Fourth-rank Tensor Research Articles

Active Actuating of a Simply Supported Beam with the Flexoelectric Effect.

Piezoelectric materials with the electro-mechanical coupling effect have been widely utilized in sensors, dampers, actuators, and so on. Engineering structures with piezoelectric actuators and sensors have provided great improvement in terms of vibration and noise reduction. The flexoelectric effect—which describes the coupling effect between the polarization gradient and strain, and between the strain gradient and electric polarization in solids—has a fourth-rank order tensor electro-mechanical coupling coefficient, and in principle makes the flexoelectricity existing in all insulating materials and promises an even wider application potential in vibration and noise control. In the presented work, a flexoelectric actuator was designed to actuate a simply supported beam. The electric field gradient was generated by an atomic force microscopy probe. Flexoelectric control force and moment components could be induced within the flexoelectric control layer. As flexoelectricity is size-dependent, the key parameters that could affect the actuating effect were examined in case studies. Analytical results showed that the induced flexoelectric control moment was strongly concentrated at the probe location. The controllable transverse displacement of the simply supported beam was calculated with the modal expansion method. It was found that the controllable transverse displacement was dependent on the probe location as well.

Tensor-to-matrix mapping in elasto-optics

A matrix method for tensor transformations in Voigt notation known from the elasticity calculations has been applied to elasto-optical calculations. Using invariant tensor-to-matrix mapping, the second- and fourth-rank Cartesian tensor transformations and basic operations can be performed by means of matrix multiplication. This approach brings what we believe is a new method of correct tensorial operations in Voigt notation, replacing a well-known approach that allocated specific constants to some matrix elements to even up the difference between the tensor transformations and 6 × 6 matrix operations. This general approach also simplifies the use of elasto-optical calculations for an arbitrary crystal class in an arbitrary orientation.

Damage and healing mechanics in plane stress, plane strain, and isotropic elasticity

The present paper presents a theoretical formulation of different self-healing variables. Healing variables based on the recovery in elastic modulus, shear modulus, Poisson's ratio, and bulk modulus are defined. The formulation is presented in both scalar and tensorial cases. A new healing variable based on elastic stiffness recovery in proposed, which is consistent with the continuum damage-healing mechanics. The evolution of the healing variable calculated based on cross-section as function of the healing variable calculated based on elastic stiffness is presented in both hypotheses of elastic strain equivalence and elastic energy equivalence. The components of the fourth-rank healing tensor are also obtained in the case of isotropic elasticity, plane stress, and plane strain. It is found that the healing variable calculated based on elastic stiffness reduction is greater than the one calculated based on cross-section reduction in the case of the hypothesis of elastic energy equivalence. It is also shown that the healing tensor fits the boundary conditions of the healing variable in the case of scalar formulation.

Compliance contribution tensor of an arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic elastic material

This paper focuses on the analysis of possibilities to find approximate closed form solution for an isolated ellipsoidal inhomogeneity embedded in an orthotropic matrix and to calculate overall elastic properties of heterogeneous materials with orthotropic matrix. The approach is based on approximation of an orthotropic material by one with the special type of symmetry, when fourth rank tensor of elastic stiffness can be expressed in terms of a second rank tensor. Such a symmetry is called in literature “elliptic orthotropy” or “ellipsoidal orthotropy”. If the accuracy of the approximation is sufficient, appropriate affine transformation of coordinates yields the required solution.

Space-Time Curvature Mode Quanta

Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a model of space-time curvature modes and associated curvature quanta in slightly warped space-time generated by a light Photon is derived. Both a Schr?dinger and a Second Quantized representation of the space-time curvature mode quanta are calculated and are fourth rank tensors. The eigenvalues of these equations are radii of curvature, not energy. The Eigenfunctions are linear functions of the components of the tensor that describes the curvature of space-time.

Stress concentration in frictional polymeric composites reinforced by oriented fibers

Multicomponent frictional polymer composites based on an epoxyphenol binder reinforced with E-glass fibers and dispersed inclusions of rubber, alumina, graphite, and barite are considered. It is believed that glass fibers are oriented along the X and Y axes of a rectangular coordinate system and have the same volume contents in these directions, which corresponds to the reinforcement of the material with fabric. Numerical calculations of the values of the components of the stress concentration operator (a fourth-rank tensor connecting the local stress values at an arbitrary point in an inhomogeneous medium with external stresses applied to the composite) were carried out for the considered frictional polymer composites, taking into account changes in the volume contents of their fillers.

