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The mathematics of the physical properties of crystals

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... Tilting the c-axis around the x-axis (tilt angle ) keeps the mirror plane m y intact, and we obtain a rigidity tensor for a monoclinic symmetry. In this system we name the tensors (at constant E) for the rigidity, for the piezoelectricity, and for the dielectric constants [6,7]. Where, A and M are the following matrices: (6) In the longitudinal mode, the parallel resonance frequency (f p ) and in the shear mode the series resonance frequency (f s ) are the linear functions of the respective sound velocities, and we can write: ...
... In this system we name the tensors (at constant E) for the rigidity, for the piezoelectricity, and for the dielectric constants [6,7]. Where, A and M are the following matrices: (6) In the longitudinal mode, the parallel resonance frequency (f p ) and in the shear mode the series resonance frequency (f s ) are the linear functions of the respective sound velocities, and we can write: ...
... When the interface is oblique to the grid lines or arbitrarily passes through the FD cell ( Fig. 3), assuming that the angle of the interface passing through the cell is θ, we obtain an elasticity matrix after considering the equivalent medium by utilizing a simplified Bond transformation (Bond 1943;Zhu & Dorman 2000) ...
... When an arbitrary interface crosses the 3-D grid, the approximate tangential planar interface has a dipping angle of θ and a strike direction of φ. The elasticity matrix after considering equivalent medium parametrization can be obtained by utilizing the Bond transformation (Bond 1943;Zhu & Dorman 2000) ...
Article
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In recent years, many higher-order and optimized schemes have been developed to reduce the dispersion error with the use of large grid spacing in finite-difference seismic waveform simulations. However, there are two problems in the application of coarse grids for heterogeneous media: the generation of artefact diffraction from the stair-step representation of non-planar interfaces and the inaccuracy of the calculated waveforms due to the interface error. Several equivalent medium parameterization approaches have been proposed to reduce the interface error of the finite-difference method (FDM) in heterogeneous media. However, these methods are specifically designed for the standard (2,4) staggered-grid scheme and may not be accurate enough for coarse grids when higher-order and optimized schemes are used. In this paper, we develop a tilted transversely isotropic (TTI) equivalent medium parameterization method to suppress the interface error and the artefact diffraction caused by the staircase approximation under the application of coarse grids. We use four models to demonstrate the effectiveness of the proposed method, and analyse the accuracy of each seismic phase related to the interface. The results show that our method can be used with higher-order and optimized schemes at 3 points per wavelength (PPW) and produce satisfactory results.
... Fortunately, this could be done based on our knowledge in descriptions of the VTI media [18], [26], [27] with a stiffness matrix of hexagonal crystal solid. A stiffness matrix of a TTI medium could be obtained from a stiffness matrix of the VTI medium through the Bond transformation [29] in rotational coordinates. An inverse Bond transformation could transform a TTI matrix back to a VTI matrix. ...
... A stiff matrix for a TTI medium may be obtained through Bond transformation [29], C (T ) = GC (V ) J −1 , by rotating the vertical axis with the tilting anle φ, as shown in Fig. 1, which yields ...
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We report analytical polarization coefficients for inhomogeneously refracted P-wave in the post-critical incident-angle region, induced at the interfaces between different types of anisotropic rocks. For the A-shale/T-sandstone interface, the phase velocity of SV-wave in a large T-sandstone is smaller than that of the P-wave in A-shale. For the A-shale/O-shale interface, the phase velocity of SV-wave in O-shale is larger than that of P-wave in A-shale in some directions but smaller in other directions. In both cases, critical incident-angles do not exist related to the refracted SV-wave. Applying the widely reported rock parameters, we have obtained typical results of slowness that provide a logical explanation for the long holding scientific puzzle that some vertical axis of symmetry-tilt axis of symmetry (VTI-TTI) interface systems appear to have a critical incidence angle but it is not real. We have obtained reflection and refraction coefficients, polarization coefficients, particle displacement of the induced homogeneous wave, the polarization state of the inhomogeneously refracted P-wave, and the elliptical polarization trajectories. We conclude that the polarization coefficients of the induced waves are determined only by the physical nature of the media, the geometric structure of the interface, and the incident-angle. The analytical polarization coefficients provide a base for accurate calculations of the reflection coefficients at the anisotropic-rocks interfaces that are widely used for amplitude variations with offset (AVO) analysis and inversion interpretation of seismic exploration data. The analyses reported in this article apply not only to the VTI-TTI interfaces but also to the generic TTI-TTI interfaces through Bond transformation.
... Due to the stiffer direction in x−direction, tile 2 and 3 show a tetragonal effective material symmetry with C 11 , C 22 , C 44 , C 55 , C 12 and C 23 as independent entries: The material tensors C T i for the second changing parameter -the rotation around the z−axis -can be computed by a coordinate transformation, and thus require no homogenization simulations. The Bond-Transformation matrices [8] can be used to rotate the effective elasticity tensor by a matrix-matrix multiplication. Assume the following ordering of the macroscopic stresses σ M ij and strains ε M ij in the Voigt notation ...
... The following polar diagrams depict the independent entries C ii of the three material tensors (of the unit tiles) for an arbitrary rotation around the z−axis. The values are computed with the Bond transformation matrices [8], according to Equation (23). Thus, at an angle of 0 • the value equals the corresponding entry of the respective unrotated material tensor C * T i of Example 3.3.4. ...
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This paper proposes an extension of the nite cell method (FCM) to V-rep models, a novel geometric framework for volumetric representations. This combination of an embedded domain approach (FCM) and a new modeling framework (V-rep) forms the basis for an efficient and accurate simulation of mechanical artifacts, which are not only characterized by complex shapes but also by their non-standard interior structure. These types of objects gain more and more interest in the context of the new design opportunities opened by additive manufacturing, in particular when graded or micro-structured material is applied. Two different types of functionally graded materials (FGM) are considered: The rst one, multi-material FGM is described using the inherent property of V-rep models to assign different properties throughout the interior of a domain. The second, single-material FGM { which is heterogeneously micro-structured { characterizes the effective material behavior of representative volume elements by homogenization and performs large-scale simulations using the embedded domain approach.
... The 2D stress field , , is transformed counterclockwise by an angle ∈ [0 , 360 ] to the stress field , , . To transform the material elastic properties from the ( − ) coordinates to the ( − ) coordinates we resort to the Bond-stress and strain transformation matrices [48,49,[64][65][66]. The linearly elastic stress field in the ( − ) coordinates can be related to the strain field in the same coordinate system by Hooke's law [54]. ...
... The linearly elastic stress field in the ( − ) coordinates can be related to the strain field in the same coordinate system by Hooke's law [54]. As demonstrated by Bond [64], the stress field in the ( − ) coordinates, , can be transformed to a stress field in the ( − ) coordinates, , using the Bond-Stress Transformation matrix, , such that: = (48) The 2D Bond-Stress Transformation matrix is given by: Similarly, the corresponding strain field in the ( − ) coordinates, , can be transformed to a strain field in the ( − ) coordinates, , using the Bond-Strain Transformation matrix, , such that: = (50) Inverting the matrices in Equations (48) and (50) and substituting the results into stress-strain relation (Hooke's law), gives: (51) is utilized to compute the transformation of the stiffness properties of LMs through an angle ∈ [0 , 360 ]. The LM relative density is set to unity, i.e., = 1. Figure 15 to Figure 20 illustrate the polar diagram of the Young's modulus (for the and directions as well as the shear modulus in the direction) for the 13 topologies studied in this paper. ...
