Showing papers on "Transverse isotropy published in 2008"
TL;DR: In this article, the authors used elastic wave velocity measurements in the laboratory to assess the evolution of the microstructure of shales under triaxial stresses, which are representative of in situ conditions.
Abstract: Elastic wave velocity measurements in the laboratory are used to assess the evolution of the microstructure of shales under triaxial stresses, which are representative of in situ conditions. Microstructural parameters such as crack aperture are of primary importance when permeability is a concern. The purpose of these experiments is to understand the micromechanical behavior of the Callovo-Oxfordian shale in response to external perturbations. The available experimental setup allows for the continuous, simultaneous measurement of five independent elastic wave velocities and two directions of strain (axial and circumferential), performed on the same cylindrical rock sample during deformation in an axisymmetric triaxial cell. The main results are (1) identification of the complete tensor of elastic moduli of the transversely isotropic shales using elastic wave velocity measurements, (2) assessment of the evolution of these moduli under triaxial loading, and (3) assessment of the evolution of the elastic ani...
140 citations
09 Jun 2008
TL;DR: In this article, a new approximate partial differential equation for qP-waves in transverse isotropy with a vertical symmetry axis (VTI) media is proposed, which combines efficient implementation and low artifacts.
Abstract: SUMMARY We propose a new approximate partial differential equation for qP-waves in transverse isotropy with a vertical symmetry axis (VTI) media. We analyse its relationship to two other published "pseudo-acoustic" VTI equations. All three pseudo-acoustic VTI wave equations are coupled systems of second-order PDEs in time, derived from the same dispersion relation for qP waves by introducing different auxiliary functions. The new method combines efficient implementation and low artifacts. Modeling and reversetime migration are shown to validate the wave equation.
120 citations
TL;DR: The constitutive relations and field equations for anisotropic generalized thermoelastic diffusion are derived and deduced for a particular type of anisotropy, i.e., transverse isotropy as mentioned in this paper.
108 citations
TL;DR: In this article, a fully three-dimensional continuum damage model is proposed to predict both the intralaminar and the interlaminar failure mechanisms that occur in laminated fiber-reinforced polymer composites.
Abstract: This article proposes a fully three-dimensional continuum damage model, developed at the sub-ply level, to predict in an integrated way both the intralaminar and the interlaminar failure mechanisms that occur in laminated fiber-reinforced polymer composites The constitutive model is based on the assumption that the composite material is transversely isotropic, and accounts for the effects of crack closure under load reversal cycles The damage model is implemented in an implicit finite element code taking into account the requirement to ensure a mesh-independent computation of the dissipated energy The comparison between the model predictions and published experimental data indicates that the model can accurately predict the effects of transverse matrix cracks on the residual stiffness of quasi-isotropic laminates, the interaction between transverse matrix cracks and delamination, and final failure of the laminate
107 citations
TL;DR: In this article, a two-step methodology is proposed in order to estimate from numerical simulations the permeability of deformed woven fabrics, assuming yarns behave as hypoelastic materials with transverse isotropy.
103 citations
TL;DR: In this article, the problem of a functionally graded, transversely isotropic, magneto-electro-elastic circular plate acted on by a uniform load is considered and the displacements and electric potential are represented by appropriate polynomials in the radial coordinate.
98 citations
TL;DR: In this paper, the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr k (k is zero or a finite even number).
97 citations
TL;DR: In this paper, a micromechanical model was used to describe the macroscopic effective elastic properties and anisotropy of the rock in terms of its microscopic features, such as intrinsic anisotropic and crack/pore geometry.
