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Isostatic networks are minimally rigid and therefore have, generically,nonzero elastic moduli. Regular isostatic networks have finite moduli in thelimit of large sizes. However, numerical simulations show that all elasticmoduli of geometrically disordered isostatic networks go to zero with systemsize. This holds true for positional as well as for topological disorder. Inmost cases, elastic moduli decrease as inverse power-laws of system size. Ondirected isostatic networks, however, of which the square and cubic latticesare particular cases, the decrease of the moduli is exponential with size. Forthese, the observed elastic weakening can be quantitatively described in termsof the multiplicative growth of stresses with system size, giving rise to bulkand shear moduli of order exp{-bL}. The case of sphere packings, which onlyaccept compressive contact forces, is considered separately. It is argued thatthese have a finite bulk modulus because of specific correlations in contactdisorder, introduced by the constraint of compressivity. We discuss why theirshear modulus, nevertheless, is again zero for large sizes. A quantitativemodel is proposed that describes the numerically measured shear modulus, bothas a function of the loading angle and system size. In all cases, if a densityp>0 of overconstraints is present, as when a packing is deformed bycompression, or when a glass is outside its isostatic composition window, allasymptotic moduli become finite. For square networks with periodic boundaryconditions, these are of order sqrt{p}. For directed networks, elastic moduliare of order exp{-c/p}, indicating the existence of an "isostatic length scale"of order 1/p.

Elastic collapse in disordered isostatic networks