Effective Elastic Parameters Research Articles

Micro-failure process and failure mechanism of brittle rock under uniaxial compression using continuous real-time wave velocity measurement

In this study, the micro-failure process and failure mechanism of a typical brittle rock under uniaxial compression are investigated via continuous real-time measurement of wave velocities. The experimental results indicate that the evolutions of wave velocities became progressively anisotropic under uniaxial loading due to the direction-dependent development of micro-damage. A wave velocity model considering the inner anisotropic crack evolution is proposed to accurately describe the variations of wave velocities during uniaxial compression testing. Based on which, the effective elastic parameters are inferred by a transverse isotropic constitutive model, and the evolutions of the crack density are inversed using a self-consistent damage model. It is found that the propagation of axial cracks dominates the failure process of brittle rock under uniaxial loading and oblique shear cracks develop with the appearance of macrocrack.

Progressive damage analysis for multiscale modelling of composite pressure vessels based on Puck failure criterion

This study proposed a method for progressive failure analysis to investigate the failure modes of 35 MPa Type III composite pressure vessels during hydraulic burst tests. Representative volume elements (RVEs) were introduced, and the degraded elastic parameters of composite, which include fibre and matrix failure modes, were calculated by the finite element analysis (FEA) method to obtain the macroscopic stiffness degradation parameters. The Puck failure criterion was adopted, and the effective elastic parameters calculated from RVEs was introduced by the UDSLFD subroutine to simulate the multiscale progressive damage of composite layers during hydraulic burst test. Combined with acoustic emission detection, composite microscopic photos and hydraulic burst test, the burst pressure and failure modes of the vessels were analysed. The results showed that meso-macroscopic finite element models could effectively predict the progressive damage of composite pressure vessel, such as matrix crack, interlaminar failure and fiber fracture under internal pressure loading. Finally, the predicted burst pressure was about 5.4% average difference of the actual experimental results. Besides, cylinder section of the vessels was the predicted blasting position, which was close to the actual tests. Thus, the proposed method can provide theoretical reference for the design and optimisation of composite pressure vessels.

Numerical modelling of thin anisotropic membrane under dynamic load

ABSTRACTThis work aims to describe a mathematical model and a numerical method to simulate a thin anisotropic membrane moving and deforming in 3D space under a dynamic load of an arbitrary time and space profile. The anisotropic continuum medium model described in the article can be used to model a membrane made of composite material using its effective elastic parameters. The model and the method allow the consideration of problems when the quasi-static approximation is not valid and elastic waves caused by the impact should be calculated. The model and the method can be used for numerical study of different processes in thin composite layers, such as shock load, ultrasound propagation, non-destructive testing procedures and vibrations. The thin membrane is considered as a 2D object in 3D space, an approach that allows a reduction in the computational time compared with full 3D models, while still having an arbitrary material rheology and load profile.

Bayesian inference as a tool for improving estimates of effective elastic parameters of wood

A simple approach to the identification of geometrical and material uncertainties of wood is presented. This stochastic mechanics problem combines classical micromechanics, computational homogenization and experimental measurements with Bayesian inference to estimate the model parameters including the characteristics of errors in macroscopic elastic properties of wood caused by randomness of microstructural details on the one hand and the experimental errors on the other hand. The former source of uncertainty includes, for example, variability in microfibril angle and growth ring density. Even such limiting consideration of random input illustrates the need for combined computational and experimental approach in a reliable prediction of the desired material properties. Tying the two approaches in the framework of Bayesian statistical method proves useful when addressing their limitations and as such giving better notion on the credibility of the prediction. This is demonstrated here on one particular example of spruce wood.

Estimating the Effective Elastic Parameters of Nodular Cast Iron from Micro-Tomographic Imaging and Multiscale Finite Elements: Comparison between Numerical and Experimental Results

Herein, we describe in detail a methodology to estimate the effective elastic parameters of nodular cast iron, using micro-tomography in conjunction with multiscale finite elements. We discuss the adjustment of the image acquisition parameters, address the issue of the representative-volume choice, and present a brief discussion on image segmentation. In addition, the finite-element computational implementation developed to estimate the effective elastic parameters from segmented microstructural images is described, indicating the corresponding computational costs. We applied the proposed methodology to a nodular cast iron, and estimated the graphite elastic parameters through a comparison between the numerical and experimental results.

