## What is/are Transversely Isotropic?

Transversely Isotropic - In this paper, differential evolution algorithm (DEA) has been utilized to quantitatively analyze the piezoelectric properties of transversely isotropic piezoelectric mediums via PFM experiment.^{[1]}We apply this method to polycrystalline ceramics with an intergranular weak plane and fiber structures with transversely isotropic crack resistance.

^{[2]}Further, the constitutive equations that describe the mechanical behavior of the domain in [Formula: see text] under plane stress conditions are derived, assuming that the material is transversely isotropic in 3D.

^{[3]}In this paper, the consistent second-order plate theory is developed for transversely isotropic plates.

^{[4]}The failure and mechanical behavior of transversely isotropic rock are significantly affected by the original bedding planes.

^{[5]}The paper deals with the finite bending analysis of transversely isotropic hyperelastic slender beams made of a neo-Hookean material with longitudinal voids.

^{[6]}It indicates that the MAE behaves as a transversely isotropic material in the presence of an external magnetic field.

^{[7]}Regional and strain-dependent constitutive parameters were finally proposed for the human lateral meniscus, suggesting an isotropic behavior of both the horns, and a transversely isotropic response of the pars intermedia.

^{[8]}The tool, along with a unique processing scheme, enables the determination of horizontal and vertical resistivity as well as the dip angle and the azimuth of the formation based on an assumption of transversely isotropic (TI) formation models (Graciet and Shen, 1998) while drilling in real time.

^{[9]}Moreover, the HM is reinforced with transversely isotropic macroscale carbon fibers in which the Halpin–Tsai scheme is used for multiscale homogenization procedure.

^{[10]}Compared with isotropic media, at least two extra parameters are involved in common P-wave seismic data processing and interpretation for transversely isotropic media.

^{[11]}Thus, this work proposes to determine the optimized fiber orientation of a fiber reinforced composite structure by using the SPIMFO method with a constitutive equation in fully nonlinear range based on transversely isotropic neo-Hookean model.

^{[12]}In this work, we present the mathematical formulation and the numerical implementation of a new model for initially straight, transversely isotropic rods.

^{[13]}Borehole acoustic logging plays an important role in inverting the five Thomsen parameters of many formations characterized as a transversely isotropic medium with a vertical axis of symmetry (VTI).

^{[14]}The article is devoted to solving the problem of cylindrical bending of a flat panel made of a transversely isotropic incompressible composite material with finite deformations.

^{[15]}Comparison with existing study indicates that the proposed model has a good performance in simulating the typical pre and post strain localization behaviors of transversely isotropic geomaterials.

^{[16]}To further understand the non-linear, tension-compression asymmetric and transversely isotropic properties of the SSTPS, inner configurations were investigated by X-ray computed tomography and scanning electron microscopy.

^{[17]}Exact solution of axisymmetric wave propagation problem in radially and functionally graded circular cylinder made from combination of isotropic and transversely isotropic materials is obtained.

^{[18]}By incorporating the range of results derived from the inversions with advanced interpretations such as transversely isotropic constants, these uncertainties can be further used in stochastic models in downstream workflows.

^{[19]}Transversely isotropic rock is a typical anisotropic rock type.

^{[20]}We illustrate the implementation of the model by showing reasonable predictions on the transversely isotropic behaviour of the heterogeneous pavement.

^{[21]}For the sedimentary genesis, this rock can be considered a transversely isotropic geomaterial, as also documented in the experimental literature.

^{[22]}Transversely isotropic fibers obey linear elastic rule while isotropic polymeric matrix follows both standard solid and Kelvin-Voigt viscoelastic models.

^{[23]}As a result, the material properties cannot be described within the framework of the transversely isotropic medium model.

^{[24]}Combining the mechanic theory of transversely isotropic material, the anisotropy parameters of the tight sandstone are analyzed, as well as the influence on the hydrofracturing technology for tight sandstone in the field.

^{[25]}According to the transversely isotropic characteristics of layered rock mass, the failure approach index (FAI) was selected to evaluate the damage and failure of thin shale.

^{[26]}Aiming at the vibration isolation in transversely isotropic soil, a T-shaped partially embedded periodic barrier for surface waves is proposed, and its shielding performance is explored by using fi.

