Anisotropic Elastic Behavior Research Articles

Dynamic elastic anisotropy and nonlinearity in wood and rock

Ultrasonic techniques are used to characterize the anisotropic and nonlinear elastic behavior of wood and rock and to interrogate the structural properties of these materials. For elastic anisotropy two types of experiments are performed, namely qP-wave velocity measurements on spherical samples in about 100 directions of propagation rather regularly sampled in space, together with S-wave birefringence measurements in a few directions on additional samples. For nonlinear elasticity, acoustoelastic experiments were conducted, consisting in P and S-wave velocity measurements under controlled confining pressure. The experimental results show that wood exhibits much larger elastic anisotropy, but much weaker elastic nonlinearity than rock. For instance, the deviations from isotropy in wood can reach 70%, whereas in rock, typically, it can hardly exceed 20%. In contrast, regarding nonlinearity the increase of P or S-wave moduli per unit confining pressure in wood is always smaller than 30, in the radial direction, or 10, in the longitudinal and transversal directions, whereas it can reach roughly one to a few hundreds in rock. These contrasted behaviors can be simply explained by structural considerations. Thus, the exceptionally strong elastic anisotropy of wood is due to the strict structural alignment of its constituents, that is to say to the preferential orientation of the anatomical elements (tracheids, fibers, ray cells, vessels etc.) for ‘textural’ anisotropy, and to the cellular wall organisation for ‘microstructural’ anisotropy. In comparison, rock only exhibits a rough statistics of the orientation distribution function of its constituents, mainly the grain minerals, the pores and the cements. In contrast, the strinkingly strong nonlinear elastic response of rock, a wellestablished classical observation, is due to the presence of compliant mechanical defects (cracks, microfractures, grain-joints etc.). Such features are practically nonexistent in wood which explains its weak nonlinear response.

Comparison between prediction and measurement of viscoelastic moduli in composite materials versus temperature using ultrasonic immersion technique with oil

An oil immersion ultrasonic method used for measuring elastic and viscoelastic properties of materials versus temperature is presented. From velocity and amplitude measurements, the complex viscoelastic tensor can be entirely determined from room temperature to temperatures near the glass transition temperature of the material. This tensor permits the computation of velocities and attenuations of heterogeneous or homogeneous modes in any direction of wave propagation. A micromechanics model is used to compare experimental results with theoretical ones and to analyze anisotropic elastic and viscoelastic behavior of composite materials.

Qualitative analysis of strain localization. Part I: Transversely isotropic elasticity and isotropic plasticity

While the conditions for the onset of strain localization in inelastic materials with isotropic elasticity have been extensively studied in recent years, few attempts have been devoted to investigate the effect of anisotropy and in particular the influence of the anisotropic elastic behavior on the localization phenomenon. In Part I of this paper, the localization analysis will be performed for an elastoplastic von Mises material with transversely isotropic elasticity subjected to uniaxial tension. Numerical and analytical results showing the influence of the deviation from isotropic elasticity on the onset of strain localization are presented. The anisotropic elastic behavior may substantially trigger the shear band initiation and modify the corresponding failure pattern. These observations become critical for highly damaged materials. Further developments concerning both transversely isotropic elasticity and plasticity are given in Part II of the paper.

Laboratory investigation of the mechanical behaviour of Tournemire shale

This paper presents laboratory investigations of the mechanical behaviour of a typical anisotropic rock: Tournemire shale. Hydrostatic compressibility tests allowed characterization of the structural anisotropy of the material. Triaxial compression tests, including unloading cycles, were performed. Anisotropic elastic response, plastic deformation and failure behaviour have been investigated. The elastic behaviour of the shale is non-linear and the shale exhibits large anisotropic plastic deformation. The failure behaviour of the shale is anisotropic and strongly depends on confining pressure and loading orientation.

