Elasticity-controlled jamming criticality in soft composite solids

Soft composite solids are made of inclusions dispersed within soft matrices. They are ubiquitous in nature and form the basis of many biological tissues. In the field of materials science, synthetic soft composites are promising candidates for building various engineering devices due to their highly programmable features. However, when the volume fraction of the inclusions increases, predicting the mechanical properties of these materials poses a significant challenge for the classical theories of composite mechanics. The difficulty arises from the inherently disordered, multi-scale interactions between the inclusions and the matrix. To address this challenge, we systematically investigated the mechanics of densely filled soft elastomers containing stiff microspheres. We experimentally demonstrate how the strain-stiffening response of the soft composites is governed by the critical scalings in the vicinity of a shear-jamming transition of the included particles. The proposed criticality framework quantitatively connects the overall mechanics of a soft composite with the elasticity of the matrix and the particles, and captures the diverse mechanical responses observed across a wide range of material parameters. The findings uncover a novel design paradigm of composite mechanics that relies on engineering the jamming properties of the embedded inclusions.

Dispersing nano-to-micron-sized particles within a soft polymeric gel forms soft composite solids that are widely used in various engineering materials, including synthetic tissue [1], wearable biomedical devices [2,3], and soft robots [4].In addition to reinforcing the polymer matrix [5], the dispersed particles can enable diverse functional features such as anisotropic elasticity [6], shape-memory effects [7,8], and stimuli-responsive behaviors [9,10].Due to the great compliance of soft polymeric gels, the embedded particles can undergo moderate displacement within the matrix without causing internal fractures [9].This particle rearrangement may alter both the strain couplings among neighboring inclusions [11] and the stress fields over a large length scale [9,12].Compared with classical stiff composite materials [13], the current understanding of the multi-scale interactions within soft composites remains very limited.
The complexity of composite mechanics increases exponentially with the volume fraction of the inclusions.In a dilute composite, the mechanics is solely determined by the interactions between an isolated inclusion and the surrounding matrix, which allows the effective shear modulus to be described by the classical Eshelby theory [14].Further, modified effective medium theories have been extended to systems with finite-density inclusions, where neighboring particles interact via their induced strain fields [11,15].However, this assumption of matrix-mediated, short-range interactions breaks down in the dense limit, where the overall stress response may involve networks of direct contacts [16,17] or long-range rearrangements of dispersed particles [12].Due to the inherently disordered and heterogeneous microstructures of dense soft composites, predicting their mechanics is challenging for the classical composite theories.
To address these issues, we systematically investigated the strain stiffening of soft elastomers containing a high volume fraction of stiff microspheres.Inspired by the concepts of both granular jamming [18][19][20][21] and rigidity transitions in disordered systems [22][23][24], we demonstrate that the mechanical responses of soft composites are governed by elasticity-controlled scalings near a continuous phase transition.In the absence of matrix elasticity, the transition coincides with shear-jamming of the included particles.This novel criticality framework captures the stiffening responses for a variety of material parameters where the classical theories break down.The results provide a new approach to understand the nonlinear mechanical responses of various multi-phase soft materials.

Strain-stiffening responses of soft composite solids
We prepared compliant polydimethylsiloxane (PDMS) elastomers filled with stiff polystyrene (PS) microspheres having an average diameter of 30 µm (Fig. 1(a) and Fig. S6).While the shear modulus of the PS spheres is G p = 1.6 GPa (Fig. S1), the shear modulus of the PDMS matrix was systematically varied from G m = 0.04 kPa to 4 kPa by tuning the crosslinking density (Ref.[25] and Fig. S5).The mechanical properties of the soft composites were characterized using a rheometer equipped with a parallel-plate shear cell (Fig. 1(b)).The top plate controls the gap size (d) and applies axial compressive strains (ε) (Fig. S7).The volume of the sample remains unchanged under an axial compression (Fig. S9), which gives rise to a pure shear.At each given ε, the shear modulus G of the composites was measured using an oscillatory shear with a small amplitude (δγ a = 0.01 %) and a low angular frequency (ω = 0.1 rad/s).The resulting G(ε) represents the linear elastic response of the soft composites in differently sheared states (Fig. S8).
