# Geometric Theory of Mechanical Screening in two-dimensional solids

Noemie Livne  Amit Schiller  Michael Moshe<EMAIL_ADDRESS>Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem,
Israel 9190

###### Abstract

Holes in mechanical metamaterials, quasi-localized plastic events in amorphous
solids, and bound dislocations in a hexatic matter are different mechanisms of
generic stress relaxation in solids. Regardless of the specific mechanism,
these and other local stress relaxation modes are quadrupolar in nature,
forming the foundation for stress screening in solids, similar to polarization
fields in electrostatic media. We propose a geometric theory for stress
screening in generalized solids based on this observation. The theory includes
a hierarchy of screening modes, each characterized by internal length scales,
and is partially analogous to theories of electrostatic screening such as
dielectrics and Debye-Hückel theory. Additionally, our formalism suggests that
the hexatic phase, traditionally defined by structural properties, can also be
defined by mechanical properties and may exist in amorphous materials.

##  I Introduction

The concept of screening, which refers to the reduction of energy density
through a material’s local responses, is central to many physical systems.
Examples include dielectrics and ionic liquids, in which induced dipolar or
monopolar charge densities respond to the background electric field. As a
result, the effective electric field is modified either quantitatively or
qualitatively [1]. Previous research has successfully applied the concept of
screening to mechanical systems. For instance, the onset of buckling in 2D
defective membranes has been interpreted as the screening of structural
defects by curvature [2]. Additionally, studies have shown that mechanical
stresses in curved self-assembled crystals can be screened through the
nucleation of structural defects [3, 4, 5, 6].

The duality between curvature and defects as entities that screen and are
screened is reflected in the first Föppl–von Kármán equation for the stress
potential $\chi$

$\displaystyle\frac{1}{Y}\Delta\Delta\chi=K_{D}-K_{G}\;.$ (1)

In this equation, the Gaussian curvature of the actual deformed configuration
is represented by $K_{G}$, and singular or distributed defects are represented
by $K_{D}$ [2]. This equation demonstrates that when the curvature $K_{G}$ is
fixed, stresses can be reduced by distributing defects through $K_{D}$, and
vice versa.

Physical phenomena that can be explained by geometric screening include the
shape of virus capsids [7] and defect patterns on curved colloidal crystals
[5, 4]. Another example is the theory of linear and nonlinear screening by
imaginary quadrupoles, which was systematically derived to describe the
emergent mechanics in Kirigami [8] and planar elastic meta-materials
containing arrays of holes [9]. In Fig. 1(a) we demonstrate a state in which
imaginary quadrupoles interact nonlinearly, leading to a spontaneous breaking
of symmetry with an alternate pattern [9].

Previous works on mechanical screening have been largely influenced by an
early discovery of mechanical screening within the statistical theory of 2D
crystalline matter, which led to the concept of two-step melting of a solid
through an intermediate hexatic phase to a liquid state [10, 11]. In this
theory, the three phases are distinguished by their structural properties, and
the transitions from solid to hexatic and hexatic to liquid correspond to a
sequential destruction of translational and rotational quasi-long-range order.

From a mechanical perspective, the low, intermediate, and high temperature
phases form elastic solids supplemented by thermally induced tightly-bounded
dislocation-pairs, tightly bounded disclination-pairs (dislocations), and free
disclinations, respectively. The free element in each phase forms a potential
screening mechanism. In the intermediate hexatic phase, for example,
dislocations can form in pairs and unbind to screen out external loads, and
are the key mechanism behind its vanishing shear modulus and the screened
interactions between disclinations [10, 11, 12, 13]. This is illustrated in
Fig. 1(b) where a bubble-raft model of a 2d crystalline matter shows the
unbinding of dislocations due to external shear.

The ever-growing list of systems that contain screening mechanisms is not
limited to ordered systems. Examples include granular amorphous solids, where
local quadrupolar particle rearrangements are induced in response to external
loads [14] (shown in Fig. 1(c)), epithelial tissue [15, 16], and wrinkles and
crumples in strongly confined thin sheets, where local out-of-plane
deformations are also of quadrupolar nature [17, 18] (shown in Fig. 1(d)).

Figure 1: Mechanical Screening. Top panel -Stress relaxation mechanisms: (a)
Nonlinear quadrupole screening in holey metamaterials, established in [9], (b)
Screening by dipoles via dislocation unbinding in a 2d crystal bubble-raft
model [19, 20], (c) Quadrupolar Eshelby plastic event in a model of amorphous
solid, adapted with permission from [14], (d) Screening by local quadrupolar
wrinkles, adapted with permission from [17]. Bottom panel - diagrammatic
description of the different screening modes. The linear and nonlinear
quadrupole screening theory was established in [9]. Here we focus on linear
dipole and monopole screening theories, extending linear quadrupole screening
in analogy with the extension of dielectrics to Debye-Hückel screening.

Motivated by the wide range of screening mechanisms found in solids, a linear
continuum theory was developed to describe various modes of screening in
elastic materials [21]. Specifically, two distinct screening regimes were
predicted: a quasi-elastic regime and an anomalous one. It was suggested that
a transition between these different screening modes can be achieved, for
example, in granular solid by decreasing the confining pressure.

Indeed, the theory’s predictions, including the emergence of anomalous
mechanics, have been validated through a series of numerical and experimental
studies on the mechanics of granular and glassy materials in both two and
three dimensions [22, 23, 24, 25, 26]. Despite its success in predicting the
mechanics of granular and glassy materials, the theory presented in [21, 22,
23, 24, 25, 26] is derived based on ad hoc assumptions on the general nature
of screening. In addition, we identify three main drawbacks of the theory: (i)
It is written in a specific coordinate system. (ii) It assumes a geometrically
linearized strain measure. (iii) The analytic methods available within the
current displacement-formulation are limited.

In this paper, we derive a hierarchy of screening theories from (geometric)
first principles. We address the limitations of previous theories by
developing a covariant geometric formulation of screened elasticity. Our
theory reveals three distinct screening regimes, controlled by quadrupole,
dipole, and monopole screening mechanisms. Additionally, we develop a
generalized Airy potential theory, in which the governing equations take
different forms in each of the regimes

$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi=\bar{K}^{0}\quad$ Quadrupole
(2)
$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi+\frac{1}{\tilde{Y}}\ell_{P}^{-2}\Delta\chi=\bar{K}^{0}\quad$
Dipole
$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi+\frac{1}{\tilde{Y}}\ell_{M}^{-4}\chi=\bar{K}^{0}\quad$
Monopole

Our study demonstrates that the different screening regimes are characterized
by different length scales, $\ell_{P}$ and $\ell_{M}$, which act as new moduli
that extend classical elasticity. The theories of Dipole and Monopole
screening predict non-affine deformations in response to uniform external
loads and are expected to be relevant to any solid whose mechanics is
controlled by local relaxation mechanisms, such as local rearrangements in
amorphous solids, wrinkles in confined thin sheets, and T1 transitions in
living cellular tissue.

The possible extensions of continuum mechanical screening are summarized in
bottom panel of Fig. 1. In this work we focus on the yellow-colored boxes
representing linear dipole and monopole screenings, in which an unusual or
anomalous mechanical behavior is predicted.

Our theory allows studying new problems that the non-geometric formulation in
[21, 22] could not address. For example, we show that a monopole elastic
charge screened by dipoles is mechanically equivalent to a disclination
screened by dislocaitons in the Hexatic phase. Furthermore, we study how
screened defects interact via the screening field. These and other predictions
are proposed as test measurements for identifying mechanical screening.
Surprisingly, the geometric approach to mechanical screening uncovered an
explicit link between the mechanics of the hexatic phase within the theory of
melting, and the mechanics of screened solids, even in the absence of
underlying order.

The structure of this paper is as follows: We start with introducing an
electrostatic analog in Sec.(II), where we derive electrostatic screening
theories from energy functional minimization, an approach that is more natural
when athermal mechanical systems are considered. In Sec.(III) we develop the
general framework of geometric screening in elastic-like solids. In Sec.(IV)
we derive equilibrium equations for the different screening modes, followed by
the development of generalized screened Airy stress function approach in
Sec.(V). In Sec.(VI) we study the implications of mechanical screening on
basic physical properties such as the Green’s function associated with each
screening mode, and the interactions between sources of stresses in the
presence of screening. In Sec.(VII) we conclude by discussing the future road
map towards a general theory of screening in solids.

##  II The electrostatic Analog

A familiar implementation of screening theory is within electrostatics of
continuous media. As such, we find it instructive to start with the
electrostatic analog and later implement the same ideas, with the necessary
adjustments, to elastic solids. The main idea behind the analogy is the
hierarchical structure of linear and nonlinear electrostatic screening as
summarized in Fig. 2.

