Appendix A: Crystal Symmetries and Elastic Constants. A.1 Crystal Classes, Point Groups, and Laue Groups

Transcription

1 Appendix A: Crystal Symmetries and Elastic Constants In anisotropic materials, constitutive relations and corresponding material coefficients depend on the orientation of the body with respect to the reference coordinate system. This Appendix discusses symmetry operations for point groups and Laue groups comprising the seven crystal systems first introduced in Chapter. equisite terminology and definitions are given in Section A.. Listed in Section A. are matrix forms of generic polar tensors (i.e., material coefficients) of orders one, two, three, and four for each Laue group, and forms for individual crystal classes comprising each Laue group for polar tensors of odd rank. Second- and third-order elastic constants are given particular attention in Section A.. The content generally applies to materials whose elastic behavior is independent of deformation gradients of orders higher than one, more specifically hyperelastic materials of grade one. The discussion and tabular results of Appendix A summarize a number of previous descriptions (Hearmon 946, 956; Bond et al. 949; Schmid and Boas 950; Landau and Lifshitz 959; Brugger 964, 965; Truesdell and Noll 965; Thurston 974; Teodosiu 98). A. Crystal Classes, Point Groups, and Laue Groups Under an orthogonal transformation of Cartesian reference coordinates X QX, deformation gradient F= x/ X and right Cauchy-Green T strain E= ( F F )/ transform as F FQ T and E Q EQ, respectively. Consider the free energy density in the first of (5.6) for an elastic crystal at fixed temperature, i.e., the strain energy density. The specific form of the elastic mechanical response function, in this case the strain energy density, respects the symmetry, or lack thereof, of the material: T ψ QEQ = ψ E. (A.) ( ) ( ) The set of all orthogonal tensors Q that leave the mechanical response function unaffected, i.e., those operations for which (A.) is satisfied, is J.D. Clayton, Nonlinear Mechanics of Crystals, Solid Mechanics and Its Applications 77, DOI 0.007/ , Springer Science+Business Media B.V. 0 54

2 544 Appendix A: Crystal Symmetries and Elastic Constants called the symmetry group or isotropy group for the material. For two transformations Q, Q, their product QQ. Furthermore, Q T implies Q = Q. Every isotropy group contains the identity map Q= and the inversion Q=. Subgroups consisting of all proper orthogonal tensors ( det Q = + ) of the isotropy groups comprise the rotation groups +. Any member of particular group + can be created by a product of rotation matrices called generators of that group, labeled. + Any entry of for that group can then be obtained from entries of via use of the inversion. The group of all rotations, reflections, and translations that leave a structure invariant is called the space group. A space group can include combinations of translations, rotations, and reflections (i.e., screw and glide operations) individually not contained in that space group (Chaikin and Lubensky 995). The set of all rotations and reflections that preserve a structure at a point is called the point group. Crystals sharing the same point group are said to belong to the same crystal class. In three dimensions, the number of distinct space groups is 0, and the number of distinct point groups or crystal classes is thirty-two. Every crystal class falls into one of eleven Laue groups. Each Laue + group corresponds to a different set of rotations to which elastic mechanical response functions in the context of (A.) are invariant. Table A. lists Laue groups, point groups, and generators for all crystal classes, n following Thurston (974). The notation N denotes a right-handed rotation by angle π /n about an axis in the direction of unit vector N. Boldface indices i, j, and k denote unit vectors of a right-handed orthonormal / basis for X, and m= ( i+ j + k ). Also denoted by asterisk in Table A. are the eleven crystal point groups one per Laue group with an inversion center or center of symmetry (i.e., those that are centrosymmetric). The seven crystal systems of Fig.. whose symmetry elements are listed in Table A. are recovered from the eleven Laue groups, since two Laue groups comprise each of the cubic, tetragonal, rhombohedral, and hexagonal crystal systems. The alpha-numeric notation used to label each point group gives details about symmetries of crystals in that group, and a number of different notational schemes exist in crystallography for labeling point (and space) groups. The reader is referred to Thurston (974), ohrer (00), and references therein for a more detailed explanation. Modern Hermann-Mauguin international symbols are used in column five of Table A., following ohrer (00). Columns six and seven of Table A. follow the scheme of Thurston (974), who in turn cites Groth (905). n N

3 A. Crystal Classes, Point Groups, and Laue Groups 545 Table A. Laue groups, generators, point groups, and crystal classes Crystal Laue # Generators Point Crystal class # system group group Triclinic N triclinic asymmetric triclinic pinacoidal* Monoclinic M monoclinic sphenoidal k m monoclinic domatic 4 /m monoclinic prismatic* 5 Orthorhombic O i, orthorhombic disphenoidal 6 j mm orthorhombic pyramidal 7 mmm orthorhombic dipyramidal* 8 Tetragonal TII tetragonal pyramidal 0 k 4 tetragonal disphenoidal 9 4/m tetragonal dipyramidal* Tetragonal TI 5 4 k, 4 tetragonal trapezohedral i 4mm ditetragonal pyramidal 4 4m tetragonal scalenohedral 4/mmm ditetragonal dipyramidal* 5 hombohedral II 6 trigonal pyramidal 6 k trigonal rhombohedral* 7 hombohedral I 7 k, trigonal trapezohedral 8 i m ditrigonal pyramidal 0 m ditrigonal scalenohedral* Hexagonal HII hexagonal pyramidal k 6 trigonal dipyramidal 9 6/m hexagonal dipyramidal* 5 Hexagonal HI 9 6 k, 6 hexagonal trapezohedral 4 i 6mm dihexagonal pyramidal 6 6m ditrigonal dipyramidal 6/mmm dihexagonal dipyramidal* 7 Cubic CII 0 i, j, m cubic tetartoidal 8 m cubic diploidal* 0 Cubic CI i, j, k 4 cubic gyroidal 9 4m cubic hextetrahedral mm cubic hexoctahedral* Transversely isotropic** ϕ k, 0< ϕ < π Isotropic** all proper rotations *Denotes a point group with an inversion center **Not a natural crystal system

