Effective Elastic Stiffness Research Articles

Detection of Natural Tilted Fractures From Azimuthal Seismic Amplitude Data Based on Linear-Slip Theory

Tilted transverse isotropy (TTI) is common for naturally fractured reservoir when the fractures exist a tilted axis of symmetry. Seismic characterization of natural tilted fractures from observable azimuthal reflected amplitude data, however, is quite difficult. We focus on the detection of natural tilted fractures from azimuthal seismic amplitude data, especially for low- and high-angle tilted fractures. Using the linear-slip theory, we first express the effective elastic stiffness matrix of a TTI medium as a function of background elastic moduli, fracture density, and three angles, including the incident angle, the azimuth angle, and the tilt angle. Based on the scattering function and the first-order perturbations in stiffness components across a weak-contrast reflection interface separating two weak-anisotropy TTI media, we then derive a weak-anisotropy and linearized PP-wave reflection coefficient containing the tilt angle and fracture density. For low- and high-angle aligned fracture sets, we formulate the low- and high-angle approximate PP-wave reflection coefficient equations, respectively, and propose a Bayesian seismic inversion approach to estimate the tilt angle and the corresponding fracture densities using the azimuthal differences in seismic reflected amplitude data. Synthetic and real datasets are used to demonstrate the feasibility and reliability of the proposed inversion approach. Test results make us believe that our proposed inversion approach allow us to obtain the tilted fracture properties of hydrocarbon reservoirs in a more accurate manner than previous cases of vertically transverse isotropy (VTI) or horizontally transverse isotropy (HTI).

Characterization of Existing Steel Racks via Dynamic Identification

Steel storage racks are widely used in logistics for storing materials and goods. Rack design is carried out by adopting the so-called design-assisted-by-testing procedure. In particular, experimental analyses must be carried out by rack producers on the key structural components in order to adopt the design approach proposed for the more traditional carpentry frames. For existing racks, i.e., those in-service for decades, it is required to evaluate the load carrying capacity in accordance with the design provisions currently in use. The main problem in several cases should be the appraisal of the key component performance, owing to the impossibility to obtain specimens from in-service racks without reduction or interruption of the logistic flows. To overcome this problem, a quite innovative procedure for the identification of the structural unknowns of existing racks has been proposed in the paper. The method is based on in-situ modal identification tests combined with extensive numerical analyses. To develop the procedure, cheap measurement systems are required, and they could be immediately applied to existing racks. A real case study is discussed, showing the efficiency of the procedure in the evaluation of the effective elastic stiffness of beam-to-column joints and base plate connections, that are parameters which remarkably affect the rack performance. The structural unknowns have been determined based on four sets of modal tests (two configurations on the longitudinal direction and two in the transversal direction) plus 9079 iterative structural analyses. The results obtained were then directly compared with experimental component tests, showing differences lower than 9%.

Ellipsoidal inhomogeneity with anisotropic incoherent interface. Multipole series solution and application to micromechanics

The ellipsoidal inhomogeneity with transversely isotropic incoherent interface is considered in the conductivity and elasticity context. The general imperfect interface is modeled by the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure to the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface moduli. The rigorous analytical solution to the conductivity and elasticity problems has been obtained by the multipole expansion method in terms of ellipsoidal solid harmonics. An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths. These results apply equally to the inhomogeneities with anisotropic interphases and nano level incoherent interfaces. The obtained solutions are valid for the non-uniform far loading and are readily incorporated in the many-particle (finite cluster or representative unit cell) model of heterogeneous solid with anisotropic incoherent interface. The tensors of effective conductivity and elastic stiffness of composite with ellipsoidal inhomogeneities are evaluated using Maxwell homogenization scheme. The obtained accurate numerical data indicate that the effective properties of composite may vary widely due to shape of inhomogeneities and the interface anisotropy ratio. Taking the incoherency of interface into account may increase reliability of predicting the behavior of nanostructured solids.

