Simple average expression for shear-stress relaxation modulus

Abstract

Focusing on isotropic elastic networks we propose a novel simple-average expression G(t) = µA − h(t) for the computational determination of the shear-stress relaxation modulus G(t) of a classical elastic solid or fluid and its equilibrium modulus Geq = limt→∞ G(t). Here, µA = G(0) characterizes the shear transformation of the system at t = 0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stressτ (t) as a function of time t. While investigating sampling time effects we also discuss the related expressions in terms of shear-stress autocorrelation functions. We argue finally that our key relation may be readily adapted for more general linear response functions.

Shear-stress relaxation in free-standing polymer films

“…Incidentally, using the Lebowitz-Percus-Verlet transformation rules this provides one way to elegantly demonstrate the stress-fluctuation formula Eq. (1) within a couple of lines [8,10]. Interestingly, the expectation values, i.e.…”

“…The indicated ∆t-dependences naturally arise since the averages for µ A and µ F are commonly and most conveniently done by first "timeaveraging" over time windows of length ∆t of the stored * Electronic address: joachim.wittmer@ics-cnrs.unistra.fr data entries of a given configuration and only in a second step by "ensemble-averaging" over completely independent configurations (Appendix B 2). Assuming the time-translational invariance of the time series it can be demonstrated (Appendix C) that the ∆t-dependence can be traced back to the stationarity relation [9,11,14] µ(∆t) = 2 ∆t 2 ∆t 0 dt t 0 dt G(t ).…”

“…(2) the meaning of a generalized quasi-static modulus also containing information about dissipation processes associated with the reorganization of the particle contact network. This has been extensively tested for self-assembled transient networks [11].…”

Dynamics of Vitrimers: Defects as a Highway to Stress Relaxation

“…This gives a detailed picture of a single exchange event, but does not provide large enough time and length scales to assess macroscopic properties. To get to macroscopic scales, a coarse-grained model that captures the network-topology aspects of the exchange reactions is needed.In recent years, scientists have developed different numerical models to study exchange materials [11][12][13], embedding Monte Carlo hopping moves into hybrid molecular dynamics or Monte Carlo (MD,MC) simulations to reproduce bond swaps. In this letter, we study a vitrimer model consisting of associative star-polymers using a three-body potential to reproduce bond exchange dynamics with a controllable rate [14], avoiding the need for hybrid features.…”

“…The stress autocorrelation function is often assumed to be equal to the stress relaxation G(t), but it was recently pointed out that the equality holds only in liquids [27,28]. Still, for self-assembled networks, C(t) on average converges to G(t) [11]. The correct way to define the stress relaxation would bewhere G eq is the shear modulus and C ∞ is the long-time asymptote of C(t) (so C ∞ ∝ σ 2 ).…”