggested that
such a nontrivial soft mode space would come paired with any generic mechanism
even outside the dilational limit Rocklin  _et al._ (2017); Czajkowski and
Rocklin (2021); Zheng  _et al._ (2021); Czajkowski  _et al._ (2022), and some
indicative nonlinear examples of this phenomenon have since been identified
Zheng  _et al._ (2021). However, by the introduction of a space of stress-free
continuum responses, these mechanisms must also necessarily be eliminating
stress-bearing response patterns that might have been supported in these
systems. Therefore, the question presses: how is the overall space of
supported deformations necessarily changed by mechanism design?

Here, rather than focus on a particular microstructure or type of mechanism,
we explore how the presence of an arbitrary mechanism necessarily determines a
two-dimensional structure’s non-uniform equilibrium response, in much the same
way that conventional translational and rotational symmetries necessarily give
rise to elastic waves as Goldstone bosons Chaikin and Lubensky (2000). We find
that the impact of an arbitrary mechanism is to generate spatial patterns of
stress-free strains that are dual to the system’s permitted stresses. Both are
analytic in a particular set of sheared coordinates, which we introduce here,
in which the shearing passes from a real transformation for mechanisms with
conventional positive Poisson ratios ($\nu>0$) to complex coordinates for
auxetic mechanisms as in the purely dilational case Czajkowski  _et al._
(2022).

In Sec. I we establish the alteration which a mechanism produces on the
elastic energy, introducing appropriate new strain variables in a
generalization of Voigt notation. In Sec. II we explore the impact of this
energy, identifying a closed-form solution for the deformation patterns which
are stress-free. We show that these deformations, which we term “sheared
analytic modes”, obey a duality with the force-balanced patterns of stress.
These spaces of modes are completely determined by the Poisson’s ratio and
principal axes of the mechanism, and in Sec. III, we show the existence of an
exceptional point in the Poisson’s ratio separating categories of spatial
response in these systems. We illustrate the qualitative division between
auxetic and anauxetic mechanism response using two illustrative analytic
examples, further using the latter, a long strip geometry, to indicate a
possible new device for mechanical computing.

Figure 2:  Unimode mechanics in lattice metamaterials (a - d) Minimal loading
which is compatible with soft (stress-free) motions yields mechanism dominated
soft strain patterns. (a) The systems deform due to a set of additional
springs at the system left boundary which are constrained to extend according
to a smoothly varying function through space. (b) The mechanism strain
$\varepsilon_{m}$ exhibits significant variation through space. (c, d) This
mechanism strain dominates over the nonmechanism strain to an increasing
degree, and the fitting to the global analytic form for soft strains improves
as the lattice structure becomes finer. (e-h) More strict loading which is not
compatible with a soft motion may still be deciphered using sheared analytic
functions. (e) Loading is generated using a known force balanced mode,
generated from a sheared analytic function, along with known nonaffine
relaxation, applied to all boundary nodes and then numerically relaxed in the
interior. (f) The nonmechanism stress is finite and varies through space. (g,
h) This nonmechanism stress dominates increasingly, and the fitting to a
sheared analytic mode improves as the continuum limit is approached just as
with the soft strains.  Figure 3:  Spatial distribution of static unimode
response near an open boundary (a - d) Stress-free continuum deformations of
the unimode metamaterial in the semi-infinite plane, with arbitrarily oriented
mechanism principal strain axes (large black arrows) will change character as
an exceptional point is crossed at Poisson’s ratio $\nu=0$. (a) For the
maximally auxetic system ($\nu=-1$), an oscillatory set of displacements
(black lines) along the open boundary (bottom edge) will oscillate paralell to
the boundary (purple arrow) while decaying into the bulk along the
perpendicular. (b) For a more generic auxetic system, bulk oscillation will be
determined by a special direction (purple arrow) set by the specific boundary
orientation and by $\nu$. (c, d) On the other side of the exceptional point,
the soft displacements of the anauxetic mechanism will oscillate into the bulk
without decay, and the boundary conditions on the transverse waves may change
with the addition of more modes when the longitudinal component is fixed.
(e-h) The same unimode metamaterial, subjected to stressed boundary conditions
will display identical spatial patterns in the stress distribution due to the
duality.

##  I Elasticity theory for generic planar mechanism metamaterials

Consider an elastic solid undergoing a deformation such that matter initially
located at material coordinates $\mathbf{R}=(x,y)$ is displaced by
$\mathbf{u}(\mathbf{R})$. Because the system is translationally invariant, the
energy depends on displacement gradients, rather than the bare displacements.
In addition, because it is rotationally invariant, the energy depends only on
the symmetrization of these gradients, the small strains
$\varepsilon_{ij}\equiv(\partial_{i}u_{j}+\partial_{j}u_{i})/2$ Landau  _et
al._ (1986); Chaikin and Lubensky (2000). The local energy then takes the
general and well-known form $C_{ijkl}\varepsilon_{ij}\varepsilon_{kl}/2$, in
terms of the three strain components
$\varepsilon_{11},\varepsilon_{12},\varepsilon_{22}$.

In contrast, consider an elastic structure containing a _mechanism_. In a
coarse description, such a system is defined to contain a particular strain
pathway that, as was the case with rotations, does not contribute to the
elastic energy density. As shown in Appendix A.2, it is still always possible
to construct orthonormal components of strain which separate this _mechanism
strain_ $\varepsilon_{m}$ from the non-mechanism strains
$\varepsilon_{1},\varepsilon_{2}$. Such variables inherently span all possible
energy costly strains and we may write the elastic energy in general form as

$\displaystyle
E=\frac{1}{2}\int\mathrm{d}^{2}\mathbf{R}\left(G_{11}\varepsilon_{1}^{2}+G_{22}\varepsilon_{2}^{2}+2G_{12}\varepsilon_{1}\varepsilon_{2}\right)\,.$
(1)

in terms of these variables. The stiffnesses $G_{ij}$ also define the
constitutive relationship between the strains and the corresponding stresses,
which is written compactly,

$\begin{bmatrix}\sigma_{m}\\\ \sigma_{1}\\\
\sigma_{2}\end{bmatrix}=\begin{bmatrix}0&0&0\\\ 0&G_{11}&G_{12}\\\
0&G_{12}&G_{22}\end{bmatrix}\begin{bmatrix}\varepsilon_{m}\\\
\varepsilon_{1}\\\ \varepsilon_{2}\end{bmatrix}\,,$ (2)

using the Voigt convention of treating stress and strain tensors as vectors of
orthonormal components. Energy conservation requires the symmetry of this
matrix, so that the presence of the mechanism eliminates three of the six
independent stiffnesses. Thus, the defining property that the mechanism strain
alone cannot generate stress also implies that mechanism stress with the same
tensorial form cannot be supported $\sigma_{m}=0$.

It is always possible to choose coordinate axes such that the mechanism strain
is proportional to a diagonal tensor:

$\displaystyle\mathbf{\hat{e}}_{m}\equiv A\begin{bmatrix}1&0\\\
0&-\nu\end{bmatrix},$ (3)

where the mechanism Poisson’s ratio $\nu$ is the negative of the ratio of the
two principle strains and $A\equiv\sqrt{2/(1+\nu^{2})}$ is a normalization
factor. Note that a quarter-turn of the coordinate system inverts the Poisson
ratio, and therefore the mechanism strain can switch from auxetic ($\nu<0$) to
anauxetic $\nu>0$ by passing through the through uniaxial strains where the
Poisson’s ratio either vanishes or diverges.

Arbitrary mechanisms can be generated via a system of rigid rotating
parallelograms joined at their corners by ideal frictionless hinges, as shown
in the generalization of the canonical “Rotating Squares” structure shown in
Fig. 1a,b. Here, the mechanism motion is traversed by rotating the rigid
parallelogram blocks in opposite fashion to their neighbors, around the ideal
frictionless hinges that connect them. This generates finely detailed
rearrangements within each unit cell, as well as an overall displacement of
each unit cell’s position (i.e. center of mass) which varies smoothly through
space. The mechanism elasticity theory above then applies to a coarse
description of the material deformations in terms of these smoothly varying
unit cell positions, in which the mechanism strain (Eq. 3) describes the
transformation of the lattice vectors (Fig. 1a,b pink arrows) which link these
unit cells to their neighbors. As shown in Fig. 1d, varying the parallelogram
angle $\psi$ and mechanism rotation $\theta$ spans the possible values of the
Poisson’s ratio $\nu$ and, paired with a rotation of the coordinate system,
any desired linear mechanism strain may be probed.

##  II Sheared analytic modes

For the special case of pure-dilational mechanisms, zero-energy deformations
are those that disallow shear and hence preserve angles in the material. As is
well-known, maps with this property are complex-analytic. That is, when points
in the real plane are mapped to the complex plane,
$(z,z^{*})\equiv(x+iy,x-iy)$ with the equivalent map for displacements
$(u,u^{*})\equiv(u_{x}+iu_{y},u_{x}-iu_{y})$ (see, e.g., England (2003)), the
zero-energy (i.e. stress-free) deformations are precisely those that satisfy
_complex analyticity_ , $\partial_{z^{*}}u=\partial_{z}u^{*}=0$, which has
yielded tremendous insight into dilational metamaterials Sun  _et al._ (2012);
Czajkowski  _et al._ (2022); Zheng  _et al._ (2021).

