Voigt Notation Research Articles

Parameterization of anisotropic media by A-parameters

AbstractMost common parameterization of anisotropic media is by twenty one independent elements aijkl of the density-normalized stiffness tensor or by twenty one independent elements Aαβ of the density-normalized matrix of elastic parameters in the Voigt notation. These parameters are commonly of significantly different sizes, are dimensional, in (km/s)2, often appear in combinations. We are offering an alternative parameterization by twenty one A-parameters (anisotropic parameters), which removes the mentioned disadvantages and possesses some additional useful properties. For example, axes or planes of coordinate systems, in which A-parameters are defined, need not be related to symmetry axes or planes of the considered anisotropy symmetry as required in other similar parameterizations. In combination with the first-order weak-anisotropy approximation, in which anisotropy is considered as the first-order perturbation of reference isotropy, parameterization by A-parameters yields insight into the role of individual A-parameters in the wave propagation problems. For example, it turns out that in the first-order weak-anisotropy approximation, P- and S-wave velocities are each controlled by fifteen A-parameters. A set of six of them appears only in the expression for P-wave velocity, a set of other six A-parameters appears only in S-waves velocity expressions. Remaining set of nine A-parameters is common for both waves. We present transformation of A-parameters, analogue to Bond transformation, and useful formulae for the weak-anisotropy approximation for anisotropy of any symmetry and arbitrary tilt.

MATERIAL CONSTANTS OF TRIGONAL SYNGONY PYROPIEZOELECTRICS UNDER THE INFLUENCE OF PHYSICAL FIELDS OF DIFFERENT NATURE

The article studies the previously proposed model of a prestressed thermoelectroelastic medium of class 3m trigonal syngony under the action of initial mechanical stresses, an electrostatic field at a given temperature. The model is based on linearized constitutive relations constructed within the framework of sequential linearization of nonlinear equations of continuous medium electrodynamics. Linearization is carried out within the framework of superimposing small deformations on final ones with preservation of high-order terms in the equation of state. When constructing a model based on the use of linearized constitutive relations, it is taken into account that it is impossible to use the Voigt notation, which is usual for the natural state, in this case. Within the framework of this model, the behavior of the material constants of a thermoelectroelastic medium of class 3m trigonal syngony is studied. It is shown that, unlike electrical and mechanical effects, the thermal factor linearly affects the constants: preheating leads to an increase in the elastic constants, cooling – to their decrease. The effect on the piezoelectric constants is also linear, but more complex: some constants increase, others decrease. In this case, the possibility of using the Voigt notation, which is usual for the natural state, is preserved. Mechanical and electrical effects lead to the separation of constants coinciding in the natural state; in this case, the use of the Voigt notation, which is usual for the natural state, is impossible. The decomposed constants depend nonlinearly on mechanical stresses, but linearly on the electrostatic field. The nature and magnitude of the change in the constants depend both on the type of mechanical action and on the polarity of the electrostatic field. It should be noted that in this work, the effect of minor thermal effects was investigated. The effect of high initial temperatures significantly changes the properties of the material, but goes beyond the linearized theory and requires the use of nonlinear relationships.

Study on crack propagation by eigenvalue/eigenvector of atomic elastic stiffness: Interaction with Ni/Co/Zr/Mo obstacles in Fe

Molecular dynamics simulations are implemented on the interaction between mode I crack in Fe single crystal and cylindrical Ni/Co/Zr/Mo obstacles, substituting Fe atoms with those elements in a periodic slab cell. The crack penetrates the Ni and Co obstacles while it bypasses the Mo one by the slip and void nucleation at the Fe-Mo interface. The crack also cuts in the Zr obstacle; however, it is caught by the amorphized Zr atoms in Fe matrix, showing remarkable resistance against crack propagation. Then these behaviors are discussed with the eigenvalue/vector of the atomic elastic stiffness coefficient, Bαij=Δσiα/Δεj, where σiα and εj are the atomic stress and strain in the Voigt notation. Atoms in the Ni, Co, and Zr obstacles show lower eigenvalue, ηα(1), the solution of BαijΔεj=ηαΔεi, against Fe matrix while those in the Mo obstacle have higher one than Fe. That is, the crack cuts in the “soft” Ni, Co and Zr obstacles and bypasses the “hard” Mo one. The unstable modes of crack tip, slip in the obstacles and at the Fe-Mo interface are visualized from the maximum shear direction of the eigenvector of {Δε1,Δε2,･･･,Δγ6}T={Δεxx,Δεyy,･･･,Δγxy}T for unstable ηα(1)< 0 atoms.

