Elastic heterogeneity of soft random solids

Spatial heterogeneity in the elastic properties of soft random solids is examined via a semi-microscopic model network using replica statistical mechanics. The elastic heterogeneity is characterized by random residual stress and Lamé coefficient fields, and the statistics of these quantities is inferred. Correlations involving the residual-stress field are found to be long ranged and governed by a universal parameter that also gives the mean shear modulus. Copyright c © EPLA, 2007 Introduction. – As a consequence of randomness incorporated at synthesis, random solids (e.g. polymer networks, glasses, α-Si) are heterogeneous. For example, the mean positions of the constituent particles exhibit no apparent long-range order, and every particle inhabits a unique spatial environment. Particularly for soft random solids, such as rubber, in which the particle positions undergo large thermal fluctuations, heterogeneity also manifests itself via the RMS particle displacements, which are random and continuously distributed [1–3]. The elasticity of rubber, and especially its softness with respect to shear deformations, have been studied for many years via the classical theory, developed by Kuhn, Flory, Wall, Treloar and others [4] and based on a microscopic picture of Gaussian polymer chains. While the classical theory has proved highly successful, it is a homogeneous theory, preserving no information about the random structure. Thus, it is incapable of describing consequences of the heterogeneity, such as random spatial variations in the local elastic parameters, or nonaffine local strain response to macroscopic stress [5–7]. In this, letter we present a theoretical development that goes beyond the classical theory of rubber elasticity by accounting for the heterogeneity. What emerges is an elasticity theory featuring spatially fluctuating Lamé coefficients and residual stresses, together with a statistical characterization of these quantities in terms of their mean values and spatial selfand cross-correlations. In particular, we find that not only is the stress-stress correlation long ranged —behavior that can be argued for on general grounds— but so are the cross-correlation between the residual stress and, e.g., the shear modulus. By contrast, we find the self-correlation of the shear modulus to be short ranged. Furthermore, we find that the long-ranged correlations and the average shear modulus are governed by a common universal parameter that is independent of microscopic details. To obtain our statistical characterization of soft random solids we take the following route. First, we examine a nonlocal phenomenological model of a random elastic medium, which we subsequently derive from a semimicroscopic model. We then determine the state to which it relaxes when randomness is present, and re-expand the elastic free energy around this new equilibrium reference state. This relaxed state is, however, still randomly stressed [9]; nevertheless, the stress in the relaxed state —the so-called residual stress— satisfies the mechanical equilibrium condition ∂jσjk(x) = 0. In its local limit, the proposed phenomenological model reproduces a version of Lagrangian elasticity theory that features random Lamé coefficients and residual stresses. Second, we consider the statistical mechanics of a minimal semi-microscopic model of a random-solid-forming system —the randomly linked particle model (RLPM) [10]. Via the replica method, applied to the RLPM to deal with its structural randomness, followed by an analysis of Goldstone fluctuations 1A similar construction was employed in ref. [8].