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

1.4 Strains, stresses, and the elastic wave equation 13
σ z
σyz
σxz
dz
σ xy
σy
σzy
σzx
σyx
σx
Figure 1.4: Stresses (and strains) in Cartesian coordinates.
dx
dy
1.4 Strains, stresses, and the elastic wave equation
1.4.1 Cartesian coordinates
Strains
ε x = ∂u x
∂x ,
ε yz = ∂u z
∂y + ∂u y
∂z ,
ε y = ∂u y
∂y ,
ε zx = ∂u x
∂z + ∂u z
∂x ,
ε z = ∂u z
∂z
ε xy = ∂u y
∂x + ∂u x
∂y
ε zy = ε yz , ε zx = ε xz , ε yx = ε xy
(1.46)
In matrix notation, this can be written as
u = [u x u y u z] T = displacement vector (1.47)
ε = [ε x ε y ε z ε yz ε xz ε xy] T = strain vector (1.48)
ε = Lu = strain–displacement relation (1.49)
⎧
⎫
∂
∂ ∂
0 0 0
∂x
∂z ∂y
⎪⎨
L T ∂ ∂ ∂
⎪⎬
= 0 0 0
(1.50)
∂y ∂z ∂x
∂ ∂ ∂
⎪⎩ 0 0
0 ⎪⎭
∂z ∂y ∂x
which can be abbreviated as
∂
L = L x
∂x + L ∂
y
∂y + L ∂
z
∂z
(1.51)