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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.[1] 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.[2]


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.

Input Value (e.g.) Description
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
rho0 4.000e+03 reference density
visc 1.000e+24 dynamic viscosity
grav 1.000e+01 gravitational acceleration
texpn 2.500e-05 volume coefficient of thermal expansion
tcond 4.000e+00 thermal conductivity
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

Comparison study between ISV model and Power-law creep 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.[3] 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. TERRA2D mantle convection simulation using ISV material model


  1. Dziewonski, A. M., Anderson, D. L., “Preliminary reference Earth model”, Physics of the Earth and Planetary Interiors, Vol. 25 (1981), pp. 297-356.
  2. Baumgardner, J. R., “A Three-dimensional Finite Element Model for Mantle Convection”, Ph.D. thesis, UCLA
  3. 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.


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