Effective stiffness and period-dependent seismic response modification factors for flexure-dominated fully-grouted reinforced masonry rectangular shear walls

The wall effective elastic stiffness, displacement ductility, and seismic force modification factors are important force-based seismic design (FBD) parameters for reinforced masonry (RM) shear walls. The effective stiffness, ke of reinforced masonry shear walls (RMSW) is crucial in computing the natural period and thus the elastic forces, and essential also in computing the displacements corresponding to the design seismic forces. This study analyzes previously reported test results of forty-three flexure-dominated fully-grouted rectangular RMSWs subjected to quasi-static cyclic load to evaluate the FBD parameters adopted by Canadian and American standards. In this study, based on the experimental results of 43 tested walls, a new stiffness reduction factor is proposed considering the effect of axial stress, vertical reinforcement ratio, and horizontal reinforcement ratio. Moreover, the seismic force modification factors are compared to the Canadian and US codes. The results showed that ductility related reduction factors should be dependent on the wall natural frequency and there is a room to relax the current proposed code values. In addition, incremental dynamic analysis was performed using a simplified numerical model to study the effect of dynamic loading on response modification factors and stiffness characteristics of the walls.

Damage and sensitivity analysis of a reinforced concrete wall building during the 2010, Chile earthquake

Buildings with Reinforced Concrete (RC) walls are commonly used to resist lateral forces in seismic countries because they provide high lateral stiffness and strength. In recent earthquakes, shear wall buildings have shown good behavior in general; however, a small percentage underwent severe damage localized typically in lower stories. Several numerical models have been developed and proposed to simulate the failure mechanism and behavior of RC walls. From the existing models, only those denoted as micro-models can accurately simulate the stress and strain distributions. The aim of this research is double: (i) to validate a nonlinear finite element wall model and the associated material stress-strain constitutive relationship using the behavior of a real building during the 2010 Chile earthquake; and (ii) to analyze the uncertainty of the response of the building due to changes in model parameters. To validate the response of the wall model, four experimental benchmark RC wall specimens were studied, and model accuracy was evaluated using five parameters: initial stiffness, peak base-shear, ultimate base-shear, maximum displacement, and dissipated energy. A sensitivity analysis was carried out to study the influence of material parameters in the wall response and its damage. The case-study is a 18-story building with 1 basement, which suffered severe damage during the 2010 Chile earthquake, which has been studied by non-linear response-history analysis. Uncertainty in the building response due to three important modeling assumptions was considered: Rayleigh’s damping model parameters; effective elastic bending stiffness of the structural elements; and effect of the vertical ground motion component. Results showed that the proposed model can predict the seismic response of the building with reasonable accuracy by identifying correctly the damage location. This case-study enabled us to assess also the effect of damping in non-ductile structures, the important influence of the slab stiffness in the response, and the effect of the vertical ground motion component in the sequence of damaged walls.

Effective stiffness and thermal expansion of three-phase multifunctional polymer electrolyte coated carbon fibre composite materials

Multifunctional composites including polymer electrolyte coated carbon fibres and polymer matrix systems gained recent interest in light-weight design related research areas. Compared to classical fibre reinforced plastics, the interphase, made by electropolymerisation on the fibre surface, represents a new, third material phase. The coating serves as ion-conducting separator in structural batteries and as insulating layer in energy transmitting multifunctional composites. The importance of this study is related to the fact, that multifunctional applications, based on such composites, are exposed to temperature changes in many cases. The coating material, acting as thin interphase, shows a significant temperature dependant Young’s modulus, determining the overall macroscopic behaviour under thermal loads. The new influences on the effective elastic properties of the composite are determined in this work in a 3D microstructural simulation approach based on a unit cell geometry. For the first time, the resulting effective properties are discussed towards the state of research and future work. First, the effective elastic stiffness is computed by isothermal virtual material testing, applying unit strain modes on the unit cell. Second, a uniform temperature change is applied and the effective thermal expansion coefficients are computed. The results show that a change of stiffness in the coating domain has a great influence on the effective stiffness in the transversal isotropic plane. The effective thermal expansion of the composite is also highly sensitive to the thermal expansion behaviour of the coating phase. Main conclusions are drawn towards multiphysical material simulation: Influences of the coating material properties have to be taken into account to compute effective properties. In particular, it is necessary to include the temperature dependant stiffness and the coefficients of thermal expansion of the interphase, which affect effective properties significantly. A thermo-mechanic coupled microscale model is needed to represent in-situ properties of such composites for applications with heat exposure.

