Molecular Dynamics Simulation of Nanoscale Elastic Properties of Hydrated Na-, Cs-, and Ca-Montmorillonite

The knowledge of nanoscale mechanical properties of montmorillonite (MMT) with various compensation cations upon hydration is essential for many environmental engineering-related applications. This paper uses a Molecular Dynamics (MD) method to simulate nanoscale elastic properties of hydrated Na-, Cs-, and Ca-MMT with unconstrained system atoms. The variation of basal spacing of MMT shows step characteristics in the initial crystalline swelling stage followed by an approximately linear change in the subsequent osmotic swelling stage as the increasing of interlayer water content. The water content of MMT in the thermodynamic stable-state conditions during hydration is determined by comparing the immersion energy and hydration energy. Under this stable hydration state, the nanoscale elastic properties are further simulated by the constant strain method. Since the non-bonding strength between MMT lamellae is much lower than the boning strength within the mineral structure, the in-plane and out-of-plane strength of MMT has strong anisotropy. Simulated results including the stiffness tensor and linear elastic constants based on the assumption of orthotropic symmetry are all in good agreement with results from the literature. Furthermore, the out-of-plane stiffness tensor components of C33, C44, and C55 all fluctuate with the increase of interlayer water content, which is related to the formation of interlayer H-bonds and atom-free volume ratio. The in-plane stiffness tensor components C11, C22, and C12 decrease nonlinearly with the increase of water content, and these components are mainly controlled by the bonding strength of mineral atoms and the geometry of the hydrated MMT system. Young’s modulus in all three directions exhibits a nonlinear decrease with increasing water content.

Random Stiffness Tensor of Particulate Composites with Hyper-Elastic Matrix and Imperfect Interface

The main aim of this study is determination of the basic probabilistic characteristics of the effective stiffness for inelastic particulate composites with spherical reinforcement and an uncertain Gaussian volume fraction of the interphase defects. This is determined using a homogenization method with a cubic single-particle representative volume element (RVE) of such a composite and the finite element method solution. A reinforcing particle is spherical, located centrally in the RVE, surrounded by the thin interphase of constant thickness, and remains in an elastic reversible regime opposite to the matrix, which is hyper-elastic. The interphase defects are represented as semi-spherical voids, which are placed on the outer surface of this particle. The interphase is modeled as hyper-elastic and isotropic, whose effective stiffness is calculated by the spatial averaging of hyper-elastic parameters of the matrix and of the defects. A constitutive relation of the matrix is recovered experimentally by its uniaxial stretch. The 3D homogenization problem solution is based upon a numerical determination of strain energy density in the given RVE under specific uniaxial and biaxial stretches as well as under shear deformations. The analytical relation of the effective composite stiffness to the input uncertain parameter is recovered via the response function method, using a polynomial basis and an optimized order. Probabilistic calculations are completed using three concurrent approaches, namely the iterative stochastic finite element method (SFEM), Monte Carlo simulation and by the semi-analytical method. Previous papers consider the composite fully elastic, which limits the applicability of the resulting effective stiffness tensor computed therein. The current study voids this assumption and defines the composite as fully hyper-elastic, thus extending applicability of this tensor to strains up to 0.25. The most important research finding is that (1) the effective stiffness tensor is sensitive to random interface defects in its hyper-elastic range, (2) its resulting randomness is not close to Gaussian, (3) the semi-analytical method is not perfectly suited to stochastic calculations in this region of strains, as opposed to the linear elastic region, and (4) that the increase in random dispersion of defects volume fraction has a much higher effect on the stochastic characteristics of this stiffness tensor than fluctuation of the strain.

Failure in granular materials based on acoustic tensor: a numerical analysis

We investigate localization in granular material with the support of numerical simulations based upon DEM (Distinct Element Method). Localization is associated with a discontinuity in a component of the incremental strain over a plane surface through the condition of the determinant of the acoustic tensor to be zero. DEM simulations are carried out on an aggregate of elastic frictional spheres, initially isotropically compressed and then sheared at constant pressure p0. The components of the stiffness tensor are evaluated numerically in stressed states along the triaxial test and employed to evaluate the acoustic tensor in order to predict localization. This occurs in the pre-peak region, where the aggregate hardens under the circumstance to be incrementally frictionless: it is a regime in which the tangential force does not change as the deformation proceedes and, consequently, the deviatoric stress varies only with the normal component of the contact force.

Modeling Cylindrical Inhomogeneity of Finite Length with Steigmann–Ogden Interface

A mathematical model employing the concept of energy-equivalent inhomogeneity is applied to analyze short cylindrical fiber composites with interfaces described by the Steigmann–Ogden material surface model. Real inhomogeneity consists of a cylindrical fiber of finite length, and its surface possessing different properties is replaced by a homogeneous, energy-equivalent cylinder. The properties of the energy-equivalent fiber, incorporating properties of the original fiber and its interface, are determined on the basis of Hill’s energy equivalence principle. Closed-form expressions for components of the stiffness tensor of equivalent fiber have been developed and, in the limit, shown to compare well with the results available in the literature for infinite fibers with the Steigmann–Ogden interface model. Dependence of those components on the radius, length of the cylindrical fiber, and surface parameters is included in these expressions. The effective stiffness tensor of the short-fiber composites with so-defined equivalent cylindrical fibers can be determined by any homogenization method developed without accounting for interface.

