Axis Of Material Symmetry Research Articles

Ultrasonic scattering from polycrystals with transversely isotropic material texture

During laser-based metal additive manufacturing (AM), the melted powder is subjected to rapid cooling and the resulting microstructures often have material texture. In many cases, the texture is uniaxial (i.e., transversely isotropic) such that one symmetry axis defines the material response. Therefore, ultrasonic inspection methods that exploit the scattering from the microstructure are complicated by the resulting texture which affects the coherent propagation and scattering. In this presentation, a generalized approach is described in which the covariance of the elastic modulus tensor for a uniaxial ensemble of cubic crystals is expressed in terms of a fundamental set of constants. These constants are determined for an arbitrary texture from synthetic polycrystals created using DREAM.3D. With this information, calculations for wave velocity, attenuation, and diffuse scattering can be made efficiently for any wave type and propagation direction relative to the material symmetry axis. Results are compared with analytical expressions for simplified cases and then more generalized textures are examined. Finally, prospects for characterization of components created using metal AM are discussed within the context of input data from electron backscatter diffraction measurements. These results are expected to provide insight regarding the inversion of measurement data for texture characterization.

Near simultaneous multiple fragment impacts on aramid fabrics: Effects of velocity, dispersion and time intervals

Protection against fragment impacts is a critical concern across various applications. Conventional standardised methods use single-impact tests as a simplified approach for evaluating material resistance against impacting fragments. An alternative, near simultaneous triple impact test was implemented for testing protective equipment and materials against fragmentation, effectively simulating the impact of a dense cluster of fragments. Aramid fabrics, the most common protective measure against fragments, exhibit up to 13 % reduced ballistic resistance under triple impacts opposed to single impact tests signifying the importance of a multi-hit test method when evaluating protection against fragments.In this article, a finite element model is developed to conduct a detailed analysis of the dynamic interactions within a triple impact test. This analysis decouples and isolates the various contributing factors, including: the impacting velocity, the dispersion between impacts, the impact positions with respect to the main axes of material symmetry and the time intervals between the impacting projectiles. The model describes 15 layers of aramid fabric, in a detailed discrete mesoscale format, subjected to single, dual and triple impacting 1.102 g Fragment Simulating Projectiles (FSP) in different configurations using the finite element package LS-Dyna.As anticipated, penetrability increases when the distance between two impacting projectiles decreases. Additionally, on-axis impacts are more likely to result in perforation due to increased stress wave destructive–constructive interferences. Furthermore, the time interval between impacts is found to play a major role on the target's eventual ballistic performance. The oscillating phase caused by the excitation from the preceding impacting projectile, can amplify or diminish the relative impacting velocity of the subsequent projectile by as much as 40 %, resulting in either an over-perforation with residual velocity 55 % of the initial impacting velocity, or no-perforation with 0 % penetration depth.

Phase and group velocities for shear wave propagation in an incompressible, hyperelastic material with uniaxial stretch

Objective. Determining elastic properties of materials from observations of shear wave propagation is difficult in anisotropic materials because of the complex relations among the propagation direction, shear wave polarizations, and material symmetries. In this study, we derive expressions for the phase velocities of the SH and SV propagation modes as a function of propagation direction in an incompressible, hyperelastic material with uniaxial stretch. Approach. Wave motion is included in the material model by adding incremental, small amplitude motion to the initial, finite deformation. Equations of motion for the SH and SV propagation modes are constructed using the Cauchy stress tensor derived from the strain energy function of the material. Group velocities for the SH and SV propagation modes are derived from the angle-dependent phase velocities. Main results. Sample results are presented for the Arruda–Boyce, Mooney–Rivlin, and Isihara material models using model parameters previously determined in a phantom. Significance. Results for the Mooney–Rivlin and Isihara models demonstrate shear splitting in which the SH and SV propagation modes have unequal group velocities for propagation across the material symmetry axis. In addition, for sufficiently large stretch, the Arruda–Boyce and Isihara material models show cusp structures with triple-valued group velocities for the SV mode at angles of roughly 15° to the material symmetry axis.