Influence of fiber spatial distribution in unidirectional composite cross-section on homogenized elastic parameters

The aim of this work is to analyze the influence of spatial distribution of fibers in unidirectional longfiber composite material on the mechanical behavior of a lamina using finite element models of unit cells (micromodels). The dependencies of effective elastic parameters of a lamina on degree of irregularity of spatial fiber distribution are investigated, proved and described.Degree of irregularity of real composite material cross-sections and the unit cells is described by a proposed parameter. This parameter expresses deviation of a given composite cross-section (2D geometry) from idealized regular hexagonal fiber distribution.Effective elastic parameters of 200 micromodels with geometries having different degrees of irregularity are calculated. A dependence between the parameter evaluating the cross-section irregularity and the spread (interval between extreme values) of the resulting effective parameters is obtained.

Quantitative inversion of azimuthal anisotropy parameters from isotropic techniques

Exploration and development of unconventional reservoirs, where fractures and in situ stress play a key role, calls for improved characterization workflows. In this work, we present a method for quantitative estimation of anisotropic parameters related to stress and fracture detection that makes use of standard isotropic modeling and inversion techniques in anisotropic media. Based on the Rüger reflectivity equations for horizontal transverse isotropic media, we build a set of transforms that map the elastic parameters used in prestack inversion into effective anisotropic elastic parameters. When used in isotropic forward modeling and inversion, these effective parameters accurately mimic the anisotropic reflectivity behavior of the seismic data, thus closing the loop between well-log data and seismic inversion results in the anisotropic case.

Experimental verification of effective anisotropic crack theories in variable crack aspect ratio medium

ABSTRACTPhysical modelling of cracked/fractured media using downscaled laboratory experiments has been used with great success as a useful alternative for understanding the effect of anisotropy in the hydrocarbon reservoir characterization and in the crustal and mantle seismology. The main goal of this work was to experimentally verify the predictions of effective elastic parameters in anisotropic cracked media by Hudson and Eshelby–Cheng's effective medium models. For this purpose, we carried out ultrasonic measurements on synthetic anisotropic samples with low crack densities and different aspect ratios. Twelve samples were prepared with two different crack densities, 5% and 8%. Three samples for each crack density presented cracks with only one crack aspect ratio, whereas other three samples for each crack density presented cracks with three different aspect ratios in their composition. It results in samples with aspect ratio values varying from 0.13 to 0.26. All the cracked samples were simulated by penny‐shaped rubber inclusions in a homogeneous isotropic matrix made with epoxy resin. Moreover, an isotropic sample for reference was constructed with epoxy resin only. Regarding velocity predictions performed by the theoretical models, Eshelby–Cheng shows a better fit when compared with the experimental results for samples with single and mix crack aspect ratio (for both crack densities). From velocity values, our comparisons were also performed in terms of the ε, γ, and δ parameters (Thomsen parameters). The results show that Eshelby–Cheng effective medium model fits better with the measurements of ε and γ parameters for crack samples with only one type of crack aspect ratio.

Effective elastic moduli of a heterogeneous oolitic rock containing 3-D irregularly shaped pores

We have characterized the microstructure of a heterogeneous oolitic rock (limestone from Lavoux, France) with X-ray nano-computed tomography. This rock comprises an assemblage of porous grains (oolites), irregularly shaped three-dimensional pores, and inter-oolitic crystals (cement). To model the effect of this microstructure on the macroscopic behavior of the rock, we approximate the porous oolites by spheres, and the irregularly shaped pores by ellipsoids. This approximation is performed based on the principal component analysis PCA, which provides the geometrical properties such as length of semi-axes and orientation of resulting ellipsoids. The sphericity of the approximated oolites was calculated and the value close to 1 allows us to consider oolites as spheres. To verify the approximation in the case of pores, we evaluated the contribution of these irregularly shaped three-dimensional pores to the overall elastic properties. Thus, compliance contribution tensors for 3D irregular pores and their ellipsoidal approximations are calculated using the finite element method. These tensors were compared to the compliance tensors for ellipsoids obtained using analytical solutions based on Eshelby's theory and a relative error is estimated to evaluate the accuracy of the approximation. This error produces a maximum discrepancy of 4.5% between the two solutions respectively of pores and ellipsoids which verifies the proposed approximation procedure based on principal component analysis. The FEA numerical method is verified by comparing the numerical solution of compliance contribution tensors of the ellipsoids to the known analytical solution of these same shapes based on Eshelby's theory. The difference between these two solutions does not exceed 3%. Compliance contribution tensors are finally used to compare effective elastic parameters of a material containing irregular pores via the Maxwell homogenization scheme. These elastic parameters (bulk modulus and shear coefficient) coincide with a maximum deviation of 5% with ones for a material with ellipsoidal pores.

