Anisotropic Elasticity Theory Research Articles

Elastic models of symmetrical 〈001〉 and 〈011〉 tilt grain boundaries in diamond

The disclination-structural units model (DSUM), which was previously applied to grain boundaries in metals and a limited number of structures in the diamond cubic lattice, is extended to treat more complicated metastable structures of 〈001〉 and 〈011〉 symmetric tilt boundaries in diamond. These structures are described in terms of flat and faceted disclination dipole walls and screw dislocation dipole walls, with the energies of these defects calculated from anisotropic elasticity theory. Disclination-dislocation models are constructed for 〈001〉 tilt boundaries in the complete misorientation range and for 〈011〉 tilt boundaries in the range $0<~\ensuremath{\theta}<~70.5\ifmmode^\circ\else\textdegree\fi{},$ and the energy versus misorientation curves are calculated using input structures and energies of key grain boundaries obtained from atomistic modeling. Quantitative agreement with atomistic predictions is obtained for the most stable grain-boundary structures, while predictions of the DSUM for higher-energy metastable structures are qualitatively consistent with atomic studies.

Dislocation Interaction with Point Defects in Transition-Metal Disilicides

ABSTRACTEnergetics of the formation of jog-pairs and kink-pairs on a straight dislocation are analyzed using anisotropic elasticity theory for the equivalent slip systems of seven transition-metal disilicides. While glide loops of the active slip systems are stable in all cases, having positive line tension, the interaction energies of two opposite segments in kink-pairs and jog-pairs are found to be very anisotropic with respect to dislocation orientation. The anisotropic interaction plays an important role in the glide resistance due to dislocation-point defect interactions. A dislocation model is proposed for the glide resistance on edge and near edge dislocations based on jog-pairs resulting from the contact interaction between dislocations and intrinsic point defects. The available data of yield strength anomaly and dislocation structures of disilicide crystals are discussed in view of the proposed model for jog-pair pinning and dynamic breakaway.

Weak beam observations of dissociation of pure screw dislocations in Ni3Al

The dissociation of superdislocations in Ni3Al, which have a pure screw character, has been recorded on weak beam images at the g(5g) imaging condition. All screws were observed to dissociate on an {001} habit plane bounding APB faults. By computer modelling, the dissociation distance (dCSF) of the Shockley partial dislocations that bound a complex stacking fault is shown to be less than 1.5 nm. This is the first reported case of fourfold dissociation bounding CSF + APB + CSF faults with a non-planar configuration and pure screw character. The energy values deduced from the measured spacing show quite large difference from the results obtained from dissociation with a planar system. It is questionable to use the linear anisotropic elasticity theory for such small splitting and nonplanar configuration. In addition our simulations demonstrate that a g(ng) condition is a wise choice for imaging partial separation.

Thermoelasticity of spatially reinforced composite plates

The objective of this paper is to obtain the appropriate applied theory describing the thermoelastic behaviour of the spatially reinforced composite plates. The hypothesis of free thermal expansion in the direction orthogonal to the reference surface has been introduced in order to derive the governing equations. The special averaging procedures were implemented in order to reduce three-dimensional equations of the anisotropic theory of elasticity to the two-dimensional model. A similar approach is used to convert three-dimensional equations of thermal conductivity to the two-dimensional ones and obtain equations for the specific coefficients characterising thermoelastic behaviour of thin-walled composite components with spatially oriented reinforcement. These coefficients can be calculated as functions of the structural parameters of spatially reinforced composite material provided that the mechanical and physical properties of the elementary reinforcing layer are determined from the corresponding mechanical and physical tests.

