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Unformatted text preview: CHAPTER To further our understanding of the origin and signif
icance of the folds, foliations, and lineations discussed
in the lasr four chapters, we need 1‘0 become more fa
miliar with the nature of strain, as manifested in rocks.
We introduced some concepts of strain in Chapters 7,
9, 12, and 14, but we need a more thorough and sys
tematic understanding in order to evaluate theoretically
the models proposed for formation of ductile structures,
as well as to test these models against observations of
natural deformation. Our approach is largely geometric and qualitative,
because our intent is to provide intuition into the phys
ical characteristics of deformation, and strain lends itself
easily to geometric description. The quantirative anal—
ysis of the ideas discussed in this chapter requires a
rigorous mathematical treatment of strain, which we
introduce in Box 15.1, and which is developed in depth
in more advanced boo s on continuum mechanics and
its geologic plicatio 8 (see the list of readings at
the end this. chap er). Readers interested in this mation.
size e srrain is homogeneous if the changes in
shape are proportionately identical for each
small part of the body and for the body as a whole
(Figure 15.1A, B). A consequence of these c0nditions is that for any homogeneous strain, planar surfaces remain
planar, straight lines remain straight and parallel planes
and lines remain parallel. The strain is inhomogeneous
(Figure 15.1A, C) if the changes in size and shape of
small parts of the body are proportionately different
from place to place and different from that of the body
as a whole. Straight lines become curved, planes become
curved surfaces, and parallel planes and lines generally
do nor remain parallel after deformation. The strain must be inhomogeneous during folding,
because in such a deformation, planes and lines do not
generally remain planar, straight, or parallel. Within
very small volumeelements, however, the strain is sta
ristically homogeneous, and we describe an inhomo
geneous strain as a variation of homogeneous strain
from place to place in the Structure. We discuss how
big such a “small” volumeelement musr be in Section
15.7. The progressive deformation of a body refers to
the motion that carries the body from its initial unde
formed state to its ﬁnal deformed state. The strain states
through which the body passes during a progressive
deformation deﬁne the strain path. The state of strain
of a body is the net result of all the deformations the
body has undergone. Although all states of strain are
the result of progressive deformation, the ﬁnal state of
strain provides no information about the particular
strain path that the body experienced. A. Undeformed state B. Homogeneously deformed state C. inhomogeneously
deformed state Figure 15.1 Homogeneous and inhomogeneous plane defor
mation of a material square. A and B. Homogeneous Strain.
The small black square is strained in exactly the same way
as the whole square and as all the other squares. Lb is the angle
of shear. A and C. Inhomogeneous strain. The small black
square is sufﬁciently small that its srrain is essentially ho
mogeneous, but it is not identical to the strain of the whole
square or to that of any of the other small squares. Strain in general must be described in three di
mensions, because the size and shape of a body are
threedimensional characteristics. In much of our dis
cussion, however, we consider only a twodimensional
deformation called plane strain, in which the strain is
completely described by changes in size and shape in a
single orientation of plane through the body, and no
deformation occurs normal to that plane. Although
plane strain is commonly used to analyze deformation,
its application to many situations in natural rock de—
formation is, strictly speaking, unjustiﬁed. Nevertheless,
the geometry of two—dimensional deformation is intu
itively easier to understand, and the generalization to
three dimensions adds considerable complexity but lirtle
insight into the geometric characteristics of deforma—
tion. For these reasons we concentrate on the properties
of twodimensional strain. In discussing the geometry of strain, we refer to
geometric objects such as lines, planes, circles, and elm
lipses. Such geometric objects are called material objects
if they are always deﬁned by the same set of material
particles. A bedding plane, for example, is a material
plane because no matter how it moves and deforms, it is always deﬁned by the same set of material particles.