4 546 Appendix A: Crystal Symmetries and Elastic Constants Particular aspects of each Laue group are discussed next. The reader is referred to Bond et al. (949), Landau and Lifshitz (959), and Teodosiu (98) for additional details. Formal standards on terminology and assignment of coordinate systems for crystalline solids are given by Bond et al. (949). The triclinic crystal system contains Laue group N or. This system has the lowest material symmetry; a triclinic crystal has no symmetry axes n or planes. Though the coordinate system corresponding to N in Table A. may be chosen arbitrarily (Landau and Lifshitz 959), standard conventions do exist for selecting i, j, and k for a given specimen (Bond et al. 949). Monoclinic crystals belong to Laue group M or in the numbering scheme of Table A.. Crystals of this system possess at any point, an axis of symmetry of second order, a single plane of reflection symmetry, or n both. The coordinate system corresponding to N is chosen such that k is aligned along the axis of symmetry or is perpendicular to the plane of symmetry. The orthorhombic system includes Laue group O or. It describes crystals with three mutually perpendicular twofold axes, two mutually perpendicular planes of reflection symmetry, or both. The coordinate system corresponding to n N is chosen such that i and j are aligned along the twofold axes or perpendicular to the planes of symmetry. If in this case k is also a plane of symmetry, then the orthorhombic crystal is also orthotropic. The tetragonal system includes Laue groups TII (4) and TI (5). Crystals of group TII have a single axis of fourfold symmetry. The k axis is chosen parallel to this axis. Crystals of the tetragonal subsystem TI possess, in addition to this fourfold axis, an axis of twofold symmetry along vector i. The rhombohedral system includes Laue groups II (6) and I (7). Crystals of these groups have an axis of threefold symmetry, conventionally called the c-axis when the Bravais-Miller system is used. For this reason, sometimes the rhombohedral system is referred to as the trigonal system. Furthermore, sometimes groups 6 and 7 are labeled as belonging to the hexagonal system, even though crystals of these groups do not possess an axis of sixfold symmetry. The unit vector k is taken parallel to the axis of symmetry, that is, parallel to the c-axis. Crystals of group I also possess an axis of twofold symmetry parallel to unit vector i. The hexagonal system contains Laue groups HII (8) and HI (9). These crystals have an axis of sixfold symmetry, conventionally labeled the c- axis. According to the notation in Table A., unit vector k is aligned parallel to the axis of sixfold symmetry. Crystals of group HI also possess an axis of twofold symmetry parallel to unit vector i.

5 A. Generic Material Coefficients 547 The cubic crystal system includes Laue groups CII (0) and CI (). Cubic crystals have three twofold axes of symmetry. The coordinate system is chosen such that i, j, and k are each parallel to one of these cube axes. Groups CII and CI are distinguished by their generators as is clear from Table A.. Two other forms of symmetry not corresponding to any natural crystal + class arise often: transverse isotropy and isotropy. The rotation group for transverse isotropy, i.e., group in Table A., includes the unit tensor and all rotations n k where k is normal to the plane of isotropy and + < n <. When consists of all proper orthogonal tensors, the material is isotropic; otherwise, it is anisotropic (i.e., aleotropic). The rotation group for isotropy is labeled. Isotropic materials are always centrosymmetric. Table A. Symmetry elements of crystal systems, after Thurston (974) Crystal system Triclinic Monoclinic Orthorhombic Tetragonal hombohedral Hexagonal Cubic Essential symmetry None One -fold axis or one plane Three orthogonal -fold axes or planes intersecting via a -fold axis Either a 4-fold rotation axis or a 4-fold rotation-inversion axis One -fold axis but no 6-fold rotation or 6-fold rotation-inversion axis Either a 6-fold rotation axis or a 6-fold rotation-inversion axis Four -fold rotation axes which also implies three -fold axes A. Generic Material Coefficients Material coefficients are listed in matrix form in Tables A.-A.7 for each of the thirty-two point groups, following Thurston (974), who in turn refers to Mason (966). The point group numbers used in Tables A. and A.6 correspond to those in the rightmost column of Table A.. The Laue group numbers used in Tables A.4, A.5, and A.7 correspond to those in the third column of Table A.. Tensors of rank (i.e., order) one are given in Table A., of rank two in Tables A.4 and A.5, of rank three in Table A.6, and of rank four in Table A.7. These are all so-called polar tensors of Thurston (974), defined as derivatives of thermodynamic potentials with respect to state variables measured in some thermoelastically undistorted state. This undistorted state could be the stress-free reference configuration of thermoelasticity (Chapter 5 and Chapter 0), or the stress-free intermediate configuration(s) of finite elastoplasticity (Chapter 6-Chapter 9). State variables in these contexts of geometrically nonlinear mechanics