Article
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This paper investigates the macroscopic anisotropic behavior of periodic cellular solids with rigid-jointed microscopic truss-like architecture. A theoretical matrix-based procedure is presented to calculate the homogenized stiffness and strength properties of the material which is validated experimentally. The procedure consists of four main steps, namely, (i) using classical structural analysis to determine the stiffness properties of a lattice unit cell, (ii) employing the Bloch's theorem to generate the irreducible representation of the infinite lattice, (iii) resorting to the Cau-chy-Born Hypothesis to express the microscopic nodal forces and deformations in terms of a homogeneous macroscopic strain field applied to the lattice, and (iv) employing the Hill-Mandel homog-enization principle to obtain the macro-stiffness properties of the lattice topologies. The presented model is used to investigate the anisotropic mechanical behavior of 13 2D periodic cellular solids. The results are documented in three set of charts that show (i) the change of the Young and Shear moduli of the material with respect to their relative density; (ii) the contribution of the bending stiffness of microscopic cell elements to the homogenized macroscopic stiffness of the material; and (iii) polar diagrams of the change of the elastic moduli of the cellular solid in response to direction of macroscopic loading. The three set of charts can be used for design purposes in assemblies involving the honeycomb structures as it may help in selecting the best lattice topology for a given functional stiffness and strength requirement. The theoretical model was experimentally validated by means of tensile tests performed in additively manufactured Lattice Material (LM) specimens, achieving good agreement between the results. It was observed that the model of rigid-joined LM (RJLM) predicts the homogenized mechanical properties of the LM with higher accuracy compared to those predicted by pin-jointed models.
... The material tensors C Ti for the second changing parameter-the rotation around the z−axis-can be computed by a coordinate transformation, and thus require no homogenization simulations. The Bond-Transformation matrices [86] can be used to rotate the effective elasticity tensor by a matrix-matrix multiplication. Assume the following ordering of the macroscopic stresses σ M ij and strains ε M ij in the Voigt notation ...
... The following polar diagrams depict the independent entries C ii of the three material tensors (of the unit tiles) for an arbitrary rotation around the z−axis. The values are computed with the Bond transformation matrices [86], according to Eq. (23). Thus, at an angle of 0 • the value equals the corresponding entry of the respective unrotated material tensor C * Ti of Example 3.3.4. ...
Article
Full-text available
This paper proposes an extension of the finite cell method (FCM) to V-rep models, a novel geometric framework for volumetric representations. This combination of an embedded domain approach (FCM) and a new modeling framework (V-rep) forms the basis for an efficient and accurate simulation of mechanical artifacts, which are not only characterized by complex shapes but also by their non-standard interior structure. These types of objects gain more and more interest in the context of the new design opportunities opened by additive manufacturing, in particular when graded or micro-structured material is applied. Two different types of functionally graded materials (FGM) are considered: The first one, multi-material FGM is described using the inherent property of V-rep models to assign different properties throughout the interior of a domain. The second, single-material FGM—which is heterogeneously micro-structured—characterizes the effective material behavior of representative volume elements by homogenization and performs large-scale simulations using the embedded domain approach.
... The fracture-oriented system is denoted by ðx 0 1 ; x 0 2 ; x 0 3 Þ and will be referred to as the fracture eigencoordinate system. Through the Bond transformation (Bond, 1943;Mavko et al., 2009), the eigencoordinate system can be transformed back into the observation coordinate system and the fractures can be azimuthally dependent (Appendix A). The effective compliance matrix can be written as ...
... (A-1) and the Bond transformation matrix is given by (Bond, 1943;Mavko et al., 2009) β 33 β 31 β 31 β 32 2β 21 β 31 2β 22 β 32 2β 23 β 33 β 22 β 33 þ β 23 β 32 β 21 β 33 þ β 23 β 31 β 22 β 31 þ β 21 β 32 2β 31 β 11 2β 32 β 12 2β 33 β 13 β 12 β 33 þ β 13 β 32 β 11 β 33 þ β 13 β 31 β 11 β 32 þ β 12 β 31 2β 21 β 11 2β 12 β 22 2β 13 β 23 β 22 β 13 þ β 12 β 23 β 11 β 23 þ β 13 β 21 β 22 β 11 þ β 12 β 21 ...
Article
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The linear slip theory is gradually being used to characterize seismic anisotropy. If the transversely isotropic medium embeds vertical fractures (VFTI medium), the effective medium becomes orthorhombic. The vertical fractures, in reality, may exist in any azimuth angle which leads the effective medium to be monoclinic. We apply the linear slip theory to create a monoclinic medium by only introducing three more physical meaning parameters: the fracture preferred azimuth angle, the fracture azimuth angle, and the angular standard deviation. First, we summarize the effective compliance of a rock as the sum of the background matrix compliance and the fracture excess compliance. Then, we apply the Bond transformation to rotate the fractures to be azimuth dependent, introduce a Gaussian function to describe the fractures' azimuth distribution assuming that the fractures are statistically distributed around the preferred azimuth angle, and average each fracture excess compliance over azimuth. The numerical examples investigate the influence of the fracture azimuth distribution domain and angular standard deviation on the effective stiffness coefficients, elastic wave velocities, and anisotropy parameters. Our results show that the fracture cluster parameters have a significant influence on the elastic wave velocities. The fracture azimuth distribution domain and angular standard deviation have a bigger influence on the orthorhombic anisotropy parameters in the ( x 2 , x 3 ) plane than that in the ( x 1 , x 3 ) plane. The fracture azimuth distribution domain and angular standard deviation have little influence on the monoclinic anisotropy parameters responsible for the P-wave NMO ellipse and have a significant influence on the monoclinic anisotropy parameters responsible for the S1- and S2-wave NMO ellipse. The effective monoclinic can be degenerated into the VFTI medium.
... Consider two Cartesian co-ordinate frames x-y-z (old) and x '-y'-z' (new). The relative orientations between each may be described using a matrix N of direction cosines (Bond, 1943;Auld, 1990;Walker and Wookey, 2012 cos(x, x ') cos(x, y ') cos(x, z ') cos(y, x ') cos(y, y ') cos(y, z ') cos(z, x ') cos(z, y ') cos(z, z ') ...
... Similarly, the general form of the Bond (1943) ...