A coordinate plane deﬁned by two reference axes, on
the other hand, is a nonmatetial plane because as a body
deforms, its material particles can move through the
coordinate plane and, consequently, different sets of
material particles occupy the coordinate plane at dif—
ferent times. This distinction is important in the sub
sequent discussion. Lz' ear Strain ' e of a body is measured by its volume, which in
turn is proportional to the product of three characteristic
lengths of the body. For example, the volume V of a
rectangular block that has edges of lengths [1, f2, and
{’3 is V = {16263, and the volume of an ellipsoid that
has semiaxes of lengths r1, r2, and Q is V = [4/3] rt
r1r2r3. In Cartesian coordinates, the description of the
change in size requires speciﬁcation of the change in
length of line segments in the three coordinate
direcrions. The change in absolute length is an inadequate
measure of the deformational state of a line segment,
because for a given change in length, the intensity of
the change is much greater for a short line segment than
for a long one. Thus the lengthening is expressed as a
propOrtion of the original line length. Two measures in
common use are the stretch 5,, and the extension e,,,
which was int the beginning of Chapter 9. f I; to the
rs when rferring (152) value for extension measures a lengthening, whereas a positive value
for stress measures a compression. We thus end up with a positive
stress Causing a negative extension. i'his incompatibility does no:
arise with the engineering sign convention for stress, which is why
it is generally used in analytic applications of continuum mechanics. Geometry of Homogeneous Strain 293 294 m A More Quantitative View of Strain A homogeneous transformation of any material point
from the undeformed state to the deformed state is
represented mathematically by a linear relationship
between the coordinates of any point in the unde
formed state (X1, X3) and its coordinates in the de
formed state (11, x3), where we use uppercase letters
to describe the undeformed state and lowercase let
ters to de5cribe the deformed state. If we restrict our
analysis to plane deformation, the general form of
such a transformation is x1=AX1+BX3+C I3=DX1+EX3+F (l5.l.l) where A, B, C, D, E, and F are constants. The parts
of the transformation deﬁned by C and F are the same
for all particles, and therefore these constants describe
a rigidbody translation. If any or all of these constants
vary with time, then these equations describe the mo
tion of the material particles. The equations say that given the original lo cation
of any material particle in the undeformed state (X 1,
X3), we can calculate its ﬁnal location in the deformed
state (x1, x3). The equations may be solved for X1 and
X3 so that given the deformed location of a material
particle (11, 13), we can also calculate its original
location (X1, X3). These equations deﬁne the inverse
transformation. X1=axl+bx3+c X3=dx1+ex3+f (15.1.2) where
E —B BF—CE
as b5 :5—
AE—BD AE—BD AE—BD
d= —D e— A f_DC—AF
_AE—BD _AE—BD _AE—BD
(15.1.3) and where, again, (2 and f describe a rigid body trans
lation. As examples of such a transformation and its
inverse, the following equations describe a pure shear,
which transforms a square with sides parallel to the
principal coordinates into a rectangle (Figure 1598) x1=AX1
X1=(1/A)11 x3 =(1/AlX3
X3 = AX3 (15.1.4) A simple shear, which transforms a square into a
parallelogram (Figure 15.11B), and its inverse are de
scribed by xl=XJ+BX3 I3=X3 (15.1.5)
X1=x1—Bx3 X3 =13 When the constants A in Equation (15.1.4) and
B in Equation (15.1.5) are linear functions of time,
the motions are steady and these equations describe DUCTILE DEFORMATION progressive pure shear and progressive simple shear,
respectively (see Section 15.4). With Equations (15.1.2), it is easy to show that
a homogeneous deformation transforms a circle into
an ellipse. A circle of unit radius in the undeformed
state is represented by the equation (X02 + (X3)2 =1 (15.1.6) If we substitute for X1 and X3 from Equations (15. 1.2),
we ﬁnd the locus in the deformed state of all material
particles that lie on the circle in the undeformed state.
Because a rigidbody translation does not contribute
to the strain, we assume c = f: 0. Then, making the
substitution, we ﬁnd (a2 + d2)(x1)2 + 2(ab + de)x1x3 + (b2 + el)(x3)2 =1
(15.1.7) Equation (15.1.7) is the equation of an ellipse with
its principal axes tilted with respect to the coordinate
axes, and it is, in fact, the strain ellipse. The components of the strain tensor are related
to the displacement vectors for the material particles.
A displacement vector connects the position of a par
ticle in the undeformed state to its position in the
deformed state. The vector and its components (U1,
U3) parallel to the X1 and X3 coordinate axes are
(Figure 15.1.1A) st—x (15.1.3) U1 =X1 —X1 U3=X3—X3 (15.19) When a material deforms, the displacement vec
tors for two neighboring material points are different.
If they were the same, the ”deformation” would be
a rigid body motion. The difference in these displace
ment vectors therefore describes the deformation.