6 548 Appendix A: Crystal Symmetries and Elastic Constants must have components referred to the coordinate system corresponding to the undistorted state, and not referred to the deformed spatial configuration, for example. Obviously, the state variables and resulting material coefficients must also be referred to a single consistent configuration; e.g., two-point tensors of material coefficients may not always exhibit the symmetries evident in Tables A.-A.7. Material coefficients as defined in the present context are constants at an undistorted reference state wherein vector- and tensor-valued independent thermodynamic state variables such as electric polarization and strain vanish, and wherein the material exhibits the full symmetry of its original crystal structure. Tangent material coefficients do not necessarily obey the definition of a polar tensor used here in Appendix A and may have different symmetries than those listed in the forthcoming tables. For example, the matrix of isotropic tangent elastic moduli in (5.9) depends on up to eight independent scalar functions and does not generally exhibit the same symmetries as the matrix formed from two independent second-order elastic constants in (5.) and (5.). Examples of polar tensors of rank one include pyroelectric coefficients of (0.97), (0.06), and (0.). Examples of polar tensors of rank two include thermal stress coefficients of (5.68), (5.85), and (5.0); secondorder thermal expansion coefficients of (5.0); Gruneisen s tensor entering (5.); dielectric susceptibilities of (0.89) and (0.05); and the dielectric permittivity tensor of (0.08). Examples of polar tensors of rank three include piezoelectric coefficients of (0.9), (0.05), and (0.09). Polar tensors of rank four include the second-order elastic stiffness tensor of (5.65), (5.85), and (5.90), as well as its inverse, i.e., the tensor of second-order compliance constants measured in an undistorted reference state. Coefficients of polar tensors of odd rank vanish identically for all eleven classes of centrosymmetric crystals. Coefficients of polar tensors of rank three also vanish for non-centrosymmetric crystal class 4 (point group 9) because of other symmetry properties of this cubic structure (Thurston 974). Tensors of rank one, e.g., pyroelectric coefficients, vanish in all but ten of the twenty-one crystal classes lacking an inversion center. Indices of coefficients listed in Tables A.-A.7 are referred to rectangular Cartesian axes X, X, X. Care must be taken to properly account for the relationship between these axes and the crystallographic axes, especially when interpreting experimental data for elastic constants of materials of less than cubic symmetry. Standards exist for various crystal classes (Bond et al. 949), though in some cases ambiguities arise leading to different sign conventions for the coefficients (Winey et al. 00). Methods for identifying material symmetry of a substance given the values of second-order elastic constants referred to a known but otherwise arbitrary

7 A. Generic Material Coefficients 549 with respect to material structure coordinate system have been developed (Cowin and Mehrabadi 987). Tensors of rank two are written first in Table A.4 without presuming symmetry a priori; symmetric forms are then indicated in Table A.5, corresponding for example to thermal stress or thermal expansion coefficients. Notably, symmetric polar tensors of rank two are diagonal in form for crystals of all Laue groups that are not triclinic or monoclinic. Furthermore, for cubic crystals (and for isotropic solids), polar tensors of rank two are spherical with only one unique entry. Table A. Forms of rank one polar tensors (Thurston 974) Point groups: 4 7,0,4,6,0,,6 Others Components No. constants: 0 Table A.4 Forms of rank two polar tensors (Thurston 974) Laue groups*: 4,6,8 5,7,9, 0,, Components No. constants: 9 5 *Includes transversely isotropic () and isotropic () Table A.5 Forms of symmetric rank two polar tensors (Thurston 974) Laue groups*: 4,6,8 5,7,9, 0,, Components No. constants: 6 4 *Includes transversely isotropic () and isotropic ()

8 550 Appendix A: Crystal Symmetries and Elastic Constants Tensors of orders three and four, on the other hand, are written for conciseness by assuming additional symmetries. Specifically in Table A.6, the last two indices are reduced to one according to the Voigt (98) notation, assuming that these indices correspond to differentiation of the thermodynamic potential with respect to a symmetric variable. For example, the nine components of a symmetric second-order tensor reduce to six according to the correspondence ~, ~, ~, (A.) = ~ 4, = ~ 5, = ~ 6. In Table A.6, the first index spanning,, which varies among the three rows of that matrix corresponds to the component of the polar tensor arising from differentiation of a thermodynamic potential with respect to a vector. The second index spanning,, 6 which varies among six columns corresponds to differentiation with respect to a symmetric second order tensor. When this polar tensor describes piezoelectric-type coefficients, for example, the vector is the electrical variable and the tensor is the symmetric elastic strain or stress. The convention used here is consistent with that of Section Table A.6 Forms of rank three polar tensors (Thurston 974) Point groups: , Components No. constants:

9 A. Generic Material Coefficients 55 Table A.6 (Continued) Point groups:,4 4,6* , Components No. constants: 6 4 *Includes transversely isotropic The coefficients in Table A.7 are written assuming symmetry with respect to each individual pair of indices, consistent with Section This symmetry reduces the maximum number of independent coefficients from 4 =8 to 6 6=6. For example, for the case of elastic moduli, each index spanning,, 6 corresponds to differentiation of the strain energy density with respect to an independent component of strain. However, the matrices listed in Table A.7 are more generic than elastic moduli that result from the assumption of hyperelasticity, and do not require that the 6 6 matrix of coefficients is symmetric. Thus Table A.7 admits coefficients defined via differentiation of a thermodynamic potential with respect to two different symmetric, second-order tensors. The additional symmetry that would arise from differentiation with respect to the same symmetric second-order tensor further reduces the maximum number of independent coefficients to. This is discussed by example in the context of secondorder elastic constants in Section A..

10 55 Appendix A: Crystal Symmetries and Elastic Constants Table A.7 Forms of rank four polar tensors (Thurston 974) N M O 4 TII 5 TI 6 II 7 I 8 HII 9 HI 0 CII CI t-i iso A* A* A* A* A* A* A* A* * A = The discussion in Section A. pertains to a geometrically nonlinear response, as implied by (A.). For geometrically linear or small-strain representations, the same symmetries in material coefficients listed in Tables A.-A.7 still apply, following from appropriate linearization of the finite deformation descriptions.

