Effective Elastic Shear Modulus Research Articles

Absolute effective elastic constants of composite materials

The objective is to determine the mechanical properties of the composites formed in two types, theoretically. The first composite includes micro-particles in a matrix while the second involves long, thin fibers. A fictitious, homogeneous, linear-elastic and isotropic single material named as effective material is considered during calculation which is based on the equality of the strain energies of the composite and effective material under the same loading conditions. The procedure is carried out with volume integrals considering a unique strain energy in a body. Particularly, the effective elastic shear modulus has been calculated exactly for small-particle composites by the same procedure in order to determine of bulk modulus thereof. Additionally, the transverse shear modulus of fiber reinforced composites has been obtained through a simple approach leading to the practical equation. The results have been compared not only with the outcomes in the literature obtained by different method but also with those of finite element analysis performed in this study.

Elastic-plastic behavior of cyclotrimethylene trinitramine single crystals under spherical indentation: Modeling and simulation

A nonlinear anisotropic elastic-plastic model is developed for single crystals of the energetic material cyclotrimethylene trinitramine (RDX). Numerical simulations of spherical indentation on oriented single crystals are performed; predictions are compared with experimental data and observations from the literature. Model predictions for elastic response and initial yield using elastic constants obtained from resonant ultrasound spectroscopy agree with experimental data; predicted forces using constants obtained from Brillouin scattering tend to exceed experimental data. Influences of elastic anisotropy and elastic nonlinearity are significant. Predicted slip system activity is in reasonable agreement with that deduced from experimental surface profiles when a uniform strength of G/20 is assigned to all six slip systems, with G an effective elastic shear modulus. Predicted indentation forces in the post-yield regime exceed those observed in experiments, suggesting that surface and possibly subsurface fractures may contribute to a loss of stiffness in experiments at larger indentation depths.

Large scale computations of effective elastic properties of rubber with carbon black fillers

A general method, based on a multi-scale approach, is proposed to derive the effective elastic shear modulus of a rubber with 14\% of carbon black fillers from finite element (FE) and fast Fourier transform (FFT) methods. The complex multi-scale microstructure of such material was generated numerically from a mathematical model of its morphology which was identified from statistical moments out of TEM images. For FE computations, the simulated microstructures were meshed from three-dimensional reconstruction of isosurface using Marching Cubes algorithm with a special attention to the quality of the topology and the geometry of the mesh. To compute the shear modulus and to determine the representative volume element, homogeneous boundary conditions were prescribed on meshes and combined with a domain decomposition method. Regarding parallel computing, specific difficulties related to the highly heterogeneous microstructures and complex geometry are pointed out. Hereby, the experimental shear modulus (1.8MPa) obtained from DMA (dynamic mechanical analysis) was estimated by the Hashin-Shtrikman lower bound (1.4MPa) and the computations on simulated microstructures (2.4MPa). The shear modulus was determined for two materials with the same volume fraction but different distribution of fillers. The current model of microstructures is capable of estimating the relative effect of the mixing time in processing associated with change in morphology on the elastic behavior. The computations also provide the local fields of stress/strain in the elastomeric matrix.

LARGE-SCALE COMPUTATIONS OF EFFECTIVE ELASTIC PROPERTIES OF RUBBER WITH CARBON BLACK FILLERS

A general method, based on a multiscale approach, is proposed to derive the effective elastic shear modulus of a rubber with 14% carbon black fillers from finite element and fast Fourier transform methods. The complex multiscale microstructure of such material was generated numerically from a mathematical model of its morphology that was identified from statistical moments out of transmission electron microscopy images. For finite element computations, the simulated microstructures were meshed from three-dimensional reconstruction of the isosurface using the marching cubes algorithm with special attention to the quality of the topology and the geometry of the mesh. To compute the shear modulus and to determine the representative volume element, homogeneous boundary conditions were prescribed on meshes and combined with a domain decomposition method. Regarding parallel computing, specific difficulties related to the highly heterogeneous microstructures and complex geometry are pointed out. The experimental shear modulus (1.8 MPa) obtained from dynamic mechanical analysis was estimated by the Hashin-Shtrikman lower bound ( 1.4 MPa) and the computations on simulated microstructures ( 2.4 MPa). The shear modulus was determined for two materials with the same volume fraction but different distribution of fillers. The current model of microstructures is capable of estimating the relative effect of the mixing time in processing associated with change in morphology on the elastic behavior. The computations also provide the local fields of stress/strain in the elastomeric matrix.