Thus motivated, we seek to extend this analyticity to generic mechanisms
outside of the pure-dilational limit. A single transformation applied
identically to both the material coordinates and the displacements cannot
achieve this. However, this can be achieved via a pair of related but distinct
transformations on the material coordinates and the displacements:

$\displaystyle\begin{bmatrix}w\\\ \bar{w}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}1&\frac{1}{\gamma}\\\
1&-\frac{1}{\gamma}\end{bmatrix}\begin{bmatrix}x\\\ y\end{bmatrix},$ (4)
$\displaystyle\begin{bmatrix}u\\\ \bar{u}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}1&-\gamma\\\
1&\gamma\end{bmatrix}\begin{bmatrix}u_{x}\\\ u_{y}\end{bmatrix},$ (5)
$\displaystyle\gamma$ $\displaystyle\equiv\frac{1}{\sqrt{\nu}}.$ (6)

Note that $\gamma$ is real for anauxetic mechanisms and imaginary for auxetic
ones. For dilational mechanisms, $\gamma=-i$ recovers the known conformal case
in which displacements and material coordinates transform identically. These
transformations also determine the gradients in the transformed coordinates,
since we require $\partial_{w}\bar{w}=\partial_{\bar{w}}w=0$ and
$\partial_{w}w=\partial_{\bar{w}}\bar{w}=1$.

The utility of this transformation is seen in the identification of zero-
energy deformations. As shown in Appendix B.1, it transforms the requirement
that the two non-mechanism strains vanish, into the requirement that two of
the derivatives vanish:

$\displaystyle\partial_{\bar{w}}u=0$ $\displaystyle\quad\rightarrow\quad
u=f_{1}(w)$ (7) $\displaystyle\partial_{w}\bar{u}=0$
$\displaystyle\quad\rightarrow\quad\bar{u}=f_{2}(\bar{w})\,.$ (8)

Hence, for a continuum stress-free deformation, the transformed fields are
each analytic functions of just one of the transformed coordinates.

In simply-connected domains, these functions may then be generated by simple
series expansions, e.g. $f_{1}(w)=\sum_{n=0}^{\infty}C_{n}w^{n}$. The
requirement that $u_{x},u_{y}$ be real-valued (simply taking the real value of
arbitrary complex functions would violate analyticity) enforces nontrivial
restrictions on the functions $f_{1},f_{2}$. On the auxetic side, where
coordinates $w,\bar{w}$ are complex-valued, we require that
$f_{1}(w),f_{2}(\bar{w})$ be complex conjugates of one another. For the
anauxetic side, $w,\bar{w}$ are real-valued and we simply require that
$f_{1},f_{2}$ be real-valued functions. As this recipe for generating energy-
free continuum unimode deformations relies on sheared analytic functions of a
sheared coordinate system, we refer to them as “sheared analytic modes”.
Again, the exact mathematics of the conformal soft maps from Czajkowski  _et
al._ (2022) is easily recovered in the limit $\nu\rightarrow-1$.

In addition to such stress-free displacements, there are patterns of stress
that satisfy the bulk equilibrium condition $\partial_{i}\sigma_{ij}=0$ Landau
_et al._ (1986). Upon making a similar transformation similar to Eq. 4

$\displaystyle\begin{bmatrix}\sigma\\\ \bar{\sigma}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}A&-\gamma\\\
A&\gamma\end{bmatrix}\begin{bmatrix}\sigma_{1}\\\ \sigma_{2}\end{bmatrix}\,,$
(9)

these conditions for force balance may be rewritten simply as
$\partial_{\bar{w}}\sigma=\partial_{w}\bar{\sigma}=0$. It is clear that this
is again the same set of equations that were governing the stress-free
displacement patterns. Beyond the immediate analytic insight this generates,
Eq. 9 indicates that the responses to “force-free” (i.e. stress-free) versus
“force-bearing” loading are mathematically dual spaces; solutions to one
problem can readily be transformed into a solution to a dual problem in the
other. To confirm this suggestive result, we examine numerically force
balanced states of our unimode material in both force-free (Fig. 2a-d) and
force-bearing (Fig. 2e-h) load situations as the unit cell structure becomes
finer. In both cases, following the analyses described in Appendix. C, our
analytic framework captures the vast majority ($>95\%$) of observed
deformation and stress, and the quality ubiquitously improves as the system
size is increased. In other words, sheared analytic modes take hold and
control response as the continuum limit of these materials is approached. Note
that “stress-free” here refers to the deformations themselves while “force-
free” refers to the loading conditions that generate them, and similarly for
the “stress-free” patterns and loading conditions.

While the duality is most clearly illustrated in the stress variables of Eq.
9, the $w,\bar{w}$ formalism also allows the nontrivial integrations involved
in determining force-balanced displacement fields from stresses to be
surpassed. In Appendix B.2 the general form for a force-balanced displacement
field is derived, and is again composed purely from sheared analytic
functions, which showcases the power of this formalism.

##  III Spatial character of generic unimode response

Figure 4:  Auxetic Unimode for Filtering of Mechanical Signals (a) For an
auxetic mechanism, a long strip of material will amplify displacement loads
with polarization to the right. Polarization of input signal is the direction
of travel in which the displacements would appear to rotate counterclockwise.
(b) Shows the opposite exponential suppression of the opposite polarization
input.

As a mechanism is tuned from the auxetic to the anauxetic, the sheared
coordinate systems $w$ and $\bar{w}$ briefly converge, becoming equal at
$\nu=0$ and then real for $\nu>0$. This defines an exceptional point
separating auxetic and anauxetic metamaterial mechanisms. Rather than
controlling the more common energetic spectra Miri and Alù (2019) or phase
transitions Fruchart  _et al._ (2021) for nonconservative systems, this is an
exceptional point in a spatial coordinate transform, and it therefore
distinguishes between spatial patterning types. On the auxetic side, the
components of the displacement for a sheared analytic mode obey elliptic
partial differential equations (see e.g. Stone and Goldbart (2009)) and are
harmonic conjugate functions of the sheared coordinates. On the anauxetic
side, these components remain conjugate, but as real-valued functions outside
of the complex analytic setting; these components obey partial differential
equations of hyperbolic character.

To illustrate these response patterns, we consider the infinite half-plane, of
arbitrary orientation, and with the component of displacement along the
boundary fixed to an oscillatory function. As shown in Fig. 3a,b, the auxetic
response decays into the bulk while simultaneously oscillating in a direction
determined by both $\nu$ and orientation of the mechanism principal axes. As
$\nu\rightarrow 0$ approaches the exceptional point from the negative side,
the lengthscale of spatial decay diverges, eventually leading to persistent
bulk oscillatory response in the anauxetic case. We note that, in this case
the component of displacement along the edge is not sufficient to completely
constrain the motion: the perpendicular displacement component at the boundary
is free, and there is an infinite space of bulk modes which do not alter the
boundary constraint which may be superimposed here, as described in greater
detail in Appendix B.3.

This behavior may be exploited in a long strip geometry. Here, an arbitrary
displacement input on one boundary may be decomposed into two polarizations,
which rotate in opposite directions along the boundary. As derived in Appendix
B.4 and shown in Fig. 4, one polarization decays exponentially into the bulk,
while the other polarization (which was ruled out of the soft modes in the
half-plane) will be amplified. As such, the unimode material acts not only as
a mechanical amplifier but also as a filter that polarizes an initially
generic static response. Note that, because of the duality, similar
amplification and filtering will persist for stressed boundary conditions.
Furthermore, as the structure traverses the mechanism motion, undergoing
uniform large deformations, the Poisson’s ratio itself changes sign as shown
in Fig. 1d. This filtering property may therefore be switched on and off, with
uniform applied strains acting as a “gating voltage” for mechanical signal
processing in analogy with transistors. As suggested by the phrasing, these
properties may prove useful in mechanical circuitry and logic.

##  IV Discussion

We have shown that the presence of a single mechanism (unimode) in two-
dimensional elasticity, as can be achieved in mechanical metamaterials,
ubiquitously confines static response to dual spaces of stress-bearing and
stress-free deformation. This spanning space of available response patterns is
determined solely by the orientation and Poisson’s ratio of the principal
mechanism strain, thereby unifying response across microctructures and loading
conditions. The tunable Poisson’s ratio $\nu$, and particularly the
exceptional point at $\nu=0$, open the door to switchable elastic behavior
which may become useful in metamaterial devices to amplify and filter signals
in mechanical computing and circuitry.

These properties seem to hinge on a type of mechanical criticality present in
two-dimensional unimode materials. Mechanism strain and rotation constitute
two fields whose spatial variation must satisfy the compatibility condition
that the induced two-dimensional displacement vanish around a loop (Appendix A
and Czajkowski and Rocklin (2021); Czajkowski  _et al._ (2022); Zheng  _et
al._ (2021)). The two stress fields in bulk (away from the loading) likewise
are constrained by the requirement that the two components of external force
vanish. It is an open question, then, how such general principles extend to
three-dimensional flexible mechanical metamaterials, either bulk ones or
curved two-dimensional surfaces Bertoldi  _et al._ (2017); Schenk and Guest
(2013); Aharoni  _et al._ (2014); Nassar  _et al._ (2017, 2022); Schenk and
Guest (2013); Overvelde  _et al._ (2017); Griniasty  _et al._ (2019).

The duality between the spatial patterns of stresses and strains joins an
impressive contingent of dualities in mechanics. Energy conservation and
reciprocity imply a general conjugacy between uniform stresses and strains,
Maxwell-Cremona dualities Baker  _et al._ (2013); Behringer and Chakraborty
(2018) exist between force balance and position compatibility, and certain
dilational metamaterials have recently been shown to possess self-dual phonon
dispersions Fruchart  _et al._ (2020), while general elasticity is itself dual
to a tensor gauge theory Pretko and Radzihovsky (2018).