11 A. Elastic Constants 55 As implied in the discussion of polar tensors above, (A.) can be extended to apply when the free energy density (or another thermodynamic potential such as internal energy density) depends on other state variables, for example temperature in thermoelastic bodies or electric polarization in dielectric bodies. However, as noted by Thurston (974), when polar tensors of odd rank are involved (e.g., piezoelectric constants in electromechanical theories), the number of operations needed to describe all kinds of symmetry in the response functions requires extension beyond those generators shared by all crystal classes within a given Laue group listed in Table A.. This is evident from Table A.6, wherein sixteen different forms of rank three polar tensors are required to address the non-centrosymmetric point groups. Put another way, polar tensors of odd rank require precise consideration of which of the thirty-two point groups a crystal structure belongs, not just which of the eleven Laue groups it falls into. Delineation of symmetries of magnetic properties, outside the scope of this text, evidently requires introduction of ninety magnetic crystal classes (Thurston 974). A. Elastic Constants Consider a hyperelastic body with a strain energy density function expressed in polynomial form as ABCD ABCDEF Ψ0( EAB ) = ρψ 0 = EABECD + EABECDEEF!! (A.) ABCDEFGH + EABECDEEF EGH ! Strain energy per unit reference volume is Ψ 0, strain energy per unit mass is ψ, mass density is ρ 0, and elastic constants at null elastic strain (i.e., in the undistorted state) satisfy ABCD Ψ0 =, (second-order elastic constants); (A.4) EAB ECD E= 0 ABCDEF Ψ0 =, (third-order constants); (A.5) EAB ECD EEF E= 0 4 ABCDEFGH Ψ0 =, (fourth-order constants); (A.6) EAB ECD EEF EGH E= 0 and so forth for elastic constants of orders higher than four. ecall from (5.0) that in a hyperelastic material of grade one, the second Piola- Kirchhoff stress tensor satisfies

12 554 Appendix A: Crystal Symmetries and Elastic Constants ψ Ψ Ψ = = = = AB 0 0 BA Σ ρ0 Σ EAB EAB EBA, (A.7) leading to AB ABCD ABCDEF Σ = ECD + ECDEEF, (A.8) where terms of orders four and higher in the strain E are dropped from (A.) henceforth. From the symmetry of strain tensor E and the hyperelastic definitions of the elastic moduli in (A.4) and (A.5), ABCD ( AB)( CD) ( CD)( AB) = =, (A.9) ABCDEF ( AB)( CD)( EF ) ( AB)( EF )( CD) = =..., implying that the tensor of second-order elastic constants contains at most independent entries, and the tensor of third-order elastic constants at most 56 independent entries. In the compact notation of Brugger (964), Thurston (974), and Teodosiu (98), components of the stress and moduli are re-written to take advantage of these symmetries: ( AB ) A ( AB)( CD Σ ~ Σ, ) AB ( AB)( CD)( EF ~, ) ABC ~. (A.0) Barred indices span,, 6 and correspond to unbarred pairs of indices as indicated in (A.). Consistent with (A.0), strains and second-order elastic compliances are re-written as E( ) = ( + δ AB ) E, 4S AB A ( )( ) = ( + δab )( + δcd )S, (A.) AB CD AB recalling from (5.48) that elastic moduli and compliances are inverses of each other: CDEF E F F E BC C SABCD = δ. Aδ. B + δ. Aδ. B, S = δ. (A.) AB. A As a consequence of hyperelasticity, the elastic coefficients exhibit the remaining symmetries AB ( AB) ABC BAC ACB CAB =, = = =, S = S AB ( AB). (A.) Using (A.0) and (A.), the strain energy density of (A.) is AB ABC Ψ 0 ( E ) = E E + E E E +..., (A.4) A A B A B C!! and the stress tensor of (A.8) is written compactly in Voigt s notation as A AB ABC Σ = E + E E. (A.5) B B C Other condensed notations exist for elastic coefficients (Birch 947; Murnaghan 95; Hearmon 95; Toupin and Bernstein 96); transformation formulae among several notations are given by Brugger (964).

13 A. Elastic Constants 555 The number of independent elastic coefficients for a given substance may be further reduced because of material symmetry associated with the structure of the substance. Specifically, the strain energy density Ψ 0 for crystals of a particular Laue group is invariant with respect to rotations of + reference coordinates belonging to the proper rotation group of that Laue group. This requires that the strain energy density depend only on certain scalar functions, called invariants, of E (or of the symmetric deformation tensor C= F F) that leave the energy unchanged with respect T to such rotations (Smith and ivlin 958). Scalar invariants of E are labeled as I, I,... I P. The stress (A.7) can then be found as P AB Ψ0 Ι λ Σ = λ = Ι λ E. (A.6) AB Lists of invariants for each Laue group are given by Truesdell and Noll (965) and Teodosiu (98) and are not repeated here. Consideration of the list of invariants for each Laue group enables deduction of the independent elastic coefficients for that group, as explained for example by Teodosiu (98). Alternative arguments providing the independent elastic constants for various Laue groups are provided by Landau and Lifshitz (959). Tables A.8 and A.9 list independent second- and third-order elastic constants, respectively, for the eleven Laue groups of crystals, for transversely isotropic materials, and for isotropic bodies. The bottom row in each column provides the total number of independent coefficients. The reduced notation of Brugger (964) explained in (A.0)-(A.5) is used in Tables A.8 and A.9. Table A.8 applies for (second-order) elastic compliance as well as elastic stiffness constants, while Table A.9 applies only for (thirdorder) elastic stiffness constants. As a result of (A.), differences in factors of two arise among some entries of Tables A.7 and A.8. Notice that footnotes for Table A.9 continue onto the following page. Second- and third-order elastic constants are often obtained from experimental measurements of sound velocities in stress-free and homogeneously stressed crystals, respectively (Thurston and Brugger 964; Thomas 968; Thurston 974; Hiki 98). Constants of orders four and higher, not listed here, may be important in some shock compression events (Graham 99) and can be estimated from temperature dependence of lower-order elastic coefficients or deviations from a linear relationship between sound velocity and initial stress (Markenscoff 977; Hiki 98).