The potential energy density stored in the electric field is
$\mathcal{U}=\tfrac{1}{2}\varepsilon_{0}\mathbf{E}^{2}$, and the work done on
the system by assembling a charge density $\rho_{f}$ is
$\mathcal{W}=\rho_{f}\phi$. The mechanical free energy in a domain
$\mathcal{M}$ is therefore

$F=\int_{\mathcal{M}}\left(\mathcal{U}-\mathcal{W}\right)\mathrm{d}S=\int_{\mathcal{M}}\left(\frac{1}{2}\varepsilon_{0}\mathbf{E}^{2}-\rho_{f}\phi\right)\mathrm{d}S$
(3)

with $\mathbf{E}=-\nabla\phi$ the electric field derived from a potential, and
$\varepsilon_{0}$ the vacuum permittivity. If the domain $\mathcal{M}$ is
filled with matter, atoms and molecules may polarize in response to electric
field, creating electric dipoles that modify the electric field. At the
continuum level the dipoles are described by the polarization density
$\mathbf{P}$ [1]. The self interaction energy of a dipole, or the work
required for its nucleation, is material dependent and reflects the
microscopic origin of the charge separation within the atom or the molecule.
To account for this effect we note that the energetic cost is quadratic in the
polarization, and that dipoles interact with each other via the total electric
field, so

$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{0}\mathbf{E}^{2}+\mathbf{E}\cdot\mathbf{P}$
$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2\varepsilon_{0}\chi_{e}}\mathbf{P}^{2}+\rho_{f}\phi\;.$
(4)

Here $\chi_{e}$ is the electric susceptibility, and as before, $\mathcal{U}$
quantifies the energy stored in the electric field and $\mathcal{W}$ the work
done on the system by assembling the monopole and dipole densities $\rho_{f}$
and $\mathbf{P}$. The equilibrium equations are then

$\displaystyle\mathbf{P}$ $\displaystyle=$
$\displaystyle\varepsilon_{0}\chi_{e}\mathbf{E}$
$\displaystyle\nabla\cdot\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{0}}\left(\rho_{f}-\nabla\cdot\mathbf{P}\right)$
(5)

Upon substituting the first relation in the second we get

$\displaystyle\nabla\cdot\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{1}{\varepsilon_{0}(1+\chi_{e})}\rho_{f}=\frac{1}{\varepsilon}\rho_{f}$
(6)

Figure 2: Diagrammatic representation of screening hierarchy in electrostatic
media. The equation for the electric field depends on the induced polarization
$P$ which depends on the electric field via a constitutive relation,
illustrated here for each screening regime.

Thus, we see that the permittivity constant is renormalized by the induced
dipoles. These equations are the basis for linear dielectrics.

An important observation is that the form of $\mathcal{W}$ in Eq.(4) is not
the most general one. Upon assuming that $\mathcal{W}$ is an analytic function
of $\mathbf{P}$ and its derivative, the most general form that preserves the
symmetries to rotations and translations is

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\rho_{f}\phi+\frac{1}{2}\alpha_{2}\mathbf{P}^{2}+\frac{1}{24}\alpha_{4}\mathbf{P}^{4}+\dots$
$\displaystyle+$
$\displaystyle\frac{1}{2}\beta_{2}(\nabla\cdot\mathbf{P})^{2}+\frac{1}{24}\beta_{4}(\nabla\cdot\mathbf{P})^{4}+\dots$
$\displaystyle+$
$\displaystyle\frac{1}{2}\gamma_{2}(\nabla\times\mathbf{P})^{2}+\frac{1}{24}\gamma_{4}(\nabla\times\mathbf{P})^{4}+\dots$

Within a linear theory, only three terms contribute, with nonzero
$\alpha_{2},\beta_{2},\gamma_{2}$, and perhaps additional terms in quadratic
higher order derivatives. However, from a physical perspective, the
interpretation of $\mathbf{P}$ as a polarization field , together with the
multipole expansion

$\displaystyle\rho=\rho_{f}+\nabla\cdot\mathbf{P}+\nabla\nabla
Q+\dots\noindent\color[rgb]{0,1,1},\normalcolor$ (7)

imply that $\nabla\times\mathbf{P}$ does not contribute to the charge
distribution. Hence, in electrostatic systems we expect $\gamma_{2}=0$. For
the same reason, higher order derivatives of $\mathbf{P}$ are irrelevant,
leaving the general form

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\rho_{f}\phi+\frac{1}{2}\alpha_{2}\mathbf{P}^{2}+\frac{1}{2}\beta_{2}(\nabla\cdot\mathbf{P})^{2}.\
$ (8)

The term proportional to $(\nabla\cdot\mathbf{P})^{2}$ represents the
nucleation cost associated with effective monopoles, created by non-uniformly
distributed dipoles. The two coefficients correspond to an inherent length
scale $\ell\equiv\sqrt{\frac{\beta_{2}}{\alpha_{2}}}$. When compared with
system size, the dielectric state corresponds to $\ell\ll L$. In the other
limit, $L\ll\ell$, the term $\mathbf{P}^{2}$ is negligible, and Eq.(4) takes
the form

$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{0}\mathbf{E}^{2}+\mathbf{E}\cdot\mathbf{P}$
$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell_{0}^{2}}(\nabla\cdot\mathbf{P})^{2}+\rho_{f}\phi\;.$
(9)

Upon minimizing
$F=\int_{\mathcal{M}}\left(\mathcal{U}-\mathcal{W}\right)\mathrm{d}S$ the
equilibrium equations are

$\displaystyle\nabla(\nabla\cdot\mathbf{P})$ $\displaystyle=$
$\displaystyle-\varepsilon_{0}\ell_{0}^{-2}\mathbf{E}$
$\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon_{0}}\left(\rho_{f}-\nabla\cdot\mathbf{P}\right)\;.$
(10)

The first equation can be written as

$\displaystyle\nabla(\nabla\cdot\mathbf{P}-\varepsilon_{0}\ell_{0}^{-2}\phi)=0$
(11)

implying that the expression in brackets is constant which can be set to zero
using the potential gauge freedom

$\displaystyle\nabla\cdot\mathbf{P}-\varepsilon_{0}\ell_{0}^{-2}\phi=0\;.$
(12)

Since the gauge is fixed, from this point onward we should no longer expect
the equations to be invariant under gauge transformations. Upon substituting
Eq.(12) in Eq.(10) we find

$\displaystyle\Delta\phi-\ell_{0}^{-2}\phi=-\frac{1}{\varepsilon_{0}}\rho_{f}$
(13)

This is the Helmholtz equation from the Debye-Hückel theory, describing
screening by mobile monopole charges in an ionic liquid. We emphasize that in
both dipole and monopole screenings, the fundamental fields with respect to
which the energy is minimized are the electric potential and the polarization
field. In the monopole screening case, the variation with respect to the
polarization enforces the conservation of total charge. Eq.(13) is
traditionally derived from the Poisson-Boltzman equation using a detailed
microscopic theory, which gives an explicit expression for the Debye-screening
length $\ell_{0}$ in terms of temperature, ionic strength etc.

Our minimization approach avoids the microscopic statistical picture, and thus
provide no details on the parameter $\ell_{0}$. Despite this weakness, such an
approach is advantageous in this work, since the systems we are interested in
are mostly athermal and disordered.

##  III Pure and Screened Elasticity

One challenge in writing a screening theory for solids is the identification
of the basic screening element, which arises naturally from a geometric
approach to elasticity [27]. In this formulation the reference state of a
solid $\mathcal{M}$ is defined by the rest distances between material
elements, and quantified by the reference metric $\bar{g}^{0}$ via
$\mathrm{d}l_{0}^{2}=\bar{g}^{0}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$.
A configuration is described by the metric $g$, quantifying the actual
(potentially deformed) distances between material elements given by
$\mathrm{d}l^{2}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$. Contrary to
the reference metric, the actual one is induced from an embedding
$\phi:\mathcal{M}\to\mathbb{R}^{2}$ describing the material configuration with
$g=\nabla\phi^{T}\nabla\phi$. The strain is defined as the deviation of $g$
from its rest state $u=\tfrac{1}{2}(g-\bar{g}^{0})$. A key property in this
formulation is the curvature associated with the reference metric. A stress
free configuration is available if the reference Gaussian curvatures
$\bar{K}^{0}$ associated with $\bar{g}^{0}$, vanishes. Therefore $\bar{K}^{0}$
is a measure of geometric incompatibility, and consequently for sources of
residual stresses. Singular sources of stresses are described by singular
$\bar{K}^{0}$, exhibiting a natural multipolar hierarchy, as shown in Table 1.

Type | $\bar{K}$ | Realization
---|---|---
Monopole | $m\,\delta(\mathbf{x})$ | Disclination
Dipole | $\mathbf{p}\cdot\nabla\delta(\mathbf{x})$ | Dislocation
Quadrupole | $(\nabla^{T}\cdot\mathbf{q}\cdot\nabla)\delta(\mathbf{x})$ | Dislocation-pair, Interstitial
Table 1: Reference curvatures multipoles and possible realizations.

In a continuum limit, the reference curvature describes distributed multipoles

$\displaystyle\bar{K}^{0}=M(\mathbf{x})+\bar{\nabla}_{\alpha}P^{\alpha}(\mathbf{x})+\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}(\mathbf{x})+\dots$
(14)

with $M$, $P$ and $Q$ distributions of disclinations, dislocations, and
quadrupoles [28]. Singular multipoles are materialized via anelastic
deformations which modify the reference metric. The simplest anelastic
deformation is a local change in the reference state,

$\bar{g}_{\alpha\beta}=\bar{g}^{0}_{\alpha\beta}+\delta^{(n)}(\mathbf{x})\,q_{\alpha\beta}\;.$
(15)

The trace of $q$ corresponds to an area change, and the trace-less symmetric
part corresponds to local shear. This type of metric deformation describes a
wide variety of screening mechanism, as illustrated in Fig. 1. For small
anelastic deformations the leading order of the reference curvature associated
with $\bar{g}$ is

$\bar{K}=\bar{K}^{0}+{Q}^{\alpha\beta}\bar{\nabla}_{\alpha\beta}\delta(\mathbf{x})$
(16)

with
${Q}^{\alpha\beta}=\bar{\varepsilon}^{\alpha\mu}\bar{\varepsilon}^{\beta\nu}q_{\mu\nu}$
and $\bar{\varepsilon}$ are the Levi-Civita tensors with respect to
$\bar{g}^{0}$ [29]. In light of the multipole expansion in Eq.(14) we find
that a local material rearrangement induces a localized quadrupolar elastic
charge.

