4.2 Linear Elasticity—Generalized Hooke’s Law
103
Also, due to orthotropic symmetry, the number of material constants for a linear
elastic orthotropic material reduces to 9 as shown below.
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11 D 12 D 13 0 0 0
D 22 D 23 0 0 0
D 33 0 0 0
D 44 0 0
D 55 0
D 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.16)
Now, in the case of a transversely isotropic material, the material exhibits a rationally elastic symmetry about one of the co-ordinate axes, x, y and z. In such case, the
material constants reduce to 8 as shown below.
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11
D 12 D 13
0
0 0
D 22 D 23
0
0 0
D 33
0
0 0
1
2
(D 11 − D 12 ) 0 0
Symmetry
D 55 0
D 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.17)
Further, for a linearly elastic material with cubic symmetry for which the properties along the x-, y- and z-directions are identical, there are only 3 independent
material constants. Therefore, the matrix form of the stress–strain relation can be
expressed as:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
σ y
σ z
τ xy
τ yz
τ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
D 11
D 12 D 12 0 0 0
D 11 D 12 0 0 0
D 11 0 0 0
D 44 0 0
Symmetry
D 44 0
D 44
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
ε x
ε y
ε z
γ xy
γ yz
γ zx
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(4.18)
Isotropy
For a material whose elastic properties are not a function of direction at all, only
two independent elastic material constants are sufficient to describe its behaviour
completely. This material is called “isotropic linear elastic”. The stress–strain relationship for this material is thus written as an extension of that of a transversely
isotropic material as shown below.