[1]Michael Moshe

[1]Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem,
9190401, Israel

2]Department of Mechanical Engineering and Materials Science, Yale University,
New Haven, 06520, Connecticut, USA

3]Van der Waals-Zeeman Institute, Institute of Physics, University of
Amsterdam, Science Park 904, Amsterdam, 1098XH, The Netherlands

# Odd mechanical screening in disordered solids

Yael Cohen<EMAIL_ADDRESS>Amit Schiller
<EMAIL_ADDRESS>Dong Wang<EMAIL_ADDRESS>Joshua A. Dijksman
<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [  [

###### Abstract

Disordered solids, straddling the solid-fluid boundary, lack a comprehensive
continuum mechanical description. They exhibit a complex microstructure
wherein multiple meta-stable states exist. Deforming disordered solids induces
particles rearrangements enabling the system to transition between these meta-
stable states. A dramatic consequence of these transitions is that quasistatic
deformation cycles modify the reference state, facilitating the storage and
release of mechanical energy. Here we develop a continuum theory that
generalizes classical elasticity by accounting both for the absence of
reference state, and the lack of conserved potential energy. Our theory
includes a new modulus describing non-conservative mechanical screening. We
derive predictions in closed analytic form for the deformation field for
various perturbations and geometries. While our theory applies to general
disordered solids, we focus on a two-dimensional disordered granular system
and predict accurately the non-affine displacement fields observed in
experiments for both small and large deformations. The new proposed moduli
satisfy universal relations that are independent of the specific experimental
realization. Our work thus forms the basis of an entirely new family of
continuum descriptions of the mechanics of disordered solids.

###### keywords:

Mechanical Screening, Anomalous Mechanics, Elastic Charges, Granular Matter

Solid materials are capable of supporting external stresses, while liquids
flow indefinitely in response to similar external perturbations. This clear
distinction between the mechanical behaviors of different phases of matter is
disrupted by disordered solids. Examples such as granular and glassy materials
exhibit a combination of fluid-like and solid-like properties simultaneously
[1, 2, 3, 4]. Shear perturbation uncovers the complex and distinctive
mechanics of such disordered materials. This includes the phenomena of shear-
jamming [5, 6, 7, 8], material failure via shear-banding [9, 10, 11, 12, 13,
14], and non-affine anomalous responses [15, 16, 17, 18]. It is generally
accepted that these complex phenomena are governed by the interactions between
localized plastic events, realized as local material rearrangement [19, 20,
21]. Their corresponding flow and induced elastic-like fields are of
quadrupolar nature and are modeled as Eshelby inclusions [22], force dipoles
[23] and more [31].

Plastic events have two primary consequences. First, they modify the reference
state to relieve strains [25, 26]. Second, they transition the system from one
stable state to another, suggesting the presence of multiple meta-stable
states [27, 28]. Because different meta-stable states may possess distinct
energies, the inclusion of such effect in a continuum theory would lead to a
violation of energy conservation. More specifically, in addition to
dissipating mechanical energy as heat, plastic events introduce another
mechanism for releasing or storing mechanical energy, by plastically modifying
the reference state, a mechanism that is at the heart of our work. While there
has been a substantial body of work focused on plasticity in disordered solids
[29, 30, 31], and despite an ongoing effort to develop a continuum theory of
disordered solids [32], current continuum theories of matter fall short in
accurately predicting the intricate anomalous deformation fields observed in
disordered solids.

Here we present a new continuum theory that describes disordered solids via
the introduction of a new elastic constant that captures the amount mechanical
work involved in a distribution of plastic events. The theory generalizes
classical elasticity by incorporating the primary implications of plastic
events by capturing both the alteration of the rest state, and the violation
of energy conservation due to multi-stability. We test our theory against
experiments and find quantitative agreement.

We draw inspiration from recent work on active solids, where “Odd-Elasticity”
theory is developed to describe elastic solids whose equations of motion do
not originate from a potential energy [33, 34]. A core idea of this odd-
elasticity is the inclusion of anti-symmetric (odd) moduli in the elastic
tensor
$\mathcal{A}^{\alpha\beta\gamma\delta}\neq\mathcal{A}^{\gamma\delta\alpha\beta}$,
which relates stress with deformation. These new moduli modify the classical
equation of motion, and are shown to quantify the amount of work that can be
extracted or stored during a closed deformation cycle.

Figure 1: The basic mechanical screening modes, their realizations, and the
corresponding elastic charges acting as sources of stresses when implemented
in a material. (a) Strain-induced particles rearrangement and the resultant
quadrupole of angle deficit/excess marked by $\pm$. The quadrupole magnitude
is proportional to the inducing strain, as indicated by the color-bar. On the
right, a heat map displays induced charge, with dark/bright colors
representing negative/positive charges. (b) A visual representation of the
linear relation between the inducing strain modes and the induced quadrupole
charges. (c) Displacement-induced particle rearrangement results in non-
uniformly distributed quadrupoles. On average, deformation description
necessitates a combination of a quadrupole and a dipole, as shown in the
asymmetric charges heatmap on the right. (d) A visual representation of the
linear relation between the inducing displacement and the induced dipole
charge $\mathbf{P}=\Gamma\mathrm{\mathbf{d}}$. In the example shown in (c) the
displacement and dipole are parallel, corresponding to $\kappa_{o}=0$.

