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Achieving adjustable elasticity with non-
affine to affine transition
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Supplementary information
https://doi.org/10.1038/s41563-021-01046-8

Supplementary Information for “Achieving adjustable elasticity
with non-affine to affine transition”
Xiangying Shen1,2,3†, Chenchao Fang1,3†, Zhipeng Jin1,3, Hua Tong4,5, Shixiang Tang1,
Hongchuan Shen1, Ning Xu6, Jack Hau Yung Lo1∗, Xinliang Xu2,7∗, Lei Xu1,3∗
1Department of Physics, The Chinese University of Hong Kong, Hong Kong, China
2The Beijing Computational Science Research Center, Beijing, China
3Shenzhen Research Institute, The Chinese University of Hong Kong, Shenzhen, China
4School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China
5Department of Physics, University of Science
and Technology of China, Hefei 230026, China
6Hefei National Laboratory for Physical Sciences at the Microscale,
CAS Key Laboratory of Soft Matter Chemistry, and Department of Physics,
University of Science and Technology of China, Hefei 230026, China
7Department of Physics, Beijing Normal University, Beijing 100875, China
(Dated: May 27, 2021)
1

I. PACKING CONFIGURATION AND PACKING DERIVED NETWORK
To avoid crystallization, the jamming diagram is produced by bi-dispersed particle pack-
ing [1–3] where particles are repulsive spheres mixed in equal numbers with radii R2= 1.4R1.
A square box of linear size Lis filled with Nsoft repulsive particles randomly. Denoting
the radii of each pair of contacted particles are Riand Rj, the dimensionless overlapping
parameter is defined as δij = 1 −rij /(Ri+Rj). The interaction potential is
Vij = 0, δij <0,
Vij =kδθ
ij , δij ≥0,(1)
where kis the spring constant, and in our work harmonic interaction potential is adopted
leading to power θ= 2. Once the contact potential is specified, one can generate packing
configurations by a variety of protocols. We adopt the commonly used molecule dynamics
(MD) simulation. In MD, the process starts from a completely random configuration. Then,
as packing fraction φ=Nπ(R2
1+R2
2)/2L2is raised, we wait for the force balanced on
every constituent so that the total energy is minimum. By minimizing the energy, a packing
configuration is obtained. In order to further convert the packing configuration into network,
each particle center is replaced by a node and each contact is replaced by a bond or spring.
When ∆z=z−zC= 0, there is no stress between particles. When ∆z > 0, in a real packing
system the contact force fij should be taken into account when the energy variation under
stain or stress is calculated:
Ve=1
2
i,j
k

δ−→
r∥
ij 


2
−fij
krij 
δ−→
r⊥
ij 

2,(2)
where rij is the distance between two contacted particle centers and 
δrij =
δrj−
δriis the
relative displacement of two particles under stress. δ−→
r∥
ij is the projection along the bond,
and δ−→
r⊥
ij is the projection perpendicular to the bond. However, the pre-stresses, i.e., the
second term on the right, are always neglected in our network model, even when ∆zis high.
Therefore, our network model only contains the configuration and contact information of
packing systems but without pre-stress information.
2

II. COMPARISON BETWEEN OUR AFFINE TRANSITION AND THE PREVI-
OUS STUDIES AT HIGH PACKING DENSITIES
Before our study, it is well known that at high packing densities jamming systems exhibit
very similar affine displacement fields as ours at zaff = 6 [4, 5]. One question natuarlly
arises: is the non-affine to affine transition already discovered in these previous studies? We
compare our study with these previous high-packing-density studies in this section.
Fig.5 in Ref. [4]
Fig.2 in our study
Fig.SI- 1: Comparing the previous high-packing-density studies with our affine transition and affine
regime. The left panel shows the behaviors of elastic moduli in the previous study Ref.[4], and the right
panel shows the behaviors in our study. Apparently there is a new affine regime for ∆z≥2 in our study, in
which Kand Gvary with the same power law and K/G is a constant.
Instead of comparing the displacement field images, which only give a qualitative overall
picture that may not be precise, we illustrate it by comparing the quantitative moduli data as
shown in Fig.SI-1. In the side by side comparison, the left panel shows the moduli Kand G,
and their ratio, G/K in Fig.5 of Ref.[4]; while the right panel shows similar quantities (K/G
3

instead of G/K in our case) in our study (Fig. 2(a) and (b) in the main text). Apparently,
there is a fundamental difference between the two sets of data. In the left column of previous
study, Kand Gvary with distinct power laws, making their ratio keep changing with ∆z
and never reach a constant value. This is a typical non-affine behavior. In the right column
of our study, however, we reveal a new affine regime for ∆z≥2 (i.e., z≥6), in which K
and Gvary with the same power law and their ratio remains a constant. This is a typical
affine behavior. This fundamental difference in the two sets of data unambiguously shows
that our discovery of the affine regime at ∆z≥2 is new and has not been discovered in
these previous studies.
Why the affine regime discovered by us is so difficult to reach by the previous packing
research? Because in conventional packing systems, particles can only touch their nearest
neighbors and cannot penetrate through these nearest neighbors to contact the next nearest
neighbors. As a result, the cross bonds that are essential to our affine regime cannot appear in
those conventional studies. In our study, however, we use packing derived network instead of
particle packing system itself and have the freedom of adding cross bonds absent in previous
studies, thus enabling the new discovery of affine regime at ∆z≥2.
III. AFFINE AND NON-AFFINE RESPONSES
It is convenient to express a network in matrix form [6, 7]. For the central force network
without bending energy, define |F⟩as a vector in dN dimensional space that indicates the
force at each node, and let |T⟩be a vector with NCdimensions representing the tensions
along with the bonds. Note that all the components in these vectors are scalars with plus
or minus signs. The force vector |F⟩is coupled to the tension vector |T⟩by a dN ×NC
dimensional matrix, Q, which is called the equilibrium matrix mapping the tension space to
the force space:
Q|T⟩=−|F⟩.(3)
Similarly, we can define a dN dimensional vector |δr⟩as the displacement field of nodes and
aNCdimensional vector |E⟩describing the elongations of bonds. The relationship between
these two vectors is:
C|δr⟩=|E⟩,(4)
4

