elastic constants of crystals

by Hillard Bell Huntington

Publisher: Academic Press in New York and London

Written in English
Published: Pages: 139 Downloads: 905
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  • Crystals,
  • Elasticity

Edition Notes

StatementH. B. Huntington
SeriesSolid state reprints
The Physical Object
Pagination139 p. :
Number of Pages139
ID Numbers
Open LibraryOL14807680M

Young’s modulus, bulk modulus, and Rigidity modulus of an elastic solid are together called as Elastic constants. When a deforming force is acting on a solid. It results in the change in its original dimension. In such cases, we can use the relation between elastic constants to understand the magnitude of deformation. Elastic constant formula. Note that in general C ≠ E. and the elastic relationship between stresses and strains in crystals must be stated in a more generalized manner: ij C ijkl kl and kl S klij ij (4) In Eq. (4), C ijkl are stiffness constants of the crystal and S klij are the compliances of the crystal and both are a fourth rank tensor (Wooster, ; Nye, ). Crystal Binding and Elastic Constants Figure 1 Dependence of potential energy on interatomic distance. The calculated values of the cohesive energy are compared with experimental results, which can be obtained by measuring the latent heat of sublimation at various low temperatures, and extrapolating to zero Kelvin. Elastic constants and homogenized properties of two monoclinic structures (gypsum and tobermorite) were investigated by first-principles method. The gypsum (chemical formula of CaSO4•2H2O) is an evaporite mineral and a kind of hydration product of anhydrite. Besides, the 11 Å tobermorite model (chemical formula: Ca4Si6O14(OH)42H2O) as an initial configuration of C-S-H structure is.

ELASTIC CONSTANTS. Different types of stresses and their corresponding strains within elastic limit are related which are referred to as elastic constants. The three types of elastic constants are: Modulus of elasticity or Young’s modulus (E), Bulk modulus (K) and; Modulus of . Interest in quartz’s elastic constants continues for three reasons: First, quartz’s elastic constants remain incompletely understood, mainly because quartz is piezoelectric and we lack a good theory for the elastic constants of piezoelectric crystals.7 Second, quartz occupies a central place in crystal. Since the values of v 2 for different directions of propagation and corresponding directions of displacement are also known in terms of the elastic constants c 11, c 12 and c 44, we obtain, by. where c 11, c 12, and c 44 are the independent single-crystal elastic constants of a crystal having cubic symmetry. Zener’ s anisotropy factor yields unity when the crystallite is isotropic.

Elastic is a set of python routines for calculation of elastic properties of crystals (elastic constants, equation of state, sound velocities, etc.). It is a fifth version of the in-house code I have written over several years and is implemented as a extension to the ASE system and a script providing interface to the library not requiring.   Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C 14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. . Single Crystal Elastic Constants and Calculated Aggregate Properties. A Handbook. Second Edition by Simmons, Gene, Wang, Herbert and a great selection of related books, art and collectibles available now at This paper reports the measurement of the five independent elastic constants of a transversely isotropic liquid crystal elastomer. We express the elastic constants Ε L, Ε T, G 12, k 12, and G 23 in terms of strains and stresses, then measure these and determine .

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& Chang, Y.A. and a great selection of related books, art and collectibles available now at Elastic Constants of Crystals [Huntington, H.B.] on *FREE* shipping on qualifying offers. Elastic Constants of CrystalsAuthor: H.B. Huntington.

Data on the elastic properties of single crystals have increased dramatically since Professor Simmons completed his first compilation in While this book is a consolidation of his earlier work, it has been extensively updated and revised to include new material and references, is far more complete, and is presented in a more useful is actually a handbook consisting of computer.

At present it is impossible to calculate the elastic properties of a random, macroscopically isotropic aggregate of crystals from the single crystal elastic constants, but bounds may be obtained for the aggregate properties from the single crystal constants.

Accordingly the book tabulates the Voigt and Reuss averages for all materials for which Cited by: The elastic constants of crystals. [Hillard Bell Huntington] Home. WorldCat Home About WorldCat Help.