Article
A borehole existing in any geologic formation concentrates the far-field tectonic and overburden stresses amplifying the magnitudes of certain stress components near the borehole. It is important to understand the magnitudes and patterns of this stress concentration because these lead to damage and can even collapse the borehole if sufficiently strong. The solution of the stress distributed near a borehole can be complicated considering the elastic anisotropy of rocks. We have developed programs (ASCIB3D) in MATLAB and Python to model the stress distribution around an inclined borehole in an arbitrarily oriented anisotropic medium. The program is built on the Lekhnitskij-Amadei solution. The input orientation of the far-field stresses and the elastic stiffness matrix of the medium into the program are geology angles instead of the rotation angles shown in previous studies, making the code more convenient for users. The sign convention for the inverse function, which is ignored in previous studies, is discussed in detail. The results indicate that the program ASCIB3D is a useful tool for modeling the stress distributed around an inclined borehole in the anisotropic formation and analyzing the effect of anisotropy and borehole inclination on stress distribution. The inclination and azimuth of the borehole and the anisotropy of the rocks affect the orientation and strength of the stress concentration.
... ,大角度入射的地震 数据的精确时-深转换 [3] , 以及声波测井中声波传播速度 的准确估算 [4] ,极化系数在研究各向异性岩石界面的反 射和折射以及模式转换波的极化状态中也至关重要。 虽然地震波(声波)和光波(电磁波)之间存在差 异,但它们的特性也有许多相似之处。例如,波的极化 是所有波的重要属性。不仅是地震波(声波),关于光 波(电磁波)的极化研究也已有很多报道 [5 , 6] 。许多研 究人员对无限大各向异性固体中声波的极化状态的研 究进行了报道。Crampinet 等 [7] 和 Hosten [8] ,Lanceleur 等 [9] 对在各向异性岩石中传播的 P-波的极化方向偏离 其传播方向的问题进行了讨论, Helbig 等 [10] 报道了无限 大各向异性岩石中传播的 P-波和 SV-波的异常极化现象。 Carcione [11] 给出了在无限大 VTI 介质中传播的准 P-波和 准 SV-波的极化系数解析表达式。应用著名的实验测量 岩石各向异性参数 [12][13][14] ,Fa 等人研究了岩石各向异性 对在无限大 VTI 介质中传播的准 P-波和和准 SV-波的极 化方向的影响 [15] 。基于对准 P-波和和准 SV-波的极化的 先验知识,Daley 和 Hron 给出了一个计算 VTI-VTI 介 质界面反射/折射系数的矩阵方程 [16] ,并仅对具有椭圆 各向异性的 VTI-VTI 介质界面的反射系数和折射系数 进行了数值计算 [17] 。Fa 等人提出了计算 VTI-VTI 岩石 界面的反射/折射系数的快速算法 [18] ,并建立了非均匀 折射 P-波的椭圆极化方程, 讨论了岩石各向异性对椭圆 极化状态的影响 [19][20][21] [11,22,23] , 利用邦德变换矩阵法则实现旋转坐标下的各向异性岩 石的刚度系数矩阵变换 [24] ,从 VTI 岩石介质的刚度矩 阵可以得到 TTI 岩石介质的刚度矩阵, 也可将 TTI 岩石 介质的刚度矩阵转换为 VTI 岩石介质的刚度矩阵, 从而 保证了我们在数学上能够建立严格的和具有实际物理 意义的 TTI-TTI 岩石介质界面的模型。 由于通过邦德变换矩阵法则 [24] 可以实现 TTI-TTI 12 13 sin cos ...
... ,大角度入射的地震 数据的精确时-深转换 [3] , 以及声波测井中声波传播速度 的准确估算 [4] ,极化系数在研究各向异性岩石界面的反 射和折射以及模式转换波的极化状态中也至关重要。 虽然地震波(声波)和光波(电磁波)之间存在差 异,但它们的特性也有许多相似之处。例如,波的极化 是所有波的重要属性。不仅是地震波(声波),关于光 波(电磁波)的极化研究也已有很多报道 [5 , 6] 。许多研 究人员对无限大各向异性固体中声波的极化状态的研 究进行了报道。Crampinet 等 [7] 和 Hosten [8] ,Lanceleur 等 [9] 对在各向异性岩石中传播的 P-波的极化方向偏离 其传播方向的问题进行了讨论, Helbig 等 [10] 报道了无限 大各向异性岩石中传播的 P-波和 SV-波的异常极化现象。 Carcione [11] 给出了在无限大 VTI 介质中传播的准 P-波和 准 SV-波的极化系数解析表达式。应用著名的实验测量 岩石各向异性参数 [12][13][14] ,Fa 等人研究了岩石各向异性 对在无限大 VTI 介质中传播的准 P-波和和准 SV-波的极 化方向的影响 [15] 。基于对准 P-波和和准 SV-波的极化的 先验知识,Daley 和 Hron 给出了一个计算 VTI-VTI 介 质界面反射/折射系数的矩阵方程 [16] ,并仅对具有椭圆 各向异性的 VTI-VTI 介质界面的反射系数和折射系数 进行了数值计算 [17] 。Fa 等人提出了计算 VTI-VTI 岩石 界面的反射/折射系数的快速算法 [18] ,并建立了非均匀 折射 P-波的椭圆极化方程, 讨论了岩石各向异性对椭圆 极化状态的影响 [19][20][21] [11,22,23] , 利用邦德变换矩阵法则实现旋转坐标下的各向异性岩 石的刚度系数矩阵变换 [24] ,从 VTI 岩石介质的刚度矩 阵可以得到 TTI 岩石介质的刚度矩阵, 也可将 TTI 岩石 介质的刚度矩阵转换为 VTI 岩石介质的刚度矩阵, 从而 保证了我们在数学上能够建立严格的和具有实际物理 意义的 TTI-TTI 岩石介质界面的模型。 由于通过邦德变换矩阵法则 [24] 可以实现 TTI-TTI 12 13 sin cos ...
Preprint
The rock layers in nature are often not wholly horizontal, nor are they purely vertical. For the interface between rock layers, the most realistic model is the TTI-TTI media interface. Under this umbrella, the subcases include the VTI-VTI interface, the TTI-VTI interface, and the VTI-TTI interface. The analysis of the TTI-TTI media interface is genetically complex. However, upon Bond transformation, the TTI-TTI rock interface may be analyzed using the VTI-TTI media interface. This paper reports the analytic expressions of polarization coefficients for waves induced at the interface between two different types of anisotropic rocks. Primarily, we report the analysis of the inhomogeneously refracted P-wave in the post-critical incident-angle region, achieved by establishing a fourth-order polynomial for solving the reflection angles in the VTI side of the VTI-TTI interface and eighth-order polynomials for solving the refraction angles in the TTI rock side the VTI-TTI interface. We also present the physically and mathematically rigorous analytical relations between the polarization coefficients of the mode-conversion waves, the reflection and refraction coefficients of the interface, the physical and anisotropic parameters of the interfacial media, and the incident angle. The polarization coefficient of the mode-conversion waves and the reflection and refraction coefficients from the interface depends on the media's physical and anisotropic parameters on both sides of the interface and the incident angle. The polarization coefficients of the mode-conversion waves are closely related to the reflection and refraction coefficients at the interface. They are validating each other from the law of energy conservation and the interfacial boundary conditions. The analysis results derived in this report provide a theoretical basis for studying and calculating the reflection and refraction properties of the anisotropic rock interface, the polarization status of mode-conversion waves generated at this interface, and the physical interpretation of some produced phenomena at the interface. It also provided a base for the exact AVO inversion analysis and time-depth conversion of seismic exploration data measured in anisotropic rock formations.
