Graphical Representation of the Generalized Hooke's Law Izei®Ei H

TECHNISCHE MECHANIK, Band 21, Heft 2, (2001), 145-158 Manuskripteingang: 30. Januar 2001

Graphical Representation of the Generalized Hooke’s Law

T. Böhlke, C. Brüggemann

The anisotropic linear elastic behavior of single crystals can be described equivalently by a 4th-0rder elasticity tensor or two functions E(d) and K These functions represent Young’s modulus and a generalized bulk modulus as functions of the tensile direction d in a tension test. In the present paper three— and two—dimensional graphical representations of Young’s modulus and the generalized bulk modulus are given for single crystals belonging to one of the following symmetry groups: monoclinic, rhombic, trigonal, tetragonal, hemagonal, and cubic symmetry.

1 Introduction

The generalized Hooke’s law is the geometrical and physical linear relation between stress and strain of anisotropic elastic solids. It is one of the oldest and best known constitutive relations in continuum mechanics. One dimensional formulations have been discussed ﬁrst in the 17th century. Over a long time the scientiﬁc community agreed to differ about the most general form in linear elasticity. Finally Voigt (1850-1919) answered the open question how many constants have to be determined in the isotropic and anisotropic case. He proved by experiments Green’s (1793-1841) hypothesis that in the isotropic case two and in the general anisotropic hyperelastic case 21 constants have to be determined.

Based on Hooke’s law structures are analyzed in all branches of mechanical and civil engineering. In most cases the isotropic version of Hooke’s law is used. But in the last decades more and more anisotropies have been taken into account, which are especially important if the component parts are laminated or made from single crystals. Also the change of elastic properties of polycrystals during metal forming operations is of importance in order to estimate spring—back effects. Independently from the special application the engineer requires methods to illustrate or to visualize the anisotropic linear elastic properties of materials. In the present paper a visualization based on Young’s modulus and a generalized bulk modulus is developed.

The outline of the paper is as follows: In Section 2 the basic properties of Hooke’s law are discussed and the elasticity tensor is speciﬁed for the eight different symmetry classes relevant in the context of linear elasticity. In Section 3 the harmonic decomposition of elasticity tensors is introduced. This decomposition delivers a tool to decompose a 4th-order elasticity tensor uniquely into isotropic and anisotropic parts. In Section 4 characteristic components of the elasticity tensor are discussed, which are useful scalar functions describing the anisotropic linear elastic behavior (Böhlke, 2001). It is shown that Young’s modulus and a generalized bulk modulus depend only on a normalized vector specifying the tensile direction in a tension test. Both scalar functions uniquely determine the elasticity tensor (He and Curnier, 1994). In Section 5 three— and two-dimensional graphical representations of Young’s modulus and the generalized bulk modulus are given for single crystals belonging to one of the following symmetry groups: monoclinic, rhombic, trigonal, tetragonal, hexagonal, and cubic symmetry.

Notation. Throughout the text a direct tensor notation is preferred. In order to avoid additional formal deﬁnitions, the index notation is applied in some cases using the summation convention. A linear mapping of a 2nd—order tensor is written as A : (C[B]. The composition, the scalar product, the dyadic product, and the Frobenius norm of 2nd—order tensors are denoted by AB, A - B, A ® B, and = (A - A)1/2, respectively. Irreducible, i.e., completely symmetric and traceless tensors are designated by a prime, e.g., A’ and C’. The basic isotropic 2nd—, 4th-, and 6th—0rder tensors (Zheng and Betten, 1995) are denoted by I, II, and J, respectively

Izei®ei H=i9i®ej®(ei®ej+ej®ei) (1)

145 J: %(ez‘®ej+ej®ei)®(ej®ek+6k®e_j)®(ek ®ei+ei®ek) (2) where {ei} is an arbitrary but ﬁxed orthonormal basis. The Rayleigh product of a 2nd-order tensor A and a 4th-order tensor C = Cijklei ® Ej 69 e], (8 e; is deﬁned by

A * C : C'mnopAem ® Aen ® Aeo ® AeZ7 (3)

If (3) is written in components with respect to {e,} then one obtains

(A * (C)ijkl : AimAjnAkoAlpCmnop

If C exhibits a symmetry in the ﬁrst or second pair of indices (e.g., Cm“ = Cjikl) or the major symmetry (Cijk; = CH“), then A * C shows the same symmetry properties for all A. Furthermore, the associativity

