Elastic Properties of Metals and Alloys, I. Iron, Nickel, and Iron-Nickel Alloys

Cite as: Journal of Physical and Chemical Reference Data 2, 531 (1973); https://doi.org/10.1063/1.3253127 Published Online: 29 October 2009

H. M. Ledbetter, and R. P. Reed

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Journal of Physical and Chemical Reference Data 2, 531 (1973); https://doi.org/10.1063/1.3253127 2, 531

© 1973 The U. S. Secretary of Commerce on behalf of the United States. Elastic Properties of Metals and Alloys, I. Iron, Nickel, and Iron-Nickel Alloys

H. M. Ledbetter and R. P. Reed

. Cryogenics Division, Institute for Basic Standards, National Bureau of Standards, Boulder, Colorado 80302

A comprehensive compilation is ~ven of elastic properties of iron-nickel alloys. When sufficient data exist, preferred values are recommended_ This compilation covers, besides pure iron and pure nickel, the entire binary composition range, both b.c_c. and f.c.c. phases. Elastic constants included are: Young's modulus, shear modulus, bulk modulus (reciprocal compressibility), Poisson's ratio, and single­ crystal elastic stiffnesses, both second-order and higher-order. Data are compiled for variation of ela~tic com;tant<;. with rompo!>.ition, tpmpprlltllre, prp!>.!>.nrp., magnetic neld. mechanical deformation. annealing, and crystallographic transitions. An overview is given f~om the vantage points of the electron theory ·of m.etals, elasticity theory, and crystallographic theory. Also included are discussions of iso­ thermal and adiabatic elastic constants, interrelationships among engineering elastic constants, com­ putation of the latter from single-crystal elastic stiffnesses, and similar topics. Where key data have not been measured, they were generated if possible from existing data using standard formulae. Other gaps, both theoretical and experimental, in the elastic properties of iron-nickel alloys are indicated. A few theoretical results are included where experimental data are nonexistent or scarce. A semantic scheme is proposed for distinguishing elastic constants of solids.

Key words: Bulk modulus; compressibility; elastic constants; Debye temperatures; iron; iron alloys; Lame constants; nickel; nickel alloys; Poisson's ratio; shear modulus; single-crystal elastic coefficients; Young's modulus. Contents

Page Page List of Tables_ .... _._...... 531 13. Temperature Dependence of Elastic Con- List of Figures ...... 532 stants ...... ; .... . 546 Principal Notations ...... 532 14. Pressure Dependence of Elastic Constants .. . 548 15. Magnetic Field Dependence of Elastic Con- 1. Introduction ...... •...... 533 stants ...... 549 2. Perspective on Elastic Constants •...... 534. 16. Effects of Crystallographic Transitions on 3. Terminology ...... 535 Elastic Constants ...... 550 4. Definitions of Elastic Constants ...... 535 17. Theoretical Calculation of Elastic Constants. 551 5. Relationships Among Elastic Coefficients 17.1. Fundamental Models ...... _. 551 and Engineering Elastic Constants ...... 537 17.2. Hard-Sphere Model...... 552 5.1. Isotropic Case (A = 1) ...... 537 18. Relationship of Elastic Constants to Other 5.2. Anisotropic Case (A ::;61) ...•.•.•.•.•••.. 537 Physical Parameters ...... 553 6. Measurement Methods and Errors ...... 538 19. Concluding Remarks ...... 556 6.1. Measurement Methods •...... 538 20. References for Text...... 556 6.2. Mea8un~IIlellL En·ur:; ...... 539 21. Bibliography of Reviews and Compilations .. . 560 7. Isothermal and Adiabatic Elastic Constants .. . 539 22. Older References ...... 560 8. Elements Iron and Nickel...... 540 22.1. Iron, Young's Modulus ...... 560 9. Iron-Nickel Phase Diagram ...... 540 22.2. Iron, Shear Modulus ...... 561 10. Compositional Dependence of Elastic Con- 22.3. Iron, Poisson's Ratio ...... 561 stants ...... 541 22.4. Nickel, Young's Modulus ...... 561 11. Effects of Mechanical Deformation, Amieal- 22.5. Nickel, Shear Modulus ...... 561 ing~ Recovery and Lattice Defects ...... 542 22.6. Iron-Nickel Alloys, Young's Modu- 11.1. Mechanical Deformation ...... 542 lus ...... , ...... 561 11.2. Annealing ...... 543 23. Notes on Tables .. , ...... 561 11.3. LaLLice Defect:; ...... 543 24. TaLle:5 aud Figule:5 ...... ri62 12. Higher-Order Elastic Coefficients ...... 544 25. References for Tables and Figures ...... 613

list of Tables

1. Connecting Identities X = X (Y,Z) for Elastic Con­ Elastic Compliances Sij for Cubic Crystals stants of Quasi-Isotropic Solids 4. Selected Properties of Iron and Nickel 2. Expressions for Engineering Elastic Constants of 5. Second-Order Elastic Stiffnesses Cij of Iron Cubic Single Crystals 6. Second-Order Elastic Stiffnesses Cij of Nickel 3. Conversions Between Elastic Stiffnesses Cij and 7. Second-Order Elastic Stiffnesses Cij of Iron-Nickel Copyright © 1973 by the U.S. Secretary of Commerce on behalf of the United States. This Alloys copyright will be assigned to the American Institute of Physics and the American Chemical Society. to whom all requests regarding reproduction should be addressed. 8. Young's Modulus E of Iron

531 J. Phys. Chern. Ref. Data, Vol.. 2, No.3, 1973 532 H. M. lEDBEnER AND R. P. REED

9. Young's Modulus E of Nickel 20. Third-Order Elastic Stiffnesses Cijk of Iron 10. Young's Modulus E of Iron-Nickel Alloys 21. Third-Order and Fourth-Order Elastic Stiffnesses 11. Shear Modulus G of Iron Cijk and Cijkl of Nickel 12. Shear Modulus G of Nickel 22. Pressure Derivatives dCij/dP of Elastic Stiffnesses of 13. Shear Modulus G of Iron-Nickel Alloys Iron and Nickel 14. Bulk Modulus B and Compressibility B-1 of Iron IS. Bulk Modulus B and Compressibility B-1 of Nickel 23. Relative Elastic Constants Based on a Hard-Sphere 16. Bulk Modulus B and Compressibility B-1 of Iron- Model, B = 1 Arbitrarily Nickel Alloys 24. Debye Characteristic Temperatures of Iron, Nickel, 17. Poisson Ratio lJ of Iron and Iron-Nickel Alloys Calculated From Single­ 18. Poisson Ratio lJ of Nickel Crystal Elastic Data 19. Poisson Ratio lJ of Iron·Nickel Alloys 25. Conversions Among Commoply Used Stress Units

List of Figures 1. Schematic Interconnectivity of Elastic Parameters 19. Temperature Variation of Young's Modulus E of, of Solids Nickel 2. Chronological Variation of Observed Elastic Prop­ 20. Temperature Variation of Young's Modulus E of ertielS of Iron II-on-Nickel Alloy:'5- Face-CtuttlCd Cubk 3. Iron-Nickel Phase Diagram 21. Temperature Variation of Young's Modulus E of 4. Compositional Variation of Elastic Stiffnesses Cij of Iron-Nickel Alloys-Body-Centered Cubic Iron-Nickel Alloys 22. Relative Temperature Variation of Young's Modulus 5. Compositional Variation of Young's Modulus E of E of Some Iron-Nickel Alloys Iron-Nickel Alloys 23. Temperature Variation of Shear Modulus G of Iron 6. Compositional Variation of Shear Modulus G of Iron­ 24. Temperature Variation of Shear Modulus G of Nickel Nickel Alloys 25. Temperature Variation of Shear Modulus G of Iron­ 7. Compositional Variation of Compressibility B-1 of Nickel Alloys Iron-Nickel Alloys 26. Relative Temperature Variation of Shear Modulus G 8. Compositional Variation of Poisson ratio 2-' of Iron of Some Iron-Nickel Alloys Nickel Alloys 27. Temperature Variation of Compressibility B-1 of 9. Cold-Work D~pendence of Young's Modulus E of Iron Iron 28. Temperature Variation of Compressibility B-1 of 10. Cold-Work Dependence of Shear Modulus G of Iron Nickel 11. Cold-Work Effect on Young's Modulus E of Nickel 29. Temperature Variation of Compressibility B-1 of 12. Cold-Work Effect on Shear Modulus G of Nickel Iron-Nickel Alloys 13. Cold-Work Effect on Young's Modulus E uf Fact­ 30. Tempel ature Valiatiuu of Puh;~uJl naliu II uf 11011 Centered Cubic Iron-Nickel Alloys 31. Temperature Variation of Poisson Ratio v of Nickel 14. Annealing Effect on Young's Modulus E of Nickel 32. Temperature Variation of Poisson Ratio v of Iron­ 15. Tp.mpp.ratnrp. Variation of F.la~ti~ Stiffnp.~~p.~ Cij of Nickel Alloys Iron 33. Magnetic Field and .Temperature Effects on Yol.mg's 16. Temperature Variation of Elastic Stiffnesses Cij of Modulus E of NiCkel Nickel 34. Magnetic Field and Temperature Effects on Young's 17a. Temperature Variation of Elastic Stiffnesses Cij of Modulus E of Iron-Nickel Alloys Iron-Nickel Alloys 35. Magnetic Field and Temperature Effects on Shear 17b. Temperature Variation of Elastic Stiffness Cll of Modulus C of an Iron-Nickel Alloy Iron-Nickel Alloys 36. Magnetic Field and Temperature Effects on Bulk 17c. Temperature Variation of Elastic Stiffness C12 of Modulus B of an Iron-Nickel Alloy Iron-Nickel Alloys 37. Magnetic Field Effect on Ynung's Modulus E of 17d. Temperature Variation of Elastic Stiffness C44 of Nickel und Iron-Nickel Alloye Iron-Nickel Alloys 18. Temperature Variation of Young's Modulus E of 38. Phase Transformation (bee to fcc on Heating) Effect Iron on Younv;'f! Mortulus E oflron-Nickel Alloys

Principal Notations

A elastic anisotropy (after Zener) B hulk lnUfluJll10 A* elastic anisotropy (after Chung and Bues­ .uH'mHj·ord(~r clastic stiffnesses (contracted, sem) full)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 533

Cijkl, Cijklmn third-order elastic stifInesses (contracte~, Sij, Sijkl elastic compliances (contracted, full) full) S entropy ClI, C12, C44 three independent elastic stiffnesses fPr T temperature (degrees Kelvin) cubic symmetry u displacement v velocity C =C44 v,V specific volume, volume C' = (Cll -c12)/2 a linear thermal expansion coefficient specific heat at constant x ex f3 volume thermal expansion coefficient E Young's modulus shear strain F Helmholtz energy 'Y Griineisen's constant G shear modulus 'Y ~ij Kronecker's delta G Gibbs energy E strain h Planck's constant strain H enthalpy "fJ 8 Debye characteristic temperature Ii tensor in variants A Lame constant k Boltzmann's constant A magnetostriction constant l, m,,'f, M urnaghCLIl '::I third-urder 4. uCL::Ii-isutru»ic Lame constant (= G) elastic constants JL v Poisson's ratio M general elastic modulus p mass density N Avogadro's number (T stress p pressure T shear stress

1. Introduction ing shows that elastic constants are key design param­ eters. Indeed, few stress-bearing members .can be designed adequately in ignorance of the elastic proper~ In this century, solid-state physics has advanced ra­ ties of the constitutive material. matically. Essential to this advance was accumulat on Engineering materials that are macroscopically isot­ of vast quantities of data describing properties of soli s. ropic are completely. defined elastically by two param­ As much as any set of properties, elastic properties w re eten;. Parameter 5election is arbitrary, varying with central in this advance. Similarly, technologies suc as application. As examples: pressure vessel design metallurgy and ultrasonics have matured only throug a requires knowing the bulk modulus, design of rotating necessary knowledge and understanding of elastic p p­ shafts requires knowing the rigidity modulus, and design erties of solids that provide their basis. of flexed beams or support columns requires knowing Most solids are crystalline, therefore elastically ni­ Young's modulus. sotropic, and are either single crystals or polycryst 'ne aggregates. Thus. crystalline studies form a large art For some special scientific purposes it is sufficient to . of solid-state physics, which attempts generally to re ate know the elastic properties of a material in some refer­ properties of atoms and atom groupings to macrosc pic ence state, for example-a pure, annealed, defect-free, properties of solids at various temperatures~ pressu es, single crystal at 0 K and 1 atmosphere. However, for etc. ordinary purposes the elastic properties of a material While most phenomenological crystal elasticity must he known in other, non-reference, states. Depart­ developed by late 19th century, systematic studies ure from the state of reference usually involves varying practical applications of crystalline elasticity emer ed one or more of-composition, temperature, pressure, only recently. Uses of crystals in solid-state devices nd mechanical deformation, magnetic field, or degree of engineering applications of elastic solids increased ub­ poly crystallinity. Thus, an understanding is sought as to stantially since about 1945. ·Measurement of ela tic how these variables affect elastic properties. These properties by ultrasonic techniques facilitated t ese topics are discussed in sections 10-11 and 13-16. applications~ The purpose' of this paper is to present a compre­ With the view that elasticity of solids will be incr as­ hensive compilation and a critical overview of the ingly important in both science and t~chnology, a be 'n­ elastic properties of alloys of the binary system iron­ ning is made here toward a comprehensive compila ion nickel. These alloys hold much interest for both en­ and critical review of elastic properties of ~elected ys­ gineers and scientists, for both metallurgists and terns that hold high interest for both of these comm ni­ geophysicists. Iron-nickel allOYS also provide fertile ties. For examples of practical applications of el tic ground for probing relationships between elasti,c con­ properties one need only consult any standard refere ce stants and phase transitions, an area of physics and on strength of materials. Perusal of various formula for metallurgy now ripe for both theoretical and experi­ describing states of stress such as compression or b nd- mental study. Since both iron and nickel are ferro-

J. Phys. Chern. Ref. Data, Vol. 2, No.3, J973 534 H. M. LEDBETIER AND R. P•. REED magnetic, their alloys show a variety of curious magnetic properties of metals and alloys are described in general effects, many of which are important technologically; terms. Much of the discussion necessarily applies also to and magnetic effects are frequently coupled to elastic other types of solids such as ionic or covalent crystals. constants, as for example in magnetostriction. Similarly, However, readers should make such extrapolations only lnvar 1 is an alloy of Fe-36Ni 2 where magnetic effects cautiously. combine with thermal expansion such that virtually no thermal expansivity is exhibited over a wide temperature 2. Perspective on Elastic Constants range, one of the curious exceptions to the almost uni­ versal thermal expansion of solids. Invoking three realms of knowledge of solids-pure Other materials such as Covar (thermal expansion science, phenomenology, and engineering-elastic coefficient similar to that of glass) and Permalloy parameters of solids are distributed among these realms as shown schematically in figure 1. As far as elastic prop­ (exceptionally. high magnetic permeability) are based on iron-nickel binary alloys. Metallic meteorites, also erties of solids are concerned, objects of interest in these scientific curiosities, are iron alloys containing as much realms are, respectively- discrete atoms or molecules, as 60 percent nickel. Extensive meteorite metallography anisotropic continua, and quasi-isotropic' continua. . The a and f3 can be taken to represent extensions (or has been done to deduce the thermal-mechanical history of meteorites to elucidate the problem of planetary contractions) and bendings of valence bonds between genesis. Finally, the earth's core is generally assumed atoms in solids; subscript i denotes the various sets of to be an alloy of iron and nickel, and elastic constants atomic neighbors. The Ci/S represent elastic stiffness enter many geophysical calculations, notably those coefficients that relate stresses to strains; both stress dealing with seismic-wave propagation. The many tech­ and strain are specified with respect to a set of axes nological applications of Fe-Ni alloys are discussed by denoted by indices i and j and usually chosen to coincide with crystallographic axes. E, G, B, and v denote the Chickazumi [1~,3 Rosenberg [2], and Everhart [3]. This review collects, for the first time, all available Young's modulus, shear modulus, bulk modulus, and data on the elastic properties of iron-nickel alloys. Poisson ratio, parameters arising naturally in char­ acterizing, respectively, uniaxial loading, shear loading, Data are discussed from the viewpoints of the modern theory of metals, elasticity, and crystallography in an hydrostatic loading, and transverse strain under uniaxial attempt to understand, as II:J.uch as possible, in a unified loading. (In section 4 all of. the elastic constants are defined. In section 5 relationships between single-· way the elastic properties of iron-nickel alloys. This is crystal and polycrystal c()!lstants are discussed.) useful because: (1) most of the literature on the subj~ct Many properties of solid~ are related to elastic coeffi­ is purely experimental, and (2) an overview facilitates cients. The most important of these properties is Debye's semi-quantitative extrapolation of existing information characteristic temperature (J. Several techniques exist to conditions where neither experimental nor theoretical for computing () frum the Cij. In turn, () relates directly data are available. to such properties of solids as heat capacity, intensities Most data are available as engineering elastic con­ of Bragg diffractions, Mossbauer emission, thermal con­ stants: Young's modulus, shear modulus, bulk modulus, nnctivity, electrical resistivity, superconducting transi­ and Poisson's ratio. Single-crystal data exist also, but tion temperature, etc. Many of thes~ relationships ~re less abundantly. Relationships between single-crystal discussed in section 18. data and engineering data are discussed in section 5. Elastic coefficients are central in considering defects Besides the compositional variation of the elastic con­ in solids such as vacancies, interstitials, substitutional stants, data on the effects of temperature, pressure, mag­ impurities, dislocations, twin boundaries, and grain netic field, mechanical deformation, and annealing are boundaries. Being related to the second spatial derivative also included and discussed. This review is intended to of the interatomic potential, ela.:stic coefficient:"! relate provide a convenient source of information on the elastic intimately to the problem of cohesion in solids, an properties of iron-nickel alloys and to stimulate further especially important. problem for metals. In this regard, research, both theoretical and experimental. Gaps in pure-shear elastic coefficients allow for volume effects the klluwledg,e aud undensli:l.uding, uf the daIS lie pW,lJep to be separated from non-volume effects. Many crystalline ties of iron-nickel alloys are delineated. phenomena- thermal expaup,jnll, temperature and pres­ Since the present article is seen as the first of a series, the discussion herein is more general and more extensive sure derivatives of the 8(~cond·()rder elastic coefficients, than is necessary for treating only the elastic properties thermal condudivity 01 inRulntufs, sound wave attenua­ of iron-nickel alloys. Remarks special to iron-nickel are tion, ete. - are llTllwrmonir: effects related directly to so designated whenever appropriate. Mainly, the elastic the existenee of .hircl·ont(~r, and higher-order, elastic GOIl}!tantA. Tl"HI. Iwttm' undt~rst.unding, both experimental 1 Trade names are used to facilitate understanding of work presented; no approval. endorse­ and Ilwort'lkul. of dn~'ic cunstants of solids pays ment, or recommendation by NBS is implied. 2Throughout, compositions are expressed as weight percent nickel. Because iron llnd c1ivicl(~ndl'l in cHvc'nu' wuys, often unsuspected. (Section nickel have similar atomic weights, weight and atomic percentages always differ by Ie," 1 I.:~ I n'nt~ IId..,ly Illt' topic of lattice defects and elastic thu.n lwu Vc.rCt:J1L. 3 Numerals in brackets indicate literature references al the end of Ihis paper. ('oIlMlnntl',)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 535

3. Terminology Equation (4.1) means that the strain response of an elastic material is instantaneous, independent of Despite several proposals, there is no generally rate or history. Removal of force restores the original accepted, rational and consistent terminology for reference configuration; no strain exists· in the un­ describing the elastic parameters of solids that form stressed state and vice versa. If non-mechanical the subject of this review. This problem becomes (electrical, magnetic, thermal, etc.) effects are incorpo­ especially acute when semantic distinctions are required rated, then stress and strain may not vanish simultane­ among: (1) the general class of all elastic parameters, ously unless the non-mechanical effects are subtracted (2) the sub-class of the parameters used commonly to from the total stress and strain. Metals approximate describe quasi-isotropic or engineering solids, and (3) Hookean behavior within their elastic limit; plastic the sub-class of parameters used to describe anisotropic deformation frequently begins before non-linear elastic single crystals. Therefore, a scheme of usage is proposed behavior becomes significant. here that corresponds closely to that usually accepted Except for third-order elastic constants, the elastic but makes useful semantic distinctions: constants described herein are either those Cijkl in eq (4.1) or can be derived simply from them. The Cijkl are components of a fourth-rank tensor and an~ (;all~d Lhtj dalSLic 8Lifi'ne88 coefficitjnL8. Gtjlltjr­ E = Young's. mOd.uluS } (polycrystal) ally, there are 34 = 81 such components. However, G = shear modulus ela~i~ engineering mo U elastic thermodynamic and symmetry considerations show for B = bulk modulus } elastic ~on!ltant!l all crystal systems that p - f'OiIOIOOll'lO liltio cunsUmts eij = elastic stiffnesses } (single·crystal) (4.4) SI} = elastic compliances elastic coefficients Complete com mutability of indices occurs only when Adoption of this, or a similar, semantic scheme in the Cauchy's relations, based on central forces between literature of elastic phenomena of solids should allow atoms and inversion symmetry, hold. In metals these for a more logical and a more lucid description of the relations are broken because of the conduction electrons subject than has resulted previously. In much of the acting through electron-electron and electron-ion interactions. existing literature the Cij are referred to simply as "elastic constants". Thus, an alternative to the present From eq (4.4) it follows that in general only 21 Cijkl are 4 proposal would be to substitute "constants" for "coeffi­ independent. If Voigt's contracted notation is invoked, then the Cijkl can be represented as a symmetric 6 X 6 cients" and perhaps "parameters" for "constants". matrix, and Hooke's law becomes Also, some authors prefer "technical", "practical", or "bulk" to "engineering". Ua = CalJ EIJ (4.5)

4. Definitions of Elastic Constants where repeated indices are summed from 1 to 6 and Elastic materials obey Hooke's law CalJ = Cijkl (4.1) (T,,= (Ttj

where repeated indices are summed from 1 to 3, Uij are EIJ = Ekl, f3 = 1, 2, 3 stress components given by

= 2Ekl, f3 = 4, 5, 6. (4.2) For cubic symmetry, which is the case for iron, nickel, where Ji are the components of a force actin~ internally and all iron-nickel alloY5, the 21 CalJ (;umpulltjnls reduce on an imaginary plane (with normal n) of a body, and Ekl to three: are the (infinitesimal) strain components given by

(4.6) (4.3)

(4.7) and Uk, Ul are components of the displacement. While Hooke's law emerged experimentally, it can and be derived in various ways. For example, Lanczos [4] (4.8)

showed that Hooke's law· is the necessary stress-strain • The Voigt contraction scheme is summarized ~s follows: relationship when Hamilton's variational principle ij, kl: II 22 33 23,32 13,31 12, 21 is invoked. a,13: 1 2 3 4 5 6

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 536 H. M. LEDBEnER AND R. P. REED all other Ca /3 are zero. (4.19) . The elastic compliance coefficients Sijkl have the same symmetry as the Cijkl and their relationship is inverse The constant C= C44 relates a shear stress on a {100} plane to the shear strain in any direction in that plane. (4.9) For a shear stress u acting along (IOO) [010] and effect­ ing a shear angle I' where I denotes the identity matrix. t'ormulae for con­ verting Ca.~ to Sa.~ and vice versa are collected in table 3. CT6= CT, (4.20) For cubic symmetry, Hooke's law in matrix form is and E6= y. (4.21) CTt Cll C12 C12 0 0 0 El All other stress and strain components are zero. Thus, eq (4.1) becomes CT2 C12 Cll Cl2 0 0 0 E2 C6S= CT/Y= C44= C. (4.22) CTa C12 C12 Cll 0 0 0 E3 = (Cll (4.10) The constant C' -c12)/2 relates a shear stress on a {1l0} plane to a shear strain along (110). This is 0"4 0 0 0 C44 0 0 e4 equivalent to the C44 case just described', but rotated ± 45° about [001]. In this case the non-vanishing stress CTs 0 0 0 0 C44 0 Es components are

CT6 0 0 0 0 0 C44. E6 (4.23) The choice of three independent elastic constants is and the non-vanishing strains are not unique. For example, a useful alternative set advo­ cated by Zener [5] is El =- E2 = 1'/2 (4.24)

:50 that eq (4.1) become:5 in thi:5 ca:5e

(4.12) (Cll -c12)/2= u/Y= C' C44 rotated ±45° about [001]. and (4_25)

(4.13) For isotropic materials there are only two independent elastic constants. The Lame constants A and fJ. were the A set arising frequently in ultrasonic experiments is C, first set to be used. For isotropic media, Hooke's law C', and in a redu.ced matrix form ,is

c,. = (CII +C12 + 2C44)/2. (4.14) o 0 0 El

It can be shown that B is the bulk modulus for cubic o 0 0 E2 symmetry. Thus, when a hydrostatic pressure CTa (4.26) o o o is applied to a cubic crystal the strains are o o o o J.L 0 E5 El = E2 = E3 =-ilV/3V(E4= E5 =Es= 0), (4.16) o o o and eq (4.1) becomes

B=-PI(l~VIV). (4.17) Clearly, for isotropic media

For cubic symmetry the compressibility (4.27)

(4.28) (4.18) and is given by (4.29)

J. Phys. Chern. Ref. Data, Vol. 2, No. 3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 537

Much recent work on isotropic media uses, instead of waves in an infinite medium, that is to the so-called A and JL, such constants as the bulk modulus B, Young's body or volume waves. modulus E, the shear modulus G, and Poisson's ratio v. The operational definitions of E, B, G, and v for B is defined by eqs (4.13) and (4.17). isotropic media apply also to' single crystals. If these Young's modulus is defined as the ratio of uniaxial measurement operations are applied to single crystals, stress (tensile or compressive) to strain measured results depend on the crystal axis (hkl) along which the along the same axis. Thus, eq (4.1) becomes for stress crystal is tested. Elastic constants so obtained are along [100] designated Ehkl, Ghkl , and Vhkl and are related to the Cij and Sij by formulae summarized in table 2 for cubic

E == 0"1=...!.o • symmetry. (Note that Bhkl=B for all possible hkl.) El Sl1 (4.30) Thus, by measuring engineering constants on single crystals along three independent directions (100, 110, There can be only one shear modulus in the isotropic and 111 are the simplest) Cij and/or Sij can be deter­ case, so that mined. Before the advent of ultrasonic measurement methods, this method was used extensively to determine (4.31) co; it is still used occasionally. Poisson's ratio is defined as the negative of the ratio 5. Relationships Among Elastic CoeHicients and of transverse strain to longitudinal strain for the case Engineering Elastic Constants of uniaxial stress. For stress alon~ fOOl], As described above, cubic crystals are characterized by three elastic coefficients while isotropic and quasi­ (4.32) isotropic solids are characterized elastically by two con­ :stantl5. The relatiolll5hip between the con:stantl5 E, G, B, Zener [S] introduced an additional parameter, a and v of a quasi-isotropic material and the Cij of a single dimensionless ratio crystal of the same substance is considered here. 5.1. Isotropic Case (A 1 ) (4.33) = The isotropy condition for cubic crystals is called the elastic anisotropy factor. 5 Clearly, A = 1 for isotropy since all shear moduli, including C and C', are (S.l) equal. Aluminum and tungsten are the only cubic metals with known elastic coefficients where A = 1. Zener's When this relationship is satisfied, identities in table 1 anisotropy concept has been very useful in discussing hold among various constants for both single crystals b.c.c. lattice instabilities. In fact, Barrett [7] was led and polycrystals. by this consideration to discover low-temperature martensitic transformations in both lithium and sodium. 5.2. Anisotropic Case (A #= 1) Iron-nickel alloys have moderate to high elastic anisot­ In computing two constants from three, the problem is ropy. As discussed below, marteilsitic transitions in generally overdetermined. The overdeterminancy was iron-nickel alloys can be characterized by a high value eliminated in the case A = 1 by eq (S.I). Many different of A even though the parent phase is f.c.c. as opposed approaches have been suggested for the anisotropic case. to b.c.c. for lithium and sodium. A general review of this subject has apparently not been Energetic considerations show that B, E, G (both C made, although comparisons of many of the methods and C' for cubic crystals) are all positive. Bounds on were made by Ledbetter [8]. Two of the proposed meth­ v are 1/2 and -1. No negative values ofv have eyer been ods are discussed here since they are important histor­ observed for isotropic media. Typically, v ranges from ically ~ and their basic assumptions establish upper 'and 0.25 to 0.4S for metals. Lacking any data, the best gues's lower limits for the correct result. A compromise aver­ for metals is v = 1/3. aging method is then discussed that gives good results Connecting identities among E, G, B, and v are with quite simple formulae. given in table 1. Other variables included there are The problem is one of averaging a tensor property over Lame's constants A and JL, longitudinal and transverse all possible spatial orientations~ that is, sound velocities VI and Vt, quasi-isotropic elastic stiff­ nesses C~j' and quasi-isotropic elastic compliances S~j' (Cij) =-21 11T i1T f(cp,O) sin 0 dO dcp (S.2) The sound wave velocities considered herein refer to 7r 'P=O 8=0

'There is no unique definition of crystalline elastic anisotropy. For example, Chung and where f(cp,O) contains the directional dependence of -Buessem [6J proposed an improved form, although slightly more complicated. Their factor the Cij. Many schemes have been proposed for solv~ng A* is zero for isotropy, gives better relative magnitudes of elastic anisotropies than does A, and, is especially useful in interpreting A < 1 cases, which occur frequently for non·metals eq (S.2) or its equivalent, but these have not· been and also for metals such as vanadium and chromium. reviewed critically.

