Axis Of Material Symmetry Research Articles

Determination of the symmetries of an experimentally determined stiffness tensor: Application to acoustic measurements

For most materials, the symmetry group is known a priori and deduced from the realization process. This allows many simplifications for the measurements of the stiffness tensor. We deal here with the case where the symmetry is a priori unknown, as for biological or geological materials, or when the process makes the material symmetry axis uncertain (some composites, monocrystals). The measurements are then more complicated and the raw stiffness tensor obtained does not exhibit any symmetry in the Voigt's matrical form, as it is expressed in the arbitrarily chosen specimen's base.A complete ultrasonic measurement of this stiffness tensor from redundant measurements is proposed. In a second time, we show how to make a plane symmetry pole figure able to give visual information about the quasi-symmetries of a raw stiffness tensor determined by any measurement method. Finally we introduce the concept of distance from a raw stiffness tensor to one of the eight symmetry classes available for a stiffness tensor. The method provides the nearest tensor (to the raw stiffness tensor) possessing a chosen symmetry class, with its associated natural symmetry base.

Guided elastic waves in orthotropic surface layers

Propagation of guided waves along the interface of an elastic surface layer of uniform thickness overlying an elastic homogeneous half-space is examined. The structure is made of materials with arbitrary strain energy functions. To gain understanding of the propagation characteristics and their dependence on the elastic constants and mass densities, the mathematically tractable case of material orthotropy is considered. For propagation along a material axis of symmetry, the dispersion equation is obtained in explicit form when the axes of symmetry of the two materials coincide and one of them is normal to the plane separating the surface layer from the underlying half-space. Analysis of the dispersion equation reveals the propagation characteristics of interfacial waves and their dependence on the material parameters. Propagation occurs either in single or multiple modes, depending on the material parameters of both the surface layer and the underlying half-space. The low-frequency phase speed is obtained from the dispersion equation in terms of wavelength and layer thickness, elastic constants and mass densities. The influence of the two materials on phase speed as it deviates from its value in an orthotropic half-space is, thus, given explicitly. Parameter conditions are defined under which guided waves are not allowed to propagate in certain frequency regimes. Numerical results complement the analysis.

Optimal determination of the material symmetry axes and associated elasticity tensor from ultrasonic velocity data

A simultaneous identification of the angular parallax locating the higher symmetry coordinate system and the associated optimal stiffness tensor from wave speed measurements of obliquely ultrasonic bulk waves in an arbitrarily oriented coordinate system is presented. The property used in classifying a material with regard to its elastic symmetry is the existence and the number of planes of reflective or mirror symmetry. That leads to considering the problem of determining the symmetry class and the directions of the elements of symmetry. To consider the uncertainties of the experimental data, the wave speed measurements are only used to determine the symmetry frames and the optimal stiffness tensor. The proposed inverse propagation algorithm consists of minimizing a functional where the unknowns are the elasticity constants and the Euler angles between the geometric coordinate system and the frame of higher symmetry. Stability of the used least-square algorithm to the initial guesses and to the noise in the wave speed measurements is shown by using simulated data for materials with the most general anisotropy. The applicability of the method is demonstrated using experimental data for arbitrarily oriented but known composite materials.

Remarks on Coaxiality of Strain and Stress in Anisotropic Elasticity

Two different consequences of the problem of the extremization of the strain energy by varying the orientation of the material symmetry axes relative to the principal axes of stress are discussed. These two different consequences depend upon whether the stress state is considered as arbitrary and general or as fixed and specific.

Elastic interfacial waves in orthotropic interlayers.

Elastic interfacial waves propagating along one of the planar boundaries separating an orthotropic interlayer of arbitrary uniform thickness from an orthotropic infinite surrounding solid are studied. The axes of material symmetry of the two materials are aligned with one of the axes coinciding with the propagation direction and another being perpendicular to the interfaces. The dispersion equation is derived in explicit form yielding the interfacial phase or group speed in terms of frequency, nondimensionalized with respect to the interlayer thickness, and the elastic constants and mass densities of the interlayer and the surrounding solid. Limiting cases of the dispersion equation give the secular equation for interfacial (Stoneley) waves in two semi‐infinite orthotropic materials and the frequency equation for an orthotropic plate. Analysis of the dispersion equation reveals several features. Under parameter conditions which are defined propagation at low frequencies cannot occur. Also material parameter combinations are found for which interfacial waves of arbitrary wavelength as compared to the interlayer thickness cannot propagate. Finally, the existence of standing waves as solutions of the bifurcation equation, a special case of the dispersion equation, is defined with respect to the parameters of the interlayer and the host material.

Optimal design of biaxial tests for structural material characterization of flat tissues.

A rational methodology is developed for optimal design of biaxial stretch tests intended for estimating material parameters of flat tissues. It is applied to a structural model with a variety of constitutive equations and test protocols, and for a wide range of parameter levels. The results show nearly identical optimal designs under all circumstances. Optimality is obtained with two uniaxial stretch tests at mutually normal directions inclined by 22.5 deg to the axes of material symmetry. Protocols which include additional equibiaxial tests provide superior estimation with lower variance of estimates. Tests performed at angles 0, 45, and 90 deg to the axes of material symmetry provide unreliable estimates. The optimal sampling is variable and depends on the protocols and model parameters. In conclusion, the results indicate that biaxial tests can be improved over presently common procedures and show that this conclusion applies for a variety of circumstances.