A*(B*C) :(AB)*

2 Hooke’s Law: Basic Properties

In classical linear elasticity, the stress tensor T is given as a linear map of the (inﬁnitesimal) strain tensor E, and vice versa

T z on] E : err] (6) where C is the stiffness tensor and S is the compliance tensor. S : C*1 holds on the space of symmetric 4th—order tensors. This formulation of Hooke’s law can be applied if both the strains and rotations are small. For hyperelastic materials a strain energy and a complementary energy exist. They represent potentials for the stresses and strains, respectively. In the hyperelastic case the elasticity tensors possess the major symmetry, i.e., Gig-kl = Cklij. The symmetry in the ﬁrst and last pair of indices can be assumed for convenience, e.g. 0,ij : Cjikl, since non-symmetric parts do not affect the stress-strain response. For a general discussion of the symmetries of elasticity tensors, see Bertram (1985). If the strain energy density is required to be positive for all non—zero strains, then the elasticity tensors have to be positive deﬁnite.

The 4th—order stiffness tensor C and the corresponding compliance tensor S are speciﬁed by the symmetry group S of the material being a subgroup of the special orthogonal group, e.g., in terms of stiffnesses (C:H*C VHeSgOrth (7)

Two stiffness tensors C1 and Q are called equivalent if there is a rotation Q E Orth (proper orthogonal group) such that S(

Depending on the number of rotations or reflections of symmetry, Voigt (1910) classified crystals into 32 classes. All kinds of physically possible 2- and 3—dimensional symmetry groups have been classified by Zheng and Boehler (1994). Forte and Vianello (1996) classified all different forms of linear operators C due to 3-dimensional symmetry groups and showed that in the context of 4th-order operators only 8 different symmetry classes can be distinguished (see Lemma 2 and the corresponding remarks on pp. 98—99 in the aforementioned article). The same result has been obtained by Khatkevich (1961); Federov (1968); Cowin (1995); Ting (1996). Note that e.g. by Gurtin (1972) ten different matrices have been given, which contain redundancies. Khatkevich (1961) and Cowin (1995) give an explicit proof that two of the matrices given by Gurtin (1972, pp. 88—89) are redundant, whereas Federov (1968) mentioned the redundancies without proof. Forte and Vianello (1996) discuss the redundancies in a group theoretical context and give an historical overview concerning the representation of linear elastic laws due to different types of anisotropy.

146 It is convenient to represent elasticity tensors by six by six matrices, the components of which refer to the orthonormal basis Ba(a : 1_. . .6) of symmetric 2nd-order tensors, i.e., Gag : Ba - (C[Bß]‚ where

B1 : 81 ® e1 B4 2 «Ti (e2 ®e3 +63 <362)

B2 1‘ 82 ® 82 Bs 1 g (61 ®ea +93 (Eel)

B3 Z e3 ® '33 B6 = 12—5 (61 ®e2 +92 (3361)

This modiﬁed Voigt notation has been proposed, e.g., by Federov (1968) and Cowin (1989). The representation of elasticity tensors by six by six matrices with respect to the basis Ba has the advantage that invariants, eigenvalues, and eigentensors can be determined by means of the square matrix Gag. Note that this is not the case if Voigt’s original notation is used.

In the following we list the component forms of the stiffnesses corresponding to the eight different symmetry classes (Ting, 1996). The matrices are given together with the number of independent elastic constants Nc and the number of planes of symmetry Np:

triclinic symmetry (NC : 21, NP 2 0)

01111 C‚1122 01133 @01123 ﬂcllw «501112

C2222 C2233 flc2223 «502213 flC2212 C3333 flossza flcazna flcaalz Ba®Bß 202323 2C2313 202312 sym. 201313 201312 201212 monoclinic symmetry (NC = 13, NP 2 1)

C1111 C1122 01133 £01123 Ü Ü C2222 02233 @02223 0 O 03333 «503323 0 Ü 0 0 Ba 8) B5 (10) 202323 sym. 201313 2C1312 201212

orthotropic or rhombic symmetry (NC = 9, NP : 3)