Microplane fatigue model MS1 for plain concrete under compression with damage evolution driven by cumulative inelastic shear strain

Mathematical modeling of fatigue behavior of concrete under compressive loading presents a challenging topic that attracts an increasing interest of the scientific community. The present paper proposes a new formulation of the microplane model referred to as MS1 aiming to capture the fundamental inelastic mechanisms that can reflect the tri-axial stress redistribution within a material zone upon cyclic loading at subcritical load levels. The key idea of the proposed approach is to link the fatigue damage evolution to the cumulative inelastic shear strain. In this way the accumulation of fatigue damage under compressive loading owing to internal shear/sliding between aggregates at subcritical load levels can be reflected. The present model is developed within the context of the microplane theory using a homogenization scheme based on the principle of energy equivalence with a direct tensorial representation of the effective elastic stiffness. The paper presents elementary studies of the model behavior accompanied with the evaluation of its capability to capture the response of concrete fatigue under compressive loading. A calibration and validation procedure of the model response is provided based on an accompanying experimental program including normal- and high-strength concretes subjected to several loading scenarios. The model shows the ability to reproduce the concrete behavior under monotonic, cyclic and fatigue loading with consistent set of material parameters.

Engineered bi-material lattices with thermo-mechanical programmability

Lightweight meta-structures can reveal combinations of properties that do not occur in conventional structures. In this paper, a 2D bi-material unit cell is proposed based on a re-entrant architecture. Two 3D unit cells, so-called star and cubic architectures, are then constructed using the proposed 2D unit cell. Finally, lattice-based unit cells are periodically arranged to develop a meta-structure. The thermo-mechanical functionalities of the developed meta-lattices can be programmed by tailoring either architectural parameters (i.e., strut length, diameter, shape, and orientation) or the material combination of struts. The influence of the architectural parameters and material combinations on the effective elastic stiffness, Poisson’s ratio (PR), and coefficient of thermal expansion (CTE) is systematically investigated. A finite element (FE) model is established to perform a goal-driven multi-objective optimization and to attain an uncommon combination of PR and CTE (e.g. negative PR up to −1.8 or near-zero CTE) with uncompromised stiffness. Despite the complicated design (many struts are diagonal with respect to the printing bed), the low-cost additive manufacturing process (i.e., fused filament fabrication) parameters are optimized to fabricate a selected number of bi-material meta-lattices with a reasonable time and quality. By employing the FE model, the optimized architectural parameters are determined for a series of applied engineering applications such as structurally robust aerospace structures, energy absorbable structural elements, and shape transforming structures. The developed lightweight meta-lattices with tunable thermo-mechanical properties can offer the material selection charts with more economical material options for engineering applications such as antennas and precision instruments.

Azimuthal Fourier coefficient inversion for horizontal and vertical fracture characterization in weakly orthorhombic media