Abstract: Thisstudywasdevotedtotheinterpretationoftheevolutionof elastic wave velocities in anisotropic shales that are subjected to deformation experiments in the laboratory. A micromechanical model was used to describe the macroscopic effective elastic properties and anisotropy of the rock in terms of its microscopic features, such as intrinsic anisotropy and crack/pore geometry. The experimental data reported in Part 1 were compared quantitatively with the micromechanical model predictions to gain someinsightintothemicrostructuralbehavioroftherockduring deformation. The inversion of the experimental data using the micromechanicalmodelwascarriedoutbymeansofanumerical minimization of the least-squares distance between data and model in terms of effective compliances. Under isotropic mechanical loading, the overall behavior of the dry shale is consistent with the closure of crack-like pores, which are aligned in the plane of symmetry of the transversely isotropic background matrix. Those cracks represent a low fraction of the total porosity, but they have a strong effect on elastic wave velocities. The data are consistent with an initial horizontal crack density of 0.07. Crack closure also is evidenced at early stages of axial loading applied perpendicular to the shale bedding plane, whereas crack density increases significantly as axial stress is increased. Interpretation of the wet experiment is less straightforward, although some preliminary conclusions could be drawn. Under isotropic stress, crack closure also is evidenced, whereas crack density remainsconstantattheearlystagesofdeviatoricloading.Whenaxial peak stress is approached, crack density increases drastically, which likely indicates onset and development of vertical cracking. Wet experiments probably are more complex because water islikelytobeexpelledfromcrack-likeporestowardequantpores inresponsetothemechanicalloading.
90 citations
TL;DR: In this article, the local problems and corresponding homogenized coefficients of the elastic, dielectric, magnetic permittivity, piezoelectric and magneto-electric tensors are explicitly described.
Abstract: Summary In this work, based on the periodic unfolding homogenization technique, the limiting equations modelling the behaviour of three-dimensional magneto-electro-elastic periodic structures are rigorously established. The local problems and the corresponding homogenized coefficients of the elastic, dielectric, magnetic permittivity, piezoelectric, piezomagnetic and magneto-electric (ME) tensors are explicitly described. The homogenization model is exemplified for laminated composites and a unified general formula for all effective properties of periodic multilaminated magneto-electro-elastic composites is obtained. This formula is applied to investigate the global behaviour for the important case of transversely isotropic constituents and any finite number of layers in each periodic cell. Examples that provide theoretical evidence of the presence of both a product property and the ME effect are given.
84 citations
TL;DR: Compared methods for derivation of the anisotropic elastic properties of the mineralized collagen fibrils as well as the detailed structure of a lamella can only be accomplished with further model assumptions with respect to the hierarchical structure of cortical bone.
Abstract: Several high-frequency ultrasound techniques have been developed during the last decade with the intention of assessing elastic properties of bone at the tissue level. The basic measurement principles can be divided into: 1) measurement of the compressional wave velocity in thin tissue sections; 2) measurement of surface acoustic wave velocities in thick sections; and 3) derivation of the acoustic impedance from the confocal reflection amplitude in thick sections. In this paper, the 3 principles are described with example measurements given in the frequency range from 50 MHz to 1.2 GHz. The measurements were made with 2 microscopes operating in the pulse-echo mode, either with frequencies up to 200 MHz and time-resolved detection or between 100 MHz and 2 GHz and amplitude detection. The methods are compared and their application potentials and limitations are discussed with respect to the hierarchical structure of cortical bone. Mapping of the confocal reflection amplitude has superior capabilities for deriving quantitative elastic and structural parameters in the heterogeneous bone material. Even at low frequencies (50 MHz), the mineralized tissue matrix can be separated from the larger pores (Haversian canals), and the elastic coefficient in the probing direction can be measured in 2 dimensions. Depending on the type of sample surface preparation (flat or cylindrically shaped), local distribution of a single elastic coefficient or the average transverse isotropic stiffness tensor can be derived. With frequencies in the GHz range, the lamellar bone structure can be analyzed. However, at one GHz, the acoustic wavelength is still one order of magnitude larger than the individual mineralized collagen fibrils. Although the thickness of a lamellar unit can easily be assessed from the acoustic image, the derivation of the anisotropic elastic properties of the mineralized collagen fibrils as well as the detailed structure of a lamella can only be accomplished with further model assumptions.
80 citations
TL;DR: In this paper, the authors proposed a critical plane approach to describe the strength anisotropy of transversely isotropic rock masses subjected to a three-dimensional stress state.
TL;DR: In this paper, an efficient and accurate analytical derivation of the fundamental Green's functions for a transversely isotropic elastic half-space subjected to an arbitrary, time-harmonic, finite, buried source is presented.
TL;DR: In this paper, the simulation results showed that the shear modulus is relatively independent of the out-of-plane stress component, which can be revealed by the indistinctive change in the contact normal distribution and the normal contact forces on that plane in the DEM simulations.