A micromechanics-based methodology for evaluating the fabric of granular material

A micromechanics approach can successfully model the stress–strain–strength relation of a granular material once the microcharacteristics of the material have been obtained. However, it is usually difficulty to determine the fabric of natural granular materials. Using a stress-dependent micromechanics model, the elastic properties can become a function of the geometric and kinetic fabric. Also, the wave velocity can be related to the elastic properties. Consequently, this paper proposes a methodology for evaluating the fabric of a granular assembly from a set of measured wave velocities. The methodology contains three elements: (a) a stress-dependent micromechanics elastic model, (b) an anisotropic elastic wave propagation theory and (c) an optimization procedure. It is verified by calibrating available wave velocity data of a glass assembly and washed mortar sand. The methodology is further applied to study the microstructural evolution of the washed mortar sand under biaxial stresses. Two aspects of fabric change can be observed: (a) concentration of contact normals in the major principal direction and (b) a residual fabric after subsequent loading/unloading. Une approche micromécanique permet de faire une maquette de la relation tension-allongementrésistance ďun matériau granulaire une fois que ses micro caractérisdques ont W obtenues. Cependant, il est habituellement difficie de déterminer la structure des matériaux granulaires naturels. Si ľon utilise une maquette micromécanique tributaire de la tension, les propr!étés élastiques peuvent devenir fonction de la structure géométrique et cinédque. De plus, la vitesse des ondes pent Être apparentée aux propriétés é1astiques. C'est pourquoi, dans cette étude, nous proposons une méthodologie permettant ďévaluer la structure &caorn;un assemblage granulaire à partir &acorn;un lot de vitesses ďondes mesurées. La méthodologie contient trois éléments: (a) une maquette élastique micromécanique tributaire de la tension, (b) une théorie de propagation des ondes élastiques et anisotropes et (c) une procé& dure ďoptimisation. Nous vérifions en les calibrant les données disponibles sur les vitesses ďondes ďun assemblage de boule de verre et ďun sable de morder lavé La méthodologie sert ensuite à étudier ľévolution microstructurale du sable de morder lavé soumis à des tensions biaxiales. On observe alors deux genres de changements de structure: (a) une concentration des perpendiculaires de contact dans la direction majeure principale et (b) une structure résiduelle aprés des charges/décharges consécutives.

Recent developments in anisotropic elasticity

Anisotropic elasticity has been an active research subject for the last thirty years due to its applications to composite materials. There are essentially two formalisms for two-dimensional deformations of a general anisotropic elastic material. The Lekhnitskii formalism [Lekhnitskii, S.G., 1950. Theory of Elasticity of an Anisotropic Elastic Body. Gostekhizdat, Moscow (in Russian)] has been the favorite among the engineering community, while the newer Stroh formalism is well-known in the material sciences, applied mathematics and physics community. The Stroh formalism (Stroh, A.N., 1958. Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3, 625–646.) is mathematically elegant and technically powerful. It began to be noticed by the engineering community in recent years, specially among the younger researchers. A comprehensive treatment of both formalisms and applications of the theory have been presented in a book by Ting. Since the appearance of the book in 1996, there have been several new developments in the theory and applications of anisotropic elasticity. We present here new results that have appeared since 1996. Only linear anisotropic elasticity is considered here; for nonlinear elasticity, the reader is referred to the book by Antman (Antman, S.S., 1995. Nonlinear Problems in Elasticity. Springer–Verlag, New York).

Failure of orthotropic plates containing a circular opening

Failure of composite plates containing a circular opening is investigated and results are given for plates subjected to uniaxial loading. Stresses and strains in the composite were analyzed using the Lekhnitskii anisotropic elasticity theory. A failure criterion based on the strain energy density function is presented for predicting the externally applied load that will initiate failure and to determine the location of the failure at the opening. Typical results are presented for boron/epoxy, graphite/epoxy and glass/epoxy composites.

Gradient Elasticity Theory for a Mode III Crack in a Functionally Graded Material

Anisotropic strain gradient elasticity theory is applied to the solution of a mode III crack in a functionally graded material (FGM). The theory includes both volumetric and surface energy terms, and a particular form of the moduli variation. The crack boundary value problem is solved by means of Fourier transform and a hypersingular integral equation of the Fredholm type. The solution naturally leads to a cusping crack, which is consisteut with Barenblatt's cohesive zone theory, but without the assmuption regarding existence of interatomic forces. The numerical implementation is discussed and examples are given, which provide insight into the cracking phenomenon in FGMs governed by strain gradient elasticity with characteristic lengths associated to volumetric and surface strain energy terms.

