The nonlocal elastic response function is crucial for understanding many properties of soft solids. This may be obtained by measuring strain-strain autocorrelation functions. We use computer simulations as well as video microscopy data of superparamagnetic colloids to obtain these correlations for two-dimensional triangular solids. Elastic constants and elastic correlation lengths are extracted by analyzing the correlation functions. We show that to explain our observations displacement fluctuations in a soft solid need to contain affine (strain) as well as nonaffine components.Elasticity theory [1,2] is based on the assumption that the displacement field u(r) is smooth and continuous for all po sitions r within a volume V. It is expected that this assump tion breaks down [3] below a limiting length g el . How small is g el in practice, and what are the consequences of the break down? What implications does this have on the mechanical response of the material? While for most conventional engi neering problems such issues are practically irrelevant, they do become important when investigating microelastic and microrheological properties of soft solids [6]. We answer these questions in the context of the nonlocal mechanical response of a soft solid in two dimensions (2D). Nonlocal elastic effects are important whenever strain gradients are large, e.g., near crack tips, grain boundaries, and interfaces [4]. They should also be important if the linear dimension of the solid is not much larger than g el . Both of these criteria are satisfied in colloidal solids under usual experimental condi tions [5].Recently, there has been a lot of interest in obtaining the elastic moduli of soft solids [6,7] from fluctuations of par ticle coordinates [8][9][10][11][12]. These methods have several advan tages. First, no specialized equipment is necessary other than standard video microscopy [7]. Second, no external forces need to be applied which may change the very properties that are being measured. This is especially convenient when try ing to obtain elastic constants near the dislocation unbinding transition [1] where the solid is sensitive to external pertur bations [11]. Lastly, these techniques are general enough so that they may be applied to a wide variety of systems [13]. Here we extend this procedure a step further to study nonlo cal [4] elasticity for solids in two dimensions far from the melting transition.The nonlocal elastic response function, or compliance X ij (r , r') (i = x , y), is defined as the strain e ij (r') produced at position r' due to a stress C ij (r) at r. This is given by X ij = (k B T) −1 G ij , where, for a homogeneous solid, G ij (r') = (e i (0)e ( j r')> is the strain-strain correlation function. The (¯> denote a thermal average (and one over the choice of origin) and k B T is the Boltzmann constant times the tempera ture [1]. The linear combinations of components of the strain tensor e y , x ( u derivatives of the displacement field =1,2,3 i ( i e relevant to a 2D solid are ) defined as ) calcu...
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