C1111 01122 01133 0 O Ü C2222 02233 Ü 0 0 03333 Ü ÜO 0 Ba <29 Bß

sym. 201313 0 201212

147 trigonal symmetry (NC : 6, Np = 3)

01111 01122 01133 @0112?! 0 0 C1111 C1133 ‘ﬁcnza 0 Ü (C : C3333 0 0 0 B, ® Bß (12)

Sym- 202323 2C1123 01111 — C1122 tetragonal symmetry (NC = 6, Np : 5)

01111 01122 01133 Ü 0 Ü C1111 C1133 0 0 0 c : 03333 0 0 0 Ba «29 Bß (13) 202323 0 0

sym. 2C2323 Ü

2C1212

transversely isotropic or hexagonal symmetry (NC : 5, Np : 1 + oo)

C1111 01122 01133 Ü Ü 0 C1111 01133 Ü 0 O 03333 0 0 0 (C = Ba ® B5 (14) 202323 Ü Ü

sym. 202323 0 C1111 '- C1122

cubic symmetry (NC : 3, Np : 9)

01111 01122 01122 Ü 0 0 01111 C1122 0 0 0 01111 Ü 0 0 (c 2 Ba 09 Bg (15) 202323 Ü 0 Sym- 202323 Ü 202323

isotropic symmetry (NC = 2, Np : oo)

C1111 C1122 01122 0 0 0

01111 C1122 0 0 0 (C : C 1111 Ü Ü O Ba ® Bß (16) C1111 - 01122 Ü 0 sym. C1111 — C1122 0

C11111 " CV1122

The orientation of the crystal is such that its symmetry planes coincide with some of the coordinate

148 planes. Therefore, in all but the triclinic and isotropic case, in addition to the Nc coefﬁcients, the orientation of the anisotropy axes has to be supplemented in terms of three parameters. Otherwise the elasticity tensor is not determined uniquely. In the transversely isotropic case only two additional parameters have to be given.

3 Harmonic Decomposition of Elasticity Tensor-s

It is possible to decompose the 4th—order elasticity tensor of arbitrary symmetry into a direct sum of orthogonal subspaces on which the action of Orth is irreducible which means that there are no proper invariant subspaces. The harmonic decomposition has the form

C:h11P{+h21Pg+H'1®I+I®H'1+4.)[Hg)+IHI’ (17)

where

Pl : ®I : H— I (Aimöjn + Ainöjm + 511mAer + 6inAjm) ei ® ej ® em ®

(Schouten, 1924; Spencer, 1970; Cowin, 1989; Boehler et al., 1994). A review concerning this representation is given by Forte and Vianello (1996). In and h2 are called the ﬁrst and second isotropic parts; H’1 and Hg are the ﬁrst and second deviatoric part, respectively; H’ is the harmonic part. The symmetry group of (C is the intersection of the symmetry groups of its deviatoric and harmonic parts. The tensors H’l, H’27 and llil’ are irreducible, i.e., completely symmetric and traceless

Hlij Z Hlji Z 0 Héij : Hijz' Héii : 0 (19)

Hijkl : Hjl'ikl Z Him = Him: : Him Z 0 (20)

Irreducible 2nd—order tensors have ﬁve and irreducible 4th—order tensors have nine independent components.

The ﬁrst and second isotropic parts h1 and h2 are determined by a projection of (C onto the space of isotropic 4th—order tensors, i.e.

PI 7 WW

(7 : 1,2). h1 and h2 are the eigenvalues of the isotropic part of (C. Hence h1 : 3K and h2 2 2G hold, where K and G are the bulk and shear modulus of the isotropic part of (C, respectively. The scalars h1 and h2 of the stiffness tensor (C correspond to the (isotropic) Voigt estimate (Voigt, 1910) of the polycrystals bulk modulus h1 : 3K VI and shear modulus h2 : 20‘”. The isotropic parts of the compliance tensor S correspond to the bulk modulus 1 /h1 = 3K1” and shear modulus 1/h2 : ZGRI of the Reuss estimate (Reuss, 1929). In the following the bulk modulus 3KRI and Young’s modulus ERI : 9KRIGRI / (3K RI + GRI) are used to normalize directional dependent quantities.

