Showing papers on "Transverse isotropy published in 2000"
TL;DR: In this article, the authors use the linear slip theory of Schoenberg and co-workers and the models developed by Hudson and Thomsen for pennyshaped cracks to relate the anisotropic parameters to the physical properties of the fracture network and to devise fracture characterization procedures based on surface seismic measurements.
Abstract: The simplest effective model of a formation containing a single fracture system is transversely isotropic with a horizontal symmetry axis (HTI) Reflection seismic signatures in HTI media, such as NMO velocity and amplitude variation with offset (AVO) gradient, can be conveniently described by the Thomsen‐type anisotropic parameters e(V), δ(V), and γ(V) Here, we use the linear slip theory of Schoenberg and co‐workers and the models developed by Hudson and Thomsen for pennyshaped cracks to relate the anisotropic parameters to the physical properties of the fracture network and to devise fracture characterization procedures based on surface seismic measurements Concise expressions for e(V), δ(V), and γ(V) linearized in the crack density, show a substantial difference between the values of the anisotropic parameters for isolated fluid‐filled and dry (gas‐filled) penny‐sh aped cracks While the dry‐crack model is close to elliptical with e(V)≈δ(V), for thin fluid‐filled cracks e(V);≈0 and the absolute value
474 citations
TL;DR: In this paper, a numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus.
Abstract: This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).
294 citations
TL;DR: In this paper, the linear slip theory was used to obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media with a horizontal symmetry plane for naturally fractured reservoirs, under the assumptions of weak anisotropy of the background medium and small compliances of the fractures.
Abstract: Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between t...
242 citations
TL;DR: The ventricular laminar architecture may give rise to anisotropic material properties transverse to the fibers with greater resting stiffness within than between myocardial laminae, though it remains to be seen if active stress is also orthotropic with respect to the laminationar architecture.
Abstract: Recent morphological studies have demonstrated a laminar (sheet) organization of ventricular myofibers. Multiaxial measurements of orthotropic myocardial constitutive properties have not been reported, but regional distributions of three-dimensional diastolic and systolic strains relative to fiber and sheet axes have recently been measured in the dog heart by Takayama et al. [30]. A three-dimensional finite-deformation, finite element model was used to investigate the effects of material orthotropy on regional mechanics in the canine left ventricular wall at end-diastole and end-systole. The prolate spheroidal model incorporated measured transmural distributions of fiber and sheet angles at the base and apex. Compared with transverse isotropy, the orthotropic model of passive myocardial properties yielded improved agreement with measured end-diastolic strains when: (1) normal stiffness transverse to the muscle fibers was increased tangent to the sheets and decreased normal to them; (2) shear coefficients were increased within sheet planes and decreased transverse to them. For end-systole, orthotropic passive properties had little effect, but three-dimensional systolic shear strain distributions were more accurately predicted by a model in which significant active systolic stresses were developed in directions transverse to the mean fiber axis as well as axial to them. Thus the ventricular laminar architecture may give rise to anisotropic material properties transverse to the fibers with greater resting stiffness within than between myocardial laminae. There is also evidence that intact ventricular muscle develops significant transverse stress during systole, though it remains to be seen if active stress is also orthotropic with respect to the laminar architecture.
241 citations
TL;DR: In this paper, a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method is introduced. But this approach is not suitable for the case of 3D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.
Abstract: We introduce a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method. The technique solves a weak formulation of the wave equation, which is discretized using a high-order polynomial representation on a finite element mesh. For isotropic media, the spectral element method is known for its high degree of accuracy, its ability to handle complex model geometries, and its low computational cost. We show that the method can be extended to fully anisotropic media. The mass matrix obtained is diagonal by construction, which leads to a very efficient fully explicit solver. We demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2-D transversely isotropic test case, and by comparing its predictions against those based upon a finite difference method for a 2-D heterogeneous, anisotropic medium. We show its generality and its flexibility by modeling wave propagation in a 3-D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.
214 citations
TL;DR: A model of transverse isotropy for cancellous bone in the jaw, where the symmetry axis is along the infero-superior (weakest) direction is suggested.