#### Acknowledgements

We acknowledge helpful conversations with Corentin Coulais and Martin van
Hecke.

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##  Appendix A  Elasticity theory for generic planar mechanism metamaterials

Here we expand on the argument in the main text, constructing the most general
possible continuum theory governing elastic response in planar mechanism-based
(i.e. unimode) metamaterials. In this section we develop a formalism to
describe the coarse deformations of mechanism-based metamaterials. While an
example class of mechanisms is given and analyzed as an example, the analytic
approach will be generic and independent of the particular microscructure
unless otherwise indicated. Further, we will focus on lattice mechanisms,
meaning that the microstructure repeats periodically upon translation by
either of a pair of lattice vectors $\mathbf{l}^{(1)},\mathbf{l}^{(2)}$.
Working in the continuum-limit, where the microstructure of the unit cell
becomes infinitesimally fine, means that strain gradient terms will be
neglected in favor of the elastic energy terms coupling strains only.

###  A.1 Mechanism motion in the continuum

Consider the mechanism depicted in main text Fig. 1a. The mechanism motion is
traversed by rotating each light grey (rigid) quadrilateral by an angle
$\theta$, and simultaneously rotating the neighboring dark grey quadrilaterals
by the opposite angle $-\theta$ about the hinges at the corners where the
blocks are attached. This motion, analogous to that of the canonical Rotating
Squares lattice, will cost zero energy when the hinges are frictionless (i.e.
perfectly flexible). More importantly, this motion generates a change in the
lattice vectors. Undergoing a mechanism counter-rotation
$\theta_{0}\rightarrow\theta$ changes these vectors from initial values
$\mathbf{l}_{0}^{(1)},\mathbf{l}_{0}^{(2)}$ to final values
$\mathbf{l}^{(1)},\mathbf{l}^{(2)}$. For a large finite system composed of
$N_{1},N_{2}$ unit cells in the respective lattice directions, the same
uniform application of mechanism motion will change the macroscopic system
shape according to
$\\{N_{1}\mathbf{l}_{0}^{(1)},N_{2}\mathbf{l}_{0}^{(2)}\\}\rightarrow\\{N_{1}\mathbf{l}^{(1)},N_{2}\mathbf{l}^{(2)}\\}$.
The change in the lattice vectors, as well as the change in the overall system
shape are captured by a linear (affine) transformation
$\mathbf{l}^{(i)}=\mathbf{\Lambda}(\theta)\cdot\mathbf{l}^{(i)}_{0}$, where
$i$ here indexes which lattice vector is being mapped, rather than the
component. Note that this transformation tensor must depend implicitly on the
choice of initial counter-rotation $\theta_{0}$ so that
$\mathbf{\Lambda}(\theta_{0})=\mathbb{1}$.

Any lattice mechanism may be parameterized in the manner of the previous
paragraph and, for generality, we refer to a generic mechanism via a scalar
$m$ in place of $\theta$. In general, such a material may be described using
coarse coordinates so that a unit cell initially located at $\mathbf{R}$ will
be located at $\mathbf{r}(\mathbf{R})$ after the mechanism motion. This is the
approach of finite strain theory in the continuum. Here, $\mathbf{R}$ captures
a continuum of initial material coordinates, and is insensitive to the fine
details of the lattice microstructure. The final continuum positions after
deformation $\mathbf{r}(\mathbf{R})$, as well as the displacements
$\mathbf{r}(\mathbf{R})-\mathbf{R}$ are then smooth continuous vector fields
defined over this reference position space $\mathbf{R}$. The affine
transformation $\mathbf{\Lambda}(m)$, in this continuum context, is a uniform
instance of a quantity known as the deformation gradient tensor. This tensor
controls the transformation of the “material infinitesimals” which arbitrarily
close points of the continuum material via
$\mathrm{d}\mathbf{r}=\mathbf{\Lambda}\cdot\mathrm{d}\mathbf{R}$. In this case
of slowly varying continuum deformations, the lattice vectors take on the role
of the material infinitesimals.

It is well-known in finite strain theory that the deformation gradient tensor
may always be decomposed into
$\mathbf{\Lambda}=\mathbf{\mathcal{R}}\cdot\mathbf{U}$, where
$\mathbf{\mathcal{R}}$ is a rotation and $\mathbf{U}$ is a symmetric
transformation known as the right stretch tensor. As we are considering a
specific mechanism motion, rather than some generic deformation,
$\mathbf{\Lambda}(m)=\mathbf{\mathcal{R}}(\phi(m))\cdot\mathbf{A}(m)$ must be
parametrized by the mechanism. Here, we have distinguished the right stretch
tensor of the mechanism $\mathbf{A}(m)$ via a particular tensor function, and
$\phi(m)$ is the coarse rotation generated by the mechanism motion. As the
mechanism motion is uniform, we may choose to rotate our coordinate frame
along with the mechanism motion to eliminate $\phi(m)$ (i.e. we can always mix
the mechanism with an energy-free rotation). It is clear, then, that tensor
function $\mathbf{A}$ controls the strain of the metamaterial induced by the
mechanism transformation. By a further choice of the orientation of the
material reference (initial) coordinate frame, we may diagonalize the right
stretch tensor induced by the mechanism
$\mathbf{A}(m)\rightarrow((\lambda_{1}(m),0),(0,\lambda_{2}(m)))$. These
diagonal components are known as the principal stretches. While $\mathbf{A}$
will not stay diagonal for all mechanism final states $m$, the parallelogram
mechanisms explored in the main text have orthogonal lattice vectors which
stay orthogonal all along the mechanism motion, and therefore the right
stretch tensor also stays diagonal. The lattice vectors in this case may be
written in compact form as $\mathbf{l}^{(1)}=2\cos(\psi+\theta)\hat{x}$ and
$\mathbf{l}^{(2)}=2\cos(\theta)\hat{y}$. Then the principal stretches, given
the starting mechanism point of $\theta_{0}$, are
$\lambda_{1}(\theta)=\cos(\psi+\theta)/\cos(\psi+\theta_{0})$ and
$\lambda_{2}(\theta)=\cos(\theta)/\cos(\theta_{0})$.

Note also that the mechanism right stretch is not an arbitrary tensor
function, but must obey $\mathbf{A}(m_{0})=\mathbf{\mathbb{1}}$. Further, from
knowledge of the right stretch induced by the mechanism, we may construct the
metric of deformation (also known as the right Cauchy-Green deformation
tensor)
$\mathbf{g}(m)=\mathbf{\Lambda}^{T}\cdot\mathbf{\Lambda}\equiv\mathbf{A}(m)\cdot\mathbf{A}(m)$
and the Lagrange strain
$\mathbf{\varepsilon}^{(\text{Lagrange})}(m)\equiv\frac{1}{2}(\mathbf{g}(m)-\mathbf{\mathbb{1}})$
each generated by the mechanism motion.

The linear incrementation of the mechanism parameter by a small amount $\delta
m=m-m_{0}$ from an arbitrary starting state $m_{0}$ will then induce a
symmetric linear strain
$\varepsilon_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2$ of the coarse
system. Again choosing to consider an oriented initial coordinate system which
diagonalizes this generic strain, this may written in terms of the principal
stretch functions as

$\displaystyle\begin{split}\mathbf{\varepsilon}(\delta m)=&\delta
m\begin{bmatrix}\lambda_{1}^{\prime}(m_{0})&0\\\
0&\lambda_{2}^{\prime}(m_{0})\end{bmatrix}\\\ &=\delta
m\lambda_{1}^{\prime}(m_{0})\begin{bmatrix}1&0\\\
0&-\nu\end{bmatrix}\,,\end{split}$ (10)

where the primes denote derivatives with respect to the mechanism argument.
Here, we have implicitly defined the mechanism-induced Poisson’s ratio from
the main text
$\nu\equiv-\lambda_{2}^{\prime}(m_{0})/\lambda_{1}^{\prime}(m_{0})$. This
corresponds to the conventional Poisson’s ratio, which is defined from the
ratio of the strain components $\varepsilon_{11}$ and $\varepsilon_{22}$,
simplified using the knowledge that $\lambda_{1}(m_{0})=\lambda_{2}(m_{0})=1$.
Again, for the example of the parallelograms this simplifies to
$\nu(\theta_{0})=-\tan(\theta_{0})/\tan(\psi+\theta_{0})$.

