12.005 Lecture Notes 20
Plates
Rock rheology → function of T; T increases with depth.
Thin region near surface remains elastic on geologic times. Below this, mantle behaves as
a viscous fluid.
Plate theory:
Assume plate thickness h L, with L a characteristic length scale
3-D equations simplify
Applications:
Seamount Loading
Subduction Zone
Fore-arc bulge
Trench
Folding
λ
Laccolith
Magma
Figure 20.1
Figure by MIT OCW.
Main Feature
y
x
h
z
Fibers compressed
}
Neutral surface
}
Fibers extended
Figure 20.2
Figure by MIT OCW.
“Fiber stresses” large compare to tractions applied to surfaces.
Neutral surface – smooth and centered
Kirchoff’s assumption – approximate, but useful.
σ yy = σ yx = σ yz = 0
h
2
σ xx , σ zz linearly through − ≤ y ≤
h
2
or
every straight line originally perpendicular to neutral
plane remains straight and perpendicular
More accurate formulations lead to nonlinear coupled equations, not solvable analytically.
To derive plate equations – consider the following figures:
Moments
Surface Loads
q
M
Shear Forces
End Loads
V
Figure 20.3
Figure by MIT OCW.
x
x+dx
q(x)
x
w+dw
w
P
M
P
V+dV
V
Figure 20.4
Figure by MIT OCW.
V – shear force / unit length
P – horizontal force / unit length
M – moment / unit length
q – force / unit area
M+dM
Force balance – vertical
qdx + dV = 0 ⇒
dV
= −q
dx
Torque balance (+ counterclockwise)
dM − Pdw − Vdx = 0
dM
dw
=V + P
dx
dx
Next – relate M to w.
y
M
h/2
o
-h/2
Figure 20.5
Figure by MIT OCW.
h/2
M=
∫
σ xx ydy
−h / 2
For plane stress,
σ xx =
E
exx
(1 −ν 2 )
Bending the plate causes fiber strains
δl y
exx = − =
l
R
φ
R
y
δl = -yφ = -y l
R
l
φ
Figure 20.6
Figure by MIT OCW.
Now – let l get small
d 2w
ε xx = − y 2
dx
dα
R
α+dα
l
α = - dw
dx
dx
Figure 20.7
Figure by MIT OCW.
dw
Then, substituting
h/2
E d 2w
M =−
y2d y
2
2
∫
(1 −ν ) dx − h / 2
M =−
Eh3 d 2 w
12(1 −ν 2 ) dx 2
or
M = −D
where D ≡
d 2w D
=
dx 2 R
Eh3
12(1 −ν 2 )
(flexural rigidity)
Substituting
D
d 4w
d 2w
=
q
(
x
)
−
P
dx 4
dx 2
This equation is called plate equation.