^{[27]}However, even though the theory for transversely isotropic (TI), spherically symmetric, models has been known since the late 1960s, readily available programs for traveltime calculations are restricted to isotropic models.

^{[28]}This study investigated the transversely isotropic properties of water-bearing shale by performing laboratory experiments on shale specimens with different bedding angles of 0°, 45°, and 90° under different saturation conditions.

^{[29]}The aim of this paper is to extend recent works devoted to the study of the effect of 3 D pores of concave shape embedded in isotropic matrix to the case of transversely isotropic (TI) matrix.

^{[30]}The use of isotropic and transversely isotropic elastic theories was explored, as well as the implementation of stress relaxation in the plastic regime of the material.

^{[31]}We concentrate our attention on the state of a permeability tensor is transversely isotropic with the isotropy axis in the vertical direction of gravity and the permeability ratios of vertical to horizontal are different in the macropores and micropores.

^{[32]}A transformation matrix based on Rodrigues’ rotational formula for transversely isotropic Micropolar-Cosserat lamina has been introduced; which reduces it to the well-known non-classical (classical and couple-stress) elastic formulation.

^{[33]}It is represented as a mixture of intact myocardium and a transversely isotropic scar structure.

^{[34]}They are used to identify the elastic constants of the fibres, assuming a transversely isotropic behaviour, by minimising a cost function between measured and estimated values.

^{[35]}In this paper, the nonlocal elasticity theory is applied to study the propagation of plane wave and Rayleigh-type surface wave in an incompressible, rotating and transversely isotropic material.

^{[36]}The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media.

^{[37]}The influence of the implantation of transversely isotropic mechanical models was also studied, by comparing the basilar membrane with isotropic and transversely isotropic mechanical properties.

^{[38]}To describe the averaged properties of rubber-cord, the elastic potential of a transversely isotropic or orthotropic material can be used.

^{[39]}A method proposed earlier, relying on the use of harmonic Cartesian polynomial and rational functions, is extended here to find a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a system of long parallel, non-intersecting, transversely isotropic elastic cylindrical electrical conductors.

^{[40]}This paper investigates the Mode-I crack problem that a three-dimensional infinite space of transversely isotropic multi-ferroic composite medium is weakened by multiple coplanar penny-shaped cracks under uniform mechanical, electric and magnetic loadings.

^{[41]}

^{[42]}The purpose of this work is to analyze the dynamic response of transversely isotropic saturated media under circular moving loads, in which the vertical and tangential loads can be simultaneously considered.

^{[43]}A technique is proposed for solving a 3D problem of elasticity theory for a transversely isotropic half-space.

^{[44]}A systematic and comprehensive assessment of popular 3D macroscopic failure criteria for predicting pure inter-fiber fracture of transversely isotropic unidirectional (UD) composites is performed in the present study.

^{[45]}A numerical approach based on the particle discrete element theory is adopted to study the fracture evolution of transversely isotropic rocks with a pre-existing flaw under compression tests.

^{[46]}Using the SEG advanced modeling Barrett Unconventional model, we have assessed the influence of errors in the anisotropy parameters by conducting a sensitivity analysis in three types of 3D models: transversely isotropic with a vertical symmetry axis, transversely isotropic with a horizontal symmetry axis, and orthorhombic media.

^{[47]}We construct the analytic solution of a three-dimensional problem of thermoelasticity for a transversely isotropic space with internal stationary heat sources.

^{[48]}This work presents a method to estimate longitudinal-wave anisotropy in transversely isotropic tissues.

^{[49]}This study conducts uniaxial compression creep tests to experimentally investigate the transversely isotropic creep characteristics and the damage mechanism of layered phyllite samples having bedding angles of 0°, 22.

^{[50]}

## boundary element method

The object of the Field Boundary Element Method is the transversely isotropic plane plate.^{[1]}This paper studies the stress intensity factors of square cracks horizontally placed in one-layered transversely isotropic halfspaces using the dual boundary element method (DBEM).

^{[2]}An indirect boundary element method (IBEM) is proposed to solve the scattering of qP-, qSV- and SH-waves by a three-dimensional (3D) arbitrary-shaped alluvial valley embedded in a layered transversely isotropic (TI) half-space.

^{[3]}The case study of this Field Boundary Element Method is the transversely isotropic plane plate.

