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... Thus, all the acoustic modes in the isotropic material media are nothing but the pure modes of the first kind (i.e., all the directions represent the longitudinal normals) [12]. Notice that two kinds of special directions can be distinguished [12,13]: the first kind allows propagation of one longitudinal and two transverse modes (a longitudinal normal), while one transverse and two mixed modes propagate along the directions of the second kind (a transverse normal). A degenerated case for the glass media appears due to their high symmetry and the equality C 44 = (C 11 -C 12 )/2. ...

... If the AWs propagate in the ZY plane at the angle X with respect to the Y axis, the angle becomes equal to 90 deg. In this case we have M 12 = M 31 = 0 and M 23 = 24 ' C in the matrix given by Eq. (13). However, since the AWs propagate inside the principal crystallographic plane, the equality M 23 = 24 ' C = 0 holds true and the non-orthogonality angle is zero. ...

... However, the angle of AW propagation at which they acquire pure polarizations differs for different crystallographic planes. For the XZ plane, this angle with respect to the X axis can be written as 1/ 2 13 11 55 13 33 55 ' ' 2 ' a tan ' ' 2 ' ...

... This question has been extensively studied, from both the theoretical and numerical points of view, since the pioneering work of Gazis, Tadjbakhsh and Toupin [30], and subsequent works in the 90s [8,7,26,28]. Most works focus on the elasticity tensor [30,24,26,27,33,28,46,39,23], a few ones on the piezoelectricity tensor [63]. So far, we are not aware of some similar studies for the Hill plasticity tensor or the combination of several constitutive tensors. ...

... Even if some analytical attempts exist [60,56,6], the distance to an elasticity symmetry class problem is most often solved numerically, following [28], using the far from being injective parameterization of a symmetry class by its normal form A (for instance (7) for cubic symmetry [24]) and a rotation Q ∈ SO(3), ...

... In our application the tensor E 0 , taken from [28] (refer to [40,8,7,27,21,16] for measurements), is the elasticity tensor of a Nickel-based single crystal superalloy. In Voigt notation: (24) [ ...

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickelbased CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It is known that closed symmetry strata (for any tensorial representation of the rotation group) are semialgebraic sets, defined by polynomial equations and inequalities. It has been recently shown that the closed cubic elasticity stratum is moreover algebraic, which means that it can be defined by polynomial equations only (without requirement to polynomial inequalities). We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gr{\"o}bner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).

... If the degree of the anisotropy is sufficiently small, then, obviously, the properties of a crystal will slightly differ from the properties of an isotropic medium and the analysis of the properties of a crystal can be replaced by a more simple analysis of the properties of a model isotropic medium, which parameters are selected in an optimal way. In [2] it is proposed to find the elastic moduli of an approximating isotropic body from the condition of minimum of the quantity ...

... This approximation is quite reasonable for a qualitative, and in many cases quantitative, analysis of the integral properties of a crystal and those effects that are not related to the existence of distinguished directions in crystals and can be described in the approximation of an isotropic medium. In [2] such an approach was developed for the theory of elastic waves in crystals. A similar approach can be used when nonlinear effects are taken into account in crystals, for example, to simplify the calculations of the matrix elements of the interaction between phonons in crystals of any system [3,4]. ...

... These formulas coincide with the formulas (3) obtained on the basis of another criterion introduced in [2]. As was also shown here, such approximation proves to be valid at an arbitrary temperature. ...

When analyzing thermodynamic and kinetic properties of crystals whose anisotropy is not large and the considered effects do not relate to the existence of singled-out directions in crystals, one may use a more simple model of an isotropic medium with a good accuracy, after having chosen its parameters in an optimal way. Based on the quantum mechanical description it is shown that the method of approximation of the moduli of elasticity of a crystal by the model of an isotropic medium, proposed earlier in [2], follows from the requirement of the minimal difference between the free energies of a crystal and an approximating isotropic medium. The two-parametric Debye model is formulated, which, in contrast to the standard model where the average speed of phonons is introduced, takes into account the existence in an isotropic medium of both longitudinal and transverse phonons. The proposed model contains, except the Debye energy, an additional dimensionless parameter and, consequently, the law of corresponding states for the heat capacity being characteristic of the standard model does not hold. With taking account of the two phonon branches the structure of the density of phonon states proves to be more complex as compared to the standard model and has a singularity that resembles Van Hove singularities in real crystals. As an example, an application of the two-parametric Debye theory to such crystals of the cubic system as tungsten, copper, lead is considered. It is shown that the calculation of the low-temperature heat capacity of these crystals by means of the approximated moduli of elasticity within the framework of the two-parametric model leads to a considerably better agreement with experiment than in the case of the standard Debye model.

... The quadratic dependence on temperature was obtained in [15], and subsequently the form of the coefficient before T 2 was refined in other works [4]. In [16 -18], there was obtained a formula for the surface heat capacity in various approaches, which in the notation of (17), (19), (20) can be represented in the form [6,16] ...

... When analyzing thermodynamic and kinetic properties of crystals whose anisotropy is not large and the considered effects are not associated with the existence of singled-out directions in crystals, it is possible to use with a good accuracy a more simple model of an isotropic medium after choosing its parameters in an optimal way [20]. It was shown in [2] that the previously proposed method of describing the elastic properties of crystals on the basis of a comparison with an isotropic medium [20] follows from the requirement of the maximal closeness of the free energies of a crystal and an isotropic medium. ...

... When analyzing thermodynamic and kinetic properties of crystals whose anisotropy is not large and the considered effects are not associated with the existence of singled-out directions in crystals, it is possible to use with a good accuracy a more simple model of an isotropic medium after choosing its parameters in an optimal way [20]. It was shown in [2] that the previously proposed method of describing the elastic properties of crystals on the basis of a comparison with an isotropic medium [20] follows from the requirement of the maximal closeness of the free energies of a crystal and an isotropic medium. The two-parameter Debye model for an isotropic medium with effective elastic moduli [2] can be a good approximation for describing the properties of crystals. ...

A quantum description of the surface waves in an isotropic elastic body without the use of the semiclassical quantization is proposed. The problem about the surface waves is formulated in the Lagrangian and Hamiltonian representations. Within the framework of the generalized Debye model, the contribution of the surface phonons (rayleighons) to thermodynamic functions is calculated. It is emphasized that the role of the surface phonons can be significant and even decisive in low-dimensional systems, granular and porous media, and that their contribution to the total heat capacity increases with decreasing temperature.