Discrete mechanical growth model for plant tissue.

We present a discrete mechanical model to study plant development. The method is built up of mass points, springs and hinges mimicking the plant cell wall’s microstructure. To model plastic growth the resting lengths of springs are adjusted; when springs exceed a threshold length, new mass points, springs and hinges, are added. We formulate a stiffness tensor for the springs and hinges as a function of the fourth rank tensor of elasticity and the geometry of the mesh. This allows us to approximate the material law as a generalized orthotropic Hooke’s law, and control material properties during growth. The material properties of the model are illustrated in numerical simulations for finite strain and plastic growth. To solve the equations of motion of mass points we assume elastostatics and use Verlet integration. The method is demonstrated in simulations when anisotropic growth causes emergent residual strain fields in cell walls and a bending of tissue. The method can be used in multilevel models to study plant development, for example by coupling it to models for cytoskeletal, hormonal and gene regulatory processes.

Coherent two-dimensional electronic and infrared crystallography.

The two-dimensional electronic and infrared spectroscopy of oriented single crystals is sensitive to structure and point group symmetry. The third order response of crystals is generally different from measurements of isotropic solutions because each coherence path that contributes to the measured field scales to the ensemble average of the four-point correlation functions of the four field-dipole interactions involved in the respective Feynman paths. An analytical evaluation of 2D optical crystallography which depends on the crystal symmetry, laboratory orientation, and the orientation in the crystallographic frame is presented. Applying a symmetry operator in the basis of the allowed polarised radiation modes provides a method for evaluation of non-zero fourth rank tensor elements alternative to direct inspection methods. Uniaxial and biaxial systems are distinguished and the contributions to the rephasing and non-rephasing directions are evaluated for isolated and coupled oscillators. By exploiting coordinate analysis, the extension of non-linear electronic and infrared crystallography for coupled oscillators demonstrates the structural, directional, and symmetry dependent selection of coherences to the four-wave mixing signal.

Group theory description of third harmonic generation in diamond and zincblende lattice

We provide a description that shows the importance in finding a susceptibility tensor in nonlinear optics and apply group theory to derive a fourth rank tensor of Oh point group related to the bulk symmetry of monoatomic diamond semiconductor such as Silicon. It is well known that the fourth rank tensor is closely related to the third order nonlinear susceptibility tensor in nonlinear optics through Neumann’s principle and is crucial to understand third harmonic generation (THG) and electric field induced second harmonic generation (EFISH). Using a general Rodriguez rotation, we were able to obtain a three-dimmensional rotation matrix required to perform the symmetry operations, whereas their reflection matrixes are derived using a Householder fromula. We also show how a Td point group corresponding to zincblende can be obtained by breaking the symmetry of the Oh tensor rather than repeating the tedious procedure when obtaining the Oh tensor. Assuming a driven frequency far from the resonance frequency, we can apply Kleinman symmetry and demonstrate that the number of independent components in the Oh and Td fourth rank tensor is reduced from four to just two.

Aggregate Elasticity and Tabletability of Molecular Solids: a Validation and Application of Powder Brillouin Light Scattering.

Describing the elastic deformation of single-crystal molecular solids under stress requires a comprehensive determination of the fourth-rank stiffness tensor (Cijkl). Single crystals are, however, rarely utilized in industrial applications, and thus averaging techniques (e.g., the Voigt or Reuss approach) are employed to reduce the Cijkl (or its inverse Sijkl) to polycrystalline aggregate mechanical moduli. With increasing elastic anisotropy, the Voigt and Reuss-averaged aggregate moduli can diverge dramatically and, provided that drug molecules almost exclusively crystallize into low-symmetry space groups, warrants a significant need for accurate aggregate mechanical moduli. This elasticity data, which currently is largely absent for pharmaceutical materials, is expected to aid understanding how materials respond to direct compression and tablet formation. Powder Brillouin light scattering (p-BLS) has recently demonstrated facile access to porosity-independent, aggregate mechanical moduli. In this study, we extend our previous p-BLS model for obtaining mechanical properties and validate our approach against a broad library of molecular solids with diverse intermolecular interaction topologies and with previously determined Cijkl which permits benchmarking our results. Our Young's and shear moduli determined with p-BLS strongly correlate, with limited bias (i.e., a near 1:1 relation), with the Voigt-averaged Young's and shear moduli determined using the Cijkl. Through follow-on tabletability studies, we introduce initial classifications of tabletability behavior based on the results of our p-BLS studies and the apparent elastic anisotropy. With further development, this approach represents a robust and novel method to potentially identify materials for optimum tabletability at early developmental stages.

