Elastic properties of powders during compaction. Part 1: Pseudo-isotropic moduli

The elastic moduli of powdered materials undergoing uniaxial compaction was investigated, paying particular attention to effects of solid phase material properties and initial particle shape. Elastic properties were characterised by the isotropic elastic moduli Poisson’s ratio and Young’s modulus, calculated from elastic wave speeds measured in the axial (pressing direction). To isolate material property effects, three different ductile metal powders (copper, stainless steel, and aluminium) with equivalent particle shape (spheroidal) were tested. Comparison with similar measurements for a brittle spheroidal powder (glass) illustrated that solid phase yield mechanism affects the evolution of pore character, and hence bulk elastic properties of the powder compact. Pore character was also studied separately by comparing copper powders with differing particle shapes (spheroidal, irregular, and dendritic). For all powders, Young’s modulus increased monotonically with compaction (reducing porosity). For the ductile spheroidal powders, differences in evolution of Young’s modulus with compaction were accounted for by solid phase elastic properties. The different morphology copper powders showed an increase in compact compliance as particle (pore) ruggedness increased. Poisson’s ratio followed a concave porosity dependence: decreasing in the initial stages of compaction, then increasing as porosity approached zero. Comparison between powders indicated the initial decrease in Poisson’s ratio was insensitive to solid phase material properties. However, as the compact approached solid phase density, the Poisson’s ratio—porosity locus diverged towards corresponding solid phase values for each particle material, indicating an influence of solid phase elastic properties.     

On a Possible Approximation of Changes in Elastic Properties of a Transversely Isotropic Material due to an Arbitrarily Oriented Crack

The present paper addresses an approximate analytical model for contribution of an arbitrarily oriented circular crack into effective elastic compliance of a transversely isotropic material. We numerically examine the bounds of applicability of the hypothesis that change in elastic potential due to an arbitrarily oriented circular crack in a transversely-isotropic material can be approximated by the change calculated for a certain isotropic environment. In particular, we obtained that the error of such an approximation is less than 20% if the extent of anisotropy is moderate – the ratio of Young’s moduli in transverse and in-plane directions is less than 1.87. The obtained result can be used for development of a simple model for microcracked transversely-isotropic materials with mild-to-moderate extent of anisotropy.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.     

Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects

Summary We consider the macroscopic behavior of two-phase fibrous piezoelectric composites. The fibers are of circular cross-section with the same radius. Along the interfaces between the fibers and the matrix we consider the effects of surface stress and surface electric displacement. The constituents are transversely isotropic and exhibit pyroelectricity. We find that the overall thermoelectroelastic moduli of these solids must comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides five constraints between the six axisymmetric overall electroelastic moduli. The second set relates the effective coefficients of thermal stress and pyroelectric coefficients to the effective electroelastic moduli, in analogy with Levin’s formula. In contrast to their conventional counterparts, i.e., without surface effects, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.     

Determination of the Elastic Symmetry of a Monolithic Ceramic using Bulk Acoustic Waves

A nondestructive optimal determination of elastic properties from ultrasonic bulk wave velocity measurements on a monolithic ceramic plate immersed in water is presented. This procedure, that is applicable to flat plates with unknown material properties, is based on already established methods and includes discussions, using experimental data, on the reliability of the elastic property identification, such as the stiffness tensor and the material symmetry. By solving inverse propagation problems deduced from the Christoffel equation and depending on wave speed measurements, we show that the studied sample can be described by twenty-one dependent stiffness constants and that its intrinsic elastic material symmetry was hexagonal (or transversely isotropic).     

Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.     

Elastic properties of single- and polycrystalline LaFeAsO, SrFe2As2, and LiFeAs basic phases for new FeAs superconductors

The elastic coefficients (C _ij ), bulk compression moduli (B), and shear moduli (G) of layered tetragonal LaFeAsO, SrFe₂As₂, and LiFeAs crystals, which are potential basic phases for new superconductors of the 1111-, 122-, and 111-FeAs groups, respectively, have been calculated from first principles using the full-potential linearized augmented plane wave (FLAPW) method with an exchange-correlation potential in the generalized gradient approximation (GGA). Numerical estimates of the bulk moduli, shear moduli, Young’s moduli, and Poisson’s ratios of the corresponding polycrystalline ceramics have been obtained for the first time.     