Simple shearing flow of a dry Kelvin soap foam

Simple shearing flow of a dry soap foam composed of identical Kelvin cells is analysed. An undeformed Kelvin cell has six planar quadrilateral faces with curved edges and eight non-planar hexagonal faces with zero mean curvature. The elastic-plastic response of the foam is modelled by determining the bubble shape that minimizes total surface area at each value of strain. Computer simulations were performed with the Surface Evolver program developed by Brakke. The foam structure and macroscopic stress are piecewise continuous functions of strain. Each discontinuity corresponds to a topological change (T1) that occurs when the film network is unstable. These instabilities involve shrinking films, but the surface area and edge lengths of a shrinking film do not necessarily vanish smoothly with strain. Each T1 reduces surface energy, results in cell-neighbour switching, and provides a film-level mechanism for plastic yield behaviour during foam flow. The foam structure is determined for all strains by choosing initial foam orientations that lead to strain-periodic behaviour. The average shear stress varies by an order of magnitude for different orientations. A Kelvin foam has cubic symmetry and exhibits anisotropic linear elastic behaviour ; the two shear moduli and their average over all orientations are G min = 0.5706, G max = 0.9646, and G = 0.8070, where stress is scaled by T/V 1/3 , T is surface tension, and V is bubble volume. An approximate solution for the microrheology is also determined by minimizing the total surface area of a Kelvin foam with flat films.

Grain shape effects on interface-controlled diffusional creep under multiaxial stresses

Interface control of the emission and absorption of vacancies reduces, but does not eliminate, the dependency of diffusional creep on grain size and shape and it increases the sensitivity to stress. Previous analyses of interfacially controlled diffusional creep of polycrystals with equiaxed grains under a uniaxial stress are extended and further developed to determine approximately the behaviour of material with anisotropic grain shapes under multiaxial stresses. Whilst all details of this creep process are not fully understood, it is shown that, by noting a set of conditions to be fulfilled, the form of equation required to describe anisotropic creep strength can be identified. Although interfacial control implies a power law dependance on stress, it is noted that the directionality in creep deformation is governed by grain shape orientation and by linear functions of the applied stresses and so anisotropic aspects of creep response under multiaxial stresses can be described by “creep compliance coefficients” analogous to those representing anisotropic elastic behaviour. Since anisotropy is influenced by three parameters, representing the principal orthogonal grain dimensions, only three of these coefficients are independent. From this approach a Von Mises type function can be formulated through which anisotropic interface-controlled diffusional creep strength may be evaluated.

Anisotropic elasticity and strength of glutaraldehyde fixed bovine pericardium for use in pericardial bioprosthetic valves.

Uniaxial tensile tests were performed on glutaraldehyde fixed bovine pericardial strips prepared from chemically modified pericardial samples. These samples originated from an area which demonstrated anisotropic mechanical properties in the native material and which is suitable for the construction of leaflets for pericardial bioprostheses. After glutaraldehyde fixation the tissue had retained its anisotropicity in stiffness and strength in two orthogonal directions. In the range of the functional stresses for a heart valve leaflet (< 1 MPa) the unconstrained fixation regime had modified the initial anisotropic elastic behavior into a more isotropic one. The implications of these findings are that leaflets manufactured from bovine pericardium can be made to resemble, to a degree, the well known anisotropy found in two orthogonal directions in natural human heart valve leaflets, or porcine bioprosthetic heart valve leaflets.

A time integration algorithm for elasto‐viscoplastic cubic crystals applied to modelling polycrystalline deformation

AbstractAn algorithm for integrating the constitutive equations for an elasto‐viscoplastic cubic crystal is presented which is shown to be easily employed in a polycrystalliné analysis. Anisotropic elastic behaviour is incorporated into the standard constitutive equations for ductile single crystals. The algorithm is shown to be efficient, robust and general. The primary advantage of this algorithm is that is provides an implicit integration of the plastic deformation gradient while including the elastic response. This permits taking large time steps while maintaining accuracy and stability. Several polycrystalline examples are presented to demonstrate the effect of the time step on the solution. Examples also are presented which compare the algorithm described herein to an algorithm which neglects the elastic part of the deformation. In addition, the effect of the anisotropic component of the elasticity is investigated by comparing the results with those obtained assuming isotropic elasticity.

Effective stress paths and pore-pressure responses during undrained shear along the bedding planes of varved Fort William Clay

Previous studies of the varved sensitive clays from the Thunder Bay area indicated that the geotechnical properties were influenced by their layered structure. When sheared along the bedding planes, double strength envelopes were apparent, with strength parameters depending on the stress levels applied. Thus, a series of consolidated isotropical undrained triaxial tests with pore-pressure measurements was carried out on specimens with the varves inclined to the vertical axis. A number of unusual results were observed: (i) low B-values for presumably saturated samples; (ii) failure of the specimens along the clay seams at stress levels above precompression load, exhibiting highly compressive behaviour and considerable cohesion; (iii) failure of the specimens in the silt seams at stress levels below the precompression load, exhibiting dilatancy and a lower cohesion than in the normally consolidated range; (iv) slightly anisotropic elastic deformation behaviour almost up to failure for testing in the precompression range; however, strongly anisotropic, largely nonelastic behaviour in the normally consolidated stress range, indicating larger stiffnesses vertically than horizontally. The stress paths in the normally consolidated stress range suggested also that consolidation of the clay seams occurred during undrained shear owing to internal dissipation of pore pressures into dilating silt seams. Key words: varved sensitive cemented clay, pore-pressure response, undrained shear, internal pore-pressure dissipation, anisotropic elastic behaviour, critical-state conditions.