14 556 Appendix A: Crystal Symmetries and Elastic Constants Table A.8 Second-order elastic constants (Brugger 965; Teodosiu 98) N M O 4 TII 5 TI 6 II 7 I 8 HII 9 HI 0 CII CI t-i iso B* B* A* A* A* A* A* A* * A : = ; S = (S S ) : ; S44 (S S ) B = =

15 A. Elastic Constants 557 Table A.9 Third-order stiffness constants (Brugger 965) N M O 4 TII 5 TI 6 II 7 I 8 HII 9 HI 0 CII CI t-i iso A* A* A* A* L* B* M* C* C* D* D* D* D* M* M* E* E* F* M* G* L* H* H* I* I* I* I* M* M* M* M* J* J* J* J* J* L* K* K* K* K* K* N*

16 558 Appendix A: Crystal Symmetries and Elastic Constants Table A.9 (Continued) * A= + E = 4 4 I = ( )/ 4 M = ( )/ 4 B= (5 + 5)/ F = 5 5 J = ( )/ N = ( + )/8 C = (4 + 4)/ G = (5 5)/ K = (44 55)/ D= ( + )/ 4 H = (4 4)/ L= ( )/ Highly symmetric materials are given special consideration in the text that follows. Specifically considered are cubic crystals, transversely isotropic materials, and isotropic materials. A.. Cubic Symmetry Cubic crystals include Laue group numbers 0 and in the notation of Table A.. Both groups exhibit three independent second-order elastic constants. Group 0 exhibits eight independent third-order elastic constants, while group exhibits six independent third-order elastic constants. All second rank polar tensors (e.g., thermal expansion coefficients) are spherical or diagonal in cubic crystals, as is clear from Table A.4. Neglecting third- and higher-order elastic coefficients, and when the coordinate system is chosen coincident with the cube axes, strain energy density (A.) can be written as (Thurston 974) A 44 AB Ψ0 = ( E. A) + EABE 44 + ( ) ( ) ( ) ( ) E + E + E A 44 AB = K( E. A) + E ABE µ ( ) ( ) ( ) E + E + E A 44 = K( E. A) + ( E) ( E) ( E) + + (A.7) ( + ) ( E ) + ( E ) + ( E ) A 44 = K( E. A) + ( ) ( ) ( ) E + E + E AB + µ { E ABE ( E ) + ( E ) + ( E ) } A AB = K( E. A) + µ E ABE + µ ( E) + ( E) + ( E), C where E AB = EAB δ ABE. C / is the deviatoric part of the strain tensor, satisfying the identity

17 A. Elastic Constants 559 AB E E AB = ( E) + ( E) + ( E) + ( E) + ( E) + ( E). (A.8) Second-order bulk modulus K and shear moduli µ and µ are defined, respectively, in terms of cubic second-order elastic constants as ( K = + ), ( ) 44 µ =, µ = ( ). (A.9) When µ = 0, the material becomes isotropic in the materially linear approximation of (A.7). An anisotropy ratio A (Zener 948) measuring the deviation from isotropy is typically defined according to the formula A = /( ) = / µ. When anisotropy ratio A =, the solid is perfectly isotropic. To ensure that the elastic strain energy density of (A.7) remains positive for any nonzero strain, the second-order cubic 44 elastic constants are constrained by K > 0, µ > 0, and > 0. When the deformation is spherical, the deformation gradient becomes a / a / F = J δ in Cartesian coordinates with strain E = (/ )( J ) δ,. A. A A E A / so that. = (/ )( J ) and E AB = 0. In that case, all but the first term vanish on the right side of the final equality in (A.7), and the stress state is purely hydrostatic. The second Piola-Kirchhoff stress tensor becomes Σ = ( K/ )( J ) δ for spherical deformation, the Cauchy AB / AB ab / / ab stress tensor becomes σ = ( K/ )( J J ) δ, and the Cauchy / / pressure becomes p= ( K/ )( J J ). Elastic stress wave propagation (i.e., acoustic waves) and methods of determination of elastic constants in cubic crystals from ultrasonic measurements are discussed in detail by Thurston (974). A procedure for determining cubic elastic constants at high pressures from sound wave velocities is outlined by Dandekar (970). AB AB A.. Transverse Isotropy Materials with transverse isotropy, belonging to rotation group of Table A., exhibit five independent second-order elastic constants. Thus, the second-order moduli of transversely isotropic solids exhibit the same form as those of hexagonal crystals of Laue groups 8 and 9, as is clear from Table A.8. However, transversely isotropic materials have only nine independent third-order elastic constants. This is in contrast to hexagonal crystals which exhibit twelve independent constants (group 8) or ten constants (group 9). Transverse isotropy does not correspond to naturally occurring, single crystal Bravais lattices, but can emerge in polycrystalline samples

18 560 Appendix A: Crystal Symmetries and Elastic Constants via texturing. It is also commonly used to describe properties of certain kinds of fiber-reinforced composite materials, for example those with randomly or periodically distributed cylindrical fibers oriented perpendicular to a plane of symmetry. A.. Isotropy Isotropic solids exhibit two second-order elastic constants and three thirdorder elastic constants. The second-order elasticity tensor is constructed from two independent constants in Cartesian reference coordinates as ABCD = µ ( δ AC δ BD + δ AD δ BC ) + λδ AB δ CD, (A.0) where µ is the shear modulus and λ is Lamé s constant. The third-order elastic constants of an isotropic elastic body can be written as (Toupin and Bernstein 96; Teodosiu 98) ν =, ν =, ν =, (A.) leading to the following representation of the third-order elastic moduli: ABCDEF AB CD EF = ν δ δ δ AB CE DF CF DE CD AE BF AF BE + ν δ ( δ δ + δ δ ) + δ ( δ δ + δ δ ) EF AC BD AD BC + δ ( δ δ + δ δ ) (A.) AC BE DF BF DE BD AE CF AF CE + ν δ ( δ δ + δ δ ) + δ ( δ δ + δ δ ) AD BE CF BF CE BC AE DF AF DE + δ ( δ δ + δ δ ) + δ ( δ δ + δ δ ). In the absence of material nonlinearity (i.e., omitting the third-order elastic constants), the strain energy density and stress-strain relations are simply 0 ( A. ) AB Ψ = λ E A + µ EABE, (A.) Σ AB = λe C AB. AB Cδ + µ E. (A.4) Three other elastic constants are often introduced to describe the isotropic mechanical behavior demonstrated in (A.): elastic modulus E, Poisson s ratio ν, and bulk modulus K. elationships among the five isotropic second-order elastic constants are listed in Table A.0. To ensure that the elastic strain energy density of (A.) remains positive for all non-zero strains, isotropic elastic constants are restricted to µ > 0, E > 0, K > 0, and < ν /. When ν = /, then µ = E /, / K = 0, K, and the material is elastically incompressible. Using definitions in Table A.0, stress-strain relations (A.4) become