where Cis named as compatible matrix mapping displacements space to the elongations
space. It has been proved that the compatible matrix and the equilibrium matrix are
transposes of each other according to the work-energy principle, i.e., C=QT.
Since all the bonds in our model comply with Hook’s law, the energy variation of the
whole system under infinitesimal distortions should be written as the summation of elastic
energies induced by bonds’ elongations:
Ve=1
2⟨E|k|E⟩=1
2⟨δr|D|δr⟩,(5)
where kis NC×NCdiagonal matrix of springs’ constants, and D=CkCT=QTkQ is the
dN ×dN dimensional matrix. If the mass of particles in the system is normalized, Dis
the dynamical matrix. Its eigenvectors are vibrational normal modes of all particles, and
the eigenvalues are squared angular frequencies of these modes. The elastic energy density
associated with the strain is:
Ve
V=del =1
2Klξκχεlξ εκχ,(6)
where the V,Klξκχ are the volume and an element of the rank four stiffness tensor of the
network. The above equation follows the Einstein convention that the summation acts on
repeated indices. lξκχ label the Cartesian components of the imposed strain field. For
a shear deformation εxy in the xy plane, we have Kxyxy. As a result, the corresponding
modulus is obtained:
Klξκχ =∂2del
∂εlξ ∂εκχ
.(7)
In two-dimensional system under Voigt notation, the strain tensor [8] can be expressed as a
three-dimensional vector:
εlξ = (εxx, εyy,2εxy ).(8)
On the other hand, the stress tensor can be written in the form:
σlξ = (σxx, σyy, σxy ).(9)
The stiffness tensor is a 3 ×3 matrix following the generalized Hooke’s law





σxx
σyy
σxy





=




Kxxxx Kxxyy Kxxxy
Kxxyy Kyyyy Kyyxy
Kxxxy Kyyxy Kxyxy










εxx
εyy
2εxy





.(10)
5

The response of the network can be expressed in terms of the changes of the bonds’ rest
lengths [9]. Considering a packing derived network under a given strain, it is convenient to
study the network’s elasticity by assuming that the imposed change of rest length on each
bond equals to its affine elongation |T⟩=|Eaff ⟩, which is equivalent to forcing every node
in the network moving affinely then letting them relax. The final energy after relaxation is:
Ve=k
2min
δrna ⟨QTδrna −Eaff |QTδrna −Eaff ⟩,(11)
where |δrna⟩is the relaxation displacement field of nodes and |Eaff ⟩=|nij,l ·εlξ ·rij,ξ ⟩is the
affine elongations of bonds induced by the strain. Here, nij is a normalized vector showing
the orientation of the bond connecting nodes iand j, and rij is its length vector. If QT=Cis
not a full rank matrix, there always exists a |δrna ⟩that exactly offsets the bonds’ rest length
changes |Eaff ⟩, forming a floppy mode that corresponds to the z < 2dsituation in Maxwell
isostatic theorem. For z > 2d, if QTδrna = 0, Vereaches its maximum Vaff
e= (k/2)⟨Eaff |Eaff⟩,
which means the network is completely affine. Vecan reach its maximum only under either
of the following two conditions: first, Q= 0; second, |Eaff⟩is in the null space of Qleading
to no residual force on each node so that Q|Eaf f ⟩=−|F⟩= 0. As a result, the relaxation
displacement δrna = 0.
The first condition, Q= 0, can be achieved in lattices with one site per unit cell. Fourier
transforms of Qin periodic lattices are defined in terms of wavenumbers qin the first
Brillouin zone. As q= 0, all the entries in Q(q) are zero. Hence, this type of crystalline
network is completely affine. For disordered networks, however, in general Q= 0, and it is
too coincidental to find that |Eaff⟩is in the null space of Q, so the forces at some sites induced
by the affine displacement field are nonzero. These sites will relax to positions of zero force,
which indicates the common presence of the non-affinity in disordered networks [7].
There are many disordered networks or lattices with multiple sites per unit cell and we
cannot exhaust all to show the common presence of non-affinity. In this section, we only
show two examples to demonstrate it. We start from a two-dimensional triangular lattice,
which is a one site per unit cell lattice with bonds of the same length. We can numerically
calculate the non-affinity, ∆E= (Vaff
e−Ve)/V aff
e. When ∆E= 0, the network system is
completely affine. Before any perturbation is applied, ∆Eequals zero for such a network,
as shown in Fig.SI-2(a).
We then either distort it into a disordered system in (b) or cut the bonds randomly to
6