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.

On the orientation relationships between A15 precipitates and the Nb-rich matrix in Nb-Al(-X) alloys

The preferred orientation relationships for A15 Nb3Al precipitates in an A2 Nb matrix have been calculated on the basis of anisotropic elasticity theory using the assumptions of coherence and homogeneous moduli. The calculated orientation relationships are in good agreement with most of those observed experimentally for binary and ternary Nb-Al(-X) alloys.

Nonplanar interfacial cracks in anisotropic bimaterials

This paper presents a general approach for the two-dimensional elastic problem of a crack lying along an elliptical interface seperating two dissimilar anisotropic materials. The analysis is based upon the use of the Eshelby–Stroh formalism of anisotropic elasticity theory and a special conformal mapping technique devised by Lekhniskii. The resulting elastic fields are fully described by a pair of function vectors whose components are holomorphic functions. These function vectors define the two-phase potentials of the bi-material. The associated expressions are universal in the sense of being applicable to any applied load. As in the case of a planar interface crack, the crack tip stress field is free of oscillation if the bimaterial matrix H is real. The general results are applied to specific examples and explicit forms of solutions are obtained.

Poroelastic solutions in transversely isotropic media for wellbore and cylinder

This paper presents closed-form solutions for the pore pressures and stress fields for the inclined borehole and the cylinder, induced by boundary stress perturbation in an anisotropic poroelastic medium. The governing equations for the transversely isotropic poroelastic materials under the cylindrical coordinate system are presented. The solution for the inclined borehole, subjected to a three-dimensional far-field anisotropic stress field, is derived with the borehole generator coinciding with the material axis of symmetry ; while the solutions for the cylinder are obtained under various loading conditions that are encountered in laboratory testing procedures. The analysis shows, in addition to the effect of time on stress and pore pressure variations, that poromechanical anisotropic material coefficients also play an important role in calculating the in-plane stress fields, which is not the case using the classical anisotropic theory of elasticity.

The contact behavior between laminated composites and rigid impactors

A general method for analyzing the contact behaviour between a composite laminate and a rigid impactor is presented. The method developed is based on anisotropic elasticity theory of plate by taking into account the normal stress σ z and the interlaminar shear strains ε yz and ε xz. The contact pressure is assumed to be represented by a double Fourier series. The relationship between the force F and the indentation a is then obtained by constructing a Green's function of the laminate. The solution of the problem is performed by using the redundant field point method (RFP). The predictions of the present study agree very well with published experimental results. The method can be applied to general symmetric composite laminates to determine the force-indentation relationship, contact pressure and contact area.