A computational framework for meso and macroscale analysis of structural masonry

In this study a computational framework is outlined for modeling the mechanical response of structural masonry at both meso and macroscale. The mesoscale approach accounts for the presence of distinct constituents (i.e., bricks and mortar joints) and their geometric arrangement. A constitutive law with embedded discontinuity, combined with the level-set approach, is used to model the onset and discrete propagation of localized damage in these constituents. The approach is verified against a range of experimental data published in the literature. It is shown that the proposed framework can adequately predict the load-deformation response, as well as the fracture pattern under combined loading conditions. The macroscale approach incorporates the notion of anisotropy parameter whose value depends on the orientation of the principal stress axes in relation to the axes of material symmetry. The material parameters/function appearing in this approach are identified from the ‘virtual data’ generated by a mesoscale analysis of masonry panels subjected to biaxial tension–compression at different orientations of the bed joint. Thus, the mesoscale considerations serve as a bridge for upscaling to the macrolevel.

Transient Green's functions of a buried pulse in a constrained transversely isotropic half-space

In this paper, transient surface Green's functions (GFs) excited by a general dynamic pulse buried in a constrained transversely isotropic (TI) half-space are derived in analytical forms. The constrained transverse isotropy is defined by an explicit relation that expresses one of the five elastic constants in terms of the other four independent constants so that no coupled P-SV wave exists. Use is made of the solutions available for general TI materials in Hankel–Laplace transformed domain in order to extract the corresponding response functions for the constrained material. Under this constraint, the coupled fourth-order wave operator, which represents the convolved motion of the quasi-longitudinal and quasi-transverse waves, is decoupled into two separated second-order wave operators describing the motion of the pure dilatational (P) and pure shear (SV) waves. The involved Laplace transform inversion is carried out using the novel Cagniard scheme so that the final displacement GFs are expressed in terms of the elementary line integrals over a finite interval. The GF solution is valid for the entire range of the constrained TI material including material isotropy as its special case, and clearly illustrates the disturbances due to different stress waves including, P, SV, SH, Rayleigh and diffracted SP waves. Moreover, the GFs show that in this constrained TI material, the compressional and horizontally polarized waves, excited by a source located at an arbitrary depth, propagate towards the surface with direction-dependent velocities. The explicit formulas for these velocities of the waves are obtained in terms of the inclination angle of the wave normal with respect to the axis of material symmetry. On the other hand, the vertically polarized wave and the diffracted SP wave propagate with eigen velocity, independent from the direction of the propagation. Two critical epicentral distances, at which mode conversion in ray diagram occurs, are detected for this constrained TI material. One critical distance marks the phenomena of total reflection of SV-waves where the incident SV-wave is totally reflected and the diffracted SP-wave travels along the free surface. The other marks the distance where the SH- and SP-waves become coalescent, and furthermore, before this epicentral distance the SH-wave is faster, and after this distance, the SP-wave is faster. To show the relevance of this constrained TI material in engineering and material modeling applications and its relation to the physics of TI materials, we also demonstrate that, in addition to its mathematical simplification, this reduced anisotropy can well approximate the elastic coefficients of some real TI materials, particularly those encountered in exploration geophysics and seismology.

Generalized solution to the anisotropic Mandel's problem

SummaryExisting solutions to Mandel's problem focus on isotropic, transversely isotropic, and orthotropic materials, the last two of which have one of the material symmetry axes coincide with the vertical loading direction. The classical plane strain condition holds for all these cases. In this work, analytical solution to Mandel's problem with the most general matrix anisotropy is presented. This newly derived analytical solution for fully anisotropic materials has all the three nonzero shear strains. Warping occurs in the cross sections, and a generalized plane strain condition is fulfilled. This solution can be applied to transversely isotropic and orthotropic materials whose material symmetry axes are not aligned with the vertical loading direction. It is the first analytical poroelastic solution considering mechanical general anisotropy of elasticity. The solution captures the effects of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction on the responses of pore pressure, stress, strain, and displacement. It can be used to match, calibrate, and simulate experimental results to estimate anisotropic poromechanical parameters. This generalized solution is capable of reproducing the existing solutions as special cases. As an application, the solution is used to study the responses of transversely isotropic and orthotropic materials whose symmetry axes are not aligned with the vertical loading direction. Examples on anisotropic shale rocks show that the effects of material anisotropy are significant. Mandel‐Cryer's effects are highly impacted by the degree of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction.

