STRUCTURAL GEOLOGY
Lectures 35-36
Models for Ductile Failure
(Power law creep)

Ductile deformation occurs if the rock under stress does not loose its strength by means of a brittle failure.  This behavior is illustrated using stress-strain curves from rock deformation experiments (Fig. 35-1).  Each test is run at constant strain rate which means that in a triaxial test the piston is advanced into the cylindrical rock sample at a constant rate.  The initial behavior of the rock is elastic for which a linear stress-strain curve is shown.  Brittle failure causes a complete lose of strength.  Ductile flow shows that the strength is maintained during continuos straining of the sample.

(Fig. 35-1)

Percent ductility is a measure of the amount of strain that a rock undergoes before losing strength.  Ductility varies with lithology.  The strongest and most brittle of the rocks is a quartzite or silica cemented sandstone.  In contrast, halite is very weak and will undergo large amounts of ductile flow without brittle failure.  Figure 35-2 shows a variety of rocks and their relative ductilities as a function of depth of burial within the earth.  Starting with the most brittle there is silica cemented sandstone, dolomite, calcite-cemented sandstone, shale, limestone, and halite.
Initially constant strain-rate tests were most convenient for laboratory experiments.   However, conditions within the crust of the earth closely resemble constant stress tests such as that shown in Figure 35-3.  This is so because the differential stress within the crust does not change rapidly with time.
The most interesting characteristic of constant stress tests is that steady state creep is achieved.  This is a state where the rock exhibits no change of strain with time.  Steady state creep occurs during the linear portion of the strain-time curve shown in Figure 35-3.

(Fig. 35-2)

(Fig. 35-3)

Various mechanisms of ductile flow were introduced during the previous lecture.  Each of these mechanisms can be dominate during the creep of rocks.  The dominate mechanism depends on the temperature and differential stress affecting the rock.  Figure 35-4 is a plot of the temperature of deformation verses stress.  The temperature T is mormalized to the melting temperature (Tm ) by the ratio T/Tm.  The stress of deformation is normalized by the shear modulus of the rock (µ).
Various creep mechanisms include the following:
Nabarro-Herring Creep - bulk diffusion of point vacancies down a stress gradient.  Recall that a point vacancy is a single missing atom.
Anelastic Creep -  below a critical shearing stress for large dislocation movement mechanisms as Coble Creep take place.
Low-Temperature Creep - includes multiplication and glide of dislocations.  Stresses have to be reasonably high to cause this type of creep.
High-Temperature Creep  -  at higher temperatures edge dislocations can climb and screw dislocations can cross slip.

(Fig. 35-4)

An equation for Nabarro-Herring Creep gives the strain rate (é) in terms of stress s

	é  =  (aDVas)/kTL2.                                                               

a is a geometric factor;  L is the diameter of the grain;  D is the diffusion coefficient;  Va is the atomic volume;  T is the temperature;  s is stress; and k is the Boltzman number.  Steady creep flow of rock materials can also be modeled using the Weertman Equation

	é  =  Aexp(-Qc/RT) f(s)                                                  

where T = temperature,  Qc = creep activation energy, and R = gas constant.  This equation can be evaluated using a plot of  log s versus -log (é) by rearranging the above equation

	log(é/A)  =   Qc/RT  +   flogs                                         

where f is the slope of the lines given in Fig. 35-5 and Qc is determined as the slope of the plot of log é versus 1/T at constant stress.    Experiments show that creep rate at high temperature is a strong function of stress.  Figure show the functional relationship between é and s in terms of three power law equations.  Power law creep is non linear flow

	é  =  asn                                                                      

whereas Newtonian viscosity is linear flow

	é =  as

Figure 35-7 shows a deformation mechanisms map for calcite (limestone).  Given a stress and temperature the diagram in Fig.  35-7 shows which of six mechanisms are favored.  These mechanisms include cataclasis, pressure solution, dislocation glide, dislocation climb, Coble creep, and Nabarro-Herring creep

(Fig. 35-5)

.

(Fig. 35-6)

(Fig. 35-7)