Fig.SI- 2: The sketch shows the networks of (a) a triangular lattice; (b) the distorted triangular lattice; (c)
the network obtained by randomly cutting off bonds from the triangular lattice. The calculation results of
∆Evaring with the perturbed degrees are presented in (d) and (e).
approach the percolation threshold in (c). We calculate the corresponding non-affinity ∆E
with respect to such operations, as shown in panels (d) and (e). In the first situation, as we
distort the periodic lattice with larger and larger perturbation magnitude d[10], the system
becomes more and more non-affine with increasing ∆E, as shown in (d). The distortion to
regular lattice confirms our expectation that disordered systems should be non-affine.
In the second situation, as we cut more and more bonds, the system also deviates pro-
gressively from a periodic lattice to a disordered system. Once again, the non-affinity ∆E
in panel (e) keeps increasing as the system becomes more disordered and approaches the
percolation threshold (z= 3.92) [11]. This result is again consistent with our expectation
that disordered systems should be non-affine. However, we also note one important differ-
ence between the two situations: the former distortion situation makes different influences
to the shear and compression ∆E, while the latter bond-cutting situation makes the same
influence to ∆Evariation. This discovery calls for further studies in this area.
Although disordered systems should be non-affine in general, we find that the disordered
jamming system can achieve almost ideal affine elasticity at zaf f = 6. We believe it is
7

the high local uniformity of the jamming configuration that causes this special feature.
To measure the uniformity, we use the number fluctuation variance, σ2(r), of the particle
numbers encircled by a circle with radius rlocating at various random positions, as shown in
Fig.SI-3. To obtain a systematic picture, we compare the variance in four typical systems:
(a) the triangular lattice, (b) the random system, (c) the jamming system, and (d) the
distorted jamming system.
To quantitatively compare their uniformity, we renormalize the variance by the circle
area, σ2(r)/S(r), and plot the four systems data together in Fig.SI-3(e). Apparently, the
renormalized variance, σ2(r)/S(r), quantifies the uniformity very well. For the trangular
lattice, σ2(r)/S(r) is the lowest and decays rapidly towards zero, indicating the highest
uniformity; and for the random system, σ2(r)/S(r) is the highest and never decays to zero,
indicating the least uniformity. For the jamming system and its distortion, they are in the
middle and both systems decay towards zero at large r, because both are hyperuniform
systems which should decay to zero at infinity [12, 13].
Next, we correlate their uniformity with their affine and non-affine properties at the
complete triangulation point. As we naturally expect, the most uniform crystalline system is
affine and the least uniform random system is non-affine, as confirmed by their non-affinity
value ∆E. However, the two hyperuniform systems in the middle exhibit an interesting
difference: the jamming system is affine while its distortion is non-affine. Therefore, only
the hyperuniform feature itself, which only describes the large scale uniformity, is not enough
to determine the affinity and the local uniformity must be considered.
Apparently, at large rthe two jamming-related systems approach each other, and their
major difference exists at small r, which is determined by the local uniformity. Because
the jamming system is more uniform at small r, it reaches the affine-like state while its
distortion does not. Therefore, to achieve the affine-like state, the system not only needs to
be uniform at large scale, it also needs to be uniform at small scale. How uniform locally
is enough to achieve the affine-like state? We suspect that the uniformity of the jamming
system is a good criterion because perturbations to this system leads to larger and larger
non-affinity (see Fig.1f in the main text). To fully understand this issue, however, extensive
future studies are still required.
8

Fig.SI- 3: Comparing the uniformity with renormalized number variance, σ2(r)/S(r), in (a) triangular
lattice, (b) random system, (c) jamming system, and (d) distorted jamming system. (e) shows the data for
the four situations. Apparently, the triangular lattice has the smallest variance and it is most uniform, and
the random system is the least uniform. The jamming system and its distortion are hyperuniform and
locate in the middle.
9

IV. THE NON-AFFINE RELAXATION AND THE CHARACTERISTIC LENGTH
As aforementioned, under a certain strain εlξ, the elastic energy of a non-affine network
is smaller than the completely affine hypothesis so that it can be written as the energy
calculated under the completely affine assumption minuses the non-affine correction term:
Ve=Vaff
e−Vna
e.(12)
Although the bonds’ lengths in our system are different, for simplicity, we still suppose all
the connections are equal in length and stiffness. Since our network is derived from the
marginal jamming state of bi-disperse system, in which the distance between each pair of
contacted particles ranges from R1to 1.4R1, the hypothesis is roughly valid. Thus, assuming
the stiffnesses and lengths of springs are normalized, the unbalanced forces of nodes can be
expressed in vector form:
|−→
Fi⟩=|Σj(nij,l ·εlξ ·nij,ξ)nij ⟩.(13)
The non-affine displacement is along the direction of the relaxation force, and according to
Hooke’s law, the energy is the product of the non-affine displacement of each node |
δrna⟩
and the force vector:
Vna
e=1
2⟨
Fi|
δrna⟩=1
2⟨
Fi|D−1|
Fi⟩,(14)
where D−1is the inverse dynamical matrix, so the non-affine term can be expressed as:
Vna
e=1
2Σω
1
ω2⟨
Fi|
δri(ω)⟩⟨ 
δri(ω)|
Fi⟩.(15)
The |
δri(ω)⟩is the normalized normal mode with eigenvalue ω2.
Eq. (15) is hard to solve analytically. With the typical mean-field approach, we first
calculate the average values and then add them up. The non-affine energy then becomes:
Vna
e=1
2Σω
1
ω2⟨Σi
δri(ω)
Fi2
⟩.(16)
Here ⟨...⟩means averaging operation. The mean value is expanded as:
⟨
δri(ω)
Fi2
⟩= Σi⟨
F2
i
δr2
i(ω)⟩+ Σi=j⟨
Fi
Fj
δri(ω)
δrj(ω)⟩.(17)
In the absence of any orientational correlations as in the random network model, the 
Fi,
Fj,