The dense soft composites exhibited characteristic strain-stiffening responses under the axial compressions (Fig. 1(c)).The stiffening degree was determined by both the particle volume fraction ϕ and the shear modulus of the elastomer matrix G m .First, at a fixed G m = 1.28 kPa, the relative shear modulus, G r = G/G m , grows more rapidly with ε as ϕ increases from 0.44 to 0.67.Second, at a fixed ϕ = 0.60, the strain stiffening becomes more pronounced while G m decreases from 1.28 kPa to 0.04 kPa.
We define G r,max as the relative shear modulus at the maximally stiffened states and G r,0 as the relative shear modulus at ε = 0. Within experimental uncertainty, G r,max appears at approximately ε = 0.2 regardless of ϕ and G m .Therefore, we estimated G r,max for all the samples using the values of G r at ε = 0.2.For ε > 0.2, G r decreases with ε, and the composites were unable to fully recover their original shapes after the compressions were removed.This plasticity was likely caused by internal fractures between the elastomer and the particles [26].In contrast, the plots of G r (ε) appear to be highly reproducible when the compressions are released at ε < 0.2.
Hence, we focus exclusively on the stiffening regime between ε = 0 and 0.2.
Figure 1(d) shows both G r,max (solid points) and G r,0 (hollow points) as a function of ϕ as G m varies between 0.04 kPa and 1.28 kPa.For ϕ < 0.4, only G r,0 was reported since no strain-stiffening was found.For comparison with the classical theories of composite mechanics, we plotted the predictions from the Eshelby theory [14] and the Mori-Tanaka approximation scheme [15], which align well with the G r,0 measured in the dilute limit (ϕ < 0.2).However, for dense composites (ϕ > 0.4), the classical theories significantly deviate from the measured G r,max and G r,0 , and also fail to describe the strain-dependent shear modulus G r (ε).These mismatches suggest that potential mechanisms, such as direct contact between inclusions [16,17], were overlooked in the classical models of the mechanics of dense soft composites.

Signatures of jamming-controlled elasticity
We re-examined the super-exponential rise of G r,max in Fig. 1(d).As G m decreases, the growth of G r,max becomes increasingly more divergent near ϕ ≈ 0.6.Since a soft composite solid will asymptotically become a granular suspension as the matrix elasticity approaches zero, we hypothesize an underlying connection between the shear-jamming of dense suspensions and the strainstiffening of soft composites in the limit of G m → 0.
To validate this assumption, we first characterized the shear rheology of a concentrated PS suspension in the PDMS base solution without any crosslinkers (Fig. S2).

Suspensions
We define the relative viscosity (η r ) as the ratio of the viscosity of the suspension (η) to that of the PDMS base (η s = 1.0 Pa•s): η r = η/η s .The left panel in Fig. 2(a) shows η r measured under steady shear flow in the frictional regime.The results are well described by the Krieger-Dougherty relation [27] η with a fixed exponent γ = 2 and a fitted jamming volume fraction ϕ J = 0.594 ± 0.003.For ϕ > ϕ J , we did not observe homogeneous steady shear flow at any shear rate.In this regime, as shown in the right panel of Fig. 2(a), nonzero shear moduli of the PS-PDMS suspensions (G s ) were measured at ε = 0.2.Since no significant change in G s was found when ε was further increased, ϕ J = 0.594 represents the shear-jamming volume fraction of the PS-PDMS suspensions in the large strain limit.In Fig. 2(b), we plot η r (ϕ) from Eq. 1 together with G r,max (ϕ) of the composites for a comparison.The traces of G r,max gradually converge to η r as G m decreases, suggesting that G r,max ≈ (1 − ϕ/ϕ J ) −γ for ϕ < ϕ J as G m approaches zero, and the actual shear modulus G max scales linearly with G m in this limit.In contrast, for ϕ > ϕ J , G max becomes independent of G m (Fig. 2(c)) and is close to the value of G s measured independently from the jammed suspensions (Fig. 2(a)), suggesting a particle-dominated response.Considering the contrasting mechanical behaviors exhibited for the ranges ϕ < ϕ J and ϕ > ϕ J , it is likely that the shear-jamming point of the suspensions controls a crossover of the mechanical properties of the composites.

