Shear-strain and shear-stress fluctuations in generalized Gaussian ensemble simulations of isotropic elastic networks

Shear-strain and shear-stress correlations in isotropic elastic bodies are investigated both theoretically and numerically at either imposed mean shear-stress τ (λ=0) or shear-strain γ (λ=1) and for more general values of a dimensionless parameter λ characterizing the generalized Gaussian ensemble. It allows to tune the strain fluctuations $$\mu _{\gamma \gamma } \equiv \beta V\left\langle {\delta \hat \gamma ^2 } \right\rangle =(1 - \lambda )/G_{eq}$$ with β being the inverse temperature, V the volume, $$\hat \gamma$$ the instantaneous strain and G eq the equilibrium shear modulus. Focusing on spring networks in two dimensions we show, e.g., for the stress fluctuations $$\mu _{\tau \tau } \equiv \beta V\left\langle {\delta \hat \tau ^2 } \right\rangle$$ ( $$\hat \tau$$ being the instantaneous stress) that μ ττ | λ =μ A − λ G eq with μ A =μ ττ | λ=0 being the affine shear-elasticity. For the stress autocorrelation function $$C_{\tau \tau } (t) \equiv \beta V\left\langle {\delta \hat \tau (t)\delta \hat \tau (0)} \right\rangle$$ this result is then seen (assuming a sufficiently slow shear-stress barostat) to generalize to C ττ (t)| λ =G(t) − λ G eq with G(t)=C ττ (t) | λ=0 being the shear-stress relaxation modulus. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015