A component-free Lagrangian finite element formulation for large strain elastodynamics

Standard finite element formulation and implementation in solid dynamics at large strains usually relies upon and indicial-tensor Voigt notation to factorized the weighting functions and take advantage of the symmetric structure of the algebraic objects involved. In the present work, a novel component-free approach, where no reference to a basis, axes or components is made, implied or required, is adopted for the finite element formulation. Under this approach, the factorisation of the weighting function and also of the increment of the displacement field, can be performed by means of component-free operations avoiding both the use of any index notation and the subsequent reorganisation in matrix Voigt form. This new approach leads to a straightforward implementation of the formulation where only vectors and second order tensors in {mathbb {R}}^3 are required. The proposed formulation is as accurate as the standard Voigt based finite element method however is more efficient, concise, transparent and easy to implement.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin\u2013Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

A finite element model of the Japanese koto constructed from computed tomography scans.

This paper presents a method for using computed tomography (CT) scan images of a Japanese koto as the basis of a mesh of a finite element model. The CT images gave access to previously inaccessible internal geometry. Images of grain structure and orientation in all three dimensions of the koto's Paulownia wood was important to the model's development. Obtaining the wood's physical properties for the Voigt notation and using curvilinear coordinates to rotate it to account for the grain was also a critical step. Following the model construction, studies were conducted to investigate the koto's fundamental vibrational behavior. First, the model predicted the koto's mode shapes and frequencies. Simulations in the frequency and time domains also identified unreported aspects of the koto's acoustics. Second, five physical experiments were conducted including transducer studies, Chladni patterns, an acoustic camera tests, and the response of the koto when played. The acoustic camera in particular gave insights into the koto's sound radiation. The combined results advance the limited knowledge of the koto including identifying 100 Hz as the first eigenmode and 85 Hz as a major air mode.

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.

Turbulent flows as generalized Kelvin–Voigt materials: Modeling and analysis

We perform a new modeling procedure for a 3D turbulent fluid, evolving towards a statistical equilibrium. This will result to add to the equations for the mean field (v,p) the term −α∇⋅(ℓ(x)Dvt), which is of the Kelvin–Voigt form, where the Prandtl mixing length ℓ=ℓ(x) is not constant and vanishes at the solid walls. We get estimates for mean velocity v in Lt∞Hx1∩Wt1,2Hx1∕2, that allow us to prove the existence and uniqueness of regular-weak solutions (v,p) to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to consider Reynolds averaged equations and pass to the limit in the quadratic source term, in the equation for the turbulent kinetic energy k. This yields the existence of a weak solution to the corresponding Navier–Stokes Turbulent Kinetic Energy system satisfied by (v,p,k).

Density Functional Study on Enhancement of Modulus of Confined Fluid in Nanopores

Elastic modulus in fluid is a crucial thermodynamic property due to its close relation to the viscoelastic behaviors in dynamics and the pore size analysis in porous materials. Especially, elasticity in nanoconfined fluid, which exhibits significant deviation from that of bulk fluid, has gained a multitude of applications in industrial and engineering fields. In this work, general expressions of bulk and shear moduli for inhomogeneous fluids have been derived based on Hooke’s law. In the bulk phase, the obtained moduli reduce exactly to those obtained from thermodynamic fluctuation theory. In addition, both moduli have been expressed as functionals of fluid density functions. Besides, the Cauchy relation and Voigt notation for inhomogeneous elasticity have been first obtained. The validity of the obtained moduli has been verified through their comparison with grand canonical Monte Carlo simulation. Our further calculations suggest that the bulk modulus of confined fluid is inhomogeneous across the pore an...

Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

A locking-free face-centred finite volume (FCFV) method for linear elastostatics

A face-centred finite volume (FCFV) method is proposed for linear elastostatic problems. The FCFV is a mixed hybrid formulation, featuring a system of first-order equations, that defines the unknowns on the faces (edges in two dimensions) of the mesh cells. The symmetry of the stress tensor is strongly enforced using the well-known Voigt notation and the displacement and stress fields inside each cell are obtained by means of explicit formulas. The resulting FCFV method is robust and locking-free in the nearly incompressible limit. Numerical experiments in two and three dimensions show optimal convergence of the displacement and the stress fields without any reconstruction. Moreover, the accuracy of the FCFV method is not sensitive to mesh distortion and stretching. Classical benchmark tests including Kirch’s plate and Cook’s membrane problems in two dimensions as well as three dimensional problems involving shear phenomenons, pressurised thin shells and complex geometries are presented to show the capability and potential of the proposed methodology.

A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor

This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element post-process procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

SummaryThe first superconvergent hybridisable discontinuous Galerkin method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element by element a superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent displacement field for low‐order approximations. The resulting method is robust and locking‐free in the nearly incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its superconvergent properties in two and three dimensions and for different element types.

Local lattice instability analysis on mode I crack tip in [formula omitted]-SiC: Characteristics in binary covalent crystal

Local lattice instability analysis based on the atomic elastic stiffness (AES), Bijα=Δσiα/Δεj (i,j=1–6 in Voigt notation), is applied to mode I crack tip in covalent binary crystal β-SiC of Tersoff interatomic potential. The application limit of the AES is disclosed at the discontinuous local energy surface of disordered broken bonds in the discrete binary system; however, we also captured interesting results with the 1st eigenvalue ηα(1) of Bijα as follows, (1) the atomic stress of C atoms coincides with the stress singularity in the linear fracture mechanics, when the minimum eigenvalue of C atom shows the first sudden drop or the first instability, (2) but the crack doesn’t propagate nor there is no remarkable change in the stress–strain curve, (3) bond breaking and reconstruction occurs in the vicinity of crack tip at the second instability in the eigenvalue, (4) C atoms in the disordered configuration show extraordinary large negative eigenvalue (possibly due to discontinuity of energy surface), while Si atoms show third drop in the eigenvalue just before unstable crack propagation, (5) linear fracture mechanics fails to predict the stress singularity at the crack propagation, (6) the principal axes of the eigenvector of ηα(1)<0 atoms reveal the deformation mode for local slip and cracking on the multiple (111) planes in the unstable domain at the forefront of the propagating crack.

A viscoelastic constitutive law for hydrogels

A viscoelastic constitutive law is developed for a hydrogel to encompass deviatoric deformations where the viscosity of the solvent contributes to the deviatoric stress of the system. The law is obtained in Kelvin–Voigt form by adding together the stress in the polymer network, the osmotic and solvent pressures and the viscous stress induced in the solvent. Transient behavior of the hydrogel is presented in terms of an example where transport of the solvent through the gel and viscosity due to deviatoric deformation interact to enable response that would otherwise be precluded, showing the significance of including the deviatoric viscosity. In addition, a simple relaxation problem is solved for a swollen hydrogel where volume of the hydrogel is preserved, namely small strain shear as induced in a rheometer.

Analytic calculation of phase and group velocities of P-waves in orthorhombic media

We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

BFree Finite Element Approach for Saturated Porous Media: Consolidation

TheBfree finite element approach is applied to the governing equations describing the consolidation process in saturated poroelastic medium with intrinsically incompressible solid and fluid phases. Under this approach, where Voigt notation is avoided, the finite element equilibrium equations and the linearization of the coupled governing equations are fully derived using tensor algebra. In order to assess theBfree approach for the consolidation equations, direct comparison with analytical solution of the response of a homogeneous and isotropic water-saturated poroelastic finite column under harmonic load is presented. The results illustrate the capability of this finite element approach of reproducing accurately the response of quasistatic phenomena in a saturated porous medium.