We first introduce a first-principles-based theory of odd mechanical screening
(OMS), for passive systems that possess internal degrees of freedom of non-
conservative strain relaxation. The elastic content of OMS assumes a symmetric
elastic tensor,
$\mathcal{A}^{\alpha\beta\gamma\delta}=\mathcal{A}^{\gamma\delta\alpha\beta}$,
thus distinguishing it from odd-elasticity. Nevertheless, it introduces a
novel (odd) screening modulus $\kappa_{o}$ next to the usual even screening
modulus $\kappa_{e}$. We show that $\kappa_{e}$ quantifies strain relaxation
as recently proposed in [26], and the odd modulus $\kappa_{o}$ quantifies the
amount of work that can be harnessed or stored within a closed deformation
cycle, as attributed to plastic events. We quantitatively test the theory
against unexplained features in the continuum strain fields in an experimental
model passive amorphous material [35]. As we will see, OMS quantitatively
predicts an experimentally observed anomalous coupling between orthogonal
components of the displacement field. Importantly, the values of the new
moduli display physically meaningful behavior. At small strains, $\kappa_{e}$
and $\kappa_{o}$ emerge as well-defined constants that characterize the
material, as expected from a linear continuum theory. As we extend our
analysis to finite strains, we find that the moduli adhere to a universal,
strain-dependent relationship. Remarkably, this relationship remains
independent of initial conditions or packing fraction. This revelation
provides new impetus to the search for the underpinnings of plasticity in
granular matter, and offers many tantalizing perspectives in a wider
application of OMS to the broader field of disordered materials. Most
importantly, we confirm that continuum theories of matter can describe large
deformations in disordered materials experiencing plastic events.

Mechanical Screening  One ingredient of the theory is the assumption that the
primary function of localized plastic events is to suppress (or screen)
elastic fields by modifying the rest state relative to which elastic strains
are measured [26]. A mechanical screening theory based on this concept has
already been successful in describing certain types of anomalous deformation
modes in amorphous systems [36, 37]. The natural relaxing degrees of freedom
have been shown to be quadrupolar elastic charges, which unify the description
of force-dipoles, Eshelby inclusions, holes, and more [38, 39, 40]. In the
current context of disordered solids these charges, denoted $Q^{\alpha\beta}$,
describe plastic events and are induced by an imposed strain as illustrated in
Fig. 1(a). A typical particles rearrangement is shown to induce a quadrupole
of angle excess or deficit, thus acting as a source of stress when implemented
in an elastic solid [39, 41]. The linear relation between the inducing strain
and the induced charge, shown in Fig. 1(b), establishes the basis for a theory
of quadrupoles mediated mechanical screening. This relation accompanies the
force balance equation, which in the presence of screening dresses the stress
field and reads
$\partial_{\alpha}(\sigma^{\alpha\beta}+\tfrac{Y}{4(1+\nu)}Q^{\alpha\beta})=0$.
When the quadrupole degrees of freedom are integrated out, an elastic-like
theory is recovered [26], see _methods_ for more details. This approach to
mechanical screening is strongly inspired by electrostatic screening, and the
regime of quadrupole screening is analogous to electrostatic dipole screening
as in dielectric materials.

Figure 2: The displacement induced by a body shear as obtained by solving (1)
without (a) or with (b) odd dipole screening. Arrows indicate the displacement
direction and background colors indicate displacement magnitude.

A key property in this class of models is the dipole field $\mathbf{P}$ which
quantifies non-uniformly distributed quadrupoles
${P}^{\alpha}=\partial_{\beta}Q^{\alpha\beta}$. This is illustrated in Fig.
1(c) where we show that particles displacement induces non-uniform
quadrupoles. Consequently, the deformation description of a collection of
particles requires a combination of a quadrupole and a dipole. Then, the
equilibrium equation reflecting force balance is accompanied by a screening
constitutive relation which relates inducing displacement with induced dipoles
${P}^{\alpha}=\Gamma^{\alpha\beta}\mathrm{d}_{\beta}$, with $\Gamma$ the
screening tensor, as illustrated in Fig. 1(c,d). In homogeneous, isotropic,
conservative systems the screening tensor can be expressed as
$\Gamma=\kappa_{e}I$ with $I$ the identity. In this case the inducing
displacement is parallel to the induced dipole, as in Fig. 1(c).

