ABSTRACT
The strains induced in an orthotropic material subjected to a general 3-D stress tensor, a uniform change in temperature, and/or a uniform change in moisture content was described in Chapter 4. The strain response is summarized by Equation 4.32, repeated here for convenience:
ε
ε
ε
γ γ γ
0 0 0
=
S S S
S S 2 23
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S
S S S
S
S
S
+
σ
σ
σ
τ
τ
τ
α
α
α
3 ∆T
0 0 0
0 0 0
+
∆M
(4.32) (repeated)
Equation 4.32 is valid for any orthotropic material, as long as the strain tensor, stress tensor, and material properties are all referenced to the principal 1−2−3 coordinate system. Consider the strains induced by a state of plane stress. Assuming that σ33 = τ23 = τ13 = 0, Equation 4.32 becomes
ε
ε
ε
γ γ γ
0 0 0
=
S S S
S S 2 23
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S
S S S
S
S
S
+
σ
σ
τ
α
α
α
330 0 0
0 0 0
∆T
+
∆M
0 0 0
(5.1)
Note that Equation 5.1 shows that in the case of plane stress, the out-ofplane shear strains are always equal to zero (γ23 = γ13 = 0). It is customary to write the expressions for the remaining four strain components as follows:
ε
ε
γ
σ
σ
τ
0 0
0 0
=
S S
S S
S
+
+
∆ ∆T M
α
α
β β
0 0
(5.2a)
and
ε σ σ α β33 13 11 23 22 33 33= + + +S S T M∆ ∆ (5.2b)
Equations 5.2a and 5.2b are called reduced forms of Hooke’s law for an orthotropic composite. They are only valid for a state of plane stress, and are called “reduced” laws because we have reduced the allowable stress tensor from 3-dimensions to 2-dimensions. The 3×3 array in Equation 5.2a is called the reduced compliance matrix. Note that despite the reduction from 3-to 2-dimensions, we have retained the subscripts used in the original compliance matrix. For example, the element that appears in the (3,3) position of the reduced compliance matrix is labeled S66. The definition of each compliance term is not altered by the reduction from 3-to 2-dimensions, and each term is still related to the more familiar engineering constants (E11, E22, ν12, etc.) in accordance with Equations 4.17.