^{[4]}This paper provides an indirect boundary element method (IBEM) to elaborate elastic wave propagation in a layered anisotropic medium (simplified as transversely isotropic, TI) with three-dimensional (3-D) irregular free surfaces.

^{[5]}

## extended precise integration

In this paper, we investigate the response of multilayered transversely isotropic poroelastic media under vertical rectangular moving loads by utilizing the extended precise integration method (PIM).^{[1]}To investigate this problem, a time-dependent analysis method for the interaction between retaining piles and stratified saturated soils with transversely isotropic property due to pre-excavation dewatering (PED) is established by the coupling of the extended precise integration method (PIM) and the finite element method (FEM).

^{[2]}Based on the extended precise integration method, we investigate the multi-dimensional consolidation problem of transversely isotropic viscoelastic saturated soils.

^{[3]}

## finite element method

Then, using the finite element method (FEM), PD due to the ARFI excitation PSFs in 42 elastic, incompressible, transversely isotropic (TI) materials with shear moduli ratios of 1.^{[1]}The formula for the model I fracture toughness of the transversely isotropic material is obtained on the basis of the finite element method (FEM) together with the J-integral.

^{[2]}In this paper, a numerical model for transversely isotropic layered shale with transition zone was established by utilizing the extended finite element method (XFEM) based on cohesive zone model (CZM).

^{[3]}

## semi infinite medium

The initially stressed composite structure is comprised of a transversely isotropic fluid saturated porous layer over a foundation with dry sandy elastic stratum and functionally graded substrate (semi-infinite medium) in which the bonding between the interfaces of the layers and semi-infinite medium are considered to be imperfect.^{[1]}The assumed model consists of tri-mediums resting over a viscoelastic semi-infinite medium and the considered tri-mediums are transversely isotropic, porous, and heterogeneous respectively under the impression of initial stress.

^{[2]}

## Layered Transversely Isotropic

Considering the large offset coverage of modern seismic acquisitions, we propose new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis (VTI).^{[1]}This paper presents an analytical solution to the mixed boundary value problem for a dynamic rigid rectangular plate on the surface of a layered transversely isotropic half-space.

^{[2]}This paper studies the stress intensity factors of square cracks horizontally placed in one-layered transversely isotropic halfspaces using the dual boundary element method (DBEM).

^{[3]}In layered transversely isotropic media with a vertical symmetry axis, the accuracy of traditional traveltime approximations is limited to near offsets.

^{[4]}An indirect boundary element method (IBEM) is proposed to solve the scattering of qP-, qSV- and SH-waves by a three-dimensional (3D) arbitrary-shaped alluvial valley embedded in a layered transversely isotropic (TI) half-space.

^{[5]}This paper proposed a reflection-transmission matrix method to solve the time-history response of a layered transversely isotropic (TI) saturated site subjected to obliquely incident seismic waves.

^{[6]}This model can describe separate flows of two immiscible fluids in layered transversely isotropic unsaturated poroelastic media subjected to harmonically dynamic loads.

^{[7]}

## Homogeneou Transversely Isotropic

We validate the method by comparing the numerical results with the exact solutions for a homogeneous transversely isotropic model and a two-layered model.^{[1]}This method reduces the problem of a layered material with isotropic layers to the problem of a homogeneous transversely isotropic medium.

^{[2]}The constitutive relations and basic governing equations of motion for homogeneous isotropic elastic semiconductor (n-type) and homogeneous transversely isotropic ( class) piezoelectric elastic media, in the absence of body forces and electric sources are made non-dimensional in order to reduce the mathematical complexity.

^{[3]}In this paper, the three-dimensional solutions of homogeneous transversely isotropic coated structure under spherical contact loadings are derived based on the three-dimensional general solutions of the transversely isotropic material.

^{[4]}The proposed algorithm is validated using transmission tests for a homogeneous transversely isotropic model with a vertical symmetry axis that contains a Gaussian anomaly in the shear-wave vertical attenuation coefficient.

^{[5]}We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry.

^{[6]}

## Tilted Transversely Isotropic

To solve these problems, we have formulated the factored topography-dependent anisotropic eikonal (FTDAE) equation in tilted transversely isotropic (TTI) media using the factorization principle.^{[1]}For our numerical tests, we specify the operators for a mildly anisotropic tilted transversely isotropic (TTI) medium.