Crystallographic approach to obtain intensive elastic parameters of k33 mode piezoelectric ceramics

This paper presents an advanced piezo-material characterization method to obtain intensive elastic parameters of the k33 vibration mode from the k31 geometry through crystallographic approach with canted polarization. Three effective k31 bars with 0, γ, and π/2-γ degree angled polarization is needed where 0<γ<π/4. Eliminating s13E,* and s55E,* components from the effective elastic parameters, the k33 mode elastic parameters could be derived. Pb{(Zr0.56Ti0.44)0.99Nb0.01}O3 with 0, 15, 30, 45, 60, 75° angled polarization were prepared and characterized under constant vibration velocity to obtain the effective elastic parameters for the k31 mode. The intensive elastic compliance s33E and elastic loss tanϕ′33 are derived and compared to the measured value from the conventional methods Due to the low structural impedance of the measured geometry (i.e., large sample capacitance), the reliability on the proposed method is extraordinary, and the implicit relative error in the conventional methods could be avoided.

A Superposition Procedure for Calculation of Effective Diffusion and Elastic Parameters of Sparsely Porous Materials

Effective material parameters for diffusion and elastic deformation are calculated for porous materials using a continuum theory-based superposition procedure. The theory that is limited to two-dimensional cases, requires that the pores are sufficiently sparse. The method leads to simple manual calculations that can be performed by, e.g. hospital staff at clinical diagnoses of bone deceases that involve increasing levels of porosity. An advantage is that the result relates to the bone material permeability and stiffness instead of merely pore densities. The procedure uses precalculated pore shape factors and exact size scaling. The remaining calculations do not require any knowledge of the underlying field methods that are used to compute the shape factors. The paper establishes the upper limit for the pore densities that are sufficiently sparse. A cross section of bovine bone is taken as an example. The superposition procedure is evaluated against a full scale finite element calculation. The study compares the pore induced change of the diffusion coefficient and elastic modulus. The predictions differ between superposition and full scale calculations with 0.3% points when pore contribution to the diffusion constant is 3–7%, and 0.7% points when the pore contribution to the modulus of elasticity is 4.5–5%. It is uncertain if the error is in the superposition method, which is exact for small pore densities, while the full scale finite model is not.

Moment Tensor for Seismic Sources on a Bimaterial Interface: A Hyperfunction Approach

Abstract In this article, we consider the problem of finding the effective moment tensor for shear slip along a bimaterial interface. When the slip occurs at the jump of the values of elastic parameters, it is not clear how the elasticity tensor of each side should be combined in a unique way to obtain the effective elasticity tensor that then can be used in the moment tensor expression. Moreover, the Green’s function of the medium is not differentiable at the bimaterial interface, which prevents writing down the moment tensor form of the representation theorem, therefore leading to an ambiguity. In this article, we first show that the derivative of the Green’s function can be obtained once the Green’s function is considered as a hyperfunction, which is an equivalent concept to generalized functions or distributions. The resulting Green’s function at the bimaterial interface is the average of the Green’s function derivatives evaluated on either side. Finally, to obtain the effective elastic parameters, we used the equivalence of the unambiguous potency form and the moment tensor form of the representation theorems constrained with the boundary conditions, which are the continuity of the traction acting on the fault and the tangential strain across the fault. A unique solution is obtained for the isotropic case as well as transversely isotropic and orthotropic media if the symmetry axes of the tensors are aligned with the fault orientation. However, for general anisotropy or arbitrarily oriented structures, the solution is nonunique, for which certain physical constraints such as positive definiteness of the elasticity tensor can be used to confine the solution.