Abstract: The elastic moduli have not been reported for cancellous bone from the edentulous mandible. Accurate values are needed for finite element modeling of the mandible. The aim of this study was to determine elastic modulus values in three orthogonal directions for cancellous bone taken from an edentulous jaw and to relate these values to apparent density and volume fraction. Seven samples were obtained from the edentulous mandible of a 74-year-old female. Young's modulus was determined by compression testing of cubes cut with the faces aligned with the anatomic axes. Bone volume fraction averaged 0.33 (SD 0.14) and apparent density averaged 0.55 g/cc (SD 0.29). Young's modulus was greatest in the mesio-distal direction (mean 907 MPa, SD 849 MPa), followed by the bucco-lingual (mean 511 MPa, SD 565 MPa) and infero-superior direction (mean 114 MPa, SD 78 MPa). The infero-superior direction was less than the bucco-lingual (P = 0.03) and mesio-distal (P = 0.002). The mesio-distal and bucco-lingual directions could not be shown to be different (P = 0.32). This suggests a model of transverse isotropy for cancellous bone in the jaw, where the symmetry axis is along the infero-superior (weakest) direction.
169 citations
TL;DR: In this paper, failure criteria for geomaterials are formulated in terms of the stress state and a microstructure tensor for orthotropy and transverse isotropy.
157 citations
TL;DR: In this article, the authors have made sets of independent compressional and shear wave velocity measurements, which with density, allow them to completely characterize the transverse isotropy of samples from five metamorphic belts: the Haast schist terrane (South Island, New Zealand), Poultney slate, Chugach phyllite, Coldfoot schist, and Pelona schist (United States).
Abstract: We have made sets of five independent compressional and shear wave velocity measurements, which with density, allow us to completely characterize the transverse isotropy of samples from five metamorphic belts: the Haast schist terrane (South Island, New Zealand), Poultney slate, Chugach phyllite, Coldfoot schist, and Pelona schist (United States). These velocity measurements include compressional wave velocities for propagation parallel, perpendicular, and at 45° to the symmetry axis, shear wave velocity for propagation and particle motion perpendicular to the symmetry axis, and shear wave velocity for propagation parallel to the symmetry axis. Velocity measurements were made up to pressures of 1 GPa (∼35-km depth) where microcracks are closed and anisotropy is due to preferred mineral orientation. Our samples exhibit compressional wave anisotropy of 9–20% as well as significant shear wave splitting. Metamorphic terranes that are anisotropic to ultrasonic waves may also be anisotropic at the scale of active and passive seismic experiments. Our data suggest that a significant thickness (10–20 km) of appropriately oriented (steeply dipping foliation) schist in the crust could contribute as much as 45% of observed shear wave splitting. Our data set can also be used to model the effects of crustal anisotropy for active source seismic experiments in order to determine if the anisotropy of the terrane is significant and needs to be taken into account during processing and modeling of the data.
150 citations
TL;DR: In this article, a two-stage inversion of the waveforms of surface waves was used to determine the pattern of polarization anisotropy in the Australasian region, and the results from 1584 Love and Rayleigh wave seismograms were combined in a tomographic inversion to provide a representation of three-dimensional structure for wavespeed heterogeneities.
147 citations
TL;DR: In this article, the elastic and viscoelastic properties of unidirectional reinforced composites are modeled by a transversely isotropic approach at small and at finite strains respectively.
134 citations
TL;DR: In this paper, the authors derived the Green's functions for anisotropic bimaterials based on Stroh formalism and two-dimensional Fourier transforms, where the full-space Green's function is in an explicit form, while the complementary part is expressed in terms of simple regular lineintegrals over [0, 2pa] that are suitable for standard numerical integration.
TL;DR: In this paper, a weak-link fractal scaling theory was extended to include a pre-exponential factor and obtained an exact expression for relationship between the Young's modulus and the volume fraction of solids.
