Page 77 - Microtectonics

3.14 · Flow Laws and Deformation Mechanism Maps 65

Box 3.11 Flow laws
–1
–1
The following flow laws are commonly quoted in the literature, R – gas constant [J mol K ]
–1
and have been used to construct deformation mechanism maps k – Boltzmann constant [J K ]
–2
in Fig. 3.43. µ – shear modulus [Nm ]
For one of the simplest models, bulk diffusion-controlled dis- d – grain size [m]
location creep (also known as Weertman creep) the flow law is: W – grain boundary thickness [m]
–1
3
V – molar volume of the solid [m mol ]
σ/µ – normalised shear stress [dimensionless number]
Parameters for Fig. 3.43
For Coble creep:
For the deformation mechanism maps in Fig. 3.43 the following
parameters have been used:
= 1550 K (melting temperature of quartz in the presence
T m
and for Nabarro-Herring creep: of water)
–1
R = 8.3143 J mol K –1
–1
k = 1.38062 × 10 –23 Jmol K –1
3
–5
V =2.6 × 10 m mol –1
b =5 × 10 –10 m
9
Notice that dislocation creep is non-Newtonian and that the µ =42 × 10 Nm (Sosman 1927)
–2
diffusion creep flow types are Newtonian (Sect. 2.12). The sym- A c = 141 (grain boundary sliding possible)
bols in the equation have the following significance (units in A NH = 16 (grain boundary sliding impossible)
3
–1
square brackets): H L =243 × 10 Jmol for grain boundary diffusion used in the
flow laws for Weertman creep and Nabarro-Herring creep
e – exponential number (2.718281) at 450–590 °C and a mean stress of 100 MPa (Farver and
Ü – shear strain rate [s ] Yund 1991a)
–1
σ – shear stress [Nm ] D L =2.9 × 10 m s Bulk oxygen self-diffusion in the pres-
–2
2 –1
–5
T – temperature [K] ence of water for Weertman creep and Nabarro-Herring
b – Burgers vector [m] creep at 450–590 °C and mean stress = 100 MPa (Farver
– numerical factor for Coble creep depending on grain shape
A c and Yund 1991a)
3
–1
and boundary conditions H G =113 × 10 Jmol for grain boundary diffusion used in the
A NH – numerical factor for Nabarro Herring-creep depending flow law for Coble creep (Farver and Yund 1991b)
on grain shape and boundary conditions D W =3 × 10 –17 m s Bulk oxygen self-diffusion in the presence
3 –1
G
–1
H – molar activation enthalpy for self-diffusion [J mol ] of water for Coble creep at 450–800 °C and 100 MPa mean
2 –1
D – diffusion constant for self-diffusion [m s ] stress (Farver and Yund 1991b)
example, they are defined for only one mean stress value cannot show all deformation mechanisms to advantage.
(100 MPa in the case of Fig. 3.43). This is useful for ex- Pressure solution, a very important mechanism in quartz
perimental purposes where mean stress is usually kept (and probably in feldspar and other minerals; Wintsch
constant, while temperature and strain rate are varied. and Yi 2002) is difficult to include because a deformation
However, in nature, mean stress and temperature increase mechanism map is only valid for a specific mean stress.
together with increasing depth and this effect is usually Since fluid pressure is important in pressure solution (but
not shown in an ordinary deformation mechanism map. possibly also in dislocation creep; Tullis and Yund 1991),
We should not imagine a situation where one deforma- it is difficult to show exact boundaries for pressure solu-
tion mechanism takes over abruptly from another at a set tion in deformation mechanism maps. However, a field
temperature, pressure or other variable. In this sense, the of pressure solution should plot on the low stress and tem-
fields in a deformation mechanism map can be slightly perature side, in the lower left-hand corner of Fig. 3.43g,h
misleading; they indicate dominant deformation mecha- where it replaces the Coble creep field for quartz, calcite,
nisms – other deformation mechanisms may also be ac- feldspar and micas in most geological situations where
tive in these fields, and towards a boundary one mecha- water is present. Finally, flow laws and consequently de-
nism will gradually take over from the other. formation mechanism maps are valid for steady state flow;
Like most geological diagrams, deformation mecha- deformation of a recrystallising aggregate with porphy-
nism maps suffer from the disadvantage that too many roclasts can therefore not be shown on deformation
parameters must be shown in just two dimensions. The mechanism maps.
effect of grain size on rheology is also strong and has to An alternative diagram to deformation mechanism
be shown on separate maps (Fig. 3.43g,h). Another dis- maps is the depth-strength diagram commonly used to
advantage of deformation mechanism maps is that they show a strength profile of the lithosphere (Figs. 3.42, 3.44)