On the dilation of diamond by nitrogen impurity aggregated in A defects

Lattice parameter measurements by modern methods of photography of transmitted divergent–beam X–ray diffraction patterns are reported. A small (less than 10 μm diameter) source of divergent CuKα X–rays is produced by an electron beam impinging on a thin copper film evaporated on to one surface of the diamond, which is placed in a scanning electron microscope. Patterns of high dispersion and high resolution are obtained by use of a 700mm camera length and an Ilford L4 nuclear emulsion as the recording medium. A synthetic diamond volume containing only about one atomic part per 10 6 nitrogen impurity served as a pure–diamond standard. This was compared with a natural diamond of nearly pure IaA spectral type, unusually rich and uniform in content of nitrogen impurity aggregated in A–defect form (N concentration circa 1135 atomic parts per 10 6 ). In both specimens, volumes free from dislocations and bounded by well–polished damage–free surfaces were probed. Advances in pattern–measuring methods included a film–stacking technique to improve signal–to–noise ratio, digitized microdensitometry and the measurement of pattern shifts relative to invariant datum points in the pattern provided by angularly sharp features arising from coherent multiple diffraction effects. Procedures were developed to correct for diffraction–line–profile asymmetry (which depends upon lattice perfection as well as other diffraction parameters). An absolute lattice parameter determination on the pure diamond yielded , a 0 = 0.356710(4)nm, considered to be a good value. After subtracting the calculated dilatation contribution (24%) due to a small population of {001} platelet defects in the nitrogen–rich diamond, the dilatation due to A defects derived is Δa 0 /a 0 = (0.6 ± 0.25) x 10 –6 [μ A(1282cm–1) /cm –1 ], where μ A(1282cm–1) is the infrared absorption at 1282cm –1 due to A defects, in units of cm –1 . In terms of c N(A) , the fractional atomic concentration in the diamond of nitrogen impurity present in A defects, and subject to more uncertainty, Δa 0 /a 0 = 0.036c N(A) , which indicates that a nitrogen atom in an A defect occupies circa 1.11 times the volume of the normal carbon atom it replaces. This dilatation is compared with those expected from computed structures of the A defects.

Surface instability in gradient elasticity with surface energy

Biot's theory of plane strain surface instability of an isotropic elastic body under initial stress in finite strain is extended to include higher order strain-gradients. Higher order straingradients are properly introduced in the definition of the strain energy density, leading to an anisotropic gradient elasticity theory with surface energy. Accordingly, the present theory includes two material lengths characterizing the volume strain energy and the surface energy of the elastic body. The consideration of these two material lengths leads to the occurrence of a boundary layer. This in turn, gives rise to interesting phenomena related to the stability of the half-space, i.e. extra surface instability modes, thin skin effects and significant weakening of the half-space. It is also shown that the appearance of surface instability is associated with the vanishing velocity of propagation of Rayleigh waves. Furthermore, results derived in the context of the present theory on the dependence of the critical buckling stress of the layer on the thickness, suggest that it can be used effectively for the homogenization of elastic bodies containing periodic arrays of collinear Griffith cracks.

Gradient elasticity with surface energy: Mode-I crack problem

The solution of the mode-I crack problem is given by using an anisotropic strain-gradient elasticity theory with surface energy, extending previous results by Vardoulakis and co-workers, as well as by Aifantis and co-workers. The solution of the problem is derived by applying the Fourier transform technique and the theory of dual integral and Fredholm integral equations. Asymptotic analysis of the solution close to the tip gives a cusping crack with zero slope of the crack displacement at the crack tip. Cusping of the crack tips is caused by the action of “cohesive” double forces behind and very close to the tips, that tend to bring the two opposite crack lips in close contact. Consideration of Griffith's energy balance approach leads to the formulation of a fracture criterion that predicts a linear dependence of the specific fracture surface energy on increment of crack propagation for such crack length increments that are comparable with the characteristic size of material's microstructure. This important theoretical result agrees with experimental measurements of the fracture energy dissipation rate during fracturing of polycrystalline, polyphase materials such as rocks and ceramics. The potential of the theory to interpret the size effect, i.e. the dependence of fracture toughness of the material on the size of the crack, is also presented. Also, the theory predicts an inverse first power relation between the tensile strength and the size of the pre-existing crack which is in accordance with experimental evidence. Furthermore, it is shown that the effect of the volumetric strain-gradient term is to shield the applied loads leading to crack stiffening, hence the theory captures the commonly observed phenomenon of high-effective fracture energies of rocks and ceramics; the effect of the surface strain-energy term is to amplify the applied loads leading to crack compliance and essentially captures the development of the “process zone” or microcracking zone around the main crack in a brittle material. Thus, the present anisotropic gradient elasticity theory with surface energy provides an effective tool for understanding phenomenologically main crack-microdefect interaction phenomena in brittle materials.