In states exhibiting elastic-like behavior, for example disordered matter
under strong confinement, quadrupoles will form only sparsely. Consequently,
no dipoles will form, corresponding to $\kappa_{e}=0$. Conversely, in
situations where the cost associated with quadrupole formation is negligible,
as in disordered solids with lenient confining conditions, quadrupoles form
abundantly, corresponding to a finite $\kappa_{e}$. In this case the screening
relation modifies the form of the force balance equation. Upon integrating out
the screening degrees of freedom, the force balance equation in $2D$ reduces
to

$\displaystyle\Delta\mathbf{d}+r\nabla\left(\nabla\cdot\mathbf{d}\right)=-{\Gamma}\mathbf{d}\;.$
(1)

with $r_{\mathrm{2D}}=\tfrac{1+\nu}{1-\nu}$ and $\nu$ the Poisson’s ratio;
this can be generalized to 3D [42]. Equation (1) is a mechanical analogue of
Helmholtz equation in Debye-Hückel theory of ionic liquids.

Odd Mechanical Screening Eq. (1) is derived from a potential energy, which
imposes a symmetry condition on the screening tensor
$\Gamma^{\alpha\beta}=\Gamma^{\beta\alpha}$. However, in realistic (yield
stress) materials, energy is not conserved due to particles rearrangements
[28]. During quasi-static shear, particles rearrange, and upon completing a
closed loop in strain space, e.g. by sequentially applying different shear
strain modes, it is likely that the system will not end up in the same state.
This basic observation from many amorphous materials reflects the existence of
multiple meta-stable states, and even fractal-like energy landscape [27].
Consequently, during this work cycle net work can be extracted or stored in
the system.

This suggests that Eq. (1) with $\Gamma=\kappa_{e}I$ cannot provide a complete
description of disordered solids. Therefore, as a second ingredient of the
theory, we need to remove the assumption on the existence of potential energy
while retaining the successful modeling picture of mechanical screening
theory. To do so, we allow $\Gamma$ to contain an anti-symmetric (odd)
screening modulus

$\Gamma=\begin{pmatrix}\kappa_{e}&-\kappa_{o}\\\
\kappa_{o}&\kappa_{e}\end{pmatrix}\equiv\kappa\,\mathrm{Rot}(\theta_{\kappa})$
(2)

Here $\kappa=\sqrt{\kappa_{e}^{2}+\kappa_{o}^{2}}$ quantifies the screening
magnitude, and $\mathrm{Rot}(\theta_{\kappa})$ is a rotation operator with
$\tan\theta_{\kappa}=\kappa_{o}/\kappa_{e}$, which we term the odd screening
phase. Geometrically, the odd screening modulus $\kappa_{o}$ quantifies the
rotation angle between an inducing displacement field and the induced dipole.
From a mechanical perspective, $\kappa_{o}$ quantifies the amount of work
extracted or stored during the work cycle due to plastic events. Explicitly,
$\Delta W=\oint f^{\alpha}d_{\alpha}\mathrm{d}^{2}x\propto\kappa_{o}$, see
_methods_ for detailed calculation.

Figure 3: Odd-displacement coupling in disordered granular matter Left panel:
Theory (solid lines) and experimental data (dots) for anomalous displacement
$d_{x}(x)$ induced by an affine shear $d_{y}=u_{xy}\,x$. The displacement is
averaged along the $y$ direction and fitted to Eq. (3) with respect to
$\kappa$ and $\theta_{\kappa}$. In (a), we present the experimental and
theoretical displacement fields for various packing fractions at a fixed
imposed shear, and in (b), we display them for various shear loadings at a
fixed packing fraction. Right panel: $\theta_{\kappa}$ as function imposed
shear $u_{xy}$. In (c), we illustrate $\theta_{\kappa}$ for various initial
realizations with a fixed packing fraction. The variance is shown in red, with
virtually no variance observed up to strains of 10%. In (d) we depict the
ensemble-averaged screening phase $\left<\theta_{\kappa}\right>$ for a range
of initial packing fractions, highlighting its independence not only from
initial conditions but also from the initial packing fraction.

Anomalous displacement coupling An immediate implication of OMS is the new
coupling between the different displacement components in Eq. (1). This
anomalous coupling is absent from other theories of matter in homogeneous and
isotropic solids, in is crucial for describing disordered solids. To
illustrate its implications we solve Eq. (1) for the two most studied loading
protocols of shearing a material either in an annular or in a rectangular
domain, as studied in an immense volume of works, see for example [3, 43, 44,
45, 46] and references therein. Our predictions for the case of a rectangular
domain are given in detail and compared to experimental measurements; the case
of an annular domain is presented in the SI.