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.

Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog.

We have studied the three-dimensional arrangement of ventricular muscle cells and the associated extracellular connective tissue matrix in dog hearts. Four hearts were potassium-arrested, excised, and perfusion-fixed at zero transmural pressure. Full-thickness segments were cut from the right and left ventricular walls at a series of precisely located sites. Morphology was visualized macroscopically and with scanning electron microscopy in 1) transmural planes of section and 2) planes tangential to the epicardial surface. The appearance of all specimens was consistent with an ordered laminar arrangement of myocytes with extensive cleavage planes between muscle layers. These planes ran radially from endocardium toward epicardium in transmural section and coincided with the local muscle fiber orientation in tangential section. Stereological techniques were used to quantify aspects of this organization. There was no consistent variation in the cellular organization of muscle layers (48.4 +/- 20.4 microns thick and 4 +/- 2 myocytes across) transmurally or in different ventricular regions (23 sites in 6 segments), but there was significant transmural variation in the coupling between adjacent layers. The number of branches between layers decreased twofold from subepicardium to midwall, whereas the length distribution of perimysial collagen fibers connecting muscle layers was greatest in the midwall. We conclude that ventricular myocardium is not a uniformly branching continuum but a laminar hierarchy in which it is possible to identify three axes of material symmetry at any point.

Petrov-Galerkin finite elements in time for rigid-body dynamics

The problem of rigid-body rotational dynamics is examined in the context of a novel Petrov-Galerkin mixed finite-element-in-time formulation. The present approach crucially differs from other formulations in the choice of the test functions that weight the constitutive equations in an integral average sense. Within this framework, two second-order accurate time-stepping algorithms are derived: the first attains a slightly greater accuracy than the second when compared with the analytical solution of a free-tumbling rigid body with an axis of material symmetry, but in general it does not preserve energy for force-free motion; the second is marginally less accurate, but it is associated with a discrete conservation law of the kinetic energy. Numerical simulations are presented that confirm the results of the analysis and illustrate the excellent performance of the proposed numerical approach.

Wood viscoelastic compliance determination with special attention to measurement problems

A new experimental technique has been developed for the determination of wood viscoelastic compliance in tension. It associates tensile creep tests parallel to, and off the material symmetry axes with a strain measurement system using a rosette gauge. Particular attention has been paid to accurate strain measurement, and to hydrothermal and mechanical disturbance factors. Application is restricted to the determination of the four compliances which characterize the viscoelastic behaviour of spruce in the LR (radial) plane, at 21°C and 12% equilibrium moisture content. The time-dependent strain induced by load level up to 58% of the ultimate strength, and applied during 4 to 10 days, is largely recoverable. As proposed by Schniewind and Barrett, the compliances are represented by time power functions. The experimental results support earlier observations that the creep level is much higher in tension in the radial direction than parallel to the grain, and that the creep level in shear over the LR plane is an intermediate between the above two cases. Finally, the experimental results illustrate the validity of the superposition principle under the test conditions examined, which is the basis of the linear viscoelastic model introduced here.

Crack trajectories influenced by mechanical and thermal disturbance in anisotropic material

This work is concerned with the application of the volume energy density criterion for predicting the crack trajectories as influenced by mechanical and thermal disturbance in an anisotropic material. Two-dimensional linear thermoelasticity is employed in conjunction with the well-known complex potentials such that a linear relationship is obtained for the boundary conditions across the crack or line of discontinuity. Boundary collocation is then used to determine the unknown coefficients from which the contours of the volume energy density in the cracked plate can be obtained. The crack path is assumed to coincide with the loci where dilatation would dominate. This corresponds to the locations of relative minimum energy density which can be found by visual inspection. An equal and opposite mechanical stress and thermal gradient are applied on the cracked plate. The former and latter enhance symmetric and asymmetric crack growth, respectively. They would complete depending on the magnitude of the mechanical and thermal load. Numerical results are presented for three (3) different cases of a plate whose principal axes of material symmetry are tilted to the crack plane. The influence of anisotropy on crack path is found to be secondary.

Axisymmetric Scattering of a Plane Longitudinal Wave by a Circular Crack in a Transversely Isotropic Solid

The scattering of elastic waves by a circular crack situated in a transversely isotropic solid is studied here. The axis of material symmetry and the axis of the crack coincides. The incident wave is taken as a plane longitudinal wave propagating perpendicular to the crack surface. A Hankel transform representation of the scattered field is used, and after some manipulations using the boundary conditions this leads to an integral equation over the crack for the displacement jump across the crack. This jump is expanded in a series of Legendre polynomials which fulfill the correct edge condition and the integral equation is projected on the same set of Legendre polynomials. The far field is computed by the stationary phase method. A few numerical computations are carried out for both isotropic and anisotropic solids. Results for the isotropic solid compare favorably with those available in the literature.

