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Department of Earth SciencesKFUPM
Gravity Modeling
Introduction to GeophysicsIn
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Highest peaks on the planet
Previous Lecture
Models of IsostasyAiry Model Pratt Model
Local Isostasy Lithostatic Pressure
Mathematical Background of Airy ModelMathematical Background of Pratt Model
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Recall: Elasticity
Elasticity: ability of materials to return to original shape, size and position after removal of stressThe lithosphere behaves elastically, therefore it bends when forces (loads) are applied to it.This behavior is called flexure, which is a means of distributing a mass excess (a load) over a large area in order to achieve isostatic equilibrium.
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The type of isostatic compensation depends on the flexural rigidity of the supporting material.
The models (Pratt/Airy) are based on the local isostasy where no rigidity, compensation is directly below the load.
Local Isostasy & flexureIn
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Regional Isostasy & flexure
Material with rigidity are flexed, distributing the load over a broader region.
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Local I sostasy
Regional I sostasy
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Flexural Rigidity (D)
A thin diving board (small elastic thickness) has low flexural rigidity. A thin, weak board (small D) bends greatly, especially near the diver.
D is the resistance to bending., thus depends on the elastic thickness of each board.
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Flexural Rigidity (D)
A thick board (large elastic thickness) of the same material behaves more rigidly (D high); the diver causes a smaller deflection.In
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Flexural Modeling
X=horizontal distance from the load to a point on the surface of the plate
Ρa = density of the material above the plate
Ρb = density of the material below the plate
q(x)=load applied to the top of the plate at x.
Parameters for 2 dimensional model of a plate flexed by a linear load. Both plate and load extend infinitely in and out of the page.
)(4
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xqgwdx
wdD ab
Turcotte and Schubert (1982)
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D =the (flexural) rigidityW= vertical deflection of plate at x
Flexural Modeling
If the plate has no rigidity, D=0 and we get
Here the load q(x)=ρ g h
We’ll also set P=0The flexural equation is:
This is just the expression for Airy isostasy
So if the flexural rigidity is zero, we get isostasy
lab hw )(
ghgwdx
wdD lab
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Positions of depressions and bulges formed on the surface of a flexed plate. A strong plate has shallow but wide depression.
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Large D (Strong Plate) ~ Small Amplitude Deflection
The depressions and peripheral bulges have larger amplitudes on a weak plate, but are closer to the load.
Small D (Weaker Plate) ~ High Amplitude Deflection
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A very weak plate collapses into local isostatic equilibrium.
D=0 ~ Collapse for Local IsostasyIn
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3. Plates with no strength collapse into local isostatic equilibrium.
1. A strong lithospheric plate (large D) will have a small amplitude deflection (small w), spread over a long wavelength;
2. A weak lithospheric plate (small D) has large deflection (large w), but over smaller wavelength;
)(4
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xqgwdx
wdD wm
Some important concepts from the above equation:
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Examples of Lithospheric Flexure
A flexural bulge and depression (trench) develop as the downgoing plate is flexed at a subduction zone.
The weight of a mountain range causes adjacent depressions that fill with sediment (foreland basins).
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Example: Application of Flexure Modelling
Homework, Due to May 20
Given the following exercises of your handout:8.28.38.8Try to use one of those provided Excel Programs if possible in solving your problems from the link:http://www.mtech.edu/clink/Home/Classes/Geop3020/chapter6.htm
Then, e-mail your homework which is solved under Excel to me: [email protected].
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