infinitesimal strain and their accumulation

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Chapter 2

Infinitesimal Strains and Their

Accumulations

In this chapter we will study infinitesimal deformation and the rate of deformation. Most

geologic structures are the integral of infinitesimal deformations, so that it is important

to understand how infinitesimal and finite deformations are related.

2.1 Infinitesimal strain

The lifetime of a deformation zone or an orogenic belt can be over millions of years, during which

time rock masses undergo large deformations. Tectonism is fairly slow and the incremantal defor-

mation in our usual time-scale is very small. Therefore, the deformations detected by a geodetic

survey are usually very small. Geology detects the result of long-term deformations.

Geological measurement is much less accurate than geodetic one. Thus, it is usually difficult

for geology to detect such small strains as geodesy detects. However, it is possible in some cases1

that geologic structures provide clues to quantify those strains. In addition, it is necessary to know

the theories of infinitesimal strains to understand the following sections in which the dynamics of

tectonic deformatoins is discussed.

2.1.1 Definition

Infinitesimal strainis a very small strain. If a deformation gradient tensor is nearly equal to the iden-

tity tensor (F I), it is calledinfinitesimal deformation. Let us first rewrite Fwith the displacement2,

u = x .1Note that strain is defined as the change of length compared to the original one. Given a tiny strain, the associated length

change can be large enough for detection by geological measurement.2The famous textbook by Turcotte and Schubert [245] defines the displacement u =

x in order to deal with contraction

and compression as possitive strain and stress, respectively.

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2.4. PILE UP OF INFINITESIMAL DEFORMATIONS 39

is obtained (Exercise 2.2) as the time rate of the Jacobian (Eq. (1.7)), Therefore, if

v = 0,

then we have J = 0. This is a differential equation indicating that Jhas no temporal change. The

equation has the solution J = const. The constant should be 1, otherwise J= 0 when a volumechange occurs. Consequently, a rock mass has no volume change if v = 0. If the velocity fieldsatisfies the equation

v =0, (2.29)

the deformation is called incompressible. In that case, density does not change, either. Incom-

pressibility is often assumed in tectonic models, but is not applicable to phenomena with significant

density changes associated, for example, with metamorphic reactions from low to high pressure

phases.

How fast are actual tectonic movements? A million years is often used as the unit to measure

geologic ages. The lifetime of an orogenic belt is on the order of ten to a hundred million years

during which rock masses deform by hundreds of %. A million years is approximately equal to

3.16 1013 s, because a year is approximately 60 seconds 60 minutes 24 hours 365.25 days= 31,557,600 s. If a rock mass deforms by 100 % in a million years, the strain rate is 1 /(3.16 1013s) 3.17 1014 s1.

Tectonics of island arcs is often faster than that of continents [263]. Extensional deformations

in the Early Miocene was as fast as 1013 1014 s1 in western Northeast Japan (12.2). TheNortheast Japan arc has been subject to compressional tectonics since the Pliocene, and deforma-

tion was accelerated in the Pleistocene. The strain rate representative for Northeast Japan has beenestimated by several methods. A seismological estimate from moment tensor accumulation (2.7)

indicates the rate to be about 1 1016 s1 [258]. This is the rate averaged over a few decades. Thatover the last100 years was determined by triangulation at (3 6) 1015 s1 [160]. Northeast

Japan has undergone compressional deformations, mainly in the Quaternary [202]. Folds and faults

have shortened the crust under the arc. They were combined with chronostratigraphy to infer the

strain rate averaged over the last 1.3 million years [201]. The result was on the order of 10 14 s1.The discrepancy between the estimated rates may shed light to non-steady movements.

Stable continents are not rigid, but are very slowly deforming. As an example, earthquakes

rarely occur in Fenoscandia. The stress field responsible for such seismic activity is consistent with

the combination of plate boundary forces with the plate flexure accompanied by Holocene glacial

rebound [229]. The average strain rate of cratonic lithosphere is on the order of 1015 s1 [37]. With

a strain rate below the order of, say, 1017, the region is too slow, even for geology, to detect and

may seem rigid.

2.4 Pile up of infinitesimal deformations

Most tectonic structures are the result of finite deformations that are an accumulation of infinites-

imal deformations. Therefore, it is important for tectonics how finite deformations are related to

infinitesimal deformations.


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44 CHAPTER 2. INFINITESIMAL STRAINS AND THEIR ACCUMULATIONS

Figure 2.4: Physical interpretation of the additive decomposition L = D + W. An infinitesimaldeformation in an infinitesimal time interval can be resolved to D and W, indicating the coaxial

strain and rotation, respectively. The sides of this rectangular solid is parallel to the principal strain

axes. Di stands for the ith principal strain. W indicates the infinitesimal rotation, the angular

velocity of which isw .

is known as thevorticity. From Eqs. (2.25) and (2.42) the components ofWare obviously

W= 0 1

2

v1x2

v2x1

1

2

v1x3

v3x1

12 v2

x1 v1

x2 0 1

2 v2

x3 v3

x2

12

v3x1

v1x3

1

2

v3x2

v2x3

0

= 0 1

23

12

2

12

3 0 12 1

12

212

1 0

. (2.43)

Thus, the vorticity is twice the vector axial vector ofW. Letw be the axial vector, then

=2w. (2.44)

According to the discussion on the rotation tensor and rotation vector in 2.1.3, we see that w rep-

resents angular velocity. The vector points the direction of rotation axis like a right-hand screw, and|w| indicates the scalar angular velocity. Since a deformation within an infinitesimal time interval isalso infinitesimal, strain and rotation are additive and commutative (Fig. 2.4).

