elastic anisotropy of hcp metal crystals and polycrystals
elastic anisotropy of hcp metal crystals and polycrystals
elastic anisotropy of hcp metal crystals and polycrystals
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IJRRAS 6 (4) ● March 2011 Tromans ● Elastic Anisotropy <strong>of</strong> Hcp Metal Crystals <strong>and</strong> Poly<strong>crystals</strong><br />
force. Consequently, referring to Fig. 1b, <strong>and</strong> (i.e. kk) are tensile strains (fractional extensions) parallel<br />
to the X1, X2 <strong>and</strong> X3 axes, respectively. The shear strains kl are due to a rotation towards the Xk axis <strong>of</strong> a line<br />
element parallel to the Xl axis. For example, 23 indicates a rotation towards the X2 axis <strong>of</strong> a line element parallel to<br />
the X3 axis, which obviously involves a rotation about the third axis X1<br />
.<br />
X 3<br />
23<br />
23= 32 = <br />
32<br />
X 3<br />
<br />
X2 X2 (a) (b)<br />
Figure. 2. Relationship between (a) tensor shear strain pairs <strong>and</strong> (b) engineering shear strain .<br />
463<br />
= 23+ 32 The shear strains are angles measured in radians. For example, if a pure shear stress (torque) is applied to cause a<br />
rotation about the X1 axis the resulting pure shear strains 23 <strong>and</strong> 32 are shown in Fig. 2. Note that the engineering<br />
shear strain is the sum <strong>of</strong> the shear strains 23 + 32. The consequences <strong>of</strong> this are discussed later in relation to<br />
compliances.<br />
2.2. Elastic Stiffness <strong>and</strong> Compliance.<br />
Elastic materials exhibit a proportional relationship between an applied stress <strong>and</strong> the resulting tensile strain ,<br />
provided the strains are smallThe resulting linear relationship is known as Hooke’s Law. In engineering, the<br />
constant <strong>of</strong> proportionality is known as the tensile <strong>elastic</strong> modulus E (Young’s Modulus) <strong>and</strong> the usual form <strong>of</strong> the<br />
relationship is given by Eq. (1) where is a uniaxial stress <strong>and</strong> is the strain elongation in the direction <strong>of</strong> the<br />
applied stress:<br />
= E 1)<br />
In fact, Equation 1 describes a uniaxial stress situation with three dimensional strains (elongation strain plus lateral<br />
strains dependent upon Poisson’s ratio) <strong>and</strong> is more formally stated in <strong>elastic</strong>ity in terms <strong>of</strong> the compliance S with <br />
as the dependent variable:<br />
= S (2)<br />
where S is the reciprocal Young’s Modulus (1/E).<br />
In analogous manner, a three dimensional stress situation with uniaxial strain is expressed in terms <strong>of</strong> the stiffness C<br />
with the stress in the direction <strong>of</strong> uniaxial strain being the dependent variable:<br />
= C <br />
2.3. Tensors <strong>and</strong> Matrices.<br />
Note that in general C ≠ E. <strong>and</strong> the <strong>elastic</strong> relationship between stresses <strong>and</strong> strains in <strong>crystals</strong> must be stated in a<br />
more generalized manner:<br />
C <strong>and</strong> S <br />
(4)<br />
ij ijkl kl<br />
kl klij ij<br />
In Eq. (4), Cijkl are stiffness constants <strong>of</strong> the crystal <strong>and</strong> Sklij are the compliances <strong>of</strong> the crystal <strong>and</strong> both are a fourth<br />