Abstract: The elastic modulus of a colloidal aggregate network is dependent on the amount and spatial distribution of mass, as well as particle properties including size, shape, and particle-particle interactions. At high volume fractions, the elastic properties of a network of close-packed particle flocs is dependent on the strength of the interfloc links. A previously developed weak-link fractal scaling theory relates the elastic constant ~K! of the network to the volume fraction of solids ~F!, namely K;F. In this paper, we extend this theory to include a pre-exponential factor and obtain an exact expression for relationship between the Young’s modulus and the volume fraction of solids.
TL;DR: In this article, a boundary element analysis of linear elastic fracture mechanics in three-dimensional cracks of anisotropic solids is presented, which can model the solids with multiple interacting cracks or damage.
Abstract: This paper presents a boundary element analysis of linear elastic fracture mechanics in three-dimensional cracks of anisotropic solids. The method is a single-domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions.
The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack-front elements are introduced to capture the crack-tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny-shaped or a square-shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed-form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.
TL;DR: Huang et al. as mentioned in this paper derived the integral representation for the piezoelectric Eshelby tensor for cylindrical inclusions aligned along the axes either parallel or perpendicular to the axis of the anisotropy in a transversely isotropic material.
TL;DR: In this paper, the transversely isotropic elastic moduli of plasma-sprayed coatings are calculated in terms of microstructural parameters, and the dominant features of the porous space are identified as strongly oblate pores, that tend to be either parallel or normal to the substrate.
TL;DR: In this article, the authors developed a method which allows one to calculate the complete elastic field (stress field unit displacements) of layered materials of transverse and complete isotropy under given load conditions.
Abstract: This paper develops a method which allows one to calculate the complete elastic field (stress field unit displacements) of layered materials of transverse and complete isotropy under given load conditions. It is assumed that the layered body consists of at infinite half-space and various infinite planes which are all ideally bonded to each other. Thus, the interfaces are parallel to the surface of the resulting coated half space. The approach is based on the method of images in classical electrostatics. The final solution for an arbitrary load problem can be presented as a series of potential functions, where corresponding functions may be interpreted as image loads the analogous to image charges. The solution for the elastic field for any arbitrary stress distribution on the surface of the coated half space can be obtained in a relatively straightforward manner by using the method described here as long as the corresponding solution for the homogeneous half space is known. Further, if this solution of the homogeneous case may be expressed in terms of elementary functions, then the solution for the coated half space is elementary, too, Explicit formulas for the stress fields for some particular examples are given.
TL;DR: In this paper, the tensor of elastic moduli of a single test specimen of mudrock subjected to an anisotropic stress field is determined from ultrasonic group velocity measurements involving point-like transducers.
Abstract: An experimental technique for determining the in‐situ elastic properties of mudrocks with (horizontal) alignments in the microstructure is used to study the accuracy of a set of three nested scalar anisotropic approximations for transversely isotropic (TI) media. Each subsequent approximation adds one more velocity parameter and includes the previous as a special case. These approximations are convenient and robust because of their close relationship to standard geophysical measurements. There exists no good theory to predict the effects of an imposed stress on the elasticity of mudrocks. In this study, the tensor of elastic moduli of a single test specimen of mudrock subjected to an anisotropic stress field is determined from ultrasonic group velocity measurements involving pointlike transducers. The mechanical characterization (performed at constant pore pressure) is accompanied by detailed microscopic observations and analysis. The method was used to obtain accurate elastic constants for five well‐defi...
TL;DR: In this article, the authors derived the extended Boussinesq and Cerruti solutions for point forces and point charge acting on the surface of a transversely isotropic piezoelectric half-space, aiming at a series of common three-dimensional contact including spherical contact, a conical indentor and an upright circular flat punch.
TL;DR: Pan and Yuan as mentioned in this paper proposed a method based on the Stroh formalism and two-dimensional Fourier transforms in combination with Mindlin's superposition method for the analysis of three-dimensional Green's functions for anisotropic piezoelectric bimaterials.