Earthquake nucleation on rate and state faults – Aging and slip laws

We compare 2‐D, quasi‐static earthquake nucleation on rate‐and‐state faults under both “aging” and “slip” versions of the state evolution law. For both versions mature nucleation zones exhibit 2 primary regimes of growth: Well above and slightly above steady state, corresponding respectively to larger and smaller fault weakening rates. Well above steady state, aging‐law nucleation takes the form of accelerating slip on a patch of fixed length. This length is proportional to b−1 and independent of a, where a and b are the constitutive parameters relating changes in slip speed and state to frictional strength. Under the slip law the nucleation zone is smaller and continually shrinks as slip accelerates. The nucleation zone is guaranteed to remain well above steady state only for values of a/b that are low by laboratory standards. Near steady state, for both laws the nucleation zone expands. The propagating front remains well above steady state, giving rise to a simple expression for its effective fracture energy Gc. This fracture energy controls the propagation style. For the aging law Gc increases approximately as the square of the logarithm of the velocity jump. This causes the nucleation zone to undergo quasi‐static crack‐like expansion, to a size asymptotically proportional to b/(b−a)2. For the slip law Gc increases only as the logarithm of the velocity jump, and crack‐like expansion is not an option. Instead, the nucleation zone grows as an accelerating unidirectional slip pulse. Under both laws the nucleation front propagates at a velocity larger than the slip speed by roughly μ′/bσ divided by the logarithm of the velocity jump, where μ′ is the effective elastic shear modulus. For this prediction to be consistent with observed propagation speeds of slow slip events in subduction zones appears to require effective normal stresses as low as 1 MPa.

The temperature dependence of the effective elastic shear modulus of snow

In order to determine the temperature dependence of the effective elastic properties of snow, dynamic torsional shear experiments at a frequency of 1 Hz were performed in a cold laboratory with a stress-controlled rheometer. Two types of natural snow samples of small rounded or faceted grains with partly decomposed and fragmented particles (mean size 0.4–0.5 mm) with densities in the range of 220–250 kg m −3 were tested. Results indicate that snow is a rheologically simple material, so that the data can be normalized for analysis. The mean effective elastic shear modulus G′ at −10 °C was found to be about 0.75 MPa for the density range and snow type tested. The modulus decreases with increasing temperature following an Arrhenius relation below −6 °C with an apparent activation energy of about 0.2 eV. Between −6 °C and the melting point, the decrease is much stronger showing that the modulus is highly temperature-sensitive in the range of interest for snow slab stability evaluation. An empirical relation is fitted to the data over the whole range of temperatures to provide a single smooth relation for engineering applications.

Micropipette aspiration on the outer hair cell lateral wall

The mechanical properties of the lateral wall of the guinea pig cochlear outer hair cell were studied using the micropipette aspiration technique. A fire-polished micropipette with an inner diameter of approximately 4 microm was brought into contact with the lateral wall and negative pressure was applied. The resulting deformation of the lateral wall was recorded on videotape and subjected to morphometric analysis. The relation between the length of the aspirated portion of the cell and aspiration pressure is characterized by the stiffness parameter, K(s) = 1.07 +/- 0.24 (SD) dyn/cm (n = 14). Values of K(s) do not correlate with the original cell length, which ranges from 29 to 74 microm. Theoretical analysis based on elastic shell theory applied to the experimental data yields an estimate of the effective elastic shear modulus, mu = 15.4 +/- 3.3 dyn/cm. These data were obtained at subcritical aspiration pressures, typically less than 10 cm H2O. After reaching a critical (vesiculation) pressure, the cytoplasmic membrane appeared to separate from the underlying structures, a vesicle with a length of 10-20 microm was formed, and the cytoplasmic membrane resealed. This vesiculation process was repeated until a cell-specific limit was reached and no more vesicles were formed. Over 20 vesicles were formed from the longest cells in the experiment.