The deviatoric parts can be computed by

5 4 3 2

where C1 and 0; represent the dilatational modulus and the Voigt tensor (Cowin, 1989)

C1 = Ciiklek ® el Cz = Cikilek ® ez (23)

Based on eqns (21) and (22), the harmonic part llil’ can be determined as H’ z (C) — gm ® I) + émoxr ® I) (24)

149 where

c z c:1 + 202 (25)

The bracket formula is deﬁned by (A,B E Sym)

1 (AijBkl) = 6(AijBkl + Aikle + AilBkj + BijAkl + BikAjl + BuAkj) (26) respectively. Note that (C has the major symmetry. Therefore, (Gig-kl) = CHM + Cum + Cm”- holds. The formulae (21)-(24) show that all parts of the harmonic decomposition are linear functions of the components of C.

4 Characteristic Components of the Elasticity Tensor

In order to determine the elastic anisotropy experimentally, Young’s modulus E is measured in most cases in tensile tests. A tensile test is deﬁned by an uniaxial stress state, i.e., T : ad ® d with the tensile direction d. The corresponding strain tensor is given by E : S[T] : aS[d® d]. The tensile stress 0’ and the corresponding strain E are the components of the stress and strain tensor with respect to the base tensor d ® d. Hence we have a = T - d (8 d and e : E - d (8) (:1, respectively. Young’s modulus E(d) is deﬁned by the ratio of tensile stress a and tensile strain a, i.e., E = 0/5. Combining the aforementioned facts one derives m : d®d-S[d®d]

= §h1+ghg+2H1-d®d+4H;-d®d+d®d-H’[d®d] (27) 1 : EW+(2E;+4H1)-d®d+d®d-H’[d®d]

(Böhlke, 2001). Here and in the following, h1, h2, H’l, H’z, and H’ denote the parts of the harmonic decomposition of S.

In the isotropic case the bulk modulus is defined by one third of the ratio of the traces of the stress and the strain tensor. He and Curnier (1994) extended the definition of the bulk modulus to the anisotropic case. They defined a generalized bulk modulus by one third of the ratio of the tensile stress and the trace of the strain tensor. Again we have a : T - d ® d and tr(E) : E - I, respectively. Note that tr(E) represents the relative change of volume (dV — dV0)/ dVg. The strain tensor corresponding to T 2 ad ® d is given by E : US [d ® d]. Combining the aforementioned facts one derives

1 3K(d) H I-S[d®d] 4 h1+\/§H’1-d®d+§\/§H;-d®d (28)

1 4 I w+(\/§Hi+§\/§HIZ)-d®d

(Böhlke, 2001). It is concluded that the quantity 1/(3K) represents the relative change of volume per tensile stress 0’ in direction d. The formula for the generalized bulk modulus shows that K (d) can exhibit only one of the following symmetries: isotropy, transverse-isotropy or orthotropy. This is due to the fact that it depends only on the 2nd-order deviatoric parts and does not depend on the 4th—order harmonic part. He and Curnier (1994) have shown that E(d) depends generally on 15 and K (d) on 6 components of S, and that both functions uniquely determine the tensor S. As a result, elasticity tensors can be determined from tensile tests alone. For cubic crystals K is independent of d since the deviatoric parts vanish.

Similar to Young’s modulus and the bulk modulus other characteristic components of the elasticity

150

tensors may be expressed by the parts of the harmonic decomposition (He and Curnier, 1994; Rychlewski, 1995) but these quantities are not deﬁned on the unit sphere in R3. The shear modulus G, in a plane which is speciﬁed by the direction d and the normal n through M = \/2(d ® n + n ® d) /2 is given by

1 2G(d,n) Z M'SM = h2+2H'1-(d®d+n®n)+M-IHI’[M] (29)

= ä+2HQ-(d®d+n®n)+M-H’[M]

Another characteristic component is Poisson’s ratio V in direction n under stretch in direction d

_l/(d,n) E(d) z d®d-S[n®n]

= ä—ﬂn-h2)+H1-(d®d+n®n)+d®d-H[n®n] (30)

„R1 = —W+H'1-(d®d+n®n)+d®d-llil’[n®n]

5 Graphical Representation of E(d) and K ((1)

Inspection of eqns (27) and (28) shows that in the isotropic case Young’s modulus and the generalized bulk modulus are given by

1 1 2 —ERI Z ghl + ähg ——’— 3 h1

In the following the quantities E(d) and K (d) are normalized by ERI and KRI , respectively. The dimensionless quantities E(d)/Em and KM / K (d) can be computed by the formulae presented in the preceding sections for each direction in R3. All directions in R3 can be parameterized by spherical coordinates {7330,19} with r = 1.