^{[2]}In this paper, we develop a tilted transversely isotropic equivalent medium parametrization method to suppress the interface error and the artefact diffraction caused by the staircase approximation under the application of coarse grids.

^{[3]}We introduce the multi-GPU scheme by using the acoustic wave propagation and then describe the implementation of RTM in tilted transversely isotropic (TTI) media.

^{[4]}

## Dimensional Transversely Isotropic

The present research deals with the mathematical model for study of the disturbance due to mechanical (horizontal or vertical) and thermal source in two dimensional transversely isotropic magneto thermoelastic solid under initial stress due to time-harmonic source with generalized Lord–Shulman (LS) theory of thermoelasticity with two temperatures.^{[1]}We present a three-dimensional transversely isotropic velocity model of the crust and upper mantle beneath the East Asia Continent and Western Pacific subduction zone based on the full waveform inversion (FWI).

^{[2]}In this paper, we study a penny-shaped crack model with electrically and thermally semi-permeable boundary conditions in a three-dimensional transversely isotropic piezoelectric semiconductor.

^{[3]}This paper presents the general solutions of three-dimensional transversely isotropic multilayered media subjected to a vertical or horizontal rectangular dynamic load by utilizing the analytical layer-element method, and the solutions can be used for both time-harmonic loads and moving ones.

^{[4]}

## Vertical Transversely Isotropic

Herein, qP-wave propagation is simulated in vertical transversely isotropic (VTI) media using the one-way wave equation in the ray-centred coordinate system (15°), which combines the flexibility of ray theory and accuracy of wave theory to describe wave propagation.^{[1]}We apply this methodology to develop an efficient 3-D FSM to compute the first-arrival traveltimes for qP waves in 3-D vertical transversely isotropic (VTI) media.

^{[2]}Characterizing the kinematics of seismic waves in elastic vertical transversely isotropic (VTI) media involves four independent parameters.

^{[3]}In this study, the characteristics of Love waves in viscoelastic vertical transversely isotropic layered media are investigated by finite-difference numerical modeling.

^{[4]}

## Incompressible Transversely Isotropic

The present paper proposes a new Strain Energy Function (SEF) for incompressible transversely isotropic hyperelastic materials, i.^{[1]}Recently, we introduced a nearly incompressible transversely isotropic (NITI) model based on two independent shear moduli determining tensile and out-of-plane shear behavior.

^{[2]}In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled based on the strain energy density function proposed based on a novel framework.

^{[3]}The strain-energy density W $W$ for incompressible transversely isotropic hyperelastic materials depends on four independent invariants of the strain tensor.

^{[4]}

## Elastic Transversely Isotropic

The composites are assumed to be constituted by a non-aging, isotropic viscoelastic matrix reinforced by square or hexagonal arrangements of elastic transversely isotropic long and short fibers, the latter being cylindrical inclusions.^{[1]}The analysis of the results demonstrated that the purely elastic transversely isotropic material model is adequate for predicting behavior, at least before nonlinearities occur.

^{[2]}In this paper, the 3D solutions of elastic transversely isotropic coated structure under conical contact are derived based on the general solutions of the elastic transversely isotropic material.

^{[3]}

## Vertically Transversely Isotropic

We develop a generalized reflection and transmission coefficient method (GRTM) for generating six-component (6-C) synthetic seismograms in horizontally layered vertically transversely isotropic (VTI) media.^{[1]}To solve the eikonal equation in vertically transversely isotropic (VTI) or tilted transversely isotropic (TTI) models with irregular geometry, we have formulated a new iterative fast sweeping method (FSM) on unstructured triangular meshes.

^{[2]}The stratification of the medium is taken into account using the vertically transversely isotropic model.

^{[3]}

## Multilayered Transversely Isotropic

An efficient theoretical solution is proposed to investigate the time history response of an elastic circular thin plate bonded on multilayered transversely isotropic soils.^{[1]}The foundation is modeled as a rigid disk, with either permeable or impermeable and smooth contact surface, whereas a multilayered transversely isotropic poroelastic half-space is defined for the supporting soil.

^{[2]}In this paper, we investigate the response of multilayered transversely isotropic poroelastic media under vertical rectangular moving loads by utilizing the extended precise integration method (PIM).