... That is, the wave's vibration direction is neither parallel nor perpendicular to the direction of travel. If, however, the direction of travel is an elastic symmetry axis, then, with some unlikely exceptions, the wave must indeed be either a P-wave or an S-wave (Fedorov 1968). If also the relevant elastic map T has for its symmetry group one of the reference subgroups of Section 12, then in most cases both the vibration direction and the speed of the wave are simply related to the intrinsic parameters for T. (We do not treat these topics here.) ...

... Treatments of elasticity can be found in Fedorov (1968); Nye (1957Nye ( , 1985; Auld (1973); Musgrave (1970); Helbig (1994); Chapman (2004); Slawinski (2015) and many others. A reference for linear algebra is Hoffman & Kunze (1971). ...

... Consistent with Theorems 11 and 12, Fedorov (1968) recognized that the distinctions between what are effectively our matrices T4 and Ttet, and between our T3 and Ttrig, are only distinctions in orientation; it is a matter of where the 2-fold axes fall. (See Table 4 for Ttet and Ttrig.) ...

The elastic map, or generalized Hooke’s Law, associates stress with strain in an elastic material. A symmetry of the elastic map is a reorientation of the material that does not change the map. We treat the topic of elastic symmetry conceptually and pictorially. The elastic map is assumed to be linear, and we study it using standard notions from linear algebra—not tensor algebra. We depict strain and stress using the “beachballs” familiar to seismologists. The elastic map, whose inputs and outputs are strains and stresses, is in turn depicted using beachballs. We are able to infer the symmetries for most elastic maps, sometimes just by inspection of their beachball depictions. Many of our results will be familiar, but our versions are simpler and more transparent than their counterparts in the literature.

... obtaining the set of feasible slowness vectors from a given unique ray direction vector) in anisotropic media has been studied by several researchers (e.g. Musgrave 1954aMusgrave , 1954bMusgrave , 1970Fedorov 1968;Helbig 1994;Vavryčuk 2006;Grechka 2017;Zhang & Zhou 2018). A brief overview of these studies is provided in the first part of Appendix A. proposed a new efficient approach to the slowness inversion in polar anisotropy based on the inherent property of this symmetry class, where the three vectors: the phase velocity (or the slowness), the ray velocity and the medium axis of symmetry are coplanar. ...

... The Hamiltonian in this approach is the unit eigenvalue of the Christoffel matrix (i.e. the solution of the Christoffel equation, rather than the equation itself). This method leads to a set of three polynomial equations of degree six, which means a higher algebraic complexity, 6 × 6 × 6 = 216, than the classical approach suggested and used by Musgrave (1954a, 1954b, 1970), Fedorov (1968, Helbig (1994) and Grechka (2017), with the algebraic complexity 5 × 5 × 6 = 150. In a particular case of coupled qP-qSV waves in polar anisotropy, and under a commonly accepted assumption that the angle between the slowness and the ray direction vectors, n and r, respectively, cannot exceed 90 o , 0 < n · r = v phs /v ray ≤ 1 (where v phs is the phase velocity magnitude), the method suggested by Vavryčuk (cited above) reduces to a set of two quartic equations (algebraic complexity 16) and a constraint that cuts off one half of the roots. ...

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.

... Any non triclinic Elasticity tensor has a normal form. An orthonormal frame in which the matrix representation of this tensor belongs to such a normal form is called a proper or natural basis for E [14]. For instance, consider a cubic Elasticity tensor which is given in an arbitrary frame by its Voigt representation as (1.1) [E] = ...

... Remark 6.14. Recall that, following [21,14], an additional zero can be placed in the normal form̂︀ E = E Z 2 of a monoclinic tensor. This is due to the fact that any rotation around the third axis = 3 of the normal form ( ...

We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor, once we know its symmetry class. In other words, we produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants.

... x r co-exist.) In this study, we apply the Hamiltonian-based approach, exploiting the collinearity of the Hamiltonian gradient (wrt the slowness vector) and the ray direction (e.g., Musgrave, 1954;Fedorov, 1968;Grechka, 2017), along with the condition for the vanishing Hamiltonian, ...

... Given the medium elastic properties at a given location () Cx , and the ray velocity direction r , we compute the corresponding slowness vector by solving the Hamiltonian-based nonlinear set of three polynomial equations (e.g., Musgrave, 1954;Fedorov, 1968), for a given wave mode (equation set 18). For compressional waves, the magnitude of the ray velocity is uniquely defined by its direction, while for shear waves up to 18 solutions may co-exist (Grechka, 2017 The cross-product in this equation set represents three scalar equations, but only two of them are independent. ...

We present a new ray bending approach, referred to as the Eigenray method, for solving two-point boundary-value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where convergence to the stationary paths, based on conventional initial-value ray shooting methods, becomes challenging. The kinematic ray tracing solution corresponds to the vanishing first traveltime variation, leading to a stationary path, and is governed by the nonlinear second-order Euler-Lagrange equation (Part I). In Part II we elaborate on theoretical aspects of the proposed method and validate its correctness for general anisotropy. In Part III we use a finite-element approach, applying the weak formulation. In Part IV we propose an efficient method to compute the geometric spreading of the entire stationary ray path using the global traveltime Hessian. In Part V we formulate the dynamic ray tracing, considering the second traveltime variation, which leads to the linear second-order Jacobi equation, and in Part VI we relate the proposed Lagrangian approach to the commonly used Hamiltonian approach. The solution is provided in Part VII, where we implement a similar finite element approach applied for the kinematic problem. In both kinematic and dynamic problems, in between the nodes, the values of the ray characteristics are computed with the Hermite interpolation, which we find most natural for anisotropic media. We distinguish two types of stationary rays, delivering either a minimum or a saddle-point traveltime (due to caustics).

... Many studies have focused on the calculation of the directions of wave propagation, either in Elasticity [30,14,6] or in the case of coupled phenomena, such as Piezo-electricity or Piezomagnetism [1,4,43]. Indeed, any symmetry plane ν ν ν ⊥ of an Elasticity tensor C gives rise to a propagation direction ν ν ν of a longitudinal wave [48,32,14,38], also called an acoustic axis. ...

... Many studies have focused on the calculation of the directions of wave propagation, either in Elasticity [30,14,6] or in the case of coupled phenomena, such as Piezo-electricity or Piezomagnetism [1,4,43]. Indeed, any symmetry plane ν ν ν ⊥ of an Elasticity tensor C gives rise to a propagation direction ν ν ν of a longitudinal wave [48,32,14,38], also called an acoustic axis. ...

In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order [Formula: see text]. These conditions are effective and of degree [Formula: see text] (the tensor’s order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an elasticity tensor, a piezo-electricity tensor or a piezo-magnetism pseudo-tensor.