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 538 H. M. LEDBEnER AND R. P. REED

Voigt [9] assumed constant strain and averaged the elastic constants while derived. methods measure Cij with the result nonelastic parameters. Secondly, static and dynamic methods are distin­ (5.3) guished. Static methods are characterized by time­ independent or, at least, slowly-varying (quasi-static) Reuss [10], on the other hand, assumed constant stress applied stresses. Slow deformation rates ensure heat and averaged the Sij with the result exchange with surroundings. Thus, static methods are constant temperature, or isothermal. Dynamic methods G = ___...:;5 __ _ are characterized by time-dependent, rapidly-varying R (5.4) 4(Sl1 - S12) + 3S 44 ' stresses that preclude heat transfer between a specimen and its environment. Thus, dynamic methods are isent­ For both cases, the bulk modulus is given by eq '(4.13). ropic or adiabatic. Static methods have been important Thus, all engineering elastic constants can be deter­ historically, but have been largely pre-empted by mined from G and B by using equations in table 1. Hill dynamic methods. (Distinctions between isothermal and [11) showed on thermodynamic grounds that eqs (5.3) adiabatic elastic constants are discussed in section 7.) and (5.4) represent upper and lower limits on the shear Emergence of high-frequency (ultrasonic) experimental modulus and propo~ed either Rn Rrithmetic or a geo­ methods has heen accompRniell naturRlly by smaller metric average, that is, specimen sizes, about one centimeter. For efficiency and accuracy in the case of single crystals, specimens are (5.5) cut and oriented on specific crystallographic planes. or Thirdly, relaxed and unrelaxed moduli are distin­ (5.6) guished. Usually, measuring an elastic constant involves imposing a vibrational frequency on a specimen. Relaxa­ where subscripts V and R refer to Voigt's and Reuss's tion processes due to interstitial impurities, disloca­ approximations. Hill's method, owing to both its sim­ tions, grain boundaries, residual stresses, etc. can occur plicity and its reliability, is used widely and was used within the specimen. A measurement frequency lower here. While Hill's average has no theoretical or physical than the natural frequency of the relaxation events meas­ basis, it agrees closely (particularly for low elastic ures a relaxed elastic constant. Conversely, a measure· anisotropy) with averages that have such bases. As ment frequency exceeding the relaxation frequency discussed by Landau and Lifshitz [12], the problem of measures unrelaxed elastic properties. If super-imposed averaging the Cij to obtain R, (;., v, etc. cannot be solved and relaxation frequencies are about equal, then consid­ uniquely. erable internal friction or energy dissipation results, and measured elastic constants fall somewhere between 6. Measurement Methods and Errors relaxed and unrelaxed values. Differences between relaxed and un relaxed constants are usually less than a 6. 1. Measurement Methods few percent. Relative merits of various experimental methods for The subject of measuring elastic constants has been specific materials, conditions, and elastic constants are reviewed extensively by several authors, [13]-[22]. Thus, discussed in references [13]-[22]. only those experimental aspects essential to under­ ~tanding Hnd interpreting the data herein will be Aside from magnetic effects, which are discussed described briefly. below, there are two additional problems in measuring Direct, indirect, and derived methods are distin­ elastic properties of iron-nickel alloys. quished first. Direct methods are based on definitions of First, impurities. Effects of substitutional impurities elastic constants. For example, measuring Young's can be estimated from t~e work of Speich, et aI. [23] who modulus directly requires measuring simultaneously determined the variation of E and G for Fe due to alloying stress and strain along a uniaxial loading direction. elements. Carbon is the principal interstitial impurity, Indirect methods are based on calculations using other an,.) ~mR 11 concentrations have negligible effect on elas­ measured elastic constants as input. Thus, an indirect tic properties of iron-nickel alloys; Whether large con­ measurement of Young's modulus could be made by centrations of carbon increase [24] or decrease [25] measuringB and v and using the formulaE= 3B/(l-2v). elastic stiffnesses is still unresolved. Most theory pre­ A derived method involves a physical relationship dicts an increase. But most experiments reveal a betwee'n ,an elastic constant and some nonelastic param­ decrease due to carbon [25]. eters, the latter being measured and the elastic Secondly, b.c.c. phases that contain twins as a result constant then being calculated. For example, Young's of the f.c.c. to b.c.c. transition may show spuriously low modulus is a simple and well· known function of longi· elastic stiffnesses due to a twin-boundary contribution to tudinal and transverse sound-wave velocities and mass the strain. This problem becomes acute near -35Ni density. Both dir~ct and indirect method~ meR~llre whp.re thp. m::Jrtp.nsite is heavily twinned, but it can be

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 539 overcome by using high-frequency measurement where A.ij, A.kl= the temperature coefficients of stress at methods. constant strain, and C£ = heat capacity per unit volume at constant strain. 6.2. Measurement Errors For cubic symmetry The emergence of dynamic methods, particularly all = a22= a33 = a, (7.3) ultrasonic techniques in the megahertz region, has allowed such high precision that elastic constants are au = al3 = a23 = 0, (7.4) now among the most accurately measurable properties of solids. Static methods are limited mainly by the im­ A. u = A.22 = A.33 = A., (7.5) precision of measuring stress and strain, and extensive and calibrations are usually required. Dynamic methods A.J2 = A.I3 = A. aa = 0; (7.6) employ relatively small stresses and strains, namely thus those produced by a piezoelectric transducer. Whereas A. = 3aBr , (7.7) static methods claim accuracies of about one percent, dynamic methods can detect relative elastic modulus so that in the cubic case, in compact notation: changes as small as 10-8 (see Holder [26]). Absolute values of elastic moduli are restricted primarily by transducer-specimen bond corrections, phase-shift cor­ a 2 T rections (see McSkimin [27]), non~flatness and mis­ S11 S - Sl1 r = S12 S - Sl2 r = - Cp , (7.8) orientation of specimen faces (see Watermnn [2B]), and transducer misorientation. Careful specimen prepa­ ration and proper experimental techniques can result in S44s - S44r = 0, (7.9) absolute errors being as small as 10-4• Chronological variations of the engineering elastic Cll - = Cl2S - 2 constants of iron are shown in figure 2. While more S CUr C12r = Bs - Br= 9a Bf TIC v, recent values are more self-consistent, none of the older (7.10) values can be excluded on the basis of deviations from and the mean. (7.11) 7. Isothermal and Adiabatic Elastic Constants For completeness, relationships between isothermal Distinctions between isothermal and adiabatic elastic and adiabatic engineering elastic constants are given constants are discussed here. (These constants are also: denoted herein by subscripts T and S denoting constant

temperature and constant entropy, respectively.) The _ Er _ 2 Tf32 distinction arises naturally from measurement methods. Es- l-Er (Tf32/9C p) - Er+ (Er) 9Cp' Slow or static loading gives isothermal constants while (7.12) rapid or dynamic loading gives adiabatic constants. For most engineering purposes, differences between the two cases are negligible, being a few percent or less. (7.13) However, for detailed comparisons between elastic constants, this difference becomes important. Using thermodynamic relations generalized to include elastic strain energy it can be shown that [29]

(7.1) (7.14)

where Sijkl = p.la!'ltlp. p.ompli:mces; a;j, akl = thermal and expansion coefficients; T = temperature, and C cr = heai Gs = Gr. (7.15) capacity per unit volume at constant stress (J". Since solids have, almost invariably, positive thermal expan­ where T= temperature, f3 = volume thermal expansion sion coefficients, adiabatic compliances are smaller than coefficient, V = volume, and C p = heat capacity per unit isothermal compliances. Differences were computed for volume at constant pressure. Derivations of these rela­ it few cases by Hearmon [29]. tionships were given by Landau and Lifshitz [31] and Bhatia [32], for example. Approximations in eqs (7.12) A similar relationship exists between the isothermal and (7.14) assume thatETf32/C p is small. Isothermal and and adiabatic stiffnesses, Cijkl. Thus, after Mason [30]; adiabatic shear moduli are equivalent since shear at constant volume, and entropy, leaves temperature (7.2) unaffected.

J. Phys. Chem. Ref. Data, Vol. ~ No.3, 1973 54'"

Finally, the relationship of the isothermal and adia­ phase diagram was made OIi the basis of meteorite batic moduIi to the thermodynamics of elastic deforma­ m~tallography [39]. A currently accepted phase diagra~ tion is indicated. The first law of thermodynamics is, is shown in figure 3, adapted from Hansen· and Anderko including an elastic energy term, [40], Elliott [41], and Shunk [42]. This diagram is rela~ tively simple, reflecting similar atomic sizes, melting (7.16) points, heats of fusion, etc. of Fe and Ni. Hume-Rothery'~ criteria [43] for solid solubility are favorable: atomic The increment of Helmholtz free energy F = U - TS is size difference· is small, electronegativities and valences are identical. Difference in crystal structures at room (7.17) and lower temperatures precludes complete solid sol~ ubility. Complete mutual miscibility does occur at so that the stress tensor is given by higher temperatures where both crystal structures are f.c.c. (7.18) Nickel is f.c.c. at all temperatures and is compli­ cated only by a paramagnetic-to-ferromagnetic transi­ for the isothermal and adiabatic cases, respectively. And tion, the Curie temperature being 627 K. Iron has two hn::llly, allotropic phase transitions: h.c.c. (a)-to-f.c.c. ('y) at 1183 K, and f.c.c. (1')-to-b.c.c. (8) at 1663 K. Also, (7.19) Fe undergoes a paramagnetic-to-ferromagnetic transi;. tion at 1043 K. The lower allotropic transition in iron is where € denotes constancy of all strains except €k so that unusual since a b.c.c. phase becomes stable at lower temperatures. Because of vibrationBl entropies~ b.c.c. (7.20) phases are stable usually at higher temperatures. The upper allotropic transition 'Y-to-S is also unusual. As and first explained by Seitz [44], f.c.c. l' would be ex·­ pecled to be stable up to its melting point since its (7.21) Debye temperature is lower than that of the b.c~c. phase. Debye temperatures are measures of lattice 8. Elements Iron and Nickel vibrational energies. According to Seitz, the 1'-to-8 transition occurs because the electronic energy becomes The purpose here is to collect data for both Fe and Ni large, on the order of kT. This happens because the elec­ that relate to their elastic properties and to the elastic tronic specific heat, which increases Ii.nearly with T, be­ properties of their alloys. Table 4 summarizes the data. Both elements are transition metals (incomplete 3d comes relatively large due to partly. fillt~·a. d-sheiIs; -thed electronic shells) and occur in the first long period of shells have ·a high density of states; ther-efore a large Mendeleev's table, separated in that row only by Co. As number of electrons are available at the top of the shown in table 4, Fe and Ni are quite similar in most of unfilled band to absorb thermal energy. their properties. As discussed below. this similarity is Polymorphic transitions of iron were interpreted reflected in many aspects of the elastic properties of differently by Zener [5]. He ascribed the upper transition: Fe~Ni alloys. S~o-1' to a low value of C' = (ell - C12)/2, the {l!Oh Properties of solids that are determined lan?;ely by <110)6 shear modulus. Abscnce of elastic data of any orbital electrons must vary periodically with atomic num­ kind for 8 Fe precludes experimental confirmation of ber according to Mendeleev's table of elements. This this proposal. He ascribed the lower transition 1'-to-a subject is an important part of the phenomenology and to a spontaneous magnetization (ferromagnp.ti~m) of science of solids. However, since this review treats only b.c.c. Fe. The difference hetween the Curie temperature two elements that are closely related in both the periodic and the allotropic transition temperature was ascribed table and in their basic properties, periodic variation of to a local correlation of electron spins, ferromagnetism elastic and related properties will not he discussp.d hp.fp. being a long-range correlation. in any detail. Interested readers should see Mack [33], Dorn and Tietz [34], Vereshchagin and Lichter [35], - The 1'-to-a tr3n~iti()n that occurs on cooling has Ryahinin [36], Koster and Franz [37], and Gschneider martensitie ehnracter in the range of ahout 18 to 34 Ni [38]. and "massjvt~" dlUfOC!ter for compositions up 10 about ]8 Ni f4SJ. Some fwidence suggests that very rapid 9. Iron-Nickel Phase Diagram cooJillg of l' prr.Vfmltlo a diffusion mechanism and gives a martentlhk IfUntlformation in alloys of lower Ni 1 Since elastic properties of alloys are related inti­ onn1enl. ftV4Hl Fc;·I46I. mately to the corresponding constitution diagram, some CurietcmJ~rutures representing paramagnetic-to­ cursory considerations of Fe-Ni phase equiHbrin Uf't~ ft~rtt)nm"n~ti(~ tront~itions are shown in figure 3 as dotted appropriate. One of the first proposals for an Fe·Ni UUt,,!: dUIlt wt!rrtuken from the recent comn-il<>tion of

J.Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 541

":onnolly and Copenhaver [47]. Possible relevance of shown by Reed and Schramm [52], lattice parameters Curie temperatures to structural transitions was dis­ of both Fe and Ni are reduced only slightly by alloying; cussed by Davies and Magee [48]. thus, the lattice paramenter effect is small and possibly An order-disorder transition occurs in the FeNia negligible. The interatomic potential may change signifi­ region, the critical temperature at stoichiometry being cantly; 'this effect has apparently never been estimated. about 776 K. Some evidence also exists that order­ An estimation could be made by assuming a Born­ disorder transitions occur near the stoichiometric com­ Mayer, or similar, ion-core repulsive potential and positions Fe3Ni and FeNi [49]. Because of strong evaluating' the Born-Mayer parameters from the Cij. similarities between Fe and Ni atoms, occurrence of Similarly, contributions from 3d and 4s band structure Fe-Ni superlattices is surprising and not understood energies are undoubtedly important but apparently completely, though magnetic interactions between have never been evaluated. Thus, the theory of alloying atoms are surely important. effects on the elastic properties of transition metals is Thermodynamic equilibrium in Fe-Ni alloys is not presently inadequate for treating the problem even established readily, and the final form of the equilibrium qualitatively. Since both valence and lattice param­ diagram has probably not yet emerged. For present eter effects are negligible, the Fe-Ni system provides purposes 7 metastaple stHtes corresponning to continuous an opportunity for stllnying effer.ts of hand structure heating and cooling are more important than the and/or effects of changing interatomic potential on equilibrium states since man-made Fe-Ni alloys are elastic properties. metastable. In figure 3 the two long-short dashed curves An alternative model by Zener [53] explains lowering are so-called realization curves representing boundaries of elastic moduli by alloying such as shown in figures of the a and 'Y phases' for Fe-rich alloys. Thus, in 5 and 6. Zener's model is based on residual strain practice the b.c.c. phase exists at room temperature up energy arising from the atomic size difference of solvent to about 30 Ni, beyond the 10 Ni predicted by "equi­ Cl.IH.J. ~ulule alulll~. Since: Idative: i51ze:i5 al~o affect the librium" phase boundaries. mutual solubility of species, the effect is related crudely Recent experimental and theoretical work on meteor­ to the limits of maximum solubility; a lower solubility ites has suggested modifications of existing Fe-Ni giving a larger effect on the modulus per unit solute. phase diagrams; for example, see Goldstein and Ogilvie It can be shown simply that strain energy associated [50] and Kaufman and Ringwood [51]. with uniaxial tension is largely shear-strain energy For further details on Fe-Ni phase equilibria readers rather than dilatation-strain energy. For the shear mod­ can consult refs. [40, 41, and 42J. ulus Zener was able to relate the composition coefficient 10. Compositional Dependence of Elastic to the temperature coefficient. Thus, Constants I/G(dGldx) = 4/Nk(6r/r)2(dG/dT) , (10.1) Compositional variation data, including iron and where x=compositional variable, N=Avogadro's num­ nickel, are given in figures 4-8 and in tables 5-19. b~r, k=Boltzmann's constant, r=radius of solvent Many earlier data (pre-1945) are based on poorly char­ atom, and 6r= radius difference between solvent and acterized materials. and data variation is due probably solute atoms. As shown in section 13, (dG/dT) is usually to non·identical specimens and test conditions. Trace negative so that (dG/dx) is negative also. Since be­ impurities influence' elastic constants through an haviors of E and G are usually parallel, compositional impurity-dislocation interaction; dislocations contribute dependence of E is also accounted for, albeit crudely. an additional strain upon stress application. Similarly. Magnetic effects are involved in the minima of elastic small residual stresses affect measured elastic con­ stiffnesses near 30 Ni; these effects are discussed in stants. Thus. measurements made on low-impurity. section IS. well-annealed specimens are much preferred. Both Poisson's ratio v is a poorly characterized elastic resid uaJ stresses and dislocation density are reduced constant, both experimentally and theoretically. Espe­ by annealing. As discussed in section 11. presences cially, the effect of alloy composition on v cannot be of some types of defects are useful for minimizing predicted a priori. Observed variations of v with com­ dislocation effects. position for Fe-Ni alloys are shown for both b.c.c. and Changes in elastic constants due to alloying are usu- f.c.c. phases in figure 8. While data are sparse, certain ally considered to consist of three parts: (1) change in trends seem to be established. The b.c.c. data, ob­ valency or electron/atom ratio; (2) change in inter­ tained by indirect methods, increase monotonically atomic spacing, and (3) change in interatomic potential. with increasing Ni content, extrapolating to v = 1/2 at A fourth contribution - change in band structure about 40 Ni. indicating that b.c.c. Fe-Ni is unstable energy-is compli~ated. not generally understood. and at higher Ni contents. The large magnitude of dvJdx is treated usually either by simple approximations or for the b.c.c. phase is surprising. and reflects a sig­ by neglecting it. nificant change in the interatomic interaction as Ni For Fe-Ni alloys valence is invariant if d electrons is alloyed with Fe. This may be the largest change inv are neglected; both elements have two 4s electrons. As now known for any primary solid solution and reflects

J. Phys. Chem. Ref., Data, Vol. 2,f'lo. 3, 1973 542 H. M. LEDBETTER AND R. P. REED undoubtedly the transition-metal aspect (unfilled 3d terms of macroscopic mechanical states and explicitly electronic shell) of the two species. For comparison, v in terms of defect contributions per sea is constant within experimental error for alloys of eu, 11.1. Mechanical Deformation 0-38 percent Zn [37]. For the f.c.c. phase (the open symbols in figure 8), Since mechanical loading beyond the elastic limit v changes in a more complicated manner with composi­ causes plastic deformation and increases a solid's tion, having a maximum at about 40 Ni. Koster and Franz volume, one expects a lowering of elastic stiffnesses by [37] interpreted the compositional variation of v for the cold-working. Few data exist for testing this hypothesis f.c.c. phase in terms of magnetostriction, which is for Fe-Ni alloys. Most available experimental data are discussed in section 15. given in figures 9 and 10 for iron, figures 11 and 12 for Decreases in E and G with increasing Ni content nickel, and in figure 13 for their alloys. Properties of shown in figures 5 and 6 are relatively large, about 1 deformed metals are difficult to understand because of percent modulus change per atomic percent of solute preferred orientations that may be introduced by work­ ing, that is, a mechanically induced anisotropy whose up to ~ 35 Ni. Drastic changes of E and! or G with alloy­ ing are typical of systems that have limited mutual degree and nature are usually unknown. For example, solid solubility and that tend to form intermetallic such anisotropy may account for the non-parallel phases. Strong compositional variations in Fe-Ni alloys behavior of E and G of Ni with plastic deformation as are unexpected because Fe and Ni have similar proper­ shown in figures 11 and 12. Usually, highly anisotropic ties. Magnetic effects are probably important. This mechanical states arise from severe plastic deformation. subject is ripe for theoretical study. Additional refer­ Effects of mechanical deformation ~n elastic properties ences on alloy effects were given by Speich, Schwoeble, are difficult problems both experimentally and theoretic­ and Leslie [23]. ally. As discussed below, full understanding is inacces­ sible since it depends on detailed interactions among Studies of effects of alloying on elastic constants in various species of lattice defects that are deformation systems other than iron-nickel were summarized by induced. These interactions vary with material and de­ Hearmon [54]. Some additional studies include: pend on composition, crystal structure, mechanical and Cu, Ag + many solutes Hopkin, Pursey, Markham [55] thermal histories, etc., that is, on any variable which af­ Cu-AI Cain, Thomas [56] fects the character and/or number of deformation-in­ Cu-AI, Cu-Sn Moment [57] duced defects. An obviously important variable is method Mg-Ag, -Sn, -In Eros, Smith [58] of deformation; whether loading is tensile, compressive, Fe-AI Leamy, Gibson, Kayser [59] hydrostatic, torsional, slow, impulsive, etc. In short, Ni-Co Leamy, Warlimont [60] the relationship between plastic deformation and Ni-Cu Sakurai, et al. [61] elastic properties is complicated and will remain so. Pd-Rh, Pd-Ag Walker, et al. [62] Additional well-characterized experimental dats would Cu. Ag. Au + B-metals Koster [63] illumine the subject somewhat. Interested readers may find some solace in Zener's [69] study where the latent Friedel [64] correlated deviations from Vegard;s law energy of cold-work is related to the lattice expansion wilh difference in compressibilities of solute and solvent. accompanying lattice distortion. (Vegard's law states that, in substitional alloys, lattice A good example of effects of preferred orientations parameters are related linearly to concentration.) As on elastic properties is shown in figure 8 where the discussed by Mott [6S], Friel'1el [66] extended these upper curve gives the compositional variation of v for considerations to elastic constants. His correlations are 44 percent cold-worked alloys. In several cases, values most appropriate when atomic volume changes rapidly of v exceeding 1/2 were found. As described in section with composition but Fermi energy is constant. Oriani 4, 1/2 is the upper limit for an isotropic aggregate. Thus, [67] criticized some aspects of the elastic approach. v can provide a fairly sensitive index for presence of Munoz [68] showed that if 8r/r exceeds about 1 percent strong preferred orientations. The problem of correcting then second-order· terms become important but that elasticity measurements for slightly anisotropic ag­ second-order theory breaks down if Sri r exceeds about gregates was discussed in detail by Bradfield and 7 percent. . Pursey [70] and by Pursey and Cox [71]. 11. Effects of Mechanical Deformation, Effects on elastic constants due to mechanical defor­ mation can be correlated to a certain extent with effects Annealing, Recovery, and Lattice Defects due to radiation damage since both methods introduce Crystals and crystal aggregates described above were large numbers of lattice point defects. Many funda­ assumed tacitly to be perfect except for substitutional mental studies, both experimental and theoretical, impurities. thermal vibrations. free surfaces. and (in have been made on radiation damage effects. For a the case of aggregates) grain boundaries. In this section review see Seitz and Koehler [72]. Additional data on the small but important effect on elastic constants due to deformation and elastic constants are included in figures variou~ lattice defect:!> is di15cu:s:s~J, LuLh implicitly in 14, 18, 19, 26,30, and 32.

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 543

11.2. Annealing of the defect j, gii = partial Gibbs energy of defect,

nj = number of defects of species j, and superscript 0 When deformed materials are heated, lattice defects denotes the defect-free case. For thermal monovacan­ and distortions induced by cold-work are eliminated; cies eq (11.1) becomes that is, lattices recover to un strained and unstressed states. If both amount of cold-work and annealing (11.2) temperature are sufficiently high, recrystallization occurs; a new set of strain-free grains grows from the deformed matrix material. Preferred orientations. can since (aV(P, T)/aPh is small; va=atomic volume. arise also from recrystallization and subsequent grain Thus, the bulk modulus B is decreased by thermal growth. Thus, interpretation of the elasticity of a: re­ vacancies. crystallized material can also be complicated by an Conversely, interstitials increase mass density, and anisotropic· aspect or annealing texture just as in the higher elastic stiffnesses are expected. Theoretical case of a material with a deformation texture. Some calculations by Bruggeman [73] and by Dienes [75] also annealing data for nickel are given in figure 14. For predict this effect. Considering copper and sodium, nickp.l, thp. Young's modulus for a heavily deformed Dienes showed that a 1 percent vacancy concentration specimen is close to that of the magnetically saturated decreases all of the elastic stiffnesses by about 1 percent, state; and higher annealing temperatures produce while a 1 percent interstitial concentration increases lower moduli. As also shown in figure 14, rapid cooling all elastic stiffnesses by about 10 percent. The large from the annealing temperature to room temperature difference between effects of vacancies and interstitials (quenching) tends to increase the modulus, presumably is related directly to relaxation of lattices around because of stresses induced by temperature gradients defects. Dienes found for copper that the percent change during quenching. in interatomic distance upon relaxation was 2 and 9 percent for vacancies and interstitials, respectively. Dienes also concluded that interatomic relaxation is 11 .3. Lattice Defects much larger in a b.c.c. (more open) crystal structure than in an f.c.c. (close·packed) crystal structure. Since both deformation and annealing processes must Melngailis [78) considered how Frenkel defects (vacancy­ be described ultimately by creation and annihilation interstitial pairs) affect elastic properties of copper and of lattice defects, effects on elastic constants due to concluded that an elastic softening results. This is con­ four types' of lattice imperfections-vacancies, inter­ sistent with the observation that Frenkel defects de­ stitials, dislocations, and grain boundaries - will be dis­ crease mass density. Point defects can alter elastic cussed briefly. Effects due to presence of lattice defects stiffnesses by another mechanism, dislocation pinning. are expected to be especially strong in systems where This phenomenon is discussed next. core-core repulsion energies contribute significantly to In discussing effects of dislocations on elastic con­ elastic constants, for example in noble and in transition metals. Of various energy terms that contribute to stants, two types of dislocations are distinguished - those elastic constants (see section 17), exchange repulsion which move freely upon stress application and those terms are probably most affected by displacements of which are immobile or pinned. That dislocations should atoms from equilibrium positions. lower elastic stiffnesses was· noted apparently first by Since creation of vacancies decreases mass density, Eshelby [791- Subsequently the problem was treated one expects vacancies to lower elastic stiffnesses. theoretically by Koehler and De Wit [80] for the case Theoretical verification of this was obtained by Brugge­ of pinned dislocations in f.c.c. crystals. Elastic stiff­ man [73] and by Mackenzie [74], and by Dienes [75] who nesses are altered because reversible dislocation used the Fuchs [76]. extended Wigner-Seitz approach motions contribute a reversible elastic strain component with a Morse potential for sodium and a Born-Mayer to the total strain. Thus, total strain is increased for a potential for copper. given stress and observed elastic stiffnesses are lower. Koehle.l" iiull DeWit found approximately that In the presence of thermal or static defects, Schok­ necht and Simmons [77] showed that aE/E=-KpL2, (11.3)

where E= Young's modulus, K= constant, p= disloca­ tion density, a~d L = average dislocation loop length. For annealed copper they concluded that pinned dis­ _ ~[ .(avi) (P, T») .. (.ani) ] locations contribute only a few percent to elastic ~ n; aP +Vt; aP , (11.1) t,; T T constants, but that for slightly deformed materials (where dislocation densities are much higher) the where vij= (agij/aPh is the free volume of formation contribution to elastic constants can be 10 percent or

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 544 H. M. LEDBETTER AND R. P. REED

2 higher. Because of dilatational components in their (a Ula(x - XO).2)x=xo strain fields, edge dislocations are about ten times as = 2a =second-order elastic coefficient (12.2) effective as screw dislocations (which have only a shear strain field) in decreasing elastic stiffnesses. (a3Ula(x- XO)3)x=xo Granato and Lucke (811 gave a detailed vibrating third-order elastic coefficient (12.3) string model for pinned dislocations where damping is = 6b = related to elastic modulus change; this work was and so on. initiated by Koehler [82]. Identifying elastic stiffnesses with coefficients in a A related subject-anisotropic dislocation theory­ series expansion of energy in tenDS of strains was first has become within recent years a flourishing area of discussed in detail by Birch [85]. More recently, Brugger solid-state physics with many intriguing applications [86] gave definitions of nth order isothermal and adia­ for single·crystal elastic data. Interested readers batic elastic coefficients: should see chapter 13 of the text by Hirth and Lothe ~8~1. (12.4) If thermal oscillations (phonons) are considered crystalline defects. then phonon contributions to elastic (12.5) stiffness can be computed from statistical thermo­ dyna~ics. Holder [84] did such a calculation for the (12.6) bulk modulus with the result for large T that and (12.7) aB == 3NkT {_ I' ti_ V ~ (.!)} (11.4) B BV V aV V where (12.8) where B = modulus, N·= Avogadro's number, k = are Lagrangian strain components. F = Helmholtz Boltzmann's constant, V= volume, T= temperature, elastic energy, S = entropy, H = enthalpy, G = Gibbs and 1'= Griineisen's constant. According to Holder, elastic energy, T= temperature, ai and Xi are Cartesian substitution of appropriate experimental values into components of a material particle in un strained and eq (11.4) predicts approximately the observed thermal strained states, and po = mass density of undeformed variation of the bulk modulus. Extension of this, or a state. The tij represent thermodynamic tensions con­ similar, approach to dB/dT would be quite useful. jugate to the variables 'Y/ij; for example Grain boundaries can affect elastic constants at higher temperatures when a static or slowly-varying (12.9) load is applied such that stress relaxation occurs across grain boundaries [5]. At higher temperatures, the Currently there is much activity, both theoretical viscous aspect of grain boundaries becomes more and experimental, on third-order elastic constants of important. grain·boundary sliding occurs as a Le IIleta15. Fourth-order and higher-order elastic con· Chatelier accommodation, and the elastic moduli are stants have been explored only slightly. effectively lowered. This effect is often seen as a The primary significance of third-order elastic coeffi­ departure from linear temperature dependence and cients is that they relate directly to anharmonic prop­ varies with the stress frequency. erties of lattices such as thermal expansion, thermal 12. Higher-Order Elastic Coefficients conductivity, differences between adiabatic and iso­ thermal elastic constants, and temperature and pressure Deviations from Hooke's law (stress is proportional coefficients of elastic constants. Anharmonic crystal linearly to strain) require the concept of elastic coeffi­ properties were discussed by Leibfried and Ludwig [87]. cients higher than second-order. These coefficients Anharmonic properties of solids can be described arise naturally from a Taylor expansion of the elastic conveniently by invoking Griineisen'g ')I. Brngger [RR] internal energy about the un strained or reference gave a generalized isothermal Griineisen parameter state. Thus, 1'1.'1= ____I (awo)I (12.10) i Wi a'YJkl T}=O' U = U 0+ a (x - xo) 2+ b (x -xo) 3+ c (x - xo) 4+ . where Wj= angular frequency of ith normal mode. By differentiating the wave equation Brugger related this (12.1) parameter to second-order and third-order elastic stiffnesses C,;llIItI and Cklmnop: and the nth order elastic constant is the nth derivative of the energy evaluated at the reference spacing. For I'~ol (n) =- (4wd [2W;UkUI+ (Ckllll1l+CklmnopuOup)nmnn] example (12.11 )

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 545 where and therefore only six independent third-order elastic coefficients. Both b.c.c. and f.c.c. crystal structures have 0 h point symmetry. and u = polarization vector. The Cauchy (central-force) conditions for cubic Griineisen's parameter figures prominently in third-order elastic constants are: equations of state for solids. For example C1l2 = C166, (12.22) P+ dU/dV=yEvlbr/V (12.12) and Cl44 = Cl23 = C4560 (12.23) is the Mie-Griineisen equation of state, which is a special case (all Yi are equal) of the more general form Thus, cubic Cauchy crystals have three independent third-order elastic constants. Like second-order con­ stants, Cauchy conditions for third-order constants are _1 {.V 2 hVi} P+dU/dV -17"'2; Yi h i/ + exp (hVi/kT) -1 (12.13) not expected to hold in metals because of free-electron effects. However, for Cu, Ag, and Au, Hiki and Granato that follows simply from the thp.rmodynamic partition [93] showed that Cauchy conditions are more closely function for an array of oscillators. followed for third-order than for second-order con­ Griineisen's model succeeds for two reasons. First, stants. This is because ion-ion overlap forces. which while the Yi are not equal the assumption describes are central forces, contribute more strongly to higher­ macroscopic thermodynamic properties fairly well. order elastic constants. Similar conside:r:ations should Secondly, while Y depends on volume the dependence apply also to the transition metals. is slight; thus, treating y as a constant, independent For isotropy, there are three relationships among of temperature, leads to theoretical predictions that third-order constants corresponding to P.q (5.1) for are largely verified by observation. Section 18 discusses second-order constants: a few anharmonic properties. Interested readers should see Griineisen [89] and Slater [90] for further discussion Cl12 = Cl23 + 2C144, (12.24) of y and its applications to theory of solids. For tetrahedral cubic symmetry (point groups T= 23, C166 = Cl44 + 2C456, (12.25) T h = m3) there are eight independent third-order elastic and coefficients as shown by Birch [85], Fumi [91], and CIII = Cl23 + 6Cl44 + 8C456. (12.26) Hearmon [92].