For a rectangular domain subjected to a body shear, as studies in [7], the
equilibrium equation (1) is supplemented with the conditions
$d_{y}(x)=u_{xy}x$ and $d_{x}(\pm L/2)=0$. The deformation protocols and the
corresponding solutions are illustrated in Fig. 2. In the absence of OMS
$\kappa_{o}=0$, the response is purely affine $d_{y}(x)=u_{xy}x$, with
$d_{x}(x)=0$, as illustrated in Fig. 2(a). In the presence of OMS, we find a
non vanishing solution

$\displaystyle\mathrm{d}_{x}(x)=\tau\,\left(x-\frac{\sin\left(\zeta\,x\right)}{\sin\left(\zeta\right)}\right)\;,$
(3)

with $\tau=u_{xy}\,\tan\theta_{\kappa}$,
$\zeta=\sqrt{\cos\theta_{\kappa}\,\kappa/(1+r)}\,L/2$, and $x$, and $d$ are
measured in units of $L/2$. This solution is illustrated in Fig. 2(b).

Results The anomalous displacement coupling predicted by OMS align with
observations from recent experiments on two-dimensional granular model system
subjected to simple body shear [35]. In this experiment nonzero displacement
fields $d_{x}\neq 0$ emerged, and thus serves as the ultimate test for the
theory of OMS. We investigate a disordered system of 2D shear-jammed granular
matter composed of bi-dispersed frictional disks. The body shear protocol is
applied quasistatically by uniformly deforming the underlying table (see
_method_ for further details). We explore a range of imposed strains, up to
22%, and initial packing fractions within the range of $0.692<\phi<0.816$. To
assess sensitivity to initial conditions, each experiment is repeated five
times. In the experiment, we measure the displacement field of each particle
and, due to the system’s symmetry, we average it along the $y$ direction
before comparing it with the prediction of (3). For each measurement of the
displacement field, we fit the parameters $\eta$ and $\tau$, from which we
extract the screening moduli $\kappa$ and $\theta_{\kappa}$. In left panel of
Fig. 3, we present plots of the induced transverse displacement $d_{x}$ as a
function of position $x$ together with our predictions. In (a) we present
plots for various values of packing fractions with a fixed imposed strain, and
in (b) we present plots for various values of imposed strains with a fixed
packing fraction. We observe that (3) fits very well and perfectly reproduces
the functional form of the anomalously induced displacement.

The success in recovering the functional form of the displacement field by
fitting the screening moduli is a promising step. To further demonstrate that
that the screening moduli are proper macroscopic quantifiers for the state of
jammed granular matter, in the _method_ section, we show that for small
strains, both screening moduli remain independent of the initial conditions
and assume constant values as a function of strain, independent of the initial
configurations. Surprisingly, our analysis reveals that these explicit moduli
are also unaffected by variations in the initial packing fraction, with values
of $\eta\approx 11L^{2}$ and $\theta_{\kappa}\approx 0.94\pi$. Notably, the
value of the odd screening phase $\theta_{\kappa}$ indicates that the
screening effect is predominantly even $\kappa_{o}/\kappa_{e}\approx-0.2$.

Another surprising discovery is that the theory, initially designed for small
deformations based on linearized strain-stress and dipole-displacement
relations, describes the displacement field accurately even at substantial
strains exceeding 20%. Drawing an analogy to non-linear optics, where the
permittivity can explicitly depend on the field’s magnitude, it is plausible
that at large deformations, the screening moduli might exhibit a dependence on
the strain. As expected, we observe that at finite strains $\theta_{\kappa}$
and $\kappa$ deviate from their constant values at low strains (see _methods_
for plots of $\kappa$). In Fig. 3(c) we show plots of $\theta_{\kappa}$ for
five different realizations of identical initial packing fraction
$\phi=0.736$. Remarkably, we find that in all experiments the dependence of
$\theta_{\kappa}$ on the imposed strain $u_{xy}$ is identical, with a variance
that is practically zero, up to strain of $~{}10\%$. At larger strains
$\theta_{\kappa}(u_{xy})$ still acquire a clear functional form, though with
different functional form for different initial conditions. This observation
suggests that the system is self averaging up to strains as high as $10\%$,
thus allowing its description within a continuum theory even for large
deformations. The origin of the precise functional form of
$\theta_{\kappa}(u_{xy})$ is an exciting avenue for much future work, while
the large strain effectiveness of the continuum modeling directly reduces the
need for expensive mesh-free methods to predict material behavior at large
strains. The situations becomes even more surprising when
$\theta_{\kappa}(u_{xy})$ is computed for different initial packing fraction,
where we discover that all initial conditions and all packing fractions have
exactly the same form, as shown in Figure 3(d). This universal form indicates
a universal underlying nonlinear screening mechanism independent of the
packing fraction.