Third harmonic generation in ZnO semiconductor using the simplified bond hyperpolarizability model

We describe the fourth rank tensor and the related third harmonic generation (THG) light intensity profile in a (0002) wurtzite structure using the simplified bond hyperpolarizability model (SBHM). We show that the resulting THG intensity is isotropic e.g., does not depend on the azimuthal rotation angle of the material. Assuming that THG inside wurtzite structures are dominated solely by bulk dipoles, only one fitting parameter in terms of the effective THG hyperpolarizability is required to generate the Rotational Anisotropy THG (RATHG) experiment result.

Bimodal or quadrimodal? Statistical tests for the shape of fault patterns

Abstract. Natural fault patterns formed in response to a single tectonic event often display significant variation in their orientation distribution. The cause of this variation is the subject of some debate: it could be noise on underlying conjugate (or bimodal) fault patterns or it could be intrinsic signal from an underlying polymodal (e.g. quadrimodal) pattern. In this contribution, we present new statistical tests to assess the probability of a fault pattern having two (bimodal, or conjugate) or four (quadrimodal) underlying modes and orthorhombic symmetry. We use the eigenvalues of the second- and fourth-rank orientation tensors, derived from the direction cosines of the poles to the fault planes, as the basis for our tests. Using a combination of the existing fabric eigenvalue (or modified Flinn) plot and our new tests, we can discriminate reliably between bimodal (conjugate) and quadrimodal fault patterns. We validate our tests using synthetic fault orientation datasets constructed from multimodal Watson distributions and then assess six natural fault datasets from outcrops and earthquake focal plane solutions. We show that five out of six of these natural datasets are probably quadrimodal and orthorhombic. The tests have been implemented in the R language and a link is given to the authors' source code.

Increasing contribution of grain boundary compliance to polycrystalline ice elasticity as temperature increases

ABSTRACTMeasured elastic stiffnesses of ice polycrystals decrease with increasing temperature due to a decrease in grain boundary stiffness with increasing temperature. In this paper, we represent grain boundaries as imperfectly bonded interfaces, across which traction is continuous, but displacement may be discontinuous. We express the additional compliance due to grain boundaries in terms of a second-rank and a fourth-rank tensor, which quantify the effect on elastic wave velocities of the orientation distribution as well as the normal and shear compliances of the grain boundaries. Measurement of the elastic stiffnesses allows determination of the components of these tensors. Application of the method to resonant ultrasound spectroscopy measurements made on ice polycrystals enables determination of the ratio BN/BS of the normal to shear compliance of the grain boundaries, which are found to be more compliant in shear than in compression. The ratio BN/BS is small at low temperatures, but increases as temperature increases, implying that the normal compliance increases relative to the shear compliance as temperature increases.

Effective Elastic Coefficients of an Inhomogeneous Solids

The first special boundary value problem (SBVP) of the theory of elasticity for an inhomogeneous body is considered. The effective elasticity coefficients are found from the solution of the SBVP. They form a fourth-rank tensor, namely the tensor of effective elasticity moduli that makes it possible to express the volume average stresses via the mean deformations. It is shown that the solution of the first SBVP and hence the effective coefficients of elasticity are expressed in terms of the integrals of the Green tensor. The integrals of the Green tensor with respect to one of the variables are called the structural functions. The auxiliary equations, the solutions of which are determined by the functional dependence of the elastic characteristics on the coordinates, have been obtained for these structural functions. It is shown that in the case when the elastic moduli are periodic functions of one, two, or three coordinates, then the structural functions, far from the body boundary, are also periodic functions of the same coordinates. The structural functions are transformed to be equal to zero on the all body border approaching the boundary. In other words, in an inhomogeneous body with a periodic structure, it is possible to distinguish the boundary layer, which separates the regions of periodic values of structural functions from non-periodic ones. The thickness of this layer is of the order of the characteristic size of the periodicity cell. The effective tensors are found due to the structural functions. It is proved that the tensor of effective elastic moduli satisfies all the conditions of symmetry and positive definiteness. The case of an infinite plate with non-uniform thickness is considered in detail.

