EN0175
11 / 07 / 06
Chap. 6
Boundary value problems in linear elasticity
Equilibrium equations:
i
i
j
ij
u
f
&
&
ρ
σ
=
+
,
(1)
Kinematic equations:
(
i
j
j
i
ij
u
u
,
,
2
1
+
=
ε
)
(2)
Hooke’s law:
ij
kk
ij
ij
E
E
δ
σ
ν
σ
ν
ε
−
+
=
1
or
ij
kk
ij
ij
δ
λε
με
σ
+
=
2
(3)
For the most general problems of linear elasticity, you have to solve a system of 15 independent
equations with 15 unknown variables.
There are two kinds of solution techniques:
1. Displacement based solution methods
2. Stress based solution methods
1. Displacement based method (Navier displacement equation)
Eliminate strain by inserting (2) into (3):
(
)
ij
k
k
i
j
j
i
ij
u
u
u
δ
λ
μ
σ
,
,
,
+
+
=
(4)
Eliminate stress by inserting (4) into (1):
(
)
(
)
i
i
ki
k
jj
i
i
i
ij
kj
k
ij
j
jj
i
u
f
u
u
u
f
u
u
u
&
&
&
&
ρ
λ
μ
μ
ρ
δ
λ
μ
=
+
+
+
⇒
=
+
+
+
,
,
,
,
,
(5)
Equation (5) is called Navier displacement equation. The vector form of this equation is
(
)
u
f
u
u
&
&
v
v
v
v
ρ
λ
μ
μ
=
+
⋅
∇∇
+
+
∇
2
Similar to the concept of splitting stress/strain into a volumetric part and a deviatoric part, the
displacement field can also be decomposed into a dilatational part plus a distortional part,
ψ
ϕ
v
v
×
∇
+
∇
=
u
where
ϕ
∇
is the dilatational term and
ψ
v
×
∇
is the distortional term.
The volume change
u
u
V
V
k
k
kk
v
⋅
∇
=
=
=
Δ
,
ε
. If
ψ
v
v
×
∇
=
u
,
0
=
×
∇
⋅
∇
=
⋅
∇
ψ
u
v
, which
means the distortional part causes no volume change.
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 Spring '06
 HUAJIANGAO
 airy stress function, elastic wave, Navier displacement equation

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