... where R is the bond transformation matrix (Bond, 1943). ...
Conference Paper
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Since the computational time of FDM is proportional to the number of grid points, many higher-order and optimized schemes were developed to increase the spatial grid size while maintaining the dispersion error below predefined levels. However, mentioned techniques are applicable only to homogeneous media since large error due to the interfaces may arise when the large grid spacing is used. To solve this problem, various effective medium parameterization methods have been proposed. In our work, We develop a new TTI effective media parameterization method and analyze existing different effective media parameterization methods. Our tests show that the proposed TTI effective media parameterization method is better than other effective media parameterization methods and can be used with the higher-order scheme to utilize 3-4 PPW to produce satisfactory results.
... where C* is the averaged tensor and C(x,y,z) is the subvolume original rock tensor, both in 6x6 Voigt notation. The transformation matrix M(x,y,z) and its transpose M T (Bond, 1943) are 6x6 rotation operators whose elements contain directional cosines that define local structural tilt (see Auld, 1973;Okaya and McEvilly, 2003). For each structure illustrated in Figure 7, we calculated its SGA effective medium for a unit shape based on a characteristic length or height, assuming the structure is filled with our representative fabric whose foliations conform to the shape. ...
Article
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A macroscopic geological structure can geometrically map a local rock material anisotropy into a larger volume that may have different net anisotropic properties on a scale to which seismic waves respond. The bulk structure’s anisotropy intensity, symmetry type and orientation of symmetry axes will generally be different from the local rock; a typical crustal rock with material fabric showing slow-axis transverse isotropy can be converted, for example, into a bulk structure that is weaker fast-axis orthorhombic or lower symmetry. We define this modification as “structural geometric anisotropy” (SGA). The seismic anisotropy signals produced by this structure are influenced by the length scale of seismic waves: shorter wavelengths respond to each larger part of the structure (path integration) whereas longer wavelengths respond to just the bulk average of all parts (effective medium). We present a tensor formulation that under certain conditions can decompose an anisotropy-filled structure into its macroscale structural geometry separated from infilling rock types. When a single representative rock material can be substituted for local rocks with fabric, the orientation operators that describe the structure’s geometry can be separately volume averaged to produce a unique “structural geometry operator” that can then be used to define the equivalent structure’s effective medium. We illustrate these principles using common geometrical structures and show as an example the progressive modification of seismic anisotropy produced by cylindrical folding. Due to the widespread distribution of crustal tectonic structures, their effects on seismic anisotropy should be incorporated into interpretations of seismic anisotropy. The assumption of slow-axis transverse isotropy in crustal volumes is not always valid.
... where R is the bond transformation matrix (Bond 1943). Muir et al. (1992) proposed that using TTI equivalent medium method in rectangular grids simulation of sloping interface model can eliminate the diffracting staircase problem. ...
... In this example we choose to adopt the first approach where the computations are directly performed in the global frame. Applying the rotation technique suggested by Bond (1943) (2000), Section 3.6.6, as a factorized anisotropic inhomogeneous medium (FAI). ...
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The form of the Lagrangian proposed in Part I of this study has been previously used for obtaining stationary ray paths between two endpoints in isotropic media. We extended it to general anisotropy by replacing the isotropic medium velocity with the ray (group) velocity magnitude which depends on both, the elastic properties at the ray location and the ray direction. This generalization for general anisotropy is not trivial and in this part we further elaborate on the correctness, physical interpretation, and advantages of this original arclength-related Lagrangian. We also study alternative known Lagrangian forms and their relation to the proposed one. We then show that our proposed first-degree homogeneous Lagrangian (with respect to the ray direction vector) leads to the same kinematic ray equations as the alternative Lagrangians representing first- and second-degree homogeneous functions. Using different anisotropic examples, we further validate/demonstrate the correctness of the proposed Lagrangian, analytically (for a canonical case of an ellipsoidal orthorhombic medium) and numerically (including the most general medium scenario: spatially varying triclinic continua). Finally, we analyze the commonly accepted statement that the Hamiltonian and the Lagrangian can be related via a resolvable Legendre transform only if the Lagrangian is a time-related homogeneous function of the second-degree with respect to the vector tangent to the ray. We show that this condition can be bypassed, and a first-degree homogeneous Lagrangian, with a singular Hessian matrix, can be used as well, when adding a fundamental physical constraint which turns to be the Legendre transform itself. In particular, the momentum equation can be solved, establishing, for example, the ray direction, given the slowness vector.
... Translating the effective stiffness tensor into the global domain, the strain concentration tensors for each inclusion A E , and A I is calculated in the local domain and equivalent strain concentration tensors are calculated in the global system using a 4th rank transformation tenor, Q i j kl (Koay [16], this tensor is function of the transformation matrix Q i j [9] defined in the previous section). The "Bond transformation" [17] is a tensor that accounts for the inclusion's 3-dimensional attributes. ...
... where C* is the averaged tensor and C(x,y,z) is the subvolume original rock tensor, both in 6x6 Voigt notation. The transformation matrix M(x,y,z) and its transpose M T (Bond, 1943) are 6x6 rotation operators whose elements contain directional cosines that define local structural tilt (see Auld, 1973;Okaya and McEvilly, 2003). For each structure illustrated in Figure 7, we calculated its SGA effective medium for a unit shape based on a characteristic length or height, assuming the structure is filled with our representative fabric whose foliations conform to the shape. ...
Article
A macroscopic geological structure can geometrically map a local rock material anisotropy into a larger volume that may have different net anisotropic properties on a scale to which seismic waves respond. The bulk structure's anisotropy intensity, symmetry type and orientation of symmetry axes will generally be different from the local rock; a typical crustal rock with material fabric showing slow-axis transverse isotropy can be converted, for example, into a bulk structure that is weaker fast-axis orthorhombic or lower symmetry. We define this modification as "structural geometric anisotropy" (SGA). The seismic anisotropy signals produced by this structure are influenced by the length scale of seismic waves: shorter wavelengths respond to each larger part of the structure (path integration) whereas longer wavelengths respond to just the bulk average of all parts (effective medium). We present a tensor formulation that under certain conditions can decompose an anisotropy-filled structure into its macroscale structural geometry separated from infilling rock types. When a single representative rock material can be substituted for local rocks with fabric, the orientation operators that describe the structure's geometry can be separately volume averaged to produce a unique "structural geometry operator" that can then be used to define the equivalent structure's effective medium. We illustrate these principles using common geometrical structures and show as an example the progressive modification of seismic anisotropy produced by cylindrical folding. Due to the widespread distribution of crustal tectonic structures, their effects on seismic anisotropy should be incorporated into interpretations of seismic anisotropy. The assumption of slow-axis transverse isotropy in crustal volumes is not always valid.
... where C B is the stiffness coefficient matrix for a transversely isotropic medium with a vertical symmetry axis, I is the 6 × 6 unit matrix, and Δ 1 and Δ 2 are the fracture compliance matrices for fracture set 1 (being normal to the x-axis) and fracture set 2 being azimuthally rotated on azimuth angle ϕ 0 , respectively. The term Rðϕ 0 Þ is the corresponding Bond (1943) matrix. The details on the computation of equation 41 are found in Appendix B. ...