Horizontally and vertically aligned fractures (joints) permeated in an isotropic background can give rise to a long-wavelength equivalent orthorhombic medium, which is common for naturally fractured reservoirs. We have developed a feasible approach to characterize the horizontal and vertical fractures using observed azimuthal seismic reflection data. For weak anisotropy, we assume that the horizontal and vertical fractures are decoupled. Using a linear-slip orthorhombic model, we first obtain analytic expressions for the effective elastic stiffness matrix and the corresponding perturbed matrix of an effective orthorhombic anisotropic elastic medium. Combining the scattering function and the perturbed matrix of the orthorhombic medium, we then derive a linearized PP-wave reflection coefficient of effective orthorhombic medium in terms of compressional-wave (P-wave) and shear-wave (S-wave) moduli, density, and horizontal- and vertical-fracture-induced normal and tangential weaknesses. To handle the inverse problem for multiple model parameters in an orthorhombic medium, we reexpress the linearized PP-wave reflection coefficient as a Fourier series. According to the sensitivity analysis of Fourier coefficients (FCs) to isotropic background elastic parameters and four fracture weaknesses, we finally establish a three-step inversion approach to describe the orthorhombic model, which involves (1) estimation of the FCs at different incidence angles from observed azimuthal seismic data and determination of the symmetry axis azimuth, (2) an iterative Bayesian inversion for isotropic background elastic parameters and fracture weaknesses of horizontal fractures from the zeroth-order FC, and (3) an iterative Bayesian inversion for fracture weaknesses of vertical fractures from the second-order FC. The proposed approach is validated by tests on synthetic and field data sets, which demonstrates that the inversion results of P- and S-wave moduli, density, and four fracture weaknesses are robust and reasonable for gas-bearing fractured reservoir characterization with orthorhombic symmetry.

Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.

Elastic properties of nanocrystalline materials of hexagonal symmetry: The core-shell model and atomistic estimates

Anisotropic core-shell model of a nano-grained polycrystal is extended to estimate the effective elastic stiffness of several metals of hexagonal crystal lattice symmetry. In the approach the bulk nanocrystalline material is described as a two-phase medium with different properties for a grain boundary zone and a grain core. While the grain core is anisotropic, the boundary zone is isotropic and has a thickness defined by the cutoff radius of a corresponding atomistic potential for the considered metal. The predictions of the proposed mean-field model are verified with respect to simulations performed with the use of the Large-scale Atomic/Molecular Massively Parallel Simulator, the Embedded Atom Model, and the molecular statics method. The effect of the grain size on the overall elastic moduli of nanocrystalline material with random distribution of orientations is analyzed.

Effect of saturation on the elastic properties and anisotropy of cortical bone

This paper focuses on the modeling of the effect of saturation on the overall elastic properties of cortical bone. We first use micromechanical model of Salguero, Saadat & Sevostianov (2014) to model anisotropic effective elastic stiffness of drained cortical bone and then apply replacement relations (see review of Sevostianov, 2020) to evaluate effect of the saturation. The model is verified by comparison with the experimental data of Granke et al. (2011). It is shown that accounting for the saturation makes the model consistent with the experiment data. We also compared the extents of anisotropy of the saturated and drained bones and show that presence of the biological fluid in pores reduces the overall anisotropy.

Analytical model for determining effective stiffness and mechanical behavior of polymer matrix composite laminates using continuum damage mechanics

The present paper proposes a new analytical model for predicting the effective stiffness of composite laminates with fiber breaks and transverse cracks. The model is based on continuum damage mechanics and the classical laminate theory. We derived damage variables describing stiffness reduction due to fiber breaks and its maximum value during ultimate tensile failure from the global load-sharing model. Furthermore, a simplified analytical model is presented for obtaining two damage variables for a cracked ply subjected to transverse tensile loading or in-plane shear loading. This model was developed assuming that the displacement field of the longitudinal direction can be expressed in the form of a quadric function by loosening the boundary condition for the governing differential equation. For verifying the developed model, the elastic constants of damaged composite laminates were predicted for cross-ply and angle-ply laminates and compared with the finite element analysis results. As for the appropriate expression of the effective elastic stiffness matrix of the damaged ply, we verified four types of effective compliance/stiffness matrices including the Murakami, Yoshimura, Li, and Maimí models. We found the Maimí model to be the most appropriate among these four models. Moreover, we successfully simplified the expressions for damage variables in the complicated infinite series obtained in our previous study. We also proved that this could contribute toward improving the accuracy of our analysis. After verifying the present model, the stress–strain response and failure strength of carbon- or glass-fiber-reinforced plastic cross-ply laminates were predicted using Maimí’s compliance model and the simplified damage variables.