For a two-dimensional representation of E(d) /ERI and KM /K(d) it is possible to project these functions by, e.g., a stereographic projection into a plane which contains the point with r = 0. Note that E(d) = E(—d) and K (d) : K(—d) hold. The two—dimensional representation could be called quasi pole figure in analogy to the projection technique commonly used in texture analysis (see, e.g., Bunge, 1993; Kocks et al.7 1998). The quasi pole figure representation of elastic properties allows for a graphical representation of anisotropic linear elastic properties in two figures. In the following the normal of the projection plane is equal to e3. Then the spherical coordinates {90,19} can be expressed by the polar coordinates {77, 95}

(p : (,5, 19 = 2 arctan(7“') (32)

The projection is described in Figures 1. Figure 2 shows the function E(d) for Gold which has a cubic crystal symmetry.

In the Figures 3 — 20 the projections of the functions E(d)/EM and KM /K (d) are given for different materials belonging to one of the following symmetry groups: monoclinic7 rhombic, trigonal, tetragonal, hexagonal, and cubic symmetry. In all Figures lines of equal moduli are plotted. The difference between two lines amounts to A(E (d)/ ERI) : 0.1 and A(KRI/K = 0.1, respectively. The thick line denotes the ratios of E(d)/ER] :: 1 and K RI/ K (d) : 1. Hence the number of lines allows to quantify the amount of anisotropy in terms of the isotropic parts. For the dark—shadowed areas the ratios are larger than one. For the light-shadowed areas the ratios are less than one.

Monoclinic symmetry: In Figures 3 and 4 the three-dimensional and two—dimensional representations of Young’s modulus and the bulk modulus are given for Hornblende and Feldspar. Both materials exhibit a

151

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\ .‘u- u: ....‚. W“ \ .-.'/„x\‘ e E(E6 )‚ Km) cfiiiiia‘i-‘Eiggfiitiil‘n‘fi I m N I! I x ligiifl'li‘t‘i‘i‘t

[24" ~ \‘ i I .5

Figure 1: Stereographic Projection of Young’s Figure 2: Young’s Modulus of Gold (Cubic Sym- Modulus and the Generalized Bulk Modulus. metry). significant anisotropy in terms of the functions E(d) and K (d), respectively. In the case of a monoclinic symmetry one plane of symmetry exists (Np z 1). The pole figures indicate that for the considered set of elastic constants the normal of the plane of symmetry is given by n 2 e1. Rhombic symmetry: In Figures 5 — 8 the three— and two-dimensional representations are given for Uranium, Olivine, Rochelle Salt, and Topaz. All the materials but Topaz show a significant anisotropy. In the case of a rhombic symmetry three planes of symmetry exist (Np : 3). Their normals are given by the base vectors e,- = 1,2, 3). Trigonal symmetry: For this symmetry group the representations for Aluminium—Phosphate and Quartz are given in Figures 9 and 10. In the trigonal case three planes of symmetry exist (Np : 3). They are given by the base vectors

n1 : cos(30°)el + sin(30°)e2 n2 : e2 11;; : cos(150°)el + sin(150°)e2 (33)

The pole ﬁgure of the bulk modulus shows a transverse isotropic symmetry. Tetragonal symmetry: Here the representations for Weissblech and Zircon are given (Figures 11 and 12). The Figures show the following ﬁve planes of symmetry (Np 2 5)

n1_2,3 : e133 n4 : cos(45°)e1 + sin(45°)e2 n5 : cos(135°)e1 + sin(135°)e2 (34)

Note that similar to the trigonal case the bulk modulus exhibits a transverse isotropic symmetry. Hexagonal symmetry: The Figures 13 and 14 represent the elastic properties of Magnesium and Tita— nium. In this case the plane with the normal 11 : (33 is a plane of symmetry. Furthermore all planes which contain the eg-axis are planes of symmetry (Np : 1 + oo). Cubic symmetry: The Figures 15 — 20 show the elastic properties of Copper, Aluminium, Gold, Iron, Iron—Aluminium, and Pyrite. In the case of a cubic symmetry the material has nine planes of symmetry (NP 2 9). Their normals are given by the three base vectors ei. Furthermore all vectors n : (e,- iej)/\/§ with i 75 j are planes of symmetry. Note that in the cubic case the bulk modulus does not depend on d.