Body waves propagation in a fluid-saturated transversely isotropic poroelastic solid with a potential method

This study is focused on the propagation of plane harmonic body waves in a transversely isotropic linear poroelastic fluid-saturated medium, where the material symmetry axes for both solid and fluid are coincide. Simplified formulation is considered as the framework. A set of two scalar potential functions is employed to decouple the coupled fluid continuity equation and equations of motion. The velocities and corresponding attenuation coefficients of both longitudinal and transverse waves are extracted from the presented acoustic equations for the body waves. To show the validity of the analytical solution given in this paper, degeneration to the case of a single phase transversely isotropic and consequently isotropic solid is presented to provide interesting comparisons with the solutions reported in the literatures. The incompressible solid and fluid are degenerated from the general slowness obtained in this study for some special cases. In addition, the effects of mechanical and hydraulic parameters of materials on the velocity of propagation and attenuation coefficient of the waves are investigated in more detail. To this end, various synthetic poroelastic transversely isotropic materials are defined and the dependency of wave motion to these parameters is illustrated by plotting the graphs.

Analysis of multiple shear wave modes in a nonlinear soft solid: Experiments and finite element simulations with a tilted acoustic radiation force

Tissue nonlinearity is conventionally measured in shear wave elastography by studying the change in wave speed caused by the tissue deformation, generally known as the acoustoelastic effect. However, these measurements have mainly focused on the excitation and detection of one specific shear mode, while it is theoretically known that the analysis of multiple wave modes offers more information about tissue material properties that can potentially be used to refine disease diagnosis. This work demonstrated proof of concept using experiments and finite element simulations in a uniaxially stretched phantom by tilting the acoustic radiation force excitation axis with respect to the material's symmetry axis. Using this unique set-up, we were able to visualize two propagating shear wave modes across the stretch direction for stretches larger than 140%. Complementary simulations were performed using material parameters determined from mechanical testing, which enabled us to convert the observed shear wave behavior into a correct representative constitutive law for the phantom material, i.e. the Isihara model. This demonstrates the potential of measuring shear wave propagation in combination with shear wave modeling in complex materials as a non-invasive alternative for mechanical testing.

Tractable calculation of the Green’s tensor for shear wave propagation in an incompressible, transversely isotropic material

Assessing material properties from observations of shear wave propagation following an acoustic radiation force impulse (ARFI) excitation is difficult in anisotropic materials because of the complex relations among the propagation direction, shear wave polarizations, and material symmetries. In this paper, we describe a method to calculate shear wave signals using Green’s tensor methods in an incompressible, transversely isotropic (TI) material characterized by three material parameters. The Green’s tensor is written as the sum of an analytic expression for the SH propagation mode, and an integral expression for the SV propagation mode that can be evaluated by interpolation within precomputed integral functions with an efficiency comparable to the evaluation of a closed-form expression. By using parametrized integral functions, the number of required numerical integrations is reduced by a factor of 102–109 depending on the specific problem under consideration. Results are presented for the case of a point source positioned at the origin and a tall Gaussian source similar to an ARFI excitation. For an experimental configuration with a tilted material symmetry axis, results show that shear wave signals exhibit structures that are sufficiently complex to allow measurement of all three material parameters that characterize an incompressible, TI material.

Study of a rhombohedral graphite X-ray filter using the sphere-in-contact model

We have studied with the sphere-in-contact model the structure of rhombohedral graphite and found that there are molecular channels 22.7° with respect to the material symmetry axis. We show via theoretical simulations with the Laue equation, Kramers equation and Duane-Hunt law that X-ray radiation (λ < 1.34 Å) that is traveling along these molecular channels would have a larger mean-free-path. This suggests that stacks of rhombohedral graphite can become effective filters for X-ray sources, such as the ones used in Laue photography and other X-ray techniques that use Bremsstrahlung.