TL;DR: In this paper, the authors present constitutive models for nearly incompressible, transversely isotropic materials in finite hyperelasticity, particularly for reinforced rubber-like materials, which are of essential engineering interest.
TL;DR: In this paper, a general solution of the three-dimensional equations of transversely isotropic piezothermoelastic materials (crystal class, 6 mm) was derived using the operator theory.
Abstract: This paper derives a general solution of the three-dimensional equations of transversely isotropic piezothermoelastic materials (crystal class, 6 mm). Two displacement functions are first introduced to simplify the basic equations and a general solution is then derived using the operator theory. For the static case, the proposed general solution is very simple in form and can be used easily in certain boundary value problems. An illustrative example is given in the paper by considering the symmetric crack problem of an arbitrary temperature applied over the faces of a flat crack in an infinite space. The governing integro-differential equations of the problem are derived. It is found that exact expressions for the piezothermoelastic field for a penny-shaped crack subject to a uniform temperature can be obtained in terms of elementary functions.
TL;DR: In this article, the authors investigated residual stresses and dimensional changes in the compression moulded glass-mat reinforced thermoplastic (GMT) parts and showed that the elastic material model is sufficient for predicting geometrical changes if the temperature history is symmetric through the thickness of the specimen, but insufficient for prediction of residual stresses when the temperature field is not symmetric.
Abstract: Residual stresses and dimensional changes in the compression moulded glass-mat reinforced thermoplastic (GMT) parts are investigated. A heat transfer and crystallisation model with temperature dependent matrix properties is used to obtain input to the subsequent thermal stress analysis. Both an isotropic viscoelastic and a transversely isotropic elastic material model are investigated through finite-element calculations. The results show that the elastic material model is sufficient for prediction of geometrical changes if the temperature history is symmetric through the thickness of the specimen, but insufficient for prediction of residual stresses and dimensional changes if the temperature field is not symmetric. In this case a viscoelastic material model is necessary.
Journal Article•
TL;DR: In this article, two independent state equations with variable coefficients were derived from the three-dimensional theory equations of piezoelasticity for transverse isotropy, and a laminated approximation was used to transform the state equations to those with constant coefficients in each sub-layer.
TL;DR: In this paper, a quasi-static, normal indentation of a transversely isotropic, linear elastic, piezoelectric half-space by a rigid spherical indenter was analyzed.
Abstract: The present paper deals with theoretical and computational analysis of quasi-static, normal indentation of a transversely isotropic, linear elastic, piezoelectric half-space by a rigid spherical indenter. The contact is axisymmetric, nonconforming, monotonically advancing with load, frictionless and adhesionless. The indenter was modeled either as perfect conductor or as perfect insulator. The mechanical and electrical fields below the surface were examined. The issues of mechanical and dielectric strength due to indentation were examined using Weibull statistics of surface imperfections. The particular cases of PZT-4, PZT-5A, BaTiO 3 , and (Ba 0.917 Ca 0.083 )TiO 3 indented by rigid punches having either zero electrical potential or zero electric charge were solved with finite element analysis.
TL;DR: The model was able to fit the material response to rapid loading and equilibrium indentation test data to approximately 50 percent strain, and suggested even higher percentage of stress supported by the fluid phase of cartilage than given earlier by small deformation theories of biphasic cartilage.
Abstract: Articular cartilage is known to behave nonlinearly for large deformations, Mechanical properties derived from small strain experiments yield excessively large deformations in finite element models used in the study of severe blunt impact to joints. In this manuscript, a method is presented to determine the nonlinear elastic properties of biphasic cartilage based on a transversely isotropic hypo-elastic model. The elastic properties were estimated by fitting two force-displacement curves (in rapid loading and at equilibrium) obtained from large deformation indentation relaxation tests on cartilage using a nonporous spherical indentor. The solid skeleton of the cartilage was modeled as a transversely isotropic hypo-elastic material and a commercial finite element program was employed to solve the problem of a layer indented by a rigid sphere. Components of the hypo-elasticity tensor were made dependent on deformation according to the variations defined by a transversely isotropic hyperelastic formulation given earlier by others. Material incompressibility was assumed during the initial stage of rapid loading. The analysis was utilized for the determination of in situ properties of rabbit retropatellar cartilage at large deformations. The model was able to fit the material response to rapid loading and equilibrium indentation test data to approximately 50 percent strain. This material model suggested even higher percentage of stress supported by the fluid phase of cartilage than given earlier by small deformation theories of biphasic cartilage.