Failure of micron scale Single Crystal Silicon bars due to torsion developed by MEMS micro instruments

AbstractWe present an experimental study on a single crystal silicon (SCS) bar subjected to pure torsion using MEMS micro instruments. The bar is in the form of a pillar, anchored at one end to the silicon substrate. It is attached to a lever arm at the other end. The pillar has a minimum cross sectional area at its mid height. The cross section coincides with the (100) plane of SCS. Torsion is generated by applying two equal forces on the lever arm on either side of the pillar. Two micro instruments apply the forces. Each consists of an electrostatic actuator and a component that calibrates it. The actuator generates high force (≈ 200 µN at 50 V) and is capable of developing large displacements (≈ 10 μm). Calibration involves determination of the force generated by the actuator at an applied voltage, as well as the linear and higher order spring constants of its springs. Each microinstrument is thus calibrated independently.With the application of forces by the two micro instruments, a torque is generated which twists the pillar. The angle of twist at different applied voltages are recorded using an angular scale. The corresponding torques are determined from the calibration parameters of the actuators. Torque is applied until the pillar fractures. Two such sample pillars, samples 1 and 2, are tested. There cross sectional areas are 1 and 2.25 µm2. We find that both the pillars behave linearly until failure. The stresses prior to fracture are evaluated based on anisotropic theory of elasticity. Samples 1 and 2 fail at shear stresses of 5.6 and 2.6 GPa respectively. The fracture surfaces seem to coincide with the (111) plane of SCS.

The Influence of Solid Solutions on Flow Behavior in -γ-TiAl

ABSTRACTModifications of alloy chemistry are often used to tailor the intrinsic flow behavior of structural materials. Models of solution strengthening, high temperature yield stress and creep must relate the effects of chemistry to the mechanisms that influence these material properties. In ordered alloys, additional information regarding the crystallographic site occupancy of ternary elements is required. In this study relaxed structures and energies for intrinsic and substitutional point defects are calculated using a first principles plane-wave-pseudopotential method. Calculated defect energies are used to predict the density and site preferences of solid-solutions (Si, Cr, Nb, Mo, Ta and W) in ³TiAl. Size and modulus misfit parameters are calculated and the interaction of these defects with a dissociated ordinary screw dislocation evaluated within anisotropic elasticity theory. The derived interaction strength is then related to solid-solution strengthening for these defect centers. Predicted solid-solution effects are in good agreement with experimental observations for the binary alloy.

The formulation of linearized boundary integral equations of the anisotropic theory of elasticity and their application in geometrical inverse problems

An elastic bounded anisotropic solid with an elastic inclusion is considered. An oscillating source acts on part of the boundary of the solid and excites oscillations in it. Zero displacements are specified on the other part of the solid and zero forces on the remaining part. A variation in the shape of the surface of the solid and of the inclusion of continuous curvature is introduced and the problem of the theory of elasticity with respect to this variation is linearized. An algorithm for constructing integral representations for such linearized problems is described. The limiting properties of the linearized operators are investigated and special boundary integral equations of the anisotropic theory of elasticity are formulated, which relate the variations of the boundary strain and stress fields with the variations in the shape of the boundary surface. Examples are given of applications of these equations in geometrical inverse problems in which it is required to establish the unknown part of the body boundary or the shape of an elastic inclusion on the basis of information on the wave field on the part of the body surface accessible for observation.

Mixed Crack Type Problem in Anisotropic Elasticity

The paper deals with three-dimensional mixed boundary value problem of the anisotropic elasticity theory when the elastic body under consideration has a cut in the form of an arbitrary non-closed, two-dimensional, smooth surface with a smooth boundary: on one side of the cut surface the Dirichlet type condition (i.e., the displacement vector) is given, while on the other side the Neumann type condition (i.e., the stress vector) is prescribed. Applying the potential method and invoking the theory of ΨDEs uniqueness, existence and regularity results are proved in various function spaces. The asymptotic expansion of the solution of the corresponding system of boundary ΨDEs is written.