Elastic properties of transversely isotropic rocks containing aligned cracks and application to anisotropy measurement

Currently, most rock physics models, used for evaluating the elastic properties of cracked or fractured media, take into account the crack properties, but not the background anisotropy. This creats the errors of in the anisotropy estimates by using field logging data. In this work, based on the scattered wavefield theory, a sphere-equivalency method of elastic wave scattering was developed to accurately calculate the elastic properties of a vertical transversely isotropic solid containing aligned cracks. By setting the scattered wavefield due to a crack equal to that due to an equivalent sphere, an effective elastic stiffness tensor was derived for the cracked medium. The stability and accuracy of the approach were determined for varying background anisotropy values. The results show that the anisotropy of the effective media is affected by cracks and background anisotropy for transversely isotropic background permeated by horizontally aligned cracks, especially for the elastic wave propagating along the horizontal direction. Meanwhile, the crack orientation has a significant influence on the elastic wave velocity anisotropy. The theory was subsequently applied to model laboratory ultrasonic experimental data for artificially cracked samples and to model borehole acoustic anisotropy measurements. After considering the background anisotropy, the model shows an improvement in the agreement between theoretical predictions and measurement data, demonstrating that the present theory can adequately explain the anisotropic characteristics of cracked media.

Effective elastic stiffness of 2D materials containing nanovoids of arbitrary shape

This paper aims to determine the in plane effective elastic properties of two-dimensional (2D) materials containing nanovoids of arbitrary shape. To achieve this objective, the complex variable and the associated conformal mapping techniques are used to solve the heterogeneity problem of a single nanovoid with arbitrary shape embedded in an infinite matrix. In this particular problem, to capture the edge and size effects of nanovoids, line elasticity model is used for the void boundary. The results of the heterogeneity problem are then used to determine the elastic properties of 2D nanoporous materials by applying the dilute and Mori–Tanaka schemes. Applications to the case of aluminum and the study the shape and size effects are also presented.

The impact of different clay minerals on the anisotropy of clay matrix in shale

ABSTRACTAlthough clay is composed of disconnected anisotropic clay platelets, many rock physics models treat the clay platelets in shale as interconnected. However, the clay matrix in shales can be modelled as anisotropic clay platelets embedded within a soft isotropic interplatelet region, allowing the influence of disconnected clay platelets on the elastic properties of the clay matrix to be analysed. In this model, properties of the interplatelet region are governed by its effective bulk and shear moduli, whereas the effective properties of the clay platelets are governed by their volume fraction, aspect ratio and elastic stiffness tensor. Together, these parameters implicitly account for variations in clay and fluid properties, as well as fluid saturation. Elastic stiffnesses of clay platelets are obtained from the literature, including both experimental measurements and first‐principles calculations of the full anisotropic (monoclinic or triclinic) elastic stiffness tensors of layered silicates. These published elastic stiffness tensors are used to compile a database of equivalent transverse isotropic elastic stiffness tensors, and other physical properties, for eight common varieties of layered silicates. Clay matrix anisotropy is then investigated by examining the influence of these different elastic stiffnesses, and of varying model parameters, upon the effective transverse isotropic elastic stiffness tensor of the clay matrix. The relationship between the different clay minerals and their associated anisotropy parameters is studied, and their impact on the resulting anisotropy of the clay matrix is analysed.