... [53] are presented in Table 1. For calculation of the properties of acoustic waves in YX LiNbO 3 plate one should recalculate all material constants in a new coordinate system [54] by using Euler's angles [55]. Obtained values are presented in Table 1 in corresponding cell after slash. ...
Article
A new method for detecting the forward and backward acoustic waves in the piezoelectric plates by using a system of the acoustically isolated interdigital transducers (IDT) with the different spatial period is developed. This method was tested on the piezoelectric acoustic modes A1 and SH1 propagating in Y-X LiNbO3 plate. Theoretical analysis has shown that the SH1 mode in the entire frequency range represented a forward wave. At that the dispersion dependence of the A1 mode near the cutoff frequency has a smooth transition from a forward wave to a backward one with decreasing frequency. In order to experimentally observe this transition 19 IDTs with different periods were deposited on a single lithium niobate plate. The measurement of the frequency dependence of the real part of the electrical impedance of these IDTs showed that for SH1 mode the resonance frequency decreased monotonically with a growth in the spatial period. This behavior corresponded to a forward wave. For A1 mode the resonance frequency initially decreased with the growth of the IDT period and then began to increase after reaching a value of ZGV frequency. This behavior is explained by the smooth transition from the region of the forward wave to the region of the backward wave. The calculation of the frequency dependences of the real part of the electrical impedance of each transducer for the waves considered, carried out by the finite element method, turned out to be in good agreement with the experiment.
... After coordinate system rotation, we can express the rotated stiffness matrix as C 0 ¼ MCM T , where C and C 0 are the stiffness matrices in the intrinsic coordinate system and survey coordinate system, respectively, and M is the Bond transform matrix (Bond 1943 ...
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Reservoirs with vertically aligned fractures can be represented equivalently by horizontal transverse isotropy (HTI) media. But inverting for the anisotropic parameters of HTI media is a challenging inverse problem, because of difficulties inherent in a multiple parameter inversion. In this paper, when we invert for the anisotropic parameters, we consider for the first time the azimuthal rotation of a two-dimensional seismic survey line from the symmetry of HTI. The established wave equations for the HTI media with azimuthal rotation consist of nine elastic coefficients, expressed in terms of five modified Thomsen parameters. The latter are parallel to the Thomsen parameters for describing velocity characteristics of weak vertical transverse isotropy media. We analyze the sensitivity differences of the five modified Thomsen parameters from their radiation patterns, and attempt to balance the magnitude and sensitivity differences between the parameters through normalization and tuning factors which help to update the model parameters properly. We demonstrate an effective inversion strategy by inverting velocity parameters in the first stage and updates the five modified Thomsen parameters simultaneously in the second stage, for generating reliably reconstructed models.
... In this example we choose to adopt the first approach where the computations are directly performed in the global frame. Applying the rotation technique suggested by Bond (1943) (2000), Section 3.6.6, as a factorized anisotropic inhomogeneous medium (FAI). ...
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This part of the study is dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in Part I. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e., the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point‐source and plane‐wave. For the proposed point‐source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in Part I, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyze these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach. This article is protected by copyright. All rights reserved
... While defining the intrinsic coordinate system with the x-axis parallel to the symmetry axis, the survey coordinate system is azimuthally rotated by a constant angle. The elastic coefficients (c ij and c ′ ij ) that are defined in these two coordinate systems, respectively, can be connected through the Bond transform (Bond 1943). ...
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In seismic waveform inversion, selecting an optimal multi-parameter group is a key step to derive an accurate subsurface model for characterising hydrocarbon reservoirs. There are three parameterizations for the horizontal transverse isotropic (HTI) media, and each parameterization consists of five parameters. The first parameterization (P-I) consists of two velocities and three anisotropy parameters, the second (P-II) consists of five elastic coefficients and the third (P-III) consists of five velocity parameters. The radiation patterns of these three parameterizations indicate a strong interference among five parameters. An effective inversion strategy is a two-stage scheme that first inverts for the velocities or velocity-related parameters and then inverts for all five parameters simultaneously. The inversion results clearly demonstrate that P-I is the best parameterization for seismic waveform inversion in HTI anisotropic media.
... For viscoelastic anisotropic TTI media, it is possible to apply the Bond transformation (Bond, 1943) with an inclination angle θ 0 and an azimuthal angle φ 0 to the VTI medium given in Table 1; one obtains a tilted transversely isotropic (TTI) medium that is often applied to defining more complex structures of anisotropic rocks with variable angles. For example, by setting up θ 0 ¼ 30°and φ 0 ¼ 0, we obtain a similar figure (not shown), but just like the rotation of 30°from the VTI results ( Figure 2). ...
Article
In a viscoelastic anisotropic medium, velocity anisotropy and wave energy attenuation occur and are often observed in seismic data applications. Numerical investigation of seismic wave propagation in complex viscoelastic anisotropic media is very helpful in understanding seismic data and reconstructing subsurface structures. Seismic ray tracing is an effective means to study the propagation characteristics of high-frequency seismic waves. Unfortunately, most seismic ray-tracing methods and traveltime tomographic inversion algorithms only deal with elastic media and ignore the effect of viscoelasticity on the seismic raypath. We have developed a method to find the complex ray velocity that gives the seismic ray speed and attenuation in an arbitrary viscoelastic anisotropic medium, and we incorporate them with the modified shortest-path method to determine the raypath and calculate the real and imaginary traveltime (wave energy attenuation) simultaneously. We determine that the complex ray-tracing method is applicable to arbitrary 2D/3D viscoelastic anisotropic media in a complex geologic model and the computational errors of the real and imaginary traveltime are less than 0.36% and 0.59%, respectively. The numerical examples verify that the new method is an effective and powerful tool for accomplishing seismic complex ray tracing in heterogeneous viscoelastic anisotropic media.
... where matrix M (m) is the Bond transformation matrix (Bond 1943;Slawinski 2014), Table 1. The rock is saturated with water. ...
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The monoclinic medium with a horizontal symmetry plane is gradually being studied for seismic anisotropy characterization. The principle goal of this paper is to investigate the effect of fracture parameters (azimuth angle, density, aspect ratio, scale) on the exact and approximate monoclinic anisotropy parameters. We derive the monoclinic porous media based on the Chapman model which accounts for the wave-induced fluid flow and give the expressions of the Thomsen-style anisotropy parameters (nine orthorhombic anisotropy parameters: VP0, VS0, ε1, ε2, γ1, γ2, δ1, δ2, δ3, three exact monoclinic parameters: ζ1, ζ2, ζ3, and three approximate monoclinic parameters:$\widetilde{\zeta _{1}}, \widetilde{\zeta _{2}}, \widetilde{\zeta _{3}}$). The dependence of Thomsen-style anisotropy parameters associated with azimuth angle between two fracture sets is analyzed. The orthorhombic anisotropy parameters and monoclinic anisotropy parameters have the same period (π) on the azimuth angle between two fracture sets. The exact and approximate monoclinic anisotropy parameters responsible for the rotation of the P-wave NMO ellipse have a similar trend versus the azimuth angle, while those responsible for the rotation of the S1- and S2-wave NMO ellipses have significant discriminations. The influence of fracture density, aspect ratio, and scale on the monoclinic parameters are also analyzed. The monoclinic anisotropy parameters responsible for the rotation of the P-wave NMO ellipse decrease with fracture density and aspect ratio increasing from 0 to 0.1, while those responsible for the rotation of S1- and S2-wave NMO ellipses increase with the fracture parameters. The fracture density has a bigger influence on the monoclinic anisotropy parameters than the fracture aspect ratio. When saturated with different fluids (water and CO2), the monoclinic parameters have a similar behavior versus the azimuth angle between two fracture sets.
