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### hw5

Course: GEOL 446, Fall 2008
School: Maryland
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work Home 5: Rock and mineral physics Assigned: 11/05/2007 Due: 11/16/2007 1. The constitutive equation for rocks undergoing viscous creep can be generally expressed as: =A n ar 2 O H dp exp Q , RT (1) where is the strain rate, is the stress in 106 Pa, d is grain size in 106 m (micron), Q is the activation enthalpy for creep, T is the temperature in K, R is the universal gas constant, and n and p are...

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work Home 5: Rock and mineral physics Assigned: 11/05/2007 Due: 11/16/2007 1. The constitutive equation for rocks undergoing viscous creep can be generally expressed as: =A n ar 2 O H dp exp Q , RT (1) where is the strain rate, is the stress in 106 Pa, d is grain size in 106 m (micron), Q is the activation enthalpy for creep, T is the temperature in K, R is the universal gas constant, and n and p are dimensionless exponents. The values of this parameters are supplied in the table. Assume the grain size (average size of the crystals) of the aggregate has a uniform value of 15 microns, T = 1573 K, and R = 8.314 J/mol/K. Such viscous deformation of rocks is called creep. If the rock creeps by the motion of point defects alone, the deformation mechanism is called diusion creep. If the creep takes place by motion of dislocations, its called dislocation creep. (a) The creep mechanism producing a faster strain rate for any given stress is called a dominant creep mechanism. Show that for a dry rock the cross-over stress at which the dominant creep mechanism switches is given by: Adif 1 0 = p exp Adis d Qdis Qdif RT 1 ( n1 ) , (2) where subscripts dif and dis indicate the relevant parameters from equation (1) for diusion and dislocation creep constitutive relations, m is the stress exponent for dislocation creep, and p is the grain size exponent for diusion creep. 10 pts (b) Calculate the cross-over stress from the parameters given in the table, for a grain size of 15 microns at a of temperature 1573 K 10 pts 2. The viscosity of clinopyroxenite (a lower crustal rock) is given by: = 0 d d0 1 3 exp Q , RT (3) Table 1: Parameters for Problem 1 Diusion creep Dislocation creep A (micronp /Mpan /s) n p r (dry) r (wet) Q (J/mol) 6.97 108 1 3 0 1 375 103 5.11 104 3.5 0 0 1 530 103 where 0 is a constant, d is the grain size of the rock, d0 is a constant reference grain-size, Q is activation energy for creep, R is the universal gas constant, and T is the temperature of deformation. We can use this value as a constrain for the height of topography that can be supported by the rocks. For a topography of height h and a slope h/x, we dene the strength of the rock c as the stress at which the strain rate assumes a very small constant value . This strength can be shown to be: c = c gh h , x (4) where c is the density of the crustal rock. (a) Show that if density and grain size of the rock remains constant, then at a constant temperature, for a constant slope, the maximum heights of elevations corresponding to a dry and a wet crust are related to the viscosities as: dry hdry = . hwet wet 5pts (5) (b) For the parameters listed in Table 2, for a grain size of 3 mm, at a temperature of 1373 K, calculate the ratio of maximum heights of topography that can be supported by a dry and wet crust, given by equation (5). 5 pts 2 Table 2: Parameters for the viscosity temperature relationship in Problem 2. o (Pa s) 1020 107 do (m) 10 10 Q (kJ/mol) 760 340 Dry Wet 3
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