Elasticity-controlled criticality near jamming
Since the plots of G max (ϕ) in Fig. 2(c) resemble the critical behaviors near a continuous phase transition [23,24,28,29], we next investigated the scalings of the composite shear modulus (G max ) near (ϕ = ϕ J , G m = 0).Motivated by the observation that G r,max approaches η r as G m → 0 (Fig. 2(b)) and the classical analogy between the effective shear modulus and the shear viscosity in multi-phase systems [30][31][32], we conjectured the scaling law with γ = 2 and ϕ J = 0.594.To demonstrate the validity of this scaling assumption, we plot G r,max against ϕ J − ϕ in Fig. 3(a) with different G m values, where the results show the best agreement with Eq. 2 for the softest matrix.We further considered how G max varies with G m at ϕ = ϕ J .In Figure 3(b), G max is plotted at ϕ = 0.59 ≈ ϕ J against G m , which can be fitted to the power-law scaling with a fitted exponent 1/δ = 0.6 ± 0.1.Considering the scalings in Eqs. 2 and 3, we compared the soft composites near ϕ J with a ferromagnetic system near the Curie temperature (T c ).The material parameters of the soft composites (G max , G m , ϕ−ϕ J ) are directly analogous to (M, H, T − T c ) in the Ising model.By assuming a scale-invariant free energy at the critical point (ϕ = ϕ J , G m = 0), we propose a universal scaling form where β = γ/(δ − 1) = 3.0 ± 0.7 and ∆ = δβ = 5.0 ± 1.1, and the crossover scaling functions f + and f − apply to the regimes of ϕ > ϕ J and ϕ < ϕ J , respectively.The derivations of Eq. 4 and the relationships between the exponents are described in Methods.A similar scaling was previously applied to study fiborous networks near central force rigidity transitions, where the bending rigidity plays a similar role as G m in soft composites [23,29].
To test our scaling ansatz (Eq.4), the mechanical responses (G max ) measured for different G m and ϕ are plotted in Fig. 3(c) using the rescaled variables The range of G m spans two orders of magnitude, from 0.04 kPa to 4.0 kPa, while ϕ increases from 0.45 to 0.67 around ϕ J = 0.594.Consistent with Eq. 4, the data points for ϕ > ϕ J and ϕ < ϕ J are nicely collapsed onto two distinct branches.
The ϕ < ϕ J branch exhibits a slope close to 1, indicating that G max ∼ G m .The ϕ > ϕ J branch reaches a plateau independent of G m , suggesting that G max is dominated by the particle phase.In the limit of ϕ ≈ ϕ J , a critical regime emerges where the two branches become indistinguishable and both follow the same scaling,
Assuming that the critical exponents (β = 3 and ∆ = 5) and the material parameters (c 1 = 1.4 and c 2 = 1.3) remain constant for different ε, Eq. 5 is rewritten as where hε For each ε, we search for an optimal ϕ J (ε) that allows the composite shear modulus G(ε) measured with different G m and ϕ to be collapsed onto Eq. 6 (the dashed gray line in Fig. 4(a)).As a consequence, we are able to overlay G(ε) measured within the range of 0 ≤ ε ≤ 0.2 by plotting The resulting ϕ J (ε) in Fig. 4(b) can be fitted to a form that describes the shear-jamming phase boundary of granular materials [35,38,41] with ϕ 0 = 0.688 ± 0.004, ϕ m = 0.594 ± 0.002, and a characteristic strain scale ε * = 0.035±0.003.While ϕ m agrees with ϕ J = 0.594 measured under the steady-state rheology of the PS-PDMS suspensions shown in Fig. 2(a), ϕ 0 is consistent with the simulated random close packing of spheres having the same size distribution as our samples (Fig. S4).Although Eq. 7 was obtained from the scaling The solid black and gray curves represent the best fits of ϕJ(ε) to Eq. 7 for these two material systems, respectively.The pink area indicates the shear-jammed phase of the PS-PDMS suspensions.(c) Plots of the shear modulus of PS-PDMS composites (G) as a function of both ε and ϕ.The blue and pink connected points represent the experimental results for Gm = 1.28 kPa and 0.12 kPa, respectively.The blue and pink surfaces represent the theoretical predictions from Eq. 8 for these two Gm values.
behaviors of soft composites, it effectively predicts the line of rigidity transitions for the PS-PDMS suspensions in our experiments (Fig. S3).