δri(ω) and 
δrj(ω) are independent, so the second term in the above equation vanishes due
10

to the symmetry. Hence in the random network model, only the first term is left, which is
caused by those contacts that share at least one common node:
⟨
δri(ω)
Fi2
⟩= Σi⟨
F2
i
δr2
i(ω)⟩
=⟨
F2
i⟩
=z⟨(nij,l ·εlξ ·nij,ξ)2⟩+
z(z−1)
2⟨(nij,l ·εlξ ·nij,ξ)nij (nip,l ·εlξ ·nip,ξ )nip ⟩.
(18)
Obviously, for a large packing derived network, the component in the first term,
⟨(−→
nij ·εlξ ·−→
nij )2⟩, is determined by the bonds’ orientations and the applied strain tensor,
which is not associated with the frequencies of the normal modes. Thus, we use a constant,
Alξκχ =⟨(−→
nij ·εlξ ·−→
nij )2⟩, to represent it. The second term shows the effect of bonds that
have one node in common. Due to their mutual excluded volume [14] introduced by the
packing process, they do have contributions to the relaxation energy. However, since the
nodes’ positions in our packing derived network never change as ∆zrises, the extent of the
excluded volume is also unchanged. Therefore, the term ⟨(nij ·εlξ ·nij)nij (nip ·εlξ ·nip)nip ⟩
is independent of ∆zor ωand can be represented by another constant Blξκχ.
Now we are in a position to further reduce the non-affine energy
Vna
e=1
2Σω
1
ω2zAlξ κχ +z(z−1)
2Blξκχ
=N
2zAlξ κχ +z(z−1)
2Blξκχdω D(ω)
ω2,
(19)
where D(ω) is the density of states. For small ∆z, because a long plateau exists in the
density of states of the packing derived network, D(ω) can be regarded as a constant for the
most time when ωvaries. Therefore, Eq. (19) gives:
Vna
e≈N
2zAlξ κχ +z(z−1)
2Blξκχω>ω∗
dω 1
ω2∼1
ω∗∼l∗,(20)
In Eq. (20), for small ∆z, the plane-wave like modes taken place at ω < ω∗, which is
negligible due to small ω∗. As a result, the non-affine energy correction is scaled by the
characteristic length, l∗[9, 15, 16], which is the size of a patch cut out from the system
in order to create a floppy mode. This argument is reasonable because if ∆z→0 then
l∗/L →1, with Lthe size of the entire system, the non-affine term will be comparable to the
affine component, leading to the zero elastic energy for the floppy condition. On the other
hand, if l∗/L →0, the non-affine component is negligible and the total elastic energy equals
11

to the affine term. At this point the network becomes completely affine and zreaches the
non-affine to affine transition zaf f .
V. ESTIMATION OF MODULI IN THE AFFINE AND HIGHLY NON-AFFINE
SITUATIONS
If the network is an affine system, we obtain:
Vaff
e=k
2⟨Eaff|Eaff ⟩=kN z
4⟨(nij,l ·εlξ ·nij,ξ)2⟩,(21)
where ⟨...⟩denotes the average value. In the large system limit, we derive:
Ve=kN z
4⟨(nij,l ·εlξ ·nij,ξ)2⟩ ≡ k
4Nzε2π
0
Plξκχ(α) cos2αdα, (22)
where Plξκχ(α) is the distribution function of the angle α= arccos(nij ·δrij /∥δrij ∥) between
the relative displacement vector of a bond and its orientation, and εis the magnitude of
the strain. For such an affine network, the corresponding modulus as an element in stiffness
tensor can be estimated as:
Klξκχ =2Ve
V ε2∼kz π
0
Plξκχ(α) cos2αdα. (23)
Although the above estimation is for the affine situation, we now show that it is also
valid for the highly non-affine situation, in which Plξκχ(α)→δ(π/2). Due to the strain,
each bond is stretched by eij = (eij cos βij , eij sin βij ), and the relaxed displacement δrna
ij =
(δrna
ij cos θij , δrna
ij sin θij ). Thus, the relative displacement of the bond’s two ends is δrij =
δrna
ij −εij . Plugging these into Eq. (11), we get the energy change under stress:
Ve=k
2Σij eij −δrna
ij cos(θij −βij )2.(24)
Since the network is highly non-affine, eij =δrna
ij cos(θij −βij ) + oij ,oij →0. Therefore, the
energy change can be approximated to an affine part minus an non-affine part:
Ve≈k
2Σij e2
ij −k
2Σij (δrna
ij )2cos2(θij −βij ).(25)
According to the simple geometric relationship as shown in Fig.SI-4, we obtain:
(δrna
ij )2≡e2
ij sin2αij
sin2γij
.(26)
12

Fig.SI- 4: The sketch shows the geometry of the bond ij’s stretching vector εij , the relative relaxed
displacement δrna
ij and the relative displacement δrij between nodes i,j.
Because Plξκχ(α)→δ(π/2), we get sin2γij ≈cos2(θij −βij ). Thus, the relaxed displacement
of each bond can be expressed as:
(δrna
ij )2≈e2
ij sin2αij
cos2(θij −βij ).(27)
We substitute the δrna
ij for the above relationship in Eq. (25) and the energy term becomes:
Ve≈k
2Nz
2⟨e2
ij ⟩−⟨e2
ij ⟩Σij sin2(αij )
=k
4Nz⟨e2
ij ⟩1−π
0Plξκχ(α) sin2αdα=k
4NzAlξκχ ε2π
0Plξκχ(α) cos2αdα,
(28)
where Alξκχ is a coefficient associated with the applied strain. As a result, we get the stiffness
tensor elements for the highly non-affine network as:
Klξκχ =2Ve
V ε2∼kz π
0
Plξκχ(α) cos2αdα. (29)
Eq. (29) also has very clear physical meanings: the kz term reflexes the affine effect and has
been well interpreted by EMA theory while the integration term characterizes the non-affine
rotational effect, and their combination describes the overall system behavior.
To summarize, the estimation in Eq. (29) is applicable for both highly affine and highly
non-affine situations that can be used to predict an arbitrary modulus for the entire packing
process, as demonstrated in the main text Fig. 2. However, if the non-affinity and affinity
13