Characterization of anisotropic elastic wave behavior in media with parallel fractures and aligned cracks

Ultrasonic experiments were performed to investigate the anisotropic properties of an isotropic background medium with a system of long closely spaced parallel fractures. Immersion experiments at frequencies from 0.1‐0.7 MHz were performed on Plexiglas that was cut into thin sheets and then pressed together. Measurements of phase and group velocity were obtained for compressional and shear waves both normal to and parallel to the “fractures” and for quasicompressional and quasishear waves at a variety of oblique angles. The data were analyzed to determine the anisotropy parameters due to the fracturing, based on modeling the fractures as a set of infinite, linear, although not necessarily rotationally symmetric, slip interfaces [M. Schoenberg, J. Acoust. Soc. Am. Suppl. 1 82, S88 (1987)]. For the most part, the experiments fit the theory quite well even though losses that can occur in the fractures have been ignored in this first attempt to match theory and experiment. The anisotropy attributable to the fracture system depends on the geometry of the fractures, the roughness, the static closing stress across the fractures, and the nature of the infilling medium, if any. From a knowledge of the anisotropic elastic moduli of the fractured medium and the isotropic background moduli, the average compliance matrix may be deduced. In turn, this gives the average behavior of each fracture plane.

Characterization of anisotropic elastic wave behavior in media with parallel fractures and aligned cracks

Characterization of anisotropic elastic wave behavior in media with parallel fractures and aligned cracks

On the solution for the elastic response of involute bodies

A recent model to treat the elastic response of bodies of revolution possessing axisymmetric material structure, but otherwise arbitrary heterogeneous anisotropic elastic behavior, is examined in detail. Of particular interest is the derivation of the end conditions appropriate to the particular numerical solution technique employed (finite difference) because of the difference between the numbers of field equations and boundary conditions. It is felt that the procedure adopted may have much wider application than the present usage. The response of involute bodies is featured in this work as the illustrative problem of a frustrum of an involute cone under a constant temperature change is considered. A thorough comparison is made between the predictions given by the present model and those from the PATCHES-III finite element code, with generally favorable results.

Brillouin scattering and ultrasonic investigations in the 1D-ferromagnet CsNiF3

The elastic constants of CsNiF3 are derived versus temperature from Brillouin spectroscopy (5–300 K) and ultrasonic propagation (100–300 K). The results show a marked anisotropic elastic behaviour. A softening of C44 below 170 K could arise from magnetic correlations. No effect of magnetic field is observed upon the longitudinal modes.

A plastic-fracture model for concrete

The paper summarizes recent efforts in formulating an elastic-plastic-fracture model for the finite-element analysis of concrete structures. Based on the geometrical considerations, a four-parameter fracture (or yielding) criterion was proposed which embraces some of the simpler existing models. Isotropic elastic and anisotropic elastic behaviors were proposed for the initial loading and the post-failure behaviors. A plastic model displaying mixed hardening was proposed to describe material behaviors between the initial yielding and the fracture failure. Incremental stress-strain relationships were derived based on the associated flow rule and Ziegler's kinematic hardening rule. Three different types of failure modes were considered. A simple crushing coefficient was defined based on a dual criterion to identify the crushing type, the cracking type and the mixed type of failure. Material parameters required for each element of the plastic-fracture model were determined. An important feature of the paper is that matrix formulations for all the constitutive equations were derived and are available for finite-element implementations.