... Ainsi, la contrainteà proximité d'un coquillage dépend de l'angle θ entre la direction e 1 et la direction définissant l'orientation de l'inclusion e θ . Pour chaque orientation, la contrainte au voisinage de l'inclusion doitêtre calculée selon l'équation (V.24) en utilisant une transformation de Bond [185] pour < σ > puisque les autreséléments de l'équation (V.24) sont ecrits dans la base (e r , e θ , e z =e 2 ). Enfin, la répartition de la contrainte principale positive la plus importante dans la matrice σ I p ,m (x) θ décrit comment la contrainte se concentre près de l'inclusion (coquillage ou granulat) par rapport aux différents angles θ. ...
Thesis
Les sédiments de dragage, classés aujourd'hui comme déchets, semblent être une alternative prometteuse aux matériaux conventionnels. Les opérations de dragage, réalisées par la Direction Territoriale du Bassin de la Seine (DTBS), génèrent un volume annuel des sédiments d’environ 150 000 m3. Dans cette thèse, nous nous intéressons tout d'abord au potentiel de valorisation de ces sédiments en tant qu'alternative aux granulats ou au ciment (comme addition). Pour ce faire, une étude de variabilité du gisement francilien et de son effet sur les propriétés du béton est réalisée. La compilation des propriétés du gisement montre que 30 % du volume dragué pourrait être considéré comme une source stable et propre de granulats, alors que les sédiments non inertes ne représentent que 6 %. L'étude expérimentale montre que la substitution de 30 % en volume des granulats ou la substitution de 10 % en volume de ciment respectivement par des sédiments grossiers ou par des sédiments fins affecte que légèrement la cinétique d'hydratation, la résistance à la compression et le retrait. Ensuite, nous avons approfondi l’effet de la matière organique des sédiments sur les propriétés rhéologiques, physico-chimiques (hydratation et retrait) et mécaniques d’une pâte de ciment. Les résultats obtenus montent que les substances humiques, matières organiques présentes dans les sédiments, ont un effet analogue à celui des lignosulfonates de calcium, un plastifiant, tant sur la cinétique d'hydratation que sur la rhéologie de la pâte cimentaire. La dernière partie de la thèse traite de l’effet de l’incorporation des coquillages sur les propriétés mécaniques et de durabilité du béton. Cette partie est scindée en deux volets ; un volet expérimental et un volet numérique. Le premier a comme objectif d’évaluer l’effet de l’incorporation des coquillages, comme graviers, sur les propriétés macroscopiques des bétons (propriétés mécaniques et propriétés de durabilité). Les résultats expérimentaux montrent qu'il n'y a aucun effet de l'incorporation des coquillages sur l'affaissement, le retrait et les propriétés de durabilité (la porosité, la diffusion des ions chlore et la carbonatation) du béton pour des substitutions inférieures à 20 %. En revanche, les propriétés mécaniques diminuent avec l'augmentation du taux de substitution; pour une substitution volumique de 20 %, la résistance à la compression et le module élastique diminuent respectivement de 20 % et 17 %. Le volet numérique a pour objectif de construire un outil numérique cohérent permettant de prédire les propriétés mécaniques et de diffusion d'un béton en fonction de sa composition et des paramètres physiques et géométriques de la microstructure. Par des observations microscopiques (MEB) et en modélisant les coquillages comme des inclusions plongées dans une matrice cimentaire, nous montrons que l'adhésion entre le coquillage et la matrice est limitée, et doit être prise en compte pour estimer les propriétés des bétons à base de coquillages
... The HR-EBSD analysis finds only the elastic components of deformation (deviatoric strain and lattice rotation) and does not measure the hydrostatic strain, which changes only the width of the Kikuchi bands and not interplanar angles [81]. However, by imposing suitable boundary conditions (i.e. the stress normal to the surface is zero) and using the crystal elastic constants [82], [83], the full three-dimensional strain and stress tensors in the observed surface layer can be accessed. In this analysis, the ferrite elastic constants (in GPa) were: 11 = 230, 44 = 117, 12 = 135 [84]. ...
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The strain fields of deformation twins in the ferrite matrix of an age-hardened duplex stainless-steel (Zeron 100: 25%Cr, 7%Ni) have been studied in situ under load, and ex situ (unloaded), using high-resolution electron backscatter diffraction (HR-EBSD). The local 2-dimensional (2D) elastic strain field acting on the twin tip was parametrised for the first time using the strain energy release rate (J-integral) and then decomposed into the mode I and mode II stress intensity factors (KI and KII). An improved method to select the strain reference was used, based on the relationship between the HR-EBSD cross-correlation peak height and mean angular error. The elastic field described by KI increased with twin thickness. The in-plane shear field, described by KII, relaxed when the load was removed. Some current limitations of the 2D analysis are discussed, which aims to provide an experimental methodology to quantify the fields that describe the local boundary conditions for twin thickening and propagation.
... Conventional rotation matrices cannot be applied to rotate a tensor in the Voigt notation. To avoid the elaborate back conversion to full subscripts, a technique proposed by Bond [153] can be used, allowing for the rotation in the Voigt notation by simple matrix multiplication with a matrix K. For a rotation by an angle of α around the z-axis, this matrix is given by: (A.11) Subsequently, the rotated elasticity tensor c rot is given by: ...