As an illustrative exercise, and to connect with the previous literature, we
consider the possible spatial patterns of pure mechanism strain that are
permitted in such metamaterials. While this setion concerns uniform
applications of the mechanism thus far, the generalization to slowly varying
strain patterns is natural, by promoting the continuum quantities to fields
over the reference coordinates just like $\mathbf{r}(\mathbf{R})$. In
particular, this is accomplished by establishing the mechanism field
$m(\mathbf{R})$ and coarse rotation field $\phi(\mathbf{R})$ and demanding
that the deformation tensor everywhere resemble
$\mathbf{\Lambda}(\mathbf{R})=\mathbf{\mathcal{R}}(\phi(\mathbf{R}))\cdot\mathbf{A}(m(\mathbf{R}))$.
Of course, not any arbitrary patterns of $m(\mathbf{R}),\phi(\mathbf{R})$ are
possible: the spatial variation will generally create geometric
inconsistencies. This problem is addressed by the well-known compatibility
relations derived by demanding that any closed path integral of material
infinitesimals be identically zero $\oint\mathrm{d}\mathbf{r}=0$ (i.e. this
integral must vanish for every path in $\mathrm{d}\mathbf{r}$ which
corresponds to a closed loop in $\mathrm{d}\mathbf{R}$) . This leads to the
both necessary and sufficient condition
$\epsilon_{jk}\partial_{k}\Lambda_{ij}=0$ which must hold for $i=\\{1,2\\}$.
Here, $\epsilon$ is the antisymmetric tensor with $\epsilon_{12}=1$ Using the
mechanism form of the deformation tensor, we arrive at a nonlinear first-order
vector PDE

$\mathbf{\nabla}\phi=\mathbf{B}(m)\cdot\mathbf{\nabla}m\,,$ (11)

with

$\mathbf{B}(m)=\mathbf{\epsilon}\cdot\mathbf{A}^{-1}(m)\cdot\mathbf{\epsilon}\cdot\mathbf{A}^{\prime}(m)\cdot\mathbf{\epsilon}\,,$
(12)

governing the soft mechanism strain patterns. As the left-hand side of this
equation is independent of $m$, we may integrate along a path from
$\mathbf{R}_{\text{init}}$ to $\mathbf{R}_{\text{final}}$ and find a recipe to
determine the rotation field from the mechanism field

$\phi(\mathbf{R}_{\text{final}})-\phi(\mathbf{R}_{\text{init}})=\int^{\mathbf{R}_{\text{final}}}_{\mathbf{R}_{\text{init}}}\mathrm{d}\mathbf{R}\cdot\mathbf{B}(m)\cdot\mathbf{\nabla}m\,.$
(13)

For $\phi$ to be a well-defined field, we will require that the integral in
Eq. 13 be path-independent. Similar to the derivation of the compatibility
relation, this means that any integral in a loop must yield zero, and using
Stokes’ law we arrive at a necessary compatibility condition on the $m$ field
alone:

$0=\epsilon_{ij}\partial_{i}\left[B_{jk}(m)\partial_{k}m\right]\,.$ (14)

Further, as any field $m$ satisfying Eq. 14 may be integrated according to Eq.
13 to define a compatible $\phi$ field (up to an overall constant), this
condition is also sufficient and either Eq. 11 or Eq. 14 may be used to
determine the compatible mechanism strain patterns.

###  A.2 Generalized Voigt notation to describe mechanism elasticity

Consider the standard procedure of constructing an energy functional for an
elastic material. The deformation is captured by the smooth field of
displacements $\mathbf{u}(\mathbf{R})$, as defined in the previous section.
Working in homogeneous space, without external fields, uniform displacements
cannot incur any energy penalty, and we must turn to displacement gradients.
In principle, the entire unsymmetrized strain
$S_{ij}(\mathbf{R})=\partial_{j}u_{i}|_{\mathbf{R}}$ is a candidate tensor
field to control the energy density. However, in isotropic space there is
another transformation in addition to the energy-free translations: the
rotations. These rotations are mixed in with the energy costly strains in
$\mathbf{S}$. To account for this, the standard procedure is to notice that
the rotation exclusively controls the anti-symmetric part of this tensor,
while the symmetric part $\varepsilon$ is independent of rotations. The energy
functional is then built from this rotation-free part of the strain and may be
generally written

$E=\frac{1}{2}\int\mathrm{d}^{2}\mathbf{R}\,\sigma_{ij}\varepsilon_{ij}\,,$
(15)

where $\varepsilon_{ij}$ and $\sigma_{ij}$ are the familiar symmetric tensors
capturing strain and stress, respectively.

The mechanism strain identified in the previous section, similar to the
rotation, cannot incur any energy, and must not appear in the elastic energy
density. We are therefore motivated to identify the portions of strain which
are orthogonal to this mechanism strain. To do this, we first must define a
notion of orthogonality, by defining an inner product for symmetric tensors in
two dimensions. Taking inspiration from the more general elastic energy Eq.
15, an appropriate inner product may be defined:

$\langle\mathbf{T}_{1}|\mathbf{T}_{2}\rangle\equiv\frac{1}{2}\mathrm{Tr}[(\mathbf{T}_{1})^{\mathrm{T}}\cdot\mathbf{T}_{2}]\,.$
(16)

Here, the factor of $\frac{1}{2}$ reflects the number of dimensions we work
in, so that the identity has unit norm.

Given this inner product, there exists an infinite variety of orthonormal
bases in which to break down stress and strain. Choosing, as in the previous
section, to work in the coordinate system in which the linear mechanism strain
is diagonal, our goal is to find a set of orthonormal unit tensors
$\\{\hat{\mathbf{\hat{e}}}\\}$, one of which is proportional to the mechanism
strain. It is straightforward to check that this is satisfied by

$\displaystyle\mathbf{\hat{e}}_{m}$
$\displaystyle=\sqrt{\frac{2}{1+\nu^{2}}}\begin{bmatrix}1&0\\\
0&-\nu\end{bmatrix}$ (17) $\displaystyle\mathbf{\hat{e}}_{1}$
$\displaystyle=\sqrt{\frac{2}{1+\nu^{2}}}\begin{bmatrix}\nu&0\\\
0&1\end{bmatrix}$ (18) $\displaystyle\mathbf{\hat{e}}_{2}$
$\displaystyle=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\,.$ (19)

As this constitutes a complete basis for symmetric tensors, we may break down
the components of the strain
$\varepsilon_{ij}=\left[\varepsilon_{m}\mathbf{\hat{e}}_{m}+\varepsilon_{1}\mathbf{\hat{e}}_{1}+\varepsilon_{2}\mathbf{\hat{e}}_{2}\right]_{ij}$
and the stress
$\sigma_{ij}=\left[\sigma_{m}\mathbf{\hat{e}}_{m}+\sigma_{1}\mathbf{\hat{e}}_{1}+\sigma_{2}\mathbf{\hat{e}}_{2}\right]_{ij}$
into the basis components. For uniform systems, $\sigma_{ij}$ is the
derivative of the elastic energy with respect to $\varepsilon_{ij}$, which
generalizes to functional derivatives for non-uniform deformations. This
mechanism basis enables a notation which resembles that of Voigt, which is
helpful for writing the constitutive relation:

$\begin{bmatrix}\sigma_{m}\\\ \sigma_{1}\\\
\sigma_{2}\end{bmatrix}=\begin{bmatrix}G_{mm}&G_{m1}&G_{m2}\\\
G_{1m}&G_{11}&G_{12}\\\
G_{2m}&G_{21}&G_{22}\end{bmatrix}\begin{bmatrix}\varepsilon_{m}\\\
\varepsilon_{1}\\\ \varepsilon_{2}\end{bmatrix}$ (20)

In this notation, similar to that of Voigt, we must have a symmetric stiffness
tensor $G$ in order to connect to an equilibrium energy. Hence $G_{1m}=G_{m1}$
and $G_{12}=G_{21}$ and so on. While the notation here is chosen based on a
mechanism, this relation is still completely general for a continuum elastic
material in isotropic two-dimensional space. To account for the mechanism, we
simply impose that the mechanism strain cannot generate any force nor energy
nor stress. We then quickly may find that $G_{mm}=G_{m1}=G_{m2}=0$ and, with
the tensor symmetry we have eliminated the row and column associated with the
mechanism, leading to the main text constitutive relation. Further, this means
that the mechanism stress $\sigma_{m}$ will always be identically zero.
Finally, this notation allows us to write the mechanism elastic energy via

$\displaystyle E=$
$\displaystyle\int\mathrm{d}^{2}\mathbf{R}\langle\mathbf{\sigma}|\mathbf{\varepsilon}\rangle$
(21)
$\displaystyle=\int\mathrm{d}^{2}\mathbf{R}\left\\{G_{11}\varepsilon_{1}^{2}+G_{22}\varepsilon_{2}^{2}+2G_{12}\varepsilon_{1}\varepsilon_{2}\right\\}\,.$

Here, conventional strains $\epsilon_{ij}$ have a different normalization such
that terms in the energy commonly appear with factors of $\frac{1}{2}$.

For completeness, the recipes to obtain these new strain variables from
explicit spatial derivatives are

$\displaystyle\varepsilon_{m}$
$\displaystyle=\frac{A}{2}(\varepsilon_{11}-\nu\varepsilon_{22})=\frac{A}{2}(\partial_{x}u_{x}-\nu\partial_{y}u_{y})$
(22) $\displaystyle\varepsilon_{1}$
$\displaystyle=\frac{A}{2}(\nu\varepsilon_{11}+\varepsilon_{22})=\frac{A}{2}(\nu\partial_{x}u_{x}+\partial_{y}u_{y})$
(23) $\displaystyle\varepsilon_{2}$
$\displaystyle=\varepsilon_{12}=\frac{1}{2}(\partial_{x}u_{y}+\partial_{y}u_{x})\,,$
(24)

where $A\equiv\sqrt{2/(1+\nu^{2})}$ is the normalization factor from Eqs. 17 &
18. Note that a similar recipe may be used to obtain the mechanism vs.
nonmechanism components of stress from $\sigma_{ij}$. Finally, this tensor
basis may still be useful in coordinate systems where the incremental
mechanism motion is not a diagonal strain, simply by rotating the basis
according to standard tensor transformation laws.

##  Appendix B Static response of continuum unimode structures

Given the elastic theory of Eq. 21, we ask here what possible nonuniform
response patterns are supported by this energy functional without generating a
bulk force density. This is explored in the continuum limit, in the absence of
strain gradient terms.