This reflects a deeper property of elastic charges: In [27] it was proved that
the lowest order elastic multipole that can be nucleated by a local material
deformation is quadrupolar. The proof relies on global geometric properties
which are impossible to change via local deformations. This geometric
conservation law makes the elastic quadrupoles analogous to electric dipoles,
which are the lowest order electric charges that can be nucleated locally
without violating conservation of charge. The inevitable conclusion is that
the quadrupolar field $Q^{\alpha\beta}(\mathbf{x})$ is, in principle, the
natural screening field in solids. Motivated by these observations we turn to
derive a screening theory of elastic-like solids by accounting for induced
quadrupoles and their nucleation cost. For that we briefly review the
geometric approach to elasticity and the possible screening modes.

###  III.1 Elasticity

For a purely elastic material the reference metric $\bar{g}^{0}$ is fixed, and
does not change in response to external loads. The elastic strain is then

$u^{\mathrm{el}}=\frac{1}{2}\left(g-\bar{g}^{0}\right)\noindent\color[rgb]{0,1,1}.\normalcolor$
(17)

The equilibrium equations is derived from a mechanical free energy

$F=\int_{\mathcal{M}}\left(\mathcal{U}-\mathcal{W}\right)\,\mathrm{d}S_{\bar{g}^{0}}-\int_{\partial\mathcal{M}}\mathcal{W}_{B}\,\mathrm{d}l_{\bar{g}^{0}}\;,$
(18)

where $\mathcal{U}$ is the elastic energy density, while $\mathcal{W}$ and
$\mathcal{W}_{B}$ encode the work density done on the system, e.g. by external
forces acting either in the bulk or on the boundary, respectively. Upon
assuming small strains, the elastic energy is Hookean

$\mathcal{U}=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\beta\gamma}\;.$
(19)

In the absence of body forces and in the presence of traction forces the work
densities are

$\displaystyle\mathcal{W}$ $\displaystyle=$ $\displaystyle 0$ (20)
$\displaystyle\mathcal{W}_{B}$ $\displaystyle=$
$\displaystyle\mathbf{t}\cdot\mathbf{d}\;.$ (21)

Here $\mathbf{d}$ is the displacement field defined relative to the ground-
state, $\mathbf{t}$ are the imposed traction forces, and $\mathcal{A}$ is the
elastic tensor encoding material properties. In a homogeneous and isotropic
material

$\mathcal{A}^{\alpha\beta\gamma\delta}=\frac{\,\nu\,Y}{1-\nu^{2}}\left(\bar{g}^{\alpha\beta}\bar{g}^{\gamma\delta}+\frac{1-\nu}{2\nu}(\bar{g}^{\alpha\gamma}\bar{g}^{\beta\delta}+\bar{g}^{\alpha\delta}\bar{g}^{\beta\gamma})\right)\,,$
(22)

with $Y$ the Young’s modulus and $\nu$ Poissons’ ratio. The stress tensor is
defined by the variation of energy density with respect to the elastic strain,
leading to Hooke’s law

$\sigma^{\alpha\beta}=\mathcal{A}^{\alpha\beta\gamma\delta}u_{\gamma\delta}\noindent\color[rgb]{0,1,1}.\normalcolor$
(23)

Upon minimizing Eq.(28) with respect to the embedding $\phi$ we obtain the
equilibrium equation $\mathrm{div}\sigma=0$, which takes the explicit form

$\bar{\nabla}_{\mu}\sigma^{\mu\nu}+\left(\Gamma^{\nu}_{\alpha\beta}-\bar{\Gamma}^{\nu}_{\alpha\beta}\right)\sigma^{\alpha\beta}=0,$
(24)

along with the boundary conditions

$n_{\alpha}\sigma^{\alpha\beta}=t^{\beta}\,.$ (25)

This form of the equilibrium equation accounts for geometric nonlinearities
and was first introduced in [30], and is given in App. A. A systematic method
for solving it nonlinearly in the case of non-euclidean reference metric was
introduced in [31].

###  III.2 Screened Elasticity

When strain relaxation mechanisms are available, the reference metric is no
longer fixed, but can evolve in response to deformations. We therefore
distinguish between the (fixed) initial reference metric $\bar{g}^{0}$, and
the temporary reference metric relative to which elastic deformations are
measured

$\bar{g}=\bar{g}^{0}+q\;.$ (26)

Here $q$ is the density of quadrupole perturbation to the reference metric
$\bar{g}^{0}$. Correspondingly, the elastic tensor $\mathcal{A}$, covariant
derivatives $\bar{\nabla}$, and the raising and lowering of indices are all
defined with the fixed reference metric $\bar{g}^{0}$. The elastic strain is
the deviation of the current metric from the updated reference metric

$\displaystyle
u^{\mathrm{el}}=\frac{1}{2}\left(g-\bar{g}\right)=\frac{1}{2}\left(g-\bar{g}^{0}-q\right)=u-\frac{1}{2}q\;,$
(27)

where $u=\tfrac{1}{2}(g-\bar{g}^{0})$ is the total strain, measuring the
deformation relative to the initial configuration. The screened elastic energy
stored in the system still has the form Eq.(19),

$F_{\mathrm{Sc}}=\int_{\mathcal{M}}\left(\mathcal{U}-\mathcal{W}\right)\,\mathrm{d}S_{\bar{g}^{0}}-\int_{\partial\mathcal{M}}\mathcal{W}_{B}\,\mathrm{d}l_{\bar{g}^{0}}\;,$
(28)

with

$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\beta\gamma}=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\beta\gamma}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u_{\alpha\beta}q_{\beta\gamma}+\frac{1}{8}\mathcal{A}^{\alpha\beta\gamma\delta}q_{\alpha\beta}q_{\beta\gamma}\;.$

This form of the energy uncovers the elastic interactions between the induced
quadrupoles: the first term in the second row represents the elastic
interaction between the quadrupole $q$ at point $\mathbf{x}$ with the
background stress and all the other quadrupoles, and the last term represents
the self-interaction elastic energy corresponding to the energy stored in the
elastic field induced by a single quadrupole. Another important contribution
to the self-interaction term is the work done on the system in order to
nucleate the quadrupole core. This material dependent property is therefore
contributing to the work term in Eq.(28)

$\mathcal{W}=\mathcal{W}[q]\;.$ (30)

Here $\mathcal{W}$ is a functional whose specific form depends on the
underlying screening mechanism and material properties.

At this point we draw inspiration from the electrostatic analogue,
specifically from Eq.(8) which builds on the multipole expansion, and write
the general form of $\mathcal{W}$ reflecting screening by quadrupoles,
dipoles, and monopoles

$\displaystyle\mathcal{W}=\frac{1}{2}\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}Q^{\alpha\beta}Q^{\gamma\delta}+\frac{1}{2}\Lambda^{\mathrm{P}}_{\alpha\beta}P^{\alpha}P^{\beta}+\frac{1}{2}\Lambda^{\mathrm{M}}M^{2}\;,$
(31)

where

$\displaystyle
Q^{\alpha\beta}=\bar{\varepsilon}^{\alpha\mu}\bar{\varepsilon}^{\beta\nu}q_{\mu\nu}\;,\,P^{\alpha}=\bar{\nabla}_{\mu}Q^{\alpha\mu}\;,\,M=\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}\;.$
(32)

From homogeneity, isotropy, and the dimensions of $\mathcal{W}$ we find

$\displaystyle\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\lambda_{Q}\bar{g}^{0}_{\alpha\beta}\bar{g}^{0}_{\gamma\delta}+\mu_{Q}\left(\bar{g}^{0}_{\alpha\gamma}\bar{g}^{0}_{\beta\delta}+\bar{g}^{0}_{\alpha\delta}\bar{g}^{0}_{\beta\gamma}\right)$
$\displaystyle\Lambda^{\mathrm{P}}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}Y\ell_{P}^{2}\bar{g}^{0}_{\alpha\beta}\;,$
$\displaystyle\Lambda^{\mathrm{M}}$ $\displaystyle=$ $\displaystyle
Y\ell_{M}^{4}\;,$ (33)

with $Y$ the Youngs modulus and $\ell_{\mathrm{P}},\ell_{\mathrm{M}}$ the
typical length scales associated with each screening multipole. The quadrupole
term in Eq.(31) represents the nucleation cost of a quadrupole field
describing a distribution of local metric perturbations to $\bar{g}^{0}$. In
this case the anelastic response of the material is quantified by the value of
$Q$, describing the average uniform Eshelby-like deformation. This is similar
to the weak screening by dislocation pairs (quadrupoles) in the solid phase of
2d crystalline materials. The second term in Eq.(31) describes the effective
nucleation cost for dipoles that emerge from non uniform distribution of
quadrupoles. In this case the anelastic response of the material is quantified
by the spatial variation of $Q$ encoded in its divergence, and is similar to
screening by dislocations (dipoles) in the hexatic phase of 2d crystalline
materials. The last term in Eq.(31) describes the effective nucleation cost
for monopoles, which is analogous to screening by disclinations (monopoles) in
a melted 2d crystalline.

The geometric realization of screening quadrupole and dipole is visualized in
Fig. 3 where the semi-transparent and opaque configurations describe the rest
states before and after the anelastic deformations, on a finite region. These
anelastic deformations are derived by calculating the displacement field
induced from uniform distribution of each multipole: The deformation induced
by a uniform $Q$ corresponds to a uniform strain and is visualized in Fig.
3(a). To interpret the dipole term we take a spatially varying quadrupole with
uniform dipole $\mathbf{P}=P_{0}\hat{y}$. The induced deformation is
visualized in Fig. 3(b), indicating a non-Eshelby deformation that is of lower
order in the multipole expansion. This is analogous to creating electric
monopole from nonuniform dipole field. As for the monopole term in Eq.(31),
this screening mechanism induces non-zero curvature, thus cannot be visualized
via a planar deformation.