6 Summary

The anisotropic linear elastic behavior of single crystals has been described by two functions E(d) and K(d) defined on the unit sphere in R3 which represent Young’s modulus and a generalized bulk modulus in the tensile direction d. E(d) is the ratio of tensile stress to tensile strain. The quantity 1 / (3K (d)) represents the relative change of volume (dV — dV0)/ dVo per tensile stress a in direction d. The elasticity tensors can be uniquely determined if the aforementioned functions are given. Since both functions are defined on the unit sphere in R3, they can be projected into a plane. The resulting plot is called quasi pole figure in analogy to the projection technique commonly used in texture analysis. Three- and two—dimensional representations of Young’s modulus and the bulk modulus have been determined for selected single crystals (Kocks et al., 1998; Simmons and Wang, 1971). The figures clearly show the type and the amount of the anisotropy of the linear elastic behavior. The approach discussed in the present paper can be used for a graphical representation of material properties which are described by a fourth—order tensor having the same index symmetries as elasticity tensors. 152

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(a) Young’s modulus (b) Bulk mo dulus (c) Pole ﬁgure of Young’s (d) Pole ﬁgure of bulk E(n)/ERI KRI/K(n) modulus E(n)/ERI modulus KRI/K(n)

Figure 3: Monoclinic Symmetry: Hornblende 42420 (011 : 130.1, 022 : 187.7, 033 : 198.4, C44 : 61.1, C55 : 38.7, 056 Z 45, 012 I 61.1, C13 : 59.5, 023 Z 61.4, C15 : 9.5, 025 = —6.9, 035 = —40.6, C43 2 ——0.9 [GPa] (Simmons and Wang, 1971))

(a) Young’s modulus (b) Bulk modulus (c) Pole ﬁgure of Young’s (d) Pole ﬁgure of bulk E(n)/ERI KRI/KÜI) modulus E(n)/ER’ modulus KRI/K(n)

Figure 4: Monoclinic Symmetry: Feldspar (74% or—19.4%Ab-1.95% An) 42414 (Cu : 61.9, C22 : 158.3, C33 2 C44 Z 055 Z 066 : 36, C12 2 C13 : 023 : 015 : -10, 025 = —1.8, 035 r: —12.1, 046 z —2.3 [GPa] (Simmons and Wang, 1971))

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(a) Young’s modulus (b) Bulk modulus (c) Pole ﬁgure of Young’s (d) Pole ﬁgure of bulk MIG/Em KIN/KW) modulus E(u)/ERI modulus KIN/Km)

Figure 5: Rhombic Symmetry: Uranium (011 = 214.8, 022 = 198.6, C33 : 267.1, 044 : 124.4, C55 : 066 I: 012 : 013 Z 21.8, 023 I (KOCkS et 31.,

153

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(a) Young 7 s modulus (b ) Bulk modulus (c) Pole ﬁgure of Young’s (d) Pole ﬁgure of bulk E ( n) /ERI KRI/K (n) modulus E(n)/ERI modulus KRI/K (n)

Figure 6: Rhombic Symmetry: 011vine 32294 (011 = 323. 7 022 = 197.6, C33 2 235. 1 1 C44 z 64.62,

C55 Z C66 : 012 Z 013 Z 023 I 75.6 [GPa] ( immens and Wang, 1971))

(a) Young’s modulus (b) Bulk modulus (C) Pole ﬁgure of Young’s ( d) Pole ﬁgure of bulk E(n) /ERI KRI/K( 11) modulus E ( n ) /ERI modulus KRI/K(n)

Figure 7: Rhombic Symmetry Rochelle Salt 32301 (011 = 25.5, C22 : 38.1, C33 Z 37. 1 C44 Z 13.4, C23 z 14.6 [GPa] (Simmons and Wang, 1971) C55 2 3.21 i 066 I 012 Z 14. 1 7 C13 Z 11 - 6, )