The weak decay of the screening phase $\theta_{\kappa}$, transitioning from
approximately $\pi$ to $0.8\pi$, signifies an important new physics of
screening within disordered media. Screening can be equivalently achieved by
either generating new charges, depending on the protocol, or releasing pre-
existing ones. In the latter scenario, the availability of pre-existing
charges is limited, resulting in an expected weakening of the screening effect
during the process. Consequently, the decline in $\theta_{\kappa}$ signifies
that screening predominantly occurs through the release of pre-existing
charges, which were trapped within the system during the jamming process. This
conclusion finds independent support in the relationship between the screening
magnitude, $\kappa$, and the applied shear stress, $u_{xy}$, which exhibits an
exponential decay, indicating the lack of existing charges to be released (see
App.A). Our theory, therefore, posits that a fundamental formulation of
screening theory, rooted in microscopic principles, necessitates a
comprehensive understanding of the initial jammed state’s pre-existing
charges. It is noteworthy that the charges induced as a response to shear
jamming and isotropic pressure may fundamentally differ, thus prompting the
need for a quantitative investigation into the connection between jamming and
induced charges.

In conclusion, our OMS theory correctly captures small and large strain
behavior of a model disordered media via the introduction of only one new
modulus. The new screening modulus has a clear physical microscopic
interpretation and is observed to be indeed universal for the experimental
model system used. Our work calls for much future work, such as applying to
the wide range of other disordered materials like colloidal glasses, foams,
emulsions and microgels. OMS can explain unusual deformation modes that can be
captured by performing new or different types of rheological experiments. It
is clearly possible expand our findings by developing a nonlinear version of
OMS to comprehensively analyze the failure modes of disordered solids. As our
theory now only works in quasi-static deformation, there is an obvious
perspective to incorporate dynamical and dissipative mechanisms. Crucially,
our theory should be integrated with other odd-elasticity principles and
applied to experimental active solids, such as epithelial tissue.

Acknowledgments This work was supported by Israel Science Foundation Grant No.
1441/19. MM thank Gil Cohen for stimulating discussions.

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##  1 Methods

Experiment Setup. We applied quasi-static simple shear to a bi-dispersed
granular system composed of elastic discs. Shear was performed in an apparatus
that suppresses shear bands allowing us to study large strains before the
onset of shear bands [47]. Particles were carried by separate slats that form
the shear box base. These slats moved affinely in accordance with the applied
shear [47, 48]. Discs were cut from elastic sheets (Vishay PSM-4), resulting
in a friction coefficient $\mu\approx 0.7$ [47, 49]. The system contained
approximately $45\times 20$ discs with a diameter ratio $1:1.25$ and a large
to small number ratio $1:3.3$ to prevent crystallization. Every run at a given
packing fraction $\phi$ was repeated five times, with the initial stress-free
state being prepared anew for each run. Shear was applied quasi-statically in
the $y$ direction to a shear box (Fig. 4(a)). Starting from a stress-free
state, the system was sheared by a strain step of $\delta\gamma=0.0027$. Then
the system was left to relax for six seconds, followed by taking an images
which reveal information on particle position. Such a process – stepwise
shearing, relaxing and imaging – was repeated until a certain amount of total
strain was achieved. The maximum shear strain $\gamma$ that studied in this
work is $0.24$ (24%). From the images, we tracked particle positions and
computed the displacement field. All tracking data are available upon request.
[50, 51, 49].

Figure 4: (a) Schematic of special apparatus for applying simple shear strain
to a collection of photoelastic discs at the different strain steps (strain
steps increase from light to dark colors). In the sketch, the slats are drawn
much larger relative to the particles than in the real experiment. The $x-y$
axes indicate the coordinate system in the lab frame, where simple shear is
applied along the y axis. (b) Example of particle displacements $d_{x}$ in a
system with initial packing fraction $\phi=0.742$, after a shear strain of
$5\%$.

(Even) Mechanical Screening This section is supplemented with a _Mathematica_
notebook available online, where all calculations, including lengthy ones, are
given in full detail.

The deformation of elastic solid is described by a displacement field
$\mathbf{d}$. The elastic strain, which measures deviations of actual lengths
from their rest values, is
$u_{\alpha\beta}=\tfrac{1}{2}(\partial_{\alpha}d_{\beta}+\partial_{\beta}d_{\alpha})$.
The elastic energy density is then

$\displaystyle\mathcal{W}=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\gamma\delta}\;.$
(4)

where $\mathcal{A}$ is the elastic tensor.

Mechanical screening is the process at which internal degrees of freedom are
available to relief strains. Then one should distinguish between the
configurational strain $u$ derived from the displacement field, and the
elastic strain $u^{\mathrm{el}}$ for which elastic energy penalizes for

$\displaystyle u^{\mathrm{el}}=u-q$ (5)

Here $q$ is the local anelastic (screening) strain. In general, it has three
independent degrees of freedom, corresponding to relaxing isotropic strain and
the two modes of shear strain. While the theory is applicable to arbitrary
form of $q$, particles rearrangements allow to relax only shear strain and not
isotropic one, thus we assume $\mathrm{trace}(q)=0$.