Figure 3: Anelastic deformations induced by (a) a uniform quadrupole and (b) a
uniform dipole on a finite region. The deformed states are superimposed on the
(semi-transparent) undeformed configuration.

According to Eq.(31) and Eq.(33), in principle all three screening mechanisms
can act simultaneously. However, elastic materials corresponds to large
$\ell_{P}$ and $\ell_{M}$, suppressing nucleation of dipoles and monopoles.
When $\lambda_{\mathrm{Q}},\mu_{\mathrm{Q}}\to 0$ the nucleation cost of
dipoles (the scale $\ell_{P}$) may become finite, and when $\ell_{P}\to 0$,
the cost of monopoles ($\ell_{M}$) may become finite as well. This hierarchy
of screening is based on scale separation of $\ell_{P},\ell_{M}$ and is in
line with the scale-separation discussed after Eq.(8) in the electrostatic
analog. It is also analogous to the hierarchy of Solid-Hexatic-Liquid phases,
where dipole and monopole screenings correspond to the unbinding of
dislocations (dipoles) and disclinations (monopoles) with finite nucleation
energy in the hexatic and liquid phases, respectively [11, 3].The mapping
between the theories is discussed in Sec. VII.

In light of this argument, in what follows we study the mechanics of the three
screening modes separately, and we assume three distinct situations in which
each of the terms in Eq.(31) dominates.

##  IV Equilibrium equations

Here we derive equilibrium equations for each of the quadrupole, dipole, and
monopole screening regimes. The detailed derivation is given in App. A.

The equilibrium equations are derived using the variation of an energy with
respect to the embedding $\phi$ describing the configuration, and the induced
quadrupole field $q$. Since $\mathcal{W}$ is independent of the configuration,
the variation with respect to $\phi$ is the same in the different screening
regimes. Explicitly, the mechanical free energy to be minimized is

$\displaystyle F$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\left(\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\gamma\delta}-\mathcal{W}[q]\right)\,\mathrm{d}S_{\bar{g}^{0}}$
(34) $\displaystyle-$
$\displaystyle\int_{\partial\mathcal{M}}\mathbf{t}\cdot\mathbf{d}\,\mathrm{d}l_{\bar{g}^{0}}\;.$

Upon defining the elastic stress

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}=\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\gamma\delta}=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}\left(g_{\gamma\delta}-g^{0}_{\gamma\delta}-q_{\gamma\delta}\right)$
(35)

we find the equilibrium equation

$\displaystyle\bar{\nabla}_{\mu}\sigma_{\mathrm{el}}^{\mu\nu}+\left(\Gamma^{\nu}_{\alpha\beta}-\bar{\Gamma}^{\nu}_{\alpha\beta}\right)\sigma_{\mathrm{el}}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle 0$ (36)

along with the boundary conditions

$n_{\alpha}\sigma_{\mathrm{el}}^{\alpha\beta}=t^{\beta}\,$ (37)

justifying our definition of the elastic stress tensor. We emphasize that from
the solutions for the stress $\sigma_{\mathrm{el}}$ and the induced charges
$q$ we can recover the actual metric through

$g_{\alpha\beta}=\bar{g}^{0}_{\alpha\beta}+q_{\alpha\beta}+2\mathcal{A}_{\alpha\beta\gamma\delta}\sigma_{\mathrm{el}}^{\gamma\delta}\;.$
(38)

Here the notation $\mathcal{A}_{\alpha\beta\gamma\delta}$ is the inverse
elastic tensor.

To recover the actual metric and configuration in equilibrium, Eq.(36) should
be supplemented with an equation for the induced screening charges, obtained
by varying the energy Eq.(34) with respect to $q$

$\displaystyle\delta_{q}F$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\left(-\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}\delta
q_{\alpha\beta}-\delta_{q}\mathcal{W}\right)\,\mathrm{d}S_{\bar{g}^{0}}\;.$
(39)

Next we perform the variation of $\mathcal{W}$, which is shown to strongly
depends on the specific screening regime.

Quadrupole screening: In this case

$\begin{split}\mathcal{W}&=\frac{1}{2}\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}Q^{\alpha\beta}Q^{\gamma\delta}=\frac{1}{2}\Lambda_{\mathrm{q}}^{\alpha\beta\gamma\delta}q_{\alpha\beta}q_{\gamma\delta}\;,\end{split}$
(40)

with $\Lambda_{\mathrm{q}}$ proportional to $\Lambda^{\mathrm{Q}}$ (see App.
C). Upon varying the the total energy with respect to $q$ we find a linear
relation between the induced quadrupole and the elastic stress

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}+2{\bar{\varepsilon}}^{\alpha\mu}{\bar{\varepsilon}}^{\beta\nu}{\Lambda}^{\mathrm{Q}}_{\mu\nu\gamma\delta}Q^{\gamma\delta}$
$\displaystyle=$ $\displaystyle 0\;.$ (41)

In analogy to models for dielectric media, such as the Maxwell-Garnett model
[32, 33], this screening regime describes a material containing a dilute
distribution of quadrupoles induced in response to external loads.

At this point we can integrate out the quadrupolar degree of freedom by
substituting $q$ either in the constitutive relation Eq. 35 or the energy in
Eq. 34. In both cases we end up with an effective elastic energy $F_{Q}$ that
only depends on the total strain

$\displaystyle F_{Q}$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\frac{1}{2}\tilde{\mathcal{A}}^{\alpha\beta\gamma\delta}u_{\alpha\beta}u_{\gamma\delta}\,\mathrm{d}S_{\bar{g}^{0}}-\int_{\partial\mathcal{M}}\mathbf{t}\cdot\mathbf{d}\,\mathrm{d}l_{\bar{g}^{0}}\;,$
(42)

where $\tilde{\mathcal{A}}$ is an effective elastic tensor given explicitly in
App. C, encoding the mechanical effect of the induced quadrupoles, leading to
a quasi-elastic theory.

This result is also similar to dielectrics, where screening by electric
dipoles re-scales the dielectric constants without otherwise modifying the
theory.

Dipole screening: In this case

$\mathcal{W}=\frac{1}{2}\Lambda^{\mathrm{P}}_{\alpha\beta}P^{\alpha}P^{\beta}=\frac{1}{2}\Lambda^{\mathrm{P}}_{\alpha\beta}(\bar{\nabla}_{\mu}Q^{\alpha\mu})(\bar{\nabla}_{\nu}Q^{\beta\nu})\;.$
(43)

Upon substituting the relation between $Q$ and $q$, and varying $\mathcal{W}$
with respect to $q$ we find

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}+\tfrac{1}{2}Y\ell_{P}^{2}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(\bar{\nabla}_{\mu}P_{\nu}+\bar{\nabla}_{\nu}P_{\mu}\right)=0$
(44)

along with the boundary condition

$\displaystyle{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(n_{\mu}P_{\nu}+n_{\nu}P_{\mu}\right)=0\;.$
(45)

Contrary to the quadrupole screening regime where a linear relation between
stress and induced quadrupoles holds, here the stress is linearly proportional
to the second gradient of the induced quadrupole field. An immediate
consequence is the relation between elastic pressure and the induced isotropic
quadrupole

$\displaystyle\mathrm{Tr}\,\sigma_{\mathrm{el}}=\bar{g}_{\alpha\beta}\sigma_{\mathrm{el}}^{\alpha\beta}=-Y\ell_{P}^{2}\bar{\nabla}_{\mu\nu}Q^{\mu\nu}$
(46)

This situation is similar to its electrostatic analog, wherein a dielectric
the induced dipoles are linearly proportional to the electric field, whereas
in Debye-Hückel theory the electric field is proportional to the second
gradient of the induced dipoles, as in Eq.(10).

Monopole screening: In this case

$\mathcal{W}=\frac{1}{2}\Lambda^{\mathrm{M}}M^{2}=\frac{1}{2}\Lambda^{\mathrm{M}}(\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})(\bar{\nabla}_{\gamma\delta}Q^{\gamma\delta})$
(47)

and from the variation of $\mathcal{W}$ we find

$\displaystyle\sigma_{\mathrm{el}}^{\rho\sigma}+Y\ell_{M}^{4}\varepsilon^{\gamma\rho}\varepsilon^{\delta\sigma}(\bar{\nabla}_{\gamma\delta}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})=0\;\noindent\color[rgb]{0,1,1}\sout{.}\normalcolor$
(48)

with the boundary condition

$\displaystyle{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(n_{\mu}\bar{\nabla}_{\nu}M+n_{\nu}\bar{\nabla}_{\mu}M\right)=0\;.$
(49)

As in the dipole screening regime, here too we will find that the pressure,
that is the trace of stress, is useful when integrating out the quadrupolar
degree of freedom, and it takes the form

$\displaystyle\mathrm{Tr}\,\sigma_{\mathrm{el}}=-\Lambda^{\mathrm{M}}(\bar{\Delta}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})\;.$
(50)

In summary, the equilibrium equations for each screening mode are

Equation Mode (51)
$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}=-{\bar{\varepsilon}}^{\alpha\mu}{\bar{\varepsilon}}^{\beta\nu}{\Lambda}^{\mathrm{Q}}_{\mu\nu\gamma\delta}Q^{\gamma\delta}\quad$
Quadrupole
$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}=-\tfrac{1}{2}Y\ell_{P}^{2}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(\bar{\nabla}_{\mu}P_{\nu}+\bar{\nabla}_{\nu}P_{\mu}\right)\quad$
Dipole
$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}=-Y\ell_{M}^{4}\bar{\varepsilon}^{\gamma\alpha}\bar{\varepsilon}^{\delta\beta}(\bar{\nabla}_{\gamma\delta}\bar{\nabla}_{\mu\nu}Q^{\mu\nu})\quad$
Monopole

##  V Potential Theory

To solve the equilibrium equations for the stress and the induced charges, we
develop a potential theory generalizing the Airy stress function approach. In
this approach a representation of the stress solving Eq.(36) is given in terms
of a scalar function

$\sigma_{\mathrm{el}}^{\mu\nu}=\frac{1}{\sqrt{|\bar{g}|}}\frac{1}{\sqrt{|g|}}{\varepsilon}^{\mu\alpha}\varepsilon^{\nu\beta}\nabla^{g}_{\alpha\beta}\chi$
(52)

A geometric compatibility condition is needed to determine the stress function
$\chi$, that is requiring the Gaussian curvature of the actual metric $g$ to
vanish. From the definition of stress and strain we get an expression for the
actual metric

$\displaystyle
g_{\alpha\beta}=\bar{g}_{\alpha\beta}^{0}+\varepsilon_{\alpha\mu}\varepsilon_{\beta\nu}Q^{\mu\nu}+2\mathcal{A}_{\alpha\beta\gamma\delta}\sigma_{\mathrm{el}}^{\gamma\delta}{,}$
(53)

which is implicit due to the complicated dependence of $\sigma_{\mathrm{el}}$
on $g$.