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(a) Young 7s modulus (b) Bulk modulus (c) Pole ﬁgure of Young’s ( d) Pole ﬁgure of bulk ) /ERI n)/ERI E( n KRI/an) modulus E ( modulus KRI/K(n)

igure 8: Rhombic Symmetry Topaz 32310 (011 2 281.36, C22 = C33 Z 294 52 C44 2 108.11, 132.98 055 Z 7 066 Z 130.89 012 : 125.82 7 (713 : 84.64, 023 : 88.15 [GPa] (Simmons and Wang, 1971))

154

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(a) Young’s modulus (b) Bulk (c) Pole ﬁgure onoung’s (d) Pole ﬁgure of bulk E(n)/ERI K1" /K(n) modulus E(n)/ERI modulus KRI/K(n)

Figure 9: Trigonal Symmetry: Aluminium-Phosphate 62835 (C11 : 105.03, C33 : 133.53, C44 : 23.14, 012 = 29.34, 6’13 = 69.27, 014 : —12.71 [GPa] (Simmons and Wang, 1971))

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Figure 10: ’I‘rigonal Symmetry: Quartz 62894 (C11 : 86.8, 0'33 : 105.75, C44 : 58.2, 012 2 7.04, C13 = 11.91, C14 : -18.04 [GPa] (Simmons and Wang, 1971))

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(a) Young’s modulus (b) Bulk modulus (c) Pole ﬁgure ofYoung’s (d) Pole ﬁgure of bulk E(n) /ERI K1"/K(n) modulus E(u)/ERI modulus KRI/K(n)

Figure 11: Tetragonal Symmetry: Weissblech 22251 (011 : 76.19, 033 = 116.24, 044 : 17.04, 056 : 19.8, 012 : 71.1, 013 = 67.68 [GPa] (Kocks et al., 1998))

155

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(a) Young’s modulus (b) Bulk modulus (c) Pole ﬁgure ofYoung’s (d) Pole ﬁgure of bulk E(n)/ERI KRI/K(n) modulus E(n)/ERI modulus KIN/Km)

Figure 12: Tetragonal Symmetry: Zircon 22251 (011 : 73.5, 033 = 46, C44 = 13.8, C66 = 16, 012 = 9, 013 : —5.4 [GPa] (Simmons and Wang, 1971))

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Figure 13: Hexagonal Symmetry: Magnesium 52582 (011 : 56.49, C33 : 58.73, C44 2 16.81, 012 : 23.16, 013 : 18.1 [GPa] (Simmons and Wang, 1971))

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Figure 14: Hexagonal Symmetry: Titanium 52743 (011 : 123.1, 0'33 : 152.9, C44 : 30.7, 012 : 99.6, 013 : 68.8 [GPa] (Simmons and Wang, 1971))

156

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Figure 15: Cubic Symmetry: Copper (011 = 168, Figure 16: Cubic Symmetry: Aluminium 10001 012 : 121.4, 044 = 75.4 [GPa] (Kocks et 31., (011 Z 25, C12 Z C44 Z 8 (Simmons 1998)) and Wang, 1971))

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_ ‚ Figure 18: Cubic Symmetry: Iron 10627 (Cu 2 Flgure 17: Cublc Symmetry: Gold 10575 (C11 : 228.09 C12 2 133.48 044 Z 110.86 [GPa] (Sim_ 185, 012 = 158, C44 2 39.7 [GPa] (Simmons and moms E’md Wang 1971)) Wang, 1971))

(a) Young’s modulus (b) Pole ﬁgure of Young’s (a) Young’s modulus (b) Pole ﬁgure of Young’s E(n)/ERI modulus E(n)/ERI E(n)/ERI modulus E(n)/ERI

Figure 19: Cubic Symmetry: Iron—Aluminium Figure 20: Cubic Symmetry: Pyrite 11391 (011 : 10676 (C11 = 170.99, C12 = 130.61, C44 2 131.7 361.88, 012 z —47.96‚ C44 : 105.49 [GPa] (Sim- [GPa] (Simmons and Wang, 1971)) mons and Wang, 1971))

157

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Address: T. Böhlke, C. Brüggemann, Otto-von—Guericke—University of Magdeburg, Institute of Mechan- ics, Postfach 4120, D-39106 Magdeburg, Germany, e—mail: boehlke®mb.uni—magdeburg.de

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