###  B.1 Identifying useful coordinate transformations

Consider, for example, the pure dilational limit of a mechanism. We know from
previous work that the standard transformation to complex coordinates
$(z,z^{*})=(x+iy,x-iy)$ along with $(u,u^{*})=(u_{x}+iu_{y},u_{x}-iu_{y})$
greatly simplifies the analysis. In this case, the nonmechanism strains
$\varepsilon_{1},\varepsilon_{2}$ (which are, in this case, the pure and the
simple shears, respectively) become independent of the complex derivative
fields $\partial_{z}u$ and $\partial_{z^{*}}u^{*}$. These are written

$\displaystyle\varepsilon_{1}^{(\nu\rightarrow-1)}$
$\displaystyle=\frac{1}{2}(\partial_{z}u^{*}+\partial_{z^{*}}u)$ (25)
$\displaystyle\varepsilon_{2}^{(\nu\rightarrow-1)}$
$\displaystyle=\frac{i}{2}(\partial_{z}u^{*}-\partial_{z^{*}}u)$ (26)

and we notice that choosing $u(z,z^{*})\rightarrow u(z)$ captures the well
known space of deformations which generate zero shear, known as the conformal
maps. This is very convenient, as $u(z)$ may be expanded as a complex analytic
function in powers of $z$, allowing this entire space of soft deformations to
be generated by choosing a single list of coefficients.

We therefore ask whether there might exist a coordinate transformation,
similar to the complex plane transformation for pure dilational mechanisms,
which simplifies the analysis of a more generic mechanism. Consider the
generic candidate transformations

$\begin{bmatrix}\bar{u}\\\ u\end{bmatrix}=\begin{bmatrix}L_{11}&L_{12}\\\
L_{21}&L_{22}\end{bmatrix}\begin{bmatrix}u_{x}\\\ u_{y}\end{bmatrix}$ (27)

and

$\begin{bmatrix}\partial_{w}\\\
\partial_{\bar{w}}\end{bmatrix}=\begin{bmatrix}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{bmatrix}\begin{bmatrix}\partial_{x}\\\
\partial_{y}\end{bmatrix}$ (28)

where it will turn out to be more convenient to work with the transformation
of the derivatives than the coordinates $w,\bar{w}$ directly.

To identify a useful transformation, we will attempt to identify which of the
possible transformations will achieve the following goals:

  1. 1.

Wherever the condition $\partial_{\bar{w}}u=\partial_{w}\bar{u}=0$ is
satisfied, we will find the non-mechanism strains are zero
$\varepsilon_{1}=\varepsilon_{2}=0$.

  2. 2.

Wherever the nonmechanism strains are zero
$\varepsilon_{1}=\varepsilon_{2}=0$, we will find the partial derivatives
$\partial_{\bar{w}}u=\partial_{w}\bar{u}=0$ also zero.

  3. 3.

$u,\bar{u}$ should be produced from linearly independent transformations.

  4. 4.

$w,\bar{w}$ should be produced from linearly independent transformations.

Searching for the conditions which allow the first two conditions to be
simultaneously met, we arrive at a linear algebra problem

$\begin{bmatrix}0\\\ 0\\\ 0\\\ 0\end{bmatrix}=\begin{bmatrix}M_{11}&-\nu
M_{12}&0&0\\\ M_{12}&-M_{11}&0&0\\\ 0&0&M_{21}&-\nu M_{22}\\\
0&0&M_{22}&-M_{21}\end{bmatrix}\begin{bmatrix}L_{11}\\\ L_{12}\\\ L_{21}\\\
L_{22}\end{bmatrix}$ (29)

which we have oriented as a problem to solve for the $L$ values given a
predetermined set of $M$ values. This has separated into two problems, and in
order to have non-trivial values of $L_{11}$ and $L_{12}$ we must have $\nu
M_{12}^{2}-M_{11}^{2}=0$. Similarly, in order to not have nonzero values of
$L_{21}$ and $L_{22}$ we require $\nu M_{22}^{2}-M_{21}^{2}=0$. Together, this
leads to the collected conditions on our transformations

$\displaystyle M_{12}$ $\displaystyle=s_{1}\gamma M_{11}$ (30) $\displaystyle
M_{22}$ $\displaystyle=-s_{1}\gamma M_{21}$ (31) $\displaystyle L_{12}$
$\displaystyle=s_{1}\gamma L_{11}$ (32) $\displaystyle L_{22}$
$\displaystyle=-s_{1}\gamma L_{21}\,,$ (33)

where the scalar $s_{1}$ may be chosen to be $+1$ or $-1$. Any transformations
satisfying these conditions will lend useful simplifications for a generic
mechanism as explored in the next sections. This allows free nonzero choice of
$M_{11},M_{21},L_{11},L_{21}$ from which we may determine the rest of the
constants. To connect with the conformal example, we make the simple choices
$M_{11}=M_{21}=\frac{1}{2}$, $L_{11}=L_{21}=1$ and $s_{1}=1$, leading to the
transformations

$\displaystyle\partial_{w}$
$\displaystyle=\frac{1}{2}(\partial_{x}+\gamma\partial_{y})$ (34)
$\displaystyle\partial_{\bar{w}}$
$\displaystyle=\frac{1}{2}(\partial_{x}-\gamma\partial_{y})$ (35)
$\displaystyle\bar{u}$ $\displaystyle=(u_{x}+\gamma u_{y})$ (36)
$\displaystyle u$ $\displaystyle=(u_{x}-\gamma u_{y})\,.$ (37)

This also implies the corresponding direct transformation of the coordinate
system

$\displaystyle w$ $\displaystyle=x+\frac{y}{\gamma}$ (38)
$\displaystyle\bar{w}$ $\displaystyle=x-\frac{y}{\gamma}\,,$ (39)

so that $\partial_{\bar{w}}w=\partial_{w}\bar{w}=0$ and
$\partial_{w}w=\partial_{\bar{w}}\bar{w}=1$. Writing our strain components in
terms of these variables

$\displaystyle\varepsilon_{m}$
$\displaystyle=\frac{A}{4}(1-\nu^{2})(\partial_{w}\bar{u}+\partial_{\bar{w}}u)+\frac{A}{4}(1+\nu^{2})(\partial_{\bar{w}}\bar{u}+\partial_{w}u)$
(40) $\displaystyle\varepsilon_{1}$
$\displaystyle=\frac{A\nu}{2}(\partial_{w}\bar{u}+\partial_{\bar{w}}u)$ (41)
$\displaystyle\varepsilon_{2}$
$\displaystyle=\frac{1}{2\gamma}(\partial_{w}\bar{u}-\partial_{\bar{w}}u)$
(42) $\displaystyle\phi$
$\displaystyle=\frac{1}{2\gamma}(\partial_{\bar{w}}\bar{u}+\partial_{w}u)\,,$
(43)

we may see immediately that the space of deformations composed purely of
mechanism strains $\varepsilon_{m}$ and rotations $\phi$ (i.e.
$\varepsilon_{1}=\varepsilon_{2}=0$) is identified with the conditions
$\partial_{w}\bar{u}=\partial_{\bar{w}}u=0$. Therefore, these are written most
generally as analytic functions $u=f_{1}(w)$ and $\bar{u}=f_{2}(\bar{w})$,
which span the space of continuum stress-free deformations that the unimode
material can support, as described in the main text.

It is important to note that any coordinate transformation which satisfies the
conditions Eqs. 30-33 will become linearly dependent at the exceptional points
$\nu=0$ and $\nu=\pm\infty$. At these points, the mechanism strain becomes
uniaxial, and a one-dimensional coordinate system becomes the natural
approach.

###  B.2 Generating analytic force-balanced modes

We now investigate the possible patterns of response that arise under static
loading. We aim beyond the stress-free analytic modes identified with the
coordinate transformations of Sec B.1 to include those which may bear stress
without generating bulk force. Continuing to employ these coordinate
transforms, we may write the energy as

$\displaystyle E$
$\displaystyle=\frac{\gamma}{2}\int\mathrm{d}\bar{w}\mathrm{d}w\Big{[}\frac{G_{11}A^{2}\nu^{2}}{4}\left(\partial_{w}\bar{u}+\partial_{\bar{w}}u\right)^{2}$
$\displaystyle+\frac{G_{22}\nu}{4}\left(\partial_{w}\bar{u}-\partial_{\bar{w}}u\right)^{2}+\frac{G_{12}A\nu}{2\gamma}\left((\partial_{w}\bar{u})^{2}-(\partial_{\bar{w}}u)^{2}\right)\Big{]}.$
(44)

We are searching for the force-balanced solutions to this energy, and
therefore require that the functional derivatives with respect to the
displacement fields, to which forces are proportional, vanish. While this is
usually done with Cartesian components $u_{x},u_{y}$, our notation allows the
use of the $u,\bar{u}$ and $w,\bar{w}$. Taking the functional derivative of
this energy is then straightforward, and with a little algebra, the equations
of force-balance are written

$\displaystyle 0$
$\displaystyle=\partial_{w}\left(B_{1}\partial_{w}\bar{u}+B_{2}\partial_{\bar{w}}u\right)$
(45) $\displaystyle 0$
$\displaystyle=\partial_{\bar{w}}\left(\bar{B}_{1}\partial_{\bar{w}}u+B_{2}\partial_{w}\bar{u}\right)$
(46)

where

$\displaystyle B_{1}$
$\displaystyle\equiv\frac{G_{11}A^{2}\nu^{2}}{2}+\frac{G_{22}\nu}{2}+\frac{G_{12}A\nu}{\gamma}$
(47) $\displaystyle\bar{B}_{1}$
$\displaystyle\equiv\frac{G_{1}A^{2}\nu^{2}}{2}+\frac{G_{22}\nu}{2}-\frac{G_{12}A\nu}{\gamma}$
(48) $\displaystyle B_{2}$
$\displaystyle\equiv\frac{G_{11}A^{2}\nu^{2}}{2}-\frac{G_{22}\nu}{2}\,.$ (49)

Note that from Eqs. 40-43 and the constitutive relation from Sec. A.2, that
this may be written quite simply in terms of stresses

$\displaystyle 0$
$\displaystyle=\partial_{w}\left(A\sigma_{1}+\gamma\sigma_{2}\right)$ (50)
$\displaystyle 0$
$\displaystyle=\partial_{\bar{w}}\left(A\sigma_{1}-\gamma\sigma_{2}\right)$
(51)

and the natural definition of two stress scalars has become evident

$\displaystyle\sigma$ $\displaystyle\equiv
A\sigma_{1}-\gamma\sigma_{2}=g_{1}^{\prime}(w)$ (52)
$\displaystyle\bar{\sigma}$ $\displaystyle\equiv
A\sigma_{1}+\gamma\sigma_{2}=g_{2}^{\prime}(\bar{w})\,,$ (53)

where we have made clear that the patterns of stress which obey force-balance
will be again described by sheared analytic functions. Here, for convenience
in the analysis that follows, we have chosen to write these functions
$g_{1}^{\prime}(w),g_{2}^{\prime}(\bar{w})$ as derivatives of analytic
functions, which, due to the analyticity itself, does not sacrifice
generality.