Thus we consider two neighboring points A and Bthat
are displaced by the deformation to a and b, respec
tively. The displacement vectors for the two points
are UV” and [1(8), and the difference between them
is dU (Figure 15.1.1B). The material line segment dX
connecting A to B is deformed into dx connecting a
to b. The change in that line segment due to the
deformation AdX is also described by the vector dU
(Figure 15.1.18). Thus, dU a U03) — UM) = Adx 2 dx i ax (15.1.10) The relationship between the ﬁrst and last terms in
this equation is just the differential of Equation
(15.1.8). ‘ . We can consider the components Xm and dX3
of the line segment dX to be two material line seg—
ments that are initially perpendicular to each other
and parallel to the coordinate axes X1 and X3 respec
tively. If we restrict our analysis to inﬁnitesimal strain, Deformed
Displacement position
vector U Undetermed
position characterized by the conditions dUl << 1 and d U3 << 1,
the displacement associated with each of these line
segments due to the deformation can be expressed
using Equation (15.1.10) and the chain rule of dif
ferentiation for dU 6U 6U Adx=AdX AdX=dU=——dx —dX
1+ 3 6X1 1+aX3 3 (15.1.11)
Thus the changes :3Xm and AdX3 in each of the line
segments due to the deformation is given in terms of the components of the displacement vect0r U by (Fig
ure 15.1.2). 6 3U
Adxl = _Us dX1= _“ de + .6231 mg
ax, 6X1 ax, (15.1.12)
5U 5U, 6U3
AdX =_dX =_ X —d
3 6X3 3 axsd 3 +ax3 X3 For each of the material line segments 01X 1 and
dX3, the extensional strains are labeled e11 and egg,
respectively, and each one is the change in length
divided by the initial length, as defined in Equation
(15.2). For in, for example, the change in length is
(am/6X1) ﬁg, and the initial length is Xm (Figure
15.1.2). Similar relations hold for dX3. Thus 1 1 av, _ 5U,
6“ ‘ dX, [6X1 dX‘] ‘ ax, _ l 5U}, EU;
833 = (EL—X3 dXs] = 5Y3
The shear strain of Xm relative to dX3 and vice versa
are labeled (213 and e31, respectively, and are deﬁned
in Equation (15.7) to be half the tangent of the shear
angle (1113 = $31: 11'; = or + 13. For very small strains,
or << 1 and ,6 << 1, and the standard trigonometric iden
tity for the tangent of the sum of two angles gives (15.1.13) tana+tanﬁ mtetm+m=mm astanct+tanfi (15.1.14) Figure 15.1.1 The displacement vector. A.
The displacement vector connects the p0v
sition of a material particle in the unde
formed state to its position in the deformed
state. B. If a material is deformed, the dis
placement vectors for two neighboring
points are different. Point A is deformed re
the position a; B is deformed to the position
b. The difference in the displacement vectors
dU describes the deformation of the mate rial. because the product tan at tan ,3 is negligibly small.
The tangent of an angle is the length of the side 0p
posite the angle divided by the length of the adjacent
side. For inﬁnitesimal strains, the side opposite the
angle at is approximately (6 U1/3X3)dX3, and the ad—
jacent side is dX3 (Figure 15.1.2). Similar relationships
hold for the angle ,8. Thus we have tmam;[ﬂlldX3]=ﬂll ng, 5X3 6X3 (15,15)
1 5U3 0U3
:— — X =__.
a” dxllax, d 1] ax,
(Continued) : ax. maxi
ax, ‘
tau, e___ 1) 3X1 Figure 15.1.2 The geometrical interpretation of the com
ponents of inﬁnitesimal strain for twodimensional strain.
For clarity, the strain is greatly exaggerated in the diagram.
The vectors dX, dx, and dU are the same as the vectors
having the same labels that appear in Figure 15.1.18. The
strain components thns are deﬁned by the change in the
displacement vector dU for two neighboring points. Geometry of Homogeneous Strain m 295 BOX 15.]. (Continued) Then using the deﬁnition of the shear strain (Equation
15.7) with Equations (15.1.14) and (15.1.15) gives 6U 6U
213 2 231: 0.5 tan (tr z 0.5(—ﬁl+——:5 6X3 5X1) (15.1.16) These relations for the extensions and shear strains
associated with the material line segments Xm and
dX3 are the components of the inﬁnitesimal strain
tensor.'In shorthand component notation, we sum
marize Equations (15.1.13) and (15.1.16) by 0U), art) a 0.5 —
8’“ (ax, + 5X), ' k,Z= 1,2, 3 (15.1.17) This expression for ck). remains exactly the same if k
and f are interchanged, which shows that ek, = 8m
and that the strain tensor is a symmetric tensor (com
pare Equation 15.12). Thus 51:; is the symmetric part
of the displacement gradient tensor 6Uk/8X). The antisymmetric part of the displacement ga
dient tensor can be shown to be the inﬁnitesimal ro
tation tensor, deﬁned by 5 a
rhEO.5(ﬂ—i), k,[=1,2,3 (15.1.18) The antisymmetric character of rk, is evident from
this equation, because interchanging the subscripts k
and 3’ gives the relation rkr="’zk (15.1.19)
Components on the principal diagonal of the matrix
rk, must therefore be zero. In twodimensional strain,
there is only one independent offdiagonal component
n3 = — r31. Thus from Equations (15.1.15) and
(15.1.18) we can see that rl3:0.5(tanot—tan ,8) (15.1.20) For very small angles, the tangent of the angle is ap
proximately equal to the angle measured in radians,