The elastic energy is

$\displaystyle\mathcal{W}_{\mathrm{el}}=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\gamma\delta}\;.$
(6)

The induced local anelastic strain $q$ creates elastic fields that relaxes the
total strain but cost energy due to its nucleation. Therefore the total energy
contain another term for the nucleation cost associated with the induced
strain relaxation. For that we note that a distribution of relaxing $q$’s
induces elastic strains whose sources (elastic charges) are
$Q^{\alpha\beta}=\varepsilon^{\alpha\mu}\varepsilon^{\beta\nu}q_{\mu\nu}$
[39]. If the $Q$ field, which will be found by energy minimization, is non
uniform, according to the multipole expansion of elastic charges a hierarchy
of multipoles will form with an effective dipole field
$P^{\alpha}=\partial_{\beta}Q^{\alpha\beta}$ and an effective monopole field
$M=\partial_{\alpha\beta}Q^{\alpha\beta}$ [39]. The nucleation cost is
therefore of the form

$\mathcal{F}=\frac{1}{2}\Lambda_{\alpha\beta\gamma\delta}Q^{\alpha\beta}Q^{\gamma\delta}+\frac{1}{2}\tilde{\Gamma}_{\alpha\beta}P^{\alpha}P^{\beta}+\frac{1}{2}\Upsilon
M^{2}\;.$ (7)

Since no isotropic quadrupole can form, the most general forms of $\Lambda$
and $\Gamma$ in isotropic and homogeneous media are

$\displaystyle\Lambda_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda\left(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}\right)$
(8) $\displaystyle\tilde{\Gamma}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\tilde{\kappa}_{e}\,\delta_{\alpha\beta}$ (9)

Monopole screening has not been observed in solid like phases and therefore is
beyond the scope of the current work, i.e. $\Upsilon=0$. Regardless of the
specific form of (7), the 2d equilibrium equation, obtained by energy
minimization with respect to the displacement is

$\displaystyle\Delta\mathbf{d}+\frac{1+\nu}{1-\nu}\nabla\left(\nabla\cdot\mathbf{d}\right)=-\mathbf{P}$
(10)

where $\nu$ is the Poisson’s ratio.

This equation is supplemented by a constitutive screening relation obtained by
variating the energy with respect to the screening degrees of freedom $Q$. For
quadrupole screening we find

$\displaystyle\mathbf{P}=-\frac{1}{1+4\tilde{\lambda}(1+\nu)}\Delta\mathbf{d}\;,$
(11)

with $\lambda=Y\,\tilde{\lambda}$. Upon substituting in (10) we find that the
form of the equation remains the same but the coefficients are renormalized by
the screening.

In the case of dipole screening, where the nucleation cost of quadrupoles is
negligible compared with that of dipoles, we find

$\displaystyle\mathbf{P}=\frac{Y}{2\tilde{\kappa}_{e}(1+\nu)}(\mathbf{d}-\mathbf{d}_{0})$
(12)

where $\mathbf{d}_{0}$ is an integration constant. This relation represent the
relation between an imposed displacement and an induced elastic dipole, as
illustrated in Fig. 1. We set $\mathbf{d}_{0}=0$, thus the constitutive
relation loses its invariance under translations and rotations. We denote

$\displaystyle\mathbf{P}=\Gamma\mathbf{d}$ (13)

with $\Gamma_{\alpha\beta}=\kappa_{e}\delta_{\alpha\beta}$ and
$\kappa_{e}=\frac{Y}{2\tilde{\kappa}_{e}(1+\nu)}$. Upon substituting in (10)
we find

$\displaystyle\Delta\mathbf{d}+\frac{1+\nu}{1-\nu}\nabla\left(\nabla\cdot\mathbf{d}\right)=-\Gamma\mathbf{d}$
(14)

that is we recovered (1).

Work cycles and odd screening When a material is deformed, the power exerted
by the elastic forces is

$\dot{W}=\intop f^{\alpha}\dot{d}_{\alpha}dS$ (15)

where $f$ the force is given by

$f^{\alpha}=\partial_{\beta}\sigma_{\mathrm{el}}^{\alpha\beta}$ (16)

with the elastic stress $\sigma_{\mathrm{el}}=\mathcal{A}u^{\mathrm{el}}$,
satisfying $\partial_{\alpha}\sigma_{\mathrm{el}}^{\alpha\beta}=0$, which is
equivalent to (10). When expressed in terms of $\sigma=\mathcal{A}u$ we find
$\partial_{\beta}\sigma^{\alpha\beta}+\tfrac{Y}{4(1+\nu)}\mathbf{P}=0$. With
these notations we rewrite the power exerted by the elastic forces

$\dot{W}=\intop\left(\partial_{\beta}\sigma^{\alpha\beta}+\tfrac{Y}{4(1+\nu)}{P}^{\alpha}\right)\dot{d}_{\alpha}dS$
(17)

we further simplify this expression by expressing $\sigma$ and $\mathbf{P}$ in
terms of the displacement field