From knowledge of the stress patterns, the typical procedure is to use the
constitutive relation to determine the strains, and then (non-trivially) to
integrate these strains to obtain the displacement patterns. However, in this
case the integration is facilitated by analyticity. Again using Eqs. 52&53 in
the constitutive relations, we may solve for the displacement derivatives

$\displaystyle\partial_{\bar{w}}u$
$\displaystyle=\frac{1}{B_{1}\bar{B}_{1}-B_{2}^{2}}\left(B_{1}g_{1}^{\prime}(w)-B_{2}g_{2}^{\prime}(\bar{w})\right)$
(54) $\displaystyle\partial_{w}\bar{u}$
$\displaystyle=\frac{1}{B_{1}\bar{B}_{1}-B_{2}^{2}}\left(\bar{B}_{1}g_{2}^{\prime}(\bar{w})-B_{2}g_{1}^{\prime}(w)\right)\,.$
(55)

These equations may then be integrated to obtain the general form of a force-
balanced displacement:

$\displaystyle u$
$\displaystyle=\frac{1}{B_{1}\bar{B}_{1}-B_{2}^{2}}\left(B_{1}\bar{w}g_{1}^{\prime}(w)-B_{2}g_{2}(\bar{w})\right)+f_{1}(w)$
(56) $\displaystyle\bar{u}$
$\displaystyle=\frac{1}{B_{1}\bar{B}_{1}-B_{2}^{2}}\left(\bar{B}_{1}wg_{2}^{\prime}(\bar{w})-B_{2}g_{1}(w)\right)+f_{2}(\bar{w})\,,$
(57)

which conveniently distinguishes between the contributions which generate
material stress $(g_{1},g_{2})$ and those which do not $(f_{1},f_{2})$. It
would seem at this point that the procedure is complete: given boundary
conditions, the response pattern observed in the bulk should be described by
the closed-form analytic solutions in Eqs. 56 & 57. However, to generate an
arbitrary member of the force-balanced modes, we must also acknowledge the
requirement that the displacement components $u_{x}=\frac{1}{2}(u+\bar{u})$
and $u_{y}=\frac{1}{2\gamma}(\bar{u}-u)$ be real-valued. Recalling the
definition of $\gamma=\frac{1}{\sqrt{\nu}}$, this requirement leads to
different conditions on the functions $u,\bar{u}$ depending on the sign of
$\nu$, which determines whether $w,\bar{w}$ are real or complex-valued. When
$\nu>0$ and $w,\bar{w}$ are everywhere real-valued, the sheared analytic
functions $f_{1},f_{2},g_{1},g_{2}$ must simply be real-valued functions (i.e.
the coefficients in the analytic expansion will be real) and may be chosen
independently, yielding geometrically valid force balanced states. However, in
the auxetic case $\nu<0$, $w$ and $\bar{w}$ are complex, and so are the
nonuniform sheared analytic functions of these variables. The realness
requirement of $u_{x}$ and $u_{y}$ then yields the nontrivial relations
$f_{1}(w)=f_{2}^{*}(\bar{w})$ and $g_{1}(w)=g_{2}^{*}(\bar{w})$. While the
functions are no longer independent, the dimension of the space of modes
remains constant (the number of real scalars required to determine the soft
mode is equal) because the modes are now defined via the real and imaginary
parts of a single complex function rather than by two real functions.

The force-balanced modes in Eqs. 56 & 57 appear to diverge when
$B_{1}\bar{B}_{1}-B_{2}^{2}=(G_{11}G_{22}-G_{12}^{2})A^{2}\nu^{3}\rightarrow
0$. This only occurs when either the reduced stiffness tensor has another zero
eigenvalue, and hence the system becomes multimodal, or when $\nu\rightarrow
0$. While the method presented here is not meant to handle the case of a
bimodal metamaterial, it is important to note that the uniaxial case is still
well-behaved (not actually divergent) when analyzed in terms of the real space
displacements at $\nu=0$.

Finally, we note that the duality of force-balanced stress and soft
deformation is enhanced when considering the soft deformations in terms of
strain rather than displacement. Similar to the nonlinear soft mode
identification procedure identified in Sec. A.1, this is a question of solving
the equation of mechanical compatibility of an unsymmetrized strain under the
assumption of vanishing nonmechanism strain
$\varepsilon_{1},\varepsilon_{2}=0$. Again, it is convenient to define
transformed variables for these strains

$\displaystyle\varepsilon$ $\displaystyle\equiv
A\varepsilon_{m}-\gamma\phi\rightarrow\partial_{w}u$ (58)
$\displaystyle\bar{\varepsilon}$ $\displaystyle\equiv
A\varepsilon_{m}+\gamma\phi\rightarrow\partial_{\bar{w}}\bar{u}\,.$ (59)

It is now evident that geometrically compatible soft strains will satisfy the
equations $\partial_{\bar{w}}\varepsilon=\partial_{w}\bar{\varepsilon}=0$. The
duality of these soft deformations with the force balanced stress-bearing
counterparts is even more direct in this notation.

###  B.3 Force-balanced modes in the half-plane

Here we describe the specific spatial patterning of force-balanced deformation
in the half-plane geometry. This is illustrative of the effect of an open
boundary in more generic settings. In this geometry, the metamaterial sample
extends infinitely far in the upper half-plane, away from an infinitely long
open surface at $y=0$. For generality, we consider that the principal axes of
the mechanism strain to be at some oblique angle $\phi$ to our open boundary,
and hence coordinate system. In this coordinate system the sheared analytic
soft modes will be functions of the variables

$\displaystyle w=$
$\displaystyle(\cos(\phi)x+\sin(\phi)y)+\frac{1}{\gamma}(-\sin(\phi)x+\cos(\phi)y)$
(60) $\displaystyle\bar{w}=$
$\displaystyle(\cos(\phi)x+\sin(\phi)y)-\frac{1}{\gamma}(-\sin(\phi)x+\cos(\phi)y)\,.$
(61)

The analytic functions constructed from these variables control the sheared
displacements $u=f_{1}(w)$ and $\bar{u}=f_{2}(\bar{w})$, which are functions

$\displaystyle u=$
$\displaystyle(\cos(\phi)u_{x}+\sin(\phi)u_{y})-\gamma(-\sin(\phi)u_{x}+\cos(\phi)u_{y})$
(62) $\displaystyle\bar{u}=$
$\displaystyle(\cos(\phi)u_{x}+\sin(\phi)u_{y})+\gamma(-\sin(\phi)u_{x}+\cos(\phi)u_{y})\,,$
(63)

of the displacement components $u_{x},u_{y}$ in this coordinate system.

In the half-plane, modes which exponentially grow or decay along the boundary
will eventually become unphysical, due to the infinite nature of the boundary
(such mechanisms will encounter nonlinear effects and eventually material
failure at large strains). Therefore we search for soft modes which display no
exponential growth along the boundary. Specifically, we will enforce the
boundary condition in which the component of displacement pointing along the
boundary follows an arbitrary oscillatory function
$u_{x}(x,y=0)=u_{x}^{0}\cos(q_{0}x+\xi_{0})$, while the transverse component
of the boundary displacement will be left free.

Matching a sheared analytic mode to these boundary conditions depends on the
geometry used. In the anauxetic case, we search for a solution of the form
$u=\mathrm{Re}[u_{0}\exp(iwq+i\xi)]$ and
$\bar{u}=\mathrm{Re}[\bar{u}_{0}\exp(i\bar{w}\bar{q}+i\bar{\xi})]$ (note that
taking the real part does not violate analyticity for the anauxetic modes). To
prevent unphysical exponential growth along the boundary we simply require
here that $q$ and $\bar{q}$ be real numbers. Matching to the boundary
condition, we find the following conditions:

$\displaystyle\bar{u}_{0}$
$\displaystyle=\frac{u_{x}^{0}}{(\frac{1}{\gamma}\sin(\phi)+\cos(\phi))}\,,$
(64) $\displaystyle u_{0}$
$\displaystyle=\frac{u_{x}^{0}}{(\cos(\phi)-\frac{1}{\gamma}\sin(\phi))}\,,$
(65) $\displaystyle q$
$\displaystyle=\frac{q_{0}}{(\frac{1}{\gamma}\sin(\phi)+\cos(\phi))}\,,$ (66)
$\displaystyle\bar{q}$
$\displaystyle=\frac{q_{0}}{(\cos(\phi)-\frac{1}{\gamma}\sin(\phi))}\,,$ (67)
$\displaystyle\xi$ $\displaystyle=\bar{\xi}=\xi_{0}\,.$ (68)

This generates a bulk oscillatory mode as indicated in the main text.

