In Earth science, as opposed to Materials
Science, Ductility refers to the capacity
of a rock to deform to large strains without
macroscopic fracturing. Such behavior may
occur in unlithified or poorly lithified sediments,
in weak materials such as halite or at greater
depths in all rock types where higher temperatures
promote crystal plasticity and higher confining
pressures suppress brittle fracture. In addition,
when a material is behaving ductilely, it
exhibits a linear stress vs strain relationship
past the elastic limit.Ductile deformation
is typically characterized by diffuse deformation
(i.e. lacking a discrete fault plane) and
on a stress-strain plot is accompanied by
steady state sliding at failure, compared
to the sharp stress drop observed in experiments
during brittle failure.
== Brittle-Ductile Transition Zone ==
The brittle-ductile transition zone is characterized
by a change in rock failure mode, at an approximate
average depth of 10–15 km (~ 6.2–9.3 miles)
in continental crust, below which rock becomes
less likely to fracture and more likely to
deform ductilely. The zone exists because
as depth increases confining pressure increases,
and brittle strength increases with confining
pressure whilst ductile strength decreases
with increasing temperature. The transition
zone occurs at the point where brittle strength
equals ductile strength. In glacial ice this
zone is at approximately 30 m (100 ft) depth.
Not all materials, however, abide by this
transition. It is possible and not rare for
material above the transition zone to deform
ductilely, and for material below to deform
in a brittle manner. The depth of the material
does exert an influence on the mode of deformation,
but other substances, such as loose soils
in the upper crust, malleable rocks, biological
debris, and more are just a few examples of
that which does not deform in accordance to
the transition zone.
The type of dominating deformation process
also has a great impact on the types of rocks
and structures found at certain depths within
the Earth's crust. As evident from Fig. 1.1,
different geological formations and rocks
are found in accordance to the dominant deformation
process. Gouge and Breccia form in the uppermost,
brittle regime while Cataclasite and Pseudotachylite
form in the lower parts of the brittle regime,
edging upon the transition zone. Mylonite
forms in the more ductile regime at greater
depths while Blastomylonite forms well past
the transition zone and well into the ductile
regime, even deeper into the crust.
== Quantification ==
Ductility is a material property that can
be expressed in a variety of ways. Mathematically,
it is commonly expressed as a total quantity
of elongation or a total quantity of the change
in cross sectional area of a specific rock
until macroscopic brittle behavior, such as
fracturing, is observed. For accurate measurement,
this must be done under several controlled
conditions, including but not limited to Pressure,
Temperature, Moisture Content, Sample Size,
etc., for all can impact the measured ductility.
It is important to understand that even the
same type of rock or mineral may exhibit different
behavior and degrees of ductility due to internal
heterogeneities small scale differences between
each individual sample. The two quantities
are expressed in the form of a ratio or a
percent.% Elongation of a Rock =
%
Δ
l
=
l
f
−
l
i
l
i
×
100
{\displaystyle \%\Delta l={\frac {l_{f}-l_{i}}{l_{i}}}\times
100}
Where:
l
i
{\displaystyle l_{i}}
= Initial Length of Rock
l
f
{\displaystyle l_{f}}
= Final Length of Rock
% Change in Area of a Rock =
%
Δ
A
=
A
f
−
A
i
A
i
×
100
{\displaystyle \%\Delta A={\frac {A_{f}-A_{i}}{A_{i}}}\times
100}
Where:
A
i
{\displaystyle A_{i}}
= Initial Area
A
f
{\displaystyle A_{f}}
= Final Area
For each of these methods of quantifying,
one must take measurements of both the initial
and final dimensions of the rock sample. For
Elongation, the measurement is a uni-dimensional
initial and final length, the former measured
before any Stress is applied and the latter
measuring the length of the sample after fracture
occurs. For Area, it is strongly preferable
to use a rock that has been cut into a cylindrical
shape before stress application so that the
cross-sectional area of the sample can be
taken.
Cross-Sectional Area of a Cylinder = 
Area of a Circle =
A
=
π
r
2
{\displaystyle A=\pi r^{2}}
Using this, the initial and final areas of
the sample can be used to quantify the % change
in the area of the rock.
== Deformation ==
Any material is shown to be able to deform
ductilely or brittlely, in which the type
of deformation is governed by both the external
conditions around the rock and the internal
conditions sample. External conditions include
temperature, confining pressure, presence
of fluids, etc. while internal conditions
include the arrangement of the crystal lattice,
the chemical composition of the rock sample,
the grain size of the material, etc.Ductilely
Deformative behavior can be grouped into three
categories: Elastic, Viscous, and Crystal-Plastic
Deformation.
