Ratio Of Effective Modulus Research Articles

Traveling Lamb wave in elastic metamaterial layer

The propagation of traveling Lamb wave in single layer of elastic metamaterial is investigated in this paper. We first categorized the traveling Lamb wave modes inside an elastic metamaterial layer according to different combinations (positive or negative) of effective medium parameters. Then the impacts of the frequency dependence of effective parameters on dispersion characteristics of traveling Lamb wave were studied. Distinct differences could be observed when comparing the traveling Lamb wave along an elastic metamaterial layer with one inside the traditional elastic layer. We further examined in detail the traveling Lamb wave mode supported in elastic metamaterial layer, when the effective P and S wave velocities were simultaneously imaginary. It was found that the effective modulus ratio is the key factor for the existence of special traveling wave mode, and the main results were verified by FEM simulations from two levels: the level of effective medium and the level of microstructure unit cell.

Rock-physics models for heterogeneous creeping rocks and viscous fluids

Rock-physics models are used to explore how small-scale heterogeneity can affect the larger scale viscoelasticity of rocks. Applications include mixtures of creeping clay and elastic quartz, mixtures of different creeping materials (e.g., clay and kerogen), or viscous fluids containing bubbles or solid fines. We have found that elastic inclusions in a Maxwell viscoelastic background change the effective viscosity and the high-frequency limiting elastic modulus. The viscosity response was similar to that observed for a Newtonian fluid, and the high-frequency elastic modulus varied as predicted by elastic effective media models. The characteristic frequency of the effective medium scales with the ratio of effective modulus and effective viscosity. Inclusions also distribute the relaxation times, converting the Maxwell material to resemble a Cole-Cole material. Elastic inclusions in a creeping background decrease the effective viscoelastic Poisson’s ratio of the composite. As with elastic media, geometric alignment of phases with contrasting properties leads to viscoelastic anisotropy. Our modeling has illustrated how the amount of heterogeneity and the microgeometry of heterogeneity affects anisotropy; for example, aligned oblate elastic inclusions can increase the amount of creep in the symmetry direction while decreasing creep normal to the symmetry direction. We have developed a suggested interpretation template for how creep function parameters vary with the amount and microgeometry of elastic phases. Interpretation also depends strongly on the material properties of the creeping phase in the absence of elastic inclusions. Extrapolating from dynamic to quasi-static viscoelastic response is intrinsically nonunique without knowledge of the material microstructure. Dominant relaxation mechanisms can be different at different measurement scales and at different measurement strain amplitudes. For example, the observed dynamic response can be fit with an infinite number of microgeometries, each of which has a different long-term behavior.

Modulus ratio of dry rock based on differential effective-medium theory

There are two points of view with regard to the relationship between the dry-framework modulus ratio of bulk to shear modulus and porosity. One view, assumed by most empirical models such as the critical porosity model, is that the dry-framework modulus ratio is independent of porosity. However, this assumption is sometimes in disagreement with experiments. Another view supported by the differential effective-medium (DEM) theory is that the dry-framework modulus ratio depends on porosity. Ordinary differential equations (ODE) for bulk and shear moduli are coupled, and there is no analytical formula for the modulus ratio of dry porous rock. By assuming that the difference between the polarization factors of dry inclusions for bulk and shear modulus within the ODE has an almost linear relationship with the effective modulus ratio, an analytical solution for the dry-rock modulus ratio is derived from the ODE. The above linear relationship characterized by intercept [Formula: see text] and gradient [Formula: see text] is confirmed by the numerical results calculated from the analytical formulas for the three specific pore shapes: spherical pores, needle-shaped pores, and penny-shaped cracks. Then the validity of this assumption is tested by integrating the full DEM equation numerically. The analytical approximation gives a good estimate of the numerical results for the cases of the three given pore shapes. The dry-frame modulus ratio is monotonic with respect to porosity, and its monotonicity depends on the mineral-grain modulus ratio and the pore shape. The analytical formula can be simplified further to some popular models such as the critical porosity model by setting constants [Formula: see text] and [Formula: see text] to specific values. Predictions of the analytical formula compare favorably with real and synthetic sandstone data and granite data measured in the laboratory.

Longitudinal waves in two-layered solid circular cylinder

The method of asymptotic expansions is used to construct a theory of wave-propagation along a solid cylinder made up of two layers of different elastic materials. Wave speed is determined by the ratio of the sum of the area times the effective moduli to that of the area times the densities. However, the effective modulus reduces to Youngs' modulus only if Poisson ratios are equal. The higher approximation, describing dispersive wave-motion, is further affected by the difference in wave-speeds for the two layers. This wave-speed is the ratio of effective modulus to density. The formal expansion is seen to give rise to secular terms. A uniformly valid approximation is given to this order, for the case of a travelling wave.