11.1 Departures from linear
viscous flow
L
aboratory determinations of steady-state rock
creep and field observations indicate that
ductile rocks may not be well-approximated
by homogeneous, isotropic, incompressible, and
linear viscous fluids of uniform viscosity. In this
chapter, we consider other constitutive relations
that broaden the range in rock behavior described
and provide a basis for understanding some of the
differences arising. Because the linear Newtonian
viscous fluid is the simplest material that undergoes large permanent deformation, the formulation and analysis of models using it and their
application to interpret a set of field observations
is always a useful first step (see Chapter 10). The
results obtained establish a benchmark from
which to understand differences in behavior associated with other constitutive relations. Whether
the new model results in a large or subtle contrast
in behavior relative to an already well-understood
viscous model, we will achieve a better understanding of the reasons for the differences and a
greater confidence in data interpretation.
For example, initiation of a regular train of
folds by selective amplification was studied for the
Newtonian viscous layer in Chapter 10. Data from
natural fold populations yield a limited range in
mean fold arc length/thickness ratios of about 4 to
6. Interpretation of the data in terms of the
folding of a Newtonian layer implies: (i) a modest
layer to host viscosity ratio, 1/R, of about 10 to 20;
(ii) a weak folding instability, so that a large layerparallel shortening is required to establish a
regular fold train, e.g. a doubling of layer thickness. On the other hand, laboratory studies of
steady-state rock creep suggest that much larger
contrasts in effective viscosity should be common
(Table 11.1), with equivalent 1/R values of about
100 to 1000 or more, and that constitutive relations for rock creep should generally be nonlinear. Do the laboratory results extrapolate to
natural rock behavior at much smaller rates of
deformation and lower temperatures? Answers to
this question not only affect conclusions as to
small-scale folding, but also the modeling of largescale crustal deformation.
In one instance (Groshong, 1975), layer thickening associated with a well-developed set of folds
in a limestone layer embedded in shale was only
ϳ10%. Interpretation of data for these folds
(Fletcher, 1974) is consistent with the type of nonlinear relation observed in the laboratory for
creep of carbonate rock. It has been suggested that
the limited range in fold arc length/thickness
ratios ϳ4 to 6 is a manifestation of highly nonlinear behavior (Smith, 1977, 1979). Further evidence for markedly non-linear behavior is offered
by examples of necking in layer-parallel extension, which cannot occur in a Newtonian layer.
Alternatively, interpretation of the relative viscosities of component rocks in a deformed conglomerate has been offered as evidence of linear
viscous behavior (Treagus and Treagus, 2002).
Motivated by these questions, we study necking
and folding of layers of non-linear power-law fluid
in this chapter.
Inhomogeneity in rock masses occurs at a wide
range in scale, from less than a grain diameter to
the tens or hundreds of kilometers appropriate to
the first-order dynamics of crustal deformation. A
useful postulate is that at the scale of interest the
rock may be treated as a continuum whose behavior is approximated by constitutive relations containing only a few rheological parameters. The
question then arises as to how these parameters
might be estimated from the three major determinants of the bulk behavior: the volume fractions of the significant mechanically distinct
components, their individual constitutive relations, and the phase geometry of the composite
material. Answers to this question would provide
a better understanding of the degree of complexity of rocks and a basis for accepting or maintaining doubts about the approximation of a locally
homogeneous continuum with spatially varying
properties. In this chapter, we use elementary
methods to analyze composites made up of two
isotropic viscous components. Composite materials must generally exhibit anisotropy in their rheological behavior, with isotropic behavior being
only a special case. We therefore include consideration of anisotropy.
Anisotropic materials such as foliated or
layered rock may have nearly uniform properties
with respect to their principal axes of anisotropy.
422
RHEOLOGICAL BEHAVIOR
viscous flow
L
aboratory determinations of steady-state rock
creep and field observations indicate that
ductile rocks may not be well-approximated
by homogeneous, isotropic, incompressible, and
linear viscous fluids of uniform viscosity. In this
chapter, we consider other constitutive relations
that broaden the range in rock behavior described
and provide a basis for understanding some of the
differences arising. Because the linear Newtonian
viscous fluid is the simplest material that undergoes large permanent deformation, the formulation and analysis of models using it and their
application to interpret a set of field observations
is always a useful first step (see Chapter 10). The
results obtained establish a benchmark from
which to understand differences in behavior associated with other constitutive relations. Whether
the new model results in a large or subtle contrast
in behavior relative to an already well-understood
viscous model, we will achieve a better understanding of the reasons for the differences and a
greater confidence in data interpretation.
For example, initiation of a regular train of
folds by selective amplification was studied for the
Newtonian viscous layer in Chapter 10. Data from
natural fold populations yield a limited range in
mean fold arc length/thickness ratios of about 4 to
6. Interpretation of the data in terms of the
folding of a Newtonian layer implies: (i) a modest
layer to host viscosity ratio, 1/R, of about 10 to 20;
(ii) a weak folding instability, so that a large layerparallel shortening is required to establish a
regular fold train, e.g. a doubling of layer thickness. On the other hand, laboratory studies of
steady-state rock creep suggest that much larger
contrasts in effective viscosity should be common
(Table 11.1), with equivalent 1/R values of about
100 to 1000 or more, and that constitutive relations for rock creep should generally be nonlinear. Do the laboratory results extrapolate to
natural rock behavior at much smaller rates of
deformation and lower temperatures? Answers to
this question not only affect conclusions as to
small-scale folding, but also the modeling of largescale crustal deformation.
In one instance (Groshong, 1975), layer thickening associated with a well-developed set of folds
in a limestone layer embedded in shale was only
ϳ10%. Interpretation of data for these folds
(Fletcher, 1974) is consistent with the type of nonlinear relation observed in the laboratory for
creep of carbonate rock. It has been suggested that
the limited range in fold arc length/thickness
ratios ϳ4 to 6 is a manifestation of highly nonlinear behavior (Smith, 1977, 1979). Further evidence for markedly non-linear behavior is offered
by examples of necking in layer-parallel extension, which cannot occur in a Newtonian layer.
Alternatively, interpretation of the relative viscosities of component rocks in a deformed conglomerate has been offered as evidence of linear
viscous behavior (Treagus and Treagus, 2002).
Motivated by these questions, we study necking
and folding of layers of non-linear power-law fluid
in this chapter.
Inhomogeneity in rock masses occurs at a wide
range in scale, from less than a grain diameter to
the tens or hundreds of kilometers appropriate to
the first-order dynamics of crustal deformation. A
useful postulate is that at the scale of interest the
rock may be treated as a continuum whose behavior is approximated by constitutive relations containing only a few rheological parameters. The
question then arises as to how these parameters
might be estimated from the three major determinants of the bulk behavior: the volume fractions of the significant mechanically distinct
components, their individual constitutive relations, and the phase geometry of the composite
material. Answers to this question would provide
a better understanding of the degree of complexity of rocks and a basis for accepting or maintaining doubts about the approximation of a locally
homogeneous continuum with spatially varying
properties. In this chapter, we use elementary
methods to analyze composites made up of two
isotropic viscous components. Composite materials must generally exhibit anisotropy in their rheological behavior, with isotropic behavior being
only a special case. We therefore include consideration of anisotropy.
Anisotropic materials such as foliated or
layered rock may have nearly uniform properties
with respect to their principal axes of anisotropy.
422
RHEOLOGICAL BEHAVIOR