... Different from the conventional approach of sputtering multi-layer tilted c-axis thin films, where the constants are partially modified [64]- [66], bonded piezoelectric thin films [67] enable the implementation of the COP platform through the integration of thin films with different orientations. For Z-cut LiNbO 3 , which can be notated as a Euler angle of (0 • , 0 • , 0 • ) in the Z-X-Z format [68], an additional layer with a Euler angle of (180 • , 180 • , 0 • ) satisfies the COP requirements for material constants via matrix rotation [68]. The rotated axis is plotted with dashed lines in Fig. 2, which will be the platform studied in this work. ...

... Different from the conventional approach of sputtering multi-layer tilted c-axis thin films, where the constants are partially modified [64]- [66], bonded piezoelectric thin films [67] enable the implementation of the COP platform through the integration of thin films with different orientations. For Z-cut LiNbO 3 , which can be notated as a Euler angle of (0 • , 0 • , 0 • ) in the Z-X-Z format [68], an additional layer with a Euler angle of (180 • , 180 • , 0 • ) satisfies the COP requirements for material constants via matrix rotation [68]. The rotated axis is plotted with dashed lines in Fig. 2, which will be the platform studied in this work. ...

In this work, we present a new paradigm for enabling gigahertz higher-order Lamb wave acoustic devices using complementarily oriented piezoelectric (COP) thin films. Acoustic characteristics are first theoretically explored with COP lithium niobate (LiNbO₃) thin films, showing their excellent frequency scalability, low loss, and high electromechanical coupling (k²). Acoustic resonators and delay lines are then designed and implemented, targeting efficient excitation of higher-order Lamb waves with record-breaking low loss. The fabricated resonator shows a 2nd-order symmetric (S2) resonance at 3.05 GHz with a high quality factor (Q) of 657, and a large k² of 21.5% and a 6th-order symmetric (S6) resonance at 9.05 GHz with a high Q of 636 and a k² of 3.71%, both among the highest demonstrated for higher-order Lamb wave devices. The delay lines show an average insertion loss (IL) of 7.5 dB and the lowest reported propagation loss of 0.014 dB/μm at 4.4 GHz for S2. Notable acoustic passbands up to 15.1 GHz are identified. Upon further optimizations, the proposed COP platform can lead to gigahertz low-loss wideband acoustic components. [2020-0127]

... We did Sensors 2020, 20, 6148 7 of 18 not add any custom equations to the model. More details of these equations are available and can be found in [31,32]. Figure 5 shows the different propagation regimes obtained for different frequencies. ...

... We did not add any custom equations to the model. More details of these equations are available and can be found in [31,32]. Figure 5 shows the different propagation regimes obtained for different frequencies. ...

In this work, we numerically investigate the diffraction management of longitudinal elastic waves propagating in a two-dimensional metallic phononic crystal. We demonstrate that this structure acts as an “ultrasonic lens”, providing self-collimation or focusing effect at a certain distance from the crystal output. We implement this directional propagation in the design of a coupling device capable to control the directivity or focusing of ultrasonic waves propagation inside a target object. These effects are robust over a broad frequency band and are preserved in the propagation through a coupling gel between the “ultrasonic lens” and the solid target. These results may find interesting industrial and medical applications, where the localization of the ultrasonic waves may be required at certain positions embedded in the object under study. An application example for non-destructive testing with improved results, after using the ultrasonic lens, is discussed as a proof of concept for the novelty and applicability of our numerical simulation study.

... The elastic stiffness tensor c describes how stress relates to strain. It comprises at most 21 independent coefficients for a triclinic crystal, the c ij coefficients (in Voigt notation), which are determined by the bonding system and the properties of the atoms (Fedorov, 1968). Knowledge of the complete tensor allows investigation of the anisotropy of the bonding chains and the derivation of numerous physical quantities, e.g. the velocity of sound waves, the bulk modulus and the Debye temperature. ...

... with frequency !, wavevector k = {k x , k y , k z } and displacement vector u, has to be solved for the given crystal symmetry (Fedorov, 1968). The scattering intensities can then be calculated by summing over the three phonon branches in equation (1). ...

The complete elastic stiffness tensor of thiourea has been determined from thermal diffuse scattering (TDS) using high-energy photons (100 keV). Comparison with earlier data confirms a very good agreement of the tensor coefficients. In contrast with established methods to obtain elastic stiffness coefficients ( e.g. Brillouin spectroscopy, inelastic X-ray or neutron scattering, ultrasound spectroscopy), their determination from TDS is faster, does not require large samples or intricate sample preparation, and is applicable to opaque crystals. Using high-energy photons extends the applicability of the TDS-based approach to organic compounds which would suffer from radiation damage at lower photon energies.

... One of the core parts of the ray bending procedure is computing the ray velocity magnitude from its direction, which, in turn, requires first to establish the slowness vector components; for compressional waves, the solution is unique. We follow the approach suggested by Musgrave (1954Musgrave ( , 1970, Fedorov (1968), and Grechka (2017), where the slownessrelated gradient of the vanishing Hamiltonian is parallel to the ray direction. A similar approach with a set of polynomial equations has been suggested by Vavryčuk (2006). ...

... The inverse problem in this study is solved by applying the Hamiltonian-based approach, exploiting the collinearity of the Hamiltonian gradient (wrt the slowness vector) and the ray direction (e.g. Musgrave, 1954;Fedorov, 1968;Grechka, 2017), along with the condition for the vanishing Hamiltonian, ...

We present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic (Part I) and dynamic (Part II) ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging.
The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler‐Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler‐Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation.
The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analyzing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter however is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics.
Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.
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... obtaining the set of feasible slowness vectors from a given unique ray direction vector) in anisotropic media has been studied by several researchers (e.g. Musgrave 1954aMusgrave , 1954bMusgrave , 1970Fedorov 1968;Helbig 1994;Vavryčuk 2006;Grechka 2017;Zhang & Zhou 2018). A brief overview of these studies is provided in the first part of Appendix A. proposed a new efficient approach to the slowness inversion in polar anisotropy based on the inherent property of this symmetry class, where the three vectors: the phase velocity (or the slowness), the ray velocity and the medium axis of symmetry are coplanar. ...