$\dot{W}=\intop\left(\mathcal{A}^{\alpha\beta\gamma\delta}\partial_{\beta}\partial_{\gamma}d_{\delta}+\tfrac{Y}{4(1+\nu)}\Gamma^{\alpha\beta}{d}_{\beta}\right)\dot{d}_{\alpha}dS$
(18)

Upon integrating by parts the first term we find

$\dot{W}=\intop\left(-\mathcal{A}^{\alpha\beta\gamma\delta}\partial_{\gamma}d_{\delta}\partial_{\beta}\dot{d}_{\alpha}+\tfrac{Y}{4(1+\nu)}\Gamma^{\alpha\beta}{d}_{\beta}\dot{d}_{\alpha}\right)dS$
(19)

Here we assumed displacement-controlled boundary conditions, thus the boundary
term vanishes. Next we calculate the total elastic work done during a closed
deformation. For the first term

$\displaystyle\Delta W_{1}$ $\displaystyle=$
$\displaystyle\oint\mathrm{d}t\,\dot{W}_{1}=-\oint\mathrm{d}t\,\intop\mathcal{A}^{\alpha\beta\gamma\delta}\partial_{\gamma}d_{\delta}\partial_{\beta}\dot{d}_{\alpha}dS$
(20) $\displaystyle=$
$\displaystyle\oint\mathrm{d}t\,\intop\mathcal{A}^{\alpha\beta\gamma\delta}\partial_{\gamma}\dot{d}_{\delta}\partial_{\beta}{d}_{\alpha}dS=\oint\mathrm{d}t\,\intop\mathcal{A}^{\gamma\delta\alpha\beta}\partial_{\beta}\dot{d}_{\alpha}\partial_{\gamma}{d}_{\delta}dS$

The third equality holds from (time) integration by parts, and the fourth
equality obtained by relabeling the indexes. We deduce that

$\Delta
W_{1}=-\oint\mathrm{d}t\,\intop\frac{1}{2}\left(\mathcal{A}^{\alpha\beta\gamma\delta}-\mathcal{A}^{\gamma\delta\alpha\beta}\right)\partial_{\gamma}d_{\delta}\partial_{\beta}\dot{d}_{\alpha}dS$
(21)

From the major symmetry of the elastic tensor this term is identically zero.

The second term in (19) is straightforward to handle. The total work is

$\displaystyle\Delta{W}_{2}$ $\displaystyle=$
$\displaystyle\oint\mathrm{d}t\,\dot{W}_{2}=\oint\mathrm{d}t\,\intop\tfrac{Y}{4(1+\nu)}\Gamma^{\alpha\beta}{d}_{\beta}\dot{d}_{\alpha}dS$
(22)

Integrating by parts the time-integral and relabeling the indexes we find

$\displaystyle\Delta{W}_{2}=-\oint\mathrm{d}t\,\intop\tfrac{Y}{4(1+\nu)}\Gamma^{\beta\alpha}{d}_{\beta}\dot{d}_{\alpha}dS$
(23)

Hence

$\displaystyle\Delta{W}_{2}=\oint\mathrm{d}t\,\intop\tfrac{Y}{8(1+\nu)}\left(\Gamma^{\alpha\beta}-\Gamma^{\beta\alpha}\right){d}_{\beta}\dot{d}_{\alpha}dS=\oint\mathrm{d}t\,\intop\tfrac{Y}{4(1+\nu)}\Gamma_{\mathrm{o}}^{\alpha\beta}{d}_{\beta}\dot{d}_{\alpha}dS$
(24)

where $\Gamma_{\mathrm{o}}$ is the odd part of $\Gamma$. In our work this
integral is proportional to $\kappa_{o}$.

##  Appendix A Values of screening moduli at small and large strains

The screening moduli in our theory, as in any other linear theory, hold
meaning only when they remain constant. However, when dealing with large
strains, there is no assurance that constant (spatially uniform) values for
the elastic and screening moduli can adequately describe the intricate and
non-affine deformation field. Nonetheless, we observe that the displacement
field accurately adheres to our theory at large strains, albeit with strain-
dependent moduli. This mirrors the behavior seen in non-linear optics, where a
material’s permittivity remains constant under weak fields but varies with the
field magnitude under strong fields.

Figure 5: The ensemble averaged screening moduli as function of strains. Top
panel shows the screening magnitude $\left<\kappa\right>$ for strain up to
$\%13$. (a) The full range, (b) zoom in on small strains regime, and (c) zoom
in on large strains regime (c). Bottom panel shows the screening phase
$\left<\kappa\right>$ for large strains (d) and small strain (e).
experimentally measured displacement fields $d_{x}(x)$ for a shear strain of
$1\%$, confirming the independence of the results from the initial packing
fraction.