rank tensor (Wooster, 1949; Nye, 1985). Figure 1 shows there are nine forms <strong>of</strong> ij <strong>and</strong> nine forms <strong>of</strong> kl so that the<br />
generalized Eq. (4) leads to 81 Cijkl stiffness coefficients <strong>and</strong> 81 Sklij compliance coefficients which form a fourth<br />
rank tensor represented by a symmetrical 9 x 9 array <strong>of</strong> coefficients. Thus, Eq. (4) becomes:<br />
11<br />
C<br />
<br />
<br />
22<br />
<br />
C<br />
<br />
33 C<br />
<br />
23<br />
C<br />
<br />
<br />
31 C<br />
<br />
12<br />
C<br />
<br />
<br />
32<br />
<br />
C<br />
<br />
<br />
13 C<br />
<br />
<br />
21<br />
<br />
C<br />
1111<br />
2211<br />
3311<br />
2311<br />
3111<br />
1211<br />
3211<br />
1311<br />
2111<br />
C<br />
C<br />
C<br />
C<br />
C<br />
C<br />
C<br />
C<br />
C<br />
1122<br />
2222<br />
3322<br />
2322<br />
3122<br />
1222<br />
3222<br />
1322<br />
2122<br />
Full tensor notation Full tensor notation<br />
C1133<br />
C1123<br />
C1131<br />
C1112<br />
C1132<br />
C1113<br />
C1121<br />
11<br />
<br />
<br />
C<br />
<br />
2233 C2223<br />
C2231<br />
C2212<br />
C2232<br />
C2213<br />
C2221<br />
22<br />
C<br />
<br />
<br />
3333 C3323<br />
C3331<br />
C3312<br />
C3332<br />
C3313<br />
C3321<br />
33<br />
<br />
<br />
C2333<br />
C2323<br />
C2331<br />
C2312<br />
C2332<br />
C2313<br />
C2321<br />
23<br />
C<br />
<br />
<br />
3133 C3123<br />
C3131<br />
C3112<br />
C3132<br />
C3113<br />
C3121<br />
31 <br />
<br />
C1233<br />
C1223<br />
C1231<br />
C1212<br />
C1232<br />
C1213<br />
C1221<br />
12<br />
<br />
<br />
<br />
C3233<br />
C3223<br />
C3231<br />
C3212<br />
C3232<br />
C3213<br />
C3221<br />
<br />
32<br />
<br />
C<br />
<br />
<br />
1333 C1323<br />
C1331<br />
C1312<br />
C1332<br />
C1313<br />
C1321<br />
13<br />
<br />
<br />
C2133<br />
C2123<br />
C2131<br />
C2112<br />
C2132<br />
C2113<br />
C2121<br />
<br />
21<br />
11<br />
S1111<br />
S1122<br />
S1133<br />
S1123<br />
S1131<br />
S1112<br />
S1132<br />
S<br />
<br />
<br />
22<br />
<br />
S2211<br />
S2222<br />
S2233<br />
S2223<br />
S2231<br />
S2212<br />
S2232<br />
S<br />
<br />
33 S3311<br />
S3322<br />
S3333<br />
S3323<br />
S3331<br />
S3312<br />
S3332<br />
S<br />
<br />
23<br />
S<br />
2311 S2322<br />
S2333<br />
S2323<br />
S2331<br />
S2312<br />
S2332<br />
S<br />
<br />
<br />
31 S3111<br />
S3122<br />
S3133<br />
S3123<br />
S3131<br />
S3112<br />
S3132<br />
S<br />
<br />
12<br />
S1211<br />
S1222<br />
S1233<br />
S1223<br />
S1231<br />
S1212<br />
S1232<br />
S<br />
<br />
<br />
32<br />
<br />
S3211<br />
S3222<br />
S3233<br />
S3223<br />
S3231<br />
S3212<br />
S3232<br />
S<br />
<br />
<br />
13 S1311<br />
S1322<br />
S1333<br />
S1323<br />
S1331<br />
S1312<br />
S1332<br />
S<br />
<br />
<br />
21<br />
<br />
<br />
S2111<br />
S2122<br />
S2133<br />
S2123<br />
S2131<br />
S2112<br />
S2132<br />
S<br />
1113<br />
2213<br />
3313<br />
2313<br />
3113<br />
1213<br />
3213<br />
1313<br />
2113<br />
S<br />
1121<br />
S<br />
S<br />
S<br />
S<br />
S<br />
S<br />
S<br />
S<br />
2221<br />
3321<br />
2321<br />
3121<br />
1221<br />
3221<br />
1321<br />
2121<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
11<br />
22<br />
33<br />
23<br />
31<br />
12<br />
32<br />
13<br />
21<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(5)
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