Abstract: In this paper, experimental studies using a true triaxial apparatus and a bender element system, and numerical simulations based on the discrete element method (DEM) were used to investigate the stress- and fabric-induced shear-stiffness anisotropy in soils at small strains. Verified by experiments and DEM simulations, the shear modulus was found to be relatively independent of the out-of-plane stress component, which can be revealed by the indistinctive change in the contact normal distribution and the normal contact forces on that plane in the DEM simulations. Simulation and experimental results also demonstrated that the shear modulus is equally contributed by the two principal stress components on the associated shearing planes. Fabric-induced stiffness anisotropy, i.e., the highest G and subscript xy or G and subscript hh, can be explained by simulation findings in which more contact normals prefer to distribute along the horizontal direction. The experiments and simulations also reveal that the fabric-induced stiffness anisotropy increases with an increasing aspect ratio of the particles. The assumption of transversely isotropic fabric in soils is valid based on the DEM simulation results; however, this assumption is not completely supported by the experimental results.
TL;DR: In this paper, the authors investigated the indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter, conical indenter and spherical indenter.
Abstract: Frictionless indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter (i.e., punch), conical indenter (i.e., cone), and spherical indenter (i.e., sphere) are investigated. Both insulating and conducting indenters are considered. The technique of Hankel transformation is employed to derive the corresponding dual integral equations for the mixed boundary value indentation problems. For the two limiting cases of infinitely thick and infinitely thin piezoelectric films, closed-form solutions are obtained. For piezoelectric films of finite thickness, a numerical method is constructed to solve the dual integral equations and semi-empirical models having only two unknown parameters are proposed for the responses of indentation force, electric charge and electric potential, and contact radius. With the two parameters inferred from the numerical results, the semi-empirical formulae are found to provide good estimates of the indentation responses for the two limiting cases of infinitely thick and thin piezoelectric films, as well as those in between. The inferred parameters in the proposed semi-empirical formulae for normalized indentation force and electric charge are checked against four different piezoelectric materials and are found to be insensitive to the selection of piezoelectric materials. It is believed that the proposed semi-empirical indentation formulae are useful in developing experimental indentation techniques to extract the material properties of piezoelectric films.
TL;DR: In this article, the authors considered the dynamic response of a transversely isotropic, linearly elastic layer bonded to the surface of a half-space of a different material under surface loads and presented an analytical formulation for the determination of the displacements and stresses in both regions in frequency domain.
TL;DR: The proposed model is based on the constitutive theory of finite elasticity and is characterized by an anisotropic strain-energy function which takes the material structure into account and polyconvex which ensures material stability.
TL;DR: In this article, the macroscopic behavior of two-phase fibrous piezoelectric composites was investigated and it was shown that the overall thermoelectroelastic moduli of these solids must comply with two sets of exact connections.
Abstract: We consider the macroscopic behavior of two-phase fibrous piezoelectric composites. The fibers are of circular cross-section with the same radius. Along the interfaces between the fibers and the matrix we consider the effects of surface stress and surface electric displacement. The constituents are transversely isotropic and exhibit pyroelectricity. We find that the overall thermoelectroelastic moduli of these solids must comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides five constraints between the six axisymmetric overall electroelastic moduli. The second set relates the effective coefficients of thermal stress and pyroelectric coefficients to the effective electroelastic moduli, in analogy with Levin’s formula. In contrast to their conventional counterparts, i.e., without surface effects, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.
TL;DR: In this paper, the authors proposed the use of ultrasonic waves in determination of the five independent elastic constants required for sedimentary shales in terms of elastic anisotropy.
Abstract: In terms of elastic anisotropy, sedimentary shales may be considered to have transverse symmetry. Experimentally determining the five independent elastic constants required for this case remains challenging. This paper proposes the use of ultrasonic waves in determination of the five constants. Arrays of specially constructed transducers with different modes of vibration were mounted on samples trimmed from natural cores to measure ultrasonic P-waves and S-waves along the horizontal, vertical and 45 ◦ -inclination axes on the samples. The elastic constants calculated from these wave velocities were compared to those determined from static tests and acoustic in situ logs. The static tests included drained triaxial compression and confined torsion tests. The acoustic logs were conducted in an open hole using monopole compressional and dipole shear transmitters and receivers. It was found that the elastic constants determined by the static tests were much lower than those determined from the ultrasonic tests and acoustic logs. Discrepancies among the elastic constants determined from these three different methods are discussed and explained.