Rigid-body rotation Let us demonstrate the correspondence between the vorticity and angular ve-

locity for the case of rigid-body rotation. Given a constanta, the velocity field v =(ax2, ax1, 0)Trepresents a rigid-body rotation about the third coordinate axis. This is proved by the observation

firstly that vx = 0 and, secondly, that |v| =

a2x22 + a

2x21 |x|. The coefficient a is the angular ve-

locity. Using the identity xi/xj =ij, we have the vorticity of this velocity field = (0, 0, 2a)T.

The magnitude is twice the angular velocity, so that the proof has been established.


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2.5. ROTATION 45

2.5.3 Spin within shear zone

Physical interpretation of the additive decomposition of velocity gradient tensor L = D +Wmay

be obtained in an analysis of simple shear deformations. Rigid-body rotation within a shear zone is

considered assuming a simple shear deformation, where the velocity field is v = (2qx2, 0, 0)T and

q=const. The velocity gradient is

L =

vi

xj

=

0 2q 00 0 00 0 0

,which is decomposed into symmetric and antisymmetric parts:

D = q

0 1 01 0 00 0 0

, W= q 0 1 01 0 0

0 0 0

. (2.45)D andWrepresent shape-change and rotation, both of which do not vanish but occur simultaneously

for simple shear (Eq. (2.45)). The characteristic equation ofD is q 0q 00 0

=3 + q2 =q( q)( + q) =0.

The eigenvalues are, therefore, q, 0 and q. E has the same eigenvalues, because D = E.Suppose a square that is deformed to a parallelogram by the instantaneous simple shear, the

time interval of which is t. Instantaneous strain is infinitesimal. The order of strain and rotation

are commutative for infinitesimal deformations, so that we consider the deformation as a two-step

operation, strain followed by rotation (Fig. 2.5). The strain in this velocity field has the principal

axes inclined at 45 to the flow direction (see 1.7). The rate of strainE has the eigenvalues q, 0

and q, so that the diagonals of the square have undergone strain qtwithin the interval t. If thesquare ABCD in Fig. 2.5 has sides with a length 2a, the diagonal OA and OB has the length

2a.

The square, first, becomesthe parallelogram ABCD. OA and OB are elongated to OA and OB,respectively. Both AA and BB have an equal length

2aqt. The inclination of the parallelogram

is represented by the angle between AB and AB. The angle is very small for t 0, so thatthe tangent of the angle is approximated by the angle itself. It is seen in Fig. 2.5(a) that the tangent

equals 2aqt/2a = qt. The inclined parallelogram ABCDis rotated by the angle. The result is aparallelogram ABCDwhose two sides are parallel to the flow direction. The angle of rigid-body

rotation is qt. It follows that the angular velocity is q.

The same quantity is obtained from the spin tensor in Eq. (2.45). We obtain the vorticity

= (0, 0,2q)T, where the minus sign indicates that the direction of rotation in this case is thecounterclockwise relative to the third coordinate axis. The axial vector ofW is w = (0, 0,

q)T,

which is identical with the angular velocity of the parallelogram in Fig. 2.5. Therefore, the axial


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54 CHAPTER 2. INFINITESIMAL STRAINS AND THEIR ACCUMULATIONS

Figure 2.12: (a) Paleomagnetic declinations [73, 79, 84, 85, 162, 165, 166, 242, 271, 273] and major

faults that were probably activated when the Japan Sea opened in the Early Miocene [271]. Note that

the forearc (Pacific side) of Northeast Japan ceased paleomagnetic rotation by 1617 Ma, earlier than

the termination of the rotation in the backarc (Japan Sea side) [84]. Major faults in Northeast Japan

(FF, Futaba Fault; HI, HachirogataIchinoseki Tectonic Line; HM, HonjoMatsushima Tectonic

Line, NT, NishikawaTakarazawa Tectonic Line; NM, NihonkokuMiomote Tectonic Line) after

[100]. OF, Oisawa Fault; TSZ, Tanakura Shear Zone; KTL, Kanto Tectonic Line. (b) Paleo-positions

of Northeast Japan and Hokkaido. (c) A simple model to account for the rotation and transtensional

deformation in the drifting NE Japan arc.

Using this moment, Eq. (2.50) is rewritten as

F= I +MgMa

V . (2.52)

The second term in the right-hand side is composed of variables that can be determined from the

fault parameters and the size of the rock mass.

The faults are numbered by their ages. That is, the number 1 is the oldest and the numberN

is the youngest. The total deformation of the mass by the fault activity is indicated by the product,

F = F(N) F(N1) F(1), where F(i) stands for the deformation gradient tensor for the ith fault.


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infinitesimal strain and their accumulation

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