Page 215 - Mechanical Behavior of Materials

216 Chapter 5 Stress–Strain Relationships and Behavior

To deal with the situation of the S ij values changing with the orientation of the x-y-z coordinate
system, it is convenient to define the values for the directions parallel to the planes of symmetry in
the material. This special coordinate system will here be identified by capital letters (X, Y, Z), as
indicated for one case in Fig. 5.14. The coefficients for Hooke’s law for an orthotropic material are

1 ν YX ν ZX
⎡ ⎤
− − 0 0 0
E X E Y E Z
⎢ ⎥
⎢ ⎥
⎢ ν XY 1 ν ZY ⎥
− −
⎢ 0 0 0 ⎥
E X E Y E Z
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ν XZ ν YZ 1 ⎥
⎢− − 0 0 0 ⎥

⎢ E X E Y E Z ⎥
S ij = ⎢ ⎥ (5.44)
⎢ 1 ⎥
0 0 0 0
⎢ ⎥
⎢ 0 ⎥
G YZ
⎢ ⎥
1
⎢ ⎥
⎢ ⎥
0 0 0 0
⎢ 0 ⎥
⎢ ⎥
G ZX
⎢ ⎥
1
⎣ ⎦
0 0 0 0 0
G XY
Examples include an orthorhombic single crystal, where α = β = γ = 90 ,but a = b = c, and
◦
fibrous composite materials with fibers in directions such that there are three orthogonal planes of
symmetry.
In Eq. 5.44, there are three moduli E X , E Y , and E Z for the three different directions in the
material. These in general have different values. There are also three different shear moduli G XY ,
G YZ , and G ZX corresponding to three planes. The constants ν ij are Poisson’s ratio constants; thus,
ε j
ν ij =− (5.45)
ε i
giving the transverse strain in the j-direction due to a stress in the i-direction. Because of the
symmetry of S ij values about the matrix diagonal,
ν ij ν ji
= (5.46)
E i E j
where i = j and i, j = X, Y,or Z. These relationships reduce the number of independent Poisson’s
ratios to three for a total of nine independent constants. It is important to remember that these
constants apply only for the special X-Y-Z coordinate system.
If the material has the same properties in the X-, Y-, and Z-directions, then it is called a cubic
material. In this case, all three E i have the same value E X , all three G ij have the same value G XY ,
and all six Poisson’s ratios have the same value ν XY . Thus, there are three independent constants.
Examples of such a case include all single crystals with a cubic structure, such as BCC, FCC, and
diamond cubic crystals. Note that there is still one more independent constant than for the isotropic
case, and the elastic constants still apply only for the special X-Y-Z coordinate system.