TL;DR: In this article, an exact analysis of a penny-shaped crack in a transversely isotropic piezoelectric medium subjected to arbitrary shear loading that is antisymmetric with respect to the crack plane is presented.
TL;DR: In this article, the authors studied the scattering of an acoustic wave incident on a transversely isotropic cylinder immersed in a fluid and found that the form function is rich in peaks whose number steadily goes down as the angle of incidence increases.
TL;DR: In this article, a mixed 3D variational principle was used to derive 2D equations for an anisotropic plate-like piezoelectric body and one-dimensional equations for a beam-like one.
Abstract: We use a mixed 3-dimensional variational principle to derive 2-dimensional equations for an anisotropic plate-like piezoelectric body and one-dimensional equations for an anisotropic beam-like piezoelectric body. The formulation accounts for double forces without moments which may change the thickness of the plate and deform the cross-section of the rod. The dependence of the bending rigidities of a transversely isotropic plate upon the angle between the normal to the midsurface and the direction of transverse isotropy is exhibited. The plate equations are used to study the cylindrical deformations of a transversely isotropic plate due to equal and opposite charges applied to its top and bottom surfaces. It is also found that a piezoelectric circular rod with axis of transverse isotropy not coincident with its centroidal axis and subjected to electric charges at the end faces is deformed into a non-prismatic body.
01 Jan 2000
TL;DR: In this paper, the authors compared strength-of-materials approaches and elasticity approaches as a means of estimating the various extensional and shear moduli, the major Poisson's ratio, and coefficients of thermal expansion of fiber-reinforced composite materials.
Abstract: Strength-of-materials approaches, which lead to rule-of-mixture models, and elasticity approaches, as well as finite-element analyses of idealized microstructures, are discussed as a means of estimating the various extensional and shear moduli, the major Poisson’s ratio, and coefficients of thermal expansion of fiber-reinforced composite materials. Numerical results for a medium modulus graphite-epoxy material are presented. Results from the various approaches are compared. It is shown that for the extensional modulus and thermal expansion coefficient in the fiber direction, E 1 and α 1 , respectively, as well as for the major Poisson’s ratio, (ν) 12 , the rule-of-mixtures results are quite accurate when compared to the more exacting finite-element results. For extension modulus E 2 , shear modulus G 12 , and coefficient of thermal expansion α 2 , properties normal to the fiber direction, the rule-of-mixtures results do not compare well with the finite-element results. However, so-called alternative and modified rule-of-mixture models for those properties are presented which lead to better agreement with finite-element results. Elasticity approaches, in the form of concentric cylinder models, lead to very good comparisons for E 1 and G 12 , and also provide a means for estimating the three-dimensional elastic properties necessary to characterize transversely isotropic composite materials.
TL;DR: For an infinite three-dimensional transversely isotropic piezoelectric material, Green's functions (which give the fuli set of electromechanical fields due to a point electric charge and an arbitrarily oriented point force) were derived in a simple way, using methods of the potential theory as discussed by the authors.
Abstract: For an infinite three-dimensional transversely isotropic piezoelectric material, Green's functions (which give the fuli set of electromechanical fields due to a point electric charge and an arbitrarily oriented point force) are derived in elementary functions in a simple way, using methods of the potential theory. For a semi-infinite transversely isotropic piezoelectric material. Green's functions are also derived, but in a limited way: a point force and a point electric charge are assumed to be applied at the boundary of the half-space. The latter solutions constitute a generalization of Boussinesqs and Cerruti's problems of elasticity for piezoelectric materials. The strength of the piezoeffect (the difference from the purely elastic case) is estimated for the example of the piezoceramics PZT-6B.