Fig.SI- 5: (a) ln(cos2α) versus ln(∆z) under shear is plotted, which indicates the scaling rule that
G∼π
0PG(α) cos2αdα ∼∆z0.5(b) ln(cos2α) versus ln(∆z) under compression is presented, and the
dependence is roughly a constant. Thus, K∼π
0PK(α) cos2αdα ∼∆z0. Therefore, we get K/G ∼∆z−0.5
as shown in Fig.2b in the main text.
magnitudes are similar, the estimation’s performance is not that excellent, with the largest
deviation around 10% to 15%.
As shown in Fig.SI-5(a) and (b), we numerically calculate the integration term for G
and Krespectively: π
0PG(α) cos2αdα ∼∆z0.5and π
0PK(α) cos2αdα ∼∆z0. As a result,
K/G ∼∆z−0.5, consistent with the simulation results in Fig.2b in the main text.
VI. LOCAL DENSITY OF STATES AND THE CROSS BONDS ADDING METHOD
The correlation between different moduli is very sensitive to the local connectivity. For
instance, to reduce the shear modulus in a specific direction to zero, it is unnecessary to cut
many bonds in a network to reach z < ziso. Instead, one only needs to remove all the bonds
14

along a hyperplane parallel to that direction [9]. The non-affinity correlates to the length
scale l∗at which the system’s response is dominated by the local packing disorder, which
is considered to be inversely proportional to ∆z. Therefore, l∗governs the crossover from
isostatic behavior at the small scale to continuum behavior at the large scale, making the
non-affinity play a crucial role in determining moduli ratios (i.e., affinity).
However, when the cross bonds are added, one may decouple l∗with ∆zand fix the affinity
or non-affinity. The method of fixing the affinity or non-affinity is to add cross bonds within
the completely triangulated (in 2D) or tetrahedralized (in 3D) regions, in which l∗reaches
its minimum. Hence, the newly added bonds will not disturb the local packing disorder,
which ensures the global non-affinity to be constant and independent of ∆z. As a result,
the affinity or non-affinity remains a constant by adding cross (i.e., intersecting) bonds in
completely triangulated or tetrahedralized regions. Here, we verify this effect by measuring
the local density of states for both the triangulated regions (in blue circles) and the “soft”
regions (in green circles) in a z= 5.275, N = 1024 network as shown in Fig.SI-6(a). The
local density of states for an arbitrary node is defined as:
ρi(ω) = 
m
δ(ωm)|⟨i|m⟩|2,(30)
where |m⟩represents the mth normal mode and ⟨i|means the amplitude of state |m⟩at the
node i. The δ(ωm) is a delta function of the corresponding vibrational frequency and m
sums up all the vibrational modes. In Fig.SI-6 (a), the well-connected regions are enclosed
by the blue circles, while the weakly connected regions are in green circles. As cross bonds
are added into the blue regions, we reach the configuration in Fig.SI-6 (b).
The local density of states for both green and blue regions are given in Fig.SI-6 (c)-(f).
For comparison, the DOS of a network at the marginal jamming state (z=ziso = 4.0) and a
completely affine one (z=zaff = 6.0) are given in the Fig.SI-6 (g), (h) as a reference. The
patterns of DOS presented in Fig.SI-6 (c) for soft regions, and Fig.SI-6 (d) for triangulated
regions are similar to (g), (h) respectively, which indicates that with the same connectivity,
the local network holds the same features as the global system. After adding cross bonds,
the local DOS and plateau behavior in the soft regions (green circles) remain unchanged,
as shown in Fig.SI-6(e) in comparison to Fig.SI-6(c), implying that adding cross bonds
makes no effect on the local environment of the soft regions. For the triangulated and well-
connected regions in blue circles, adding cross bonds does not change the basic profile of
15

Fig.SI- 6: (a) A packing derived network with z= 5.275, in which some fully connected regions are
encircled by blue circles and some weakly connected regions are encircled by green circles. (b) Cross bonds
are added to the network in (a). (c) The local DOS spectrum for the green regions in (a). (d) The local
DOS spectrum for the blue regions in (a). (e) The local DOS spectrum for the green regions in (b). (f)
The local DOS spectrum for the blue regions in (b). (g) The DOS spectrum of a network in a marginal
jamming state where z= 4.0. (h) The DOS spectrum of the network that is fully triangulated and
completely affine, where z= 6.0. The local low-z regions in (c) and (e) are similar to z= 4.0 in (g), and
the local high-z and triangulated regions in (d) and (f) are similar to z= 6 in (h).
DOS, as shown in Fig.SI-6 (h): the only change is the shift towards high frequencies due
to the increase in stiffness, as we naturally expect. To summarize, adding cross bonds into
locally triangulated regions can lock the global non-affinity since these bonds do not affect
the local packing disorder that mostly occur in green circle regions.
16