TL;DR: In this article, the authors quantify the undeformed configuration permeability of fiber-reinforced composite materials using geometrical considerations and directional averaging methods, with particular attention to the sub-cases of aligned fibres and fibres lying on a plane.
TL;DR: In this article, the authors proposed an upscaling procedure based on the effective medium theory to estimate the velocities of a wave at a given lower frequency in a layered medium, where the number of layers depends on the given frequency and logging step.
Abstract: SUMMARY
In this paper, the term ‘upscaling’ means the theoretical prediction of rock's elastic properties at lower frequency (seismic or cross-well data) using higher frequency logging data on sonic velocities (VP, VS1 and VS2), porosity and density. The mineral composition and water saturation derived from other logs are used. Due to the special treatment of sonic logging data provided by the dipole sonic probe, all the sonic velocities are obtained in the principal coordinate system of the rock's stiffness tensor. The upscaling procedure includes two steps. The first step involves the solution of an inverse problem on reconstruction of the parameters of the rock's microstructure from the logging data. The inversion is based on the effective medium theory. As a result of the inverse problem solution, the effective stiffness tensor is found for depths at which the sonic wave velocities are measured. At the second step, the velocities of waves at given lower frequencies are calculated as propagating in a layered medium. The number of layers in the medium depends on the given frequency and logging step. Each layer of the medium has the stiffness tensor found at the first step.
This upscaling procedure has been applied to a wellbore for which the dipole sonic data are available. The rocks penetrated by the well are shales. In general, the resulting medium exhibits orthorhombic symmetry at sonic frequency. This symmetry results from the preferential orientation of clay platelets and grain-related cracks and vertical cracks. The existence of the latter is indicated by the dipole sonic tool. Depending on the microstructure parameters (orientation of clay platelets and cracks, pore/crack connectivity and shale mineralogical composition), the shales, at lower frequency, have either transversely isotropic symmetry (with the vertical axis of symmetry, a.k.a. VTI) or orthorhombic symmetry.
TL;DR: In this paper, the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media is analyzed, and the authors show that the most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the energy.
TL;DR: In this paper, the relationship between the elastic moduli and compliances of transversely isotropic and orthotropic materials, which correspond to different appealing sets of linearly independent fourth-order base tensors used to cast the elastic models, are derived by performing explicit inversions of the involved fourthorder tensors, and deduced sets of elastic constants are related to each other and to common engineering constants expressed in the Voigt notation with respect to the coordinate axes aligned along the directions orthogonal to the planes of material symmetry.
Abstract: The relationships between the elastic moduli and compliances of transversely isotropic and orthotropic materials, which correspond to different appealing sets of linearly independent fourth-order base tensors used to cast the elastic moduli and compliances tensors, are derived by performing explicit inversions of the involved fourth-order tensors. The deduced sets of elastic constants are related to each other and to common engineering constants expressed in the Voigt notation with respect to the coordinate axes aligned along the directions orthogonal to the planes of material symmetry. The results are applied to a transversely isotropic monocrystalline zinc and an orthotropic human femural bone.
TL;DR: The spectral stiffness microplane model as mentioned in this paper is a general constitutive model for unidirectional composite laminates, able to simulate the orthotropic stiffness, pre-peak nonlinearity, failure envelopes, and, in tandem with the material characteristic length, also the post-peak softening and fracture.
Abstract: The paper presents the spectral stiffness microplane model, which is a general constitutive model for unidirectional composite laminates, able to simulate the orthotropic stiffness, prepeak nonlinearity, failure envelopes, and, in tandem with the material characteristic length, also the post-peak softening and fracture. The framework of the microplane model is adopted. The model exploits the spectral decomposition of the transversely isotropic stiffness matrix of the material to define orthogonal strain modes at the microplane level. This decomposition is a generalization of the volumetric-deviatoric split already used by Bauant and co-workers in microplane models for concrete, steel, rocks, soils, and stiff foams. Linear strain-dependent yield limits (boundaries) are used to provide bounds for the normal and tangential microplane stresses, separately for each mode. A simple version, with an independent boundary for each mode, can capture the salient aspects of the response of a unidirectional laminate, although a version with limited mode coupling can fit the test data slightly better. The calibration of model parameters, verification by test data, and analysis of multidirectional laminates are postponed for the subsequent companion paper. DOI: 10.1115/1.2744036
TL;DR: Keralavarma et al. as mentioned in this paper derived a yield function for materials containing spheroidal voids embedded in a perfectly-plastic anisotropic Hill-type matrix using approximate limit-analysis and a restricted set of trial velocity fields.