... The Hamiltonian in this approach is the unit eigenvalue of the Christoffel matrix (i.e. the solution of the Christoffel equation, rather than the equation itself). This method leads to a set of three polynomial equations of degree six, which means a higher algebraic complexity, 6 × 6 × 6 = 216, than the classical approach suggested and used by Musgrave (1954a, 1954b, 1970), Fedorov (1968, Helbig (1994) and Grechka (2017), with the algebraic complexity 5 × 5 × 6 = 150. In a particular case of coupled qP-qSV waves in polar anisotropy, and under a commonly accepted assumption that the angle between the slowness and the ray direction vectors, n and r, respectively, cannot exceed 90 o , 0 < n · r = v phs /v ray ≤ 1 (where v phs is the phase velocity magnitude), the method suggested by Vavryčuk (cited above) reduces to a set of two quartic equations (algebraic complexity 16) and a constraint that cuts off one half of the roots. ...

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).

... Table 1. Orthonormal base vectors of stiffness tensor; see [22]. ...

A sharp-interface model employing the extended finite element method is presented. It is designed to capture the prominent γ-γ′ phase transformation in nickel-based superalloys. The novel combination of crystal plasticity and sharp-interface theory outlines a good modeling alternative to approaches based on the Cahn–Hilliard equation. The transformation is driven by diffusion of solute γ′-forming elements in the γ-phase. Boundary conditions for the diffusion problem are computed by the stress-modified Gibbs–Thomson equation. The normal mass balance of solute atoms at the interface yields the normal interface velocity, which is integrated in time by a level set procedure. In order to capture the influence of dislocation glide and climb on interface motion, a crystal plasticity model is assumed to describe the constitutive behaviour of the γ-phase. Cuboidal equilibrium shapes and Ostwald ripening can be reproduced. According to the model, in low γ′ volume-fraction alloys with separated γ′-precipitates, interface movement does not have a significant effect on tensile creep behaviour at various lattice orientations.

... In particular, if the matrix material is isotropic , we recover known result ( Sevostianov and Kachanov, 1999 ) P 1111 = P 2222 = For inhomogeneities not aligned with the symmetry axis of the matrix, like Volkman's canals and canaliculi, we use analytical approximation for components of tensor H ijkl following approach proposed by Saadat et al. (2012) . We first find the best isotropic approximation for a transversely isotropic tensor of elastic stiffness C ijkl (see Fedorov, 1968 ) given by λ 0 δ i j δ kl + G 0 δ ik δ l j + δ il δ k j (A.11) where G 0 = ( 3 C ikik − C iikk ) / 30 , λ 0 = ( 2 C iikk − C ikik ) / 15 (A.12) Using this best fit isotropy, we can calculate the components of the compliance contribution tensor for a spheroidal inclusion with semi axis a 1 = a 2 = a , and a 3 embedded in the matrix characterized by elastic constants G 0 and λ 0 . ...

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

... separated approximately by the position of the cryofront, can explain why the variations of ultrasonic parameters during upward freezing was quite different from that during uniform freezing. Fedorov (1968) and Rokhlin et al. (1986) indicated that a major problem in understanding wave propagation concerns the reflection and transmission of elastic waves from a rigidly coupled interface between any two media. Degtyar and Rokhlin (1998) found that wave reflection and transmission are most remarkable near critical angles when elastic wave is transmitted through an solid-solid interface. ...

Common problems in engineering projects that involve artificial ground freezing of soil or rock include inadequate thickness, strength and continuity of artificial frozen walls. It is difficult to evaluate the freezing state using only a few thermometer holes at fixed positions or with other existing approaches. Here we report a novel experimental design that investigates changes in ultrasonic properties (received waveform, wave velocity Vp, wave amplitude, frequency spectrum, centroid frequency fc, kurtosis of the frequency spectrum KFS, and quality factor Q) measured during upward freezing, compared with those during uniform freezing, in order to determine the freezing state in 150 mm cubic blocks of Ardingly sandstone. Water content, porosity and density were estimated during upward freezing to ascertain water migration and changes of porosity and density at different stages. The period of receiving the wave increased substantially and coda waves changed from loose to compact during both upward and uniform freezing. The trend of increasing Vp can be divided into three stages during uniform freezing. During upward freezing, Vp increased more or less uniformly. The frequency spectrum could be used as a convenient and rapid method to identify different freezing states of sandstone (unfrozen, upward frozen, and uniformly frozen). The continuous changes in reflection coefficient rφ, refraction coefficient tφ and acoustic impedance field are the major reason for larger reflection and refraction during upward freezing compared with uniform freezing. Wave velocity Vp, wave amplitude Ah, centroid frequency fc and quality factor Q were adopted as ultrasonic parameters to evaluate quantitatively the temperature T of uniformly frozen sandstone, and their application within a radar chart is recommended. Determination of Vp provides a convenient method to evaluate the freezing state and calculate the cryofront height and frozen section thickness of upward frozen sandstone, with accuracies of 73.37%–99.23%. © 2022 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

... Generally speaking, the fourth rank stiffness tensor in Euclidean space can be rotated using Euler angles. Fortunately, VTI and HTI symmetry classes belong to a transversely isotropic (TI) medium, which has a single axis of rotational symmetry (Fedorov, 1968). Therefore, the transformation from HTI media to VTI, and vice versa, doesn't require any sophisticated operations. ...

Seismic waves may exhibit significant dispersion and attenuation in reservoir rocks due to pore-scale fluid flow. Fluid flow at the microscopic scale is referred to as squirt flow and occurs in very compliant pores, such as grain contacts or microcracks, that are connected to other stiffer pores. We perform 3D numerical simulations of squirt flow using a finite element approach. Our 3D numerical models consist of a pore space embedded into a solid grain material. The pore space is represented by a flat cylinder (a compliant crack) whose edge is connected with a torus (a stiff pore). Grains are described as a linear isotropic elastic material while the fluid phase is described by the quasistatic linearized compressible Navier-Stokes momentum equation. We obtain the frequency-dependent effective stiffness of a porous medium and calculate dispersion and attenuation due to fluid flow from a compliant crack to a stiff pore. We compare our numerical results against a published analytical solution for squirt flow and analyze the effects of its assumptions. Previous interpretation of the squirt flow phenomenon based mainly on analytical solutions is verified and some new physical effects are identified. The numerical and analytical solutions agree only for the simplest model in which the edge of the crack is subjected to zero fluid pressure boundary condition while the stiff pore is absent. For the more realistic model that includes the stiff pore, significant discrepancies are observed. We identify two important aspects that need improvement in the analytical solution: the calculation of the frame stiffness moduli and the frequency dependence of attenuation and dispersion at intermediate frequencies.

... Alternatively, the ray velocity and the slowness vectors can be related through the stiffness tensor C and the polarization vector g (e.g., Fedorov, 1968;Musgrave, 1970;Červený, 1972;Auld, 1973;Tsvankin, 2012), ...

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.