Elastic Deformation
Elastic Deformation is deformation which exhibits
a linear stress-strain relationship (quantified
by Young's Modulus) and is derived from Hooke's
Law of spring forces (see Fig. 1.2). In elastic
deformation, objects show no permanent deformation
after the stress has been removed from the
system and return to their original state.
σ
=
E
ϵ
{\displaystyle \sigma =E\epsilon }
Where:
σ
{\displaystyle \sigma }
= Stress (In Pascals)
E
{\displaystyle E}
= Young's Modulus (In Pascals)
ϵ
{\displaystyle \epsilon }
= Strain (Unitless)
Viscous Deformation
Viscous Deformation is when rocks behave and
deform more like a fluid than a solid. This
often occurs under great amounts of pressure
and at very high temperatures. In viscous
deformation, stress is proportional to the
strain rate, and each rock sample has its
own material property called its Viscosity.
Unlike elastic deformation, viscous deformation
is permanent even after the stress has been
removed.
σ
=
η
ξ
{\displaystyle \sigma =\eta \xi }
Where:
σ
{\displaystyle \sigma }
= Stress (In Pascals)
η
{\displaystyle \eta }
= Viscosity (In Pascals * Seconds)
ξ
{\displaystyle \xi }
= Strain Rate (In 1/Seconds)
Crystal-Plastic Deformation
Crystal-Plastic Deformation occurs at the
atomic scale and is governed by its own set
of specific mechanisms that deform crystals
by the movements of atoms and atomic planes
through the crystal lattice. Like viscous
deformation, it is also a permanent form of
deformation. Mechanisms of crystal-plastic
deformation include Pressure solution, Dislocation
creep, and Diffusion creep.
== Biological materials ==
In addition to rocks, biological materials
such as wood, lumber, bone, etc. can be assessed
for their ductility as well, for many behave
in the same manner and possess the same characteristics
as abiotic Earth materials. This assessment
was done in Hiroshi Yoshihara's experiment,
"Plasticity Analysis of the Strain in the
Tangential Direction of Solid Woo Subjected
to Compression Load in the Longitudinal Direction."
The study aimed to analyze the behavioral
rheology of 2 wood specimens, the Sitka Spruce
and Japanese Birch. In the past, it was shown
that solid wood, when subjected to compressional
stresses, initially has a linear stress-strain
diagram (indicative of elastic deformation)
and later, under greater load, demonstrates
a non-linear diagram indicative of ductile
objects. To analyze the rheology, the stress
was restricted to uniaxial compression in
the longitudinal direction and the post-linear
behavior was analyzed using plasticity theory.
Controls included moisture content in the
lumber, lack of defects such as knots or grain
distortions, temperature at 20 C, relative
humidity at 65%, and size of the cut shapes
of the wood samples.Results obtained from
the experiment exhibited a linear stress-strain
relationship during elastic deformation but
also an unexpected non-linear relationship
between stress and strain for the lumber after
the elastic limit was reached, deviating from
the model of plasticity theory. Multiple reasons
were suggested as to why this came about.
First, since wood is a biological material,
it was suggested that under great stress in
the experiment, the crushing of cells within
the sample could have been a cause for deviation
from perfectly plastic behavior. With greater
destruction of cellular material, the stress-strain
relationship is hypothesized to become more
and more nonlinear and non-ideal with greater
stress. Additionally, because the samples
were inhomogeneous (non-uniform) materials,
it was assumed that some bending or distortion
may have occurred in the samples that could
have deviated the stress from being perfectly
uniaxial. This may have also been induced
by other factors like irregularities in the
cellular density profile and distorted sample
cutting.The conclusions of the research accurately
showed that although biological materials
can behave like rocks undergoing deformation,
there are many other factors and variables
that must be considered, making it difficult
to standardize the ductility and material
properties of a biological substance.
== Peak Ductility Demand ==
Peak Ductility Demand is a quantity used particularly
in the fields of architecture, geological
engineering, and mechanical engineering. It
is defined as the amount of ductile deformation
a material must be able to withstand (when
exposed to a stress) without brittle fracture
or failure. This quantity is particularly
useful in the analysis of failure of structures
in response to earthquakes and seismic waves.It
has been shown that earthquake aftershocks
can increase the peak ductility demand with
respect to the mainshocks by up to 10