Elastodynamics of a multilayered transversely isotropic half-space due to the rigid motion of foundation

A transversely isotropic multilayered half-space is considered as a support of a massless circular rigid foundation undergoing forced vibrations. The problem formulation adopts the axes of material symmetry of layers with the half-space aligned depth-wise. The problem solution involves two coupled wave equations accompanied by mixed boundary conditions. Employing the influence functions, which contain the effects of the entire domain, i.e. reflecting and refracting phenomena as a consequence of the inhomogeneity of media, the governing equations reduce to a set of Fredholm integral equations of the second kind, which are then solved numerically. To illustrate the method’s accuracy, a comparison between the result of this study and the existing solutions for the vibration of a foundation on a layered isotropic half-space is given. Also, the result of the vibration of the disc on a transversely isotropic half-space is determined and compared to the available solution in the literature To explore the effect of stratification as well as material anisotropy on wave motion, dispersion curves corresponding to surface waves are shown. These curves indicate the existence of limiting values for both small and large values of wavenumbers as compared to layers’ thicknesses. Also, exploring the effect of various material properties (e.g., the negative Poisson ratio) and different layers’ arrangements, it is found that the impedance functions of the multilayered medium are closely related to the frequency of excitation while showing almost no sensitivity to properties of the half-space and, in particular, for rocking motion.

Material Spin and Finite-Strain Hypo-Elasticity for Two-Dimensional Orthotropic Media

A constitutive material spin tensor in the case of purely elastic finite-strain deformation is introduced for a two-dimensional orthotropic media using the minimizing principle applied to obtain the reloaded configuration of the material volume. This material spin explains the rotation of the orthonormal vector frame which coincides with the material symmetry axes in the initial configuration of the material volume and uniquely corresponds to a set of these axes in the current configuration although it does not coincide with the latter. The given definition is followed by the exact expression which includes the deformation gradient tensor, unit vectors of the initial material anisotropy axes and their axial parameters. This definition allows obtaining a new variant of decomposing any elastic finite-strain motion onto rigid and deformational parts and introducing the material corotational rate. The latter is used for the formulation of the anisotropic rate-type elastic law in the current configuration based on the strain measure which does not belong to the Seth-Hill family. For isotropic as well as for tetragonal media, the introduced material rotation tensor coincides with the rotation tensor from the polar decomposition of a deformation gradient.

Three-dimensional dynamic fracture in a transversely isotropic thermoelastic solid: Framework for study

A semi-infinite plane crack is at rest in an unbounded, transversely isotropic, thermoelastic solid. Point forces are applied to the crack faces, and translated with constant, subcritical speed at right angles to the initial crack edge. Fracture occurs, and a dynamic steady state is achieved: The crack edge is no longer rectilinear, but extension and translation speeds are identical. The crack plane contains the axis of material symmetry lies in that plane, and the initial crack edge is not aligned with that axis. An analytical three-dimensional solution is obtained, and a criterion for fracture based on dynamic energy release rate imposed, with kinetic energy included. A nonlinear differential equation for crack edge location and constraint equations result, and together form a framework for study of crack and contact zone geometry.