Search. Search for Library Items Search for Lists Search for Elastic constants: Document Type: Book: All Authors / Contributors: Hillard Bell Huntington.

Find more information about: OCLC Number: Notes. The Elastic Constants of Crystals | Huntington H.B. | download | B–OK. Download books for free.

Find books. ATOMISTIC THEORIES ELASTIC OF CONSTANTS The usual atomistic treatments of elastic constants can be divided into two categories. The first deals with generalized formulations of the forces between atoms and derives interrelations between such parameters as THE ELASTIC CONSTANTS OF CRYSTALS the elastic, piezoelectric, and dielectric constants.

Elastic Constants of Crystals thick was used and an average value of was obtained for C 4. Treat-ing the plate as an isotropic substance, and using the relation between the Youngs modulus and the constants Cp l and C, a value of is obtained. Appendix A: Crystal Symmetries and Elastic Constants Particular aspects of each Laue group are discussed next.

elastic constants of crystals book The reader is referred to Bond et al. (), Landau and Lifshitz (), and Teodosiu () for additional details. Formal standards on terminology and as- signment of coordinate systems for crystalline solids are given by Bond.

The elastic constants of single‐crystal NbN, VN, and TiN films were determined from surface acoustic wave (SAW) dispersion curves obtained by the use of an acoustic microscope with a line‐focus beam.

Measurements were carried out for single‐crystal nitride films grown on the ( Chapter 3: Crystal binding and elastic constants Evaluation of the Madelung constant • For a stable crystal, Madelung constant () should be positive.

• If we take the reference ion as a negative charge, the plus sign apply to positive ions and the minus. Data on the elastic properties of single crystals have been collected from the literature published through mid The elastic properties of isotropic aggregates (Young's modulus, Poisson's ratio, shear modulus, bulk modulus, compressibility, velocity of shear waves, and the velocity of compressional waves) are calculated according to the schemes of Voigt and Reuss.

constant of proportionality between them results in the number 81 being reduced to (9 + 36) = 45 constants in all. Thus, it requires 45 elastic constants to describe the elastic behaviour of a triclinic crystal. For crystals of other classes, the number of independent constants is diminished by reason of their symmetry properties.

Lecture 7 - Elasticity 3 Physics F Lect 7 13 Elasticity in Cubic Crystals • Elastic Constants C ij are completely specified by 3 values C 11, C 12, C 44 σ 1 = C 11 e 1 + C 12 (e 2 + e 3), etc.

σ 4 = C 44 e 4, etc. Pure change in volume –. Determination of the elastic and piezoelectric constants for crystals in class (3m) is complicated by' the large number of independent constants and the many possible ways in which they may be comhined. An experimental and analytical procedure has been developed to determine all the constants using primarily.

Abstract. The elastic constant matrix at room temperature and the temperature dependence of C 11 constant (30–°C) of Na Bi TiO 3 single crystal have been measured for the first time. The anomaly at the temperature, which corresponds to disappearing of ferroelectric state (∼°C), and broad minimum about temperature, which correlates to maximum of electric permittivity (∼   Values for the elastic constants of random, polycrystalline tungsten were calculated from these crystal stiffnesses.

They agree well with values determined on polycrystalline tungsten as functions of. We propose a new scheme to calculate the elastic constants of solids which is based on linear-response theory.

Elastic constants are given directly by a single self-consistent calculation (i.e., they are not obtained by numerical differentiation of total energies or stresses).

As an illustration, we apply our procedure to the determination of the equilibrium lattice constant, bulk modulus, and. @misc{etde_, title = {Crystal field effects on the elastic constants of single crystals of PrAl/sub 2/ and NdAl/sub 2/} author = {Godet, M, and Purwins, H G} abstractNote = {Measurements are reported of the sound velocity of single crystals of PrAl/sub 2/ and NdAl/sub 2/ between K and K.