To calculate the curvature of $g$ and enforce the geometric compatibility
condition we now assume that both the elastic and the total strains are small,
that is $g\approx\bar{g}\approx\bar{g}^{0}$. Within this approximation a
perturbative expansion for the stress potential is applicable. The leading
order term of the curvature takes the form

$0=\bar{K}^{0}+\nabla_{\alpha\beta}Q^{\alpha\beta}-\frac{1}{Y}\Delta\Delta\chi$
(54)

The term $\nabla_{\alpha\beta}Q^{\alpha\beta}$ represent the induced effective
monopoles, which depends on the specific screening regime. To close the
equation, and integrate out the quadrupolar degrees of freedom, we determined
the induced effective monopoles by substituting Eq.(52) in Eq. 51. We find
(see App. D for details)

$\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}=-\frac{1}{Y}\times\left\\{\begin{array}[]{ll}0&\text{Quadrupole}\\\
\ell_{P}^{-2}\bar{\Delta}\chi&\text{Dipole}\\\
\ell_{M}^{-4}\chi&\text{Monopole}\end{array}\right.$ (55)

In the third equation, corresponding to the monopole screening regime, the
induced monopole is determined up to an arbitrary function satisfying
$\bar{\nabla}_{\alpha\beta}\chi_{g}=0$, and we choose a gauge with
$\chi_{g}=0$.

Having found the explicit expression of
$\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}$ in each screening mode, equation
Eq.(54) is now closed

Screened Stress Function Mode (56)
$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi=\bar{K}^{0}\quad$ Quadrupole
$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi+\frac{1}{\tilde{Y}}\ell_{P}^{-2}\Delta\chi=\bar{K}^{0}\quad$
Dipole
$\displaystyle\frac{1}{\tilde{Y}}\Delta\Delta\chi+\frac{1}{\tilde{Y}}\ell_{M}^{-4}\chi=\bar{K}^{0}\quad$
Monopole

These equations derived based on the assumption of scale separation, discussed
in the introduction. Within this assumption we can combine them into one
equations that holds when screening is dominated by either quadrupole, dipole,
or monopole charges

$\Delta\Delta\chi+\ell_{P}^{-2}\Delta\chi+\ell_{M}^{-4}\chi=Y\bar{K}^{0}\;.$
(57)

Once the equation for $\chi$ is solved the stress tensor can be calculated and
boundary conditions enforced to uniquely determine $\chi$. However, to recover
the displacement field it is required to calculate the actual metric of the
embedding, and therefore the induced quadrupoles. For that the solution for
the elastic stress is substituted in Eq.(51) which then should be solved for
the induced quadrupoles, subjected to the boundary conditions (Eq.(45) in the
dipole regime and Eq.(49) in the monopole regime).

At this point we identify an explicit link with the theory of melting in 2d
crystals. It was recently shown that the theory of defects-mediated melting is
dual to a sine-Gordon like hamiltonian [12, 13]. Upon deriving the equilibrium
equations from the proposed hamiltonian the equation in Eq.(57) are recovered.
This observation suggests that the dipole screening regime developed in this
work forms a mechanical realization of the hexatic phase, that is
traditionally associated with structural properties.

A comment on gauge freedom is necessary at this point: One may suspect that
the explicit dependence of Eq.(57) on the value of the stress function $\chi$
violates the gauge freedom of the stress tensor. However, this only reflects
the gauge choice made when solving for the induced effective monopole in the
monopole screening regime Eq.(55). This is similar to loss of gauge freedom in
Debye-Hückel theory as in Eq.(13).

##  VI Applications

The hierarchical form of Eq.(31) suggests that solids with quadrupolar
relaxation mechanism are prone to dipole screening. This hypothesis, if true,
unifies a variety of systems that are fundamentally different from each other,
under the same screening theory. For example, cellular epithelial tissue
respond to mechanical loads by cell rearrangements [34, 15] and shape
changes[35, 36], both quadrupolar in nature. Holes in perforated (“holey”)
mechanical meta-materials release stresses by forming imaginary quadrupoles
[37, 9]. Non-uniform hole sizes, as in disordered metamaterials, will induce
spatially varying quadrupoles, potentially leading to dipole screening. Last
but not the least, screening can form in wrinkled and crumpled thin sheets.
The system shown in Fig. 1(d) demonstrate the quadrupolar nature of local
wrinkles, that can merge to form long wrinkles, as observed in other scenarios
such as [38]. If a wrinkle ends at the bulk it leaves a free dipole,
supporting the possibility of dipole screening. We therefore expect our theory
to form an effective 2d description of certain wrinkled systems, holey
metamaterials, glasses, tissue models and granular matter.

In the next subsections we study the mechanical implications of dipole and
monopole screening on prototypical mechanical scenarios such as the fields
induced by sources of stresses (defects), and the interactions between them.

###  VI.1 Screened Green’s function

A prominent manifestation of screening is the modified form of the potential
associated with a point monopole charge. This potential is of importance for
two main reasons: (i) Its functional form characterizes the nature and effect
of screening, and (ii) It forms a Green’s function for the non-homogeneous
equation Eq.(56).

Figure 4: Green’s functions associated with the inhomogeneous screened
equations Eq.(56) plotted on a semi-log scale. The blue, yellow and green
curves represent the stress function associated with a monopole screened by
quadrupoles, dipoles, and monopoles.

Monopolar elastic charges can be created by removal or insertion of an angular
section. In hexagonal crystalline structures they form 5- or 7-fold
disclinations. A metric description of defects generalizes the concept of
structural defects to solids with no underlying order, e.g. amorphous solids
[39, 27]. In analogy with the screened fundamental-solution in Debye-Hückel
theory, known as Yukawa potential, we solve Eq.(56) in each screening regime
for a monopolar source term $\bar{K}^{0}=\delta(\mathbf{x})$.

To solve the equations it is useful to define a Helmholtz operator

$\mathcal{H}^{\theta}_{\ell}=\Delta+e^{\mathrm{i}\theta}\ell^{-2}\;,$ (58)

with which Eq.(56) reads

$\displaystyle\frac{1}{\tilde{Y}}\mathcal{H}^{0}_{0}\mathcal{H}^{0}_{0}\chi=\bar{K}^{0}\quad$
Quadrupole (59)
$\displaystyle\frac{1}{\tilde{Y}}\mathcal{H}^{0}_{\ell_{P}}\mathcal{H}^{0}_{0}\chi=\bar{K}^{0}\quad$
Dipole
$\displaystyle\frac{1}{\tilde{Y}}\mathcal{H}^{\pi/4}_{\ell_{M}}\mathcal{H}^{-\pi/4}_{\ell_{M}}\chi=\bar{K}^{0}\quad$
Monopole

An important property of $\mathcal{H}$ is that the kernels of two different
operators are disjoint. Therefore the homogeneous equation in the case of
dipole and monopole screenings reduce to pairs of second order equations.

In the case of quadrupole screening the Green’s function $G^{QS}$ coincides
with the classical solution of a single disclination. To find the solution in
the case of dipole screening we write the general polar symmetric solutions of
the two equation $\mathcal{H}^{0}_{0}\chi_{D}=0$ and
$\mathcal{H}^{0}_{\ell_{P}}\chi_{D}=0$, hence

$\displaystyle\chi^{DS}(r)=c_{1}\log(r/\ell_{P})+c_{2}J_{0}(r/\ell_{P})+c_{3}Y_{0}(r/\ell_{P})+c_{4}\;,$
(60)

Similarly, the solution in the case of monopole screening is found by solving
$\mathcal{H}^{-\pi/4}_{\ell_{M}}\chi_{M}=0$ and
$\mathcal{H}^{\pi/4}_{\ell_{M}}\chi_{M}=0$, and reads

$\displaystyle\chi^{MS}(r)$ $\displaystyle=$ $\displaystyle
d_{1}J_{0}\left(e^{\tfrac{\pi\mathrm{i}}{4}}\,{r}/{\ell_{M}}\right)+d_{2}J_{0}\left(e^{\tfrac{3\pi\mathrm{i}}{4}}\,{r}/{\ell_{M}}\right)$
$\displaystyle+$ $\displaystyle
d_{3}Y_{0}\left(e^{\tfrac{\pi\mathrm{i}}{4}}\,{r}/{\ell_{M}}\right)+d_{4}Y_{0}\left(e^{\tfrac{3\pi\mathrm{i}}{4}}\,{r}/{\ell_{M}}\right)$