GENERAL SCALARIZATION METHOD OF DYNAMIC ELASTIC FIELDS IN TRANSVERSALLY ISOTROPIC MEDIA AND ITS NEW APPLICATIONS

Introduction. An efficient technique of tensor field scalarization is successfully used while investigating tensor elastic fields of displacements, stresses and deformations in the layered structures of different materials, including transversally isotropic composites. These fields can be expressed through the scalar potentials corresponding to the quasi-longitudinal, quasi-transverse, and transverse-only waves. Such scalarization is possible if the objects under consideration are tensors relating to the subgroup of general coordinate conversions, when the local affine basis has one invariant vector that coincides with the material symmetry axis of the material. At this, the known papers consider structures where this vector coincides with the normal to the boundary between layers. However, other cases of the mutual arrangement of the material symmetry axis of the material and the boundaries between layers are of interest on the practical side.Materials and Methods. The work objective is further development of the scalarization method application in the boundary value problems of the dynamic elasticity theory for the cases of an arbitrary arrangement of the material symmetry axis relative to the boundary between layers. The present research and methodological apparatus are developed through the general technique of scalarization of the dynamic elastic fields of displacements, stresses and strains in the transversally isotropic media.Research Results. New design ratios for the determination of the displacement fields, stresses and deformations in the transversally isotropic media are obtained for the cases of an arbitrary arrangement of the material symmetry axes of the layer materials with respect to the boundaries between layers. Discussion and Conclusions. The present research and methodological apparatus are successfully used in determining the stress-strain state in the layered structures of transversally isotropic materials, and in analyzing the diagnosis results of the state of the plane-layered and layered cylindrical structures under operation.

Application of Mori–Tanaka method in 3–1 porous piezoelectric medium of crystal class 6

We consider a piezoelectric medium of hexagonal crystal system of class 6 containing a uniform distribution of circular cylindrical inclusions aligned with the axis of material symmetry. We determine the piezoelectric Eshelby tensor explicitly. In the case of cylindrical holes, we use the Eshelby tensor together with the multiscale Mori–Tanaka Method to obtain analytical formulae for the effective electroelastic properties of the homogenized medium. These formulae depend upon both the electroelastic properties of the matrix material and the volume fraction of the cylindrical holes. Using electroelastic properties reported in the literature, we obtain graphs of the effective properties of the porous medium versus the volume fraction of the pores and show that these effective electroelastic properties decrease for increasing porous volume fraction, as expected. In particular, for the case of the crystal subclass 6 mm, the results obtained via Mori–Tanaka Method agree well with results obtained via Method of Asymptotic Homogenization and Finite Element Method. This work is important in the evaluation of the effective electroelastic properties of heterogeneous solids with hierarchical structures, such as bones.

Unit cell simulations and porous plasticity modelling for strongly anisotropic FCC metals

Abstract The macroscopic behaviour of anisotropic porous solids made from an aggregate of spherical voids embedded in a plastically anisotropic matrix material is investigated by means of unit cell simulations. Plastic yielding of the polycrystalline matrix is governed by the anisotropic yield criterion Yld2004-18p. Generic texture components for face-centred cubic crystals resembling those that typically emerge during rolling and annealing processes are applied in the study. A numerical method for systematic prescription of external stress states is presented and employed in the unit cell calculations. To preclude shear effects in the unit cell model, the material symmetry axes are restricted to coincide with the principal stress directions. This excludes the possibility to properly study the ductile failure mechanism and the current work is thus mainly concerned with the void growth phase. Various stress states ranging from biaxial tension to highly constrained regions in the vicinity of crack tips are employed in the study. The numerical results demonstrate that the matrix anisotropy has a marked effect on the unit cell response, both in terms of void growth and stress-strain curves. Furthermore, the void shape evolves quite differently depending upon the direction of the major principal stress relative to the material axes. A heuristic extension of the Gurson model that incorporates matrix plastic anisotropy is presented and subsequently used to describe the constitutive behaviour of the porous ductile solid. Numerical data from the unit cell analyses are used as target curves in the calibration process of the porous plasticity model. A sequential least-square optimization procedure is invoked to minimize the overall discrepancy between the unit cell calculations and the homogenized response of the plastically anisotropic porous solid for all the imposed stress states. The anisotropic porous plasticity model demonstrates predictive capabilities for the range of stress states covered in this study.

Elastic fields in rotating transversely isotropic media

Representation of elastic fields in terms of scalar functions, which permit reducing the problems of determining these fields to determining scalar potentials, are generalized to the case of transversely isotropic media rotating at a constant angular velocity. Relations for calculating the parameters of surface acoustic waves (SAW) propagating in a rotating transversely isotropic halfspace with various directions of the medium material symmetry axis with respect to the half-space surface are given.

Acoustic Radiation Force Impulse (ARFI)-Induced Peak Displacements Reflect Degree of Anisotropy in Transversely Isotropic Elastic Materials.