For the elastic constants is found at K (in 10/sup 11/N/m/sup 2/):C/sub 11/ = +- 1. The Frank elastic constants K 1, K 2, K 3 are calculated in the mean field approximation by assuming that the intermolecular force is the sum of hard rod repulsion (length L and width D) and Maier-Saupe's type main conclusions are as follows.

(1) the inequlaities K 3 ≫ K 1 > K 2 necessarily hold. (2) All the K' i s are nearly proportional to the square of the orientational.

Elastic, piezoelectric, pyroelectric, piezooptic, electrooptic constants, and nonlinear dielectric susceptibilities of crystals. [A S Bhalla;] Home. WorldCat Home About WorldCat Help. Search.

Search for Library Items Search for Lists Search for Contacts Search for a Library. Create. The elastic constants K 11 and K 33 are calculated from the threshold field H e associated with the Frederiks transition in a magnetic field of a uniform planar and a horneotropic layer, respectively.

The elastic constant K 22 is determined indirectly using a twisted planar errors associated with the determination of H c are critically evaluated.

In particular the anchoring. 30 Years after the appearance of the classic book by Born and Huang on the Dynamical Theory of Crystal Lattices, the present volume untertakes to present the field in the light of the extensive progress that has been made in the last three decades.

Chapter 2 Crystal binding and elastic constants I. Chemical bonds 1. There are two major types of chemical bonds (bonding between atoms to form a molecule: (i) ionic bond, and (ii) covalent bond.

Inoinzation energy (first) is the energy required to move an electron from a neutral isolated atom to form an ion with one positive charge. The calculated elastic constants show quite good agreement with the existing experimental data for almost all the examined systems with the exception of the relatively soft material such as α‐SiO 2 and the C 14 parameter of some trigonal crystals expressed in the hexagonal form such as in α‐Al 2 O 3.

Other structural properties derived. A first principles method is proposed to calculate the Frank elastic constants of nematic liquid crystals. These include the constants corresponding to standard splay, twist and bend deformations, and an often-ignored surface-like contribution known as saddle-splay.

The proposed approach is implemented on th. [3] Battelle-Columbus Laboratories, Engineering Property Data on Selected Ceramics, Vol. 1, Nitrides, Metals and Ceramics Information Center, Battelle's Columbus Laboratories ().Internal Report-M.C.I.C.-HB Vol I. where C ij is the tensor of elastic constants, V is the crystal volume, the Einstein summation rule over the repeated subscripts is implied, and Voigt’s notation for the strains is used.

The quadratic dependence of elastic energy on strains is usually called harmonic approximation. The total number of independent components of the elastic constant tensor depends on the symmetry of the crystal.

Single crystal elastic constants and calculated aggregate properties by Simmons, Gene,M.I.T. Press edition, in English - 2d ed. Time-resolved spectroscopy has been used to examine the elastic properties of single crystal gold nanorods with a [] growth direction.

These rods were produce by seed-mediated growth in the presence of silver ions, using both chemical and photochemical is of the experimental data yields a value of Young's modulus for the nanorods of E NR = 31 ± 1 GPa.

The Frank elastic constants of lyotropic polymer liquid crystals are formulated using an equivalent freely-jointed chain model, which can be uniquely defined by the parameters of the corresponding wormlike chain and the polymer concentration.

The contribution of the higher virial terms to the elastic constants was taken into account on the basis of the scaled particle theory used previously. Some soft matter systems behave similarly and some differently.

Understanding this is one of the significant challenges in the field. For example, while the splay and bend elastic constants for thermotropic and lyotropic nematic liquid crystals are similar, the twist elastic constant is nearly an order of magnitude smaller in lyotropic nematics.

The tables include about present it is impossible to calculate the elastic properties of a random, macroscopically isotropic aggregate of crystals from the single crystal elastic constants, but bounds may be obtained for the aggregate properties from the single crystal constants.

Accordingly the book tabulates the Voigt.