Dominant role of inverse Cotton-Mouton effect in ultrafast stimulation of magnetization precession in undoped yttrium iron garnet films by 400-nm laser pulses

Magnetization ($\mathbit{M}$) precessions driven by ultrafast laser-induced nonthermal effects are observed in undoped yttrium iron garnet (YIG) films of (100) and (111) orientations using pump-probe time-resolved magneto-optical Kerr effect. The $\mathbit{M}$ precessions show a strong dependence on the polarization direction of linearly polarized pump pulses of 400 nm. In contrast, we can barely observe any $\mathbit{M}$ precession using circularly polarized pump pulses, which indicates that the inverse Faraday effect is negligible. For the case of linear pump polarization, a phenomenological model is introduced, based on the modulation of $\mathbit{M}$ via a modulation of fourth-rank susceptibility tensors by a laser pulse. This allows one to distinguish the contributions of the inverse Cotton-Mouton effect (ICME) from those of the photoinduced magnetic anisotropy (PMA). Using the formula derived from the phenomenological model, we perform the fitting of the polarization-direction-dependent precession phase and amplitude in (100)- and (111)-oriented YIG films. The fitting results reveal that the $\mathbit{M}$-precession excitation originates from a combination effect of ICME and PMA, but the ICME plays the dominant role.

Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data

Dense suspensions of hard particles are important as industrial or environmental materials (e.g. fresh concrete, food, paint or mud). To date, most constitutive models developed to describe them are, explicitly or effectively, ‘fabric evolution models’ based on: (i) a stress rule connecting the macroscopic stress to a second-rank microstructural fabric tensor $\unicode[STIX]{x1D64C}$; and (ii) a closed time-evolution equation for $\unicode[STIX]{x1D64C}$. In dense suspensions, most of the stress comes from short-ranged pairwise steric or lubrication interactions at near-contacts (suitably defined), so a natural choice for $\unicode[STIX]{x1D64C}$ is the deviatoric second moment of the distribution $P(\boldsymbol{p})$ of the near-contact orientations $\boldsymbol{p}$. Here we test directly whether a closed time-evolution equation for such a $\unicode[STIX]{x1D64C}$ can exist, for the case of inertialess non-Brownian hard spheres in a Newtonian solvent. We perform extensive numerical simulations accessing high levels of detail for the evolution of $P(\boldsymbol{p})$ under shear reversal, providing a stringent test for fabric evolution models. We consider a generic class of these models as defined by Hand (J. Fluid Mech., vol. 13, 1962, pp. 33–46) that assumes little as to the micromechanical behaviour of the suspension and is only constrained by frame indifference. Motivated by the smallness of microstructural anisotropies in the dense regime, we start with linear models in this class and successively consider those increasingly nonlinear in $\unicode[STIX]{x1D64C}$. Based on these results, we suggest that no closed fabric evolution model properly describes the dynamics of the fabric tensor under reversal. We attribute this to the fact that, while a second-rank tensor captures reasonably well the microstructure in steady flows, it gives a poor description during significant parts of the microstructural evolution following shear reversal. Specifically, the truncation of $P(\boldsymbol{p})$ at second spherical harmonic (or second-rank tensor) level describes ellipsoidal distributions of near-contact orientations, whereas on reversal we observe distributions that are markedly four-lobed; moreover, ${\dot{P}}(\boldsymbol{p})$ has oblique axes, not collinear with those of $\unicode[STIX]{x1D64C}$ in the shear plane. This structure probably precludes any adequate closure at second-rank level. Instead, our numerical data suggest that closures involving the coupled evolution of both a fabric tensor and a fourth-rank tensor might be reasonably accurate.