Elastic Coefficients of β-HMX as Functions of Pressure and Temperature from Molecular Dynamics

The isothermal second-order elastic stiffness tensor and isotropic moduli of β-1,3,5,7- tetranitro-1,3,5,7-tetrazoctane (β-HMX) were calculated, using the P21/n space group convention, from molecular dynamics for hydrostatic pressures ranging from 10−4 to 30 GPa and temperatures ranging from 300 to 1100 K using a validated all-atom flexible-molecule force field. The elastic stiffness tensor components were calculated as derivatives of the Cauchy stress tensor components with respect to linear strain components. These derivatives were evaluated numerically by imposing small, prescribed finite strains on the equilibrated β-HMX crystal at a given pressure and temperature and using the equilibrium stress tensors of the strained cells to obtain the derivatives of stress with respect to strain. For a fixed temperature, the elastic coefficients increase substantially with increasing pressure, whereas, for a fixed pressure, the elastic coefficients decrease as temperature increases, in accordance with physical expectations. Comparisons to previous experimental and computational results are provided where possible.

Effect of variations in microstructure on clay elastic anisotropy

Rock physics provides a crucial link between seismic and reservoir properties, but it requires knowledge of the elastic properties of rock components. Whereas the elastic properties of most rock components are known, the anisotropic elastic properties of clay are not. Scanning electron microscopy studies of clay in shales indicate that individual clay platelets vary in orientation but are aligned locally. We present a simple model of the elastic properties of a region (domain) of locally aligned clay platelets that accounts for the volume fraction, aspect ratio, and elastic-stiffness tensor of clay platelets, as well as the effective elastic properties of the interplatelet medium. Variations in clay anisotropy are quantified by examining the effects of varying model parameters upon the effective transverse-isotropic (TI) elastic-stiffness tensor of a domain. Statistics of these distributions and correlations between stiffnesses and anisotropy parameters enable the most probable sets of stiffnesses to be identified for rock physics calculations. The mean of these distributions is on the order of twice the mode for in-plane stiffnesses ([Formula: see text], [Formula: see text], [Formula: see text]), but it is of the same order as the mode for out-of-plane stiffnesses ([Formula: see text], [Formula: see text], [Formula: see text]). Despite random sampling, well-defined relations emerge, consistent with similar shale relations reported in the literature. Expressing these relations in terms of [Formula: see text] for a single domain of aligned clay platelets facilitates their general application. In the limit that the volume fraction approaches unity, the elastic stiffnesses thus derived reproduce those of the clay mineral assumed as platelets. Given the elastic-stiffness tensor of a single domain of aligned clay platelets, the effective TI elastic-stiffness tensor of clay is obtained by integrating over the clay-platelet orientation-distribution function.

Single-crystal elasticity of iron-bearing phase E and seismic detection of water in Earth's upper mantle

The elastic properties of Mg2.12(2)Fe0.21(2)Ni0.01Si1.15(1)O6H2.67(8) phase E single crystals with Fe3+/ΣFe = 0.25(3) have been determined by Brillouin spectroscopy at ambient conditions. We find that that the elasticity of iron-bearing phase E is described by the six independent stiffness tensor components (all in units of GPa): C11 = 192.2(6), C12 = 56.4(8), C13 = 43.5(8), C14 = –4.3(3), C33 = 192.1(7), C44 = 46.4(3). The Voigt-Reuss-Hill averages of bulk and shear moduli are 95.9(4) and 59.6(2) GPa, respectively. The aggregate velocities of iron-bearing phase E are νP = 7.60(2) and νS = 4.43(1) km/s, markedly lower than those of major mantle minerals at ambient conditions. Modeling based on our results suggests that the presence of iron-bearing phase E may reduce the sound wave velocities in upper mantle and transition zone rocks, making it a possible target for future seismological investigations aiming to map hydration in subducting slabs.

Finite Element Analysis of Mechanical Properties of 2.5D Woven Composites

It is calculated the effective anisotropic stiffness tensor of the representative volume element in 2.5D woven composites by energy method. The Multi-point constraints are applied to periodic boundary conditions. Compared with the static tensile tests, the validity of present method is verified.

Micropolar Modeling of Auxetic Chiral Lattices With Tunable Internal Rotation

Based on micropolar continuum theory, the closed-form stiffness tensor of auxetic chiral lattices with V-shaped wings and rotational joints were derived. Representative volume element (RVE) of the chiral lattice was decomposed into V-shape wings with fourfold symmetry. A unified V-beam finite element was developed to reduce the nodal degrees of freedoms of the RVE to enable closed-form analytical solutions. The elasticity constants were derived as functions of the angle of the V-shaped wings, nondimensional in-plane thickness of the ribs, and the stiffness of the rotational joints. The influences of these parameters on the coupled chiral and auxetic effects were systematically explored. The results show that the elastic moduli were significantly influenced by all three parameters, while Poisson's ratio was barely influenced by the in-plane thickness of the ribs but is sensitive to the angle of the V-shaped wings and the stiffness of the rotational springs. There is a transition region out of which the spring stiffness does not considerably affect the auxeticity and the overall lattice stiffness.