The coefficients $c_{i}$ and $d_{i}$ are determined by boundary conditions,
and by a topological condition obtained by integrating both sides of Eq.(56)
with $\bar{K}=\delta(\mathbf{x})$ over the area. In the case of monopole
screening we also set the value of stress-function at infinity, reflecting the
gauge choice made in Eq.(55). The case of traction-free boundary conditions in
a finite systems is detailed in App. F. Green’s function is obtained by
solving the problem in an infinite system with vanishing stress at infinity.
The solutions for the three screening regimes are plotted in Fig. 4 and are
given by

$\begin{split}G^{\text{QS}}(\mathbf{x},\mathbf{x}^{\prime})&=\frac{Y\,|\mathbf{x}-\mathbf{x}^{\prime}|^{2}}{8\pi}\log\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{\ell_{P}}\;,\\\
G^{\text{DS}}(\mathbf{x},\mathbf{x}^{\prime})&=\frac{Y\,\ell_{P}^{2}}{2\pi}\log\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{\ell_{P}}\;,\\\
G^{\text{MS}}(\mathbf{x},\mathbf{x}^{\prime})&=\\\
\frac{Y\ell_{M}^{2}}{8}&\left[H_{0}\left({e^{\frac{i\pi}{4}}}\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{\ell_{M}}\right)-H_{0}\left({e^{-\frac{i\pi}{4}}}\frac{|\mathbf{x}-\mathbf{x}^{\prime}|}{\ell_{M}}\right)\right]\end{split}$
(61)

with $H_{0}$ the Hankel function defined by

$\displaystyle H_{0}(z)$ $\displaystyle=$ $\displaystyle
J_{0}(z)+\mathrm{i}Y_{0}(z)\;.$ (62)

The Green’s function screened by dipoles $G^{\text{DS}}$ in Eq.(61) is
consistent with the potential induced by a disclination in the hexatic phase,
and forms the basis for the sequential transition from hexatic to fluid phase.
Furthermore, this result provides a potential explanation for a problem
presented in a visionary study Ref. [40]. In that work the authors studied the
elastic fields induced by edge- and screw- dislocations in a Lennard-Jones
model of amorphous solid. They discovered that the stress fields of a screw-
dislocation are elastic-like, whereas those of edge-dislocation are smeared
out. In our theory an edge dislocation is dipolar and therefore is
significantly screened by dipoles as expressed by $G^{\text{DS}}$. This is
contrary to the screw dislocation which is not dipolar and therefore cannot be
effectively screened by dipoles. A systematic study of this problem from the
perspective of screening that will compare theoretical predictions with
numerical simulations of amorphous solids is left for a future work.

###  VI.2 Screened geometric charges and their interactions

In this section we highlight several key results that follow from the
fundamental solution $G^{\mathrm{DS}}$ in the dipole-screening regime.
Additionally, we study the interactions between screened geometric charges. .

It was shown in [39, 27] that defects and other sources of stresses can be
defined geometrically regardless of a specific physical model. In this theory
sources of stresses are singularities of $\bar{K}$. For example, dislocation
correspond to $\bar{K}=\mathbf{b}\cdot\nabla\delta(\mathbf{x})$, and isotropic
Eshelby inclusion corresponds to $\bar{K}=p\Delta\delta(\mathbf{x})$.

From linearity of Eq.(56), and from commutation of derivatives with the
$\mathcal{H}$ operator, taking the derivative of both sides with
$\bar{K}=\delta(\mathbf{x})$ yields new solutions for higher order sources of
stresses.

For example, the stress function of a dipole described by
$\bar{K}=\mathbf{b}\cdot\nabla\delta(\mathbf{x})$ and analogous to a
dislocation in the hexatic phase, is

$\displaystyle\chi_{\mathrm{\mathbf{b}}}=\mathbf{b}\cdot\nabla
G^{\mathrm{DS}}=\frac{Y\ell_{P}^{2}}{2\pi}\frac{\mathbf{b}\cdot\mathbf{x}}{r^{2}}\;.$
(63)

The stresses derived from this solution decay rapidly with $r$. Upon
substituting in the energy density one finds that the total energy of a
screened dislocation converges at infinite systems, and reflects only the core
energy.

The second example is that of an isotropic Eshelby inclusion, whose solution
is

$\displaystyle\chi_{\mathrm{Iso}}(r)=p\Delta
G^{\mathrm{DS}}(r)=p\delta(\mathbf{x})\;.$ (64)

This indicates that an isotropic inclusion in an infinite medium will be
completely screened by emergent dipoles. It is important to note that the
response to a localized expansion in a finite system is different (see
solution in App. F), and it exhibits spatial oscillations as previously
reported by some of the authors [21, 22, 23, 24, 25, 26].

The stress-functions of the screened dislocation and isotropic inclusion solve
the homogeneous equation Eq.(56), thus in the kernel of the relevant
differential operator. A comprehensive analysis of the kernel of
$\mathcal{H}^{\varphi}_{\ell}$ is needed in order to classify and derive all
singular solutions, and is an ongoing research topic and will be pursued in
future studies.

Next we examine the interactions between screened sources of stress. It is
well established that in the elastic regime, the energy stored in the medium
can be represented by the stress function and charge distribution [41]

$U=\int\chi\,\bar{K}\,\mathrm{dVol}\;.$ (65)

In App. E we show that this relation holds also in the screened regime, hence
we can use it to study the interactions between basic sources of stresses. For
example, it is known that isotropic inclusions do not interact in the elastic
framework [41]. However, still in the elastic framework, a disclination do
interact with an inclusion. This is seen by taking
$\bar{K}_{\mathrm{disc}}=q\delta(\mathbf{x})$ and
$\chi_{\mathrm{Iso}}=p\log(\mathbf{x}-\mathbf{x}_{0})$. From Eq.(66) we find
that the interaction between an inclusion and disclination is

$U=q\,p\log(r)\;.$ (66)

where $r$ is the distance between the two charges.

In the case of dipole screening we still have
$\bar{K}_{\mathrm{disc}}=q\delta(\mathbf{x})$, however, the screened stress
function of the inclusion is
$\chi^{DS}_{\mathrm{Iso}}=p\delta(\mathbf{x}-\mathbf{x}_{0})$. In that case
the interaction is zero, and the induced dipole field completely screen out
the interaction. The interactions between other multipoles is calculated in
the same way.

##  VII Summary and discussion

In this work, we developed a hierarchy of continuum screening theories that
generalize classical elasticity and are expected to be applicable to a variety
of different solid-like systems, such as granular materials, cellular tissue,
and mechanical metamaterials. While the traditional approach to non-mechanical
screening theories is based on statistical and thermodynamic arguments, our
theory is based on geometric arguments under the assumption that a long-
wavelength description of screened solids is valid.

Based on the conservation laws associated with the geometry of two-dimensional
Riemannian manifolds, our theory predicts three states of solid-like matter:
Quasi-elastic quadrupole-screened, anomalous dipole-screened, and monopole-
screened solids. The case of dipole screening exhibits mechanical behavior
that is similar to the hexatic phase, and thus forms an intermediate state
between a solid and liquid. The existence of dipole screening has been fully
confirmed in a series of recent works on granular and glassy matter. The
predictions from the monopole screening regime have not yet been observed in
athermal systems.

Our findings suggest that the current understanding of the jamming transition
in granular matter is incomplete. For example, it is widely accepted that upon
decreasing the pressure from a dense granular material, at a critical packing
fraction, the material undergoes an unjamming transition to a liquid-like
state that does not support shear. Instead, based on our theory, we expect a
sequential transition from a dense granular solid, to a dipole-screened solid-
like state, and then to an unjammed state described by monopole screening,
similar to the liquid state in the melting of two-dimensional crystals.

The effect of mechanical screening, in principle, is not limited to quasi-
static deformations, as studied in this work, and is expected to have
implications on the mechanics of both inertial and dissipative systems.
Furthermore, well-studied phenomena such as fracture can now be studied within
the framework of screened elasticity. These and other research questions are
left for future study.

###### Acknowledgements.

We would like to thank Mokhtar Adda-Bedia, Leo Radzihovsky and Keren
Schreiber-Re’em for stimulating discussions. The research was supported by the
Israel Science Foundation grant No. 1441/19.

##  Appendix A Derivation of Equilibrium Equation for the elastic stress

In this section we derive the nonlinear equilibrium equations for the elastic
stress and the corresponding boundary conditions.

An important quantity that will come back later is the coordinate
transformation of a vector from one coordinate system to another. Consider two
manifolds $\mathcal{M},\mathcal{N}$ on which coordinate systems are denoted
with Greek indices $\mu,\nu,...$, and roman indices $i,j,..$ respectively.
Given a mapping $\phi:\mathcal{M}\to\mathcal{N}$ the transformation of a
vector from $\mathcal{M}$ to $\mathcal{N}$ is given by

$\displaystyle v_{\mathcal{N}}^{i}=\frac{\partial\phi^{i}}{\partial
x^{\mu}}v_{\mathcal{M}}^{\mu}\;.$ (67)

The material is modeled as a manifold $\mathcal{M}$ equipped with a reference
metric $\bar{g}=\bar{g}^{0}+q$. A configuration is an embedding
$\phi:\mathcal{M}\to\mathbb{R}^{2}$ from which an actual metric is defined on
$\mathcal{M}$ as the pull-back of the euclidean metric on $\mathbb{R}^{2}$,
denoted $g$. We denoted by $\phi^{*}$ the energy minimizing configuration in
the absence of external loads. The equilibrium equations are derived from an
energy variation with respect to the embedding $\phi$ describing the
configuration. The elastic energy to be minimized is

$\displaystyle
F=\int_{\mathcal{M}}\mathcal{W}_{\mathrm{el}}(g,\bar{g})\,\mathrm{d}S_{\bar{g}}-\int_{\partial\mathcal{M}}\mathbf{t}\cdot\mathbf{d}\,\mathrm{d}l_{\bar{g}}\;.$
(68)

with $\mathbf{d}=\phi-\phi^{*}$, and

$\mathcal{W}_{\mathrm{el}}(g,\bar{g})=\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\gamma\delta}\;.$
(69)

Upon defining the elastic stress

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}=\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\gamma\delta}$
(70)

we find

$\displaystyle\delta_{\phi}F=\int_{\mathcal{M}}\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}\delta_{\phi}g_{\alpha\beta}\,\mathrm{d}S_{\bar{g}}-\int_{\partial\mathcal{M}}\mathbf{t}\cdot\delta\phi\,\mathrm{d}l_{\bar{g}}\;.$
(71)

where we used $\delta\mathbf{d}=\delta(\phi-\phi^{*})=\delta\phi$.