Fig.SI- 7: (a) The picture shows the network configuration derived from a jamming system at ∆z≈0.05.
(b) The particle inflation process is applied to the network in (a), and a ∆z= 8.06 network with same
nodes’ positions are derived from the protocol.(c) All the cross bonds (colored in blue) are added to the
triangulated regions in the original network, and rest bonds are not changed. This is the final network for
cross bond addition.
Inspired by these findings, we propose a cross-bond adding method to lock the packing-
derived networks’ affinity or non-affinity. It can be performed on an arbitrary packing
derived network to lock at any K/G value. Due to the broad range of K/G adjustable in a
jamming system, the freedom is quite large. There should exist many methods to add cross
bonds into triangulated regions, and here we give one convenient example.
First, we choose an original configuration shown in Fig.SI-7 (a) with a desired K/G at
a low density CL. Then we increase the packing density to CHby particle inflation and
reach the configuration in Fig.SI-7 (b). Next, we select the bonds belonging to CHthat can
intersect with the bonds in CLwithin the triangulated regions only. These are the cross
bonds we can add to CL, as shown by the blue ones in Fig.SI-7 (c). To find all possible
cross bonds, we can increase particle size significantly to generate a high-zconfiguration,
17

and identify all cross bonds in the triangulated regions. Cross bonds can be added to the
network to change both Kand G, while do not affect the non-affinity ∆Eor affinity K/G.
VII. THE MECHANISM OF NON-AFFINITY LOCKING
When cross bonds are added to the locally triangulated regions, l∗of the system will
not change, which makes Plξκχ(α) in Eq. (28) retains its pattern. In other words, as the
non-affinity is decoupled from ∆z, the characteristic length l∗and Plξκχ(α) will also be
independent of ∆z. This non-affinity locking effect will also lock the moduli’s ratio and the
Poisson’s ratio with respect to ∆z.
For a uniform infinitesimal strain ε, the strain tensor for bulk, shear, horizontal and
vertical Young’s mdouli should be (ε/2, ε/2,0), (0,0, ε), (ε, −νxy ε, 0), and (−νyxε, ε, 0) re-
spectively, where νlξ is the Poisson’s ratio induced by stretching. Referring to the Eq. (10),
one can easily derive:
νxy =Kxxyy
Kyyyy
νyx =Kxxyy
Kxxxx
.
(31)
According to Eq. (29), let flξκχ =cos2αPlξ κχ(α)dα denote the integration associated
with the imposed strain tensor. Then we have:
Kxxxx ∼kzfxxxx,
Kyyyy ∼kzfyyyy,
Kxxyy ∼kzfxxyy ,
Kxxxy ∼kzfxxxy ,
Kyyxy ∼kzfy yxy ,
Kxyxy ∼kzfxyxy .
(32)
When the non-affinity is locked, the corresponding Plξκχ(α) for an arbitrary strain is inde-
pendent of ∆z. Looking into Eq. (32), it is obvious that the ratios between two arbitrary
components in the stiffness tensor are independent of ∆z. Similar situation occurs for the
Poisson’s ratio:
νxy =fxxyy
fyyyy
νyx =fxxyy
fxxxx
.
(33)
18

Fig.SI- 8: (a) shows K,Gscaling behaviour of a N= 1024 network as ∆zgrows. Before cross bonds are
added at ∆z= 0.5708, the normal inflation protocol is utilized and the K/Kmax and G/Gmax curves are
separated. Once cross bonds are added, two curves of K/Kmax and G/Gmax merge together. In the inset,
the variations of K/G with ∆zis presented. Clearly with cross bonds addition, K/G is fixed at a constant
value. (b) illustrates the DOS spectrum of network in (a) as cross bonds are added from z= 4.5708.
Clearly ω∗∼1/l∗stays roughly the same and implies roughly same values of l∗at different zas cross
bonds are added. For comparison, the DOS spectrum at the same zwith normal particle inflation protocol
(i.e., adding non-cross bonds) are given in the inset and ω∗changes significantly. (c) Identical angle
distribution function, P(α), are obtained for the network in (b) with cross bonds addition. For
comparison, P(α) curves at the same zobtained by normal particle inflation (non-cross bonds) are given
in the inset and they are quite different.
19

Therefore, we theoretically show that when the cross bonds are added into the system,
z(or ∆z) will change while the ratio between two arbitrary moduli and the Poisson’s ratio
will remain fixed. Simulations verify these results in Fig.SI-8: the main panel of (a) shows
that when we initially add bonds with normal particle inflation (the first 4 points), Kand
Gchange with different rates; while later when we add cross bonds into the system, Kand
Gchange with the same rate. The inset of (a) shows that K/G varies with ∆zinitially,
but later it is locked by cross bond addition. (b) shows the density of states (DOS) for our
system at different z. In the main panel, we plot the DOS at three coordination numbers,
z= 4.5708, z= 5.1991, and z= 5.7334. Note that the increase in zis due to the addition of
cross bonds into the system. In the main panel plot, we can find a characteristic frequency,
ω∗, which is the beginning of plateau as indicated by the shaded area. ω∗remains unchanged
for different curves, which suggests that l∗is also fixed because ω∗and l∗are correlated.
This main panel plot shows that although we increase zby adding cross bonds, ω∗and l∗
do not vary. For comparison, in the inset, we also show the corresponding cases at the same
z by adding non-cross bonds. ω∗changes significantly in this case. Therefore, our data
demonstrate that adding cross bonds in the triangulated regions can increase z and moduli,
but ω∗and l∗are locked, enabling a lock-in functionality. However, adding non-cross bonds
do not have this property. Panel (c) shows the distribution of angle αunder stress for cross
bond addition (main panel) and the non-cross bond addition (inset): adding cross bonds
does not change P(α) while adding non-cross bonds with normal particle inflation process
changes P(α) significantly.
VIII. TUNE K/G AND POISSON’S RATIO BY REMOVING OR ADDING BONDS
When a packing derived network is highly non-affine, most bonds make no contributions
to the elastic energy or moduli induced by a strain because these bonds tend to rotate rather
than shrink or stretch to store energy. Therefore, only a small fraction of bonds contribute
to the elastic energy or moduli of deformations. Assuming that βij is the angle between the
bond ij and the horizontal plane. For an affine system, in Fig.SI-9 (a), (b), we show the
distributions of one bond’s contributions to compression and shear, respectively. According
to Eq. (21), for the compression of a large system, the contribution of each bond is expected
to be independent of βij , and Kij ∼ε2. For the shear strain, the single bond’s contribution
20