In Fig. 5(a), we depict the ensemble-averaged screening magnitude
$\left<\kappa\right>$ as a function of imposed shear strain, covering strains
up to 13%, presented on a semi-logarithmic scale. Each plot corresponds to a
distinct initial packing fraction. In (b), we zoom in on the small strain
region, and present $\left<\kappa\right>$ up to $1.3\%$. The constant value of
the screening magnitude justifies the assumption of employing continuum
mechanical screening theory for small strain analysis of the experimental
system. In (c), we zoom in on the region of large strains, revealing that the
exponential decay is consistent across all initial packing fractions, albeit
with different prefactors indicating potential dependence on initial
conditions.

Similarly, in Fig. 5(d), we display the ensemble-averaged
$\left<\theta_{\kappa}\right>$ for various initial packing fractions. In (e),
we focus on the small strain region, presenting $\left<\theta_{\kappa}\right>$
up to $1.5\%$. Notably, we observe that this screening modulus remains
constant in this range. The inset showcases experimentally measured
displacement fields for a shear strain of $1\%$ enclosed by the green box,
further confirming the independence of the results from the initial packing
fraction.

##  Appendix B Odd mechanical screening in sheared rectangular and annular
domains

In this section we solve (1) for rectangular and annular domain subjected to
shear strain from the boundary.

For a rectangular domain we solve (1) with the boundary conditions

$\displaystyle d_{x}(L)$ $\displaystyle=$ $\displaystyle 0\,,\quad
d_{x}(-L)=0$ (25) $\displaystyle d_{y}(L)$ $\displaystyle=$ $\displaystyle
d_{0}\,,\quad d_{y}(-L)=-d_{0}$ (26)

The solution is

$\displaystyle d_{x}(x)$ $\displaystyle=$
$\displaystyle\frac{d_{0}\kappa_{o}}{\eta}\left(\frac{\sin(x/l_{1})}{\sin(L/l1)}-\frac{\sin(x/l_{2})}{\sin(L/l2)}\right)$
(27) $\displaystyle d_{y}(x)$ $\displaystyle=$
$\displaystyle\frac{d_{0}}{2}\left(\left(1-\frac{\kappa_{e}r}{\eta}\right)\frac{\sin(x/l_{1})}{\sin(L/l1)}+\left(1+\frac{\kappa_{e}r}{\eta}\right)\frac{\sin(x/l_{2})}{\sin(L/l2)}\right)$
(28)

with

$\displaystyle l_{1}^{-1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\kappa_{e}(2+r)-\eta}{2(1+r)}}$ (29) $\displaystyle
l_{2}^{-1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\kappa_{e}(2+r)+\eta}{2(1+r)}}$ (30)
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\sqrt{\kappa_{e}^{2}r^{2}-4\kappa_{o}^{2}(1+r)}$ (31)

We see that for $\kappa_{o}=0$ we get $l_{1}=l_{2}$ and then $d_{x}=0$, that
is, the anomalous coupling appears only in the presence of odd screening.

In Fig. 6 we plot the displacement fields for different values of the
screening parameters, with $d_{0}=0.05$, $r=1$, and $L=0.5$. For large values
of $\kappa_{e}$ and small $\kappa_{o}$ we find significant deviation from the
classical affine response, with non-monotonous displacement field, though with
$d_{x}(x)=0$. For non-zero odd screening $\kappa_{o}\neq 0$ the displacement
field rotates and induces a non-zero $d_{x}$ component.

Figure 6: Solutions for (1) with boundary conditions (26), with $d_{0}=0.05$,
$r=1$, and $L=0.5$. Each block corresponds to a solution with different values
of $\kappa_{e}$ and $\kappa_{o}$.

For a rectangular domain we solve (1) with the boundary conditions

$\displaystyle d_{r}(r_{\mathrm{in}})$ $\displaystyle=$ $\displaystyle
0\,,\quad d_{r}(r_{\mathrm{out}})=0$ (32) $\displaystyle
d_{\theta}(r_{\mathrm{in}})$ $\displaystyle=$ $\displaystyle d_{0}\,,\quad
d_{\theta}(r_{\mathrm{out}})=0$ (33)

We solve this boundary value problem numerically, and in Fig. 7 we plot the
solutions for different values of the screening parameters, with $d_{0}=0.05$,
$r=1$, and $r_{\mathrm{out}}=1$ and $r_{\mathrm{in}}=0.1$. Similar to the
rectangular case, at large values of $\kappa_{e}$ and small $\kappa_{o}$ we
find significant deviation from the classical affine response, with non-
monotonous displacement field, with $d_{r}(r)=0$. For non-zero odd screening
$\kappa_{o}\neq 0$ the displacement field rotates and induces a non-zero
radial component $d_{r}$, with spiral stream lines.

Figure 7: Solutions for (1) with boundary conditions (33), with $d_{0}=0.05$,
$r=1$, and $r_{\mathrm{out}}=1$ and $r_{\mathrm{in}}=0.1$. Each block
corresponds to a solution with different values of $\kappa_{e}$ and
$\kappa_{o}$.