... For an elasticity tensor E, partial answers have been provided in [5,4,15]. The study of longitudinal waves in anisotropic media [21,14,7] has indeed allowed Cowin and Mehrabadi to derive reduced necessary and sufficient conditions for the existence of a plane symmetry for E. These conditions were first expressed in theorem 1.1, using the two independent traces of E, the dilatation tensor d and the Voigt tensor v: ...

We formulate necessary and sufficient conditions for a unit vector n to generate a plane or axial symmetry of a constitutive tensor. For the elasticity tensor, these conditions consist of two polynomial equations of degree lower than four in the components of n. Compared to Cowin-Mehrabadi conditions, this is an improvement, since these equations involve only the normal vector n to the plane symmetry (and no vector perpendicular to n). Similar reduced algebraic conditions are obtained for linear piezo-electricity and for totally symmetric tensors up to order 6.

... We expect to come back to the fascinating phenomenon of superoscillation in the very near future. It seems to open many interesting directions of research 31,32 . Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH ("Springer Nature"). ...

It is widely accepted that a signal bandlimited by σ cannot oscillate at higher frequencies. The phenomenon of superoscillation provides a refutation of that quite general belief. Temporal superoscillations have been rarely demonstrated and are mostly treated as a mathematical curiosity. In the present article we demonstrate experimentally for the first time to our best knowledge, the transmission of superoscillating signals through commercial low pass filters. The experimental system used for the demonstration is described, providing the insight into the transmission of superoscillations, or super-narrow pulses. Thus, while the phenomenon may seem rather esoteric, a very simple system is used for our demonstration.

... The acoustic wave speeds c m for a given propagation directionq may be determined from the eigenvalues and eigenvectors of the Christoffel Matrix (M ik ≡ Λ ijklqjql , where Λ ijkl are the elements of the fourth-order elasticity tensor) through the Christoffel Equations (e.g., [34][35][36][37]), ...

Nuclear resonant inelastic X-ray scattering (NRIXS) experiments have been applied to Earth materials, and the Debye speed is often related to the material’s seismic wave speeds. However, for anisotropic samples, the Debye speed extracted from NRIXS measurements is not equal to the Debye speed obtained using the material’s isotropic seismic wave speeds. The latter provides an upper bound for the Debye speed of the material. Consequently, the acoustic wave speeds estimated from the Debye speed extracted from NRIXS (Nuclear Resonant Inelastic X-ray Scattering) measurements are underestimated compared to the material’s true seismic wave speeds. To illustrate the differences, the effects of various assumptions used to estimate the Debye speed, as well as seismic wave speeds, are examined with iron alloys at Earth’s inner core conditions. For the case of pure iron, the variation of the crystal orientation relative to the incoming X-ray beam causes a 40 % variation in the measured Debye speed, and leads to 3% and 31% underestimation in the compressional and shear wave speeds, respectively. Based upon various iron alloys, the error in the inferred seismic shear wave speed strongly depends upon the strength of anisotropy that can be quantified. We can also derive Debye speeds based upon seismological observations such as the PREM (Preliminary Reference Earth Model) and inner core anisotropy model. We show that these seismically derived Debye speeds are upper bounds for Debye speeds obtained from NRIXS experiments and that interpretation of the Debye speeds from the NRIXS measurements in terms of seismic wave speeds should be done with utmost caution.

... Matrix C is a positive definite as the corresponding complete necessary and sufficient conditions for it for a trigonal crystal [37] are satisfied, that is, c 11 . c 12 j j, (c 11 + c 12 )c 33 . ...

In this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained. Boundary elements with mixed approximation of geometry and field variables with the standard nodal collocation procedure are used for spatial discretization. In order to obtain time-domain solutions, the classic Durbin’s method is applied for numerical inversion of Laplace transform. Problem of alleviating Gibbs oscillations is addressed. Dynamic boundary element analysis of the model problem involving trigonal material is performed to test presented formulation. Obtained results are compared with finite element solutions.

... The elastic constants of the undoped systems are also shown for comparison. Bohr stability criteria [27,28] states that a system is mechanically stable if C 44 > 0, C 11 > |C 12 | and C 11 + 2C 12 > 0. ...

Ab-initio density functional theory calculations have been used to explore the effect of transition metal alloying on A15 Cr–Ru intermetallic alloys. We study the structural, electronic and mechanical properties of \(\hbox {Ru}_{{\mathrm {3}}}\hbox {Cr}\) and \(\hbox {Cr}_{\mathrm {{3}}}\hbox {Ru}\) alloys doped with transition metals (\(\hbox {M}= \hbox {Mn}, \hbox {Mo}, \hbox {Pt}, \hbox {Pd}, \hbox {Fe}, \hbox {Co}, \hbox {Re and Zr}\)). Their thermodynamic and mechanical behaviours were deduced from the heat of formation, ratio of bulk to shear modulus, density of states (DOS) as well as elastic constants predictions. We find that Mn doping in these alloys leads to thermodynamic stability. These compounds also show a valence–conduction band overlap around the Fermi energy as depicted by the DOS. Furthermore, the Pugh ratio (the ratio of bulk to shear modulus) indicates the ductility character of these compounds. Their mechanical stability was illustrated by the Bohr mechanical stability criteria with all the elastic constants having a value >0. These results demonstrate that these systems can potentially be used as coating materials in high temperature structural applications.

... The elastic constants of the undoped systems are also shown for comparison. Bohr stability criteria [27,28] states that a system is mechanically stable if C 44 > 0, C 11 > |C 12 | and C 11 + 2C 12 > 0. ...

Ab-initio density functional theory calculations have been used to explore the effect of transition metal alloying on A15 Cr-Ru intermetallic alloys. We study the structural, electronic and mechanical properties of Ru 3 Cr and Cr 3 Ru alloys doped with transition metals (M = Mn, Mo, Pt, Pd, Fe, Co, Re and Zr). Their thermodynamic and mechanical behaviours were deduced from the heat of formation, ratio of bulk to shear modulus, density of states (DOS) as well as elastic constants predictions. We find that Mn doping in these alloys leads to thermodynamic stability. These compounds also show a valence-conduction band overlap around the Fermi energy as depicted by the DOS. Furthermore, the Pugh ratio (the ratio of bulk to shear modulus) indicates the ductility character of these compounds. Their mechanical stability was illustrated by the Bohr mechanical stability criteria with all the elastic constants having a value >0. These results demonstrate that these systems can potentially be used as coating materials in high temperature structural applications.

