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Matt Zhang & Esther Lee-Varisco
} “On the Applicability of a Universal Elastic Trench Profile”
◦ J.G. Caldwell, W.F. Haxby, D.E. Karig, and D.L.
Turcotte ◦ March 31, 1976
} There have been numerous citations of using thin elastic plate theory in describing lithospheric trench flexure
} Caldwell, Haxby, Karig, and Turcotte also use thin elastic plate theory in the attempts to apply a universal trench profile ◦ Central Aleutian ◦ Kuril ◦ Northern Bonin ◦ Marianas
} Deriving the solution:
◦ a - flexural parameter; characteristic length for plate bending ◦ ε - dimensionless parameter directly proportional
to S
w = A exp [- ax(1 - f
12)] sin [a
x(1 + f
12)]
} 1-D, linear deformation with hydrostatic restoring force
◦ w – vertical deflection from equilibrium depth ◦ x – horizontal coordinate ◦ S – horizontal loading force (compression positive) ◦ - hydrostatic restoring force/
unit deflection
D dx 4d 4w
+ S dx 2d 2w+ kw = 0
k = (tm - tw) g
}
} – Young’s Modulus } h – thickness of plate
} - Poisson’s ratio
D = 12 (1 - y2)Eh 3
E = fv
y = df axialdf trans
D dx 4d 4w
+ S dx 2d 2w
+ kw = 0
a4 = k4D
f =2 kD
S
w " 0; when x " 3
Dc4 + Sc 2 + k = 0 Eigen equation
c4 + DSc2 + D
k= 0
Dk= a44
DS= a24f
c4 + a24fc2 + a4
4= 0
(c 2 + a22f) 2 + a4
4- a44f 2= 0
(c 2 + a22f) 2 + a4
4 (1 - f 2)= 0
(c 2 + a22f) 2 - [ a2
2i (1 - f 2)12
] 2 = 0
(c 2 + a22f+ a22i (1 - f 2)
12
) (c 2 + a22f- a22i (1 - f 2)
12
) = 0
(c 2 + a22f + 2i (1 + f)
12 (1 - f)
12
)
(c 2 + a22f - 2i (1 + f)
12 (1 - f)
12
) = 0
[c2 - ( a-(1 - f)
12 + i (1 + f)
12
) 2]
[c2 - ( a(1 - f)
12 + i (1 + f)
12
) 2] = 0
` c1, 2 =!( a-(1 - f)
12 + i (1 + f)
12
),
c3, 4 =!( a(1 - f)
12 + i (1 + f)
12
)
when f 1 1,
w = exp ( a-x (1 - f)
12
)[C1 cos ( ax (1 + f)
12
) + C 2 sin ( ax (1 + f)
12
)]
+exp ( ax (1 - f)
12
) [C3 cos ( ax (1 + f)
12
)+C 4 sin ( ax (1 + f)
12
)]
` x & 3; w " 0
C 3 = C 4 = 0
w = [C1 cos ( ax (1 + f)
12
) + C 2 sin ( ax (1 + f)
12
)]
exp [- ax(1 - f
12)]
` x = 0; w = 0; C 1 = 0
w = A exp [- ax(1 - f
12)] sin [a
x(1 + f
12)]
C 2 = A
when f = 1
c1, 2 =! a2 i
` w = C 1 cos a2 x+ C 2 sin a
2 x
x = 0; w = 0
w = C 2 sin a2 x
when f 2 1
w + C 1 sin ax [(1 + f)
12 - (f - 1)
12 ]
+C 2 cos ax [(1 + f)
12 - (f - 1)
12 ]
+C 3 sin ax [(1 + f)
12 + (f - 1)
12 ]
+C 4 cos ax [(1 + f)
12 - (f - 1)
12 ]
when x " 3, w Y" 0
` f 2 1 has no solution
} Outer rise of trench system = max positive deflection ◦ Coordinates: , - from corrected
bathymetric profiles
w b x b
} Writing coordinates in terms of ε and a:
w b =2
A (1 + f)12
exp [- ax b(1 - f)
12 ]
x b =(1 + f)
12
aarctan ( 1 - f
1 + f)12
} Assume ε<<1 and keeping terms that linear in ε:
} To estimate ε, assume horizontal loading force (S) is equal to the mean horizontal stress across plate thickness
x b = 4ar[1 + (r
4- 1) f]
w b =212
Ae-r
4 (1 + 4rf)
S = vxxh
} Then
} Using typical values for constants and an average stress of 10kbar, ε is about 0.3
} When assuming ε=0, ◦ Less than a 5% error in getting a from xb ◦ 25% error in getting A from wb
} Thus, horizontal loads are not important and the horizontal loading force (S) cannot be found through bathymetry
f = vxx ( hkE3 (1 - y2)
)12
} Caldwell et al found that there is a good correlation between observation and theory when the limit ε = 0.
} The coordinate equations then simplify into:
x b = 4ar= 4r( k4D)14
w b = Ae-x
a sin ax
} Using non-dimensional variables:
} Universal Oceanic Lithosphere Trench Flexure Equation:
x = x bx
w = w b
w
w = 2 sin ( 4rx) exp [ 4
r(1 - x)]
} Same methodology used for bending moment (M) and shearing force (Q) at any point on a bent plate. ◦ Non-dimensionals:
M = Dw b
Mxb2
Q = Dw b
Qxb3
Q =- 322 r 3
[cos ( 4rx) + sin ( 4
rx)] exp [ 4
r(1 - x)]
M = 82 r 2
cos ( 4rx) exp [ 4
r(1 - x)]
} Took xb and wb data from the four trenches and compared to universal equation.
} Includes the outer-rise area, which goes to about 300km seaward from trench axis.
} Determined reference line of undeflected ocean floor by using bathymetric profile and correcting for sediment thickness through reflection seismology measurements. ◦ Need in order to compare with universal equation ◦ Removed sediment from data ◦ Compensated for isostatic unloading
} Also corrected for depth of ocean floor due to age ◦ Only for Aleutian – younger than others � 62.5 – 70.5 MA compared to >110 MA
} Topographic irregularities forced Caldwell et al to compare observable areas where there was no deflection on the corrected profile to the theoretical coordinates ◦ Used the best fits to determine which zero point
and coordinates they used ◦ Some regions were harder to fit than others � Bonin Trench � Still showed good correlation between theory and
actual
} Actuals fit theory pretty well ◦ Shape and amplitude similar from trench to outer
rise } Suggests lithosphere acts elastically at
stresses as high as 9 kbar (9.2 kbars at Marianas
} Nature of lithospheric bending (convex) results in normal faulting
} Cannot have fracturing of whole lithosphere or plastic bending in region max stress seaward.
} Universal equation likely to fail for young lithospheric trenches ◦ Actual likely to be less than theory in these cases
} Universal equation formed with assumption of no horizontal force ◦ Not important <10 kbar for deflection ◦ Horizontal forces might be important for
descending plate � Might also change flexural rigidity of lithosphere
(based on the xb value differences)
A global bathymetry map from the National Oceanic and Atmospheric Administration (NOAA) shows features of the ocean floor depth across the entire Western Pacific basin, from Japan to Hawaii. (Photo: NOAA via the Boston Globe)