CIII = C222 = C333, (12.14) Thus, for isotropic crystals there are three independent third-order elastic constants. (12.15) Available Cijk data for iron and nickel are collected in tables 20 and 21. No third-order elastic constants have (12.16) been measured or calculated for Fe-Ni alloys. Elastic coefficients are invaluable for testing the validity of model interatomic potentials. Such tests are. quite sensitive since second- and third-order elastic

C 144 C255 = C366 , (12.17) coefficients relate to second and third derivatives of the potential. These coefficients describe changes not (12.18) only with respect to various shear deformation, but also with respect to volume deformations. Such detailed (12.19) comparisons have borne out, for. example, the validity of pseudopotentials as applied to the simple metals, and [94]-(97]. In many cases, a knowledge of the experi­ C456. mentally determined coefficients gives directly infor­ mation on the interatomic forces. For example, if All other coefficients are zero. Cauchy relations B;re satisfied one expects a central­ !or octahedral cubic symmetry (point groups T d force type model to successfully predict the elastic =43m, 0=432, and Oh= m3m) there are two ad­ coefficients. Hiki and Granato [93] observed that the ditional relationships: third-order elastic coefficients of eu, Ag, and Au followed the Cauchy relations much more closely than C1l2=CIl3, (12.20) did the second-order elastic coefficients. From this observation they concluded that short-range central and forces, in this case arising from d-shell overlap, be­ C155= C166, (12.21) come increasingly more important as one considers

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1913 546 H. M. LEDBETIER AND R. P. REED

higher-order derivatives of the total energy. As shown is purely thermodynamic and therefore more general. in tables 5 and 6, second-order Cauchy relationships It is easy to show that temperature coefficients of elas­ are violated for both iron and nickel, as expected. tic stiffnesses, or any linear combination of elastic However, the deviations are small, about 20 percent. stiffnesses, should approach zero as T approaches zero. The data in table 21 show that for nickel the deviations Invoking Nernst's heat theorem that the entropy S of from third-order Cauchy relationships are quite small. all crystalline solids approaches zero as T approaches and also that the deviation from the condition for first­ zero, then nearest-neighbor interactions is very small. The situa­ tion for iron is more ambiguous, and additional exper­ lim (iJS/iJxdr=O, (13.1) mental third-order data would be very valuable. Several proposals have been made for averaging the where Xi is any usual thermodynamic variable that Cijk to obtain the quasi-isotropic coefficients C~jk' Most preserves crystallinity. Physically this means that the of the considerations discussed in section 5 for obtain­ entropy change in an isothermal physical process goes ing cij apply also to the Cijk case. Hamilton and Parrott to zero as T approaches zero, that is all crystalline states [98] pointed out that the difference between Voigt's and of a solid have zero entropy at zero temperature. Also, Reuss's averages are even more important for the CUk than for the Cij. Whether a Hill-type average is also lim (iJ2S/iJxiiJXj)r=lim [iJ/iJxi(iJS/iJXj)]r=O. (13.2) r-o T-6 appropriate for the Cijk has not yet been established. The problem of averaging Cijk is ripe for both theoretical and If Xi and Xj are strains Ei and Ej and if a Maxwell rela­ experimental study. Interested readers should see tion in elasticity variables Barsch [99], Nran'yan [100], Cousins [101], and Chang [102]. (l3.3) Some authors have reported quasi-isotropic third­ order elastic constants as Lame coefficients VI, V2, and , is invoked, then Va or as Murnaghan's constants I, m, and n. These are related to the Ciik as follows:

C~2:F VI = I, (12.27) and lim [d/aT(rld)] = 0, (13.5) C;44 = V2= m, (12.28) r-o since C~56= Va n, (12.29) (13.6)

C~12= VI + 2V 2, (12.30) Thus, elastic stiffnesses Cij approach constant values with vanishing slope when plotted versus tempera­ V2 2v :J. (12.31) c~oo= + ture as T approaches zero. This feature must be con­ and sidered when extrapolating elastic data at cryogenic C~I1= VI + 6V2 + 8v a. (12.32) tern peratures. Temperature dependence of elastic stiffness at higher 13. Temperature Dependence of E1astic temperatures cannot be demonstrated so simply since Constants the effect is anharmonic. Like thermal expansion, it relines to higher-order elastic coefficients, which were Temperature (and as discussed in section 14, also described in 'section 12. Any discussion of higher-order pressure) variations of elastic constants are an important or anharmonic effects is simplified by introducing part of the subject of equations of state of solids. If Griineisen's parameter equations of state were known precisely. then elastic constants and their temperature and pressure coef­ 1'=- (V/vd (dvi/dV) dIn vdd In V (13.7) ficients could be calculated immediately. Large parts of both theoretical and experimental solid-state physics where V = volume of the crystal and Vi = frequency of would then be obviated. In fact, equations of state of ith normal vihrational mode. Griineisen's l' is actually a solids are known only crudely, and such parameters as tensor property, but its isotropic form is sufficient for iJcij/iJT and iJCij/iJP are measured and used to test and many purposes. ] Il t IJf~ quasi-harmonic model, l' does to improve the equations. not depend explicitly oil temperature [105]. And it is The problem of temperature dependence of elastic related directly to various thermodynamic properties, constants of solids was first considered carefully by Born for example 11 Ofll, and co-workers [103]. Zener [1041 discussed the problem in terms of an oscillator model of solids. Discussion here 1'= f3VB r/C v (13.8)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 547

wher-e f3 = volume thermal expansion coefficient, V = wide variety of materials that dCij/dT is zero at T=O K, volume, BT= isothermal bulk modulus and Cv = specific is constant and negative at high temperatures (T > ()), heat at constant volume. Thus, y can be determined from and changes rapidly at low temperatures. The exact low a set of macroscopic thermodynamic parameters. For temperature dependence of Cij remains an unsettled metals 'Y has a range of 1 to 3 and is usually about 2 prohlem; !OIeveral suggestions. h:we been made on either [107]. This limited range of y is expected since by theoretical or empirical grounds for the low-temperature integration of eq (13.7) dependence of Cij. These include

(13.9) (13.18)

that is the volume dependence of vibrational frequencies Cij(T) =Cij(O) -AT exp (-TofT), (13.19) is weak and is roughly similar for all materials. If an and Einstein oscillator model of solids is invoked. then Cij(T) = Cij(O) - B/[exp (CIT) -1]. (13.20)

(13.10) All these functions fit selected data quite well, indicating where 0= characteristic temperature, k= Boltzmann's that the exact form of the interpolation from T > 0 to constant and h=Planck's constant. Substitution into T=O is unimportant for most purposes. eq (13.7) gives Since no fourth-order elastic stiffness have yet "been measured, it is attractive to attempt a derivation of y=-d In O/d In V=- (V/O)(dO/dV). (13.11) fourth-order constants from the temperature derivatives of second-order constants. Hiki, Thomas, and Granato . Invoking the well-known relationship [110] derived expressions among dCtj/dT. Cljk and Cijkl in the high-temperature limit of a continuum model. (13.12) This approach to the Cijkl has been made for f3 Cu-Zn [Ill], NaCI [112], and the noble metals [93] and was for quasi-isotropic solids, combining this with eq (13.8) discussed by Holder and Granato [113). and rearranging terms, Temperature variation data are given mainly in figures 15-32 and also in 33-6 and 38. The only really anomalous temperature data are shown in figure 17 where for Fe-Ni Cp/Cv= 1 +yTf3= I +yT(dV/dT)p/V. (13.13) alloys containing about 30 percent Ni, most elastic stiffnesses have positive temperature coefficients over Substituting from eq (13.11) a wide temperature range. Positive coefficients, while unusual, occur also in other systems; for example Cp/Cv= 1-T(dO/dT)p/O. (13.14) MO(c12), Pd(cll' C12, C44), and Th(cI2) [54]. However, no single theory accounts for occurrences of positive Thus, since Cp > Cl'~ the characteristic temperature de­ coefficients: separate explanation~ mu~t hp. ~ol1ght for creases with increasing temperature. The characteristic anomalies in individual cases. In the case of Fe -- 30 Ni temperature 0 can be related to the elastic moduli, alloys, apparently no explanation has been proposed in e.g., E, by invoking relationships introduced originally the literature. Usual magnetic effects can be disregarded by Madelung [108] and by Einstein [109], since Curie temperatures (see figure 3) lie well below temperature regions where anomalies occur. It is in­ ()= KEl/2f(v) , (13.15) teresting to note that alloys in this composition range show also curious "invar" effects and phase transition wheref(v) depends upon Poisson's ratio. Assuming that effects. Both these aspects are discussed below. V:F v(T), which is only crudely true, then For additional discussion on the temperature variation dO/(}= (l/2)dE/E, (13.16) of elastic constants, readers should see" refs. [87] and and [114]-[116]. (lIE) (dE/dT)p= (2/T) (I-CpICv). (13.17) At low temperatures where the internal energy is given approximately by

Thus, in thia model elal5tic :5tiffne:5~e:5 behave exactly as the characteristic temperature; their coefficients are (13.21) negative. Since elastic coefficients Cij are given by the second spatial derivative of the elastic potential energy, one it follows that a general elastic stiffness M is expects a priori a slow decrease of Cij with increasing temperature due to the interatomic potential becoming more shallow as atomic vibration amplitudes increase. In fact, it has been shown experimentally [54] for a (13.22)

J. Phys. Cham. Ref. Data. Vol. 2. Nc. 3, 1973 548 H. M. LEDBETTER AND R. P. REED where E = appropriate strain, (3 = volume expansion of Bridgman [120].6 Following Bridgman, compres­ coefficient, and the linear term .allows for thermal ex­ sibility data are expressed frequently in the form pansion. Alers [22] described the validity of eq (13.22) V=Vo(l-aP+bP2), (14.1) for several metals and pointed out that for transition metals the quadratic term, which arises from the elec­ where both coefficients a and b depend on temperature. tron gas, dominates the quartic term that arises from lattice vibrations. The compressibility is Sutherland [117] suggested eighty years ago that re­ B-l=- (l/V) (av/ap) = (a-2bP)/(l-aP+bP2). duced shear modulus G(T)/G(O) plotted versus reduced temperature T/TIII gave a universal curve for all metals (14.2) and that G becomes zero at T= Tm. While Sutherland's correlation of reduced G's remains valid and valuable, Coefficient a is the initial (zero-pressure) isothermal it is now known that G need not vanish at the melting compressibility, and b is related to the pressure deriva­ point [118]. tive of the isothermal bulk modulus by the relationship In figures 18 and 23 anomalous increases both in E and G are shown that result from crystallographic phase b= (1/2B5) (1 +aB/aP)p=o. (14.3) transitions. Generally, elastic properties are discon­ tinuous through first-order phase transitions. This topic The first expressions for the pressure dependence of is discussed in section 16. the elastic stiffnesses were given hy Birch [as):

In absence of a saturating magnetic field, ferro­ (14.4) magnetic materials may show anomalous elastic be­ havior as a function of temperature due to motion of (14.5) magnetic domains upon stress application. As shown in figures 14 and 19, mobility of ferromagnetic domain and walls leads to an effective elastic softening because domain walls can contribute an additional strain. When (14.6) materials are heated through the ferromagnetic-to­ paramagnetic transition, then they usually behave where cP is defined by normally since in the paramagnetic state there are no magnetic domain walls that can move under applied V/Vo= (1 +2cf»3/2 (14.7) stress. Applying a strong magnetic field to the ferro­ magnetic state has a similar effect; domain walls take with Vo = original volume and V = volume at press ure positions that are most favorable for minimizing the total P. Conversion of P to cf> is accomplished by eqs (14.1) energy of the system, and they are effectively immobile and (14.7). The third-order coefficients appearing in until the magnetic field is removed. These topics are eqs (l4.4}-(l4.6) are related to the third-order coef­ discussed further in section IS. ficients defined by Brugger [86] as follows: Apparently, no theory has been proposed to explain the variation of v with T. For many purposes this varia­ C~l= c1l1/6, (14.8) tion can be ignored since it is small for most solids. As discussed by Slater [119], dv/ dT should be positive Ci12= C112/2, (14.9) since the upper limit of v = 1/2 corresponds to a liquid, and heating metals increases their volume and their Ci2:J= Ct23, (14.10) behavior becomes more liquid-like. K(;ster and Franz [37] discussed some experimental and a few analytical Ci4A=2c 144, (l4.11) aspects of dv/dT. As shown in figures 30 and 31, dv/dT and is positive and small for both iron and nickel. c:6S -- 2ClSS. (14.12)

It follows that the pressure derivatives of the second­ 14. Pressure Dependence o~ Elastic Constants orrler elastic coefficients may be written as

Most existing data for the pressure variation of the (14.13) clastic constants arc for the bulk modulus (reciprocal compressibility). This is because compressibility can be determined directly as a volume change under pres­ • Much c!lnfusiou "1

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 549

parameter. The influence of, pressure on the elastic constants of (14.14) Fe-Ni alloys is shown in tables 14,15, and 22.

- (aC44/ap) p=o= (Cn + 2C12 + C44 + C144 + 2C166 )/3B, 15. Magnetic Field Dependence of Elastic (14.15) Constgnhi and Since magnetic interactions between atoms contribute to the total energy of a ferromagnet, there must be a - (aB/aP) P=o= (clll + 6C1l2 + 2C123)/9B, (14.16) corresponding contribution to the elastic constants, where the bulk modulus B is which are second derivatives of the total energy with respect to appropriate strains. Direct observation of this (14.17) contribution would elucidate the nature of the magnetic exchange energy, particularly its derivatives with By using the finite strain theory of Murnaghan [121] respect to atomic spacing. However, except in the pres­ and Birch [85], Ghate [122] neriVf~n theoretical ex­ enr.e of a saturating magnetic field. this contribution is pressions for the pressure dependence of second-, overridden by effects due to magnetic domain boundaries. third-, and fourth-order elastic constants. For the second­ In the domain. theory of ferromagnets the demag­ order case, he obtained the general form netized state corresponds to an array of domains, each of which is permanently magnetized to saturation. The (14.18) domains are oriented randomly so that a specimen has no net magnetization. Net magnetization is achieved by where 'Y/ is the Lagrangian strain and the parameters applying an external magnetic field so that magnetization A and B are linear combinations of Cij and Cijk. Ghate's vectors within domains tend to align with the external second-order equations are identical to Birch's when field. the third-order elastic coefficients are defined identically, Magnetic effects on elastic constants were first ob­ It is interesting to note that no expressions comparable served in 1902 by Honda [123] who termed the phenom­ to eqs (14.13)-(14.16) have yet been derived for the enon the "AE effect"; AE refers to the difference in temperature derivatives in terms of higher-order elastic Young's modulus Eo - Ed between saturated and de­ coefficients, even from simple lIloJ~llS. magnetized states. In the demagueli:t.t:::d :slalt:::, ele­ Both the temperature and pressure dependence of mentary magnetic domains can change their magnetiza­ the elastic constants can be deduced qualitatively tion direction under applied stress and grow at the very simply by assuming a linear anharmonic oscillator expense of neighboring domains that have different model. In the harmonic approximation the energy per orientations that are energetically unfavorable in the atom is applied stress field. Growth consists of motion of domain boundaries (Bloch walls). At magnetic saturation (14.1Y) domain walls are immobile, and a "true" or maximum value of E is obtained. where Uo=energy in unstrained (reference) state, Stress plays a role similar to magnetic field in de­ k="spring" constant, and xo=position of atom cor­ termining a specimen's magnetization. In effect, stress responding to U o. In the simplest anharmonic model, alters magnetization. which alters a ferromagnet's a single asymmetric term appears. Thus, elastic constants. If domain redistribution does not occur, then different elastic constants are observed. (14.20) Domain redistribution is prevented, for example, by superimposing a saturating magnetic field or by a high is the energy when a particle is 'displaced from Xo magnetic anisotropy energy_ to x. (The ne~i:ltive !!iigu of LIte cubic Lt:am assun::::s pmsi­ The follOWing brief analysis follows closely that given tive thermal expansion coefficients.) by Lee [124]. The second· order elastic constant is Assuming that the total strain Et consists of non­ magnetic and mllgnetic> Pllri!Ol, Kornet!llki [125] !lIhowen 2 C2 == (a U/a (x-xo)2)=k-l(x-xo). (14.21) empirically that

Thus, when the linear chain is compressed (e.g., by (15.1) increasing the pressure) x < Xo and C2 is increased. Similarly, when the chain is expanded (e.g., by increas­ where numerical subscripts indicate derivatives of ing the temperature) x> Xu and C2 is decreased. In lIE with respect to stress. In the demagnetized state thi5 model, both temperature and pressure effects are simply related to ehanges in x. the linear lattice (15.2)

J. Phys. Chem. Ref. Data, Vol~, 2, No.3, 1973 550 H. M. LEDBETTER AND R. P. REt:D

And at saturation Em=O so that E=Eo. From eq (15.1) getics and mechanisms of solid-solid transitions vary it follows that E= alE first decreases, passes through a widely. minimum, and then approaches Eo as stress is increased. If a phase transition can be characterized completely Figure 19.17 in Chickazumi [1] shows this behavior; see by lattice considerations, then clearly the lower tempera­ also figures 13.123 and 13.124 in Bozorth [126]. ture phase must be elastically harder than the high As shown in figures 18 and 19, the M effect for an­ temperature phase since it must have a higher Debye nealed Ni is several percent and for Fe less than one characteristic temperature in order to be the phase of percent. For alloys, as shown in figures 34 and 37, the lowest free energy at lower temperatures. As shown in effect is strongly composition dependent. figures 18, 23, 26, and 38, in Fe-Ni alloys, with one ex­ Clearly, the AE effect should be related to the mag­ ception at low Ni contents, low temperature phases are netostriction constant, A, the strain due to magnetic elastically softer. Thus, in this system strong electronic saturation. Chickazumi [1] has given relationships of effects are undoubtedly involved in the f.c.c.-to-b.c.c. the form phase transition. Volume increase during the f.c.c.-to­ (15.3) b.c.c. transition may also account partially for the ob­ served elastic softening. and for Fe-Ni a plot of A versus composition (his fig. Phase transitions characteristically show a marked 19.14). The AE effects shown in figures 34 and 37 increase in damping in the region of the transition temp­ correlate remarkably well with the variations of A with erature. Thus, materials should deform readily in the composition. hp.ing 7.P.TO at ahont 28 ann R2 nickel anrt temperature regions of their phase transitions since with maxima at about 40 and 100 nickel. deformation mechanisms are augmented by damping Taking AB = 0, it can be shown from table 1 that ihe mechanisms. In fact, some superplastic phenomena change in the shear modulus is related to the change in are due directly to phase-transition softening. Spurious the Young's modulus by the relationship elastic measurements .can be obtained near a phase change, and such data should be interpreted carefully. D.GIG= (3GIE) IlEIE. (15.4) It is attractive to consider a simple explanation for p.rygtRllogrRphip. trangitions, namely, mech,mip.Rl ann This effect is shown in figures 34 and· 35. Clearly, the therefore also thermodynamic instability of lattices. In bulk modulus is unaffected by magnetic fields since 1940, Born [127] derived mechanical, thermodynamic hydrostatic pressures do not cause domain boundary stability conditions that apply to all cubic crystals movements. regardless of unit-cell size and regardless of the type The importance of the IlEeffect is well-illustrated by of interatomic forces. These conditions follow from the the Elinvar alloys (Fe -- 35 Ni -- 10 Cr) where the requirement that elastic strain energy decrease in E. due to increased temperature is largely compensated for by a smaller IlEIE effect with increas­ (16.1) ing temperature. Thus, to a good approximation E is independent of T for Elinvar alloys. below their Curie is positive-definite. that is tern peratures. (16.2) 16. Effects of Crystallographic Transitions on In other words, any elastic strain or combination of Elastic Constants strains must increase elastic energy. Stability conditions When metals or alloys undergo crystallographic can be derived readily by considering the matrix array transitions, their elastic constants change for several of elastic coefficients for a cubic crystal, eq (4.10), and possible reasons: (1) change of lattice type, (2) change requiring that each principal minor of this matrix is of specific volume,· (3) change of electronic or Brillouin positive. After some algebra, it results that zone structure, (4) change of relative positions of atoms Cll > 0, (16.3) in the unit cell (even when lattice type is invariant, for example, hexagonal-to-hexagonal with a change in c 11 > /C 12/ or (Cn - C 12) > 0, (16.4) cia ratio). In most cases, there is also a non-physical effect, namely a ch~nge of co-ordinate axes. The effect (Cl1 + 2C12) > 0, (16.5) can be eliminated by a suitable co-ordinate transforma­ and tion or alternatively by considering force constants, for C44 >0. (16.6) example-the spring constant between atoms in closest­ packed direction6 in both phases. In 5hort, any param­ If any of thel::je conditione; al-e violated, then crYl5tals eter that affects vibrational spectra must also influence are unstable with respect to long-wavelength phonons elastic constants. No a priori basis exists for predicting and a transition to another crystal structure or to a liquid effects of phase transitions on elastic constants. Each phase must occur. Of course, such transitions may occur case must be considered individually since both ener- for other reasons since phase equilibria are determined

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 551 by relative stabilities of competing structures and eqs decreasing temperatures above the transformation (16~3)-(16.6) describe conditions of absolute insta­ temperature. Furthermore, data in table 7 show that bilities. These conditions may be summarized by saying the elastic anistropy of f.c.c. Fe-Ni alloys is maximum that Cll, the bulk modulus (c1l+2C12)/3, and the two at about 35 Ni, which corresponds roughly to the a - 'Y shear moduli (CU-C12)/2 and C44 must all be positive. realization phase boundary shown in figure 3. Since elastic stiffnesses for cubic crystals are found Recently, for }4'e-Ni and }4'e-Ni-C alloys, Diesburg experimentally to be always positive, eq (16.4) is [131] correlated the temperature coefficients of the eij usually the strongest stability constraint. with the morphologies of the martensite phase. Since, The following assumptions are implicit in deducing as discussed in section 13, the dCij/dT are related to eqs (16;3)-(16.6): higher-order elastic coefficients, the desirability is (a) Lattices undergo homogeneous deformations. Ac­ indicated of measuring higher-order elastic coefficients cording to Born and Huang [12B] this means that only to elucidate the problem of phase instabilities. Higher­ long-wavelength vibrational modes are important. order stability conditions corresponding to eqs (16.3)­ (b) Elastic strain energy density is expanded only to (16.6) have apparently never been published. second-order terms in-deformation parameters. Equations (lb.3)-(lb.o) also imply inequalities for the 17. Theoretical Calculation of Elastic Constants wave velocities, which are related to the Cij by the Christoffel equations (see eq (18.3»). Theory of elastic properties of solids is part of the By assuming a Mie-Griineisen form of interatomic theory of cohesion; see Jaswon [132] and Seitz [133]. potential Besides elastic properties, cohesion theories usually predict also: lattice parameters or specific volumes, a b cohesive energies, pressure-volume relationships, and, q!'(r)=--+-,rm rn (16.7) at their most ambitious, energy differences between allotropic forms. Thus, elastic properties relate funda­ where n> m for a minimum energy to exist, and m > 3 mentally to solid-state theory and therefore also to for cohesive energy to be finite, Born [127] proved that other parameters. associated with theory. (See also (a) simple cubic lattices are never stable; section lB.) (b) face centered lattices are always stable; Transition metals, which have incomplete d shells (c) body-centered lattices are unstable except for and which include both iron and nickel, pose particularly small exponents m and n in the force law. difficult problems for most theories of cohesion. It is For an interatomic potential given by eq (16.7), the known that even filled d shells, such as in noble metals, concept of mechanical instability applies only to b. c. c. can contribute significantly to cohesion. When d shells lattices. Zener [129] extended the concept of mechanical are incomplete the contribution is even larger since instability to include ideas on shear anisotropy and incomplete shells contribute to bonding in solids be­ vibrational entropy. While Zener's ideas were proposed cause the average energy of the solid's energy band originally to explain instability of higher-temperature differs substantially from the atomic energy level. In b.c.c. phases on cooling, they can be applied to any simplest models, filled energy bands contribute nothing phase instability on cooling or heating. (For example, to cohesion. as shown by Fisher and Renken [130], the hexagonal-to­ 17.1. Fundamental Models b.c.c. transitions in Ti, Zr, and Hf seem to show an un­ usual temperature depend~nce of vibrational entropy Elastic stiffnesses of monovalent metals have been near the transformation temperatures.) Despite suc­ calculated successfully by methods of Fuchs [76] and cess claimed for Zener's criteria, they are not universal. Frohlich [134]. The former gives shear constants Many phase transformations occur without shear con­ (ClI C12)/2 and C44 while the latter gives the bulk mod­ stants becoming small or elastic anisotropies becoming ulus (ell + 2cI2)/3.Fuchs considered three principal large. Conversely,. some systems with small shear contributions to shear constants: (1) electrostatic energy constants and/or high elastic anisotropies exhibit no of positive ions in a negative, charge-compensating phase transitions. electron gas, (2) exchange energy due to ion-core over­ Thus," while Zener's criteria are useful for testing lap and repulsion, and (3) Fermi or kinetic energy of for possible occurre~ces of phase transitions, they are the valence electrons. A minor contribution also con­ neither necessary nor sufficient. (As discussed above, sidered by Fuchs was the van der Waals or dipole­ the limit of vanishing Cll - C12 is sufficient reason for a dipole energy. phase transition to occur.) Recently, so-called pseudopotential methods have Iron-nickel alloys in the region of 28 to 35 Ni all proven quite ~seful for calculating several properties undergo diffusionless-shear (martensitic) transforma­ of "simple" metals [135]. These methods replace, for tions. In all cases as the transformation temperatures computational purposes, the rapidly and strongly os­ are approached, C' decreases while C is relatjvely cillating potential near" the ion-core with a smooth, unchanged. Thus, the elastic anisotropy increases with slowly-varying effective potential. With respect to

J. Phys. Chem. Ref. Data, Vol. 2:, No.3, 1973 552 H. M. LEDBETTER AND R. P. REED

elastic constants, this method has been fairly successful binding approximation) and a Born-Mayer repulsion for simple metals. But, to date, no pseudopotential term. His model accounted for the variation in Cij along calculation of elastic constants has been made for transi­ a transition series for f.c.c. and h.c.p. crystal structures. tion metals (incomplete d shells). It would be very Failure to account for the b.c.c. case was attributed to d valuable to extend pseudopotential theory to this case, band details not included in his model. For both Fe and which includes both iron and nickel. Ni, his calculated Cij agreed surprisingly well with Johnson [136] constructed an int.eratomic potential observation. for iron· based on experimentally observed elastic Lieberman [143] recently carried out a self-consistent­ constants. That is, in effect, the reverse problem; but field band calculation for zero temperature and pressure it is useful to consider the results because they decom­ with a potential incorporating both· exchange and cor­ pose the elastic constants into various contributing ener­ relation contributions. For iron a good result was gies. The Fermi energy contribution was neglected and obtained for the bulk modulus. only electrostatic and ionic terms were considered. By Rosenstock and Blanken [144] considered interatomic computing the electrostatic contribution it was deduced forces in se'veral cubic solids, including nickel, on the by difference that the ionic term makes the dominant basis of experimentally observed dispersion of lattice contribution to iron's elastic constants. This reflects vibrations. the large ion-core size of iron compared to its inter­ Zwikker [145] showed for metals that the bulk modulus atomic spacing. is given by To determine the applicability of a Morse potential mn I to studies of atomic properties of crystals, Girifalco B=-·- 9 V and Weizer [137] calculated second-order elastic co­ (17.2) efficients for several cubic metals including iron and nickel. Morse's potential for a pair of atoms is where 1= ionization energy, V= volume, and m, n= ex~ ponential factors in the Mie-Griineisen potential energy, (17.1) eq (16.7). Values of m and n were tabulated by Fiirth. where rij= eenter-to-eenter spacing of ion pair, ro However, application of a Mie-Griineiscn potential to equilibrium separation of two ions. 'P (r())= - D = dis- metals is quite approximate since it does not contain sociation energy, and a is an adjustable "hardness" non-central forces. parameter evaluated from the compressibility. These results are in only fair agreement with observa­ 17.2. Hard.Sphere Model tion considering how the three Morse parameters were Much metallurgy and crystal physics can be under­ evaluated. B = (Cll + 2C12) /3 was input, and an elaborate stood, albeit crudely, by considering a hard-sphere (1000 neighboring atoms) lattice sum was performed. model of solids. In this model, atoms are represented as And, of course, the fundamental defect of a Morse, or incompressible spheres in contact along close-packed similar, interatomic potential is that it permits only lattice directions, (n 1) b.c.c. and (nO) f.c.c., This model central pair-wise interactions, neglecting non-central assumes implicitly that only ion-ion repulsion energies and many-body forces. Central forces demand in the contribute to elastic constants. This approximation is cubic case that C12= C44, and this condition is rarely observed experimentally. Lincoln, Koliwad and Chate reasonably good for both noble metals and transition [l38] used a Morse potential to calculate third-order metals. Table 23 gives relative elastic constants for both elastic constants of several cubic metals and some of b.c.c. and f.c.c. crystals based on a hard-sphere model; their results disagree strongly with experiment. Iron B was set arbitrarily to unity. Considering the model's and nickel were not included in their calculations. Third­ crudeness, predicted relative quantities correlate sur­ order elastic stiffnesses as well as the pressure coeffi­ prisingly well with observation in many aspects. For cients of the second-order elastic stiffnesses were example: recently calculated by Mathur and Sharma [139] using a Munit: pult:Jllii:1l; lltt:ir- I-t:lSuitlS i:1rt: gi vt:u in Lablt:lS 20 and 21. All of these results should be considered cautiously since Milstein [140] recently criticized (2) B "':' E, application of Morse potentials to b.c.c. crystals since such potentials cannot predict values of Cij in the b.c.c. (3) G ~ 3/8E, case that are within stability limits imposed by Born's criteria (see section 16). Use of a Morse potential for (4) high A for b.c.c. case, and f.c.c. metals was recently criticized on experimental grounds [141]. (5) low {llO}(IIO) shear resistance for h.c.c. case. Ducastelle [142] studied theoretically the elastic coefficients of transition metals assuming the total Additional aspects of hard-sphere models were dis­ energy to consist of a d band contribution (using a tight cussed by Mott [146J.