... Figure 5 shows the P-wave (primary wave) phase velocity as a function of the phase angle of the numerical model with connected and disconnected cracks (Fig. 1), where the zero phase angle corresponds to the vertical wave propagation (along z axis). The P-and S-wave phase velocities are calculated by solving the Christoffel equation, which represents an eigenvalue problem relating the stiffness components c ij , the phase velocities of plane waves that propagate in the medium and the polarization of the waves (Fedorov, 1968;Tsvankin, 2012). Considering the plane Y -Z, the P-wave velocity is the same for phase angles of 0 and 90 • ; it changes with frequency only for phase angles between 0 and 90 • and is maximal in the high-frequency limit at phase angle of θ = 90(±90) • (Fig. 5a). ...

Understanding the properties of cracked rocks is of great importance in scenarios involving CO2 geological sequestration, nuclear waste disposal, geothermal energy, and hydrocarbon exploration and production. Developing noninvasive detecting and monitoring methods for such geological formations is crucial. Many studies show that seismic waves exhibit strong dispersion and attenuation across a broad frequency range due to fluid flow at the pore scale known as squirt flow. Nevertheless, how and to what extent squirt flow affects seismic waves is still a matter of investigation. To fully understand its angle- and frequency-dependent behavior for specific geometries, appropriate numerical simulations are needed. We perform a three-dimensional numerical study of the fluid–solid deformation at the pore scale based on coupled Lamé–Navier and Navier–Stokes linear quasistatic equations. We show that seismic wave velocities exhibit strong azimuth-, angle- and frequency-dependent behavior due to squirt flow between interconnected cracks. Furthermore, the overall anisotropy of a medium mainly increases due to squirt flow, but in some specific planes the anisotropy can locally decrease. We analyze the Thomsen-type anisotropic parameters and adopt another scalar parameter which can be used to measure the anisotropy strength of a model with any elastic symmetry. This work significantly clarifies the impact of squirt flow on seismic wave anisotropy in three dimensions and can potentially be used to improve the geophysical monitoring and surveying of fluid-filled cracked porous zones in the subsurface.

... Besides, the velocities of all the three acoustic modes do not depend on the propagation direction in the XY plane. This result agrees well with the known fact that the hexagonal crystals represent transversely isotropic media, which means that the velocities of the QL AW and the AWs QT 1 and QT 2 do not depend on the direction of their propagation within the XY plane [13]. ...

... The analysis of mechanical stability plays an important role in understanding phase transitions [35]. The mechanical stability of crystals can be determined by whether the elastic constants C ij satisfy the Born stability criteria [34,36]. For a trigonal system, the mechanical stability conditions are as follows: ...

Nickel sulfide minerals, an important type of metal sulfides, are the major component of mantle sulfides. They are also one of the important windows for mantle partial melting, mantle metasomatism, and mantle fluid mineralization. The elasticity plays an important role in understanding the deformation and elastic wave propagation of minerals, and it is the key parameter for interpreting seismic wave velocity in terms of the composition of the Earth’s interior. Based on first-principles methods, the crystal structure, equation of state, elastic constants, elastic modulus, mechanical stability, elastic anisotropy, and elastic wave velocity of millerite (NiS), heazlewoodite (Ni3S2), and polydymite (Ni3S4) under high pressure are investigated. Our calculated results show that the crystal structures of these Ni sulfides are well predicted. These Ni sulfides are mechanically stable under the high pressure of the upper mantle. The elastic constants show different changing trends with increasing pressure. The bulk modulus of these Ni sulfides increases linearly with pressure, whereas shear modulus is less sensitive to pressure. The universal elastic anisotropic index AU also shows different changing trends with pressure. Furthermore, the elastic wave velocities of Ni sulfides are much lower than those of olivine and enstatite.

... This value of λ 0 will be kept for the remainder of the study. We present in Fig. 2 the resulting shear-wave velocity V S and a measure of anisotropy defined by the ratio ||c − c iso || 2 /||c iso || 2 , where c iso is the usual isotropic projection of c (Fedorov 2013;Browaeys & Chevrot 2004) and ||.|| 2 the Euclidean (or Frobenius) matrix norm. While the original model is discontinuous and isotropic, its effective medium is smooth and anisotropic. ...

Seismic imaging techniques such as elastic full waveform inversion (FWI) have their spatial resolution limited by the maximum frequency present in the observed waveforms. Scales smaller than a fraction of the minimum wavelength cannot be resolved, and only a smoothed, effective version of the true underlying medium can be recovered. These finite-frequency effects are revealed by the upscaling or homogenization theory of wave propagation. Homogenization aims at computing larger scale effective properties of a medium containing small-scale heterogeneities. We study how this theory can be used in the context of FWI. The seismic imaging problem is broken down in a two-stage multiscale approach. In the first step, called homogenized full waveform inversion (HFWI), observed waveforms are inverted for a smooth, fully anisotropic effective medium, that does not contain scales smaller than the shortest wavelength present in the wavefield. The solution being an effective medium, it is difficult to directly interpret it. It requires a second step, called downscaling or inverse homogenization, where the smooth image is used as data, and the goal is to recover small-scale parameters. All the information contained in the observed waveforms is extracted in the HFWI step. The solution of the downscaling step is highly non-unique as many small-scale models may share the same long wavelength effective properties. We therefore rely on the introduction of external a priori information, and cast the problem in a Bayesian formulation. The ensemble of potential fine-scale models sharing the same long wavelength effective properties is explored with a Markov chain Monte Carlo algorithm. We illustrate the method with a synthetic cavity detection problem: we search for the position, size and shape of void inclusions in a homogeneous elastic medium, where the size of cavities is smaller than the resolving length of the seismic data. We illustrate the advantages of introducing the homogenization theory at both stages. In HFWI, homogenization acts as a natural regularization helping convergence toward meaningful solution models. Working with fully anisotropic effective media prevents the leakage of anisotropy induced by the fine scales into isotropic macro-parameters estimates. In the downscaling step, the forward theory is the homogenization itself. It is computationally cheap, allowing us to consider geological models with more complexity (e.g. including discontinuities) and use stochastic inversion techniques.

... Furthermore, only three planes containing a three-fold symmetry axis are available in the crystals of the symmetry group 3, such that the displacement vector and the AW wave vector both lie in these planes. The above planes are formed from so-called second-kind directions [10][11][12]. Two of the acoustic eigenwaves propagating along these directions are non-orthogonal. ...