^{[3]}

## Piezoelectric Transversely Isotropic

Electroelastic piezoelectric transversely isotropic half-plane with a homogeneous or a functionally graded coating is considered.^{[1]}The inhomogeneities and matrix materials exhibit piezoelectric transversely isotropic symmetry.

^{[2]}

## 3d Transversely Isotropic

In this work, linearized elastic isotropic continuum-kinematics-inspired-peridynamics (CPD) is further generalized to study the elastic, damage, and fracture behavior of anisotropic continua, focusing on 3D transversely isotropic and 2D orthotropic body.^{[1]}By minimising the difference between PDO and the new approximation, we compute optimal coefficients based on a least-squares method, and then derive the new time–space domain PAWEs in both 3D transversely isotropic and orthorhombic anisotropic media.

^{[2]}

## Graded Transversely Isotropic

An exact approach is used to investigate the propagation of a Love-type wave in a low-velocity piezoelectric-viscoelastic material (PV) stratum bonded to a functionally graded transversely isotropic viscoelastic (FGTIV) material substrate.^{[1]}The crux of the present study is to analyze the dispersion characteristics of the Love-type wave in a micropolar elastic stratum overlying a functionally graded transversely isotropic substrate.

^{[2]}

## Heterogeneou Transversely Isotropic

The present analysis confers the propagation characteristics of horizontally polarized shear (SH) wave through a heterogeneous transversely isotropic fluid-saturated poroelastic sandwiched layer of finite width embedded between two heterogeneous isotropic elastic half-spaces due to the impact of an impulsive line source.^{[1]}Introduction: In this paper, a mathematical model of Love-type wave propagation in a heterogeneous transversely isotropic elastic layer subjected to initial stress and rotation of the resting on a rigid foundation.

^{[2]}

## transversely isotropic medium

Considering the large offset coverage of modern seismic acquisitions, we propose new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis (VTI).^{[1]}Compared with isotropic media, at least two extra parameters are involved in common P-wave seismic data processing and interpretation for transversely isotropic media.

^{[2]}Borehole acoustic logging plays an important role in inverting the five Thomsen parameters of many formations characterized as a transversely isotropic medium with a vertical axis of symmetry (VTI).

^{[3]}As a result, the material properties cannot be described within the framework of the transversely isotropic medium model.

^{[4]}The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media.

^{[5]}If the transversely isotropic medium embeds vertical fractures (VFTI medium), the effective medium becomes orthorhombic.

^{[6]}Based on an analytical solution for the current point source in an anisotropic half-space, we study the apparent resistivity and apparent chargeability of a transversely isotropic medium with vertical and horizontal axes symmetry, respectively.

^{[7]}The one-dimensional compressibility and the elastic response of combinations of layers are derived adopting an analytical solution for the stratified, transversely isotropic medium.

^{[8]}If the transversely isotropic medium embeds vertical fractures (VFTI medium, according to Schoenberg and Helbig, 1997), the effective medium becomes orthorhombic.

^{[9]}In layered transversely isotropic media with a vertical symmetry axis, the accuracy of traditional traveltime approximations is limited to near offsets.

^{[10]}The theory of generalized thermo-magneto-electroelasticity is employed to obtain the plane wave solutions in an unbounded, homogeneous and transversely isotropic medium.

^{[11]}1 million nodes with five parameters at each node that capture velocity variations for P- and S-waves travelling at arbitrary directions in transversely isotropic media with a vertical symmetry axis (VTI).

^{[12]}This method reduces the problem of a layered material with isotropic layers to the problem of a homogeneous transversely isotropic medium.

^{[13]}The ray-based generalized Radon transform (GRT) inversion/migration strategy is studied in this paper for acoustic transversely isotropic media with a vertical symmetry axis (VTI media) represented by the P-wave normal moveout velocity, Thomsen parameter δ , anelliptic parameter η , and density.

^{[14]}Although exact solutions and approximations of the PP‐wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill‐posed.

^{[15]}Within this condition, we examine the Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media.

^{[16]}Finally, as particular cases of the general 3D form, the stress equations of motion for planar problems (plane strain and Generalized plane stress) for transversely isotropic media are formulated.

^{[17]}

## transversely isotropic material

It indicates that the MAE behaves as a transversely isotropic material in the presence of an external magnetic field.^{[1]}Exact solution of axisymmetric