In transversely isotropic (TI) materials, mechanical properties (Young's modulus, shear modulus, and Poisson's ratio) are different along versus across the axis of symmetry (AoS). In this work, the feasibility of interrogating such directional mechanical property differences using acoustic radiation force impulse (ARFI) imaging is investigated. We herein test the hypotheses that 1) ARFI-induced peak displacements (PDs) vary with TI material orientations when an asymmetrical ARFI excitation point spread function (PSF) is used, but not when a symmetrical ARFI PSF is employed; and 2) the ratio of PDs induced with the long axis of an asymmetrical ARFI PSF oriented along versus across the material's AoS is related to the degree of anisotropy of the material. These hypotheses were tested in silico using finite element method (FEM) models and Field II. ARFI excitations had F/1.5, 3, 4, or 5 focal configurations, with the F/1.5 and F/5 cases having the most asymmetrical and symmetrical PSFs at the focal depth, respectively. These excitations were implemented for ARFI imaging in 52 different simulated TI materials with varying degrees of anisotropy, and the ratio of ARFI-induced PDs was calculated. The change in the ratio of PDs with respect to the anisotropy of the materials was highest for the F/1.5, indicating that PD was most strongly impacted by the material orientation when the ARFI excitation was the most asymmetrical. On the contrary, the ratio of PDs did not depend on the anisotropy of the material for the F/5 ARFI excitation, suggesting that PD did not depend on material orientation when the ARFI excitation was symmetrical. Finally, the ratio of PDs achieved using asymmetrical ARFI PSF reflected the degree of anisotropy in TI materials. These results support that symmetrical ARFI focal configurations are desirable when the orientation of the ARFI excitation to the AoS is not specifically known and measurement standardization is important, such as for longitudinal or cross-sectional studies of anisotropic organs. However, asymmetrical focal configurations are useful for exploiting anisotropy, which may be diagnostically relevant. Feasibility for future experimental implementation is demonstrated by simulating ultrasonic displacement tracking and by varying the ARF duration.

Bi-material transversely isotropic half-space containing penny-shaped crack under time-harmonic horizontal loads

An analytical solution is presented for a two-layer transversely isotropic linear elastic half-space containing a disc-shaped crack placed at the interface of the layers. The axes of material symmetry of the top layer and the lower half-space are assumed to be depth-wise and perpendicular to the surfaces of the crack, which are parallel to the free surface of the half-space. The crack surfaces are affected by time-harmonic arbitrary distribution self-equilibrium horizontal tractions, which is related to mode II of fracture. The coupled equations of motion in each region of the domain are uncoupled with the use of two scalar potential functions, and the governing equations for the potential functions are solved with the use of Fourier expansion and Hankel integral transforms. Because of integral transforms used in this paper, the mixed boundary value problem involved here are changed to a set of coupled dual integral equations, which are analytically solved for the static case of bi-material full-space as a special case of two-layer half-space. For the dynamic problem, the integral equations are solved numerically with the aid of the finite element method. The solution is degenerated for the isotropic case, which itself is new in the literature. In addition, some numerical results are provided to show how the degree of anisotropy and thickness of the top layer affect the problem.

A Potential Method for Body and Surface Wave Propagation in Transversely Isotropic Half- and Full-Spaces

The problem of propagation of plane wave including body and surface waves propagating in a transversely isotropic half-space with a depth-wise axis of material symmetry is investigated in details. Using the advantage of representation of displacement fields in terms of two complete scalar potential functions, the coupled equations of motion are uncoupled and reduced to two independent equations for potential functions. In this paper, the secular equations for determination of body and surface wave velocities are derived in terms of both elasticity coefficients and the direction of propagation. In particular, the longitudinal, transverse and Rayleigh wave velocities are determined in explicit forms. It is also shown that in transversely isotropic materials, a Rayleigh wave may propagate in different manner from that of isotropic materials. Some numerical results for synthetic transversely isotropic materials are also illustrated to show the behavior of wave motion due to anisotropic nature of the problem.