However, we further note that any additional solution may be added to this,
satisfying the above equations, after the trade
$\bar{u}_{0}\rightarrow-\bar{u}_{0}$, as this generates no impact on the
boundary conditions. Further, the phase, amplitude, and wavenumber of this
additional mode are free to be chosen to generate such a mode from the above
conditions. Therefore, the boundary conditions on $u_{x}$ are not sufficient
to fully constrain the mode. Here, oscillation still takes place along the
directions along $w$ and along $\bar{w}$, independent of the orientation of
the boundary, as shown in main text Fig. 3c,d.

In the auxetic case, we search for a solution of the form
$u=u_{0}\exp{[iwq]}$. To again prevent unphysical decay along the boundary, we
require $q=\mathrm{Re}[q](1+\frac{\tan(\phi)}{\gamma})$ which is now a complex
number. The realness condition then determines the form of
$\bar{u}(\bar{w})=u^{*}(w)$ from complex conjugation; explicitly written as
$\bar{u}(\bar{w})=u_{0}^{*}\exp{[-i\bar{w}q^{*}]}$. Matching to the boundary
condition, the soft mode takes the form

$\displaystyle u=u_{0}$
$\displaystyle\exp{\left[iq_{0}\left(x+\frac{(1-|\nu|)\tan(\phi)}{(1+|\nu|\tan^{2}(\phi))}y\right))\right]}$
$\displaystyle\,\,\times\exp{\left[-|q_{0}|\sqrt{|\nu|}\frac{1+\tan^{2}(\phi)}{1+|\nu|\tan^{2}(\phi)}y\right]}\,.$
(69)

Here, the absolute value of $q_{0}$ must be taken in the second
(decaying/growing) exponential in order to ensure that the mode does not
exponentially grow to unphysical quantities deep in the material bulk. While
the amplitude and phase of $u_{0}$ must be determined to match precisely to
the boundary condition, there is otherwise no freedom remaining and these
boundary conditions are sufficient to completely determine soft response. The
mode decays into the bulk with a characteristic length which diverges as
$\nu\rightarrow 0^{-}$ and simultaneously oscillates in the direction of the
vector $(1,\frac{(1-|\nu|)\tan(\phi)}{1+|\nu|\tan^{2}(\phi)})$.

###  B.4 Force-balanced modes in the long strip

To illuminate the possibility of a switchable mechanical filtering device, as
suggested in the main text, we explore the geometry of the long strip. This is
related to the half-plane geometry above, with a second infinite open boundary
which is parallel and at a distance $d$ from the former, as shown in main text
Fig. 4. In this case, the sheared analytic modes available to the anauxetic
phase of the metamaterial are essentially unchanged. However, in the anauxetic
phase, modes which were divergent at long distances away from the open
boundary due to exponential growth, may now be permitted.

For an auxetic mechanism, any generic sheared analytic mode in this geometry
may be composed of the exponential analytic functions
$u(w)=u_{0}\exp\left[iq(1+\sqrt{|\nu|}\tan(\phi))w\right]$, which are well-
behaved (not divergent) along both boundaries. The complex conjugates of such
functions will determine the $\bar{u}$ counterparts. We may therefore note
that the contributions which have a positive value of $q$ will be
exponentially suppressed by a factor
$\exp\left[-\frac{2\pi\sqrt{|\nu|}(1+\tan^{2}(\phi))}{\lambda(1+|\nu|\tan^{2}(\phi))}d\right]$,
where $\lambda$ is the wavelength of the mode at the bottom boundary.
Meanwhile, those with a negative value of $q$ will be amplified by the inverse
of the same factor. These modes at the bottom boundary are differentiated by a
polarization, in which one mode sweeps out a clockwise ellipse as $x$ is
decreased (left polarized mode) while the other sweeps out a clockwise ellipse
as $x$ is increased (right polarized mode). It therefore follows that the
effect of the metamaterial filter is to amplify the portions of an input mode
with right-polarization, while suppressing those with left-polarization. The
resulting signal at the far boundary of the strip will therefore be dominated
by the right-polarized modes. This filtering behavior may be switched on and
off by applying uniform mechanism strain to the system and tuning across the
exceptional point of the system at $\nu=0$.

##  Appendix C Numerical methods

Figure 5:  Diagrammatic representation of the constitutive model used to
numerically obtain nonuniform force-balanced states of a mechanism. Blue dots
represent the nodes, while black lines connect the node degrees of freedom
with hookean springs. Grey parallelograms are for ease of viewing,
representing the effective rigid quadrilateral elements of the mechanism in
main text Fig.1.

Here, we describe the methods used to investigate force-balanced
configurations of finite-sized mechanism lattices, as were used to generate
data for main text Fig. 2. For simplicity, we employ lattices composed of
identical unstretched ideal hookean springs, so that the energy is computed
via $E=\frac{k}{2}\sum_{\mu}e_{\mu}^{2}$ where $e_{\mu}$ is the extension of
bond $\mu$. As shown in Fig. 5, the springs are placed to generate rigid
parallelogram pieces joined at free hinges to emulate the mechanisms in main
text Fig. 1: each rigid parallelogram is composed of 6 bonds and individually
may only translate and rotate without energy cost. Because there is no energy
penalty for pivoting the springs around the nodes they attach to, the
mechanism in main text Fig. 1a,b is an energy-free motion of the lattice.

To generate force-balanced data in both the “force-free” and “force-bearing”
conditions from the main text, we consider two distinct numerical processes
listed below. In both cases, the finite mechanism metamaterial is defined via
a system of identical hookean springs as depicted in Fig. 5.

###  C.1 Probing the stress-free continuum response in finite systems

The loading patterns which are compatible with one of the sheared analytic
modes $u=f_{1}(w)$ and $\bar{u}=f_{2}(\bar{w})$ should lead to precisely such
a zero-energy response in the continuum. This defines the “force-free” loading
conditions. However, for finite systems with small but not infinitesimal unit
cell size, these motions are not precisely zero-energy anymore. To examine the
force-balanced low-energy configurations generated in a finite lattice, one
may control the mechanism strain magnitude all along the left boundary of the
system, as shown in main text Fig. 2a. To approximately control the mechanism
strain in each of these boundary unit cells, a single stiff spring is added to
the unit cell, crossing an open parallelogram void. These stiff springs are
prescribed to extend according to a smoothly varying function through space.
The springs are treated as rigid constraints, and the minimum energy state
subject to these constraints is obtained. For linear mechanics, obtaining the
force-balanced state of the system is reduced to a linear algebra problem
which is readily solved in Mathematica.

In the absence of nonuniformity, extending the boundary springs would
trivially activate the mechanism alone. To probe the effect of nonuniformity,
we choose the springs to extend according to a smoothly varying function
through space. We would like to probe the trend in the effects of
nonuniformity as the continuum limit is approached. We therefore probe these
configurations as the system is made more dense with unit cells, keeping the
smoothly varying function which controls the boundary conditions fixed. This
defines our continuum limit procedure and, following the analyses in the
following sections, the fitting to our analytic soft mode theory generically
improves as this continuum limit is approached.

For the data in the main text (Fig. 2a-d), the springs are extended according
to a cubic function which is zero at the vertical midpoint and at top and
bottom edges of the system. However, this choice is arbitrary and we have
checked against various other functions which produce similar data.

###  C.2 Probing the stress-bearing continuum response in finite systems

In contrast to the stress-free simulations designed above, we also investigate
the stress-bearing configurations in the same system. To achieve this in a
controlled manner, a new set of constraints are chosen to be generically
incompatible with a sheared analytic mode. This defines the “force-bearing”
loading conditions. To this end, rather than control the extension of
additional bonds, we instead directly control the displacement of nodes all
along the system boundary. To minimize mechanical boundary effects, we choose
to prescribe the displacement of every node which has a different coordination
than in the infinite lattice system (i.e. all dangling nodes). Such a set of
nodes, along with the displacement constraints, are illustrated in main text
Fig. 2e. The applied displacements along the boundary are determined by an
arbitrary analytic force-balanced mode , known to generate finite stress even
in the continuum limit. This is achieved using only the functions
$g_{2},g_{1}$ defined in Eqs. 56 & 57. For simplicity, only the second-order
coefficients of the analytic expansions of these functions are explored here,
with random phase and unit magnitude. More explicitly, we impose
$g_{1}(\bar{w})\rightarrow\exp(i\eta_{1})\bar{w}^{2}$ and
$g_{2}(\bar{w})\rightarrow\exp(i\eta_{2})\bar{w}^{2}$, with random numbers
$\eta_{1},\eta_{2}$. These simple forms are sufficient to generate nontrivial
spatially varying stress patterns without calling into question the relative
effects of different coefficients. However, including further terms in the
analytic expansions of $g_{1},g_{2}$ is straightforward and generates
similarly well-behaved data to that shown in the main text.

To be compatible with the bulk force-balanced continuum mode, the fine motions
within the unit cell must be accounted for. Rather than apply these smooth
displacements directly to the nodes, force balance is first determined within
each boundary unit cell, subject to the local strain experienced there.
Together with the overall displacement of the unit cell itself, this
determines the applied boundary conditions. These displacement conditions are
again enforced using lagrange multipliers and the minimimum of the spring
energy is identified via straightforward linear algebra. To observe the
improving accuracy of our theory as the continuum limit is approached, the
number of unit cells in each direction $N$ is increased to generate the data
in main text Fig. 2g,h.