so we can write ruzOSW—B) (15.1.21) Thus r13 is half the difference in the components of
the shear angle, and no. is thus a measure of the net
rotation of the material line segment dX. The displacement components (U3, U3) can be
expressed solely in terms of the coordinates of the
material point in the undeformed state by substituting
Equations (15.1.1) into (15.1.9), assuming the rigid
translations are zero (C = F : 0) U1=(A —1)X1+ 8X3 U3 = ox) + (15— nx3 (15.1.22) Using Equations (15.1.22) in (15.1.17), we ﬁnd the
values of the strain components in terms of the con
stants that deﬁne the motion of the material particles: [811 943] =I: (A— i) 0.5(B + 0)] (15.1.23)
E31 (333 0.5(D + B) (E  1) As indicated above, the relationships given here
are correct only for very small strains. The analysis
of large strains is considerably more complex, al
though this geometric interpretation of the strain
components remains intuitively useful. For a line segment of arbitrary orientation in the
undeformed state, given by the angle 6 with respect
to the principal coordinate axis 21, it can be shown
that the extension and the shear strain for inﬁnitesimal
plane strain are given in terms of the principal exten
sions by en=elc0328+é3 5111119 (151 15) as = (1‘21 — €33) sin Bees 3 These equations are identical in form to Equations
(8.36), which we found for the stress components,
and the mathematical characteristics of the stress and
the inﬁnitesimal strain tensors are identical, including
the possibility of deriving a Mohr circle for inﬁnites—
imal strain. The relationships for large deformations are
somewhat more complex, but a Mohr circle that is
useful in solving strain problems can nevertheless be
deﬁned for large strains. We refer the reader to books
containing more quantitative analyses (see the works
by Means, Ramsay and Huber, and Eringen in the list
of additional readings at the end of this chapter). Comparing Equations (15.1) and (15.2} shows that these
two measures of extensional strain are related: 5 —1 (1531
Values of 5,, > 1 and of e” > 0 represent increases in the length ofrnaterial lines, and values where 0 < 5,, < 1
and en < 0 represent decreases in length (Table 15.1). 296 DUCTILE DEFORMATION Other measures are also used, including the qua
dratic elongation and the natural strain. The quadratic
elongation is simply the square of the stretch, and it is
often given the symbol 2., although some authors use
this symbol to designate the stretch. The natural strain
E”, also called the logarithmic strain, is the integral of
all the inﬁnitesimal increments of extension required to
make up the deformation, where the reference length Table 15.1 Extensional Strain of a Material Line .._s_.gr.ﬂaf..cﬁcz,s.:.~fimega Mum,  ,.__.,..,."‘._... Length Change Stretch Extension
AL 5,15 {IL 3,, a (f — L)/L
Undeformed L AL = 0 5,1 = 1 an = 0
Shortened l—Q li— AL AL=£—L<0 0<sn<1 en<0 ,
Lengthened l—£_i:ml AL = f — L > 0 s,t > 1 3,, > 0 for each increment in length (if is taken to be the in
stantaneous deformed length 5. 5r
_ d€_ fr _
5,, {7—111 (I) 1115,, (15.4) where L is the initial length, ff is the ﬁnal length, and
in indicates the natural logarithm. Notice that the nat
ural strain is the natural logarithm of the stretch. The
natural strain is sometimes convenient for discussion of
strain history (see Figure 15.7.0). It also provides a sym—
metric measure of shortening and lengthening.2 The
time derivative of the natural strain is also often used
as a measure strain rate (see Box 18.1). Volu et‘ric Srr in We can n eonsider measures of the volumetric strain,
whic e refer to as the volumetric streteh (5”) and the
vo‘ umetric extension3 (6,). If the undeforrned volume
is V and the deformed volume is U, v — V AV v V V =7=s,—1 (15.5)
A rectangular block thar undergoes only volumetric.
strain has undeformed sides (L1, L2, and L3) and de— formed sides (f1, {1, and {3). The volumetric stretch is [1’25]
51* = £1LG
5,, = 515253 =(e1+ 1)(e2 + 1)(53 +1) (155) We consider further aspects of volumetric strain in the
next sectio /
Shear Sr A dy can also change shape without changing volume.
or example, a cube can deform into a rhombohedron,
or a sphere into an ellipsoid. Changes in shape are 2 For example, for a line segment stretched to twice its initial length
and one shortened to half its initial length, 5,, = 2 and 0.5, and en = 1
and 0.5, but 3,, : 0.693 and —0.693, respectively. 3The volumetric extensiOn is commonly given the symbol A and
called the dilation, or even the dilatation. We reserve A to indicate
the change in a variable. described by the changes in the angle between pairs of
lines that are intialiy perpendicular (Figure 15.2). The
change in angle is called the shear angle (1'1, and the shear
strain e_. is defined by as E 0.5 tan If/ (15.7) As deﬁned here, a5 is the tensor shear strain. It differs
from another common measure of the shear strain, the
engineering shear strain 3;, by a faCtor of 2 (y E tan
(1'1 = 255). For two material line segments originally ori
ented along the positive co...
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