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Register a new userStress-stress fluctuation formula for elastic constants in the NPT ensemble
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Several fluctuation formulas are available for calculating elastic constants from equilibrium correlation functions in computer simulations, but the ones available for simulations at constant pressure exhibit slow convergence properties and cannot be used for the determination of local elastic constants. To overcome these drawbacks, we derive a stress-stress fluctuation formula in the $NPT$ ensemble based on known expressions in the $NVT$ ensemble. We validate the formula in the $NPT$ ensemble by calculating elastic constants for the simple nearest-neighbor Lennard-Jones crystal and by comparing the results with those obtained in the $NVT$ ensemble. For both local and bulk elastic constants we find an excellent agreement between the simulated data in the two ensembles. To demonstrate the usefulness of the new formula, we apply it to determine the elastic constants of a simulated lipid bilayer.
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We revisit the relation between the shear stress relaxation modulus $G(t)$, computed at finite shear strain $0 < gamma ll 1$, and the shear stress autocorrelation functions $C(t)|_{gamma}$ and $C(t)|_{tau}$ computed, respectively, at imposed strain $gamma$ and mean stress $tau$. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general $G(t) = C(t)|_{tau} = C(t)|_{gamma} + G_{eq}$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. $G(t)$ and $C(t)|_{gamma}$ thus must become different for solids and it is impossible to obtain $G_{eq}$ alone from $C(t)|_{gamma}$ as often assumed. We comment briefly on self-assembled transient networks where $G_{eq}(f)$ must vanish for a finite scission-recombination frequency $f$. We argue that $G(t) = C(t)|_{tau} = C(t)|_{gamma}$ should reveal an intermediate plateau set by the shear modulus $G_{eq}(f=0)$ of the quenched network.
In dislocation dynamics (DD) simulations, the most computationally intensive step is the evaluation of the elastic interaction forces among dislocation ensembles. Because the pair-wise interaction between dislocations is long-range, this force calculation step can be significantly accelerated by the fast multipole method (FMM). We implemented and compared four different methods in isotropic and anisotropic elastic media: one based on the Taylor series expansion (Taylor FMM), one based on the spherical harmonics expansion (Spherical FMM), one kernel-independent method based on the Chebyshev interpolation (Chebyshev FMM), and a new kernel-independent method that we call the Lagrange FMM. The Taylor FMM is an existing method, used in ParaDiS, one of the most popular DD simulation softwares. The Spherical FMM employs a more compact multipole representation than the Taylor FMM does and is thus more efficient. However, both the Taylor FMM and the Spherical FMM are difficult to derive in anisotropic elastic media because the interaction force is complex and has no closed analytical formula. The Chebyshev FMM requires only being able to evaluate the interaction between dislocations and thus can be applied easily in anisotropic elastic media. But it has a relatively large memory footprint, which limits its usage. The Lagrange FMM was designed to be a memory-efficient black-box method. Various numerical experiments are presented to demonstrate the convergence and the scalability of the four methods.
The shear stress relaxation modulus $G(t)$ may be determined from the shear stress $tau(t)$ after switching on a tiny step strain $gamma$ or by inverse Fourier transformation of the storage modulus $G^{prime}(omega)$ or the loss modulus $G^{primeprime}(omega)$ obtained in a standard oscillatory shear experiment at angular frequency $omega$. It is widely assumed that $G(t)$ is equivalent in general to the equilibrium stress autocorrelation function $C(t) = beta V langle delta tau(t) delta tau(0)rangle$ which may be readily computed in computer simulations ($beta$ being the inverse temperature and $V$ the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general $G(t) = G_{eq} + C(t)$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. A similar relation holds for $G^{prime}(omega)$. $G(t)$ and $C(t)$ must thus become different for a solid body and it is impossible to obtain $G_{eq}$ directly from $C(t)$.
The stress-dilatancy relation is of critical importance for constitutive modelling of sand. A new fractional-order stress-dilatancy equation is analytically developed in this study, based on stress-fractional operators. An apparent linear response of the stress-dilatancy behaviour of soil after sufficient shearing is obtained. As the fractional order varies, the derived stress-dilatancy curve and the associated phase transformation state stress ratio shift. But, unlike existing researches, no other specific parameters, except the fractional order, concerning such shift and the state-dependence are required. The developed stress-dilatancy equation is then incorporated into an existing constitutive model for validation. Test results of different sands are simulated and compared, where a good model performance is observed.
An analytical model of mechanical stress in a polymer electrolyte membrane (PEM) of a hydrogen/air fuel cell with porous Water Transfer Plates (WTP) is developed in this work. The model considers a mechanical stress in the membrane is a result of the cell load cycling under constant oxygen utilization. The load cycling causes the cycling of the inlet gas flow rate, which results in the membrane hydration/dehydration close to the gas inlet. Hydration/dehydration of the membrane leads to membrane swelling/shrinking, which causes mechanical stress in the constrained membrane. Mechanical stress results in through-plane crack formation. Thereby, the mechanical stress in the membrane causes mechanical failure of the membrane, limiting fuel cell lifetime. The model predicts the stress in the membrane as a function of the cell geometry, membrane material properties and operation conditions. The model was applied for stress calculation in GORE-SELECT.
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