Seismic performance parameters of fully grouted reinforced masonry squat shear walls

The displacement ductility, ductility-related seismic force modification factor, and the effective elastic stiffness are important force-based seismic design (FBD) parameters for reinforced masonry (RM) shear walls. In addition, stiffness degradation and equivalent viscous damping, are crucial displacement-based seismic design (DBD) parameters. This study analyzes previously reported test results of eight RM shear-dominated fully-grouted rectangular squat walls subjected to cyclic lateral excitations to evaluate the FBD and DBD parameters. The main variables of the tested walls are the level of axial compressive stress, vertical and horizontal reinforcement ratio, anchorage end detail, and the spacing of horizontal and vertical reinforcement. However, all tested walls had the same shear span to depth ratio equal to 1.25. The evaluated FBD parameters are compared to the Canadian, European and American standards. Moreover, based on the analysis, equations for the stiffness degradation of RM shear walls relative to their top drift limit or displacement ductility, are developed and presented for the examined walls. The ductility-related seismic force modification factor is computed on average 2.6, and 3.9, based on equal energy and equal displacement idealization, respectively. The US, and European standards ignore the axial stress effect on wall’s elastic effective stiffness and highly overestimate its value for the tested walls. This study presents, based on the tested walls, a guide design values for squat shear-dominated walls category that can be used in FBD and DBD.

Molecular dynamics and micromechanics study of hygroelastic behavior in graphene oxide-epoxy nanocomposites

The hygroelasticity of oxygen-functionalized graphene (GO)-epoxy nanocomposites is studied. Two different transversely isotropic nanocomposite molecular models are constructed: with uniformly distributed and interface-concentrated water molecules, respectively. The stress-strain curves and coefficients of moisture expansion (CMEs) are determined according to moisture content. The degradation of the GO/epoxy interface due to the infiltrated moisture is characterized by interfacial decohesion tests. The micromechanics model is used to derive new closed-form solutions for the effective elastic stiffness and CME of multi-phase composites with interfacial imperfections. Regardless of the moisture distribution, the overall hygroelastic behavior of nanocomposites clearly degrades upon moisture absorption.

A Quantitative Analysis of Multi-Scale Response of CMP Pad and Implication to Process Assessments

The multiscale characteristics of the pad deformation and its surface morphology control the quality and efficacy of the chemical mechanical planarization (CMP) process. The pad microstructure comprised of cell size, wall thickness and surface roughness exhibit temporal variations in the effective pad stiffness at the pad abrasive interface. An experimental and analytical approach is developed to examine a dry IC-1000 pad response at different length scales, utilizing a nano-indentation with a conical tip of 1μm radius and a flat-punch of 60μm diameter. The quasi-static load-indentation depth plots showed different characteristic trends with varying stiffness at different loading ranges. The measurements showed the role of the porous microstructure to drive a competition between the local cell-membrane indent, bending of cell membrane and the bulk response of the porous pad. These different deformation mechanisms are employed in constructing an equivalent mechanical model for the effective pad elastic stiffness. The model prediction matches well with the force-indentation depth measurements. The developed model is used to investigate the influence of pad property evolution on the propensity of scratch generation during the CMP process. Such physically based models can be utilized to optimize the pad microstructure and morphology to control the spatial and temporal modulation of the material removal rate and the propensity for scratch generation.

Spectral analysis of inhomogeneities shows that the elastic stiffness of random composites decreases with increasing heterogeneity

Investigation of inhomogeneities has wide applications in different areas of mechanics including the study of composite materials. Here, we analytically study an arbitrarily-shaped isotropic inhomogeneity embedded in a finite-sized heterogeneous medium. By modal decomposition of the influence of the inhomogeneity on the deformation of the composite, an exact relation is presented that determines the variation of effective elastic stiffness caused by the presence of the inhomogeneity. This relation indicates that the effective elastic stiffness of a composite is always a concave function of the elastic modulus of the inhomogeneity, embedded inside the composite. Therefore, as the heterogeneity of elastic random composites increases, the rate of increase in effective stiffness caused by the stiffer constituents is smaller than the rate of its decrease due to the softer constitutions. So, weakly heterogeneous random composites become softer and less conductive with increasing heterogeneity at the same mean of constituent properties. We numerically evaluated the effective properties of about ten thousand composites to empirically support these results, characterize the nonaffinity of the displacement field in random composites, characterize the sample to sample variability of the effective properties and extend the results to the thermal conductivity of composites.