Effect of Thermoelastic Characteristics of Components, Shape of Non-Isometric Inclusions, and Their Orientation on Average Stresses in Matrix Structures

The paper presents model calculations on which to predict volume-average external stress under changes of local internal stress in matrix composites with non-isometric inclusions. It is assumed that the rise of local stress owes to different coefficients of linear thermal expansion of non-isometric inclusions and matrix. The inclusions are taken as ellipsoids of rotation (disks, short fibers) and their principal semiaxes as oriented either along three mutually perpendicular directions x, y, and z of a rectangular coordinate system, only along x and y, or only along x. The average stress in the heterogeneous material and its local stress within an individual inclusion are related through a stress concentration operator (fourth rank tensor) for which an explicit expression is derived in a generalized singular approximation of random field theory. The relations obtained for external stress take into account thermoelastic characteristics of the two components as well as inclusion concentrations and orientations in the matrix. The calculation is applied to estimate the average stress along three axes in a composite consisting of an ED-20 epoxy binder and non-isometric copper inclusions.

Thermomechanical modeling of distortional hardening fully coupled with ductile damage under non-proportional loading paths

In order to capture, as accurately as possible, the complex physical effects during texture evolution of high strength materials subject to non-proportional loading paths at high temperature, a macroscale thermo-mechanical model is developed. This model accounts, in the framework of large inelastic strains, for anisotropic thermo-elasto-viscoplasticity with isotropic, kinematic and distortional hardenings strongly coupled with isotropic ductile damage. Following the footsteps of François (2001), induced anisotropy is modeled to predict the behavior under complex non-proportional loading paths. The initial plastic anisotropy and tension-compression asymmetry are considered through two different temperature dependent fourth-rank tensors. The full coupling of thermo-mechanical model with ductile damage is considered based on the total energy equivalence assumption. The proposed constitutive equations are implemented into finite element code ABAQUS/Explicit using appropriate user subroutine VUMAT. The local integration algorithm (at each Gauss point) of the developed model is based on a fully-implicit scheme. Applications have been made to three different materials (aluminum alloy AU4G T4 (2024), titanium alloy Ti-6Al-4V and magnesium alloy AZ31) to demonstrate the predictive capabilities of the proposed model.

Anisotropy in strain gradient elasticity: Simplified models with different forms of internal length and moduli tensors

Abstract Anisotropy of centro-symmetric (first) strain gradient elastic materials is addressed and the role there played by the dual gradient directions (i.e. directions of strain gradient and of double stress lever arm) is investigated. Anisotropy manifests itself not only through the classical fourth-rank elasticity tensor C (21 independent constants) in the form of moduli anisotropy, but also through a sixth-rank elasticity tensor B (171 independent constants) in a unified non-separable form as compound internal length/moduli anisotropy. Depending on the microstructure properties, compound anisotropy may also manifest itself in a twofold separable form through a decoupled tensor B = L C , consisting of a moduli anisotropy attached to C , and an internal length anisotropy attached to a symmetric positive definite second-rank internal length tensor L (6 independent constants). Tensor L confers a tensorial character to the concept of internal length and is the basis of a mutual one-to-one relationship between the dual gradient directions. Indeed, at every point of a material with separable compound anisotropy, a characteristic ellipsoid exists whereby the generic radius and the associated normal to the ellipsoidal surface constitute dual gradient directions. Therefore, at every point, there are at least three mutually orthogonal directions in each of which the dual gradient directions are collinear, but infinite in number when the ellipsoid is rotational or even spherical. With restrictions on dual gradient directions, simplified models with a reduced number of independent constants are derived, namely: i) Gradient symmetric materials which obey a reciprocity principle and generally exhibit a non-separable compound anisotropy (21 + 126 independent constants in total); ii) Materials with separable compound anisotropy, a subclass of gradient symmetric materials (21 + 6 independent constants in total). A typical boundary-value problem for materials with separable compound anisotropy is presented and a way to transform the three governing fourth-order PDEs into six second-order ones is suggested. An application to an Euler–Bernoulli beam is analytically worked out.

Hall viscosity of a chiral two-orbital superconductor at finite temperatures

The Hall viscosity known as the anti-symmetric part of the viscosity fourth-rank tensor. Such dissipationless response which appears for systems with broken time reversal symmetry. We calculate this non-dissipative quantity for a chiral two-orbital superconductor placed in a viscoelastic magnetic field using the linear response theory and apply our calculations to the putative multiband chiral superconductor Sr2RuO4. The chirality origin of a multiband superconductor arises from the interorbital coupling of the superconducting state. This feature leads to the robustness of the Hall viscosity against temperature and impurity effects. We study the temperature effect on the Hall viscosity at the one-loop approximation.