Page 216 - Mechanical Behavior of Materials

Section 5.4 Anisotropic Materials 217

For example, for a single crystal of alpha (BCC) iron, E X = 129 GPa. However, consider the
direction that is a body diagonal of the cubic unit cell—that is, along the direction of one of the
arrows in Fig. 2.23(a). The elastic modulus in this direction is E [111] = 276 GPa, which is about
twice as large as E X . Considering E values for all possible directions, these two are the largest and
smallest that occur. Polycrystalline iron is isotropic, and the value of E ≈ 210 GPa that applies is
the result of an averaging from randomly oriented single crystals. As expected, this E is between
the extreme values E X and E [111] for single crystals.
Another special case is a transversely isotropic material, where the properties are the same for
all directions in a plane, such as the X-Y plane, but different for the third (Z) direction. Here there
are five independent elastic constants: E X and ν XY for the X-Y plane, with Eq. 5.28 giving the
corresponding shear modulus G XY , and also E Z , ν XZ , and G ZX . An example is a composite sheet
material made from a mat of randomly oriented and intertwined long fibers.

5.4.3 Fibrous Composites

Many applications of composite materials involve thin sheets or plates that have symmetry
corresponding to the orthotropic case, such as simple unidirectional or woven arrangements of
fibers, as in Fig. 5.14(c). Also, most laminates (Fig. 3.26) have overall behavior that is orthotropic.
For plates or sheets, the stresses that do not lie in the X-Y plane of the sheet are usually small,
so that plane stress with σ Z = τ YZ = τ ZX = 0 is a reasonable assumption. Although strains ε Z still
occur, these are not of particular interest, so Hooke’s law can be used in the following reduced form
derived from Eq. 5.44:

1 ν YX
⎡ ⎤
− 0
⎧ ⎧ ⎫
⎫
⎪σ X ⎪
⎢ E X
⎪ε X ⎪ E Y ⎥ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎢ ⎥ ⎪ ⎪
⎨ ⎬ ⎨ ⎬
⎢ ν XY 1 ⎥
ε Y = ⎢ − 0 ⎥ σ Y (5.47)
E X E Y
⎢ ⎥
⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎢ ⎥ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎩ ⎭ ⎦ ⎪ ⎪
⎣ 1
γ XY ⎩ ⎭
0 0 τ XY
G XY
Here, capital letters still indicate that the stresses, strains, and elastic constants are expressed only for
directions parallel to the planes of symmetry of the material. (Stresses and strains in other directions
can be found by using transformation equations or Mohr’s circle, as discussed later in Chapter 6, or
in textbooks on mechanics of materials.) Equation 5.46 applies, so that
ν YX ν XY
= (5.48)
E Y E X
with the result that four independent elastic constants are being employed out of the total of nine.
Values for some composite materials with unidirectional fibers are given in Table 5.3.
Values of these constants can be obtained from laboratory measurements, but they are also
commonly estimated from the separate (and generally known) properties of the reinforcement and
matrix materials. The topic of so estimating elastic constants is rather complex and is considered in
detail in books on composite materials, such as Gibson (2004). In the discussion that follows, we
will consider only the simple case of unidirectional fibers in a matrix.