3D numerical modelling of bit‐rock fracture mechanisms in percussive drilling with a multiple‐button bit

SUMMARY This paper considers numerical modelling of rock fracture induced by dynamic bit–rock interaction in percussive drilling. The work presented here extends the author's earlier research on the topic from the axisymmetric case to 3D case. The numerical method for modelling rock fracture includes a constitutive model for rock and a contact mechanics-based technique to simulate the bit–rock interaction. The constitutive model is based on a combination of the recent viscoplastic consistency model, the isotropic damage concept and a parabolic compression cap. This model is improved here from its earlier state by calibrating the softening laws using fracture energies GIc and GIIc in tension and compression, respectively. Moreover, the viscosity modulus in tension is calibrated based on the dynamic Brazilian disc test. With these enhancements, the developed method is applied to 3D case of the bit–rock interaction problem assuming one symmetry plane. Single impact with single and multiple-button bits is simulated. In the latter case, an initial borehole is modelled in order to simulate the usual in-situ drilling conditions. The different failure types observed in the experiments as well as the interaction between the buttons resulting in chipping are realistically captured in the simulations. Copyright © 2012 John Wiley & Sons, Ltd.

Sequential multiscale modelling of SiC/Al nanocomposites reinforced with WS2nanoparticles under static loading

The mechanical behavior of SiC/Al nanocomposites reinforced with inorganic fullerene IF-${\mathrm{WS}}_{2}$ nanoparticles has been studied under static loading conditions. Density functional theory calculations, in conjunction with the finite strain method, have been undertaken to provide a complete set of elastic constants for defining the transversely isotropic behavior of a layered structure of ${\mathrm{WS}}_{2}$ bulk. The values of the elastic stiffness coefficients predicted by density functional theory are summarized in Voigt notation as ${C}_{11}$ = 236 GPa, ${C}_{12}$ = 53 GPa, ${C}_{13}=92$ GPa, ${C}_{33}$ = 42 GPa, and ${C}_{44}$ = 12 GPa. These results are used within the finite element method to yield a computationally efficient model that links static properties of inorganic fullerenes at the atomistic level with the aggregate behavior of the entire nanocomposite described as a continuous medium. The latter involves a finite element model of a unit cell of the nanocomposite, represented as an intersecting simple cubic structure. The resulting elastic properties, which have cubic symmetry, are then used in conjunction with the Voigt-Reuss-Hill model to obtain properties of the isotropic aggregate. The resulting computed Young's modulus, Poisson's ratio, and shear modulus of the SiC/Al nanocomposite reinforced with WS2 nanoparticles are in a good agreement with analytical predictions.

Molecular dynamics study on atomic elastic stiffness in Si under tension: homogenization by external loading and its limit

As a series of studies that discuss the onset of inelastic deformation based on atomic elastic stiffness (AES), we investigated the AES in silicon by the Tersoff interatomic potential. For a comprehensive discussion including the effect of structural inhomogeneity by surface and grain boundaries, we performed tensile simulations on bulk/nanowire of Si single crystal, laminate bulk/bamboo nanowire with Σ5 twist grain boundary under a very low temperature (T = 1 K). Not only the stress–strain response, but also the AESs at each atom point, , were evaluated numerically by (Voigt notation) against local strain perturbation. The deviation of vanishes/diminishes by tension both in the homogeneous case of bulk perfect lattice and inhomogeneous ones with surface and grain boundaries; however, there is a certain limit for the homogenization. That is, the subtle deviation of AES in the perfect bulk vanishes by tension but it increases again like an onset of resonance, showing precursor stress decrease just before the unstable stress drop. In the inhomogeneous cases, we demonstrated that the near-zero AESs at the initial structural defects, e.g. surface or grain boundary, do not change but the positive AESs of other stable atoms approach zero by tension. When these distributions overlap each other, the standard deviation of AES increases again as if it were the first homogenization limit. However, the real homogenization starts at that point; that is, the AES distribution changes its shape to have a single peak at the border, suggesting that the difference of initial defects and other stable part vanishes before the system instability.