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Analytical Expressions for Deformation from an Arbitrarily Oriented Spheroid in a Half- Space U.S. Department of the Interior U.S. Geological Survey Peter F. Cervelli USGS Volcano Science Center December 12, 2013

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A spheroid is an ellipsoid having two axes (b) of equal length. ProlateOblate a bb a a > bb < a

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Spheroid Geometry For an oblate spheroid, c is a complex number

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Satisfying the Uniform Internal Pressure Boundary Condition Eshelby, 1957 Yang, 1988 Centers of Dilatation Uniform Internal Pressure Double Forces Line distribution, variable strength Uniform distribution, constant strength Strength varies quadratically

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Volterra’s Equation for Displacement as Expressed by Yang et al., 1988 Center of Dilatation Mixed Double Forces Vertical Double Force Horizontal Double Force Point Force Green’s functions= Strength function for Center of Dilatation = Strength function for Double Forces = =Spheroid Dip

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Volterra’s Equation for Displacement as Expressed by Yang et al., 1988 Center of Dilatation Mixed Double Forces Vertical Double Force Horizontal Double Force Point Force Green’s functions= = = =Spheroid Dip

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Volterra’s Equation for Displacement as Expressed by Yang et al., 1988 Center of Dilatation Mixed Double Forces Vertical Double Force Horizontal Double Force Point Force Green’s functions= a b c = a c b = =Spheroid Dip a and b are the unknown coefficients that define the strength functions

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Methodology of Solution 1.Solve integral using full space Green’s functions, assuming 90°. 2.Determine parameters ( a and b ) defining strength functions by using uniform pressure boundary condition. 3.Solve integral using half space Green’s functions and combine with strength function parameters found above.

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1. Solve Integral with Full Space Green’s Functions = 90° Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012).

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2. Determine strength function parameters Pressure at x 1 = 0 and x 3 = a Pressure at x 1 = b and x 3 = 0 Values for a 1 and b 1 depend only on geometry of spheroid and elastic constants

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2. (continued) Check Boundary Condition Plugging in the values for a 1 and b 1 derived above, we can test the uniform pressure boundary condition P ( ) = n T S n Surface normal vector as function of Stress tensor as function of

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3. Solve Integral with Half Space Green’s Functions Although Yang’s 1988 paper has typos, the presented solution is otherwise correct. My work extends Yang’s by providing: Displacement derivatives, strains, and stresses Properly handled limiting cases Proof that Yang’s method works for oblate spheroids A very good approximation to the volume/pressure relationship Error-free code, validated by Mathematica

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How Well is the Uniform Pressure Boundary Condition Satisfied? Spheroid Parameters a = 3000 m b = 2000 m Depth = 10000 m P/ = 1

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How Well is the Uniform Pressure Boundary Condition Satisfied? Spheroid Parameters a = 3000 m b = 2000 m Depth = 5000 m P/ = 1 As depth decreases the departure from uniformity increases.

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How Well is the Uniform Pressure Boundary Condition Satisfied? Spheroid Parameters a = 3000 m b = 2000 m Depth = 50000 m P/ = 1 As depth goes to infinity the deviation from uniformity goes to zero.

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The Oblate Spheroid Spheroid Parameters a = 2000 m b = 3000 m Depth = 10000 m P/ = 1 Why does this work? When a < b, then c becomes complex!

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When c is Complex, the Resultant Displacements are Still Real Consider the full space version of Volterra’s equation: Re-write the equation for the oblate case: We can then push the i into the variable of integration, , and expand the result.

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Oblate Spheroid Deformation in a Full Space But, from a practical point of view, re-writing the equations is not necessary, provided your programming language supports complex numbers – the imaginary part of the output always cancels to zero ( ± rounding errors). Real Expressions

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The Oblate Spheroid as a Goes to Zero Spheroid Parameters a = 2000 m b = 3000 m Depth = 10000 m P/ = 1

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The Oblate Spheroid as a Goes to Zero Spheroid Parameters a = 1000 m b = 3000 m Depth = 10000 m P/ = 1

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The Oblate Spheroid as a Goes to Zero Spheroid Parameters a = 500 m b = 3000 m Depth = 10000 m P/ = 1

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The Oblate Spheroid as a Goes to Zero Spheroid Parameters a = 0 m b = 3000 m Depth = 10000 m P/ = 1

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Comparison to Fialko’s Penny-shaped Crack Depth / Radius ( b ) Fialko, et al. (2001) Oblate spheroid Fialko’s solution is exact Oblate spheroid is approximate Oblate spheroid with a = 0 equivalent to the solution of Sun, 1969

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The Oblate Spheroid as a Goes to Zero Spheroid Parameters a = 0 m b = 3000 m Depth = 3000 m P/ = 1

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Pressure / Volume Relationship Volume Prior to Pressurization: Volume After Pressurization: Volume Change: Partial derivatives with respect to axes: Approximate volume change: a b aa bb and

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Pressure / Volume Relationship Derive analytical expressions for a and b in a full space: Insert into: Giving the key relationships: a b aa bb

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Comparison to Other Approximations Tiampo et al., 2000, derived an approximation of volume change presented there in equation 18: Compared to: Aspect Ratio ( a / b )

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The End Please visit: volcanoes.usgs.gov/software/spheroid for Matlab code and Mathematica notebooks

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