Page 214 - Mechanical Behavior of Materials

Section 5.4 Anisotropic Materials 215

⎧ ⎫ ⎡ ⎤ ⎧ ⎫
⎪ε x ⎪ S 11 S 12 S 13 S 14 S 15 S 16 ⎪σ x ⎪
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎢ ⎥ ⎪ ⎪
⎪ε y ⎪ S 12 S 22 S 23 S 24 S 25 S 26 ⎪σ y ⎪
⎪ ⎪ ⎢ ⎥ ⎪ ⎪
⎨ ⎬ ⎨ ⎬
ε z ⎢ S 13 S 23 S 33 S 34 S 35 S 36 ⎥ σ z
= ⎢ ⎥ (5.42)
⎢ ⎥
⎪γ yz ⎪ S 14 S 24 S 34 S 44 S 45 S 46 ⎪τ yz ⎪
⎪ ⎪ ⎢ ⎥ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎣ ⎦ ⎪ ⎪
⎪γ zx ⎪ S 15 S 25 S 35 S 45 S 55 S 56 ⎪τ zx ⎪
⎪ ⎪ ⎪ ⎪
⎩ ⎭ ⎩ ⎭
γ xy S 16 S 26 S 36 S 46 S 56 S 66 τ xy
General anisotropy is considerably more complex than the isotropic case. Not only are there a large
number of different materials constants S ij , but their values also change if the orientation of the
x-y-z coordinate system is changed.
In the isotropic case, the constants do not depend on the orientation of the coordinate axes,
and most of the constants are either zero or have the same values as other ones. For example,
γ xy = τ xy /G, so that all of the S ij in the γ xy row of the matrix are zero except S 66 = 1/G.This
contrasts with the situation for highly anisotropic materials, where γ xy is the sum of contributions
due to all six stress components. The matrix of S ij coefficients, specialized to the isotropic case as
given by Eq. 5.26, is
1 ν ν
⎡ ⎤
− − 0 0 0
E E E
⎢ ⎥
⎢ ⎥
ν 1 ν
⎢ ⎥
⎢ − − 0 0 0 ⎥
E E E
⎢ ⎥
⎢ ⎥
ν ν 1
⎢ ⎥
− −
⎢ 0 0 0 ⎥
⎢

⎢ E E E ⎥
S ij = ⎥ (5.43)
⎢ 1 ⎥
0 0 0 0 0
⎢ ⎥
G
⎢ ⎥
⎢ ⎥
⎢ 1 ⎥
0 0 0 0 0
⎢ ⎥
G
⎢ ⎥
⎢ ⎥
⎣ 1 ⎦
0 0 0 0 0
G
where G is given by Eq. 5.28, so that there are only two independent constants.
In the most general form of anisotropy, each unique S ij in Eq. 5.42 has a different nonzero
value. The matrix is symmetrical about its diagonal in such a way that there are two occurrences of
each S ij , where i = j, so that there are 21 independent constants. An example of a situation with
this degree of complexity is the most general form of a single crystal, called a triclinic crystal, where
a = b = c and α = β = γ , these being the distances and angles defined in Fig. 2.9.
5.4.2 Orthotropic Materials; Other Special Cases
If the material possesses symmetry about three orthogonal planes—that is, about planes oriented
◦
90 to each other—then a special case called an orthotropic material exists. In this case, Hooke’s
law has a form of intermediate complexity between the isotropic and the general anisotropic cases.