19 A. Elastic Constants 56 A A Σ AB = µe AB, Σ. A = KE. A. (A.5) The first of (A.5) relates the deviatoric (i.e., traceless) parts of second Piola-Kirchhoff stress and right Cauchy-Green strain, and the second relates spherical parts of stress and strain measures. When the deformation a / a is spherical with F = J δ, the stress state is hydrostatic, with Cauchy. A. A / / pressure p= ( K/ )( J J ) ; the same relationship is observed for materials with cubic symmetry as mentioned already in Section A... eturning now to the materially nonlinear case, third-order elastic constants can be related to pressure derivatives of tangent bulk and shear moduli, K and µ respectively, at an undistorted stress-free state as (Teodosiu 98) K 8 K = ν + ν + ν p p= 0 µ 4, K = K + µ + ν + ν 9 p p= 0. (A.6) A more in-depth treatment of pressure derivatives of elastic coefficients, including those of anisotropic crystals, is given by Thurston (974). Table A.0 elationships among second-order elastic constants of isotropic solids E ν µ K λ E, ν - - E E Eν ( + ν ) ( ν ) ( + ν )( ν ) E, µ - E µ - Eµ µ ( E µ ) µ ( µ E) µ E EK, - K E EK - K( K E) 6K 9K E 9K E E, λ * - λ E λ + A E+ λ + A - E+ λ + A 4 6 ν, µ µ ( + ν ) - - µ ( + ν ) ( ν ) νµ ν ν, K K( ν ) - K( ν ) ( + ν ) - ν K +ν ν, λ λ( + ν)( ν) - λ( ν ) λ( + ν ) - ν ν ν µ, K 9µ K K µ - - K µ µ + K 6K + µ µ, λ µ (µ + λ) λ - µ + λ - µ + λ ( µ + λ) λ, K 9 KK ( λ) λ ( K λ) - - K λ K λ / * A= ( E + Eλ+ 9 λ )

20 56 Appendix A: Crystal Symmetries and Elastic Constants In many applications, the effect of elastic material nonlinearity on deviatoric stresses may be negligible. For example, in ductile metallic crystals, plastic yielding (i.e., dislocation glide) may take place before large deviatoric elastic strains are attained. However, because dislocation glide is isochoric as discussed in Section. of Chapter, volumetric strains cannot be accommodated inelastically in the absence of defects (e.g., in the absence of vacancy formation, void growth, or fracture). In such cases, it may be prudent discard from (A.) only terms of order greater than two in the deviatoric strains: ABCD AB CD EF Ψ0 = EABECD Kδ δ δ EABECDEEF, (A.7) where for an isotropic solid, setting EAB δ AB in (A.) and (A.7) gives 8 K = ν ν ν. (A.8) 9 The bulk modulus usually increases with increasing (compressive) pressure, in which case from (A.6) and (A.8), K > 0. The first term in (A.7) can be anisotropic; for the particular case of isotropic second-order elasticity (Clayton 005b) A AB A Ψ 0 = K( E. A) + µ E ABE K ( E. A), (A.9) resulting in the stress-strain relations Σ AB = KE C AB. AB Cδ + µ E, (A.0) where the apparent bulk modulus is A K = K K E. A. (A.) Deviatoric and spherical parts of the second Piola-Kirchhoff stress are then, respectively, AB AB A A Σ = µe, Σ. A = KE. A. (A.) The approach taken in (A.7)-(A.) is comparable to that of Murnaghan (944), who assumed a bulk modulus linearly dependent on pressure. Naturally occurring isotropic single crystals are rare, a notable exception being tungsten single crystals at room temperature and small deformations (Hirth and Lothe 98). The isotropic description is often used for aggregates of a large number of polycrystals with random texture and for amorphous solids such as non-crystalline polymers and glasses. Though typically the (isotropic) elastic constants of polycrystals are measured directly using mechanical testing or ultrasonic techniques, theoretical methods exist for estimating effective isotropic elastic constants of an aggregate from anisotropic elastic constants of its constituents. A number of methods based on various averaging or self-consistent assumptions have been de-