Note that this particular procedure of detailed applied displacements was
chosen to minimize boundary effects and observe the approach to continuum
behavior in a controlled manner. However, it may be extended successfully to
more realistic situations in which only one node is controlled per boundary
unit cell, as well as the displacement along a partial boundary.

###  C.3 Numerical estimation of strain and stress quantities from microscopic
data

Figure 6:  Useful material vectors used in the estimation of local strain from
deformed system configurations. Here, green dots represent the unit cell
centers, and the strain for a single unit cell is most accurately obtained
using the symmetric lattice vectors (pink arrows) which stretch between unit
cell centers on opposite sides.

From force-balanced configurations identified numerically in Secs. C.1 & C.2,
we wish to estimate and separate the coarse mechanism strain from the
nonmechanism strain and similarly separate the mechanism stress from the
nonmechanism stress. To do this we must first construct methods to estimate
conventional stress and strain in a coarse-grained approximation of the finite
systems.

As discussed in the main text, the coarse strain locally controls the
deformation of the lattice vectors. As shown in Fig. 6a, we estimate the
lattice vectors $\mathbf{l}^{(1)},\mathbf{l}^{(2)}$ after deformation to be
those that connect the displaced center of mass of neighboring unit cells.
Rather than constructing the symmetrized strain directly, we may estimate the
unsymmetrized strain, i.e. the deformation gradient
$S_{ij}\equiv\partial_{j}u_{i}$. In the continuum limit, this tensor controls
the changes in lattice vectors
$\Delta\mathbf{l}^{(i)}\equiv\mathbf{l}^{(i)}-\mathbf{l}^{(i)}_{0}=\mathbf{S}\cdot\mathbf{l}^{(j)}_{0}$.
The changes in two linearly independent lattice vectors are sufficient to
estimate the unsymmetrized strain by constructing a matrix equation
$(\Delta\mathbf{l}^{(1)},\Delta\mathbf{l}^{(2)})=\mathbf{S}\cdot(\mathbf{l}^{(1)}_{0},\mathbf{l}^{(2)}_{0})$,
which is solved by

$\mathbf{S}_{\text{est}}=\frac{1}{|\mathbf{l}^{(1)}_{0}\otimes\mathbf{l}^{(2)}_{0}|}\,\times\mathbf{M}\cdot\mathbf{\epsilon}\,,$
(70)

where

$M_{ij}=\left[(\Delta\mathbf{l}^{(2)}\cdot\hat{\mathbf{R}}_{i})(\mathbf{l}^{(1)}_{0}\cdot\hat{\mathbf{R}}_{j})-(\Delta\mathbf{l}^{(1)}\cdot\hat{\mathbf{R}}_{i})(\mathbf{l}^{(2)}_{0}\cdot\hat{\mathbf{R}}_{j})\right]\,,$
(71)

$\epsilon$ is the antisymmetric unit tensor with $\epsilon_{12}=1$, and
$\\{\hat{\mathbf{R}}_{i}\\}=(\hat{\mathbf{x}},\hat{\mathbf{y}})$ are the
conventional unit vectors of the Cartesian coordinate system. To reduce
spurious finite-size effects in the estimation of strain, we employ an
alternative definition of the lattice vectors which is more symmetric, as
shown in Fig. 6. It is straightforward to check that using these vectors, the
errors in the strain estimation to first order in the lattice spacing
$|\mathbf{l}^{(1)}_{0}|$ are eliminated in favor of higher-order errors.

To then obtain the mechanism strain fraction, the numerically measured
unsymmetrized strain is broken down into components

$\mathbf{S}=\varepsilon_{m}\mathbf{\hat{e}}_{m}+\varepsilon_{1}\mathbf{\hat{e}}_{1}+\varepsilon_{2}\mathbf{\hat{e}}_{2}-\phi\mathbf{\epsilon}\,,$
(72)

using the orthonormal basis of tensors introduced in Eqs.17,18,19, and the
antisymmetric tensor $\epsilon$ is the negative of the generator of rotations
(and is also of unit norm). Using the inner product
$\langle\hat{e}_{i}|\hat{e}_{j}\rangle=\mathrm{Tr}[\mathbf{\hat{e}}_{i}^{T}\cdot\mathbf{\hat{e}}_{j}]/2=\delta_{ij}$,
we obtain the strain amplitudes in this mechanism basis and are able to
measure the nonmechanism strain fraction functional
$\Delta_{\text{nonmech}}[\mathbf{\varepsilon}]$ plotted in main text Fig. 2c:

$\Delta_{\text{nonmech}}[\mathbf{\varepsilon}]=\sqrt{\frac{\left\langle\varepsilon_{1}^{2}+\varepsilon_{2}^{2}\right\rangle}{\left\langle\varepsilon_{m}^{2}+\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\phi^{2}\right\rangle}}\,,$
(73)

where $\langle\rangle$ is a spatial average taken over unit cells with a
buffer region of two unit cells excluded at the boundary to minimize boundary
effects. This quantity thus compares the average magnitude of unsymmetrized
strain to the magnitude of stress-bearing strain
$\varepsilon_{1},\varepsilon_{2}$ in bulk soft deformations.

To estimate the local stress from the stress-bearing deformations employed in
main text Fig. 2e-g, we employ the virial stress expression Batchelor (1970).
This standard formula approximates local mechanical stress from pairwise
interactions. The stress in unit cell $i$ located at $(x_{i},y_{i})$ is
estimated using a symmetric lattice form

$\mathbf{\sigma}(x_{i},y_{i})=\frac{k}{2A_{\text{unit}}}\sum_{v\in
i}\sum_{\mu\in
v}\frac{k_{\mu}(|b_{\mu}|-b_{\mu}^{0})}{|b_{\mu}|}\mathbf{b}_{\mu}\otimes\mathbf{b}_{\mu}$
(74)

where $A_{\text{unit}}$ is the unit cell area. Here the first sum is over the
vertices $v$ contained in the unit cell while the second sum is over the bonds
$\mu$ attached to the vertex $v$, with $\mathbf{b_{\mu}}$ the deformed bond
vector and $k$ the bond stiffness. With this symmetric summation convention,
bonds stretching from one unit cell to the next contribute half of their
stress to each unit cell while bonds between vertices within the unit cell are
counted twice yielding the usual contribution.

Again, the local stress obtained in Eq. 74 may be broken into mechanism and
nonmechanism pieces (there is no antisymmetric rotational piece due to the
manifest symmetry of the stress tensor). The mechanism stress fraction
functional $\Delta_{mech}[\mathbf{\sigma}]$ is then evaluated using

$\Delta_{\text{mech}}[\mathbf{\sigma}]=\sqrt{\frac{\left\langle\sigma_{m}^{2}\right\rangle}{\left\langle\sigma_{m}^{2}+\sigma_{1}^{2}+\sigma_{2}^{2}\right\rangle}}\,.$
(75)

###  C.4 Analytic fitting of sheared analytic modes and force-balanced
stresses

Finally, to estimate the validity of the analytic theory describing the global
character of the force-balanced deformations, we search for the closest
sheared analytic mode which fits with the data. This is accomplished by
minimizing the fitting error

$\delta_{\text{fit}}[\mathbf{u}(),\\{u_{k}\\}]=\sum^{N_{p}}_{k}|\mathbf{u}(\mathbf{R}_{k})-\mathbf{u_{k}}|^{2}\,,$
(76)

where $\mathbf{u}(\mathbf{R}_{k})$ is vector displacement field generated by
the candidate analytic mode at the point $\mathbf{R}_{k}$, and the sum is
taken over the $N_{p}$ bulk material data points $k$. Again, a boundary layer
of thickness two unit cells is excluded to minimize boundary effects. The
candidate analytic mode is determined by a set of scalar coeffients. We choose
to cut off these coefficients at a number $N_{c}=\text{min}(N_{p}/3,20)$ to
avoid overfitting. In the anauxetic case, there are then $N_{c}$ real scalar
numbers determining the function $u=\sum_{n}C_{n}w^{n}$ and another $N_{c}$
independent real scalar numbers determining the function
$\bar{u}=\sum_{n}D_{n}\bar{w}^{n}$. In the auxetic case, there are only
$N_{c}$ coefficients determining $u$, which then determines the function
$\bar{u}$. However, these coefficients are complex-valued, and the information
required to define each function remains the same. The error in Eq. 76 is
minimized over these coefficients, which is a straightforward linear algebra
problem. Estimating the error of these fits for main text Fig. 2d is
accomplished by the fractional error $\Delta_{\text{fit}}[u,\\{u_{k}\\}]$
(i.e. the square root of the fraction of variance unexplained)

$\Delta_{\text{fit}}[\mathbf{u}(),\\{u_{k}\\}]=\sqrt{\frac{\delta_{\text{fit}}[\mathbf{u}(),\\{u_{k}\\}]}{\delta_{\text{fit}}[\mathbf{u}(),\\{u_{k}\rightarrow
0\\}]}}\,,$ (77)

where the term in the denominator gives the square magnitude of the
displacement itself. Identifying the closest analytic fit for the dual strain
patterns, and therefore the fractional error one may utilize the exact process
above, except with the replacements $u_{x}\rightarrow A\sigma_{1}$ and
$u_{y}\rightarrow\sigma_{2}$. Analogous fitting and error estimation methods
have been used in related previous work Czajkowski  _et al._ (2022).