One-dimensional model for simulating blue-phase liquid crystal display

ABSTRACTWe proposed a one-dimensional model for simulating some performances of blue-phase liquid crystal (BPLC) and polymer-stabilised blue-phase liquid crystal (PSBPLCD). The one-dimensional model treats liquid crystal arrangement of BPLC as cholesteric liquid crystal with high and uniform tilt angle to simulate the Bragg reflection of BPLC. The simulated results agree quite well with the experimental results. The elastic and dynamical equations are formed to calculate the effective elastic parameters and the response times of PSBPLCD. The effective elastic parameter is a constant for low electric field and decreases with the increasing of electric field. The varied effective elastic parameter means the varied restoring force of liquid crystal under the electric field. The dynamical equations are proposed to calculate response times, and the response mechanism of BPLCD is discussed by researching the difference between the decay times from equation and experiment. The sub-millisecond response time results from the weak anchoring of polymer network or the increasing of rotational viscosity of PSBPLC with high-concentration monomer, and the reason of 10 µs response time is the strong anchoring and small amount of chiral dopant and monomer.

Investigation of the effective elastic parameters in the discrete element model of granular material by the triaxial compression test

The general objective of the present paper is to improve the understanding of micromechanical mechanisms in granular materials and their representation in numerical models. Results of numerical investigations on micro–macro relationships in the discrete element model of granular material are presented. The macroscopic response is analysed in a series of simulations of the triaxial compression test. The numerical studies are focused on the influence of microscopic parameters on the initial response. The effect of the contact stiffness and friction coefficient on the effective elastic moduli is investigated. Numerical results are compared with the analytical estimations based on the kinematic Voigt's hypothesis as well as with selected numerical results of other authors. The comparisons show that a better agreement between the numerical and analytical results is observed for particle assemblies with higher coordination numbers. Higher coordination numbers are related to more compact specimens and for a given specimen can be associated with low values of the contact stiffness and a higher confining pressure.

An effective medium theory for three-dimensional elastic heterogeneities

SUMMARY A second-order Born approximation is used to formulate a self-consistent theory for the effective elastic parameters of stochastic media with ellipsoidal distributions of small-scale heterogeneity. The covariance of the stiffness tensor is represented as the product of a onepoint tensor variance and a two-point scalar correlation function with ellipsoidal symmetry, which separates the statistical properties of the local anisotropy from those of the geometric anisotropy. The spatial variations can then be rescaled to an isotropic distribution by a simple metric transformation; the spherical average of the strain Green’s function in the transformed space reduces to a constant Kneer tensor, and the second-order corrections to the effective elastic parameters are given by the contraction of the rescaled Kneer tensor against the singlepoint variance of the stiffness tensor. Explicit results are derived for stochastic models in which the heterogeneity is transversely isotropic and its second moments are characterized by a horizontal-to-vertical aspect ratio η. If medium is locally isotropic, the expressions for the anisotropic effective moduli reduce in the limit η →∞ to Backus’s second-order expressions fora1-Dstochasticlaminate.ComparisonswiththeexactBackustheoryshowthatthesecondorder approximation predicts the effective anisotropy for non-Gaussian media fairly well for relative rms fluctuations in the moduli smaller than about 30 per cent. A locally anisotropic model is formulated in which the local elastic properties have hexagonal symmetry, guided by a Gaussian random vector field that is transversely isotropic and specified by a horizontal-toverticalorientationratio ξ.Theself-consistenttheoryprovidesclosed-formexpressionsforthe dependence of the effective moduli on 0 <ξ <∞ and 0 <η< ∞. The effective-medium parametrizations described here appear to be suitable for incorporation into tomographic modelling.

Uncertainty in effective elastic properties of particle filled polymers by the Monte-Carlo simulation

The main issue of this paper is to present statistical computational procedure for a determination of the effective elastic parameters for polymers filled with rubber particles. The homogenized elasticity tensor components are derived both as the variational upper and lower bounds as well as the solution for the cell problem on the composite Representative Volume Element. Probabilistic simulation is provided using both Monte-Carlo simulation technique and the generalized stochastic perturbation technique implemented in the symbolic computer algebra system MAPLE and, separately, together with the FEM-oriented code MCCEFF. Finally, up to fourth order probabilistic moments and coefficients of the homogenized tensor for the elastomers are computed as functions of the polymer’s and particles’ Young’s moduli coefficients of variation. This study gives the basis for further homogenization-based experiments, where the hyperelastic constitutive model will replace the classical Hookean one applied here.