... • the tensor is measured in a given orientation, possibly experimenter dependent, which may not coincide with an expected symmetry group, not allowing then for the recognition by eye (on the Kelvin matrix representation) of the orthotropic, tetragonal, cubic . . . wellknown expressions for C raw (the well-known normal forms of elasticity tensors [12]), • the experimental discrepancy/errors makes the material symmetries approximate. Sufficient conditions (in [8]) and necessary and sufficient conditions (in [22]) have been formulated to characterize, in an arbitrarily oriented coordinate system, the symmetry class of a three-dimensional elasticity tensor. ...

Constitutive tensors are of common use in mechanics of materials. To determine the relevant symmetry class of an experimental tensor is still a tedious problem. For instance, it requires numerical methods in three-dimensional elasticity. We address here the more affordable case of plane (bi-dimensional) elasticity, which has not been fully solved yet. We recall first Vianello's orthogonal projection method, valid for both the isotropic and the square symmetric (tetragonal) symmetry classes. We then solve in a closed-form the problem of the distance to plane elasticity orthotropy, thanks to the Euler-Lagrange method.

... The slowness vectors can be obtained from the equation set which manifests that the slowness gradient of the (vanishing) Hamiltonian / HH p p is collinear with the ray direction r (Musgrave, 1954a(Musgrave, , 1954b(Musgrave, , 1970Fedorov, 1968;Helbig, 1994;Grechka, 2017;Koren, 2021b), written in our notations as, ...

We present an original, generic, and efficient approach for computing the first and second partial derivatives of ray velocities along ray paths in general anisotropic elastic media. These derivatives are used in solving kinematic problems, like two-point ray bending methods and seismic tomography, and they are essential for evaluating the dynamic properties along the rays (amplitudes and phases). The traveltime is delivered through an integral over a given Lagrangian defined at each point along the ray. Although the Lagrangian cannot be explicitly expressed in terms of the medium properties and the ray direction components, its derivatives can still be formulated analytically using the corresponding arclength-related Hamiltonian that can be explicitly expressed in terms of the medium properties and the slowness vector components. This requires first to invert for the slowness vector components, given the ray direction components. Computation of the slowness vector and the ray velocity derivatives is considerably simplified by using an auxiliary scaled-time-related Hamiltonian obtained directly from the Christoffel equation and connected to the arclength-related Hamiltonian by a simple scale factor. This study consists of two parts. In Part I, we consider general anisotropic (triclinic) media, and provide the derivatives (gradients and Hessians) of the ray velocity, with respect to (1) the spatial/directional vectors and (2) the elastic model parameters. In Part II, we apply the theory of Part I explicitly to polar anisotropic media (transverse isotropy with tilted axis of symmetry, TTI), and obtain the explicit ray velocity derivatives for the coupled qP and qSV waves and for SH waves.

... Alternatively, the ray velocity and the slowness vectors can be related through the stiffness tensor C and the polarization vector g (e.g., Fedorov, 1968;Musgrave, 1970;Červený, 1972;Auld, 1973;Tsvankin, 2012), ...

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.
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... To quantify the anisotropy of the stiffness tensor we compare to the best approximation by an isotropic tensor (see Eq. (3.5)), i.e., we project the computed stiffness tensor onto the space of isotropic tensors of fourth order. For a detailed discussion Table 1 Parameters used for the crystal plasticity model [73,74] Cubic stiffness (in Voigt notation) [85] see the work by Federov [89] and Arts [90]. We compute the mean stiffness ...

This work is concerned with synthetic microstructure models of polycrystalline materials. Once a representation of the microstructure is generated, the individual grains need to be furnished with suitable crystal orientations, matching a specific crystal orientation distribution. We introduce a novel method for this task, which permits to prescribe the orientations based on tensorial Fourier coefficients. This compact representation gives rise to the texture coefficient optimization for prescribing orientations method, enabling the determination of representative orientations for digital polycrystalline microstructures. We compare the proposed method to established and dedicated algorithms in terms of the linear elastic as well as the non-linear plastic behavior of a polycrystalline material.

... The rotation angle depends in a complicated way on the actual components. The same result has been obtained by Fedorov (1968), Cowin (1995), Ting (1996), Forte and Vianello (1996). Gurtin (1972) still lists ten different matrices, which contain the above mentioned redundancies. ...

We here present a faithful translation of Carl Hermann's important 1934 paper "Tensoren und Kristallsymmetrie". This work, originally published in German language is transferred into English while the preceding foreword summarizes Hermann's achievements.

... These authors have definitively clarified the mathematical problem of classifying the symmetry classes of the representation of SO(3) on Ela, the 21-dimensional space of elasticity tensors. They removed the link with crystallographic point groups which was extremely confusing and lead to the false assumption that there were ten, rather than eight, symmetry classes [36,25,47]. These eight classes were confirmed in 2001, using an alternative approach [20], where symmetry planes rather than rotations play the central role. ...

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.

... Let us also note, that concerning the laws of light reflection-transmission on the media interface the complete balance of energy eliminating the possibility of any additional "longitudinal" light waves is observed [20]. An enumeration of other phenomena and effects including piezo-and thermo-electricity, mutual electro-elastic effects are described in [22,25,26]. Summarizing the outcomes set forth the following should be assigned to general properties of ether (vacuum): ...

The fundamentals of a physical model of the ethereal medium (vacuum) consisting of particles of two kinds, equal, but opposite in sign are stated. The model contains elements of the vacuum structure offered by W. Thomson, of MacGullagh continuum and conforms to the theory of electromagnetism by D. Maxwell. A uniform physical basis for an explanation of observed electromagnetic phenomena, inertia and gravitation is given.

The first-principles calculations based on density functional theory with projector-augmented wave are used to study the physical properties of monoclinic MgCO 3 at lower mantle conditions. The results show that the phase transition pressure of rhombohedral MgCO 3 (magnesite) to monoclinic MgCO 3 is between 72–79 GPa in the temperature range of the mantle. The equation of state for monoclinic MgCO 3 agrees well with recent experimental results. The elastic constants of monoclinic MgCO 3 are consistent with the latest theoretical results. The elastic modulus of monoclinic MgCO 3 , especially its shear modulus, exhibits a nonlinear pressure dependence, leading to a significant nonlinear pressure dependence of the small elastic anisotropy and the small wave velocity anisotropy, which is why carbide minerals may not be detected in the deep mantle. The thermodynamic properties of monoclinic MgCO 3 at high temperature and high pressure are predicted using the Debye-Grüneisen model.