Pressure effects on elastic and thermodynamic properties of ZrB2

A theoretical formalism to calculate the single crystal elastic constants for hexagonal crystals from first principle calculations is described. The calculated values compare favorably with recent experimental results. An expression to calculate the bulk modulus along crystallographic axes of single crystals, using elastic constants, has been derived. The calculated linear bulk moduli are found to be in good agreement with the experiments. The shear modulus, Young’s modulus, and Poisson’s ratio for ideal polycrystalline ZrB₂ are also calculated and compared with corresponding experimental values. The shear anisotropic factors and anisotropy in the linear bulk modulus are obtained from the single crystal elastic constants. The Debye temperature is calculated from the average elastic wave velocity obtained from shear and bulk modulus as well as the integration of elastic wave velocities in different directions of the single crystal. The calculated elastic properties are found to be in good agreement with experimental values when the generalized gradient approximation is used for the exchange and correlation potential. It is found that the elastic constants and the Debye temperature of ZrB₂ increase monotonically and the anisotropies weaken with pressure. The thermal properties including the equation of state, linear compressibility, ductility, and the heat capacity at various pressures and temperatures are estimated.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Measurement of the dynamic shear moduli of a transversely isotropic fibre-reinforced plastic

High frequency measurements of the two shear moduli of a transversely isotropic glass fibre-reinforced plastic are described. A theory is first developed for the torsional vibration of rectangular bars of orthotropic materials in which the effects of warping constraint (longitudinal direct stresses) are fully considered. The theory is carefully validated and used as a basis for a computer program to predict natural frequencies. Results from this program are used to deduce the moduli from measured torsional frequencies of pairs of bars, identical in size but with different orientations of the planes of material isotropy. Measurements have been made in the range 1–6 kHz. Comparisons are made with moduli deduced for the same bars from flexural vibration tests. Results from torsional tests are deemed to be superior.     

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

Low temperature-elastic moduli, Debye temperature and internal dilational and shear frictions of fused quartz

Longitudinal and transverse wave velocities, five kinds of elastic parameters (Young’s, shear and bulk moduli, Lame parameter, Poisson’s ratio), Debye temperature, and dilational and shear internal frictions for fused quartz were simultaneously measured over the temperature range from 73 to 400 K, using the ultrasonic pulse wave with frequency of 7.7 MHz. Large increase in Young’s and bulk moduli and small increase in shear modulus and Lame parameter suggest enhancement of rigidity for KI mode on heating. This would be explained by quasi-crystallization which is associated with a lateral motion of oxygen atoms and the resulting relief of macroscopic strains. The 99 and 137 K peaks and 360 K one in shear friction are probably related to dielectric loss peaks arising from Al3+–Na+ and Al3+–K+ substitutional–interstitial paired defects and to β1/β2-tridymite phase transition, respectively. This revised version was published online in November 2006 with corrections to the Cover Date.     

Modeling the effective elastic properties of nanocomposites with circular straight CNT fibers reinforced in the epoxy matrix

In the present study, the consistent effective elastic properties of straight, circular carbon nanotube epoxy composites are derived using the micromechanics theory. The CNT composites are known to provide high stiffness and elastic properties when the shape of the fibers is cylindrical and straight. Accordingly, in the present work, the effective elastic moduli of composite are newly obtained for straight, circular CNTs aligned in the specified direction as well as distributed randomly in the matrix. In this direction, novel analytical expressions are proposed for four cases of fiber property. First, aligned, and straight CNTs are considered with transverse isotropy in fiber coordinates, and the composite properties are also transversely isotropic in global coordinates. The short comings in the earlier developments are effectively addressed by deriving the consistent form of the strain tensor and the stiffness tensor of the CNT nanocomposite. Subsequently, effective relations for composites reinforced with aligned, straight CNTs but fibers isotropic in local coordinates are newly developed under hydrostatic loading. The effect of the unsymmetric Eshelby tensor for cylindrical fibers on the overall properties of the nanocomposite is included by deriving the strain concentration tensors. Next, the random distribution of CNT fibers in the matrix is studied with fibers being transversely isotropic as well as isotropic when CNT nanocomposites are subjected to uniform loading. The corresponding relations for the effective elastic properties are newly derived. The modeling technique is validated with results reported, and the variations in the effective properties for different CNT volume fractions are presented.     