TL;DR: In this article, a different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic (TI) media is explored.
Abstract: A different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic (TI) media is explored. This formulation extends Thomsen’s weak anisotropy approach to stronger deviations from isotropy without significantly affecting the equations’ simplicity. One easily recognized improvement is that the extreme value of the quasi-SV-wave speed vsv (θ) is located at the correct incidence angle θ= θex rather than always being at the position θ=45° . This holds universally for Thomsen’s approximation, although θex ≡45° actually is never correct for any TI anisotropic medium. Wave-speed magnitudes are more closely approximated for most values of the incidence angle, although there may be some exceptions depending on actual angular location of the extreme value. Furthermore, a special angle θ= θm (close to theextreme point of the SV-wave speed and also needed by the new formulas) can be deduced from the same data normally used in weak anisotropy data analy...
TL;DR: In this paper, the authors presented an application of the newly developed CAFFE model (Continuum-mechanical, Anisotropic Flow model based on an anisotropic flow enhancement factor) to the EPICA ice core at Kohnen Station, Dronning Maud Land, Antarctica.
Abstract: We present an application of the newly developed CAFFE model (Continuum-mechanical, Anisotropic Flow model based on an anisotropic Flow Enhancement factor) to the EPICA ice core at Kohnen Station, Dronning Maud Land, Antarctica (referred to as the EDML core). A one-dimensional flow model for the site is devised, which includes the anisotropic flow law and the fabric evolution equation of the CAFFE model. Three different solution methods are employed: (1) computing the ice flow based on the flow law of the CAFFE model and the measured fabrics; (2) solving the CAFFE fabric evolution equation under the simplifying assumption of transverse isotropy; and (3) solving the unrestricted CAFFE fabric evolution equation. Method (1) demonstrates clearly the importance of the anisotropic fabric in the ice column for the flow velocity. The anisotropic enhancement factor produced with method (2) agrees reasonably well with that of method (1), even though the measured fabric shows a girdle structure (which breaks the transverse isotropy) in large parts of the ice core. For method (3), we find that the measured fabric is reproduced well by the model down to ∼2100 m depth. Systematic deviations at greater depths are attributed to the disregard of migration recrystallization in the model.
TL;DR: In this paper, the authors presented an analysis of Rayleigh surface waves in a homogeneous, transversely isotropic, generalized piezothermoelastic half-space rotating with uniform angular velocity about normal to its boundary and subjected to stress free, electrically shorted/charge free and thermally insulated/isothermal boundary conditions.
TL;DR: In this paper, the authors studied the propagation and reflection characteristics of plane waves from the stress free, thermally insulated/isothermal boundary of a piezothermoelastic half space.
TL;DR: In this paper, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term, and the non-linear equations on the local and on the global level are solved by means of the Newton-Raphson scheme.
Abstract: Transverse isotropy is realized by one characteristic direction—for instance, the fibre direction in fibre-reinforced materials.Commonly, the characteristic direction is assumed to be constant, but in some cases—for instance, in the constitutive description of biological tissues, liquid crystals, grain orientations within polycrystalline materials or piezoelectric materials, as well as in optimization processes—it proves reasonable to consider reorienting fibre directions. Various fields can be assumed to be the driving forces for the reorientation process, for instance, mechanical, electric or magnetic fields. In this work, we restrict ourselves to reorientation processes in hyper-elastic materials driven by principal stretches.
The main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework. Therefore, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term. The non-linear equations on the local and on the global level are solved by means of the Newton–Raphson scheme. Accordingly, the local update of the characteristic direction and the global update of the deformation field are consistently linearized, yielding the corresponding tangent moduli. Through implementation into a finite element code and some representative numerical simulations, the fundamental characteristics of the model are illustrated. Copyright © 2007 John Wiley & Sons, Ltd.
TL;DR: In this article, a strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids.
Abstract: A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress-strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis-Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.