Geophysics includes space, atmosphere, ocean, and interior of the planet. However, investigation of the interior of the solid Earth is the heart of geophysics. The earth's mantle comprises solid rock minerals which are mainly olivine, pyroxene, garnets and their polymorphs due to phase transformations. The main exploration for the Earth's mantle has been performed since investigations of the ocean bottom. Geophysical research was mainly focused on seismicity due to limitations to access to the mantle. In 1980, Dziewonski and Anderson performed an important study using a seismological method to explore the inner earth. This PREM (Preliminary Reference Earth Model) study showed that the mantle is made of a solid material that has viscosity. Moreover, sophisticated GPS technology that allows observation of the oceanic floor demonstrated that the earth’s plates are moving with a velocity of approximately 10 cm/yr. In the study of mantle convection, computational methods are significantly important. Hence, computational methods for mantle convection problem such as Finite Element Model have been developed. One of the powerful Finite Element codes for mantle convection problem was TERRA model developed by John Baumgardner in 1983.
TERRA is one of the most sophisticated numerical model for the mantle dynamics in the world. It is able to model most physical phenomena that are expected today by geophysics community. Some examples of inputs are described in following table.
|ibc||1||boundary condition index|
|itlimit||10||max number of multigrid iterations|
|convtol||3e-2||convergent tolerance for multigrid solver|
|istrt||1||flag specifying type of initialization|
|itmax||1000||maximum number of time steps|
|step0||1.500e-02||initial time step fraction of advection limit|
|stepmax||2.500e-01||maximum time step fraction of advection limit|
|dtdump||2.738e-03||time interval in years between restart dumps|
|ieos||1||index specifying EOS type--0 for Boussinesq case|
|texpn||2.500e-05||volume coefficient of thermal expansion|
|sheat||1.000e+03||specific heat at constant volume|
|hgen||5.000e-12||specific radiogenic heat production rate|
|tb(1)||3.500e+03||temperature at bottom boundary|
|tb(2)||3.000e+02||temperature at top boundary|
|cl410||1.000e+06||Clapeyron slope (dp/dT) for 410 km phase transition|
|cl660||-1.000e+06||Clapeyron slope (dp/dT) for 660 km phase transition|
|vscmin||1.000e-11||minimum value for viscosity variation|
|vscmax||1.000e+02||maximum value for viscosity variation|
|vscscl||0.200e-00||scaling factor for horizontal viscosity activation|
|yldstrs||3.000e+08||viscous yield stress in top boundary layer|
|pwrlawn||0.000e+00||power-law exponent (zero turns off this feature)|
|pwrlawsr||0.000e-14||transition strain rate for power-law rheology|
|isvf||1||flag for turning on ISV model|
|itvisc||0||flag for tensor viscosity (1 for tensor viscosity)|
|lvz||1||flag for presence of LVZ|
|zblvz||4.000e+05||depth to the bottom boundary of LVZ|
|dvlvx||1.000e-03||viscosity reduction factor for LVZ|
|tm0||1.300e+03||surface partial melting temperature|
|dtmdz||1.000e-03||partial melting temperature gradient|
|drho||0.000e-02||density increase factor for heavy particles|
|dvis||0.000e+02||viscosity increase factor for heavy particles|
|dhgen||0.000e+02||heating increase factor for heavy particles|
|plate||0||flag specifying presence of lithospheric plate|
Internal State Variable Plasticity-Damage Model
In modeling mantle rock materials, the geophysics community mainly uses the creep model. Although creep gives approximate solutions, this method does not exactly capture the various rock’s material behavior. Alternatively, the engineering society has developed a material modeling technique to satisfy the needs of industry. Integrated Computational Material Engineering (ICME) has been developed as one of the effective material modeling methods. In many modeling studies of several materials including steels, the Internal State Variable (ISV) theory through ICME has shown good accuracy and precision. A recent study was performed by Sherburn et al. showing that the Internal State Variable model captures the rock’s material behavior much more effectively than the power-law model (creep model)(see right figure). In this context, mantle rock material modeling with ICME method is expected to produce a more accurate mantle's mechanical behavior. One superior property of ISV model is that it can track material's deformation history since it deals with internal variables in the frame of thermodynamics. Through ISV model, several internal state variables can be captured such as recrystallization effect, grain growth and damage, as well as density of statistically stored and geometrically necessary dislocations. These internal variables are now well accepted as factors play a significant role on mantle rock's mechanical behaviors when the it is deformed. Furthermore, ISV model unifies plasticity and creep behaviors in one model. This is important to capture real materials' deformations.
- ↑ Dziewonski, A. M., Anderson, D. L., “Preliminary reference Earth model”, Physics of the Earth and Planetary Interiors, Vol. 25 (1981), pp. 297-356.
- ↑ Baumgardner, J. R., “A Three-dimensional Finite Element Model for Mantle Convection”, Ph.D. thesis, UCLA
- ↑ Sherburn, J. A., Horstemeyer, M. F., Bammann, D. J., Baumgardner, J. R., “Application of the Bamman inelasticity internal state variable consititutive model to geological materials”, Geophys. J. Int., Vol. 184 (2011), pp. 1023-1036.