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 553

18. Relationship of Elastic Constants to Other (18.3) Physical Parameters which follow from the equations of motion of plane, mono­ The purpose here is to indicate briefly how elastic chromatic waves where p = mass density, Cijkl = fourth­ l)foperties of crystals relate to a wide variety of solid­ rank elastic stiffness tensor, Xi = components of unit IIlate phenomena, many of which might appear to be wave vector relative to cubic axes, and ail = Kronecker independent of elasticity. While some of the relation­ delta. ~hips described h~re are empirical and not understood For polycrystals these equations simplify to ultogether, they have proven useful in interpreting behavior of solids_ (l1J r= R + 4.G/? > (Ut4.) Elastic properties of solids are related intimately to" and tltomic vibrational, or phonon, spectra. Vibrational spec­ pV;= G, (18.5) lra link elastic properties with most other phenomena that are discussed here. where Vi = longitudinal wave velocity and Vt = transverse Frequency distributions of atomic vibrations in crys­ wave velocity. The mean velocity is obtained by averag­ taJs have intrigued scientists since Debye's parabolic ing over v-3 , that is upproximaliull tu the problem wa~ pW}Ju~ed in 1912. Despite intense efforts, theorists have failed to either (18.6) devise an exact mathematical solution for frequency spectra or to·explain why Debye's crude model is effec­ As shown in table 1, Vi and Vt can be computed from any tive; obviously, for many purposes the exact shape of two polycrystalline elastic constants together with the frequency distributions is relatively unimportant. Thus, mass density. many lattice vibrational problems in many branches of Some authors prefer to regard VI as the velocity of solid-state physics are discussed meaningfully in terms sound in solids while others prefer Vm; either concept is of Debye's theta. valid in context. While the main purpose of this section is to relate elas­ For further discussion of these relationships the reader tic properties to other solid-state phenomena, this can should consult Blackman [147], for example. be done most conveniently by invoking Debye's 8. Thus, Sound velocities for Fe, Ni, and Fe-Ni. alloys have been for present purposes {} should be considered an elastic computed from elastic coefficients by Anderson [148] stiffness parameter. and by Simmons and Wang [149]. The Debye thetas calculated by several authors from S pecijic Heats their elast~c data are given in table 24. Historically, lattice specific heats have been most fre­ Sound Velocities quently used to determine the Debye temperature ()s. Sound waves in solids differ from sound waves in Debye [150] showed that " gases or liquids in two vital ways. First, solids transmit "transverse or shear waves as well as longitudinal or (18. 7) dilatational wo.ve~. Secondly, sound waves in solids are polarized, and in the anisotropic case polarization where Cv= specific heat at constant volume, N= Avo­ vectors are not simply related (orthogonal) to the gadro's number, k= Boltzmann's constant, T= absolute propagation vector. temperature. Electronic contributions to Cv, which are Debye's () is linearly related to a mean sound velocity linear in T, must be separated from measured values of Vm, that is Cv• Measurements of 8 by specific heats have been sum­ marized by DeSorbo [151] and by Holm [152]. The rela­ 8= KVm, (l8.1) tionship between ()s and 8elastlc was discussed by Alers and Neighbors [153] and by Alers [154]. From existing where K (h/k)(3/4rrvu) 1/3, where h = Planck's data, errors in 8e are smaller than those in 8s• constant~ k= Boltzmann's constant~ and VII = atomic Many authors [155]-[157J have discussed the equival­ volume." ence of 88 and ()elastic at T= O. For single crystals~ VIII is obtained from elastic Entropies constants by the integration over all space Vibrational entropies S can be calc·ulal-ed from Boltz­ mann's relationship 3V,;;:3 = f L V;;3 dO/4rr, (18.2) a= I ,2,3 (18.8) where VI = quasi-longitudinal wave velocity, V2 and Va = quasi-transverse wave velocities, dO = increment of where w= randomness. For a system of three-dimen­ solid angle, and 41T = normalization factor. Velocities Va sional oscillators Lumsden [158] showed for a Debye are roots of Christoffel's equations frequency spectrum that

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 554 H. M. LEDBETTER AND R. P. REED

8 3 (8)2 3 (8)4 ] Since in Bloch's theory only longitudinal lattice S = Nk [4 - 3 In r+ 40 T - 2240 T +. . . . vibrations affect p, no agreement between (Jr and () would be expected; in Debye's theory there are two (18.9) transverse vibrational modes for each longitudinal Again, S calculated by eq (18.9) contains only vibra­ mode. Since (}t > (}t, it would be expected that (}r > 8. tional terms; electronic and other non-vibrational terms However, Griineisen showed that (}r = (J for a large must be determined· separately. number of cases. Blackman [163] considered the All other usual thermodynamic quantities can be cal­ problem by calculating 8z both by averaging wave velocities as v-3 , as is conventional, and as v-5 , as is culated from 8 by invoking the thermodynamic partition suggested by eq (18.12); no correlations between these function, as discussed by Fowler and Guggenheim [159], thetas and· 8 were found. A recent experimental for example. r study by Cullen [164] for Cu·Au alloys showed that 91 Zero Point Energy (from elastic coefficients) exceeds (J" but that (], and 8r Behavior of many substances at low temperatures is have the same compositional dependence, which differs influenced by zero point energy, which arises as a quan­ from that of 8. This implies that transverse phonons tum effect; the energy of a linear oscillator in its ground contribute to Or in a way not now known. state of energy at T= 0 K is hv/2 where v is the vibra­ The general problem of R,. was disr.uj:;j:;p'cI at lp.ngth by tional frequency of the oscillator. Integration over a Kelly and MacDonald [165] and by Meaden [166] Debye spectrum gives for zero point energy who gave a compilation of(}r values. Bragg Intensities 9 9 Eo=gNhvmax =gR8 (18.10) As shown early in this century by Debye and by Waner, change in intensities of 'Bragg scattering oi since hv k8. Details concerning eq (18.10) were max x-rays by crystals with increasing temperature due to given by Domb and Salter [160]. change of the atomic structure factor is given by Thermal Conductivity

I (T) = Ie-2M• (18.13) While thermal conductivity occurs by many mech­ anisms and the theories of' these are difficult and The Debye-Waller factor 2M is simply related to disputed, 8 is a pervasive parameter for describing Debye's theta: these mechanisms. For example, Klemens [161] discussed a relationship due to Leibfried and Sch.l()e­ mann for lattice thermal conductivity of non-metals (18.14) at high temperatures, T > 8: . where h = Planck's constant, k= Boltzmann's constant, 83 m = mass of atom, cJ> = Bragg angle, A. = x-ray wave­ K = constant y2T (18.11) length, and D(x) = Debye function where x = 8fT, T being absolute temperature. The factor, 1/4 allows for where y = Griineisen parameter. A similar expression zero point energy. was derived on a different basis by Dugdale and Mac· As shown by Zener and Bilinsky [167) 8M in eq (18.14) the a:veTage Donald [162]. While ')''2. is relatively constant from one i-o Qb\aineo hy material to another, (]3 changes considerably. By com­ paring theoretical and observed values of KT at room (18.15) temperatures, Klemens esta.blished the approximate validity of the 83 dependence of K. rather than the usual average

Electrical Resistivity (18.16) The most convenient point of departure here is Bloch and Griineisen's relationship that describes for many Thus, 8M is always slightly larger than 8 by a few percent. metals the temperature variation of their electrical ihis topic was diseusse(\ exlensive)y by Lonsaa)e resistivity p over a wide temperature range: [168], by James [169], and hy Ilerhsiein (170). For both Fe and Ni. Silll-!:h alld Sharma [171) recently reported Debye-Wall.·,. faclors enmputed from elastic C T5 BrIT zdz stiffnesses. p(T) =- - f - (18.12) A (]~ 0 (e z -1) (1-e-z )' Vibration Am ')/; till/('S where C = constant, A = atomic weight, and 8r = char­ dcteri~tic temperature for lattice resistivity.

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 555 in that the Debye-Walier factor can be expressed as yp (J3 = K {3B' (18.21) = 167T2 sin2

2 _ 3h2T ( ~) (18.18) (u ) - Tf2mk01 D(x) + 4 . (18.22) 2 Lonsdale [168] tabulated (u ) values for many elements. She emphasized that vibration amplitudes are anisot­ ropic even for cubic symmetry and that vibrational' anisotropy can be calculated from elastic coefficients. Atomic Diffusivity For Fe-Ni alloys, (u2 ) were computed from E and G Diffusion coefficients for metallic self-diffusion or for by Tanji and Shirakawa [172] and by Tanji [173] to substitutional diffusion of different metals are well interpret thermal expansion, electrical resistivity, and known to be given by Arrheniuss's empirical relationship abnormal volume expansion of Invar alloys. For Fe and Ni, (u2 ) were recently computed from elastic stiffnesses D = A exp (- LlHIRT), (18.23) by Singh and Sharma [171]. whereA andt.H are temperature-independent constants, Melting R == universal ga:; constant, and T ..... temperature. Assuming a vacancy mechanism for diffusion, then As discussed in section 16, if a material is heated and elastic constants change with temperature such that any (18.24) of Born'~ ~tability criteria are violated, tlum i:1 pha:st: change must occur- either to another solid phase or to where t.HJ = formation energy of a vacancy and a liquid. Of course, phase changes can occur for other t.H m = motion energy of a vacancy. Realizing from reasons- generally a lowering of free energy. Linde­ thermodynamics that mann [174] believed that melting occurred when atomic vibration amplitudes reached critical magnitudes. This t.G = AH - Tt.S, (18.25) topic was developed extensively by Pines [175] who showed that where G = Gibbs free energy and S = entropy, then elastic constants are related to diffusion coefficients kTm )1/2 through a model given by Zener [176] and independently O=c ( AVW3 ' (18.19) by LeClaire [177], which shows that

where C = constant, A = atomic weight, k= Boltzmann's I:l.G m =KE, (18.26) constant, T m = absolute melting temperature, and Vo = atomic volume. Pines concluded that melting occurs where E = Young's modulus appropriate to t.H m, and when the root-mean-square atomic displacement K has units of volume. Substitution into eq (18.23) gives becomes roughly ro/8 where 2ro= interatomic spacing. after rearrangement

Thermal Expansivity In D + KE/RT= In Do - t.GJIRT, (18.27) As is well known, thermal expansion is an anharmonic where effect inconsistent with Debye's model of solids. How­ In Do = In A - ASJ/R - t.SmIR. ever, Griineisen showed for most temperatures that vol­

ume expansivity f3 is proportional to specific heat Cp , and at low temperatures Since in single crystals E varies with direction, its choice is not unique. Reasonable choices are E III for h.c.c. lattices and E 111} for f.c.c. lattices. Along with its (18.20) directionality, the temperature dependence of E ijk must also be considered. where Cv= specific heat at constant volume, p = mass Mechanical Plasticity density, B= bulk modulus, and y= Griineisen constant with a valup. of ahont 2 fot' ::111 solids. Clearly then at low Some empirical and semi-empirical relationships temperatures between elastic and plastic properties of solids were

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 556 suggested by Pugh [178]. While this topic is now only the reader should see Partington [185], Cartz [186], embryonic, it is discussed briefly here because it has Neshpor [187], and Herbstein [170] as examples. high potential importance but has received little emphasis in the scientific literature. That elastic-plastic relationships should exist is 19. Concluding Remarks expected from microplastic theories where most all dislocation equations contain explicitly various elastic As evidenced by over 200 references to the experi­ constants. mental literature,· elastic properties of iron, nickel, and Advantages of such relationships include: (1) relating iron-nickel alloys have been much studied with the plasticity more intimately to interatomic forces; (2) result that most are now well characterized. possibility of correlating plastic properties with other Despite this intensity of effort, a few of the elastic parameters via elastic properties, for example with properties have been studied experimentally only atomic number, melting temperatures, or Debye thetas; cursorily, and the general subject would be strength­ and (3) a higher degree of correlation among plastic ened by further studies. These properties include: (I) properties themselves. temperature dependence of the bulk modulus of nickel, So~e relationships suggested by Pugh include: (1) (2) preaaure dependence of propertie~ other than ~ingle· for T < Tml3 resistance to plastic deformation is pro­ crystal coefficients or the bulk modulus, and (3) explicit portional to Gb, where G = shear modulus and b= magni­ dependence of properties on magnetic field. tude of Burgers vector; (2) fracture strength is propor­ Other properties have not yet been studied experi­ tional to Ba, where B = bulk modulus and a = lattice mentally; these include: (I) pressure derivatives! of parameter; (3) range of plasticity is proportional to alloys, (2) third-order elastic stiffnesses of alloys, and BIG, so that a high value of BIG indicates malleability (3) fourth-order elastic stiffnesses of both iron and and a low value indicates brittleness. nickel. Besides Pugh, interested readers should see also From existing single-crystal data it would be useful Crutchley and Reid [179]. to derive averaged elastic constants and sound velocities. Ripe problems for theoretical study include: (1) rela­ Diatomic Molecular Vibration Frequencies tionship of elastic constants to phase transitions, particularly martensitic transitions, (2) effects of Relationships between interatomic force constants ferromagnetism on elastic. properties, (3) contributions determined from ultrasonic wave velocities in solids and of d electrons to bonding and to elasticity, (4) thermal force constants determined spectroscopically from gas dependence of elastic properties, and (5) the existence molecules might appear at first to be vague and com­ and role of atomic ordering on elasticity. plicated. However, rough empirical relationships Since iron-nickel alloys are of much interest both scientifically and technologically, one might expect between 8solid and 8 gas were demonstrated by Baughan [180]; and Waser ~and Pauling [181] demonstrated the many of these areas to be studied intensively within relevance of Badger's rule to solids. Badger [182] the next few years. discovered for diatomic gases that k - d-3 where k= force constant and d = interatomic distance. Recent work by Haussuhl [183] and by Gilman [184] suggests Dr. C. Speich of the U.S. Steel Corp., and Drs. C. that k - d- 4 is a better· correlation for most solids, a Hausch and H. Warlimont of the Max-Planck Institut dependence first predicted theoretically by Fuchs [76]. fUr Metallforschung graciously supplied preprint These studies suggest strong correspondences versions of their work. between vibrational properties of atoms in solid and Dr. E. R. Naimon of NBS contributed grcatly.to the gaseous forms. Since better and more complete ex­ final stages of constructing the manuscript. perimental data now exist, both elastic and spectro­ This work was supported in part by the NBS Office of scopic, a re-examination of the problem would be Standard Reference Data. Dr. R. S. Marvin of OSRD appropriate to determine its synergistic aspects. Both made many valuable comments on the manuscript. iron and nickel were included in the studies of Baughan, and Waser and Pauling. 20. References for Text Other Properties [I] Chickazumi, S., Physics of Magnetism (Wiley, New York, 1964). Elastic Debye temperatures correlate with many [2] Ros~nberg, S. J., Nickel and Its Alloys, Nat. Bur. Stand. (U.S.), other solid-state phenomena that are not discussed Monogr. 106, 156 pages (May 1968). here, for example-theoretical strength, Mossbauer [3] Everhart, J. L.. Engineering Properties of Nickel and Nickel emISSIOn, sllperconducting transition temperatures, Alloys (plenum, New York, 1971). [4] Lanczos. The Variational Properties of Mechanics (Univ. of infra-red reststrahlen, diffuse x-ray scattering, and c.. Toronto Press, Toronto, 1970) p. 376. neutron scattering. For discussion of some of these f51 Zener. C, Elasticity and Anelastieity of Metals (Univ. of Chicago phenomena with respect to 8 andlor elastic constants Press. Chica!l:o, Illinois, 1948).

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 557

(61 Chung, Do' H., and Buessem, W. R., The elastic anisotropy of [35] Vereschagin, L. F., and Lichter, A. I., Dependence of the com­ crystals, J. Appl. Phys. 38,2010 (1%7). pressibility of elements on the atomic number, Dokl. Akad. [7) Barrett, C. S., A low temperature transformation in lithium, Nauk SSSR 86, 745 (1952). English translation available in Phys. Rev. 72,245 (1947). NTIS report AD 290677. [8] Ledbetter, H. M., Estimation of Debye temperatures by averag· [36] Ryabinin, Yu. N., Influence of pressure on periodic properties ing elastic .coefficients,]. Appl. Phys. 44,1451 (1973). of elements, Dokl. Akad. Nauk SSSR 104, 721 (1955). English [9] Voigt, W., Ueber die Beziehung zwischen den beiden Elastic· translation available in NTIS report AD 290676. itat $constanten isotroper Korper, Ann. Physik (Leipzig) 38, [37] Koster, W., and Franz, H., Poisson's ratio for metals and alloys, Met. Rev. 6,1 (1961). 573 (1889). [38] Gschneider, K., Physical properties and interrelationships of (10] Reuss, A., Berechnung der Fliessgrenze von Mischkristallen auf metallic and semi metallic elements, in Solid State Physics, Grund der Plastizitiitsbedingung fUr Einkristalle, Z. Angew. . vol. 16, F. Seitz and D. Turnbull (eds.), (Academic, New York, Math. Phys. 9,49 (1929). . 1964). [11] Hill, R., The elastic behavior of a crystalline aggregate, Proc. [39] Osmond, F., and Cartaud, G., Rev. Met. (Paris) 1, 69 (1904). Phys. Soc. London A65,349 (1952). Reported in reference [55]. {121 Landau, L. D., and Lifshitz, E. M., Theory of Elasticity (Per· [40] Hansen, M., and Anderko, K., Constitution of Binary Alloys (Mc· gamon, London, 1959) p. 40. Graw·Hill, New York, 1958). [13] Read, T. A., Wert, C. A., and Metzger M., in Methods of [41] Elliott, R. P., Constitution of Binary Alloys, First Supplement Experimental Physics, vol. 6A, K. Lark-Horovitz and V. A. (McGraw·Hm, New York, 1965) p. 442. . 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P. Mason (ed.), (Academic, Samsonov, G. V., Handbook of Physicochemical Properties of the New York, 1965). Elements (IFI/plenum, New York, 1968). Bechmann, R., and Hearmon, R. F. S., The third-order elastic con­ Schram, K. H., Daten zur Schallgeschwindigkeit in reinen Metallen, stants, in Landolt-Bornstein New Series, vol. 2, K.-H. Hellwege Z. Metallk. 53,729 (1962). and A. M. Hellwege (eds.), (Springer-Verlag, Berlin, 1969). Simmons, G., Hashin bounds for aggregates of cubic crystals, J. Grad. Becker, R., and miring, W., Die Temperaturabhangigkeit des Res. Cent. 36,1 (1967). Elastizitatsmoduls, in Ferromagnetismus (Springer-Verlag, Berlin, Simmons, G., Single crystal elastic constants and calculated aggre­ 1939). gates properties, J. Grad. Res. Cent. 34, 1 (1965). Bhagavantam, S., Elastic properties of single crystals and poly. Simmons, G., and Wang, H., Single Crystal Elastic Constants and Cal­ crystalline aggregates, Proc. Ind. Acad. Sci. 41,72 (1955). culated Aggregate Properties: A Handbook (The M.I.T. Press, Birch, F., Compressibility; elastic constants, in Handbook of Physical Cambridge, Mass., 1971). Constants, S. P. Clark, Jr. (ed.), (The Geological Society of America, Sundara Rao, R. V. G., Vedam, K., and Krishnan, R. S., Elastic con­ New York, 1966). stants, in Progress in Crystal Physics, vol. I, R. S. Krishnan (ed.), Boas, W., and Mackenzie, J. K., Anisotropy in metals, in Progress in (S. Viswanathan, Central Art Press, Chetput, Madras, India, 1958). Metal Physics, B. Chalmers (ed.), (Pergamon, New York, 1950). Zener, C., Elasticity and Anelasticity of Metals (Univ. of Chicago Press, Bollenrath, F., Hauk, V., and MUller, E. H., Zur Berechnung der Chicago, 1948). vielkristallinen Elastizitatskonstanten aus den Werten der ein Kristalle, Z. Metallk. 58, 76 (1967). 22. Older References Bridgman, P. W., The Physics of High Pressure (Bell. London. 1949). Bridgman, P. W., Recent work in the field of high pressures, Rev. 22.1. Iron, Young's Modulus Mod. Phys. 18, 1 (1946). Buch, A., The Mechanical Properties of Pure Metals (Scientific­ Wertheim. Pogg. Ann. Erg. E2, 1 (1848). Kupffcr, A., Mem. Imp. Acad. St. Pctcroburs (VI seriee). 6,393 (IS57), Technical Pub!.. Warsaw. 1968). Chevenard, P., Recherches experimentales sur les alliages de fer. de Kohlrausch and Loomis, Pogg. Ann. 141,481 (1870). . Streintz, H., Uber die Anderung der Elastizitat und der Large emes nickel it de chrome, Trav. Mem. Bur. Int. Poids Mes. 17,142 (1927). vom galvanischem Strome durchflorsenen Drahtes, Pogg. Ann. 150, Clark, S. P., Jr. (ed.), Handbook of Physical Constants (Geological Society of America, New York, 1966). 369(1973). Pisati. G., Sulla elasticita dei metalli a diverse temperature, Nuovo Dorn. 1. E., The modulus of elasticity-a review of metallurgical Cimento 1, 181 (1877); 2, 137 (1877); 4, 152 (1878); 5, 34, 135, factors, Metal Progr. 58,81 (1950). 145 (1879). Fedorov, F. L., Theory of Elastic Waves in Crystals (Plenum Press, Kiewiet, H., Wiedermann's Ann. 29,617 (1886). New York, 1968). Noyes, Phys. Rev_ 2,277 (18%); 3,432 (1896). Gschneider, K. A., Jr., Physical properties and interrelationships of Shakespeare. Phil. Mal!. 47,539 (1899). metallic and semimetallic elements, in Solid State Physics, vol. 16, Thomas, P. A., Disserlaliuli. Jena (1899); quoted in Keulegan, House­ F. Seitz and D. Turnbull (eds.), (Academic, New York, 1964). man (1933). Guillaume, C. -H., Recherches metrologiques sur les aciers au nickel, Gray. Blyth, and Ihllllop, I'rlH~. Roy. Soc: London 67,180. (1900); Trav. Mem. Bur. Int. Poids Mes. 17, 1 (1927). Gray, ToO EH'C'('I of hard(~llill~ (III the rigidity of steel, SCience 16,337 Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity (1902). (Oxford U. Po, London, 1961). WalkN, I'nll', B .. ", SO", Edinburgh 27,343 (1907); 28,652 (1908); Hearmon, R. F. S., The elastic constants of anisotropic materials, 31,)H(' (1'}fO), Rev. Mod. Phys. 18,409 (1946). lIarrisoll, 1"0", I'h\'<.;, Se,,', LOIH\UIl, 27,8 (1914).

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 661

Oodge, Phys. Rev. 5,373 (1915). 23. Notes on Tables Dadourian, H. M., Phil. Mag. 42,442 (1921). The following shorthand notations are used in the 22.2. Iron, She~r Modulus tables: Napiersky, Pogg. Ann. Erg. 3,351 (1853). Kupfi'er, Mem. Imp. Acad. St. Petersburg 6,397 (1857). f (A) = function of aging, Kohlrausch and Loomis, Pogg. Ann. 141,481 (1870); Am. J. Sci. 50, (1870). . f(CR) function of cooling rate, I'isati, Nuovo Cimento, Ser III. 1, 181 (1877); 2, 137 (1877); 4, 152 = (1388); 5,34,135.145 (1879). Tomlinson, Proc. Roy Soc. London 40,343 (1886). I(D) = function of plastic deformation, Katzenelsohn, BeibI. Ann. 12,307 (1888). Gray, Blyth, and Dunlop, Proc. Roy. Soc. London 67, 180 (1900). f (GS) = function of grain size, 22.3. Iron, Poisson Ratio I (H) = function of magnetic field,

Bauschinger, J., Der Zivilingenier 25,81 (1879). 1(0) = function of atomic ordering, Katzenelson, N., Inn. Diss.• Berlin (1887). Amagat, E. H., Ann. Chim. PhYs. 22,119 (1891\ Bock, A.• Weid. Ann. 52,609 (1894). f (P) = function of pressure, Stromeyer, C. E., Proc. Roy. Soc. London 55,373 (1894). Cardani. P., Z. Physik 4,449 (1903). I (Ph) = function of phase (transition), 22.4. Nickel, Young's Modulus feR) = function of recovery, Lessing, A., Uberdie Elastizitat einiger Kupfer·Nicke]·}egierungen, Dissertation, Univ. Berlin (1900). [(8) - fUllctiuu uf ~lrt:l')lS, Guillaume, C·E., Recherches sun Ie Nickel et Ses Alliages, Archives Sci. Phys. Nat., sec. 4, 5,255 (1898). I(T) = function of temperature, 22.5. Nickel, Shear Modulus f(X) = function of composition {of nickel), See references in section 22.2.

22.6. Iron-Nickel Alloys, Young's Modulus and

Barus, C., Amer. J. 34,175 (1887); Phys. Rev. 13,257 (1901). x(E) = function of composition (of element E). Mercadier, E., Sur la determination des constants et du coefficient d'elasticite de l'acier·nickel, C. R. Acad. Sci. 113,33 (1891). Howard, J. E., Nickel Steel from Springfield Armory, Tests of Metals "Best values" represent arithmetic means without of Metals (U.S. Arsenal, Watertown, Mass., No. 4816, 4817, 1893) weighting factors. "Uncertainties" represent computed p.237. standard deviations, and they include variations of Howard, J. E .. Nickel Steel from Springfield Armory, Tests of Metals. samples, experimental methods, etc. Systematic error (U.S. Arsenal, Watertown, Mass., Nos. 5520,5521,1896) pp. 60,431. Day, Electrician 39,480 (1897). is believed negligible compared to imprecision. All data Howard, J. E., Nickel steel rifle barrels, Tests of Metals (U.S. Arsenal were included in the averages with the following ex­ Watertown, Mass., 1898) pp. 391,690. ceptions. Data corresponding. to severely deformed Howard, J. E., Nickel steel, Tests of Metals, (U.S. Arsenal, Water· - specimens were omitted as being non-representative. town, Mass. 1898) p. 679. When the "same" materials were tested by different Rudeloff, M., Funfter Bericht des Sonderausschusses fur Eisen experimental methods, only a single average data point ]egierungen, Verh. Ver. Beford. Gewerb. 77,327 (1898). Stevens, J. 5., and Dorsey, H. G., Phys. Rev. 9,116 (1899). was included. ·'Different" materials measured by the Stevens,J. S., Phys. Rev. 10,111 (1900). same experimental method and by the same investigator Honda, K., and Shimizu, S.,J. SC. CoU. 16, Art. 12, 13 (1902). were included as multiple data points.