Thesis
Halbleiter-Quantenpunkte (engl. Quantum dot, QD) werden aufgrund ihrer überlegenen Leistung als effiziente Quelle von einzelnen und ununterscheidbaren Photonen als wesentlicher Bestandteil zukünftiger Quantentechnologien angesehen. Solche Einzelphotonen sind notwendig, um den Transport von Quanteninformationen über große Entfernungen durch etablierte Glasfasernetzwerke zu ermöglichen. Eine grundlegende Anforderung an Einzelphotonenquellen ist die Möglichkeit, ihre Emissionsenergie präzise abzustimmen und modulieren zu können. Da es sich bei Halbleiter-Quantenpunkten um Festkörpersysteme handelt, ist dies für diese auf Grundlage der Deformationspotentialkopplung möglich. Der Schwerpunkt dieser Arbeit liegt auf der Steuerung und Manipulation der optischen Eigenschaften einzelner Quantenpunkte durch das Deformationspotential, welches von akustischen Oberflächenwellen (engl. surface acoustic wave, SAW) induziert wird. Dabei handelt es sich um mechanische Wellen, die sich auf der Oberfläche eines Kristalles ausbreiten können und sich auf piezoelektrischen Substraten leicht durch Interdigitaltransducer elektrisch anregen lassen. Zunächst wird die Modulation der Emissionsenergie eines Quantenpunktes durch akustische Oberflächenwellen untersucht, in dem die Photolumineszenz eines Quantenpunktes unter dem Einfluss einer akustischen Oberflächenwelle analysiert wird. Es werden verschieden experimentelle Messtechniken vorgestellt, die unterschiedlich tiefe Einblicke in die zugrundeliegenden Prozesse ermöglichen und entsprechende Messdaten präsentiert. Als nächstes wird das Konzept von frequenz-gechirpten Transducern eingeführt. Das Design dieser Schallwandler ermöglicht es ihnen, akustische Oberflächenwellen nicht nur bei diskreten Frequenzen, sondern über breite Frequenzbänder hinweg, anzuregen. Diese Fähigkeit wird demonstriert, indem die optomechanische Antwort eines einzelnen Quantenpunkts über einen weiten Bereich von Oberflächenwellenfrequenzen untersucht wird. Darüber hinaus werden die Eigenschaften von frequenz-gechirpten Transducern ausgenutzt, um eine stabile Phasenbeziehung zwischen der Frequenz einer akustischen Oberflächenwelle und der Wiederholungsrate einer gepulsten Laserquelle herzustellen und somit stroboskopische SAW-Spektroskopie zu ermöglichen. Um Einzelphotonen von überlegener Qualität hinsichtlich ihrer Kohärenz und Ununterscheidbarkeit zu erhalten, wird zu einer resonanten optischen Anregung der Quantenpunkte übergegangen. Von besonderem Interesse ist dabei die resonante Fluoreszenz eines Quantenpunktes, der dynamisch durch eine akustische Oberflächenwelle moduliert wird. In diesem Regime kann die Bildung diskreter phononischer Seitenbänder im Emissionsspektrum beobachtet werden. Dies kann durch die Absorption einer diskreten Anzahl von Phononen aus dem akustischen Feld, beziehungsweise die Emission von Phononen in das akustische Feld interpretiert werden. Die Bildung dieser phononischen Seitenbänder wird sowohl in Abhängigkeit von der Modulationsfrequenz und -amplitude, als auch der optischen Verstimmung zwischen dem resonanten Lichtfeld und der Übergangsenergie des QD eingehend untersucht. Im letzteren Fall kann dabei ein parametrischer Energietransfer zwischen der optischen und der akustischen Domäne beobachtet werden. Im nächsten Schritt wird der QD nicht nur durch eine, sondern durch zwei SAWs unterschiedlicher Frequenz dynamisch moduliert. In diesem Szenario zeigt das Emissionsspektrum zusätzliche phononische Seitenbänder, deren Position der Summen- und Differenzfrequenzen beider SAWs entsprechen. Dies zeigt somit die optomechanische Wellenmischung zweier SAW-Felder und des optischen Lichtfeldes durch den QD-Übergang. Darüber hinaus wird, indem die relative Phase zwischen den zueinander kohärenten SAW-Feldern genau eingestellt wird, eine Phasenanpassung ermöglicht, wodurch eine deterministische Verstärkung und Unterdrückung einzelner Seitenbänder erreicht wird. Die hohe Stabilität dieser Phasenanpassung, die nur durch die Leistungsfähigkeit moderner Hochfrequenz-Elektronik begrenzt ist, wird experimentell gezeigt. Abschließend wird die Möglichkeit betrachtet, die Kopplung eines Quantenpunktes an photonische und phononische Felder zu verstärken, in dem diese in entsprechende angepasste Umgebungen, sogenannten photonischen und phononischen Kristallen, platziert werden. In diesem Zusammenhang wird ein gekoppeltes System, bestehend aus einem einzelnen Quantenpunkt und einem photonischen Nanoresonator, im Detail betrachtet. Dabei wird eine akustische Oberflächenwelle verwendet, um die beiden Komponenten eines solchen Systems dynamisch in und aus der Resonanz zu bringen. Dies hat aufgrund des Purcell-Effekts einen starken Einfluss auf die Emissionsrate des Quantenpunkts und wird letztendlich zur Realisierung einer akustisch ausgelösten Einzelphotonenquelle verwendet.
... However, it is easier to rotate the frame for the matrix-form presentations of the stiffness tensor, applying only matrix algebra, rather than for the actual fourth-order tensor. Rotation of a reference frame for Voigt-form stiffness tensor was originally proposed by Bond (1943). It was later 'upgraded' for more suitable Kelvin-form stiffness tensor (e.g. ...
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Consisering general anisotropic (triclinic) media and both, quasi-compressional (qP) and quasi-shear (qS) waves, in Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the elastic model parameters along ray trajectories. Ray velocity derivatives for anisotropic elastic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic media (transverse isotropy with tilted axis of symmetry, TTI) for the coupled qP waves (quasi-compressional waves) and qSV waves (quasi-shear waves polarized in the “axial” plane) and for SH waves (shear waves polarized in the “normal” plane). The acoustic approximation for qP waves is considered a special case. In seismology, the medium properties, normally specified at regular three-dimensional fine grid points, are the five material parameters: the axial compressional and shear wave velocities, the three (unitless) Thomsen parameters, and two geometric parameters: the polar angles defining the local direction (the tilt) of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, so that their spatial gradients and Hessians can be reliably numerically computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding numerical (finite difference) derivatives. Additionally, only for validation purpose, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters (five material and two geometric) into the twenty-one stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I. Since in many practical wave/ray-based applications in polar anisotropic media only the spatial derivatives of the axial compressional wave velocity are taken into account, we analyze the effect (sensitivity) of the spatial derivatives of the other parameters on the ray velocity and its derivatives (which, in turn, define the corresponding traveltime derivatives along the ray).
... However, it is easier to rotate the frame for the matrix-form presentations of the stiffness tensor, applying only matrix algebra, rather than for the actual fourth-order tensor. Rotation of a reference frame for Voigt-form stiffness tensor was originally proposed by Bond (1943). It was later 'upgraded' for more suitable Kelvin-form stiffness tensor (e.g. ...
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In Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the anisotropic model parameters, at nodal points along ray trajectories, considering general anisotropic (triclinic) media and both, quasi-compressional and quasi-shear waves. Ray velocity derivatives for anisotropic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic (transverse isotropy with tilted axis of symmetry, TTI) media for the coupled qP and qSV waves and for SH waves. The acoustic approximation for qP waves is considered a special case. The medium properties, normally specified at regular three-dimensional fine grid points, are the five material parameters: the axial compressional and shear velocities and the three Thomsen parameters, and two geometric parameters: the polar angles defining the local direction of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, where their gradients and Hessians can be reliably computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding numerical (finite difference) derivatives. Additionally, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters (five material and two geometric) into the twenty-one stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I.
... where M is the Bond transformation matrix (Bond 1943). The elastic correction for pores to the effective elastic parameters of the rock C p i j (ω) is expressed by ...