To test the universality of the scaling model, we further examined a different composite system made by dispersing glass beads in PDMS matrices.The size of these glass beads is similar to that of the PS particles but their shear modulus is ten times higher; that is, G p = 15.8GPa.The results of the glass-PDMS composites are collapsed onto the same plot in Fig. 4(a) with the same critical exponents β = 3 and ∆ = 5 but different material constants c 1 = 0.9 and c 2 = 0.8.The difference in c 1 and c 2 is likely due to the high bonding energy between glass and PDMS.The resulting ϕ J (ε) was also fitted to Eq. 7 with ϕ 0 = 0.676 ± 0.003, ϕ m = 0.613 ± 0.003, and ε * = 0.040 ± 0.07.We again found that ϕ m = 0.613 is consistent with the shear-jamming point of the glass-PDMS suspensions and that ϕ 0 = 0.676 is consistent with the predicted random close packing.
With the given parameters G m , ϕ and ε, we can calculate the shear modulus of soft composites as (8) where ϕ J (ε) is given by Eq. 7, and the functions f ± can be evaluated by numerically solving the inverse functions g ± in Eq. 6.In Fig. 4(c), we compared the measured shear moduli of two sets of PS-PDMS samples with G m = 0.12 kPa and 1.28 kPa, respectively, to the theoretical predictions from Eq. 8.
The phase diagram in Fig. 5 summarizes the fundamental aspects of our criticality framework.The G m = 0 plane represents the granular suspensions consisting of particles in uncrosslinked polymers.The solid red curve within the plane, ϕ = ϕ J (ε), denotes the boundary of the shear-jamming transition [21,33].Soft composites exist in the 3D space characterized by G m > 0, and the vertical planes in Fig. 5 represent the cross sections of this space at different strains.While there is no rigidity transition in this space with G m > 0, the mechanics is determined by the critical scalings near ϕ where C is a prefactor depending on the material parameters c 1 and Phase diagram of the mechanical responses of soft composite solids and granular suspensions.The Gm = 0 plane represents the suspensions consisting of particles dispersing in uncrosslinked polymers.The solid red line in the Gm = 0 plane signifies the shear-jamming transition (ϕJ(ε)) of dense suspensions [33].The 3D space defined by Gm > 0 represents soft composites consisting of particles dispersing in crosslinked polymeric elastomers.The mechanical properties of dense soft composites under different strains ε are controlled by the scalings (Eq.8) near the critical line ϕJ(ε).The dashed red lines Gm/Gp = |1 − ϕ/ϕJ(ε)| ∆ indicate the crossover boundary from the matrix-or particle-dominated regime to the critical regime.The solid arrow (A → B → C) illustrates a representative strain-stiffening process of soft composites with a particle volume fraction ϕm < ϕ < ϕ0.With the increase in the applied strain ε, the mechanical response of the composites crosses over from the matrix-dominated regime (ε = 0) to the critical regime (ε = ε1), and finally to the particle-dominated regime (ε = ε2).
posite is anticipated to be in the critical regime, where

Discussion and conclusions
The study revealed the essential role of shear-jamming in the mechanics of soft composites in the dense limit, a regime where the system becomes highly responsive and promises wide-ranging applications, yet remains challenging to model using conventional tools from continuous mechanics.We show that the strain-stiffening of soft composites can be interpreted as a manifestation of the criticality near a strain-dependent jamming point of dense suspensions (Fig. 5).The efficacy of our scaling model reveals the unique mechanical features of soft composites.As G m decreases to the order of 10 1 ∼ 10 2 Pa, the PDMS matrix consists of both a weakly crosslinked network and a substantial amount of uncrosslinked free chains.The characteristic pore size of the network can be estimated as a ∼ (k B T /G) 1/3 ∼ 50 nm [42].Therefore, the particles included in the PDMS matrix can po-tentially move to create direct contacts without causing fractures in the network.Consequently, the contact network within soft composites may resemble that in shearjammed granular systems as G m approaches zero.