J. Phys. Chem. Ref. Delta, Vol: 2, No.3, 1973 ~ en ." CD ~ I\) ~ n ~ ct. ? ~ ·c po ~ ~ 24. Tables and Figures z p TABLE 1. Connectir.g identities X = X (Y, Z) for elastic constants of quasi-isotIopic solids ~ .0...., Co) Y,Z X E,G E,B E, v G,B G, v B,v A.,p. cY .. cY2 sYI- sY: VI. Vt ;t 9GB JL(3A. + 2JL) (c?J -C?2) (C?l +2C?2) 1 4 E 2G(l+v) 3B(1-2v) - 3Pr1(vf - 3 v;)/(v~ - v~) ;t C+3B A.+JL C?l +CY2 -srI m -C 3BE E 3B 1-21' I I I ." G(=I-'-=PV;) 2 (cfl-CY2) = C~4 9B-E 2(l+v) 2·1+v 2(SYI -S?2) S~4 "':::f ,.,m GE E 2G l+v 2 c~I+2c~2 I B p(V2 _!V2) » 3(3G-£\ 3(1-2v) 3"·1-2v )'+31-'- 3 511 +2S12 I 3 I z c E 1 E I 3B-2G A. I ~ v I C~2 -~ 2 IVf - 2v'f)/(Vf-tlf) 2G- 2-6B 2· 3B+G 2;A.+ JL) £1J +C12 S11 ~ ;:!a -S12 m £-2G 3B E vE 2vG 3vB p(v;-2v;) m ~ 38 - B-~G £12 G 3G-E 9B-£ (1+v)(I-2v) 3 1-2v l+v (S~I -5~2) (S~l + 2s~) C I G 4(;-£ 38(3B+E) EO-v) G 2(1-v) I-v sr. +slj.z Pl~ B+~G 3Bl"+v ,\+2JL eYr 3G-E 9B-E 0+ v) (1-2v) 3 1-2v (srI - .~?2) (sr] + 2SY2) ------~

Besides the symbols already ddinedin the text, P = mass density, v,= vel[)city of long:tudinal elastic wave, and Vt= velocity of transverse elastic wave. The c~ satisfy the isotropy condition Cll -C12 = 2C44. Similarly. the s& satisfy t~_e isotropy condition 2(slI -SI2) =S44. ELASTIC PROPERTIES OF METALS AND ALLOYS 563

TABLE 2. Expressions for engineering elastic constants of cubic single TABLE 3. Conversions between elastic stiffnesses Clj and elastic crystals compliances $Ij for cubic crystals

Cu $11 Young's modulus

-512 -C12 C12 512

Shear modulus

Bulk modulus And, by derivation:

Cu - C12 = --- 511 - 512 = --­ 511- 5 12 Cu- C12 Poisson ratio 1 511 + 2512 = CII + 2C12 Vl,m=-.(~"/.'l~l =-(.'ll~+SA)/(SI1-Sn where TABLE 4. Selected properties of iron and nickel

Fe Ni r = lil~ + l~l~ + l~li where ll' l2' la = direction cosines of an arbi­ trary crystallographic direction; Atomic number ...... 26 28 Atomic wei6dtt. mk2fmol...... 55.85 58.71 Electronic structure ...... [Ar]3d6452 [Ar]3dB4s2 A = limi + l~m~ + l~m~ where m1 , m2 , m3 = direction cosines of an 0.124 arbitrary vector perpendicular to l. Ionic (Goldschmidt, 12-fold ...... 0.127 coordination) radius, nm Distance of closest approach, nm 0.24823 0.24919 Lattice parameter, nm, 293 K.... . 0.28664 0.3523(; Density, kkg/nr, 293 K ...... 7.87 8.91 Melting point, K ...... 1810 1726 Boiling point, K ...... 3106 3059 Heat of fusion, kllmo!...... 15.3 17.7 Heat of vaporization, MJ/mol. .... 0.416 0.430

TABLE 5. Second-order elastic stiffnesses Cij of iron

Investigator(s) CII C12 C44 Composition Technique Specimen, Test Conditions (Year) (1012 dyn/cm2 )

Goene, Schmid 99.85 Fe (0.025 Tranl>v. reI>. freq. Stlaiu-allll. I.aytilab. Nu fidd. 2.37 1.41 1.16 (1931) Mn, 0.03C, 0.01 P, 0.08 S, 0.06 Cu, trace Si)

Kimura,Ohno Bending, torsion opt. Recrystallized. No field. 2.41 1.46 1.12 (1934) microscope.

Kimura (1939) a) Long. res. freq. No field. 2.09 1.14- 1.11 b) Long. res. freq. Saturated field. 2.10 1.13 1.12 c) Bending, torsion. No field. 2.28 1.33 1.11

Yamamoto (1941, Fe (0.03 C, 0.01 Si, Magnetostrictive oseilla- No field. 2.34 1.35 1.176 1943) 0.04 Mn, 0.01 P, tion. Bridgman (1940) 0.04 S, 0.06 Cu) compressibility data.

Moller, Brasse Fe (0.02-0.2 Mn, Tension, induction. Hz ann., 1223 K; 3-10% strain; vac. 2.28 1.40 1.12 (1955) 0.01-0.06 Si, ann. 1153 K, 72 h. 0.015 C, 0.01 P, 0.02 S, 0.15 0)

Markham (1957) 5-20 MHz pulse-echo. One crystal. No field. 2.330 I 1.392 1.162

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 564 H. M. LEDBETTER AND R. P. REED

TABLE 5. Se(;ond-order elastic stiffnesses Cij of iron-Continued --,-- I C44 I ell CI~ Investigaror(s) .-- (Year) Composition Technique I Specimen, Test Conditions [ (1012 dyn/cm2 )

Rayne, Chand- 99.99 Fe 10 MHz pulse-echo. Strain-ann. crystal. No field, f(T) in 2.331 1.354 1.178 rasekhar (1961) figure IS.

Lord, Beshers 99.8 Fe 7,10 MHz pulse-echo. 11 kOe field (saturated). [100], [110] 2.28 1.32 1.165 (1965) crystals ann., wet H2 • 993 K, 216 h, slow cool. [111] crystal vac. ann. 1123 K, 176 h, slow cool. Strain-ann. crys- tals,J(T) in figure 15.

Truell (1965) Pulse-echo. 2.23 1.27 1.15

Rouer. Smith (l966) 99.99 Fe (Cr, Mn 10 \1.Hz pulse-echo. Specimens from crystal of Rayne, 2.314 1.346 1.164 traces) Chandrasekhar (1961); no field, f(P).

Leamy, Gibson, Fe, 1-25 Al alloys 10 MHz pulse·echo. Bridgman technique crystals, ann. 1173 2.338 1.378 1.186 Kay:ser (1967) Extrapolated from K, 72 h. extremely slowly cooled alloy data. (60 days), argon atmos;f(T) in figure 15.

Leese, L(;rd (1968) 99.8 Fe 30··60 MHz pulse·eeho. Specimens and treatment from Lord, 2.26 1.40 1.16 Beshers (1965); no field; f(T) in figure 15.

Guinan. Beshcrs 99.8 Fe 10 MHz pulse·echo. Specimens from Lord. Beshers (1965), 2.301 1.346 1.167 (1968) no field,J(P).

Dev(~r (1972) Ferrovac 40, 70 MHz phase Two crystals, no field,f(T) in figure 15. 2.322 1.356 1.170 comparison

Best values 2.29 1.34 1.15

Uncertainties 0.09 0.09 0.03

TABLE 6. Second·order elastic stiffnesses Cij of nickel

Investigator(s) I ell ell! e44 (year) Composition Technique Specimen. Test Conditions (10 12 dyn/cm2 ) I i Honda, Shirakawa Bending, opt. micro· Bridgman method crystals. 2.52 1.51 1.04 (l9~7, 1(49) scope.

Bozorth, Mason, 99.95 Ni 10 MHz pulse-echo. Bridgman method, dry H 2, crystals. 2.50 1.60 1.185 McSkimin, Walker (19;1Q)

Bozorth, Mason, 99.95 Ni 10 MHz pulse·echo. Bridgman method. dry H 2• crystals; McSkimin (1951) f(freq.); a) no field, 2.517 1.574 1.226 b) saturated field. 2.523 1.566 1.23

Neighbours, 99.9 Ni 10 MHz pulse-echo. Bridgman method crystals. annealed 2.53 1.52 1.24

Bratten, Smith 2 h in H 2 • 5 kO(' Irallsv. field. (1952)

Yamamoto (1942, Ni(0.02-0.19 Fe. Magnetostrictive Brid!;rnall IIwlh.. d t"ryslals; vae. ann. 2.44 1.58 1.02 1950.1951) 0.01 P. S, AI, oscillation. 1273 K, I It. 0.03 C, Si. Mu, 0.01-0.20 Co, eu)

Levy, Truell (1953) 99.9 Ni 27, 30 MHz pulse·echo. Bricil!,!lI:l1I 1'11"11., .. 1 .... yslals; f(H, 2.47 1.52 1.21 1"1"1"'1.): saillral,·.( fie·ld.

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 565

TABLE 6. Second·order elastic stiffnesses Cij of nickel-Continued

;::"'~

Cll C12 C44 lnvestigator(s) .1 (Year) Composition Technique Specimen, Test Conditions 1 (10 12 dyn/cm

DeKlerk, Musgrave 2-10 MHz puls~·~dlU. it) Nufidd, 2.43 1.49 1.19 (1955) b) saturated field. 2.46 1.47 1.24

Shirakawa, Ni(O.06 C, 0.02 Si, Tension, opt. micro· X(Cu) 2.55 1.69 0.902 Numakura (1957) 0.05 Fe, 0.02 Co) scope.

DeKlerk (1959) 2 -1 0 MHz pulse·echo. F(freq.). a) No field, 2.459 1.500 1.213 b) saturated field II to [001]. 2.461 '1.475 1.220

Alers, Neighbours, "Electrolytic" 10 MHz pulse-echo. Bridgman method crystals, 10 kOe 2.508 1.500 1.235 Sato (1960) field, f

Sakurai (19M) 99.95 Ni 1-5 MHz pulse-echo. Brtdgman method crystals, 10 kOe 2.51 1.53 1.24 field.

Sakurai, Fujii, 1-5 MHz pulse-echo. Bridgman method crystals, X(Cu, Fe). Cn - en =0.93 1.24 Nakamura, Takaki (1964)

Epstein, Carlson, 99.95 Ni 10 MHz pulse-echo. Bridgman method crystals; ann. 1373 K, (1965) 200h;X(Cu); a) nofield, 2.481 1.54 1.242 b) saturated field. 2.504 1.57 1.256

Vintaikin (1966) X-ray, thermal diffuse Bridgman method crystals; ann. 1173 K, 2.47 1.44 1.24 scattering). 3 h.

Salama, Alers "Pure" Change of sound velocity Saturated, 10 kOe, field. .2.516 1.544 . 1.220 (1969) under uniaxial stress.

Shirakawa, et al. Ni (0.012 Fe, 0.007 Single-crystal resonant Saturated field. 2.88 1.81 1.24 (1969) Si, 0.008 Cu, frequency. 0.003Mn)

Best values (no field) 2.49 1.55 1.14 (sat. field) 2.54 1.55 1.23

Um.::~llitilJli~1S (uu fidu) 0.04 0.07 0.12 (sat. field) 0.12 0.10 0.01

Investigator(s) CII Cn C44 (Year) Composition Technique Specimen, Test Conditions (10 12 dyn/cm 2)

AIers, Neighbours, Fe-30.0 Ni 10 MHz pulse-echo. Bridgman. method crystals. f(n in Sato (1960) figure 17, a) H=O. 1.473 0.888 1.135 b) H=10kOe. 1.463 0.881 1.132

Einspruch. Clair- Fe-73.8 Ni Pulse-echo. Bridgman method crystals, no field 2.304 1.444 1.192 borne (1964) and saturated field.

Sakurai, Fujii a) Fe-59 Ni 1-5 MHz pulse- Bridgman method crystals, 2-4 kOe CU- C 12 =0.72 1.22 Nakamura, b) Fe-75 Ni echo. . field. AE effect = 1-2%. CU-C12 =0.89 1.27 Takaki (1964) c) Fe-90 Ni Cll-Cn 1=0.95 1.25 d) 100 Ni Cll - Cn =0.94 1.24

Salama, Alers Fe-30.0 Ni 10 MHz pulse-echo. Specimens from Alers, Neighbours. 1.474- 0.894 1.134 (1968) Sato (1960); 10 kOe field; f (T) in figure 17.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 566 H. M. LEDBETTER AND R. P. REED

TABLE 7. Second-order elastic stiffnesses Cij of iron-nickel alloys- Continued

Investigalor(s) ClI CI2 C44 (Year) Composition Technique Specimen,Test Conditions i (10 12 dyn/cm2)

Bower, Claridge, a) Fe-36.1 Ni 10 MHz pulse-echo. Bridgman method crystals (quenched), 1.573 1.235 1.006 Tsong (1968) b) Fe-38.2 Ni measured at T = 4K. 1.545 1.211 0.992 c) Fe-60.8 Ni 2.283 1.501 1.176 d) Fe-61.8 Ni 2.286 1.500 1.184 e) Fe-78.4 Ni 2.476 1.512 1.277 f) Fe·89.8 Ni 2.627 1.528 1.300 g) Ni 2.614 1.548 1.309

Shirakawa, et al. a) Fe-35 Ni Single-crystal, Saturated field. 1.40 0.92 loll (1969) b) Fe·40 Ni resonant fre- 1.57 1.09 0.96 c) Fe-45 Ni quency. 1.96 1.42 0.83 d) Fe-50 Ni 2.12 1.55 0.90 e) Fe-60 Ni 2.24 1.51 1.12 f) Fe-70 Ni 2.33 1.46 1.27 g) Fe-80 Ni 2.41 1.43 1.38 h) Fe·90 Ni 2.52 1.43 1.39 i) Ni 2.88 1.81 1.24

Diesburg (1971) a) Fe-28.2 Ni 10 MHz pulse-echo- Bridgman method crystals, homog. 1.6080 0.9578 1.1598 b) Fe-30.0 Ni overlap. 1473 K 120h,f(T) in figure 17, no 1.5258 0.9157 1.1313 c) Fe·34.4 Ni field. 1.3328 0.8570 1.0591

Hausch, Warlimont a) Fe-31.5 Ni 10 MHz pulse-echo. Bridgman method crystals, 6kOe field, 1.404 0.840 1.121 (1973) f(T) in figure 17. 1.362 0.852 1.086 1.379 0.899 1.058 1.356 0.910 1.042 1.507 1.077 1.020 1.592 1.162 1.024 1.713 1.261 1.029 1.860 1.372 1.035 2.053 1.459 1.059

TABLE 8. Young's modulus Eofiron

Investigator(s) E (10 12 Composition Specimen. Test Conditions (Yo:;cu) Technique uyu/clll~)

Guillaume (1897) Spring tension. X (Fe). 1.96

Schaefer (1901) Res. freq. 1.80

Benton (1903) "Steel" Tension. F(T, 76-300 K), relative to 300 K.

Morrow (1903) "Wrought iron" Compression, opt. lever. Range of values: 2.01-2.12. 2.06

Carpenter, Hadfield, Fe(O.95 Mn, 0.17 Si, 0.47 C, Tension, extenso meter. Ann. 1023 K, x(Ni). 2.21 Longmuir (1905) 0.04 S, 0.02 P)

Griineisen (1907) 99.5 Fe(O.l C, 0.2 Si, 0.1 Mn) a) Tension, opt. lever. 2.10 b) Transv. res. freq. 2.10

Honda, Terada (1907) "Swedish steel" Tension opt. lever. Ann.,f(S), x(Nj). 2.0

Grtineisen (1908) a) "Steel" Res. freq. 2.09 b) "Iron" Res. freq. 2.13

Glilno:;il>t::11 (1910) Bemlil1~. 11,,">," .. f v"luo:;",~2.05-2.11. 2.00

Honda (1919) Bending, opt. lever. Ann. 117:1 K, x (Ni). 2.05

Honda (1919) Fc(O.29 Cu, 0.31 Mn, 0.11 Si, Dending, opt. lever. XI(:,,). 2.1 0.09 C, 0.3 P, S)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 567

TABLE 8. Young's modulus E of iron-Continued

Investigator(s) E (10 12 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Bach, Baumann (1921) F(T, 293-773 K). 2.03

Honda, Hasimoto (1921) Fe(0.18 C, 0.20 Mn+Si) Bending, opt. lever. a) Ann. 1173 K, slow cool. 2.055 b) Ann. 1173 K, quench. X (C). 1.955

Koch, Dieterle (1922) Reo. freq. F(T, 295-973 K). 1.1

Lea (1922) "Armco iron " . Tension, opt. lever. F(T, 293-573 K) . 2.0

Carrington (1924) a) "Wrought iron" Tension. Ann. 1183 K, slow cool, f(T, 293- 2.00 590K). b) Fe(0.19 C) Tension. Ann. 1183 K, slow cool, f(T, 293- 1.98 590K).

Kimball, Lovell (1925) Fe(0.15 C) Res. freq. F(T, 290-790 K).

Honda, Tanaka (1926) a) Fe(0.31 Mn, 0.29 Cu, 0.09 C, Bending, opt. lever. Ann. 1173 K, 2 h, slow cool. 2.087 0.11 Si, 0.03 P, S) Ann. 1173 K, 1/2 h, slow cool. 2.045 b) Fe(O.38 Mn, 0.1 C, 0.02 P, S) Ann. 1173 K, 1/2 h, oil quench. 1.942 X(Ni, Co, C),f(H).

Nishiyama (1929) a) Fe(O.1 C) Bending, opt. lever. Ann. 1173 K in vac., 1 h, furnace 2.12 cooled; x(Si, Y, AI, W, Mn, Cr, Co, Ni). b) "Armco iron" 2.11 c) "Electrolytic iron" 2.13

Kawai (1930) "Armco iron" Tension. Ann. 1273 K,f(D) in figure 9. 2.13

Jacquerod, Mugeli (1931) Bending. Ann., f(T, 273-390 K), relative to 2.09 273K

Everett (1931) Fe(0.35 C, 0.80 Mn, 0.10 Si, Tension, opt. lever. a) Ann. 1173 K, t h, slow cooL 2.01 0.02 P, 0.03 S) b) Unannealed. 2.11

Keulegan, Houseman Fe(0.66 C, 0.8 Mn, 0.01 P) Loaded helical springs. Detm. temp. coeff., 223-323 K. (1933) -

Bez·Bardili (1935) Long. and transv. res., 1-20 2.18 MHz.

Verse (1935) a) Fe(O.43 C, 0.86 Mn, 0.04 S, Tension, cathetometer. Ann.,f(T, 298-733 K). 2.06 0.02 P, 0.14 Si) b) Fe(0.34 C, 0.80 Mn, 0.10 Si, Long. torsional res. freq. Ann. 1173 K,f(T) in figure 18. 2.08 0.02 P, 0.03 S)

Nakamura (1935) 99.94 Fe Long. res. freq. Ann. 1273 K, 1 h, slow cool; x(Ni}; 2.107 f(H).

Cooke (1936) "Armco iron" Long. (56 MHz), torsional. a) Ann, 1200 K, 2 h, Hz atmos., slow 1.99 cool (8 h), H = O. b) Cold rolled, H=O. F(H). 1.86

Fiirster, Koster (1937) Transv. res. freq. Ann. 1200 K, ! h. alr cool. 2.13

Engler (1938) Long. res. freq. F(H) O-0.575kOe,f(T) in figure 18. 2.12

Yamamoto (1938) "Armco iron" Magnetostrictive oscillation. Ann. 1203 K,1 h;X(C);f(H). 2.11

Kimura (1939) "Armco iron" Long. res. freq. Ann., H2 atmos.;f(H),f(T) in 2.17 figure 18.

Koster (1940) "Armco iron" Transv. res. freq. X(Co, Cr, C, Ni), f(D) in figure 9. 2.12 f(T, 293-1173 K),f(GS, A, R).

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 568 H. M. LEDBErrER AND R. P. REED

TABLE 8. Young's modulus E of iron-Continued

Investigator(s) E (10 12 Composition Technique Specimen, Test Conditions (Year) dyn/cm2)

Yamamoto (1941. 1943) a) "Electrolytic" Magnetostrictive oscillation. Vac. ann. 1203 K, 1.5 h, furnace cool. 2.140 b) Fe(0.03 C, 0.01 Si, 0.39 Mn, Vac. ann. 1203 K, 1 h, furnace cool. 2.058 0.01 P, 0.04 S, 0.06 Cu) F(H).

Ki5ster (1943) Transv. res. freq. Ann. 1273 K, 1 h, H = O. a) Bec crystal structure. 2.12 b) Fcc crystal structure (extrapo- 2.20 lated). F(Ni).f(T) in figure 18, f(H)./(O).

Smith, Wood (1941) 99.95 Fe Tension, x-ray speetometer. Vac. ann. 0.050 in. sheet. 1.79

Seager, Thompson (1943) a) Fe(O.B Mn, 0.06 Si, 0.18 C, Bending, interterometer. Hot rolled, f(T, 295-500 K(, f(lm­ 1.94 0.08 S, 0.02 P) purities). b) "Armco iron" Bending, interferometer. Cold rolled, ann.,f(T) in figure 18. 1.81

Everett, Miklowitz (191.1) Fc(O.15 0 . .25 C, 0.3-0.6 Mn, Bending, opt. lever. F(T, 293-310 K). 2.03 0.04 P, 0.05 S) [SAEI0201

SeheH, Reinacher (1944) Res. freq. x (Ni),J(T) in figure 18. 2.11

Roberts, Nortcliffe (1947) Fe(0.09 Ni, 0.22 Mn, 0.21 Si, a) Transv. res. freq. Ann. 1200 K,J(T) in figure 18. 2.09 0.06 Cr, 0.09 C) b) ·'Static". Ann. 1200 K,J(T) , 293-673 1\. 2.06

Koster (1948) Transv. res. freq. Worked, ann. 1273 K;f(T) in figure. 2.11 18. Bennett, Davies (1949) 99.75 Fe Res. freq. Ann. 973 K, 6 h; f(T); relative ...... 98.83 Fe(0.25 C, 0.65 Mn) values, 273-850 K.

Andrews (1950) 99.97 Fe(O.OI C, 0.02 Mn) Transv. res. freq. 0.03 in. sheet;f(T) in figure 18. 2.12

Frederick (1947) "Armco iron" 0.5-15 MHz pulse·echo. F(T). 2.089

GCl1ufiilu, Miih::uuck., Sllliti.1 Fc(O.4ti ~II, 0.19 Si, 0.13 C, TCll~ion, opt. lever. Ann. 1173 K, l h, air cool;/(T) in 2.03 (1952) 0.01 P, 0.02 S)[SAE1015} figure 18.

Hughes, Kelly (1953) "Armco iron" Pulse-echo. F(P}. 2.1I

Yamamoto, Taniguchi Fe(0.02 AI) Magnetostrictive. Ann., x(Al,f(H). 2.152 (1954)

Burnett (1956) 99.8 Fe Res. freq., bending. F(T) in figure 18. }.98

Yamamoto (1959) Fe(O.06 Mn, 0.03 Si, C, S, Magnetostrictive oscillation. Ann. 1273 K,2 h, H2 atmos.;x(Ni); 2.091 0.02 P) f(H), a) H=O, b) H sat. 2.096

Hill, Shimmin, Wilcox Fe(0.30 Mn,0.24 C,O.Ol Si; P, a) Long. res. freq. "Warm rolled",J(T) in figure 18. 2.10 (1961) 0.04 S)[SAEI020] b) Tension, opt. strain gauges. 2.08

Voronov, Vereshchagin 99,8 Fe(O.Ol Mn, 51, 0.01 C, 10 MHz pulse-echo. Annealed. 2.09 (1961) 0.03 P, S)

Durham, et al. (1963) Fe(O.08 C, 0.3 Mn, 0.15 Si) Tension, strain gage ex- Ann. 1060 K, 1 h, oil quench, 655 K, 2.07 [SAEI075] It:IIl>OIUt::lI::r. 1 h, o.ir cool,j(T) in figure lB.

Kamber (1963) Fe(0.002 C, p, S, Mn, 0.001 1 MHz, transv. res. freq. a) 1 mm GS, H=O, 2.04 Cr, V, 0.003 Mo, AI, 0.004 b) ] mm GS, H= 1.5 kOe. Strained 2.06 Si, 0.011 Ni, 0.014 Cu, 0.040 3%, ann. 1173 K, 25 h xy) c) 30 mm GS, H=O, 1.94 d) 30 mm GS, H= 1.5 kOe. F(T) in 1.94 figure 18. Masumoto, Saito, Koba- "Electrolytic" Long. res. freq. Vac. ann. 1273 K, 1 h, slow cool; 1.97 'ashi (1963) x(Pd).

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 HASTIC PROPERTIES OF METALS AND ALLOYS 5S9

TABLE 8. Young's modulus E of iron-Continued

Investigator(s) E (10 12 Composition Technique Specimen, Test Conditions (Year) I dyn/cm2)

Shved (1964) "Armco iron " X-ray scattering, uniaxial ten- 2.21 sion.

Smith, Stern, Stephens a) Fe(0.4 C, 0.35i, 0.8 Mn) Long. transv. pulse-echo. F(P). 2.11 (1966) b) Fe(0.6 C, 0.25 i, 0.8 Mn) 2.11

Masumoto, Sawaya, Kiku- "Electrolytic" Fe(O.004 Cu, Resonance frequency, 700- Vac. ann. 1 h at 1273 K, cooled at 1.97 chi (1971) trace AI, 0.005 Mn, 0.005 900 Hz. - 573 K/h; f (T) in figure lB. Si, 0.005 C, 0.005 S, 0.004 P)

Speich, Schwoeble, Leslie Fe(0.057 C) 80 kHz pulse-echo. Austenitized 1 h at 1273 K and water 2.082 (1972) quenched.

Best value 2.05

Uncertainty I 0.12

TABLE 9. Young's modulus E of nickel

Investigator(s) 12 Composition Technique Specimen, Test Conditions E(10 (Year) dyn/cm 2 )

Guillaume (1897) Spring tension. X(Ni). 1.92

Schaefer (1901) Res. freq. 2.31

Honda, Terada (1907) a) "Pure" Tension, opt. lever. Ann.,f(S), x(Fe). 1.9 b) "Commercia]" 2.2

Griineisen (1907) 97.0 Ni(l.4 Co, 0.4 Fe. l.0 Mn. a) Transv. res. freq. Hard drawn. 1.95 0.1 Cu, Si) b) Tension, opt. lever. 2.01

Kurnakow, Rapke (1914) 99.9 Ni Tension. opt. Ann. 820-870 K, x(Cu). 1.97

Harrison (1915) Tension, opt. mic. Ann. 773 K.f(T, 293-740 K). 2.16

Koch, Dieterle (1922) Res. freq. F(]" 285-1273 K). 1.7

Honda, Tanaka (1926) Ni(0.145 C, 0.05 Si, 0.01 Cu, Bending, opt. lever. Ann. 1173 K, t h; x(Fe, Co, C);J(H). 1.93 0.15 Fe, 0.04 S)

Mudge. Luff (1928) 99.18 Ni(O.Ol C. 0.16 Cu, Tension. a) Hot rolled. 2.13 0.06 Si, Mn, 0.4 Fe, 0.01 S) b) Ann. 1030 K, 4 h. 2.15

Nishiyama (1929) Ni(O.lO Fe, 0.03 Co, 0.05 Si, Bending, opt. lever. Vae. ann. 1173 K. 2 h, furnace cool. 1.99 0.02 P,O.01 S)

Kawai (1930) Tension, opt. lever. Ann.1l73K;f(D(tension».f(T(an· 2.10 neal».

Giebe, Blechschmidt Res. freq. a) Cold worked, H=O; 2.2.')l) (1931) b) Cold worked, H~6.2 kOe; 2.2HS c) Ann. 973 K, 12 h, slow eool; 1.90S d) Ann. 973 K, 12 h, slow cool. H 2.256 =6.2 kOe; e) Ann. 973 K, 2 h quenched, H=O, 2.()63 f) Ann. 973 K. 2 h, quenched, H 2.207 =6.2 kOe.

Jacquerod, Mugeli (1931) "Pure" I Bendin~ a) As received. 2.17 b) Ann. 858 K, H2 atmos., 16 h;f(T) 2.00 I 10 fi",urecr 19.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 570 H. M. LEDBETIER AND R. P. REED

TABLE 9. Young's modulus E of nickel-Continued

Investigator(s) E(1012 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Zacharia!\ (1933) Ni(0.41 Fe, 0.13 Cu, 0.02 Mn, Long. res. freq. a) Ann. 973 K, slow cool. 2.141 0.09 Si, 0.11 C, 0.3-0.4 Co, b) Ann. 1173 K, 2 h, slow cool. 1.986 0.01 S) 99.9 Ni( < 0.003 Cu, c) Ann. 1373 K. 2 h, slow cool. 2.077 0.11 Mo) d) Ann. 1373 K, 2 h. reheated 1373 2.301 K, H20 quench. a) Single crystal from melt, ann. 2.138 1373 K, H20 quench. b) Single crystal from melt, slow 2.076 cool from 1723 K. F(T) in figure 19.

Nakamura (1935) 99.84 Ni Lonl/;. res. freq. Vac. ann .• 12n K. 1 h • .o;low cool; x (Fe). a)H=O, 1.940 b) H = 0.325 kOe. 2.275

Nakamura (1936) L~mg. res. freq. F(Cu),f(T. 288-743 K). 1.950

Siegel, Quimby (1936) 99.715 Ni(O.Ol Si, S 0.02 Cu, Long. res. freq. Ann. 1373 K. 4 h, a atmos., slow 0.11 Fe, Mg 0.05 C) figure 33. cool; GS=O.4 mm.; f(H, T) in a) H=O, 2.085 "b) H sat. 2.218 Davies, Thomas (1937) 99.2 Ni Res. freq. a) Ann. 1123 K, 45 h. 1.937 b) Unannealed. x(Fe). 1.918

Forster, Koster (1937a, Transv. res. freq. Ann. 973 K.t h,x(Fe).f(T, 293-745 2.15 1938) K).

Forster, Koster (1937b) Transv. res. freq. Ann. 973 K, 1 h, slow cool; x(Fe); 2.1193 f(vibr. ampl.).

