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

66 Three-dimensional problems in full, homogeneous spaces
Strain components (l = x, y, z = direction of load )
ε l (ii) = g il,(i) = k 2 S Aδ il H 0β,(i) + AB 0,(ii)l , i = x, y, z (4.90a)
εi l j = g il, j + g jl,i = k 2 S A(δ il H 0β,j + δ jl H 0β,i) + 2AB 0,i jl , i ≠ j (4.90b)
ε l Vol = g xl,x + g yl,y + g zl,z = A ∂
∂x l
[
k
2
S H 0β + ˆ∇ 2 B 0
]
(4.90c)
a)Strains for loads in the plane, l = x, y
(( )
2
ε l xx = γ l A
r B 2 − k 2 S k β H 1β δ xl + 1 )
r B 2 − γx 2 B 3
(( )
2
ε l yy = γ l A
r B 2 − k 2 S k β H 1β δ yl + 1 )
r B 2 − γy 2 B 3
ε l zz = γ l k 2 z AB 1
ε l xy = 2A (( 1
r B 2 − 1 2 k2 S k β H 1β
)
(δ xl γ y + δ yl γ x) − γ x γ y γ l B 3
)
ε l xz = 2i k z A(( 1
r B 1 − 1 2 k2 S H 0β
ε l yz = 2i k z A(( 1
r B 1 − 1 2 k2 S H 0β
)
δ xl − γ x γ l B 2
)
)
δ yl − γ y γ l B 2
)
εVol l = γ l A
(−k 2 S k β H 1β + k 2 z B 1 + 4 )
r B 2 − B 3
(4.91a)
(4.91b)
(4.91c)
(4.91d)
(4.91e)
(4.91f)
(4.91g)
b)Strain for axial loads, l = z
)
1
εxx z = i k z A(
r B 1 − γx 2 B 2
ε z yy = i k z A( 1
r B 1 − γ 2 y B 2
)
εzz z = i k z A ( −k 2 S H 0β + k 2 z B )
0
εxy z =−2ik z γ x γ y AB 2
(
εxz z = 2 γ x A − 1 )
2 k2 S k β H 1β + k 2 z B 1
(
ε z yz = 2 γ y A − 1 )
2 k2 S k β H 1β + k 2 z B 1
εVol z = i k z A
(−k 2 S H 0β + k 2 z B 0 + 2 )
r B 1 − B 2
(4.92a)
(4.92b)
(4.92c)
(4.92d)
(4.92e)
(4.92f)
(4.92g)