102
4 Stress–Strain Relations
∂
2 U
∂ε i ∂ε j
= D i j
(4.10)
The free index in Eq. (4.9) can be changed so that
∂U
∂ε j
= σ j = D ji ε i
(4.11)
Differentiating Eq. (4.11) with respect to ε i , then,
∂
2 U
∂ε j ε i
= D ji
(4.12)
Hence, Eqs. (4.10) and (4.12) are equal, or D i j = D ji
which implies that D i j is symmetric. Then most general form of the stiffness
matrix or array becomes
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11 D 12 D 13 D 14 D 15 D 16
D 12 D 22 D 23 D 24 D 25 D 26
D 13 D 23 D 33 D 34 D 35 D 36
D 14 D 24 D 34 D 44 D 45 D 46
D 15 D 25 D 35 D 45 D 55 D 56
D 16 D 26 D 36 D 46 D 56 D 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.13)
Or
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11 D 12 D 13 D 14 D 15 D 16
D 22 D 23 D 24 D 25 D 26
D 33 D 34 D 35 D 36
D 44 D 45 D 46
D 55 D 56
D 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.14)
Further, a material that exhibits symmetry with respect to three mutually orthogonal planes is called an “orthotropic” material. If the xy, yz and zx planes are
considered planes of symmetry, then Eq. (4.13) reduces to 12 elastic constants as
below.
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11 D 12 D 13 0 0 0
D 21 D 22 D 23 0 0 0
D 31 D 32 D 33 0 0 0
0 0 0 D 44 0 0
0 0 0 0 D 55 0
0 0 0 0 0 D 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.15)