Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions

Abstract

We revisit the relation between the shear stress relaxation modulus G(t), computed at finite shear strain 0 < γ 1, and the shear stress autocorrelation functions C(t)|γ and C(t)|τ computed, respectively, at imposed strain γ and mean stress τ . Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t) = C(t)|τ = C(t)|γ + Geq for t > 0 with Geq being the static equilibrium shear modulus. G(t) and C(t)|γ thus must become different for solids and it is impossible to obtain Geq alone from C(t)|γ as often assumed. We comment briefly on self-assembled transient networks where Geq(f ) must vanish for a finite scission-recombination frequency f . We argue that G(t) = C(t)|τ = C(t)|γ should reveal an intermediate plateau set by the shear modulus Geq(f = 0) of the quenched network.

Simple average expression for shear-stress relaxation modulus

“…We begin by presenting in Sec. II the numerical model and remind some properties of the specific elastic network investigated already described elsewhere [11,12]. Our numerical results are then discussed in Sec.…”

“…As in Ref. [11] δτ (t) has been averaged over 1000 configurations. The perfect collapse of all data presented confirms the key relation and this for all sampling times ∆t.…”

“…The long-time response yields the equilibrium shear modulus G eq = lim t→∞ G(t) as shown in panel (c). More readily, one may obtain G eq by means of equilibrium simulations performed at constant volume V and shear strain γ using the stress-fluctuation relation [5][6][7][8][9][10][11][12] G eq = g ≡ µ A − σ…”

“…with µ A being the "affine shear-elasticity" [10][11][12], a simple average characterizing the second order energy change under a canonical-affine shear strain [12]. The second contribution σ ≡ βV δτ 2 =σ − σ stands for the rescaled shear stress fluctuation with β being the inverse temperature and where we have introduced for later convenience the two termsσ ≡ βV τ 2 and σ ≡ βV τ 2 .…”

Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited

“…Examples of current interest for the determination of G eq include crystalline solids [10], glass forming liquids and amorphous solids [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], colloidal gels [27], polymeric networks [1,[28][29][30], hyperbranched polymer chains with sticky end-groups [31] or bridged equilibrium networks of telechelic polymers [32]. * Electronic address: hongxu@univ-lorraine.fr temperature T (β = 1/k B T denoting the inverse temperature) [33].…”