From the perspective of materials science, the study will benefit materials design in tissue engineering.Strainstiffening has been widely observed in both biological [43] and synthetic tissues [10,16], with the prevailing interpretations attributing them to the nonlinear mechanics of the fibrous networks in the matrix.The significance of direct contacts between inclusions and the associated jamming transition in soft matrices began attracting attention only recently [16,44].A key difference between our experiments and previous studies [10,43,44] is that the strain stiffening in our study occurs without increasing the volume fraction and thus cannot be explained by the model in Ref. [16].The connection between strainstiffening in incompressible soft composites and shearjamming in dense suspensions offers a new scheme in designing the tissue-like mechanics of soft composites.

Material preparation
Particle inclusions -Both the PS and glass particles are micron-sized spheres with size distributions that can be described by the log-normal function f ).For the PS particles, r 0 = 12 µm and σ = 0.6.For the glass particles, r 0 = 20 µm and σ = 0.5.The shear modulus of the particles, G p , was measured by compressing individual beads between two flat substrates using a nanoindenter (Bruker, Hysitron TI-980).The resulting force-displacement curves were fitted to the Hertzian contact model (see Fig. S1(b) in the Supplementary Materials).The results showed that G p = 1.6 GPa and 15.8 GPa for the PS and the glass particles, respectively.
Soft matrix -The PDMS matrix was made by mixing a silicone base vinyl-terminated polydimethylsiloxane (DMS-V31, Gelest Inc) with copolymer crosslinkers (HMS-301, Gelest Inc) and a catalyst complex in xylene (SIP6831.2,Gelest Inc).We prepared two mixture solutions, Gelest Part A and Gelest Part B, before curing.In particular, Part A consisted of a silicone base with 0.005 wt% catalyst, and Part B consisted of a silicone base with 10 wt% crosslinkers.By changing the weight ratio of A to B from 14.5:1 to 8:1, we varied G m from 0.04 kPa to 4 kPa.
Fabrication of the soft composites-We prepared diskshaped composite samples with 10 mm radius and 10 mm height an acrylic mold covered with a para-film.To fully relax the internal structures, we used a vortex mixer (BV1000, Benchmark Scientific Inc.) to vibrate the samples immediately after mixing all the components.For ϕ > 0.5, we compressed the samples using a glass plate to flatten the top surface.Each sample was then left to cure at room temperature for at least 48 hr.

Criticality analysis
Scaling form of the equations of states -We first show how the scaling form of the equations of state shown in main text Eq. 4 can be obtained by minimizing a scaleinvariant phenomenological free energy.Denote the singular part of the free energy of a dense granular suspension (G m = 0) under a given axial strain ε as F (Φ, G), where G ≡ G/G p is the dimensionless shear modulus, and Φ ≡ ϕ/ϕ J (ε) − 1 is the reduced volume fraction.For a given length scale l, we assume that the free energy is self-similar near the critical point Φ = 0, where d = 3 is the space dimension, and y Φ and y G are the scaling dimensions of Φ and G, respectively.Considering l = |Φ| − 1 y Φ , Eq. M1 can be expressed as where F+ and F− are different forms of the free energy in the regimes of Φ > 0 and Φ < 0, respectively.For a composite with G m > 0, the parameters {G, G m } are analogous to {M, H} in the Ising model.We define G m ≡ G m /G p as the dimensionless shear modulus of the elastomer matrix.To transform the variable by substituting {Φ, G} with {Φ, G m }, we minimize the following Legendre transformation function In soft composites, F (Φ, G) and L(Φ, G m ) are in direct analogy to the Helmholtz free energy and Gibbs free energy in thermodynamic systems.The explicit evaluation of Eq.M3 leads to where ∆ ≡ (d − y G )/y Φ , and β ≡ y G /y Φ .By defining f ± as the inverse functions of F ′ ± , we obtain the scaling form of the equations of state shown in Eq. 5 of the main text: (M6) Compared with Eq. 2 in the main text, we have γ = ∆ − β.