In this chapter, we consider a model of a granular medium as a rectangular lattice of rigid ellipse-shaped particles. Each particle of such a lattice possesses two translational and one rotational degrees of freedom. The space between the particles is a massless medium through which the force and coupled interactions are transmitted. In limiting cases, this model degenerates either into a chain of ellipse-shaped particles or into a square lattice of round particles. The main objectives of this chapter are to derive dynamic equations of a granular medium consisting of anisotropic particles and to identify the relationships between the physicomechanical properties of a granular material and the parameters of its microstructure. Using the results obtained in the chapter, it is possible to determine the elastic properties of an anisotropic nanocrystalline (granular) material with non-dense packing of particles by measuring the velocities of elastic waves propagating along different crystallographic directions [1].

Hooke's law and dynamic equations of motion in inhomogeneous 3‐D quaicrystals (QCs) were reduced to a symmetric hyperbolic system of the first‐order partial differential equations. This symmetric hyperbolic system describes a new mathematical model of wave propagation in inhomogeneous 3‐D QCs. Applying the theory and methods of symmetric hyperbolic systems, we have proved that this model satisfies the Hadamard's correctness requirements: solvability, uniqueness, and stability with respect to perturbation of data. Moreover, a new analytical method of solving the initial value problem for the obtained symmetric hyperbolic system which models wave propagation in vertical inhomogeneous quasicrystals was developed.

Recently, we have seen the emergence of digital and network platforms in the economic markets more and more often. For example, the taxi market in Moscow has transformed from a classical provision of material services - a passenger transportation service - into a market for the provision of information services, where the main players - digital platforms (Yandex taxi, Uber) provide information services to introduce the client-passenger to the supplier-driver. In this regard, it seems natural to consider the reasons why some firms win the market struggle in the new economic environment, while others lose it. The research is based on Frank Bass's informational approach, which assumes that the distribution of information about a new product among consumers has the main influence on the distribution of market shares (in our case, the consumer environment will be modeled using a network). \\
After setting out the basic principles of the Bass model, the model expands significantly, which allows us to describe the interaction of several firms competing within the same market for an undifferentiated information product. It is necessary to find out exactly how the specific parameters of the models affect the final stable equilibrium position, how firms can and should influence this parameter in order to win the struggle for market power. The study involves the construction of general theoretical conclusions, based on which, in the future, it will be possible to move on to the practical part of the study. \\
The work uses the mathematical apparatus from the elementary theory of differential equations, the theory of catastrophes, the theory of populations. Before we begin, let us intrigue the reader with the following thought: the market struggle of producers in a limited consumer market can be seen as a competition between two different types of predators for a limited population of prey.

The first-principles calculations based on density functional theory with projector-augmented wave are used to study the anisotropy of elastic modulus, mechanical hardness, minimum thermal conductivity, acoustic velocity and thermal expansion of magnesite(MgCO3) under deep mantle pressure. The calculation results of the phase transition pressure, equation of state, elastic constants, elastic moduli, elastic wave velocities and thermal expansion coefficient are consistent with those determined experimentally. The research results show that the elastic moduli have strong anisotropy, the mechanical hardness gradually softens with increasing pressure, the conduction velocity of heat in the [100] direction is faster than that in the [001] direction, the plane wave velocity anisotropy first increases and then gradually decreases with increasing pressure, and the shear wave velocity anisotropy increases with the increase of pressure, the thermal expansion in the [100] direction is greater than that in the [001] direction. The research results are of great significance to people's understanding of the high-pressure physical properties of carbonates in the deep mantle.

Розкрито теорію і практику пошуку ефективних шляхів поєднання дії ринкових механізмів з принципами соціальної справедливості в ринковій економіці. Досліджено економічний лібералізм як теоретичну основу формування англосаксонської моделі ринкової економіки. На основі історико-економічного аналізу розвитку корпоративного сектору економіки у США висвітлено практику державного регулювання підприємницької діяльності в умовах вільної конкуренції. Визначені основні інститути та напрями ліберальної моделі соціальної політики у США та підкреслено важливість державного втручання задля подолання значної стратифікації в доходах населення.

Theoretical estimates [1] and experimental data [2–5] show that rotational waves can exist in solids in the high-frequency field (> 109 – 1011 Hz), where it is rather difficult to carry out acoustic experiments with the technical viewpoint. The question arises: is it possible to obtain some information about the microstructure of a medium from acoustic measurements in the low-frequency range (106 – 107 Hz), when the rotational waves do not propagate in the medium? To this purpose, we will consider in this chapter the low-frequency approximation of Eqs. (2.8) and (3.6), in which the microrotations of the particles of the medium are not independent and are determined by the displacement field. Further, by comparing the obtained equations describing the propagation and interaction of longitudinal and transverse waves in a granular medium in the low-frequency approximation with the equations of the classical theory of elasticity, we will consider the problem of parametric identification of the developed models.

The mechanical interactions of C-(N-)A-S-H (Calcium-sodium-aluminum-silicate-hydrate) gel with slag and fly ash inclusions in alkali-activated materials (AAM) are quantified through image-supported grid nanoindentation. Nonuniform distributions of indent-specific indentation properties reveal that the elasticity-related domain is up to 130 times the contact indentation depth, while the hardness-related domain, in turn, is by a factor of two to three smaller. These rather large domains are consistent with the slag/fly-ash inclusions being much stiffer and harder than the surrounding C-(N-)A-S-H gel. Corresponding Hashin-Shtirkman bounds for the overall AAM stiffness consistently frame ultrasonic data characterizing this homogenized material scale. This confirms our new testing protocol.

The principles of the structural modeling method, the development of the theoretical foundations of which this monograph is devoted, are formulated in the first chapter. Moreover, the problem of the applicability of the classical mechanics laws to a theoretical description of media with micro- and nanostructure is discussed here.

Seismic wave imaging in complex media requires an accurate wavefield simulation method that can accurately describe the wave propagation in realistic media. Reverse time depth migration is the preferred method for seismic wave imaging in complex media. Although it is relatively expensive, its imaging accuracy is usually better than migrations based on the ray method. Migration of primary reflection data requires a wave propagation simulation method that can accurately describe primary reflected/scattered wave energy and incorporate anisotropy. Accordingly, we propose the simulation of wave propagation in tilted transversely isotropic media using a 15° one-way wave equation in a ray-centred coordinate system, combining the flexibility of ray theory and accuracy of wave theory. We use this equation to describe the propagation of body waves in a single ray tube, a “beam”. The wavefield along the beam, guided by its central raypath, has an angle limit defined only by the ray angle; therefore wave propagation in complex and steeply dipping media can be simulated with a 15° one-way wave equation. Numerical experiments show that the simulation results for beam propagation using the 15° equation in the ray-centred coordinate system have good accuracy. For prestack depth migration in tilted transversely isotropic media, we built a beam imaging method using this propagator, and this migration method yielded accurate images with greater efficiency than RTM.
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