Writing the metric variation in terms of the configuration and using
$\delta_{\phi}g_{\alpha\beta}=(\partial_{\alpha}\phi)(\partial_{\beta}\delta\phi)+(\partial_{\alpha}\delta\phi)(\partial_{\beta}\phi)$
we find

$\displaystyle\delta_{\phi}F$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\sigma_{\mathrm{el}}^{\alpha\beta}(\partial_{\alpha}\phi)(\partial_{\beta}\delta\phi)\,\mathrm{d}S_{\bar{g}}-\int_{\partial\mathcal{M}}\mathbf{t}\cdot\delta\phi\,\mathrm{d}l_{\bar{g}}$
(72) $\displaystyle=$
$\displaystyle\oint_{\partial\mathcal{M}}\sigma_{\mathrm{el}}^{\alpha\beta}n_{\beta}(\partial_{\alpha}\phi)\delta\phi\,\mathrm{d}l_{\bar{g}}$
$\displaystyle-$
$\displaystyle\int_{\mathcal{M}}\frac{1}{\sqrt{\bar{g}}}\partial_{\beta}\left(\sigma_{\mathrm{el}}^{\alpha\beta}(\partial_{\alpha}\phi)\sqrt{\bar{g}}\right)\delta\phi\,\mathrm{d}S_{\bar{g}}$
$\displaystyle-$
$\displaystyle\oint_{\partial\mathcal{M}}\mathbf{t}\cdot\delta\phi\,\mathrm{d}l_{\bar{g}}\;.$

In the second integral we note that integrand can be written as

$\displaystyle\mathrm{div}_{\beta}\sigma_{\mathrm{el}}^{\alpha\beta}\partial_{\alpha}\phi$
$\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{\bar{g}}}\partial_{\beta}\left(\sigma_{\mathrm{el}}^{\alpha\beta}(\partial_{\alpha}\phi)\sqrt{\bar{g}}\right)$
$\displaystyle=$
$\displaystyle\left({\nabla}_{\beta}\sigma_{\mathrm{el}}^{\alpha\beta}+\left(\bar{\Gamma}^{\nu}_{\nu\beta}-{\Gamma}^{\nu}_{\nu\beta}\right)\sigma_{\mathrm{el}}^{\alpha\beta}\right)\partial_{\alpha}\phi$

In the last integral we transform the vector $\mathbf{t}$ to the reference
manifold by setting $\mathbf{t}=t^{\mu}\partial_{\mu}\phi$. In this form the
traction forces are defined on the reference manifold, which is equivalent to
saying that the position on which forces applied are moving with the material,
as in Lagrangian coordinates. Therefore the variation takes the form

$\displaystyle\delta_{\phi}F$ $\displaystyle=$
$\displaystyle\oint_{\partial\mathcal{M}}\left(\sigma_{\mathrm{el}}^{\alpha\beta}n_{\beta}-t^{\alpha}\right)(\partial_{\alpha}\phi)\delta\phi\,\mathrm{d}l_{\bar{g}}$
(73) $\displaystyle-$
$\displaystyle\int_{\mathcal{M}}\mathrm{div}_{\beta}\sigma_{\mathrm{el}}^{\alpha\beta}\partial_{\alpha}\phi\,\delta\phi\,\mathrm{d}S_{\bar{g}}\;.$

We conclude that the equilibrium equation is

$\displaystyle\bar{\nabla}_{\mu}\sigma_{\mathrm{el}}^{\mu\nu}+\left(\Gamma^{\nu}_{\alpha\beta}-\bar{\Gamma}^{\nu}_{\alpha\beta}\right)\sigma_{\mathrm{el}}^{\alpha\beta}$
$\displaystyle=$ $\displaystyle 0,$ (74)

along with the boundary conditions

$n_{\alpha}\sigma_{\mathrm{el}}^{\alpha\beta}=t^{\beta}\,.$ (75)

##  Appendix B Derivation of Equilibrium Equation for the induced quadrupoles

Here we derive the relation between the elastic stress and the induced
quadrupoles in each screening regime.

Quadrupole screening
The variation of the work term Eq.(40) with respect to $q$ yields

$\begin{split}\int_{\mathcal{M}}\delta_{q}\mathcal{W}\,\mathrm{d}S_{\bar{g}^{0}}=\int_{\mathcal{M}}\left(-{\Lambda}_{\mathrm{q}}^{\alpha\beta\gamma\delta}q_{\gamma\delta}\delta
q_{\alpha\beta}\right)\,\mathrm{d}S_{\bar{g}^{0}}\;\sout{,}\noindent\color[rgb]{0,1,1}.\normalcolor\end{split}$
(76)

Substituting in Eq.(39) and requiring the variation to vanish we get a linear
relation between the induced quadrupole and the elastic stress

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}+2{\Lambda}_{\mathrm{q}}^{\alpha\beta\gamma\delta}q_{\gamma\delta}$
$\displaystyle=$ $\displaystyle 0\;.$ (77)

Substituting the expressions for $q$ and $\Lambda_{q}$ in terms of $Q$ and
$\Lambda^{Q}$ we obtain the first equation in Eq.(51).

Dipole Screening
The variation of the work term in the dipole screening regime reads

$\displaystyle\int_{\mathcal{M}}\delta\mathcal{W}\,\mathrm{d}S_{\bar{g}^{0}}=$
(78) $\displaystyle=$
$\displaystyle\int_{\partial\mathcal{M}}\frac{1}{2}\lambda_{P}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(n_{\mu}P_{\nu}+n_{\nu}P_{\mu}\right)\delta
q_{\alpha\beta}\,\mathrm{d}l_{\bar{g}^{0}}$ $\displaystyle-$
$\displaystyle\int_{\mathcal{M}}\frac{1}{2}\lambda_{P}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(\bar{\nabla}_{\mu}P_{\nu}+\bar{\nabla}_{\nu}P_{\mu}\right)\delta
q_{\alpha\beta}\,\mathrm{d}S_{\bar{g}^{0}}\;.$

Substituting in Eq.(39) and requiring the total variation to vanish we obtain
the second equation in Eq.(51).

Monopole Screening To perform the variation in the monopole regime two
integrations by parts are required. This is seen from the following:

$\displaystyle\delta\mathcal{W}$ $\displaystyle=$
$\displaystyle\bar{\nabla}_{\gamma\delta}\left(\Lambda^{\mathrm{M}}(\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})(\delta
Q^{\gamma\delta})\sqrt{\bar{g}_{0}}\right)$ (79) $\displaystyle-$
$\displaystyle
2\bar{\nabla}_{\delta}\left(\Lambda^{\mathrm{M}}(\bar{\nabla}_{\gamma}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})(\delta
Q^{\gamma\delta})\sqrt{\bar{g}_{0}}\right)$ $\displaystyle+$
$\displaystyle\left(\Lambda^{\mathrm{M}}(\bar{\nabla}_{\gamma\delta}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta})(\delta
Q^{\gamma\delta})\sqrt{\bar{g}_{0}}\right)\;.$

We substitute $M=\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}$, and note that the
double integration by parts performed on the first term vanishes because
$\partial\partial\mathcal{M}=0$, that is the boundary of the boundary is
closed. The variation therefore takes the form

$\displaystyle\begin{split}\int_{\mathcal{M}}\delta\mathcal{W}\,&\mathrm{d}S_{\bar{g}^{0}}=\int_{\mathcal{M}}\lambda_{M}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(\bar{\nabla}_{\mu\nu}M\right)\delta
q_{\alpha\beta}\,\mathrm{d}S_{\bar{g}^{0}}\\\
&-\int_{\partial\mathcal{M}}\lambda_{M}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(n_{\mu}\bar{\nabla}_{\nu}M+n_{\nu}\bar{\nabla}_{\mu}M\right)\delta
q_{\alpha\beta}\,\mathrm{d}l_{\bar{g}^{0}}\;.\end{split}$

Substituting in Eq.(39) and requiring the total variation to vanish we obtain
the third equation in Eq.(51).

