When are athermal soft-sphere packings jammed? Any experimentally relevant definition must, at the very least, require a jammed packing to resist shear. We demonstrate that widely used (numerical) protocols, in which particles are compressed together, can and do produce packings that are unstable to shear-and that the probability of generating such packings reaches one near jamming. We introduce a new protocol which, by allowing the system to explore different box shapes as it equilibrates, generates truly jammed packings with strictly positive shear moduli G. For these packings, the scaling of the average of G is consistent with earlier results, while the probability distribution P(G) exhibits novel and rich scalings.

We probe the onset and effect of contact changes in soft harmonic particle packings which are sheared quasistatically. We find that the first contact changes are the creation or breaking of contacts on a single particle. We characterize the critical strain, statistics of breaking versus making a contact, and ratio of shear modulus before and after such events, and explain their finite size scaling relations. For large systems at finite pressure, the critical strain vanishes but the ratio of shear modulus before and after a contact change approaches one: linear response remains relevant in large systems. For finite systems close to jamming the critical strain also vanishes, but here linear response already breaks down after a single contact change. Exciting progress in capturing the essence of the jamming transition in disordered media such as emulsions, granular matter, and foams has been made by considering the linear response of weakly compressed packings of repulsive, soft particles. When the confining pressure P approaches its critical value at zero, the resulting unjamming transition bears hallmarks of a critical phase transition: Properties such as the contact number and elastic moduli exhibit power law scaling [1][2][3][4][5][6][7][8], time and length scales diverge [5,[9][10][11], the material's response becomes singularly nonaffine [11,12], and finite size scaling governs the behavior for small numbers of particles N and/or small P [13][14][15].However, one may question the validity of linear response for athermal amorphous solids [16][17][18]. Due to disorder, one expects local regions arbitrarily close to failure, and in addition, near their critical point disordered solids are extremely fragile-even a tiny perturbation may lead to an intrinsically nonlinear response [18][19][20][21][22][23][24]. To avoid such subtleties, numerical studies of linear response have either resorted to simulations with very small deformations (strains of 10 −10 are not uncommon in such studies [25]), or have focused on the strict linear response extracted from the Hessian matrix [11][12][13]15].Here we probe the first unambiguous deviations from strict linear response: contact changes under quasistatic shear [ Fig. 1(a)]. We focus on three questions: (i) What is the mean strain γ cc at which the first contact change arises? γ cc should vanish when either N diverges or P vanishes. We find a novel finite size scaling relation for γ cc , where γ cc ∼ P for small systems close to jamming (N 2 P 1), andWhat is the nature of the first contact changes? Plastic deformations under shear have been studied extensively in systems far from jamming, which display avalanches: collective, plastic events in which multiple contacts are broken and formed and the stresses exhibit discontinuous drops [26][27][28][29][30]. A few studies have focused on what happens for hard particles, in the singular limit where even a single contact break may induce a complete loss of * deen@physics.leidenuniv.nl † hecke@physics.leidenuniv.nl rigidity [16,17,...

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