Interface cracks in anisotropic dissimilar materials : An analytic solution

Based on linear elastic fracture mechanics, analytic solutions are given for displacement and stress fields of in-plane deformation of two anisotropic half-planes, forming a composite bimaterial, with an interface crack, assuming strictly two-dimensional problems; this requires suitable orientation of the material symmetry axes to ensure decoupling of the anti-plane fracture mode from the in-plane ones. It is shown that the field equations are fully characterized in terms of four dimensionless parameters, and these parameters are expressed in terms of the twelve involved elastic constants, six for each half-plane. Analytic solutions are given for two models: (1) the fully open-crack model, involving oscillatory square-root singularities at crack tips ; and (2) the Comninou model which allows possible small contact zones close to the crack tips. Analytic expressions are obtained for the crack opening displacement, the size of the contact zone, the total force transmitted across the contact zone, and the stress field. The results are discussed and related to those for isotropic bimaterials, given by Gautesen and Dundurs.

Influence of anisotropy on the dispersion characteristics of guided ultrasonic plate modes

Dispersion curves are developed for elastic wave propagation in an anisotropic plate of monoclinic or higher symmetry. Emphasis is placed on analytic expressions for various features. Generalization of the isotropic Rayleigh–Lamb dispersion relations are derived for the cases of (a) propagation along a material symmetry axis and (b) propagation in a general direction. Examination of the high-frequency limit of the lowest symmetric and antisymmetric mode dispersion curves yields expressions for the half-space surface or Rayleigh wave velocity. It is shown that the dispersion curves for these modes can exhibit multiple crossings in approaching this limit, and an analytic solution is presented for the constant crossing interval that occurs for propagation along symmetry directions. The analytic results are illustrated by extensive numerical calculations for a variety of degrees of anisotropy with emphasis placed on the relationship between the slowness curves governing partial wave propagation and various features of the dispersion curves.

Two-Dimensional Stress-Strain Relationship for Canine Pericardium

Two-dimensional pseudoelastic mechanical properties of the canine pericardium were investigated in vitro. The pericardium was assumed to be orthotropic. The material symmetry axis was determined a priori and aligned with the stretching axis. Various biaxial stretching tests were then performed and a set of data covering a wide range of strains was constructed. This complete data set was fitted to a new exponential type constitutive model, and a set of true material constants was determined for each specimen. Using the constitutive model and the true material constants, the results from constant lateral force tests and constant lateral displacement tests were predicted and compared with experiment.

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, ϑ, θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.

Wave propagation in transversely isotropic elastic media - I. Homogeneous plane waves

In relation to transversely isotropic media, this paper presents a detailed study of those aspects of the propagation of homogeneous plane elastic waves which are essential to a basic understanding of the behaviour of surface waves. It is first shown how the ordering of the speeds of plane waves provides, directly and simply, a means of classifying the chosen materials, with the class label specifying the broad structure of the slowness surface and the location of its singular points. An examination of the shape of the outer sheet of the slowness surface follows, providing inter alia a complete account of the incidence of the various types of transonic states. The discussion turns next to exceptional waves, that is homogeneous plane waves which leave free of traction some family of parallel planes. The subset of the plane waves possessing this property is determined, after which the subset of the exceptional waves serving as limiting waves for an exceptional transonic state is picked out. Exceptional transonic states occur only when the axis of material symmetry lies either in the reference plane or at right angles to the reference vector and these orientations of the axis are referred to as α and β configurations respectively. The exceptional states are arranged in a threefold classification, one class consisting of a continuous set of α configurations and the others discrete β configurations. The paper ends with calculations of the limiting speed of the transonic state for the totality of α and β configurations.

Mixed-mode crack growth in anisotropic media

The problem of an anisotropic plate with a crack inclined with respect to the principal axes of material symmetry and the direction of the applied uniform uniaxial stress is studied. The local stress field in the vicinity of the crack tip is used in conjunction with the strain energy density theory to analyse the fracture behavior of the anisotropic plate. The dependence of the crack growth angle and the critical applied stress for unstable crack growth on the material properties of the plate and the direction of the crack with respect to the applied stress and the principal axes of material symmetry is disclosed. The relevance of these results for the study of the role of mechanical parameters in the crack propagation process of the environment-assisted cracking phenomenon is discussed.

Stresses in a constrained transversely isotropic elastic solid caused by a moving dislocation

An unbounded, transversely isotropic elastic solid whose elastic constants obey the constraint (c11–c44) (c33–c44)=(c13+c44)2 is excited by a suddenly applied moving dislocation. Explicit expressions are obtained for the plane stress field induced by the (constant speed) dislocation moving parallel to the material symmetry axis. The speed of the dislocation may be sub-, tran-, or super-sonic with respect to the material wave speeds. An unexpected result is the existence of a special dislocation speed, in the transonic range, which causes the Mach head wave to be annihilated.

Errors induced by off-axis measurement of the elastic properties of bone.

Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.