Engler (1938) 99 Ni Long. res. freq. Ann. 973 K;f(T, H) in figure 33. a) H=O, 2.06 b) H = 0.575 kOe. 2.21

Kimura (1939) "Electrolytic" Long. res. freq. Ann. 1223 K, 3 h, slow cool;!(T, H) in figure 33; a)H=O, 1.921 b) H sat. 2.174

Koster (1940) Transv. res. freq. Ann. 973 K, 1 h. slow cool; x (Fe). 2.11

Aoyama, Fukuroi (1941) 98.9 Ni(0.3 Mn, 0.05 Fe) "Dynamic". Vac. ann. 1173 K, 6 h,f(Cu); a) T=289 K. 1.70 b) T=78 K. 1.95

Yamamoto (1941,1943) "Electrolytic" Magnetoscillation. Vac. ann. 1273 K, 2 h, slow cool; 2.012 f(H).

Yamamoto (1942, 1954) 99.6 Ni(0.08 Fe, 0.32 Co, 0.01 Magnetoscillation. Vac. ann. 1173 K, 2 h, x(Cu); f (H); Si, 0.02 C) a) H=O, 1.86 b) H=0.6 kOe. 2.17

Koster (1943a) Ni(0.5 Mn) Transv. res. freq. a) Ann. 1273 K: cold rolled 80% 2.15 b) Vac. ann. 973 K 2.00 c) Vac. ann. 1173 K 1.92 d) Vac. ann. 1573 K, H = 0 1.82 e) Vac. ann. 1573 K, H sat. 2.19 F(T, GS. T anneal) in figure 14.

Koster (1943b) Ni(0.5 Mn) Transv. res. freq. Vac. ann. 913 K, 1 h, slow COlli; x(Fe);f(H);f(T,293-973 K);

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 571

TABLE 9. Young's modulus E of nickel-Continued

12 Investigator(s) Composition Technique Specimen, Test Conditions E(10 2 (Year) dyn/cm )

a) H=O, 1.87 b) H=0.5 kOe. 2.19 (Extrapolated paramagnetic fcc (2.16) phase to 293 K.)

Masumoto, Saito (1944) 99.6 Ni(0.02 Fe, 0.02 C, 0.01 "Static", helical coils. Vac. ann. 1273 K, I h, x(Cu). 1.68 Si)

Koster (1948) Transv. res. freq. Ann. 1173 K;f(T) in figure 19; a) H=O, 1.93 b) H sat. 2.18

Koster, Hauscher(1948) Transv. res. freq. X(Cu); a) H=O, 2.01 b) H sal. 2.16 Bennett, Davies (1949) 99.9 Ni, 99.2 Ni Transv. res. freq. 923 K ann. 8-29 h,f(T, 273-850 K) relative values, x(Fe).

Fukuro~, Shibuya (1950) 98.9 Ni(0.3 Mn, 0.05 Fe) Bending, interferometer. Vac. ann. 1173 K, 1 h x(Cu). 1.65

Beol"k, KOllv",lit",Q., M,..J(pp- "r.ommpT',..illlly PIIT'P" Long re!':. frell. a) Ann. at 923 K. 3 h. in HOI. 1.873 han (1951) b) Unannealed. 2.010 F(H) in figure 37. M/E data in Kouvelites, McKeehan (1952)

Frederick (1947) 0.5-15 MHz pulse-echo. F (T) in figure 19. 2.075 Yamamoto, Taniguchi Magnetostrietive oscillation. Vae. ann. 1273 K, 2 h; f (Co cone.), (1951, 1955) Ni values obtained by extrapola­ tion of Ni-Co data;f(H); a) H=O, 1.99 b) H sat. 2.24

Umekawa (1954) Transv. res. freq. Vac. ann. 1123 K,! h; x( Cu, Co). 1.72

Burnett (1956) 99.8 Ni(O.04 Fe, 0.03 Mn, 0.11 Res. freq. in bending. F(r) in figure 19. 1.89 Si,O.OI Cu, C)

Pavlov, Kirutchkov, 99.99 Ni 0.7 MHz transv. res. freq. Vac. ann. 1073 K,3 h; x(Cu); f(T) 2.00 Fedotov (1957) in figure 19.

Shirakawa, Numakura Ni(O.09 Mn, 0.01 C) Bending. a) Vac. ann. 973 K, I h, GS = 0.060 1.70 (1958) mm. b) Vac. ann. 1073 K, 1 h, GS=0.069 1.69 mm. c) Vac. ann. 1173 K, 1 h, GS = 0.082 1.68 mm. d) Vac. ann. 1273 K, I h, GS=0.111 1.66 mm. t::) Vac. <111U. 1373 K, 111, GS-o.no 1.65 mm.

Yamamoto (1959) Ni(O.1 Fe, 0.01 Si, Cu, 0.04 C, Magn~tostrictive oscillation. Ann. 1273 K, H2 atmos., 1 h; rean­ 0.02 S) nealed 1273 K, vac., Ii h; x(Fe); f(H), a) H=O, 1.749 b) H sat. 1.915

Hill, Shimmin, Wilcox Ni(O.OI C, Si, P) Long. res. freq.. F(T, 293-920 K). 2.02 (1961)

Durham, et al. (1963) Ni(O.3 Mn, 0.1 Fe, 0.06 C, 0.1 Tension, strain-gauge ex- Ann. 1213 K, i h; l( T) in figure 19. 1.96 S) ten so meter.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 572 H. M. LEDBETTER AND R. P. REED

TABLE 9. Young's modulus E of nickel-Continued

Investigator(s) E (10 12 Composition Technique Specimen, Test Conditions 2 (Year) dyn/cm )

Kamber (1963) "A" Nickel; Ni(O.1 Cu, C, 0.15 1.2 MHz transv. res. freq. a) ann. 1623 K, H2 atmos.,t h,GS= 1 1.90 Fe, 0.20 Mn, 0.05 Si, 0.005 mm; no field. S) b) ann. 1623 K, H2 at mos. ,i h, GS = 1 2.16 mm; saturated field .. c) ann. 1173 K, H2 atmos., 1 h, GS= 2.10 0.13 mm. F(T) in figure 19.

Tino, Maeda (1963) Electrolytic Ni Res. freq., 20-30 kHz. a) H=O, 2.06 b) H=320 Oe. 2.17 F(T, H) in figure 33.

Armstrong, Brown (1964) Ni(0.55 metallic, 0.05 C, 0.005 L(lng. res. freq. Ann. 1I73,/(T) in figure 19. 2.06 S+O)

Orlov, Fedotov (1966) 99.98 Ni Trans., long. res. freq. Vac. electron arc melting, x(Cu), 2.08 f(T) in figure 19.

Masumoto, Saito, Mura· Ni(0.19 Co, 0.01 Fe, AI, 0.02 Res. freq. F(T) in figure 19. 1.95 kami, Kikuchi (1968) c)

Faninger (1969) 99.99 Ni Tension, x-ray. X(Cu). f(H). 2.15

Masumoto, Saito, Sawaya Electrolytic 99.98 Ni (0.016 600-800 Hz oscillator. Vac. ann. 30 min. at 1173 K, cooled 1.96 (1970) Co, 0.001 Cu, 0.001 Fe, 300 °C/h. 0.002 S, 0.000 Si, Mn, Pb, q

Best values (no field) 1.97 (sat. field) 2.18

Uncertainties (no field) 0.15 (sat. field) 0.09

TABLE 10. Young's modulus E of iron-nickel alloys

Investigator(s) E 0012 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Guillaume (1897, 1898, 15 alloys Spring tension. X(Cr, c),f(T, 243-313 K). Figure 5 1927)

Angenheister (1903) Fe-24.1 Ni (0.36 C, 0.41 Mn) Tension. a) Ann. "non-magnetic", 1.74 b) Cooled to 76 K, "magnetic". 1.53

Carpenter, Hadfield, 6 alloys, impurities (O.40-0.5~ Tension extensometer. Ann. 1030 K, slow cool. Figure 5 Longmuir (1905) C, 0.8-0.18 Si, 0.01-0.04 S,P

Honda, Terada (1907) 6 alloys Tension, opt. lever. F(S). Figure 5

Honda (1919) 11 alloys Bending, opt. lever. Ann. lIn K. Figure 5

Muller (1922) 12 alloys, impurities (0.12- Tension, opt. microscope). Ann. H[:~- 1117:~ K. Figure 5 0.73 C, 0.34-1.24 Mn, 0.13-0.27 Si, ~ 0.02 P, ~ 0.04 S)

Carrington (1924) Fe-3.41 Ni (0.19 C, 0.55 Mn, Bending. I AIIII. [[1\0 K, ! h, slow cool; f(T) in 2.07 0.03 p, S, 0.10 Si) ti~lIn' 21.

Kimha1J, Lovell (1925) Fe-3.5 Ni (0.35 C, 0.21 Si, Transv. res. fretl. ·\1U1. 112:3 K, oil quench, drawn 923 0.02 S, P, 0.58 Mn) 1\../( 1') in figure 22.

Honda, Tanaka (1926 12 allo y s Bendin". opl. I,·v,·!. \illi. 1173 K, 2 h, slow cool; f (H). Figure 5

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 573

TABLE 10. Young's modulus E of iron-nickel alloys-Continued

Investigator(s) E(1012 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Chevenard (1927, 1943) 33 alloys Bending, torsion. Ann. 1173 K. Figure 5 also Chevenard, Crussard (1943)

Nishiyama (1929) 9 alloys from electrolytic Fe, Bending, opt. lever. Ann. 1173 K, 2 h, furnace cooled. Figure 5 Mond Ni

Kawai (1930) Fe-3 Ni (0.3 C) Tension. a) Ann. 1133 K, 2.07 b) tensile elongati~n = 2.1 %, 2.01 c) tensile elongation = 5.3%, 1.99 d) tensile elongation = 8.0%, 1.97 e) tensile elongation = 12.9%. 2.02

Honegger (1932) a) Fe-5 Ni Transv. res. freq. 1.99 b) Fe-3 Ni 1.94

Keulegan, Houseman Tension, torsion. F(T, 295-906 K) in Fe-3.5-S.0 Ni, (1933) ann. t h, furnace cooled or water quenched; Fe-35 Ni as received; I(T), temp. coeff. at 273 K.

Nakamura (1935) 10 alloys from 99.94 Fe, Ni Long. res. freq. Vac. ann. 1273 K, 1 h, slow cool, Figure 5 I(H).

Moller, Barbers (1936) Fe-0.02 C, Si, S, 0.37 Mn, a) X-ray. Vac. ann. 873 K, 1 h, slow cool. 1.79 0.05 P) b) Tension. 1.98

Forster, Koster (1937) Fe-22.4 Ni Transv. res. freq. F (T), in figure 38. 1.49

Forster, Koster (1937) 6 alloys Transv. res. freq. F(vibr. amp!.). Figure 5

Davies, Thomas (1937) Fe-48 Ni Res. freq. a) Ann. 1173 K, 7 h. 1.365 b) As received. 1.529 Doring (1938) F'e42 Ni Kes. freq. a)H=U. 1.575 b) H = 0.575 kOe. 1.62 P(T) in figure 20.

Engler (1938) 4

Scheil, Thiele (193B) Fe-22.4 Ni (0.2 Mn, 0.2B Si, Res. freq. Ann. lOB3K, Hh/(T) in ne;ure 21. 1.50 0.03 C, 0.01 P, S

Williams, Bozorth, Fe-68 Ni (0.3 Mn) Long. res. freq. a) Rolled (83% red. thick.), i)H=O, 1.8145 Christensen ii) H Sat. 1.8161 (1941) b) Ann. 1273 K, 1 h, Ii:!. i)H=O, 1.783. slow cool, ii)HSat. 1.870 c) Treatment (h), then i)H 0, 1.654 873 K, rapid cool. ii)H Sat. 1.826 d) Treatment (b), then i)H=O, 1.843 873 K, H2 cool in 0.01 ii)H Sat. 1.930 kOe long. field, e) Treatment (b); then i)H=O, 2.105 873 K, H2 , cool in 0.01 ii)H Sat. 2.181 kOe transv. field,

Chevenard, Crussard a) Fe-49 Ni Bending, torsion. Cold worked 44%; I(T) in figure 20. 1.93 (1943) b) 12 alloys Annealed. Figure 5 Seager, Thompson (1943) Fe-3.1 Ni (0.51 Mn, 0.16 Si, Bending, interferometric. Ann.,j(T) in figure 21. 1.92 0.2 C, 0.01 S, P)

Koster (1943b) 11 aHoys Transv. res. freq. Ann. 1273 K, 1 hjf(Ni). F~H)" figure 5 1 (T) in figure 20.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 574 H. M. LEDBETTER AND R. P. REED

TABLE 10. Young's modulus E of iron·nickel alloys-Continued

Investigator(s) E(10 12 Composition Technique Specimen, Test Conditions . (Year) dyn/cm2 )

SeheH, Reinacher (1944) 7 alloys (0.1-0.9 Mn, 0.1-0.3 Si) Transv. res. freq. Ann. 1173 K, H2 atmos.,! h., slow FigureS cool; J(T) in figure 20.

Fontana (1948) Fe-8.6 Ni (0.77 Mn, 0.23 Si, Tension, strain gauges. Ann. 1173 K, 1 h, air cool then 1063 K, 1.91 0.05 AI, 0.05 Ti, 0.01 C, P 1 h, air cool; then 838 K, 2 h, air 0.02 S) cool; J(T) in figure 21.

Bennett, Davies (1949) Fe48 Ni Res. Freq. Ann. 923K, 10 h; J(T) in figure 22.

Fine, Ellis (1950) 15 alloys (0-0.4 Co, 0.0.7 Mn) Long. res. freq. Cold swaged 74%; ann. 1223 K. 1 h; FigureS J(T) in figure 20. Cold swaged 41, 55%; ann. 673 K, not plotted.

Beck, Kouvelites, 5 alloys (40.2,09.1,84.9,88.5, Long. res. freq. Ann. at 923 K, 3 b, in H2. F(H) in Yigure5 McKeehan (1951) 100Ni) figure 37. ll.EIE data in Kouvelites, McKeehan (1952).

Och5cnfcld (1955) Fe-60Ni RCI!>.flCq. <1) Nu fidd, 1.76 b) Saturated field. 1.86

Markham (1957) Fe-3 Ni a) Tension. 2.01 b) 10 MHz pnJ .. e.e",ho. 2.05

Yamamoto (1959) 12 alloys from Armco iron, Magnetostrictive vibratiori. Ann. 1273 K, 1 h, H2; then vac. ann. Figure 5 Mond nickel 1273 K, H h;!(H, x).

Hill, Shimmin, Fe-35.6Ni Long. res. freq. "Recrystallized";J(T) in figure 20. 1.48 Wilcox (1961)

Durham, et al. (1963) Fe-36 Ni (0.8 Mn, ·0.4 Si, 0.2 Torsion, opt. lever. Cold drawn 12-15%,J(T) in figure 20. 1.45 Se, 0.08 C, 0.01 P, S)

Tino, Maeda (1963) 6 alloys, 26.2-100 Ni Res. freq., 20·30 kHz. F(T, H) in figure 34,/(D) in figure FigureS 13. Goldman, Robertson Fe·29.9 Ni (0.004 C) Fe·25.1 Long. res. freq. Vac. ann. 1173 K, i h; 0.035 mm GS; FigureS (1964) Ni (0.26 C) J( T) in figure 38.

Doroshek (1964) 6 alloys (0.4·0.5 Mn, 0.14-0.48 Res. freq. Figure 5 Si, 0.17·0.38 Cr, 0.01 AI, 0.02·0.03 C, 0.02·0.04 P, S)

Smith, Stem, Fe·2.5 Ni (0.4 C, 0.6 Cr, Long, transv. pulse·echo. F(P), third· order stiff'nesses. 2.10 Stephens (1966) 0.5Mo)

Eganyan, Selissikiy (1967) Fe·75 Ni Res. freq. F(H), a) H= 0, 2.06 b) H= I kOe. 2.16 F (T) in figure 20.

Kototayev, Koneva (1968) Fe-75 Ni Res. freq. F (T) in figure 22, relative values......

Khomenko, Tseytlin 9 alloys, 30.2·46.4 Ni Res. freq. Va(:. anll. at 1173 K, 2 h, cooled at (1969) lO(f/h. a) :~O.2 Ni. H = 0; 1.67 Ii) ~U.2 Ni. H = sat.; 1.67 d 46.4 Ni,H = 0; 1.33 eI) 46.4 Ni, H = sat. 1.49

Shirakawa, et at (1969) 9 alloys, 26.2·100 Ni Res. freq. /I ~at. Polycrystal and (100), (110), FigureS ( III ) single crystals.

Maeda (1971) Fe·3S Ni Res. freq. V;w. ann. at 1273 K, 10 h,

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES 01= METALS AND ALLOYS 575

TABLE 10. Young's modulus E of iron-nickel alloys-Continued

12 Investigator(s) Specimen, Test Conditions E(10 Composition Technique 2 (Year) dyn/cm )

a) H=O, 1.26 b) H= 1900e, 1.36 c) H = 1900 Oe. 1.38 F(T,H) in figure 34.

Diesburg (1971) 3 alloys 10 MHz pulse-echo-overlap. Calculated from Cij by V-R-H method. FigureS

Hausch, Warlimont (1972) 9 alloys 10 MHz pulse-echo. Calculated from Cij by V-R-H method. FigureS

TABLE 11. Shear modulus G of iron

Investigator(s) Composition Technique Specimen, Test Conditions G (10 12 2 (Ve~r) dyn/cm )

Earlier reports Range = 0.68 - 0:83. average = (1853-1900) 0.76

Schaefer (1901) Torsional res. freq. 0.719

Benton (1903) "Steel" Torsion F(T, 76-300 K), relative to 300 K.

. Horton (1905) Res. freq . F(T),f(A). 0.826

Griineisen (1908) a) "Steel" Long. res. freq. 0.812 b) "Iron" 0.831

Guye, Freedericksz Torsion pendulum. F (T) in figure 23. 0.805 (1909)

Koch, Dannecker (1915) Torsion F (T) in figure 23. 0.79

Honda (1919) Fe (0.29 Cu, 0.31 Mn, Torsion, opt. lever. X(Co). 0.84 0.11 Si, 0.09 C, 0.03 P, S)

Honda (1919) Torsion, opt. lever. Ann. 1173 K, J(X). 0.834

Honda, Hasimoto (1921) Fe (0.18 C, 0.2 Si + Mn) Torsion, opt. lever. Ann. 1173 K, a) slow cool, 0.832 b) oil quench. 0.806 X(C), J(eR).

lokibe, Sakai (1921) 99.98 Fe (0.0085 q Torsional o9cillations. Ann. 1973 K, f(T) in figure 23. 0.70

Kikuta (1921) Fe (0.3S C) Res. freq. Ann. 1173 K, J(T) in figure 23. 0.807

Honda, Tanaka (1926) Fe (0.38 Mn. 0.1 r.. 0.02 P, S, Tonion, opt. lever. X(Ni, Co, C), f(H). Fe (0.31 Mn, 0.11 Si, a) Ann. 1173 K, 1 h slow cool. 0.813 0.09 C, 0.29 Cu, 0.03 P, S) b) Ann. 1173 K, 1 h oil quench. 0.792 c) Ann. 1173 K. 2 h., slow cool. 0.834

Chevenard (1927) Torsion. X (Ni, Cr, C). 0.85

Goens (1930) a) Res. freq. 0.800 b) Torsion. 0.808

Gutenberg, Schlechtweg Res. freq., torsion. 0.78 (1930)

Everett (1931) Fe (0.35 C, 0.80 Mn, Torsion, dial gauge, mech. Ann. 1173 K, lh, slow cool F(T, 0.794 0.10 Sit 0.02 P, 0.03 S) lever. 295-773K).

Kawai (1931) "Armco iron" Torsion, opt. lever. Ann. 1273 K, J(D) in figure 10. 0.812

M'Farlene (1931) "Soft iron" Torsion pendulum. F(D) in figure 10. 0.78

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 576 H. M. LEDBETIER AND R. P. REED

TABLE 11. Shear modulus G of iron - Continued

Investigator(s) G (10 12 Composition Technique Specimen, Test Conditions 2 (Year) dyn/cm )

Keulegan, Houseman Fe (0.66 C, 0.8 Mn, 0.01 P, Helical loaded springs. Temp. coeft"., 223-323 K. (1933) 0.04 S)

Verse (1935) a) Fe (0.4C, 0.86 Mn, 0.04 S, Torsion. Ann., f(T, 298-773 K). 0.796 0.02 P, 0.14 Si) b) Fe (0.34 C, 0.80 Mn, 0.10 Si Long. torsional res. freq. Ann. 1173 K, 1 (T) in figure 23. 0.798 0.02 P, 0.03 S)

Bez-Bardili (1935) Long., transv. sound-wave 0.845 velocities, 1-20 MHz.

Brown (1936) "Armco iron" Long. (56 MHz), torsional Ann., f(H). 0.846 (39 MHz) res. freq.

Glocker, Schaaber (1938) Fe (0.06 C) Torsion, x-ray 0.798

Everett, Miklowitz (1944) Fe (0.15-0.25 C, 0.3-0.6 Mn, Torsion, opt. lever. F(T) in figure 23. 0.78 0.04 P, 0.05 S); SAE 1020

Garofalo, Malenock, Fe (0.45 Mn, 0.19 Si, 0.13 C, Torsion, opt. lever. Ann. 1170 K,! h air cooled; I(T) in 0.80 Smith (1952) 0.01 P, 0.02 S); (SAE 1015) figure 23.

Hughes, Kelly (1953) "Armco iron" Pulse-echo. F(P). 0.820

Burnett (1956) 99.8 Fe "Free-free sonic vibration", F (T) in figure 23. 0.78 res. freq. in torsion.

Hughes, Maurette (1956) "Armco iron" Pulse· echo. F(P), f(T) in figure 23. 0.814

Voronov, Vereshchagin 99.8 Fe (0.02 Mn, 0.02 Si, 10 MHz pulse-echo. Ann.; f(P). 0.812 (1961) 0.012 C, 0.03 P, S)

Durham, McClintock, Fe (0.8 C, 0.3 Mn, 0.15 Si) Torsion, opt. lever. Ann. 1056 K, 1 h, oil q~ench, temper 0.795 Reed, Warren, 655 K, 1 h, air cool; f(T) in figure Guntner (1963) 23.

Smith, Stern, Stephens a) Fe (0.8 Mn, 0.3 Si,0.4 C) Long. and transv. pulse· X (Ni), I(P), third·order polycrystal- 0.821 (1966) echo. line moduli. b) Fe (0.8 Mn, 0.2 Si, 0.6 C) 0.820

Frederick (1947) "Armco iron" 0.5-15 MHz pulse-echo. F(T). 0.808

Speich, Schwoeble, Fe (0.057 C) 80 kHz pulse-echo. Austenitized 1 h at 1000 °C and water 0.806 Leslie (1972) quenched.

BC5t value 0.31

Uncertainty 0.03

TABLE 12. Shear modulus G of nickel

Investigator(s) G (1012 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Earlier reports (1853- Range: 0.68-0.83 average 1900) 0.76 Schaefer (1901) Torsion, res. freq. 0.933

Griineisen (1908) Long. res. freq. Hard drawn. 0.770

Guye, Schapper (1910) Torsional oscillations. F(T) in figure 24. 0.762

Koch, Dannecker (1915) Res. freq. F (T) in figure 24. 0.716

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 577 TABLE 12. Shear modulus G of nickel-Continued

i i Investigator(s) G (10 12 Composition Technique Specimen, Test Conditions 2 (Year) dyn/cm )

Iokibe, Sakai (1921) Torsional oscillations under Vac. ann. 1073 K, f(T) in figure 24. 0.723 tensile lond.

Kikuta (1921) Torsional oscillations under Ann., f(T) in figure 24. 0.782 tensile load.

Honda, Tanaka (1926) 99.6 Ni (0.15 Fe, 0.14 C, 0.01 Torsion, opt. lever. Ann. 1173 K, 0.5 h; f(H)+ x(Fe). 0.750 Cu, 0.05 Si, 0.04 S}

Chevenard (1927) Torsion. X(Fe, Cr). 0.86

Mudge, Luff (1928) 99.18 Ni (0.1 C, 0.16 Cu, 0.06 Torsion. a) Hot rolled. 0.78 Si, 0.01 S, 0.4 Fe, 0.05 Mn) b) Ann. 1030 K, 4 h. 0.758

Gutenberg;, Schechtweg Res. freq. in torsion. 0.80 (1930)

Kawai (1931) 99.5 Ni Torsion, opt. lever. Ann. 1073 K, f(D, tension) in figure 0.785 1l,f(R).

Mobius (1932, 1934) Res. freq. F(H); f(T, 293-673 K).

Kikuchi (1936) Torsion. X(Cu). 0.710

Landon, Davies (1938) 99.2 Ni Res. freq. in torsion. a) Ann. 1133 K, t h, cold rolled, O.BOO Brinnell hardness = 210. b) Ann. 1133 K, 1 h, cold £oIled, ann. 0.027 1133 K, t h .

. Burnett (1956) 99.8 Ni (0.04 Fe, 0.03 Mn, Res. freq. in torsion. F (T) in figure 24. 0.738 0.11 Si, 0.01 Cu, C)

Susse (1956} 99.7 Ni Torsional res. freq. F(T) in figure 24. 0.79

Orlov, Fedotov (1966) 99.98 Ni Long., transv., torsional Ann., x(Cu), f(T) in figure 24. 0.805 res. freq.

Faninger (1969) 99.99 Ni X-ray, tension. X{Cu), f(H). 0.83

Shirakawa, et a1. (1969) Electrolytic Ni (0.012 Fe, Res. freq. H sat., 1 mm GS. 0.83 0.007 Si, 0.008 Cu, 0.003 Mnl hederick (IY47) 0-15 MHz pulse-echo. F{T). 0.808

Best value 0.785

Uncertainty 0.05

TABLE 13. Shear modulus G of iron-niCkel alloys

Investigator(s) G (10l2 Composition Technique Specimen, Test Conditions (Year) dyn/cm2 )

Angerheister (1903) Fe·24.1 Ni (0.36 C, 0.41 Mn) Tension. torsion, res. freq. a) Ann. "non·magnetic", fcc. 0.67 b) Cooled to 76K. "magnetic", bcc. O.5~

Guye, Woelfle (1907) Fe-36.11B Ni (0.02 Cn) Res. freq. F(T) in figure 25. 0.563

Honda (1919) 13 alloys Torsion, opt. lever. Ann. 1173K. Fignre6

Chevenard (1920) Fe-36 Ni Res. freq. in torsion. Ann. 973K, cold worked, tempered, ann. F(T) in figure 25.

Honda, Tanaka (1926) 12 alloys Torsion, opt. lever. Ann. 1173 K. i h;f(H). Fignre6

J. Phys. Chern. Ref. Data, Vol. 2~, No.3, 1913 578 H. M. LEDBETTER AND R. P. REED

TABLE 13. Shear modulus G of iron·nickel alloys - Continued

Investigator(s) G (lOu Composition Technique Specimen, Test Conditions (Year) dyn/cm2)

Chevenard (1927) 35 anoys Res. freq. in torsion. Ann. 1023K, x(Cr); f(T) , relative in Figure 6 tigure:lb.

Landon, Davies (1938) Fe·48 Ni Res. {req. in torsion. a) Unannealed 11 73K, 20 min., rolled 400/0. 0.601 b) Ann. 1173K, 20 min., rolled 40%, ann. 1173K, 20 min. 0.596

Burnett (1956) 66.97 Fe, 31.95 Ni (0.81 Mn, Res. freq. in torsion. F (T) in figure 25. 0.58 0.14 Si, 0.05 Cr, 0.08 C)

Markham (1957) Fe·3 Ni a) 10 MHz pulse-echo. 0.797 b) Torsion. 0.790

Bungardt, Preisendanz, a) 90.68 Fe, 9.21 Ni (0.05 Mn, Res. freq. in torsion. Ann. 1173 K, 1 h; F(T, l00-nOO K), Brandis (1962) 0.02 C, 0.01 P, S, N) relative, in figure 26. b) 82.74 Fe, 17.14 Ni (0.05 Mn. 0.02 C. 0.01 P. S. N)

Durham, McClintock, Fe·36 Ni (0.8 Mn, 0.2 Se, Torsion, opt. lever. Cold drawn 12·15%.,f(T) in figure 25. 0.565 Guntner, Warren (1963) 0.08 C, 0.01 P, S)

Goldman, Robertson a) Fe·29.9 Ni (0.004 C) Long., torsional res. freq. Vac. ann. 1173 K,! h, 0.035 mm GS; (1964) b) Fe·25.1 Ni (0.26 C) V( T) in figure 25. i) fcc 0.68 ii) bcc 0.54 i) fcc 0.72 ti) bee 0.625

Smith, Stern, Fe·2.5 Ni (0.4 C. 0.6 Cr, 0.5 Long., transv. pulse-echo. F(P), third·order stiffnesses. 0.818 Stephens (l~bb) Mo)

Roberts, Owen 6 alloys (50 ppm C) Res. freq. in torsion. Ann. 1198K, i h, 2% H2 atmos., Figure 6 (1967) quenched.

Meincke, Litva (1969) Fe-35 Ni Sound velocity measurements. Ann. 1273K. air cool. F(T) in figure 25. 0.56 Maeda (1971) Fe-35 Ni Res. freq. Vac. ann. 1273 K, 10 h, a) H=O, 0.407 b) H= 1900e, 0.458 c) H = 19000e. 0.473 F(T, H) in figure 3.

Shirakawa, et al. (1969) 9 alloys, 26.2-100 Ni Res. freq. H sat. Polycrystal and (100). (11O). Figure 6 (Ill) single crystals.