##  Appendix C The normalized elastic tensor

We firs relate $\Lambda^{\mathrm{Q}}$ with $\Lambda^{\mathrm{q}}$ as shown in
Eq.(40). Since
$Q^{\alpha\beta}=\varepsilon^{\alpha\mu}\varepsilon^{\beta\nu}q_{\mu\nu}$ we
find

$\displaystyle\mathcal{W}_{Q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}Q^{\alpha\beta}Q^{\gamma\delta}$
(80) $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}\varepsilon^{\alpha\mu}\varepsilon^{\beta\nu}q_{\mu\nu}\varepsilon^{\gamma\rho}\varepsilon^{\delta\sigma}q_{\rho\sigma}$
(81) $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\Lambda_{\mathrm{q}}^{\mu\nu\rho\sigma}q_{\mu\nu}q_{\rho\sigma}$
(82)

with

$\displaystyle\Lambda_{\mathrm{q}}^{\mu\nu\rho\sigma}=\Lambda^{\mathrm{Q}}_{\alpha\beta\gamma\delta}\varepsilon^{\alpha\mu}\varepsilon^{\beta\nu}\varepsilon^{\gamma\rho}\varepsilon^{\delta\sigma}$
(83)

Next we show how $\Lambda_{\mathrm{q}}$ normalizes the elastic tensor in
Eq.(77)

$\displaystyle
q_{\alpha\beta}=-\frac{1}{2}{\Lambda}^{\mathrm{q}}_{\alpha\beta\gamma\delta}\sigma_{\mathrm{el}}^{\gamma\delta}\;.$
(84)

Substituting in Eq. 35 we find

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathcal{A}^{\alpha\beta\gamma\delta}u_{\gamma\delta}-\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}q_{\gamma\delta}$
(85) $\displaystyle=$
$\displaystyle\mathcal{A}^{\alpha\beta\gamma\delta}u_{\gamma\delta}-\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}\left(-\frac{1}{2}{\Lambda}^{\mathrm{q}}_{\gamma\delta\mu\nu}\sigma_{\mathrm{el}}^{\mu\nu}\right)$

Noting that

$\displaystyle\sigma_{\mathrm{el}}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sigma_{\mathrm{el}}^{\mu\nu}\mathrm{Id}^{\alpha\beta}_{\quad\mu\nu}$
$\displaystyle\mathrm{Id}^{\alpha\beta}_{\quad\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta^{\alpha}_{\,\,\mu}\delta^{\beta}_{\,\,\nu}+\delta^{\alpha}_{\,\,\nu}\delta^{\beta}_{\,\,\mu}\right)$
(86)

we get

$\displaystyle\sigma_{\mathrm{el}}^{\mu\nu}\left(\mathrm{Id}^{\alpha\beta}_{\quad\mu\nu}-\frac{1}{4}\mathcal{A}^{\alpha\beta\gamma\delta}{\Lambda}^{\mathrm{q}}_{\gamma\delta\mu\nu}\right)$
$\displaystyle=$
$\displaystyle\mathcal{A}^{\alpha\beta\gamma\delta}u_{\gamma\delta}\;.$ (87)

Upon denoting

$\displaystyle\Gamma^{\alpha\beta}_{\quad\mu\nu}=\mathrm{Id}^{\alpha\beta}_{\quad\mu\nu}-\frac{1}{4}\mathcal{A}^{\alpha\beta\gamma\delta}{\Lambda}^{\mathrm{q}}_{\gamma\delta\mu\nu}$
(88)

we get

$\displaystyle\sigma_{\mathrm{el}}^{\mu\nu}$ $\displaystyle=$
$\displaystyle{\Gamma^{-1}}_{\alpha\beta}^{\quad\mu\nu}\mathcal{A}^{\alpha\beta\gamma\delta}u_{\gamma\delta}\;.$
(89)

that is

$\displaystyle\tilde{\mathcal{A}}^{\alpha\beta\gamma\delta}={\Gamma^{-1}}_{\mu\nu}^{\quad\alpha\beta}\mathcal{A}^{\mu\nu\gamma\delta}$
(90)

Note that in the absence of quadrupole screening, where all the coefficients
in Eq. 31 vanishes, $\Gamma$ reduces to the identity, and the elastic tensor
remains intact.

##  Appendix D Derivation of Induced Effective Monopoles

To derive the induced monopole charge distribution
$M_{\mathrm{ind}}=\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}$ we use the
relation between stress and induced quadrupoles given in Eq.(51). In the
quadrupole, in the first equation in Eq.(51), we take the second divergence to
express $\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}$. The divergence of the
elastic stress, and therefore its second divergence as well, vanishes in
equilibrium, hence in this regime
$\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}=0$.

In the dipole screening regime we take the trace of the second equation in
Eq.(51) and find

$\mathrm{Tr}\,\sigma_{\mathrm{el}}=-Y\ell_{P}^{2}\bar{g}^{0}_{\alpha\beta}{\bar{\varepsilon}}^{\mu\alpha}{\bar{\varepsilon}}^{\nu\beta}\left(\bar{\nabla}_{\mu}P_{\nu}+\bar{\nabla}_{\nu}P_{\mu}\right)$
(91)

Upon substituting $P$ in terms of $Q$ and
$\mathrm{Tr}\,\sigma_{\mathrm{el}}=\bar{\Delta}\chi$ we obtain the second
equation in Eq.(55)

$\displaystyle\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}=-\frac{1}{2Y\ell_{P}^{2}}\bar{\Delta}\chi$
(92)

Lastly, for monopole screening regime, substituting Eq.(52) in Eq.(48) we find

$\displaystyle\bar{\varepsilon}^{\alpha\mu}\bar{\varepsilon}^{\beta\nu}\bar{\nabla}_{\mu\nu}\left(\chi+Y\ell_{M}^{4}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}\right)=0\;.$
(93)

We conclude that

$\displaystyle\chi+Y\ell_{M}^{4}\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}=\chi_{g}\;,$
(94)

where $\chi_{g}$ is any function satisfying $\bar{\nabla}_{\mu\nu}\chi_{g}=0$,
reflecting the gauge freedom of the stress function. Upon setting a gauge such
that $\chi_{g}=0$ we find

$\displaystyle\bar{\nabla}_{\alpha\beta}Q^{\alpha\beta}=-\frac{1}{Y\ell_{M}^{4}}\chi\;.$
(95)

##  Appendix E Interactions

In this section we derive the interaction-form of the mechanical energy stored
in the screened solid. The case of quadrupole screening require no analysis
since the only effect of the induced quadrupoles is normalizing the elastic
tensor and the interaction rmeain intact apart from normalized elastic moduli.

In the case of dipole screening, the total energy is

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\left(\frac{1}{2}\mathcal{A}^{\alpha\beta\gamma\delta}u^{\mathrm{el}}_{\alpha\beta}u^{\mathrm{el}}_{\gamma\delta}-\frac{1}{2}\Lambda^{\mathrm{P}}_{\alpha\beta}P^{\alpha}P^{\beta}\right)\,\mathrm{d}S_{\bar{g}^{0}}$
(96) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\left(\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}u^{\mathrm{el}}_{\alpha\beta}-\frac{1}{2}\lambda_{P}\bar{\nabla}_{\mu}Q^{\mu\alpha}P_{\alpha}\right)\,\mathrm{d}S_{\bar{g}^{0}}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\left(\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}u_{\alpha\beta}-\frac{1}{4}\sigma_{\mathrm{el}}^{\alpha\beta}q_{\alpha\beta}+\frac{1}{2}\lambda_{P}Q^{\mu\alpha}\bar{\nabla}_{\mu}P_{\alpha}\right)\,\mathrm{d}S_{\bar{g}^{0}}$
$\displaystyle-$
$\displaystyle\int_{\partial\mathcal{M}}\lambda_{P}Q^{\mu\alpha}P_{\alpha}n_{\mu}\mathrm{d}l_{\bar{g}^{0}}$

Using the symmetry of $Q$ and substituting it in terms of $q$ we find that the
boundary term vanishes from the boundary condition in Eq.(45), and the second
and third terms in the integral cancel from the equilibrium equation Eq.(44).
We therefore conclude

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}u_{\alpha\beta}\,\mathrm{d}S_{\bar{g}^{0}}$
(97)

Upon expressing $\sigma_{\mathrm{el}}$ in terms of the stress function and
integrating by parts twice we find that at the linear approximation

$\displaystyle E$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}\frac{1}{2}\sigma_{\mathrm{el}}^{\alpha\beta}u_{\alpha\beta}\,\mathrm{d}S_{\bar{g}^{0}}=\int_{\mathcal{M}}\chi\bar{K}\,\mathrm{d}S_{\bar{g}^{0}}\;.$
(98)

##  Appendix F Complete solution for Green’s function

The Green’s function within the screened elasticity setup is the solution for
Eq.(56) with a delta-function singularity, as solved in Eq.(60) and
Eq.(LABEL:eq:GreensMonopole). The solution is first derived for a finite
domain with traction free boundary conditions. In the case of dipole screening
the constants of integration are

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle\frac{q}{2\pi
r_{\mathrm{in}}r_{\mathrm{out}}}\frac{r_{\mathrm{in}}Y_{1}\left(r_{\mathrm{in}}\right)-r_{\mathrm{out}}Y_{1}\left(r_{\mathrm{out}}\right)}{Y_{1}\left(r_{\mathrm{in}}\right)J_{1}\left(r_{\mathrm{out}}\right)-J_{1}\left(r_{\mathrm{in}}\right)Y_{1}\left(r_{\mathrm{out}}\right)}$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle\frac{q}{2\pi
r_{\mathrm{in}}r_{\mathrm{out}}}\frac{r_{\mathrm{in}}J_{1}\left(r_{\mathrm{in}}\right)-r_{\mathrm{out}}J_{1}\left(r_{\mathrm{out}}\right)}{J_{1}\left(r_{\mathrm{in}}\right)Y_{1}\left(r_{\mathrm{out}}\right)-Y_{1}\left(r_{\mathrm{in}}\right)J_{1}\left(r_{\mathrm{out}}\right)}$

and here $r_{\mathrm{in}}$ and $r_{\mathrm{out}}$ are measured in units of
$r_{s}=\sqrt{2\Lambda_{P}}$. In the limit $r_{\mathrm{out}}\to\infty$ we both
constants vanish, leading to the Green’s function $G^{\text{DS}}$ given in
Eq.(61).

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