Fig.SI- 9: (a) For an affine system, the contributions to bulk modulus of single bond with different angle β
calculated by simulations are denoted by open symbols in the graph. The corresponding moduli are
normalized to one, and a red line indicates the theoretical prediction of the contribution. The simulation
and theory agree well. (b) For an affine system, the contributions to the shear modulus of a single bond
with a different angle βare presented. For comparison, the red curve indicates the theoretical prediction,
and the two agree well. (c) For a packing derived network with large non-affinity, the distribution of a
single bond’s contribution to bulk modulus is shown, and the blue line indicates the average value. (d) For
a packing derived network with large non-affinity, the distribution of a single bond contribution to the
shear modulus. (e) For an affine network, most bonds contribute to both shear and bulk moduli. (f) For a
non-affine network, many bonds contribute to either shear or bulk modulus independently.
21

is Gij ∼ε2sin22βij . The theoretical predictions are plotted as the red curves, which agree
well with the simulation data points.
However, as the non-affinity increases in the network, the contributions of bonds change
dramatically from the affine situation, as shown in Fig.SI-9 (c), (d). The majority of bonds’
contributions are below the theoretical average value, and a small fraction of bonds make
much more contributions than the others. Because of the non-affinity, the sin22βdistribution
due to shear deformation disappears completely, which is a clear symbol for a non-affine
network. The contributions of one single bond to the bulk modulus (x-axis) and shear
modulus (y-axis) are given together in Fig.SI-9(e) for an affine network: in general, Gand
Kcontributions are coupled, which means a bond important to Gis also essential to K.
However, such contributions are decoupled in the non-affine situation, as shown in panel
(f). This property is very convenient for us to independently tune each modulus by directly
deleting the bonds important to either Gor K[17].
As bonds are removed to reduce the corresponding modulus, the Poisson’s ratio also
changes its value. For a 2D isotropic linear elastic material, the Poisson’s ratio ν=
(K/G−1)/(K/G+1). However, during the bond removal process, a packing derived network
may transform from largely-isotropic to anisotropic: as shown in Fig.SI-10 (a) and (b), re-
ducing the bulk modulus still keeps the isotropic property while reducing the shear modulus
makes the system anisotropic. It is mainly because that near the affine regime where we
start to remove bonds, the contribution of a single bond to shear modulus is highly related
to its orientations just as shown in Fig.SI-9 (b). On the other hand, for bulk modulus, the
contributions of bonds are independent of their orientations. As a result, when we remove
those bonds with high contributions to shear, a fraction of bonds with some specific orienta-
tions (around −π/4 and π/4) would be cut off, leading to more and more anisotropy in the
network. Therefore, the relation between the Poisson’s ratio νand K/G, which is only valid
in an isotropic system, no longer exists. Referring to Eq. (31), for an anisotropic material,
we have:
1
νxy
+1
νyx
+ 2 = Kxxxx +Kyyyy + 2Kxxyy
Kxxyy
.(34)
22

Fig.SI- 10: (a) shows data as we remove the bonds with the highest contributions to the bulk modulus, the
variations of Poisson ratios νxy and νyx with z. The Poisson ratios for both directions follow the same
decreasing trend, which indicates that the network is quite isotropic. (b) plots the data as we reduce the
shear modulus, the variations of νxy and νyx with z.νxy differs from νyx as zdecreases. This implies that
the system becomes anisotropic. (c) With the removal of K-important bonds, the average values of νxy and
νyx approximately equal to the isotropic expression: ν= (K/G −1)/(K/G + 1). (d) With the removal of
G-important bonds, (νxy +νyx )/2 is independent of K/G and varies within the range enclosed by the red
(upper limit) and blue (lower limit) curves.
In addition, under the affine assumption, each bond makes contribution to the moduli as:
(nij,l ·εlξ ·nij,ξ)2=1
2Kxxxxε2
xx +1
2Kyyyyε2
yy + 2Kxyxy ε2
xy +Kxxxyεxxεxy +
Kxxyy εxxεyy +Kxyyy εxy εyy
= cos4βij ε2
xx + sin4βij ε2
yy + 4 sin2βij cos2βijε2
xy+
4 cos3βij sin βij εxxεxy + 2 cos2βij sin2βijεxx εyy + 4 cos βij sin3βij εxyεy y
(35)
Apparently, for the affine case, Kxyxy =Kxxy y . For the network with high degree of affinity
23