Diesburg (1971) 3 alloys 10 MHz pulse-echo-overlap. Calculated from elj by V·R-H method. Figure 6

Hausch, Warlimont (1972) 9 alloys 10 MHz pulse-echo Calculated fwm Cij by V -R-Il lIIethud. Figure 6

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 TABLE 14. Bulk modulus B and compressibility B-1 of iron

Bs B-l=a* Br b* Investigator(s) Composition Technique SpeCimen, Test Conditions dBsldP (Year) (1012 (lQ-r (1012 (10-24 4 2 dyn/cm! cm2/dyn) dyn/cm2 ) cm /dyn )

Richards (1907) 0.40 Griineisen (190B) Res. freq., alll measure- ments. 0.62 1.61

Griineisen (190B) alit measurements. Range: 0.570-0.641. f(T) in figure 27. 0.61 1.64 m 0.61 1.64 ~ Richards, Bartlett (1915) aVIV measurements, Hg 100 < P < 500 bar. -tn batn. ..,"U 0.599 1.669 2.1B o Bridgman (1923) alii of rod under hydro- P~ lOok bar. "U 99.97 Fe 1.653 2.18 m static pressure. a) 303 K 0.605 ....., b) 348 K in 0.603 1.658 1.37 If' o 1.634 Ebert (1935) alii of rod, comparative. Single crystal, P ~ 5 kbar. 0.612 " ~ 1ft Ebert, Kussmann (1937) 99.99 Fe Press lIre dep. in magnetic X(Ni),f(H), saturated field. ~ field. en.­ 1.680 0.B2 Bridgman (1940a, 1940b) alll, lever piezometer. P~ 30 kbar, 0.595 ~ Slater (1940) correction. 0.601 1.664 0.B2 o l> a) 297 K, I""" 1.683 0.83 b) 348 K. 0.5941 ~ 1.695 0.84 3 III .... Birch (1947) Correction to Bridgman data. T=297 K. 0.590 "'Ill ~ l". Bridgman (1949) Fe(0.008 Ni. C, 0.002 i, alll, lever improved piezom- P ~ 30 kbar, T=296-299 K. o 0.001, < (J.00l Co, Cr, eler; Compared to Fe, ;- 1.641 2.7±1.6 6.093 1.641 ? Mo, Mn, (I, < 0.005 CI) Bridgman (1940). :III CD :"" Hughes. Kelly (1953) "Armco Iron" Pulsed ultrasound Simple compression and hydro- c static pressure to 9 kbar. 4.30 0.6033 1.656 0.94 7 o vel,)cities. p Hughes. Maurette (1956) "Armco iron" Pul~e-echo. Polycrystals; ~ f(T) in figure 27; 1.686 ~ isothermal data calculated; 1.683 Z P a) P= 1 bar, T=303 K; 1.735 b) P= I bar, T= 373 K; 1.712 ~ c) P=9 kbar, T=303 K; ...,00 5.13 0.606 1.650 I.H w d) P=9 kbar, T=373 K. 1.667 ~ en TABLE 14. Bulk modulus B and compressibility B-1 of iron-Continued 0) ~ C- ::r"V ~ Bs B:;.I=a* BT b* n Investigator(s) ::r Composition Technique Specimen, Test Conditions dBsldP CD (Year) 12 12 12 24 ? (10 (10- (10 (10- 2 2 2 4 2 ~ dyn/em ) cm /dyn) dyn/cm ) cm /dyn ) ;!. o Voronov, Vereshchagin 99.8 Fe(0.02 Mn, Si, 10 MHz pulse-echo. Ann. polycrystals, up to 10 kbar, iso- Q ~ (1961) 0.01 C,0_03 P, S) thermal data calculated. ~ Rotter, Smith (1966) 99.99 Fe(Cr, Mn traces) 10 MHz pulse-echo. Specimens from crystal of Rayne, ~ Chandrasekhar (1961),f(P). 1.569 5.97 0.6098 1.637 1.29 zp ~ Aliev, Lazarev, Sudovtsov Bimelallic helix. T=4.2 K. (1967) 0.58 1.72 -0 ...... Co) Guinan, Beshers (1968) "Armco iron" 10 MHz pulse-echo. Specimens from Lord, Beshers (1965), single crystals;/(P); is[)- 1.564 5.29 0.6117 1.635 1.15 thermal data calculated. !I= ~ Takahashi, Bassett, Mao High pressure x-ray diffrac- a) From Murnaghan molar volume ,.. 0.618 1.62 (1968) tiOIl, NaCl standard. eqn. mo b) From 1st-order Birch molar 01 volume eqn. 0.625 1.60 m :sm 1.17 jiIU *a and b are coefficients in the equation ilV/Vo= aP+ bPz. Best values 1.664 4.68 0.606 1.649 » z Uncertainties 0.'Jl5 1.26 0.009 0.025 0.43 o ?G :v jiIU m 1fto ELASTIC PROPERTIES OF METALS AND ALLOYS 581

TABLE 15. Bulk modulus B and compressibility B-1 of nickel

Investigator(s) Brl=a* Br b* (Year) Composition Technique Specimen, Test Conditions (10- 12 (1012 (10-24 2 2 4 cm (dyn) dyn{cm ) cm /dyn2 )

Griineisen (1908) Res. freq., alIt 0.57 1.75 measurements.

Bridgman t (1923) a) 99 Ni. alIi Fe standard, pie· Ann. 2 h. F(P, ~ 1 khar). zometer, hydrostatic P. i) T=303 K, 0.535 1.869 2.14 ii) T=348 K. 0.539 1.855 2.14 b) "Pure" Illli tensile. Drawn, ann. to "bright red" i) T=303K, 0.540 1.853 2.14 ii) T=348 K. 0.546 l.832 2.14

Ehert, Kussmann Ni(O.12 Fe, IlA/A in magnetic Saturated field. 0.542 1.845 (1937) 0.04 Mn, field. F(H),X(Ni), p~ 10 khar. Si, 0.1 Co., 0.05 Cu)

Bridgman (1949) about 99.98 Piezometer, Fe stand· Ann., H2, - 1650 K. 0.488 2.048 Ni arcL hynrostatic P, F(P, ~ .~o khllr). Ill/l. T= 296-299 K.

Aliev. Lazarev Bimetallic helix, differ· T=4.2 K. 0.49 2.02 Sudovtsov (1967) ential, compared to Ph

Tanji, et ai. (1970), Calc. from E, G data. Ann. 1273 K, 3 h. reported also in a) H=O, 0.46 2.17 Shirakawa, et al. b) H sat. 0.54 1.85 (1969)

*a andb are coefficients in the equation IlVIVo=aP+bP2. Best values 0.526 1.903 2.14 tOriginal values corrected according to Bridgman (1946, 1949). Uncertainties 0.036 0.127 -

TABLE 16. Bulk modulus B and compressibility B-1 of iron· nickel alloys

In vestigator( s) B-1 B (Year) Composition Technique Specimen, Test Conditions (10-12 (101 2 2 cm /dyn) dyn/cm2)

Ehert, Kussmann (1937) 13 alloys from 99.99 Pressure dependence in mag· X(Co, Cr, Pt). Figure 7 Fe, Ni(0.12 Fe, netic field. U.04 Mn, U.l Co, 0.05 Cu, 0.04 Si)

Takahashi, Bassett, Fe-S.15 Ni P to 300 khars, H to IS kOe, a) Using Murnaghan eqn. 0.645 1.55 Mao (1968) x-ray diffraction, NaC) b) Using Ist order Birch eqn. 0.641 1.56 standard, molar volume detn. Fe-l0.26 Ni a) Using Murnaghan eqn. 0.645 1.55 h) Using Ist order Birch eqn. 0.654 1.53

Meincke, Litva (1969) Fe-35 Ni Sound velocity measurements. Ann. 1273 K, air cool;f(T) in 0.90 1.11 figure 29.

Maeda (1971) Fe-35 Ni Flexural and torsional res;o- Vac. ann. at 1273 for 10 h. nance of a bar specimen. a) H= 190 Oe, 0.813 1.23 Calc. from E and G data. b) H= 1900 Oe. 0.847 1.18 F(T) ,J(H) in figure 36.

Diesburg (1971) 3 alloys 10 MHz pulse-echo-overlap Calculated from Clj by V-R-H Figure 7 method.

Hausch, Warlimont 9 alloys 10 MHz pulse-echo. Calculated from Cli by V-R-H Figure 7 (1972) method.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 582 H. M. LEDBETTER AND R. P. REED

TABLE 17. Poisson ratio 11 of iron

lnvestigator(s) Composition Technique Specimen, Test Conditions 11 (Year)

Earlier publica- a) "Steel" Range = 0.27 - 0.30. average 0.29 tions (1879- h) "Iron" Range = 0.26 - 0.32. average 0.28 1903)

Benton (1901) "Iron" Interference microscope. 0.29

Schaefer (1901) Torsion, res. freq. 0.247

Morrow (1903) "W rought iron" Meas. of lateral and long. Range= 0.270 - 0.289. 0.275 strains under compression.

Griineisen (1908) a) "Steel" Res. freq. 0.287 b) "Iron" 0.280

Griineisen (1910) Bending, torsion. Range= 0.28 - 0.293. average 0.284

Carrington (1924) a) "Wrought iron" Flexure. Ann. llS3 K, slow cool,f(T) in 0.23 figure 30. b) Fe(O.l9 C) Ann. 1183 K, slow cool,J(T) in 0.27 figure 30.

Honda, Tanaka Fe(0.09 C, 0.11 Si, 0.31 Mn, Bending torsion. Ann. 1173 K, 112 h,j(H), x(Ni). 0.27 (1926) 0.03 p. S, 0.29 Cu)

Keulegan, House- Fe{0.66 C, 0.8 M, 0.01 P, Loaded helical springs. Detm. temp. coeff., 223-323 K. man (1933) 0.04 S)

Verse (1935) a) Fe(0.43 C. 0.86 Mn. 0.04 Tension cathetometer Ann. f(/', 298-733 K). 0.295 S, 0.02 P, 0.14 Si) torsion. h) Fe(0.34 C, 0.80 Mn, 0.10 Long., torsional res. Ann. 1173 K,j(T) in figure 30. 0.307 Si, 0.02 P, 0.03 S) freq.

Bez-Bardili (1935) Long., transv. sound-wave Calc. from me as. E, C. 0.290 velocities, 1-20 MHz.

Smith, Wood (1941) 99.95 Fe Tension, x-ray spectrometer. Yac. ann. 0.050 in. sheet. 0.27

Everett, Miklowitz Fe(0.15-0.25 C, 0.3-0.6 Mn, Bending, torsion. a) Hot rolled, 0.313 (1944) 0.04 P, 0.05 S) [SAE 1020] h) Cold rolled. 0.286 F(T) in figure 30.

Garofalo, Fe(0.45 Mn, 0.19 Si, 0.13 C, Bending, torsion. Ann. 1170 K, 1/2 h, air cooled. F(T) 0.265 Malenock, 0.01 P, 0.02 S) [SAE 1015] in figure 30. Smith (1952)

Burnett (1956) 99.8 Fe "Free-free sonic vibration," F(T) in figure 30. 0.275 res. freq. in torsion.

·Yoronov, 99.8 Fe(O.01 C, 0.02 Si, Mn, 10 MHz pulse· echo. Ann., a) Hydrostatic P=O, 0.290 Vereshchagin 0.03 P, S) b) Hydrostatic P= 9.8 hare 2.292 (1961)

Shved (1964) "Armco iron" X-ray scattering, uniaxial 0.283 tension.

Bt:::;l va)ut:: 0.282

Uncertainty 0.019

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 583

TABLE 18. Poisson ratio II of nickel

Investigator(s) Composition Technique Specimen, Test Conditions II (Year)

Schaefer (1901) Torsion, res. freq. 0.240

Benton (1901) Interference microscope. 0.96 Diam. wire. 0.375 1.49 diam. wire. 0.271

Griineisen (1908) Static. 0.309

Honda, Tanaka (1926) Ni(0.05 Si, 0.01 Ca, 0.15 Fe, Bending, torsion. Ann. 1173 K·, 112. h; /(H); X(Fe). 0.29 0.04 S, 0.14 C)

Burnett (1956) 99.8 Ni(O.04 Fe, 0.03 Mn, 0.11 Res. freq. in bending, torsion. 0.280 Si,O.01 Cu)

Koster (1961) Calculated from KosterE, G 0.31 data.

Best value 0;296

Uncertainty 0.029

TABLE 19. Poisson ratio II of iron-nickel alloys

Investigator(s) Composition Technique Specimen, Test Conditions (Year) II

Angenheister (1903) Fe-24.1 Ni(O.36 C, 0.41 Mn) Tension, torsion, res. Ann., non-magnetic, fcc. 0.298 freq. Cooled to 76 K, magnetic, bcc. 0.358

Carrington (1924) Fe-3.4 Ni(0.19 C, 0.55 Mn, 0.03 Bending. Ann. 1180 K, Va h, slow cool; /(T) in 0.27 P, S, 0.01 Si) figure 32.

Honda, Tanaka (1926) 12 alloys Torsion, bending. Ann. 1173 K, l/a h;/(H). Figure 8

Chevenard, Crussard a) Fe-6Ni Torsion·flexure. i) Cold worked. 0.27 (1942) ii) Ann. 1073 K. 0.29 b) Fe-36Ni Ann. 1073 K; / (T) in figure 32. 0.28

Chevenard, Crussard a) Fe-49Ni Torsion-flexure. Cold worked 44%; f (T) in figure 32. 0.56 (1943) h) 12 alloys Ann. 1173 K, ('oM work",tI 44%. Figure S

Goldman, Robertson a) Fe-29.9 Ni(O.OO4 C) Long., torsional res. Vae. ann. 1173 K, 1/2 h, 0.035 mm GS;/(T) 0.28 (1964) b) Fe-25.1 Ni(0.26 C) freq. in figure 32. f('(', 293 K; 0.26 bee, 293 K; 0.23 fcc, 217 K; 0.22 bcc, 217 K. 0.20

Dieshurg (1971) 3 alloys 10 MHz pulse-echo- Calculated from Cij by V-R-H method. Figure 8 overlap.

Hausch, Warlimont 9 alloys 10 MHz pulse-echo Calculated from Cij by V-R-H method. Figure 8 (1972)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 584 H. M. LEDBETTER AND R. P. REED

TABLE 20. Third-order elastic stiffnesses Cijk of iron

Investigator(s) CUI CU2 C'23 C14.. CI66 C456 (Year) Technique I

(1012 dyn/cm2 )

Hughes, Kelly (1953)* Ultrasound velocities under tension and hydro- -3.48 -10.31 9.81 static pressure.

Seeger, Buck (1%0)* Poynting effect. -1.67 ~7.55 -14.9

Powell, Skove (1968) Measured deviations from Hooke's law of single -28.29 -8.00 -6.07 crystals. Calculated using Rotter, Smith (1966) pressure derivatives of second-order stiff- nesses.

Tietz (1969)* Ultrasound velocities under tension and hydro- 5.99 -9.44 static pressure.

Mathur, Sharma (1970) Calculated using Morse central-force potential -16.44 -2.60 -3.00 -3.00 -2.60 -3.00 and Girifalco, Weizer (1959) parameters.

*Determined isotrooic constants C~g9' c~.... ~md c~&,,: see text for relationship of these to o~her CUk.

TABLE 21. Third-order and fourth-order elastic stiffnesses CiJl,. and CiJkI of nickel

lnvestigator(s) CIlI CII2 CI23 CJ44 CI66 C456 CUll CIIl2 C1I22 C1I23 (Ye,ar) Technique (10 12 dyn/cm2 )

Rose (1966) Calc. from data in -14.37 -10.53 1.19 1.19 -10.53 1.19 102.70 58.63 65.48 -3.47 Huntington (1959) using finite strain theory and central- force potential.

Salama, Alers Change of sound -20.32 -10.43 ..,...2.20 -1.38 -9.10 -0.70 (1%9 velocity under uni- axial stress, speci- mens neutron irradiated to pin dislocations, satu- rated, 10 kUe held.

Sarma, Mathur Calc. from 9 nn Morse -17.896 -11.420 0.814 0.814 -11.420 0.814 -15.366 -0.741 -0.141 -0.083 (1969) potential.

Sarma, Reddy Change of sound -21.04 -13.45 0.59 -1.80 -7.57 -0.42 (1973) velocity under uniaxial stress, sp~cimens neutron irradiated to pin dislocations, satu· rated, 8 kOe field.

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 585

TABLE 22. Pressure derivatives dCij/dP of elastic stiffnesses of iron and nickel

CIl CI2 C44 (CII + 2Ct2)f3 (CII -Ct2)/2

Fe; Rotter, Smith (1966) 7.51 5.19 2.66 5.97 1.16

Fe: Guinan, Beshers (1968) 6.72 4.58 2.59 5.29 1.07

Ni: Salama, Alers (1967) 6.03 4.87 2.38 5.26 0.58

Ni: Sarma, MdlitUi (1969) 5.70 1.5S 2.65 4J)5 0.70

TABLE 23. Relative elastic constants based on a hard-sphere model, B= 1 arbitrarily

C~t C~2 C~4 t' Br Gr Er v A

b. c. c. 1.000 1.000 0.667 0.000 1.000 0.200 0.563 0.406 00 Fe (obs.) 1.375 0.838 0.726 0.481 1.000 0.489 1.246 0.282 2.43

f. c. c. 1.333 0.833 0.500 0.250 1.000 0.379 1.000 0.333 2.00 Ni (obs.) 1.350 0.825 0.666 0.486 1.000 0.413 1.145 0.296 2.54

TABLE 24. Debye characteristic temperatures (J of iron, nickel, and iron-nickel alloys calculated from single-crystal elastic data

Low·temperature Room-temperature

Wt. Wt. 0, K Ref. 0, K Ref. %Ni %Ni

0 472 Lord, Beshers (1965) 29.82 435 Tanji (1971) 35.3 351 Hausch, Warlimont (1972) 35.7 405 Tanji (1971) 36.1 348 Bower, et a1. (1968) 39.62 398 Tanji (1971) 37.7 358 Hausch, Warlimont (1972) 44.43 410 Tanji (1971) 38.2 346 Bower, et a1. (1968) 49.96 425 Tanji (1971) 40.0 . 369 Hausch, Warlimont (1972) 60.7 458 Tanji (1971) 42.5 383 Hausch, Warlimont (1972) 70.02 465 Tanji (1971) 45.2 396 Hausch, Warlimont (1972) 78.5 478 T anji (1971) 51.4 419 Hausch, Warlimont (1972) 89.6 475 Tanji (1971) 60.8 436 Hausch, Warlimont (1972) 99.98 474 Tanji (1971) 61.9 437 Hausch, Warlimont (1972) 78.5 463 Hausch, Warlimont (1972) 89.7 468 Hausch, Warlimont (1972) See tables 5-7 for. details. 100.0 472 Hausch, Warlimont (1972) 100.0 476 Alers, et a!. (1960)

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 !­ c.n "1:11 g: '7 ~ n ~ II ~ :lit !!- o a ~ ~ ~ Z P ~ ~..... Co) TABLE 25. Conversions among commonly used stress units

------N/m2 dyn/cm2 har khar kg/C1ll2 kg/mm2 atm psi ;z: 5 8 I Newton/sq. meter (N/m2 ) I 101 10- 10- 1.01972>< 10-5 1.01972 X 10-7 0.98692 X ]0-5 1.45038 X 10-4 ~

6 9 6 r­ I dyne/sq. em. (dyn/cm2 ) 10-1 I 10- 10- 1.01972>< 10- 1.01972 X 10-8 0.98692 X 10-6 1.45038 X 10-5 m C 01 I har 101 106 1 10-3 1.01912 1.01972 X 10-2 0.98692 1.45038 X 101 m ::sm 9 3 1 3 4 lIa I kilohar (khar) 10' 10 103 1 1.01972 X 10 1.01972 X 10 0.98692 X 10 1.45038 X 10 » Z I kilogram/sq. em. (kg/cm2 ) 0.98066 X 105 0.98066 X 106 0.98066 0.98066 X 10-3 I 10-2 0.96784 1.42233 X 101 c ?" 1 1 kilogram/sq. mm. (kg/mm2 ) 0.98066 X 107 0.98066 X 108 0.98066 X 102 0.98066 X 10- 102 I 0.96784 X 102 1.42233 X 103 :v lIa I atmosphere (atm) 1.01325 X 105 1.01325 X 106 1.01325 1.01325 X 10-3 1.03323 1.03323 X 10-2 1 1.46959 X 101 m m C I pound/sq. in. (psi) 0.68948 X 104 0.68948 X 105 0.68948 X 10-1 0.68948 X 10-4 0.70307 X 10-1 (J.70307 X 10-3 0.68046 X 10-1 I

- ~--.------~-- --

The S.I. unit for pressure and stress is the pascal (Pa), I Newton/sq. meter (N/m 2 ). ELASTIC PROPERTIES OF METALS AND ALLOYS 587

Realm: Pure Science Phenomenology Engineering

Parameter: Atomic Single-Crystal Engineering Force Constants Elastic Coefficients Elastic Constants

FIGURE 1. Schematic interconnectivity of elastic parameters of solids

1890

0.10

~ (E)

-0.10

-0.20

-0.30

0.10

M

-0.10

0.10

l::,.v <11>

-0.10

-0.20~-----~----~----~----~--~----~----~----~--~ 1890 1900 1910 1920 1930 1940 1950 1960 1970 YEAR FIGURE 2. Chronological variation of observed elastic properties of iron

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1913 588 H. M. LEDBEnER AND R. P. REED WEIGHT PERCENT Ni o 20 40 60 80 100 1 I I I I 1900 1600 I 1533° 1511° liquid 1700

1200 1500

y oU 1000 1300 ~ ... W 910° W a:: 0:: :::> Curie :::> .- 800 1100 r-

FIGURE 3. Iron-nickel phase diagram:

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND AllOYS 589

3.0xIOI2.-----.----.-----r----,...-----.----.---..-----.,-----.--~ 0' Alers.~tal. (1960) L ~ Oiesbul"g ,(1971) b ~ 6 Einspruch,Cloirborne (1964) Y II" .. t fcc, extpt,d. /. I-~,O Sakurai, et 01. (964) .. Hausch, Warlimont ~~ I~ • Salama,Alers (968)1 ~ r- 0 Shirakawa ,et 01. (1969) Y' 2.0 1---. Bce } Tabu lated -+---;:::;1'Cf~...... c...--+---+---+----+-----l ~ • Fcc Best Val ues t---.l.__ -+---+----+---f----+--fl ''', 'i" /' '... \... ~ /..011

1.0 r- dash line indicate:--+JJ;#.-..:'---_\+~__''''''---_Iy---+---+---+----+--_1 extrapolated fcc IJ---r--_ ---v ...... Dec TCC

20 40 60 80 100 WEIGHT Percent Ni

FIGURE 4. Compositional variation of elastic stiffnesses Cij of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 590 H. M. LEDBETIER AND R. P. REED

~~-+~~~~-+4-+-~-+~~'~

2AO ~~-+~+-~-+~4-~~-+~~~•

II ~~-+~+-~-+~4-+-~-+~~ 2. 32 ~t-+-+~~~~-+-4-+-~-+~+--l 0 ~t-+-+~~~~-+-4-+-~-+~+--l.

f---f---4--+-+-~-+---l V Hondo (919) - fee 2.24 ~1--+-+-'+--+--t---t=4-4-+---'I--+-+~~. - bee I!I MUlier (1922) - bee • Hondo, Tanaka {l926)-bee -t--+-+-f--I_+_-+--+-'I'--~~,,"+--+--'~I-+-+---l-\-lI" ~~~Y-+-~-+~4-+-~-+~~A -fee~~+-~-+~~~~-+~4-~~~~ 2.16 ( ~f---J,,~~+-~-+-+-4-+-t-t-+-I--+--l 0 Nishiyama 1929) - fee +-~-+-I--+-~I--+-fi!~+-~~-+-+---l-.J-l-+~ ~-I--+-+-~-+~+-~-+-I--~~~ -bec+-r;-+-+-1-+-'~~~~~~-+~+-~-+4 Bt-~+-+-I-+-+~+-+----!-+-+--I--! e Nakamura (1935) - bee -+---/--t--+-+--r----t-.~-+-~...... +_f___+_+_ ...... _+_~_+__1 2.08 Q 4-~t-t-+~+-~I__+-+~~

2.00

1.92

E, 2 1.84 d yn / em .

1.76

Koster (1943),1273 K annealed - fce 1.68 , 973 K _ bee Chevenord, Crussard (1943) - fcc Scheil, Reinaehel' (1944) - fcc I.GO • bee ;fi±±B:IlHH-+-HH--.k-+-+-H2r-+-+-I-hH-'" Fine, EJ Jis (J 950) - fee ~;rt-+~-+-+-1-M-----I-1'"1--Q-+-f-+--I--t- + Bee k, }\ouve lit es, M£ Keehan (l951).f 1.52 ~--blR-+-~..-+-Pt~I!I-I bee fr-~f;.,-t-+-IS--+-+--l~~4-+-t-t-+-I- rI ..,ill, Shimmin, Wilcox (1961)· fcc #-~tIf4-+~+--k'>!iI-+~~+-+-f-+-+- ~ Tino, Maeda (963) - fcc 1.44 -+4-1I1r-+-I-f=....fi~-+-+-1~-"--I-+-t- 0 Go 1d man, Rober t son ( 1964) • fcc • -bee ~-Ht-l-+--:I)-Jj~-+--+-+-I~-,--I-+-t- 6) Doroshek (1964) . fcc 1.36 -+--f!!'J\F\-t::-+-ftI--h+-'lH-+++-H-t- ~ Kh omen ko, Tsey f 1i n (1969) - fcc Tanji, et 01. (970). fcc niesbut'O (1971) • fcc 1.28 Speich, et a!. (972). bee Hauseh, War Ii mont (1972) • fcc 1.20 o 10 20 30 40 50 60 70 80 100 Fe Weight percent Ni Ni

FleuR.... 5. Compositional variation of Young's m"du\u,;. E of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1913 ELASTIC PROPERTIES OF METALS AND ALLOYS 591

Honda (919) - bee "v -fcc t:. Honda, Tanaka (1926) -fcc A

0.8 • -bee (J Me Cain, Maringer (1965) + Roberts,Owen (1967)· bee c Tabulated values - fcc • • bee 0

G, 2 0.70 d yn / em

iii "Meincke, Litva {1969) II Tonji, et 01. (1970)· fcc + Diesbur'g (l971)· fcc ~ Speich, et al. {\972)· bee e Hausch, Wal'limont (972)"fcc

10 20 30 40 50 60 70 80 90 100 Fe Weight percent Ni Ni

FIGURE 6. Compositional variation of shear modulus G of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 592 H. M. LEDBErrER AND R. ,P. REED

\ • Honmo (1919) 1"""--""", I X Ebert, Kussmann (1937) 1.2 I i\ .0 Takahashi, Bassett, I \ ~ \ Mao (1968) - ! 1.1 Ia Meinke, Litva (1969) ~ 1\ I Y Shirakawa, et a I (1969) \ , A Maeda (1971) 1.0 .A~ ~ Diesburg (l971} VA. \ il IVlV' \ • Speich, et al. (1972) i'" \ 0.9 ..~ ¢ Hausch, Warlimont (972) ~ ~ oIJ <\~ i rt~ ~ A Tabulated best values I~ '- \\ fC'r Fe, Ni II'" \ ~ ~- l' - _I 0.8 , , B , V \ / r\..~ :>.;~ .""- ...... cm2/dyn !~ IU ~ ...... I' 0.7 / r- r...... ~ \ -- i""II N-t-.. ,""- I -...... If' ~ NL r\. r- ...... I ...... ~ ~ "- ~ IIr. A. '- l"-I-. 0.6 ...... ~ ~ ,-~ ...... ~ - r\. - ~ ~ WI 1\ .. I'\.. ,..." v l"- ~ IIIII~ - \ t-~ ... H r-.... \ ~ 0.5 , l ~ Calculated from e , fee ~ r--... ij ...... 0.4 \ l~ bee, Calculated from E and G

i 0.3 I

i I o 10 20 30 40 50 60 70 80 90 100 Fe Weight percent Ni Ni

FIGURE 7. Compositional variation of compressibility B-1 of iron-nickel a])oys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 593

0.66 0 Chevenard, Crussard (1943), ~ ..... fcc, annealed 1173 K r :'\ 0.62 J I' • Chevenard, Crussard (1943), I ~ fcc, 44 % cold worked A Honda, Tanaka (1926), I ~ 0.58 I \ fcc, annea led II .. Honda, Tanaka (1926), bcc "' ~....., , Tabulated best values for Fe,Ni 0.54 I ~ * II , 0 Tobulated data, fcc Tabulated data, bcc upper limit for , ~ • 0.50 isotropic solid I 'I. ) " 'l. If ~~ .:...... 0.46 I ~ bee crystal strueture- ~I-a.I- U II V Calculated from cn ,fcc 1--1--1-- () Calculated from E, 'G 0.42 1/ I ~ e Speich, et 01. (1972) ttl ) + Diesburg (1971) V h ...... 0.38 7 V II r-- <> Hausch, War limont (1972) j, .I j ~ V J. II r<-. 1.,...0 ~) 0.34 V "" I~ II' ~z.. !A I";...... ;: ~'Y ~ / ~ /. ~~ r-I-- ~ "'( _/ j ... v J \ ,)A ~ ~~=' ~ ..&. II < --- ~ 0.30 __ v LpJ"" r ~ 1/1 \ -~ ...... ii ./ -- ::oJ" ...... >:" l.V Tt'" -"- "" 'J j / r-1 lt1[1 , / 0.26 -I\. -- .A ./ U\ oJ ....

0.22 - o 10 20 30 40 50 60 70 80 90 100 Weight per-cent Ni

FIGURE 8. Compositional variation of Poisson ratio v of iron-nickel alloys

2.14 X 10'z , _++J_l-4+I+l-~.H++-I-i-'4-+-H-4-++++-I-I--l-t-!-+-l--i-++ Iron LI -r- -l fi! I I I I 1 I I ITT ~r...L IKawal (1930) .. t I 1 ~+ 2 10 • \ b( ~ ~ ;. ! +. !.I .. L :\ -x ~ ~: . i t I j I- iff]... ++++-I-H-+++--I-++-H--I-++-I+-I 2.05 ~y, -~j+ f"..~;·-i 4 I .