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Accurate modeling of the frequency-dependence of seismic wave velocity related to fracture system and fluid content is crucial to the quantitative interpretation of seismic data in fractured reservoirs. Both meso-scale fractures and patchy saturation effects can cause significant velocity dispersion and attenuation in the seismic frequency band due to wave-induced fluid flow (WIFF) mechanism. Considering the coupled impact of ‘meso-scale fractures’ and ‘patchy saturation’, we derive expressions for the frequency-dependent anisotropy in partially saturated porous rock containing two fracture sets with different orientations, sizes and connectivities. Especially, we simplify the rock-physics model as an orthorhombic (ORT) media by assuming the meso-scale fractures to be orthogonal and give the explicit expressions for frequency-dependent elastic constants. Finally, we give the expressions for the frequency-dependent phase velocity in patchy saturated and fractured ORT media and investigate the effect of patchy saturation on P-wave velocity at different polar and azimuth angles. In this paper, we investigate the effects of fluid saturation and fluid pressure on frequency-dependent velocities and Thomsen anisotropy parameters. Also, the effect of the relative permeability is very noticeable. The relaxation frequency can be lower in partially saturated fractured rocks compared with the fully saturated case, which makes the rock have a larger stiffness. The non-monotonic relationships between frequency-dependent anisotropy and fluid saturation add complexity to seismic forward modeling and inversion in reservoirs with complex fracture patterns.
... The HR-EBSD analysis finds only the elastic components of deformation (deviatoric strain and lattice rotation) and does not measure the hydrostatic strain, which changes only the width of the Kikuchi bands and not interplanar angles [81]. However, by imposing suitable boundary conditions (i.e. the stress normal to the surface is zero) and using the crystal elastic constants [82], [83], the full three-dimensional strain and stress tensors in the observed surface layer can be accessed. In this analysis, the ferrite elastic constants (in GPa) were: 11 = 230, 44 = 117, 12 = 135 [84]. ...
Article
The strain fields of deformation twins in the ferrite matrix of an age-hardened duplex stainless-steel (Zeron 100: 25%Cr, 7%Ni) have been studied in situ under load, and ex situ (unloaded), using high-resolution electron backscatter diffraction (HR-EBSD). The local 2-dimensional (2D) elastic strain field acting on the twin tip was parametrised for the first time using the strain energy release rate (J-integral) and then decomposed into the mode I and mode II stress intensity factors (KI and KII). An improved method to select the strain reference was used, based on the relationship between the HR-EBSD cross-correlation peak height and mean angular error. The elastic field described by KI increased with twin thickness. The in-plane shear field, described by KII, relaxed when the load was removed. Some current limitations of the 2D analysis are discussed, which aims to provide an experimental methodology to quantify the fields that describe the local boundary conditions for twin thickening and propagation.
... For viscoelastic anisotropic TTI media, it is possible to apply the Bond transformation (Bond, 1943) with an inclination angle θ 0 and an azimuthal angle φ 0 to the VTI medium given in Table 1; one obtains a tilted transversely isotropic (TTI) medium that is often applied to defining more complex structures of anisotropic rocks with variable angles. For example, by setting up θ 0 ¼ 30°and φ 0 ¼ 0, we obtain a similar figure (not shown), but just like the rotation of 30°from the VTI results ( Figure 2). ...
Article
The real raytracing approach leads to an effective solution in the real space domain using a homogenous ray velocity vector. However, it fails to yield solutions for quasi-shear waves, which suffer triplication of the wavefronts. To address this challenging problem, a generalized real ray-tracing method and its new approximations are presented to solve the complex ray equation. The numerical results show that the generalized ray-tracing method is superior to the real ray-tracing method in the presence of triplications of the quasi-shear waves in the computation of ray velocity, ray attenuation, and ray quality factors, as well as the reflection and transmission coefficients in viscoelastic anisotropic media. Based on the assumptions of the real slowness direction and real polarization vectors, two new approximations of the generalized real ray-tracing method are developed for directly computing the homogeneous complex ray velocity vectors of three wave modes (qP, qS1, and qS2). These approximations significantly improve the computational efficiency by avoiding the iterative process required by the generalized real ray-tracing method that is inherited from the real ray-tracing method. The computational accuracies are verified through transversely isotropic models and orthorhombic models with different strengths of attenuation and anisotropy. The incorporation of the new approximation into the shortest-path method turned out to be an efficient and accurate method for seismic ray tracing in heterogeneous viscoelastic and transversely isotropic media with a vertical axis of symmetry, even in the presence of strong attenuation and anisotropy.
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In most theoretical work related to effective properties of polycrystals, the media are assumed to be infinite with randomly oriented grains. Therefore, the bulk material has absolute isotropy because each direction includes an infinite number of grains with infinite possibilities for grain orientation. However, real samples will always include a finite number of grains such that the inspection volume will have some associated anisotropy. Thus, bounds on the bulk properties are expected for a given measurement. Here, the effect of the number of grains on the variations of elastic anisotropy is studied using synthetic polycrystals comprised of equiaxed cubic grains (17 volumes with 100 realizations each). Voigt, Reuss, and self-consistent techniques are used to derive the effective elastic modulus tensor. The standard deviation of the average elastic modulus is then quantified for several materials with varying degrees of single-crystal anisotropy and is shown to be inversely proportional to the square root of the number of grains. Finally, the Christoffel equation is used to study the relevant phase velocities. With appropriate normalization, a master curve is derived with respect to the finite sample size, which shows the expected variations of phase velocity for the longitudinal, fast shear, and slow shear modes.
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I derive accurate anisotropy parameters for a monoclinic anisotropy model with a horizontal symmetry plane based on NMO (normal moveout) ellipses for P, S1, S2 and converted waves. The NMO velocity ellipse is also defined for all type of converted waves. The parameters are defined in phase domain and compared with existing approximate monoclinic anisotropy parameters. These parameters are illustrated for two benchmark models consisting of two non-orthogonal fracture sets embedded into a transversely isotropic medium with a vertical symmetry axis. The dependence of monoclinic parameters on the azimuth angle between the fracture sets is analyzed. Being linearized with respect to fracture weaknesses, the monoclinic anisotropy parameters can be decomposed into sine functions of double and quartic azimuth angle between the fracture sets with the weights given by the stiffness coefficients of background model. The discrimination between the fracture parameters computed from a given set of monoclinic parameters is dependent on background model and controlled by the azimuth angle between the fracture sets.
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The effect of substituting coarse aggregates with Corbicula shells on the mechanical and durability properties of concrete is evaluated by combining experimental investigations and micromechanical modeling. While substituting 20% of the aggregate volume with Corbicula shells does not affect slump, porosity, carbonation or chloride diffusion, a total replacement decreases slump, increases porosity and carbonation depth. Regarding mechanical properties, Young’s modulus and compressive strength decrease linearly with the incorporation rate of Corbicula shells. The shells are then modeled as non-spherical inclusions surrounded by an interfacial transition zone and embedded in a cementitious matrix, so that micromechanical models provide estimates of mechanical and diffusion properties. Combining the model and experimental results reveals that the low stiffness and drop in compressive strength of the concrete are explained by a weak adhesion between the shells and the cementitious matrix.
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