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 refer- ence coordinate system. This Appendix discusses symmetry operations for point groups and Laue groups comprising the seven crystal systems first introduced in Chapter 3. Requisite terminology and definitions are given in Section A.1. Listed in Section A.2 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.3. The content gener- ally applies to materials whose elastic behavior is independent of deforma- tion gradients of orders higher than one, more specifically hyperelastic ma- terials of grade one. The discussion and tabular results of Appendix A summarize a number of previous descriptions (Hearmon 1946, 1956; Bond et al. 1949; Schmid and Boas 1950; Landau and Lifshitz 1959; Brugger 1964, 1965; Truesdell and Noll 1965; Thurston 1974; Teodosiu 1982). A.1 Crystal Classes, Point Groups, and Laue Groups Under an orthogonal transformation of Cartesian reference coordinates XQX→ , deformation gradient FxX= ∂∂/ and right Cauchy-Green strain EFF1=−()/2T transform as FFQ→ and EQEQ→ T , respec- tively. 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 en- ergy density, respects the symmetry, or lack thereof, of the material: ψψ(QEQT ) = () E. (A.1) The set _ of all orthogonal tensors Q that leave the mechanical response function unaffected, i.e., those operations for which (A.1) is satisfied, is J.D. Clayton, Nonlinear Mechanics of Crystals, Solid Mechanics and Its Applications 177, 543 DOI 10.1007/978-94-007-0350-6, © Springer Science+Business Media B.V. 2011 544 Appendix A: Crystal Symmetries and Elastic Constants called the symmetry group or isotropy group for the material. For two transformations QQ, ∈_ , their product QQ ∈_ . Furthermore, 12 12 Q∈_ implies QQT =∈−1 _ . Every isotropy group contains the identity map Q1= and the inversion Q1= − . Subgroups consisting of all proper orthogonal tensors ( detQ = + 1) of the isotropy groups comprise the rota- tion groups _+ . Any member of particular group _+ can be created by a n product of rotation matrices called generators of that group, labeled R N . 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 transla- tions 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 1995). The set of all rotations and reflec- tions 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 230, 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 me- chanical response functions in the context of (A.1) are invariant. Table A.1 lists Laue groups, point groups, and generators for all crystal classes, n following Thurston (1974). The notation RN denotes a right-handed rota- tion by angle 2/π n about an axis in the direction of unit vector N. Bold- face indices i, j, and k denote unit vectors of a right-handed orthonormal basis for X, and mi=++3(−1/2 j k ). Also denoted by asterisk in Table A.1 are the eleven crystal point groups—one per Laue group—with an in- version center or center of symmetry (i.e., those that are centrosymmetric). The seven crystal systems of Fig. 3.1 whose symmetry elements are listed in Table A.2 are recovered from the eleven Laue groups, since two Laue groups comprise each of the cubic, tetragonal, rhombohedral, and hexago- nal crystal systems. The alpha-numeric notation used to label each point group gives details about symmetries of crystals in that group, and a num- ber of different notational schemes exist in crystallography for labeling point (and space) groups. The reader is referred to Thurston (1974), Rohrer (2001), and references therein for a more detailed explanation. Modern Hermann-Mauguin international symbols are used in column five of Table A.1, following Rohrer (2001). Columns six and seven of Table A.1 follow the scheme of Thurston (1974), who in turn cites Groth (1905). A.1 Crystal Classes, Point Groups, and Laue Groups 545 Table A.1 Laue groups, generators, point groups, and crystal classes Crystal Laue # Generators Point Crystal class # system group group Triclinic N 1 1 1 triclinic asymmetric 1 1 triclinic pinacoidal* 2 Mono- M 2 2 2 monoclinic sphenoidal 3 Rk clinic m monoclinic domatic 4 2/m monoclinic prismatic* 5 Ortho- O 3 2 2 222 orthorhombic disphenoidal 6 Ri , R j rhombic mm2 orthorhombic pyramidal 7 mmm orthorhombic dipyramidal* 8 Tetrago- TII 4 4 4 tetragonal pyramidal 10 Rk nal 4 tetragonal disphenoidal 9 4/m tetragonal dipyramidal* 13 Tetrago- TI 5 4 2 422 tetragonal trapezohedral 12 Rk , Ri nal 4mm ditetragonal pyramidal 14 42m tetragonal scalenohedral 11 4/mmm ditetragonal dipyramidal* 15 Rhom- RII 6 3 3 trigonal pyramidal 16 Rk bohedral 3 trigonal rhombohedral* 17 Rhom- RI 7 3 2 32 trigonal trapezohedral 18 Rk , Ri bohedral 3m ditrigonal pyramidal 20 3m ditrigonal scalenohedral* 21 Hexago- HII 8 6 6 hexagonal pyramidal 23 Rk nal 6 trigonal dipyramidal 19 6/m hexagonal dipyramidal* 25 Hexago- HI 9 6 2 622 hexagonal trapezohedral 24 Rk , Ri nal 6mm dihexagonal pyramidal 26 62m ditrigonal dipyramidal 22 6/mmm dihexagonal dipyramidal* 27 Cubic CII 10 2 2 3 23 cubic tetartoidal 28 Ri , R j , Rm m3 cubic diploidal* 30 Cubic CI 11 4 4 4 432 cubic gyroidal 29 Ri , R j , Rk 43m cubic hextetrahedral 31 mm3 cubic hexoctahedral* 32 Transversely iso- 12 ϕ Rk , 02<<ϕ π tropic** Isotropic** 13 all proper rotations *Denotes a point group with an inversion center **Not a natural crystal system 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. (1949), Landau and Lifshitz (1959), and Teodosiu (1982) for additional details. Formal standards on terminology and as- signment of coordinate systems for crystalline solids are given by Bond et al. (1949). The triclinic crystal system contains Laue group N or 1. This system has the lowest material symmetry; a triclinic crystal has no symmetry axes n or planes. Though the coordinate system corresponding to RN in Table A.1 may be chosen arbitrarily (Landau and Lifshitz 1959), standard con- ventions do exist for selecting i, j, and k for a given specimen (Bond et al. 1949). Monoclinic crystals belong to Laue group M or 2 in the numbering scheme of Table A.1. 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 RN 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 3. It describes crys- tals with three mutually perpendicular twofold axes, two mutually perpen- dicular planes of reflection symmetry, or both. The coordinate system cor- n responding to RN 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 RII (6) and RI (7). Crystals of these groups have an axis of threefold symmetry, convention- ally called the c-axis when the Bravais-Miller system is used. For this rea- son, sometimes the rhombohedral system is referred to as the trigonal sys- tem. 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 RI also pos- sess 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.1, unit vector k is aligned par- allel to the axis of sixfold symmetry. Crystals of group HI also possess an axis of twofold symmetry parallel to unit vector i. A.2 Generic Material Coefficients 547 The cubic crystal system includes Laue groups CII (10) and CI (11). Cubic crystals have three twofold axes of symmetry.