Damping of metallized bilayer nanomechanical resonators at room temperature

We investigate the influence of gold thin-films subsequently deposited on a set of initially bare, doubly clamped, high-stress silicon nitride string resonators at room temperature. Analytical expressions for resonance frequency, quality factor and damping for both in- and out-of-plane flexural modes of the bilayer system are presented, which allows for the determination of effective elastic parameters of the composite structure from our experimental data. We find the inverse quality factor to scale linearly with the gold film thickness, indicating that the overall damping is governed by losses in the metal. Correspondingly, the mechanical linewidth increases by more than one order of magnitude compared to the bare silicon nitride string resonator. Furthermore, we extract mechanical quality factors of the gold film for both flexural modes and show that they can be enhanced by complete deposition of the metal in a single step, suggesting that surface and interface losses play a vital role in metal thin-films.

Effective medium theory for elastic metamaterials in thin elastic plates

An effective medium theory for resonant and non-resonant metamaterials for flexural waves in thin plates is presented. The theory provides closed-form expressions for the effective parameters of arrangement of inclusions or resonators in thin plates as a function of the filling fraction of the inclusions, their physical properties and the frequency. It is shown that positive or negative effective elastic parameters are possible depending on the symmetry of the resonance but, unlike it happens for bulk elastic waves, the responsible for the negative mass density behaviour is the monopolar term, while the negative Young's modulus and Poisson's ratio is due to the combination of monopolar and quadrupolar resonances, showing also that, at least for the first order in the scattering coefficients, the dipolar resonance plays no role in the description of the effective medium. Several examples are given for both non-resonant and resonant effective parameters and the results are verified by multiple scattering theory.

Computational homogenization of regular cellular material according to classical elasticity

A two-scale computational homogenization method for deriving the effective elastic parameters of regular cell material is presented. In the present application, particle model is used as the micromechanical model and classical linear elasticity as the continuum model. The method is designed to render the same effective elastic parameters irrespective of the Representative Volume Element (RVE) used for a cell structure. This requires simultaneous fulfillment of the kinematic and kinetic conditions of computational homogenization derived in the study. Also, the relationship between the quantities of the micromechanical and continuum model needs to be invertible on a RVE. Effective elastic parameter expressions for eight planar cellular materials obtained with a typical cell as the RVE are compared to their counterparts in literature. As an application example, a new closed-form compliance expression covering e.g. the square, regular hexagon, rhombus, over-expanded hexagon, and re-entrant hexagon cell structures of literature is presented.

Gaussian uncertainty in homogenization of rubber–carbon black nanocomposites

The objective of this work is Gaussian uncertainty in material parameters of the rubber–carbon black particle reinforced nano-composite in the framework of determination of its effective elastic parameters. The analytical 3D micro-mechanical model is contrasted here with the Finite Element Method one realized with the use of the system ABAQUS® and its tetrahedral finite elements C3D4. They discretize cubic Representative Volume Element (RVE) with a centrally located spherical carbon black particle, whose deformation energies accumulated during the uniaxial and biaxial uniform tension are computed. The polynomial-based Response Function Method allows to determine analytical functions relating effective elasticity tensor with Young moduli and Poisson ratios of this composite original components via the Weighted Least Squares Method; this is done in the symbolic environment of MAPLE. These functions serve for the initial sensitivity analysis, where we detect Poisson ratio of the rubber matrix as the crucial design parameter and its Young modulus as having secondary importance, while particle Young modulus and Poisson ratio are totally irrelevant. Next, the most influential parameters are randomized according to the Gaussian distribution and are employed in the dual probabilistic scheme – by using the stochastic perturbation and, independently, semi-analytical techniques to determine up to the fourth order probabilistic characteristics of the effective tensor components. Deterministic and probabilistic hypersensitivity of the effective constitutive tensor, expected according to the rubber matrix presence, is confirmed in all computational experiments. Such a probabilistic energy-related FEM approach will allow for future applications of more advanced constitutive models for the carbon black nano-composites, for the RVEs of larger sizes – containing large agglomerations of these particles and for the imperfect interfaces simulation.