21 A. Elastic Constants 56 veloped, as can be found in texts on micromechanics (Mura 98; Nemat- Nasser and Hori 99; Buryachenko 007). The two simplest, yet still physically realistic, methods are referred to as Voigt s approximation and euss s approximation. In Voigt s approximation, all single crystals of the volume element of a polycrystalline are assigned the same strain E, and the total stress supported by the polycrystal is taken as the volume average of the stresses in each crystal. This leads to the definition of the Voigt average second-order elastic constants ABCD ABCD V = V ( X) dv, (A.) V ABCD V where the average components and the local single crystal constants ABCD, the latter listed in Table A.8 for crystals of different Laue groups, are all referred to the same global Cartesian coordinate system. The reference volume of the aggregate is denoted by V. Omitting elastic constants of orders three and higher, the average second Piola-Kirchhoff stress is AB ABCD Σ V = V ECD, (A.4) ABCD where V is assumed in the present application to exhibit isotropic symmetry of the form in (A.0). For an imposed spherical strain tensor F ECD = E. FδCD /, substituting (A.) into the trace of (A.4) gives A ACD. F F AC. ( ΣV). A = ( V). A E. FδCD = E. F. AC. dv V V (A.5) AC. F F =. AC. E. F = 9 KVE. F, since the integrand in (A.5) is a scalar invariant. The Voigt average bulk modulus K V follows in full tensor notation and reduced Voigt notation, respectively, as AB. KV =. A. B = ( ) (A.6) = ( + + ). 9 When single crystals of the aggregate are cubic, comparison with (A.9) demonstrates that (A.6) is exact. For an imposed shear strain E, relation (A.4) gives Σ V = V E = µ VE, (A.7) where the Voigt average shear modulus µ V is given by (Hill 95)

22 564 Appendix A: Crystal Symmetries and Elastic Constants AB A. B AB µ V = AB A B = 0 AB K V (A.8) ( ) ( ) ( = ). 5 For an aggregate composed of cubic single crystals with elastic constants of the form listed in (A.9), the final expression in (A.8) reduces to 44 µ V = ( ) + = µ + ( µ + µ ). (A.9) In the euss approximation, all single crystals of the element of polycrystal are assigned the same stress Σ, and the strain E supported by the polycrystal is taken as the volume average of the strains in each crystal. This leads to the definition of euss average second-order elastic compliance (S ) = V S ( X) dv, (A.40) ABCD ABCD V where average components (S ) ABCD and local single crystal constants S ABCD are all referred to the same global Cartesian coordinate system. ecall that general forms of single crystal compliance tensors are listed in Table A.8 for crystal classes of each of the eleven Laue groups. The euss average bulk modulus K is found, using a procedure similar to (A.5), as (Hill 95; Mura 98) AB. K = /S. A. B = S + S + S + (S + S + S ). (A.4) The euss average shear modulus µ is (Hill 95) µ = 4(S S S ) 4(S S S ) (A.4) + (S44 + S55 + S 66) }. For cubic single crystals, the euss average shear modulus reduces to 4 µ = + = ( ) 5 5µ 5( µ + µ ). (A.4) The euss average bulk modulus is exact, and is the same as the Voigt average bulk modulus, when single crystals of the aggregate are cubic in symmetry. The difference between polycrystalline shear moduli computed using Voigt and euss averaging when single crystals of the aggregate are cubic is 44 [ ( )] µ V µ =. (A.44) 44 5[4 + ( )]

23 A. Elastic Constants 565 It is noted that Voigt and euss averaged bulk moduli depend only on six of the up to independent single crystal elastic constants. Voigt and euss averaged shear moduli depend only on nine of up to independent single crystal elastic constants. It can be shown (Hill 95; Mura 98; Nemat-Nasser and Horii 999) that Voigt average (A.) provides an upper bound to the exact effective elastic stiffness constants of a heterogeneous polycrystal, and the inverse of the euss average (A.40) provides a lower bound for the elastic stiffness. For the true effective bulk modulus K E and shear modulus µ E, the following bounds apply, for example: K KE KV, µ µ E µ V. (A.45) Voigt and euss definitions of effective elastic constants in (A.) and (A.40) also can be prescribed when the overall behavior of the material is anisotropic, though in that case (A.4) does not strictly apply.

24 Appendix B: Lattice Statics and Dynamics A brief overview of the governing equations of classical mechanics of discrete (e.g., particle or atomic) systems is given. This overview provides insight into general methods often used in mechanics, in an analogous fashion, to describe behavior of continuous bodies, e.g., Hamilton s principle and the Euler-Lagrange equations. A description of lattice statics then offers insight into atomic-scale origins of stress tensors and elastic coefficients. This account is not comprehensive; areas not explicitly addressed are relativistic effects, lattice vibrations, electronic structure, thermal, optical, and electrical properties of matter, or topics in quantum and statistical mechanics. While terms lattice statics and lattice dynamics are used often, unless noted otherwise the description applies to any discrete particle system with conservative internal forces, regardless of whether or not such particles occupy positions on a regular lattice. B. Dynamics Governing equations of classical mechanics of discrete particle systems are now surveyed briefly. Successively discussed are Newton s equations of motion, Lagrange s equations, and equations of motion in Hamiltonian form. Much of the survey follows from Pauling and Wilson (95) and Born (960). Terms particle and atom are often used interchangeably. B.. Newton s Equations Newton s equations of motion for a discrete particle system are written in indicial notation as a ˆ a a m r = f + f, (B.) i i i i Angled brackets denote atomic labels. In (B.), m and r are the constant mass of atom i and the spatial position vector of (the nucleus) of atom i i i. Atomic nuclei are treated as ideal rigid point masses. The non-

25 568 Appendix B: Lattice Statics and Dynamics conservative force acting on particle i is written f ˆ i, and the conservative force acting on particle i is written f. elation (B.) is expressed in the i spatial (i.e., current) configuration spanned by constant orthonormal basis vectors in an inertial frame, e.g., Cartesian coordinates. The conservative force by definition is the position gradient of an energy potential: a ab Φ f = δ. (B.) i b r If instead curvilinear coordinates are used, the appropriate particle acceleration must be incorporated in (B.), and contravariant components of ab (the inverse of the) metric tensor replace δ in (B.). Potential energy Φ depends, by definition, only on nuclear coordinates, i.e., Φ = Φ r, (B.) ( i ) where N denotes the set of atoms comprising the system. Forces of mechanical, electrostatic, and/or gravitational origin can be expressed using (B.). Dependence on species of each atom i is implicit in (B.), which is applicable for systems with multiple atomic species. Non-conservative forces include time-dependent external forces and forces that depend explicitly on velocity, e.g., certain electromagnetic, viscous, and other dissipative forces. i i N B.. Lagrange s Equations Hamilton s principle can be used to obtain equations of motion for a conservative system that apply in any coordinate system (e.g., curvilinear coordinates). A scalar Lagrangian function L is introduced as L = L r, r = K Φ, (B.4) ( i i ) where K is the kinetic energy of the system. Positions and velocities can be expressed in generalized coordinates in (B.4). Hamilton s principle is stated as t t i N L dt = constant, (B.5)..a For example, denoting by { bc } the Christoffel symbols of the second kind, a a.. a b c A = D ( r ) + i i { bc} r r. i i Dt