Thermoelastic properties and conductivity of carbon/carbon fiber composites

Unidirectional carbon/carbon composites are modeled as fiber composites with cylindrically orthotropic fibers and matrix. All of the thermoelastic properties and the conductivities are evaluated on the basis of the composite cylinder assemblage (CCA) and the generalized self consistent scheme (GSCS) models. The former for axisymmetric elastic properties, axial shear modulus, thermal expansion coefficients and conductivities; the latter for transverse shear and Young's moduli and Poisson's ratio. Numerical results are given for onion skin, radial and transversely isotropic phase graphitic structures.     

On a possibility of reconstruction of Cosserat moduli in particulate materials using long waves

The paper proposes a method of reconstruction of the Cosserat elastic moduli using the measurements of velocities of the p-wave and the high-frequency twist wave as well as the low-frequency asymptotics of a shear wave dispersion relationship. It is shown that in the case of a general isotropic Cosserat continuum, the information obtained from these wave measurements is insufficient for the complete moduli reconstruction. The reconstruction is shown to be possible in the case of a 3D isotropic Cosserat continuum governed by at most four independent parameters. Such a continuum is suggested for a particulate material consisting of spherical particles connected by normal, shear and rotational links. Another case when the full reconstruction is possible consists of 2D orthotropic Cosserat continuum modelling particulate material with square packing of cylindrical particles and 2D isotropic Cosserat continuum modelling with hexagonal packing of cylindrical particles. In the 2D materials, the measurements of p-wave velocity and the shear wave dispersion relationship are sufficient for complete reconstruction of all moduli. A phase shift method and reconstruction algorithms are presented.     

Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation

We consider a system consisting of n elastic layers made of different transversely isotropic materials bonded to each other and the last layer bonded to an elastic half-space made of a different transversely isotropic material. An arbitrary tangential displacement is prescribed over a domain S of the first layer, while the rest of the layer’s surface is stress free. The tangential contact problem consists of finding the complete stress and displacement fields in this system. The Generalized Images method developed by the author is used to get an elementary solution to the problem. We first consider the case of two layers and then generalize it for the case of n layers. The same problem is solved by the integral transform method, and it is shown that an integral transform can be interpreted as a sum of generalized images. The results are valid for the case of isotropy as well.     

Heterogeneous and anisotropic long-term concrete damage of the dez arch dam using thermal inverse analysis

In this paper, an assumption used in the recent work of the author and his contributors considers the long-term concrete damage of the Dez arch dam as a homogeneous and isotropic process, was investigated in more detail and was adjusted. To this end, the vertical dam sections were divided into nine and six subsections along the thickness and height directions of the dam, respectively. In each subsection, a transversely isotropic damaged elastic constitutive law was considered for diagnostic analyses. Following the previous authors’ mentioned work, an optimization procedure (minimization of a certain error function) accompanied with an inverse thermal analysis was carried out. That analysis was performed within the framework of finite element (FE) method. Mentioned error function was defined as the sum of the squared residuals. Residuals were set as difference between nodal recorded displacements with inverse pendulums of the dam and the corresponding computed ones with the proposed model. Parameters considered as unknowns in the present inverse analysis, which have contributions in that above-mentioned error function, were the five independent elastic moduli presented in the formulation of the earlier mentioned transversely isotropic damaged elastic constitutive law: two Young’s moduli: one in the planes parallel to the up and down stream surfaces of the dam, called in-plane and the other in the perpendicular planes to those planes, named out-of-plane Young’s modulus; two Poisson’s ratios (in-plane and out-of-plane); and one shear modulus (out-of-plane). These parameters were identified performing numerous thermal inverse analyses. Obtained results revealed that the long-term damage of concrete of the Dez dam is a heterogeneous and anisotropic phenomenon because that the magnitude of the mentioned error function was obtained smaller than the corresponding value in the previous study which had been performed before based on the homogeneous and isotropic damage evolution assumption.