(z > 5) and a small amount of bonds cut off, we can treat Kxyxy ≡G≈Kxxyy, just as the
completely affine situation. Taking this approximation into Eq. (34), we get:
2
1
νxy
+1
νyx
=1
2K
G−1
≤νxy +νyx
2≤
K
G−1
K
G+ 1
.(36)
According to the above inequality, although we cannot calculate the exact value of ν, we are
still able to evaluate the upper and lower limits of the mean Poisson’s ratio, (νxy +νyx )/2,
even when the system is anisotropic. As demonstrated in Fig.SI-10 (c), the data of (νxy +
νyx)/2 obtained from the simulation of the bulk modulus decreasing case obeys the isotropic
evolution of Poisson’s ratio, (νxy +νyx)/2≈(K/G −1)/(K/G + 1). By contrast, due to the
increasing of anisotropy for the case of reducing shear modulus, the (νxy +νyx)/2 seems to
be independent of K/G and fluctuates within the range indicated by Eq. (36), as shown in
Fig.SI-10(d).
If we keep removing those bonds with high contributions to the bulk modulus, the Poisson
ratio will be negative when K/G < 1. In supplementary video 1, we show this particular
case: as we remove the corresponding bonds, the Poisson’s ratio changes from positive to
negative. The blue squares indicate the original boundary of the network. After adding the
cross bonds following the cross bond adding method, video 2 shows that the Poisson’s ratio
is locked at a particular value while the rigidity of the network is enhanced, and the effective
medium approximation estimation can predict the magnitude of the moduli.
IX. SIMPLIFIED BOUNDARY CONDITIONS IN THE EXPERIMENTS
Because the affine and non-affine tunability on moduli are not restricted to any particular
modulus, we choose to use Young’s modulus instead of the bulk modulus in the experiment
due to practical convenience. For experimental purposes, we also simplify the boundary
condition by fixing the nodes on the top and bottom boundaries, as shown in Fig.SI-11 (a).
The deformation is about 4% of the system size.
Next, we illustrate the moduli’s deviations under our simplified boundary condition from
the rigorous definition. Because the ratio K/G is an excellent parameter to characterize the
non-affinity, we perform the simulations by imposing the simplified boundary conditions to
a continuous medium in the square shape at different K/G: we fix the shear modulus and
24

Fig.SI- 11: (a) A schematic demonstrates the deformations of Young’s modulus (top row) and the shear
modulus (bottom row). The left panel is for the rigorous definition, and the right panel is the real
experimental situation. (b) The Young’s modulus is obtained from the ideal and real experimental
conditions as the non-affinity (i.e., K/G) rises with Gbeing fixed. The two agree well with at most 10%
deviation. (c) The shear modulus is obtained from the ideal and real experimental conditions as K/G
rises. The two also agree well. (d) shows E/G for ideal and real situations. The ideal and real situations
are very close, and the largest deviation is around 10%.
increase the bulk modulus, and compare the two sets of moduli values under our boundary
condition and the ideal situation. As shown in Fig.SI-11 (b-d), the two sets of values mostly
agree with each other, with the largest deviation around 10%. Therefore, we can use our
simplified boundary condition to measure the moduli reliably.
X. EXTENDING 2D RESULTS INTO 3D
Furthermore, all our results in 2D can be extended to 3D. We have shown in 2D that
the affine state is achieved when any smallest patch cut off from the network has no floppy
25

mode and it is a triangle. Similarly, in 3D the smallest patch is formed by 4 non-coplanar
nodes and we apply the Maxwell isostatic theorem, N0=dN −NC−f(d) = 0, to such a
patch. Plugging into d= 3, N= 4, NS= 0, and f(d) = 6, we get NC= 6. Therefore, the
smallest 3D patch without floppy mode is a tetrahedron with four nodes and six bonds.
Similar to 2D, when we add more bonds above zaff , cross bonds will appear and
such bonds will break the topology of Euler-Poincar´e characteristic [18], which is the
3D generalization of the 2D Euler’s characteristic. Analogous to Euler characteristic in
2D, Euler-Poincar´e characteristic is also a topological invariant and can be expressed as:
Nf−NC+N−Ncell = 1, where Nf,NC,N,Ncell are the number of faces, bonds, nodes
and Voronoi cells. When cross bonds are added inside the fully tetrahedralized region, the
Euler-Poincar´e characteristic is no longer valid, as shown in the 3D situation of Fig.5(a)
in the main text: as the red bond is added, Nf= 7, NC= 10, N= 5, Ncell = 2, then
Nf−NC+N−Ncell = 0 = 1.
XI. SUPPLEMENTARY MOVIES
Movie-1: Producing negative Poisson’s ratio in our system. Left panel: the original
network has positive Poisson’s ratio whose vertical boundaries shrink when stretched. Right
panel: after deleting some bonds the network reaches negative Poisson’s ratio.
Movie-2: Adding intersecting bonds does not change the Poisson’s ratio. Left panel:
one network with negative Poisson’s ratio. Right panel: after adding intersecting bonds in
triangulated regions, the Poisson’s ratio remains negative and unchanged.
Movie-3: Illustration of bond removal and addition operations in our 2D spring network.
Movie-4: The comparison of internal strain fields between two 3D networks at z= 7.696
and z= 9.312. Their internal strains differ significantly due to our tuning operation.
Movie-5: The comparison of internal strain fields between two 3D networks at z= 9.312
and z= 10.432. Their internal strains agree well due to our locking operation with cross
bonds addition.
Movie-6: Illustration of 3D-printed detachable bonds, which enable reversible bond addi-
tion and removal within one single system. In this 2D packing derived network, the elastic
deformation changes after some specific detachable bonds are removed.
26

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