26 B. Dynamics 569 meaning that the motion of the atoms or particles over the (arbitrary) interval t t t results a stationary value of the integrated Lagrangian, i.e., the action integral. This stationary value can correspond to a maximum, minimum, or saddle point. Equations of motion consistent with (B.5) are obtained as general solutions of the following variational problem: t δ L dt = 0, (B.6) t where δ is the first variation. Using (B.4) and integrating by parts, t t N L L δ Ldt = iδr + iδr dt i i t i t = r r i i (B.7) t N L d L = iδ r dt, i t i= r dt r i i where admissible variation δr presumably vanishes (e.g., by definition) i at endpoints t and t. Substituting (B.7) into (B.6), the resulting Euler- Lagrange equations are L d L = 0, ( i=,,... N). (B.8) r dt r i i Prescribing the total kinetic energy as the sum K = (/ ) m r ir i= i i i and using (B.) and (B.4), Newton s laws (B.) are recovered from (B.8) when forces are conservative, i.e., when f ˆ = 0 in (B.). However, particle coordinates and velocities need not always be referred to a Cartesian i frame in (B.4)-(B.8); r can represent a triplet of generalized scalar (curvilinear) coordinates of particle i as opposed to its position vector in Carte- i sian space (Pauling and Wilson 95). A more general derivation treating non-conservative and non-holonomic systems is given by Synge (960). By an analogous procedure, as discussed in Section 5.6., Hamilton s principle can be used to obtain the local balance of momentum and constitutive laws for a non-dissipative continuous body, e.g., a hyperelastic solid. N B.. Hamilton s Equations The Euler-Lagrange equations for a discrete system in (B.8) are secondorder differential equations in time. These can be expressed in so-called

27 570 Appendix B: Lattice Statics and Dynamics canonical form as a system of first-order differential equations by introducing, via a Legendre transformation, the Hamiltonian H : N ( qk pk) = pkq k L( qk q k) H,, ; k = (B.9) L q k = q k ( ql, pl), det 0. qk ql A change of notation for kinematic variables is used: triplets of generalized coordinates r, r,... r are expressed as scalars q N, q,... q N, and similarly generalized velocities are expressed as scalars q, q,... q N. Generalized momenta are defined as the scalars L pk =, ( k =,,... N). (B.0) q k Noting that in the differential N d = q L L H kdpk dqk + pk dqk k = qk q, (B.) k the inner term in parentheses vanishes by (B.0), canonical equations of motion are deduced as H H q k =, p k =, ( k =,,... N). (B.) pk qk Since for the conservative systems considered here, the Hamiltonian in (B.9) does not depend explicitly on time, N d H H H = q k + p k = 0. (B.) dt k = qk pk Noting that in conservative systems with a stationary coordinate frame, N N K pq k k = q k = K, (B.4) k= k= q k the Hamiltonian becomes H = K L = K + Φ. (B.5) Together, (B.) and (B.5) express that the sum of kinetic and potential energies of the system remains constant. This is analogous to the energy balance (4.44) for insulated, continuous bodies subjected only to conservative body forces and subjected to no surface forces. A more general derivation of the canonical equations of classical dynamics valid for nonconservative systems is given by Synge (960).

28 B. Statics 57 B. Statics Successively discussed next are governing equations, interatomic potentials, atomic stress measures, and atomic origins of elasticity coefficients. The interpretation of stress and elastic coefficients is restricted to solids that deform homogeneously without inner translations, for example centrosymmetric crystals deforming according to the Cauchy-Born hypothesis (Born and Huang 950) in the absence of lattice vibrations and ionic polarization. Example formulae for stresses and second- and third-order elastic coefficients are given for systems described by pair potentials and the embedded atom method, with Cauchy s symmetry restrictions demonstrated in the former case. The forthcoming treatment of lattice statics is restricted to coincident Cartesian coordinate systems in referential and deformed configurations of the atomic ensemble. By definition, nuclear vibrations associated with zero-point, thermal, acoustic, and/or electromagnetic phenomena are omitted in the context of molecular or lattice statics. Hence, contributions of such vibrations to the free energy are excluded in what follows. Zero-point kinetic energy (particularly of electrons) implicitly affects bonding energies and hence atomic force interactions, but such effects are incorporated implicitly in what follows via prescription of empirical interatomic force potentials. B.. Governing Equations Governing equations of motion for molecular or lattice statics are those of (B.) in the absence of atomic velocities and accelerations: ˆ a a f + f = 0, (B.6) i i where again non-conservative and conservative forces acting on particle i are respectively denoted by f ˆ and f. Statics corresponds to a system at i i null kinetic energy, as is clear from (B.4), and also corresponds to a system at null absolute temperature (i.e., θ = 0 K ) if temperature θ is assumed to depend only on kinetic energy of the nuclei in the atomic system. For example, NkBθ = K for N atoms of an ideal gas with random velocities, where k B is Boltzmann s constant. Balance relations (B.6) can be obtained from a variational principle incorporating external forces: N δ L = δφ = fˆ iδ r, (B.7) whence from (B.), i= i i