_- [·r=t -! ~~ -~i-l ;-T - ~ -1 -+ "f- ..1 ~ 1·' rlN-. j ,_·+t·-; -. - -1--1--1-1-.. + -I--I--l E 1 I-+j-.--+-+--l----If-+.+i +-!:, I i . ,.,... .f t·r i r .-t-."1 , i ~I- ;1 "~ -- -_ ... I 2 2.00 :: i :, . i'±i I ! I " -r I "'f' ; ~ ',! 1 T 1 dyn cm iff: :-;-;- ~. ; Ttl -! !l·t·i "f f';,t. -r'T --t- -t- ._.- - I+j--+ -ttL. I.··-:': Lt -rp- -t·:-tt: J '. -1 .. - -1-1l~I'~i '.' :ltf~·-(· ._.- -4-. H+-i- 1-';' L~_ .t -,-4+ H-t-t-l-,-i.-i .. J . t - I . ITi T ~fi''' . . -t· .+ ++. 1.95 ':.; :~~ ~l~ -~.+.tJ..~:UfK.i:iSl~:'-r~~9~4~T) --.~[r -H--Ff+ .[: +- titI :. : : ;; f-.J~··-i+~ -~':"Ft+-+j .J...... -- - -'++,- ,-j··-tf- f+r~'i-j- -tt·--,- t+- ~ h---Ll- I :._, ir-t-~-"lrT""··· '-,,- Tri-r -rl-t ·t-- T . I I ' 1.90 I : i : I -n-:-+- "1 -: TT T;-- - _. -:-t -t--ITiH--r~i~ I I I I i o 10 20 30 40 50 60 70 80 90 100 COLD WORK , percent

FIGURE 9. Cold-work dependence of Young's modulus E of iron

J. Phys. Chern. Ref. Data, Vol. 2, No.3, 1973 594 H. M. LEDBmER AND R. P. REED

0.90 x10'2 Iron Held I hr. ot 373 K ofter eoch strain, 0,85 -J Kawai (1931) I : j I ; 0.80 ; G, I ! M' Farlane (1931) dyn/cm'l. 0.75 1 : 0.70 I

0.65 , :

0.60 0 10 20 30 40 50 60 70 -80 90 100 COLD WORK, percent

FIGURE 10. Cold work dependence of shear modulus G of iron'

2.20xIO·2 Nickel

~ f' x ~~ ~ '7 ,J 1 2.15 il Held at 373 K I hr. -e- J prior to each measure- E, I( J ll'I~nt I Kawai (1930) -e- dyn/cm2 If J I J 2.10 t\.. """

2.05 0 10 20 30 40 50 COLD WORK , percent

FIGURE lI. Cold work effect on Young's modulus E of nickel

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 595

0.85xld 2 Nickel '- ,-r-- J ~ .... , ,...~ .. i"'oiiii; J IJ ~~ G, 1/ 0.80 ~ Prior to each 2 J dyn/cm 1/ measurement held at 375K-1/2 hi Kawai (1931)

0.75 0 10 20 30 40 50 COLD WORK, percent

FIGURE 12. Cold work effect on shear modulm. G of nickel

I I I ! I __ 0---1 I---: 0 Chevenard, Crussard (1943), 44 -'0 cold worked V ~ /. ./ 2.14 1--6 Engler (1938) , hard V / V~ ~ 0 Tino, Maeda (1963), worked ,//1/ 1.98 ); /' E, no dyn/cm2 0 1.5Z. lt7 ~ /

1.66 1\ 1 \...11

20 40 60 60 100 Weight Percent Ni

FIGURE 13. Cold work effect on Young's modulus E of face-centered cubic iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 596 H. M. LED8EnER AND R. P. REED

Nickel Koster (1943) t-H-H++-H±H-+++-I-H-+

Saturated field Cold worked.

i Cold worked t

'111191111111111\Annealed 973 K t 1.5 S~~~~~~ggffE1i~IIIIIIH'II~'~'~I~O!.0~oF9!5~m~m~gtirHaiHn§S~iZHe~~ , , ~~ Cold worked t \'COld worked, § Annealed 1573 K , Annealed 1173 K t E, §: 0.037 mm grain size tJ:::t:tm:t~~:j$:H:f:tj=++t+:t:1 O. 0105 mm grain size 2 dyn/cm

I ' ! .,

2.5 Zacharias (1933)

99 Ni ,1373 K , H20 quench

99 Ni t 973 K t Slow cool

. , 99 Ni , 1173 K , Slow cool

99 Ni, Single crystal,' ~ 1373 K t H20 quench 99.9 Ni ,Single crystal t Slow cool from 1723 K 99 Ni , 1373 K t Slow cool i 11ft-II!

500 1000 TEMPERATURE,K

FIGURE 14. Annealing effect on Young's modulus E of nickel

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 597

2.1 I---,...... +-~f-,-+-+-+-+-+-+-+--t-.....;-.f---l-I-I-'t1~+:'-~""!--1~tt1~f!,~ ....."t~If.::...... t"""1lIiI1-+--.+-- _+_-t.. --.+-r-+--+r-+----+-+-+--+--l---+--..t.::t~;:t:t::!=t:t:t~':.t.t:L_:.ij J ---~-i ~. ·+I---~·+-+'_+_+_+~r+_+_1__+_+_+__+_~,~~riT~d-+-+-+-r+~-r+-+-+-+-+-tr+-+-+-+-+~--+-~ 20 : , i./"i !""-

11 I ... Dever 1-- . ==f.tf+· J - .. l _L - ... -. ~t_ .if -. ~_·...... ·~jir'.--.t-L·-+-+-·--::.·~.CI-;t,- ... (I97~) 1-- I , +--+-+_+-+-+C II - Lord, Beshers (1965) r-I- -·,..-t- l,Ji't ,- Cij 1.9 I ~: ~n/~m218~~~:----~~~+:~---+--+-~--+-~-+--+-+H+-.~+t-t·--+-~-+-~-+-~-+-~~-++~~~-~t-+~'-t~~~~~t.t~~tt~~':.tt~=~t~~~

. ~ ~~·r.f- '~:. . ~J~--- -F'~ r-'--r:-:- -._f- .- -1=- =~,..:. -:-+=t~++-""'4,,<~,-+'-+-t-+-+-+-+-+-'+-+-I--I

I. 7 ~.'.___ ._ ----~'- - ~. + +-- ,~=-~+.-.I- f----... .. - - ---. '-++-->-' I I '" -;:-++-' ---'-' ff ' . -- --t -I--i-'- --t-- _. --+-T--+- +-f~+-++-+-+-+-+-+-IW -'-~IT----- . . - -;-, .+ __ + __ ..:... :1:_:.- -_- -_"t.-... ' . =: _--:-Y-+---I--+-+---1f-f-' IW-t-+--+-+r-r-I--+-+--I . -1--' I --;--r-- - ~ -- ~+-~-t--- j ,.

1.1 .__ f-- C44-Lord, Be~~~r.~_(~~9~~L __ i"- -r"'",,- ~~ !-- J: 1.0 --- r- --t-- .. -- --r-- ..... --1-1--'- ... 1-.. I

---t--t-t-+-t-t--t-t--+---+-t--i-+-+-+-+-+-+-+-1I-+-+-+--f--1-C44 - Dever (1972) t-l-+-+-+-i

0.9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ o 100 200 300 400 500 600 700 800 900 1000 1100 1200 TEMPERATURE, K

FIGURE 15. Temperature variation of elastic stiffnesses Cij of iron

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 598 H. M. LEDBETIER AND R. P. REED

t- . I Nickel r- 2,6 p:!:!;~

~ 2.5 r"( , lr c -Alers, ,Neighbours, 50to (l960) - .'-"'t il 2.4 ""(

2.3 I r"'t

, 2.2

2.1 I

I-- 2.0

, 1.9 C..IJ , ; 2 dyn I cm 1.8

; 1.7 H-- r-r- -- I -

1.6 l!' cI2-Aler~, Neighbours, $qto (19SQ)':

1.5

; 1.4

1.3 )0( -:-Alers, Neighbours, Soto (960) C44 1.2

.lot - .- r-. ,- . - .-f-I--- 1.1 1 i J,.(

I 1.0 o 100 200 300 400 500 600 700 800 900 1000 TEMPERATURE, K

FIGURE 16. Temperature variation of elastic stiffnesses Cij of nickel

J, Phys. Chem. Ref. Data. Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 599

I I I I I II 1 II I I I I

" 28 wi % Ni ~ (;] Alers, et al.( 1960) ~ ~ ~ ~ Saloma, Alers ( 1968) ... , 0 Diesburg (1971) ~ " p "'" :s;; ~ L;" ... ~ 1.50 p ~ ,..~~ 30

.I~ IL I,,![(- ~. I<"'l.~ .C' 1.40 ~ .~

r-+- CII J;. 34 1-1- .r.kli'J .~ 1.30

Cij , 2 1-- dyn /cm 1-- C44 1.20

IJ:'\J."~ 28

A- I- ~ 30\~ ~ LlO .... • "IE.. 34

.,L:.'" 1.00 I-r- C --I-- I2 ., w. 28 J:. IJ.: I ~ IJ;. "'" J:.. ~ ~ 30 ,J,:. 0.90 IJ.:, :..".;or .,=,~ ~ ...... 30 '='~ r. 1Ci:"'" . .,

.~ "" 34 .- ..~~ I I..:-l(.., >/ I 0.80 200 250 300 350 400 TEMPERATURE, K

FIGURE 17a. Temperature variation of elastic stiffnesses Cij of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 600 H. M. LEDBmER AND R. P. REED

12 2.20 x 10 ! 511.4 ~t % 'Ni CII r-- Hausch, Warlimont(973) 2.10 - -...... r--...... '-...... 2.00 I"-.... 45.2 ...... r-...... ~ 1.90 " I ..... ~ ...... ------.- i"...... 42.5 ...... t-- 1.80 " ~ ...... ~~ V 40.0 ...... " ~/, ~/ 1.70 ~...." f..-/ ~ ...... ~ ...... 1"- ---- // 2 V/ '/ ~ dyn/cm '-...... --/ V/ / 7/ 1.60 .,." 37.7 - ~r--... IV../- Y -r---- / ~ 1.50 -- V -- I J V 1.40 31.5 VA ~ I I ...... ::::: ..."",- V 35.3 33 8 ...... V 1.30 i / 3~.2" 1.20 1 o 100 200 300 400 500 600 700 800 TEMPERATURE, K

FIGURE 17b. Temperature variation of elastic stiffness ell of iron-nickel alloys

1.60 x 1012 CIJ 51.4~ %Ni Hausch, Wartimont (1973) 1.50 I r-- r-41·2 ...... r---...... r-- ...... 42.5 ...... 1.40 i"...... --- _I ...... "'- 40.0 .... r--.... "- .....r-...... r--.." r"...... '~ ...... r...... 1.30 ...... '" ~ '-...... r--- .L...... ?- 37.7 '-" r--...... ~~ --~ 1.20 "", ...... ~ I?"'"/ ...... ""'" ~ ~ '/L" m ~ ~"'" ...... , I~ ~ Y V 1.10 ...... -...... ;. V ~ ~ dyn/cm2 1.00 35.3 ----Tt W ...... 'V~ ...... 1: r-.... 0.90 I--~ ,,-PI 33.8 ~ 33.2- 31.5 0.80 ---

0.70

0.60 a 100 200 300 400 500 600 700 800 TEMPERATURE, K

FIGURE 17 c. Temperature variation of elastic stiffness c 12 of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 601

12 1.16 x 10 I --c-- d44 i--1_-+_1-t-- Hausch, Wal"limont (1973) 1.14

-k------1------.--~ -- 31.5 wt % Ni w--- i f- 1.12 --.---1--- I 51.4 " I I i -~ I ! 1.10 1 '"" ! \" ~ \ ~ '" ! I "",",I '\ i 1 r 1.08 ~ '" "- ! I 33.2/ /1 r".... ~ dyn/cm2 i "'! I 1.06 /"' K - -..... 45.2 ..... "' "'~ 1 ...... /' 1"- ~~~ /' - 33.8> "'i'.. 1.04 ...... ~ ~ ~ 4Q.S- ~ """"--r--., L:. r--. i ~1'. V '" " !.-- 40.0 L ,...-- ::::.- r---.:: :::::::~ t--~ 1.02 --42.0 35.3 ,. ~i- -..~ ~ .~ 37.7- 1.00 --- ~ ~ I "" ,,: -- i-" 0.9S ~ !

I I I i 0.96 I o 100 200 300 400 500 600 700 800 TEMPERATURE, K

FIGURE 17d. Temperature variation of elastic stiffness C44 of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 H. M>LEDBmER ANI)R~P.·· REEl)

2.4xld2 r---..,.-----,---~--'""T"""'"--..,..__--~-___r--_.., Iron Masumoto, et 01. (1971). I . 2.2 ~---'or-t----lf----+ KimuraI (1939) ---if----+----+----+-----t Roberts, Nortcliffe (1947) Garafalo, et 01. (1952)

2.0 1----.p.,.,..----:l~~1lt!iiiI ....----11--- H=0.575 kOe,Engler (1938)--04 I I I IDurham,et al. [20-300 K](1963), I---_-+---....-----+-~~~~'"""+Verse @00-775 KJ (1935) 1.8 I I I Kamber (1963) E, grain size = 30 mm 2 dyn/cm 1.6 Seager, Thompson (l943),cold rolled I t----+-- Burnett (1956) .--~--\\-ftf-----:-~-~~:-r-----1 1.4 I 1 Koster [76-1275 KJ (1948) 1.2 Engler [293-730 K] (l938) Schei I, Reinacher [293 -875 K] (1944) Hill, et 01. [293-920 K] (1961) I LO 1;:::====1==-~bcc~c~r:nJstrgaLI~strrry!uc~t:yuW1~e~====:t=~~~cc~crmal stuctu o 200 400 600 800 1000 1200 1400 1600 TEMPERATURE, K

FIGURE 18. Temperature variation of Young's modulus E ofiron

~ Durham ,et at. (1,963) Pavlov, Nickel KriutchkoY, r Saturated field, Koster (1948) Fedotov'l957) Burnett (1956)

2.0 Armstrong, Brown (1964) = E, 1-+-+- 2 ~~ No field, Orlov, Fedotov (1966) dyn I em ...... - Koster (1948) Cold worked, lit

Jacquerod, oJ Masumoto ,et 01. (1968) 1.5 Mugeli (1931) ~~~'--Kamber [a] (1963) Annealed ,= Jacquerod, Kamber [b] (l963) Mugeli (1931)

1.0 a 500 1000 1500 TEMPERATURE. K

FIGURE 19. Temperature variation of Young's modulus E of nickel

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 603

! I ! i II II II , I I Fe-90 Ni ° ° 2.0 x 10'2 , I Fe-80Ni 1 i"'" Fe -70 Nt .I Koster I Fe-SO Ni) Fe-42 Ni, (l94~) IS Doring (1938) Fe-50 Ni o( . ~ Fe-40 Ni 1.5 Jo.. ~ ~~ I " Fe-30 Ni ~

I I t

2.5 r , !-Fe-78 Ni, I i 1 ~ Engler (1938) Ir- Fe- 75 Ni, Egonyon, 1 III II Selisskiy (1967) I i I I ~ • • I I.... I 'I ! Fe-93 Ni, 1/ Fe-49 Ni, Chevenord I ,\ Fe-30Ni (a+y) E, h¥}gler (1938) V Crussord (1943) I I Tino, Maeda (963)7 2 , I --:r II dyn/cm 2.0 III I ,....1111 I II I I- Fe-4~ Nii I J l:jo...I. l""+- -. I- ' \ Fe-29 Ni , Scheil , Reinacher (194 4) I- Engler (1970) -:r\ ~ "... '1'\ -~e-30Ni 17 -T TTilTl Fe-40 Ni r tot 111 III J" Fe-45Ni 1 I Je-3tNi,1 . I Durham et 01. 1.5 !T(963) .. ... Ii Fe-36 Ni ~ I - Fe -52 NI Fine, Ellis Fe- 36 Ni, Hill, I Fe-40 Ni I (1950) Shimmin I Wilcox (1961)

Fe - 42 N i ° I I I I I I I I I IT TIl I lIlT o'llllllhllll II IT TT II III II I I I II I II I II I I 1 500 1000 1500 TEMPERATURE,K

FIGURE 20. Temperature variation of Young's modulus E of iron-nickel alloys-face-centered cubic

- Fe-9Ni, Fontono(1948)

Fe-3 Ni, Seager I Thompson (1943) 2.0

Et 2 Fe-3Ni, dyn / em Corrington (1924)

Fe-3Ni, Honegger (1932)

1.5 '§ Fe-:30NI~-'-Tino.

~ Maeda (1963) ~o+)' Fe-22 Ni t Scheil,Thiele (\938)

500 1000 TEMPERATURE,K

FIGURE 21. Temperature variation of Young's modulus E of iron-nickel alloys-body-centered cubic

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 604 H.··· M; .1EDBEnER A.NDR,.·· P. 'REED

1.5 I I I I J I I I II .1 I II I I I I I I Fe-75 Ni (fcc), Fe-48 NI (fcc), Kotatayev, Bennett, Koneva (1968) Davies (1949) I \ I I ~E 1.0 273K r?'

Fe-4 Ni (bec), II Kimball, Lovell (1925) I I I I I I I I Ii I 500 1000

TEMPERATURE, K

FIGUHf,; 22. Relative lell1l'el'alun~ YiuiatiuJl uf Youllg'e modulue E of &ome iron-nickel alloys

\ Iron \ 1.1 I Guye, Freedericksz (1909)

1.0 I I

I 0.9

i 0.8 I , , Burnett (l956); Iokibe, Sakai (1921)

0.7 I G , . , Kikuta (1921) 2 dyn / cm 0.6 I

0.5 Everett, Miklowitz (1944) ~

, , 0.4 I I 0.3 Verse [296-760 K ] (1930) ; 'Koch,Donneeker [295-1450K] (1915); .J. Garofolo, et 01. [295- 650 ] (1952) ; 0.2 \ Huohes • Maurette [295-575 K] (1956): Durhom ,et 01. [20 - 295 K ] (1963) 0.1 I bee crystal structure i fec crystal structure o -:- o 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

TEM PERATURE t K

FIGURE 23. Temperature variation of shear modulus G of iron

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND AllOYS 605

Nickel ~ Burnett (1956) Susse (1956) G, dyn / em 2 0.5 Iokibe, Sakai (1921)

o o 500 1000 1500

TEMPERATURE t K

I-+-I---I:::~_. - Guye, Schapper (1910) Nickel Orlov. Fedotov (l9SS)

G, dyn / em 2 0.5

Koch, Dannecker (1915) F- Kikuta (1921)

o o 500 1000 1500 TEMPERATURE, K

FIGURE 24. Temperature variation of shear modulus G of nickel

0.9 Fe-25 Ni (fcc) Fe-25 Ni (bee) Goldman, Robertson (1964) 0.8 Fe- 30 Ni (fcc) Fe- 30 Ni (bee) 0.7 G, Fe-35Ni, Meincke, 2 Fe- 32 Ni, Burnett (1956) dyn / cm 0.6 Litvo (1969) 7 . 11111 0.5 I Fe-36 Ni t 0.4 Durhom , et 01. -' (1963) 0.3

100 200 300 400 500 600 700 800 TEMPERATURE , K

FIGURE 25. Temperature variation of shear modulus G of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 606 .. H.M. lED,BEnER AND R~ P.' REED

Fe-32 Ni (fcc)anneoled ! - Fe-29Ni (fcc)annealed Chevenard Fe-26 NHfcc)annealed (1927) I"',,-~ONI (bee)

1.0

r Fe-17 Nit Bungardt t et 01. (1'362)

Fe - 36 Ni (fcc) annealed { Chevenard 0.5 Fe-36 Ni (fcc) cold worked J. (1920)

o

1.5 Fe-36Ni (fcc) annealed Fe-42 Ni (fcc) annealed ~ Fe-47 Ni(fcc)annealed l Chevenord r- Fe-50 Ni (fcc)annealed Fe-54Ni (fcc) annealed (1927) Fe-64Ni(fcc)annealed . ,-Fe-85Ni(fcc}annealed .

1.0

500 1000 1500 TEMPERATURE, K

FIGURE 26. Relative temperature variation of shear modulus G of some iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 607

0.70xlC512 0.69 !J 0.68 0.67 ,1'I I 1/ 0.66 I) 0.65 -I B l 0.64 ~'\ 2 cm /dyn 0.63 1\ "'" r-- GrUneisen (1910) ~ 0.62 I""" i---' Bridgman (l923) 0.61 l\ I 0.60 ~ ~7' 0.59 -\ Bridgman (1940) V I 0.58 ~ Aliev eta!. t z Hughes t +- i---' 0.57 (1967) I- Maurette (1956) O.SG I I I I I I I I II " I 0.55 """'/1 o 100 200 300 400 500

TEMPERATURE, K

FIGURE 27. Temperature variation of compressibility B-1 of iron

I I NICKEL 0.59 ------t-- - r---

/ 0.58 V / 0.57 V

B-! 0.56 ~ cm2 /dyn V 0.55 V 0.54 ./ Alers. Neighbours, _ ~. Sato (1970).. ~ from elj data. 0.53

0.52

o 200 400 600 800 TEMPERATURE, K FIGURE 28. Temperature variation of compressibility B-1 of nickel

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 80B H"., M. ··LEDBEnER AND. I. '. P.REED

0.9

0.8 ..... Fe-35 NI , Meincke, Litva (1969) B-I , cm2/dyn 0.7

0.6 ,

100 200 300 TEMPERATURE, K

FIGURE 29. Temperature variation of compressibility B-1 of iron­ nickel alloys

0.6 .J LIron Theoretical Annealed (Fe-O.19 Cl Cold rolled. .~ Upper Limit Everett,Miklowitz (1944)- 0.5 "'" Carrington (1924) 1 I I I Hot ro lied, I I I L Everett, 'Miklowitz (1944) I ....

0.4 i r.... : Annealed, . ~ 1/ Carrington (1924) • , . i 0.3

0.2 I Burnett (1956) Aooealed, Garofalo,Molenock, Smith (1952) Annealed Verse (1935) 0.1 ,

o o 100 200 300 400 500 600 700 800 900 1000 TEMPERATURE,K

FIGURE 30. Temperature variation of Poisson ratio II of iron

0.5 I NICKEL

0.4

Frederick (1947) - I - 0.3 -

.i1veroge tObUlotJ value

0.2

0.1

0.0 o 200 400 600 800 1000 TEMPERATURE, K

FIGURE 31. Temperature variation oCPoisson ratio II of nickel

t Phys.Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 609

0.58

0.40 .... II II ~ J,...o 0.:>6 Chevenard, Crussord (1942) 0.38 also Chevenord (1943) I-r:ii Fe·6Ni (bee) annealed t1 0.54

0.:36 . I 1 I I " fhevenard,Crussard (1942) J 0.52 Fe-6 Ni (bee) eold worked~ 0.34 1 I111I1111 1111 III I111 1 7 J !;} iT 0.50 Chevenard, Crussord (1942) ~ rf 0.32 olso Chevenard (1943) " Fe- 36 Ni (fcc) ...... 1,..000 0.48 ;t' l"- 0.30 i;' ...... 71 V ". N 0,46 ~ ~ 0.28 I' ill:::: t:;;::'" J , ~ ~ I'- Carrington (1924) 0.4'1 1/ """" 0.26 I' 1 Fe·3Ni(bcc) Goldman, Robertson (1964) I' II ro- ..... III Fe·30 Ni (fcc) 11't- 0.42 ~oldman, Rober'tson (1964) n24 i~}?~',(~Cr), 1 1 III III II. _, 1.1 I I 1 I I I I I 1111 0.40 r- Goldman, Robertson (Ir~~) 0.22 III F1!·25Ni(bcc) +-} . TTT 17 1'\ 0.~8 Chevenard, Crussard (1943) 0.20 ~ I r~' (1 ~i l(f~~\4~O(oF~ldl ~o~k~d,- I'~ .... I I 1 II 1 0.36 0.18 1 o 100 200 300 400 500 600 700 800 900 1000 1100 1200 TEMPERATURE, K

FICURE 32. Temperature variation of Poisson ratio J' of iron-niokel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 610. H.M. LEDB~TT~K ANI.) K. P~K~I:V

Nickel '=~ ~ Saturated 0.4 Saturated I H=0.320 kOe t Tino, Maeda 'rH=O ~ (1963) I=I=~ 0.6 Saturated Siegel t Quimby (l93SA:$: 2.0 I=S$ H =0.575 kOe, Ef=!=!- Engler (1938) No. field E, No field, 2 Engler (1938) dyn / em

~H =O.l kOe 1.5 Kimura (1939)

1'- H= 0.005 kOe Kimura (1939)

No field t J( imuro (1939)

.-

500 1000

TEMPERATURE, K F.IGURE 33. Magnetic field and temperature effects on Young's modulus E of nickel

I j 2.8

I 2.6 Engler (1938) Fe- 78~~ ~;'I0.4~O kOe 2.4 -l-f- .' : Egonyo .. , Selisskiy (1967) Fe- 75 Ni, 1.000 kOe ; \ 2.2 ~~ I oOI;~ IrFe-42Ni,OtO.803kOe ITino, Moedo l- jo.., L", i 1"==-'""'" r.:,;. Fe-36Ni,O,O.720 kOe (1963) l- I I -.E'" J.? 2.0 ,.;:a: ~ I- Engler' (1938) E t Goldman, Rober'tson (1964) v E=:: 2 1-1-1- dyn/cm :=:= Fe-30Ni,1.600Q.kOe l' Fe i ~3, N,i, 0.575 kOe 1-1- .... r-.... V 1.8 '\ ""'" r-.. I I 1 I I I r--"" I I II I .... I..- ~ Engler (1938) I I 1.6 r"'I I- L~ ;"';""r.,.. I Fe-50 Nit 0.464 kOe I- l,..olo- .... 17.l,..o I I i ·11 ( \.1- DOr'ing (1938) I I I I I """ Fe-42Ni,0.575kOe 1 1.4 Engler (I938) """ 1.1 Fe-42 Ni,O.575 kOe_ 1..-1-- i:::::~ r-~ I I I I I 1 """- ~ f f I I ',...... ~ -~~ '" Moeda (J971) 1 I I I 1.2 I- ~ J,...~ r'- Fe-35 Ni, 0,0.190,1.90 kOe 1-1-1- ~ I I I I I I I I I I I 1.0 o 100 200 300 400 500 600 700 800 900 1000 1100 1200 TEMPERATURE, K

FIGURE 34. Magnetic field and temperature effects on Young's modulus E of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND AllOYS 611

0.54 It 10'2 .------,,.------y-----r----.---,---.,---r------,

O.oo~-~----+---_+-~~--~A-~I--+---P~~ TC G, dyn Icm2 0.46 1.9 Maeda (1971) Fe - 35Ni. field 0.42 0.19 ~-~---_+_+--+--- in kOe --+----4

o O·;J80~----IOOL:----:2.00~---::300~--400~----:~---:~---=700~-~eoo TEMPERATUREtK

FIGURE 35. Magnetic field and temperature effects on shear modulus G of an iron­ nickel alloy

r--- Fe-35 Ni I--t'.

1.3 ~

r\'\ V H =0.019 kOe I I I B \ t " ./H= 1.90 kOe 2 1.2 ~"\ dyn / em "\ ~ Maeda (1971) 1.1 l\: ~ I" ~ II ~ ~ V 100 200 300 400 500 600

TEMPERATURE t K

Fll>UHt; 36. Magnetic field and temperature effects on bulk modulus B of an iron­ nickel alloy

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 II. M. -LEDBEnER -AND R. RRE;ED,

IrNi

Fe-84.9Ni 1 I-- 1 /" ~ Fr~ -;/ 2.00 7 7 --<.!y/ ~ -- 7 7 lV II E, dyn/cm2 1.90 /' Beck, Kouvelites, MCKeehan (l30\) 1.48 /' / / // 1.40 Fe-46.2Ni /' / -~

1.32 o 4 6 8 10 12 14 MAGNETIC FIELD, kOe

FIGURE 37. Magnetic field effect oil Young's modulus E of nickel and iron­ nickel alloys

--

1=+:::j:::I:::j:::j~f- Fe-25Ni,Goldman, Robertson (1964) . += _. .~' 1 Fe-22N~i . 2.0 Scheil , Reinacher F - 15 N' _~ :1== .-- (1944) e I tit;. ~. Fe-IONi.., fcc E, be dyn I cm 2

bee bee fcc bc~ cc

Fe -22 N i ,Fun.ter, KUliler (1537) Fe-30 Ni, Goldman, Robertson (964)-,

--;- -, -t- 500 1000 TEMPERATURE ,K

FIGURE 38. Phase transformation (bcc to fcc on heating) effect on Young's modulus E of iron-nickel alloys

J. Phys. Chem. Ref. Data, Vol. 2, No.3, 1973 ELASTIC PROPERTIES OF METALS AND ALLOYS 613

25. References for Tables and Figures [1] Alers, G. A., Neighbours, J. R., and Sato, H., Temperature [25] Carrington, H., The strength properties of wrought iron, mild dependent magnetic contributions to the high field elastic steel and nickel steel at high temperatures, Engineering constants of nickel and an Fe-Ni alloy, J. Phys. Chern. Solids (London) 117,69 (1924). 13,40 (1960). £261 Chevenard, P., Changement thermique des proprietes elastiques [2] Aliev, F. Y., Lazarev, B. G., and Sudovtsov, A. I., Compressi­ des aciers au nickel, C. R. Acad. Sci. 170,1499 (1920). bility of iron and nickel at 4.2 "f{, Ukr. Phys. J. 12, 1189 [27] Chev~nard, P., Recherches experimentales sur les alliages de (1967). Reported also in Phys. Abstr. 71,400 (1968). fer, de nickel et de chrome, Trav. Mem. Bur. Int. Poids Mes. [3] Andrews, C. W., Effect of temperature on the modulus of 27,142 (1927). elasticity, Metal Progr. 58,85 (1950). 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