In addition, Eq.M5 suggests that f ± (G m |Φ| −∆ ) ∝ (G m |Φ| −∆ ) β/∆ at the critical point Φ = 0 to prevent the divergence of free energy.Therefore, we have Compared with Eq. 3 in the main text, we obtain δ = ∆/β.Explicit form of the equations of states -We next derive the explicit form of the equation of states in Eq. 5 in the main text.Based on the scale-invariant expression of Eq.M1, the expansion of F (Φ, G) should comprise terms Φ a G b with ay Φ + by G = d.Therefore, F can be expressed as ) where a i > 0, and µ i,± are the expansion coefficients for Φ > 0 and Φ < 0. By evaluating the variation in Eq.M3, we obtain The above equation can be further simplified by including only three terms to describe the key experimental observations.First, λ = µ 0 (∆ + β)/β > 0 when a 0 = 0 to ensure that the free energy is minimum at G = 0 while Φ = 0. Second, µ ′ 1,± = µ 1,± (∆ + β − 1)/β ̸ = 0 when a 1 = 1 to ensure that ∂G(Φ)/∂Φ ̸ = 0 at Φ = 0. Finally, because G ∝ G m in the matrix-dominated regime, we have µ ′ 2,± = 2µ 2,± ̸ = 0 when a 2 = ∆ − β.As a consequence, G m can be simplified as Due to the intrinsic nature of a continuous phase transition at Φ = 0, we have µ ′ i,± = ∓µ ′ i with µ ′ i > 0 for both i = 1 and 2. By defining the reduced variables m ≡ G/|Φ| β and h ≡ G m /|Φ| ∆ , Eq. M10 can be rewritten as where c 1 = λ, c 2 = µ ′ 1 , and c 3 = µ ′ 2 .In the regime of Φ < 0, we experimentally observed h/ m = 1 as m → 0, suggesting that c 3 = 1.Therefore, we finally obtain which is Eq. 5 in the main text.
The role of the material constants c 1 and c 2 -In the particle-dominated regime, when Φ > 0 and h = 0, the nonzero solution of m from Eq. M12 gives the prefactor C in the scaling of the shear modulus G = CG p |1 − ϕ/ϕ J | β .The value of C can be obtained by solving and is thus determined by both c 1 and c 2 .
In the critical regime, as Φ = 0 and both m → ∞ and h → ∞, Eq.M12 reduces to h = c 1 m∆/β , which gives

FIG. 1 .
FIG. 1. Strain-stiffening of PS-PDMS soft composites under volume-conserving compressions.(a) Schematic of the cross section of PS-PDMS composites.For a predetermined volume fraction ϕ, polydisperse PS spheres with an average diameter of approximately 30 µm are well dispersed in a crosslinked PDMS matrix.(b) Schematic of the experimental setup used to characterize the strain-stiffening of the soft composites.The top plate moves down in a stepwise manner to apply an axial strain ε.At each ε, the linear shear modulus G was measured through an oscillatory shear with a strain amplitude of δγ = 10 −4 and an angular frequency of ω = 0.1 rad/s.(c) Plots of the relative shear modulus, Gr = G/Gm, against ε for various particle volume fractions ϕ and matrix shear moduli Gm.The blue hollow circles indicate the results for a constant Gm = 1.28 kPa as ϕ increases from 0.44 to 0.67.In addition, the hollow red squares and hollow yellow triangles represent the results of Gr(ε) at the same ϕ = 0.60 but for different matrix moduli, Gm = 0.12 kPa and Gm = 0.04 kPa, respectively.(d) Comparison between the experimentally measured Gr and the predictions from the classical theories of composite mechanics.The solid and hollow points indicate Gr,0 and Gr,max, respectively, versus ϕ for samples with varying Gm.The two dashed gray lines represent the predictions from the Eshelby theory and the Mori-Tanaka approximation.The pink area represents the range of the volume fraction where strain-stiffening was observed.

FIG. 2 .