Determination of elastic properties of very heterogeneous media with cellular automata

A cellular automaton is used to calculate the elastic properties of very heterogeneous media. The lattice gas method (designed to model viscous flow) is applied to elastic problems through the correspondence between the steady state velocity field in an incompressible Newtonian fluid at low Reynolds number and the displacement field in an incompressible elastic solid. The cellular automaton is applied to determine the elastic properties of a matrix containing randomly distributed hard inclusions. Because the equations of equilibrium are locally solved for any distribution of the inclusion phase, the method provides a means to estimate the influence of stress field perturbations on the macroscopic properties of heterogeneous systems. The effective elastic shear modulus is calculated as a function of the concentration C of the inclusion phase (0≤C≤1) for different values of R, the rheological contrast between the inclusion and the matrix. The numerical results are compared with those obtained by various micromechanical models. Estimates given by the cellular automaton agree with the analytical solution for a double periodic triangular system of discs. However, the effective shear modulus for the cellular automaton fluid does not fall within the limits of Hashin and Shtrikman. It follows the same trend as that of the lower bound but stays below the lower bound for all concentrations. Likewise, the shear modulus for a cellular automaton fluid is also systematically lower than that predicted by the differential method, although the trends are similar.

Nanorheology of Confined Polymer Melts. 3. Weakly Adsorbing Surfaces

Polymer melts confined between weakly adsorbing surfaces terminated with methyl groups (closepacked self-assembled monolayers of condensed octadecyltriethoxysilane, OTE) were studied in regard to surface forces (static forces to compress the polymer films to a given thickness) and shear rheology. The experiments involved a surface forces apparatus modified for dynamic mechanical shear oscillation. The polymers were an atactic poly(phenylmethylsiloxane), PPMS, with chain length from 31 to 153 skeletal bonds. Three principal conclusions emerged. First, the equilibrated surface force between OTE was zero down to the same small thickness (17 f 2 A) regardless of molecular weight. This decisively confirms theoretical predictions for the surface forces of polymer chains in equilibrium with a bulk reservoir. Second, enhanced effective shear moduli (measured in the linear-response regime) were observed only at this thickness of measurable surface force. This accompaniment of surface force and enhanced shear modulus was also seen in the case of strong adsorption. But in that case, these phenomena scaled with the molecular size of the polymer, approximately its radius of gyration (RG); here these phenomena appeared at a single film thickness. Third, the effective elastic shear modulus G under confinement was rubberlike in magnitude, indicating enormously slower relaxation than in the bulk fluid. This relaxation was slower, the higher the polymer molecular weight. This third conclusion is qualitatively similar to the case for these same PPMS polymers confined between strongly adsorbing surfaces. It suggests that, even in the case of weak adsorption, geometrical confinement enhances entanglement interactions between polymer chains.

Observations of viscoelastic behaviour of an elastohydrodynamic lubricant film

A novel rolling-contact experiment has been devised with the object of making direct observations of viscoelastic behaviour of elastohydrodynamic lubricant films. A ‘point contact' disk machine has been used which permits the axes of the disks to be either tilted or skewed through small angles relative to each other. Tilt subjects the film to cyclic shear while skew subjects it to unidirectional shear. By measuring and comparing the resultant shear forces developed by the film in each case, its viscoelastic properties can be deduced. Experiments at 23 °C with a high-viscosity mineral oil revealed viscoelastic behaviour above a contact pressure of about 0.5 GPa. In the pressure range 1.0 to 2.0 GPa the effective elastic shear modulus of the oil was found to vary from 0.1 to 1.0 GPa. At a pressure of 0.8 GPa the oil exhibited a viscoelastic transition in the temperature range 70 to 30 °C. The investigation was restricted to small shear strains for which the behaviour is linear.