upper half of the necking layer towards the swell
(Fig.11.19c). This number may be greater than
unity because the diffusional flux must also
counter the drag of material in the medium
toward the swell. This suggests that a contribution of slip at the layer–medium contact, which
would diminish such drag, may enhance the instability further; the effect is not studied here.
11.4 Continuum properties of
composite materials
Excluding volcanic glasses, rocks are made up of
grains or crystals, and contain cracks and pores
that in situ will be filled with liquid or gas. At a
larger scale, rock masses are heterogeneous, from
the relatively simple case of a layered sedimentary
rock to the more complex internal structure of a
strongly deformed metamorphic rock. Treatment
of these materials as continuous media, the
justification for which we have previously set
forth, still leaves open questions. How may we estimate the bulk properties of rocks relevant to
certain behavior, such as elastic deformation or
flow, from the properties of the major component
materials and their geometric configuration? As
the configuration of its mechanical components
changes in a ductily deforming rock, can the evolution in macroscopic properties be determined?
These questions relate to the correlation of
detailed observations of rock composition and
structure with their constitutive behavior when
the deformation of interest took place. For
example, it is often remarked that the rheological
behavior of a rock in ductile deformation is determined by its weakest component, such as quartz
in granite. Can we find support for this idea
through mechanical analysis of such a composite
material? Such questions may be answered by
direct laboratory experiment. But these are
sufficiently expensive in time and resources to
warrant discovering methods of estimation
whose reliability may be tested against particular
experimental results. Further, we need a conceptual basis with which to systematize experimental
results and to think about natural deformation.
In this section, we examine the simplest methods.
As a more concrete example, the sandstone in
the frontispiece is made up of larger grains,
chiefly of quartz, embedded in a fine-grained
matrix rich in phyllosilicates and quartz, two
components that might be expected to have had
distinct mechanical behavior. The larger grains
might have behaved as approximately rigid elements in a weak matrix. Evidently, though, the
large grains may be dissolved. The large grains are
elongate parallel to cleavage. Discontinuous
seams depleted in fine-grained quartz, rich in
phyllosilicates, and aligned along cleavage are a
prominent if volumetrically minor additional
component of the rock. The presence of these and
the elongate grains argue against isotropy of the
constitutive behavior, either in rheological or
elastic behavior. In shear parallel to cleavage, the
seams would possibly contribute to the deformation to a degree far outweighing their volumetric
fraction. How might we quantify these features in
producing a model for the bulk behavior?
11.4.1 Voight and Reuss estimates for
the bulk viscosity of a composite
of two viscous fluids
The simple procedures for estimating the bulk
properties of composite materials used here do
not always conform to the methodology so far presented, in which we advocate the formulation and
solution of boundary value problems. In analyzing these problems, the judicious use of approximation is useful and often necessary, but
conditions of traction continuity and continuity
of displacement or velocity, if appropriate, are
honored. In the estimation procedures introduced here, however, this use of approximation
seems to have been taken too far!
As an example, imagine a composite material
made up of two isotropic linear viscous fluids. For
definiteness, imagine a configuration like that of
sandstone, in which the quartz grains are
replaced by one viscous fluid and the fine-grained
matrix by the other. Suppose that grain contacts
are welded: no slip or separation occurs along
them. Also, set aside the processes of dissolution
and precipitation that are central to the actual
deformation of this rock. If there were a higher
fraction of quartz grains, they would form more
complex interconnected bodies of this com446
RHEOLOGICAL BEHAVIOR
(Fig.11.19c). This number may be greater than
unity because the diffusional flux must also
counter the drag of material in the medium
toward the swell. This suggests that a contribution of slip at the layer–medium contact, which
would diminish such drag, may enhance the instability further; the effect is not studied here.
11.4 Continuum properties of
composite materials
Excluding volcanic glasses, rocks are made up of
grains or crystals, and contain cracks and pores
that in situ will be filled with liquid or gas. At a
larger scale, rock masses are heterogeneous, from
the relatively simple case of a layered sedimentary
rock to the more complex internal structure of a
strongly deformed metamorphic rock. Treatment
of these materials as continuous media, the
justification for which we have previously set
forth, still leaves open questions. How may we estimate the bulk properties of rocks relevant to
certain behavior, such as elastic deformation or
flow, from the properties of the major component
materials and their geometric configuration? As
the configuration of its mechanical components
changes in a ductily deforming rock, can the evolution in macroscopic properties be determined?
These questions relate to the correlation of
detailed observations of rock composition and
structure with their constitutive behavior when
the deformation of interest took place. For
example, it is often remarked that the rheological
behavior of a rock in ductile deformation is determined by its weakest component, such as quartz
in granite. Can we find support for this idea
through mechanical analysis of such a composite
material? Such questions may be answered by
direct laboratory experiment. But these are
sufficiently expensive in time and resources to
warrant discovering methods of estimation
whose reliability may be tested against particular
experimental results. Further, we need a conceptual basis with which to systematize experimental
results and to think about natural deformation.
In this section, we examine the simplest methods.
As a more concrete example, the sandstone in
the frontispiece is made up of larger grains,
chiefly of quartz, embedded in a fine-grained
matrix rich in phyllosilicates and quartz, two
components that might be expected to have had
distinct mechanical behavior. The larger grains
might have behaved as approximately rigid elements in a weak matrix. Evidently, though, the
large grains may be dissolved. The large grains are
elongate parallel to cleavage. Discontinuous
seams depleted in fine-grained quartz, rich in
phyllosilicates, and aligned along cleavage are a
prominent if volumetrically minor additional
component of the rock. The presence of these and
the elongate grains argue against isotropy of the
constitutive behavior, either in rheological or
elastic behavior. In shear parallel to cleavage, the
seams would possibly contribute to the deformation to a degree far outweighing their volumetric
fraction. How might we quantify these features in
producing a model for the bulk behavior?
11.4.1 Voight and Reuss estimates for
the bulk viscosity of a composite
of two viscous fluids
The simple procedures for estimating the bulk
properties of composite materials used here do
not always conform to the methodology so far presented, in which we advocate the formulation and
solution of boundary value problems. In analyzing these problems, the judicious use of approximation is useful and often necessary, but
conditions of traction continuity and continuity
of displacement or velocity, if appropriate, are
honored. In the estimation procedures introduced here, however, this use of approximation
seems to have been taken too far!
As an example, imagine a composite material
made up of two isotropic linear viscous fluids. For
definiteness, imagine a configuration like that of
sandstone, in which the quartz grains are
replaced by one viscous fluid and the fine-grained
matrix by the other. Suppose that grain contacts
are welded: no slip or separation occurs along
them. Also, set aside the processes of dissolution
and precipitation that are central to the actual
deformation of this rock. If there were a higher
fraction of quartz grains, they would form more
complex interconnected bodies of this com446
RHEOLOGICAL BEHAVIOR