FIG. 2. Signatures of jamming-controlled elasticity.(a) Rigidity transition of PS particles suspended in uncrosslinked silicone oil.The black triangles show the relative viscosity ηr = η/ηs in frictional flow states for different particle volume fractions ϕ = 0.45, 0.49, 0.53, 0.55.The dashed black curve indicates the best fit of the experimental results to Eq. 1 where ϕJ = 0.594.In the regime of ϕ > ϕJ, the suspensions are shear jammed under the axial strain ε = 0.2.The red crosses represent their shear moduli (Gs) in the shearjammed states.(b) Plots of Gr,max against ϕ for different Gm values.To compare the absolute values of Gr,max with ηr, the fit to Eq. 1 obtained in panel (a) is also shown by the dashed gray line in the same plot.(c) The actual shear modulus Gmax is plotted against ϕ for different Gm values based on the results in (b).The solid lines in both panels (b) and (c) are predictions from the scaling model based on jamming criticality (Eqs.4 and 5).
FIG. Elasticity-controlled criticality near jamming.(a) Plots of Gr,max against ϕJ−ϕ for Gm = 0.12 kPa, 0.36 kPa, and 1.28 kPa, respectively.The dashed gray line indicates the scaling law of Eq. 2. (b) Plots of Gmax versus Gm for composites with ϕ = 0.59 ≈ ϕJ, where the dashed black line represents the scaling law of Eq. 3. (c) Scaling collapse of Gmax, normalized by |1 − ϕ/ϕJ| β , as a function of Gm/|1 − ϕ/ϕJ| ∆ with ϕJ = 0.594, β = 3, and ∆ = 5.The solid markers represent the experimental results obtained for ϕ > ϕJ, and the open markers represent the results obtained for ϕ < ϕJ.The data points are labeled with different colors based on ϕ.The dashed red and blue curves are the best fits to the equations of state (Eq.5) for the experimental results within ϕ > ϕJ and ϕ < ϕJ, respectively.
6  m .To model G max analytically, we derived an explicit form of the equations of stateh = g ± ( m) = c 1 m∆/β ∓ c 2 m(∆−1)/β ∓ m (5)where g ± are the inverse functions of f ± .The re-duced variables h ≡ G m G −1 p |1 − ϕ/ϕ m | −∆ and m ≡ G max G −1 p |1 − ϕ/ϕ m | −βwere used to simplify the notation.The derivation is detailed in Methods.By fitting the data in Fig.3(c) to Eq. 5, we obtained the material constants c 1 = 1.4 and c 2 = 1.3 for the PS-PDMS composites.With all the essential parameters (ϕ J , β, ∆, c 1 , and c 2 ), we can calculate G max for a given ϕ and G m .For instance, the colored solid lines in Figs.2(b) and (c) represent the theoretical predictions from Eqs. 4 and 5.

FIG. 4 .
FIG. 4. Criticality near a strain-dependent jamming transition.(a) Collapse of the rescaled composite shear modulus G(ε)/(Gp|1 − ϕ/ϕJ(ε)| β ) as a function of the rescaled matrix shear modulus Gm/(Gp|1 − ϕ/ϕJ(ε)| ∆ ) at different axial strains (ε) with two fixed critical exponents β = 3 and ∆ = 5.The data points include the experimental results obtained for two composite systems: PS-PDMS and glass-PDMS soft composites.The dotted gray (and pink) curves represent the best fit to Eq. 6 for the PS-PDMS (and glass-PDMS) composites.The vertical dashed line (Gm/Gp = |1 − ϕ/ϕJ(ε)| ∆ ) approximates the crossover boundary from the critical regime to the particle-or matrix-dominated regime.(b) Plots of the fitted ϕJ(ε) for PS-PDMS (open black circles) and glass-PDMS systems (gray uptriangles).The error bars indicate the fitting uncertainties.The solid black and gray curves represent the best fits of ϕJ(ε) to Eq. 7 for these two material systems, respectively.The pink area indicates the shear-jammed phase of the PS-PDMS suspensions.(c) Plots of the shear modulus of PS-PDMS composites (G) as a function of both ε and ϕ.The blue and pink connected points represent the experimental results for Gm = 1.28 kPa and 0.12 kPa, respectively.The blue and pink surfaces represent the theoretical predictions from Eq. 8 for these two Gm values.