On a finite element model for the analysis of through cracks in laminated anisotropic cylindrical shells

A finite element model for the analysis of laminated composite cylindrical shells with through cracks is presented. The analysis takes into account anisotropic elastic behaviour, bending-extensional coupling and transverse shear deformation effects. The proposed finite element model is based on the approach of dividing a cracked configuration into triangular shaped singular elements around the crack tip with adjoining quadrilateral shaped regular elements. The parabolic isoparametric cylindrical shell elements (both singular and regular) used in this model employ independent displacement and rotation interpolation in the shell middle surface. The numerical comparisons show the evidence to the conclusion that the proposed model will yield accurate stress intensity factors from a relatively coarse mesh. Through the analysis of a pressurised fibre composite cylindrical shell with an axial crack, the effect of material orthotropy on the crack tip stress intensity factors is shown to be quite significant.

A computer model for the analysis of ground movements in London Clay

A computer model for the stress–strain behaviour of London Clay has been developed for use in a nonlinear finite element program. The behaviour of London Clay is divided into three ranges of strain: elastic, intermediate and plastic. In the elastic range strains are very small and the material is very stiff. The intermediate range represents the larger strain behaviour normally observed in laboratory tests; it is not truly elastic, but in many respects exhibits aniso-tropic elastic behaviour. As yield is approached, plastic behaviour can accompany the intermediate phase and is governed by a flow rule and a state boundary curve. In all phases the stiffnesses of the model increase with mean normal effective stress. It is shown that the model can be used to compute the results of both laboratory and in situ tests. The movements around two excavations in London Clay, including the New Palace Yard Underground Car Park, have also been computed retrospectively with considerable success. Un modèle informatique destiné à refléter le comportement contraintes-deformations de l'Argíle de Londres a été mis au point en vue de son emploi dans un programme non linéaire à éléments finis. Le comportement de l'Argile de Londres se divise en trois domaines de déformation: élastique, intermédiaire et plastique. Dans le domaine élastique, les déformations sont très faibles et le matériau est très rigide. Le domaine intermédiaire représente les déformations plus importantes que l'on observe normalement dans les essais en laboratoire; il n'est pas véritablement élastique mais présente à bien des égards un comportement élastique anisotrope. Lorsque l'on approche de la limite d'élasticité, on peut observer un comportement plastique dans la phase intermédaire, et ce comportement est régi par une loi de fluage et une courbe de limite entre états. Dans toutes-les phases, la rigidité du modèle augmente en fonction de la contrainte efficace moyenne normale. L'article montre que le modèle peut servir à déterminer les résultats obtenus aussi bien en laboratoire qu'in situ. Le calcul rétrospectif des mouvements enregistrés autour de deux excavations effectuées dans l'Argile de Londres, dont le Parking Souterrain du New Palace Yard, a également été effectué avec beaucoup de succès.

Anisotropic elasticity of composite molecular networks formed from non‐gaussian chains

AbstractThe theory of composite networks by Berry, Scanlan, and Watson was generalized so as to include the case in which the elasticity of the chains is described by non‐Gaussian statistics. For this purpose, existing theories of non‐Gaussian networks had to be critically reviewed and a new simple treatment is presented here. This new treatment yields substantially the same results as that obtained from the more complex theory of Wang and Guth. Results obtained from the simple three‐chain model for the network, as described by Treloar, were also shown to be in satisfactory agreement with both the new treatment and that of Wang and Guth. The generalized theory of composite networks predicts anisotropic elastic behavior; the tensile force at a given strain should be generally higher for elongation in the direction parallel to orientation than in the direction perpendicular to it. However, depending upon details pertaining to the preparation of the networks (ratio of first and second stage crosslinks and elongation at which the latter are introduced), the tensile force required for elongation in the direction perpendicular to orientation may be higher. It appears that the isotropic elastic behavior, predicted by previous theories of composite networks, is merely a consequence of the use of the Gaussian approximation.

Der Einflu� der Umklapp-Prozesse auf den elektrischen Widerstand der Metalle bei tiefen Temperaturen

Blochs integral equation is solved under the assumption of a rigidly displaced electron distribution and thermal equilibrium of the lattice yielding the following results: If the Fermi surfaceE (t)=ξ periodically repeated in (t)-space consists of closed separate parts which are very near to each other (distanceq*), then transitions between two neighbouring surfaces (Umklapp-processes) give especially at rather low temperatures a large contribution to the electrical resistance. In simple cases the temperature dependence is given by ϱUP ∼ Polynomial (T/Θ*)eΘ*/T. Calculations were performed for alkali metals taking into account their anisotropic elastic behaviour. It was found that the contribution of the Umklapp-processes could be as high as 100 times that of the normal-processes. The agreement between theory and experiment is satisfactory excepting Li.