Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

14 Fundamentals
where by inspection
⎧ ⎫ ⎧ ⎫ ⎧ ⎫
1 0 0
0 0 0
0 0 0
0 0 0
0 1 0
0 0 0
⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬ ⎪⎨ ⎪⎬
0 0 0
0 0 0
0 0 1
L x =
, L y =
, L z =
0 0 0
0 0 1
0 1 0
0 0 1
0 0 0
1 0 0
⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭ ⎪⎩ ⎪⎭
0 1 0
1 0 0
0 0 0
(1.52)
Stresses
σ = [σ x σ y σ z σ yz σ xz σ xy] T = stress vector (1.53)
σ = Dε = constitutive law (1.54)
For a fully anisotropic medium, the symmetric constitutive matrix is
⎧
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
D = 13 d 23 d 33 d 34 d 35 d 36
(1.55)
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
whereas for an isotropic medium, this matrix is
⎧
⎫
λ + 2µ λ λ 0 0 0
λ λ+ 2µ λ 0 0 0
⎪⎨
⎪⎬
λ λ λ+ 2µ 0 0 0
D =
0 0 0 µ 0 0
0 0 0 0 µ 0
⎪⎩
⎪⎭
0 0 0 0 0 µ
(1.56)
in which λ = Lamé constant, and µ = shear modulus. In this case, the stress–strain relationship
is
σ j = 2µε j + λε vol ,
ε vol = ε x + ε y + ε z
σ yz = µε yz , σ zx = µε zx , σ xy = µε xy
σ zy = σ yz , σ zx = σ xz , σ yx = σ xy
(1.57)
The stresses, inertial loads, and body loads satisfy the dynamic equilibrium equation
b − ρü + L T σ = 0 (1.58)