# The finite strain ellipsoid for any strain combinations .pdf

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**The finite strain ellipsoid for any strain combinations.pdf**Titre:

**A universal method to compute the finite strain ellipsoid for any strain combinations**Auteur:

**merle**

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In: Structural Geology: New Research

Editor: S. J. Landowe and G. M. Hammler, pp.

ISBN 978-1-60456-827-1

© 2008 Nova Science Publishers, Inc.

Chapter 7

A UNIVERSAL METHOD TO COMPUTE THE FINITE

STRAIN ELLIPSOID FOR ANY STRAIN COMBINATIONS

Cécile Buisson and Olivier Merle

Laboratoire Magmas et Volcans, CNR-IRD-UBP

5 rue Kessler, 63 038 Clermont-Ferrand

ABSTRACT

Any ductile deformation in the field or within analogue models can be analyzed in

terms of principal strain axes λ1, λ2 and λ3 of the finite strain ellipsoid. In this paper, we

show how to compute the strain ellipsoid for any strain combinations. Many

deformations can be expressed as a simultaneous combination of pure shear and simple

shear. Pure shear is coaxial shortening or lengthening along X, Y and Z axes and can be

expressed by a single matrix DPS. There are at most six possibilities for simple shear

components: γXZ, γZX, γYZ, γZY, γXY and γYX, which yield six matrices for simple shear:

DXZ, DZX, DYZ, DZY, DXY and DYX. This means that, in the most complex case, there are

seven matrices to display each strain components of the finite deformation. The

simultaneous combination of these seven strain components can be expressed by the

finite deformation matrix D.

As matrix multiplication is non-commutative and the off-diagonal terms of the finite

deformation matrix a complex function of the pure and simple shear components, the

calculation of the matrix D is far from simple. Solutions proposed in the past were

complicated, known to deviate from the exact solution or applicable only for some

specific combination of strain components.

We propose a solution considering the incremental deformation matrix A, which is

the sum of incremental matrices related to each strain components:

A = APS + AXZ + AZX + …

As A = ln (D) and D = eA , the general solution is given by :

D = exp (ln DPS + ln DXZ + ln DZX + …)

2

Cécile Buisson and Olivier Merle

This new analytical solution and numerical method may be used for any combination

of simple shear and pure shear components. The new method is simple, rapid, and has

been successfully applied to a strain study within lava domes. The method is relevant for

any structural geologists who want to predict the strain pattern in any kind of geological

processes, assuming a simultaneous combination of elementary pure and simple shear

components.

INTRODUCTION

Any ductile deformation can be analyzed and interpreted in terms of principal strain axes

whose orientations and magnitudes are obtained from the finite strain

λ1, λ and λ

ellipsoid. Most finite deformations recorded in the field result from the simultaneous

combination of pure shear with one or several components of simple shear. These separate

strain components can sometimes be estimated in the field from strain methods, which are

based upon measurements of deformed strain markers (e.g. Ramsay and Huber, 1980). They

generally vary in space and time across a given geological structure.

Conceptual strain models of geological structures are important and useful in so far they

allow a better understanding of the strain pattern observed in the field. In numerical

simulations, computation of the three principal strain axes is generally achieved from the

finite deformation matrix, called D, which is supposed to take into account all strain

components acting across the studied structure. The main issue is to successfully combine

pure and simple shear components to obtain the matrix D. So far, only partial analytical

solutions have been proposed making this problem still unsolved for the most complex 3D

combinations.

In this paper, we show how it may be easy to compute the strain ellipsoid for any strain

combination. This new approach makes its possible to determine the overall stain pattern in a

geological structure , that is stretching and shortening magnitudes and directions as well as

the strain ellipsoid type.

THE FINITE DEFORMATION MATRIX

(

)

In a coordinate system 0, X,Y,Z , each point of (x0, y0, z0) co-ordinates is transformed

into a point of (x', y', z') co-ordinates after deformation. This new position can be noted x(t)

i.e. the position of the point after a time t. This change can be represented by the following

linear transformation in homogeneous deformation, which introduces the finite deformation

matrix D:

x′

x0

y ′ = D y0

z′

z

0

A Universal Method to Compute the Finite Strain Ellipsoid…

3

Many deformations can be described as a combination of pure and simple shears. Pure

shear is characterized by three specific parameters: kX, kY, kZ, which quantify lengthening or

shortening acting along the X, Y and Z axes. Pure shear is expressed by the following matrix,

DPS:

kx

DPS = 0

0

0

ky

0

0

0

kz

Six distinct simple shears can be considered although they rarely operate all together in a

single deformation. They are expressed by six specific matrices, Dij and six specific

parameters, γij. Dij and γij. refer to simple shear acting in the ij plane and along the i-direction.

As an example, DXZ describes simple shear acting in the XZ plane and along the X-direction :

DXZ

1 0 γ XZ

= 0 1 0

0 0 1

At the final deformation state, these seven strain components (one for pure shear and six

for simple shear) may have been simultaneously combined, which yields the following

deformation matrix D:

kX

D = ΓYX

ΓZX

ΓXY

kY

ΓZY

ΓXZ

ΓYZ

kZ

with Γij = f (γ ,k) where k can be any component of pure shear and γ any simple shear

component.

The non-diagonal terms of the matrix D can be called effective terms of deformation

(Tikoff and Fossen, 1993). This means that, when combining pure and simple shear, they are

not independent and interact each other. Such an interaction is the main reason explaining the

difficulty to calculate D in complex cases. There is no simple algebraic way to obtain Γij.

PREVIOUS STUDIES

Both in two and three dimensions, the solutions proposed in the literature do not apply to

the general case but to specific cases only. Even for specific cases, solutions sometimes

involve long and tedious calculations (e.g. Ramberg, 1975; Tikoff and Fossen, 1993; Soto,

1997).

Ramberg (1975a) was the first to model a deformation process as a simultaneous

combination of pure shear and simple shear. Integrating the rate-of-deformation equations, he

4

Cécile Buisson and Olivier Merle

obtained particle path equations. Applied on a set of particles defining a circle in the

undeformed state, the particle path equations allow imaging the strain ellipse after

deformation. They also give some insights into the progressive deformation through time.

Although this can be useful for a simplified approach of natural processes, these solutions,

however, remain of limited use as applied to two-dimensional deformation only. In addition,

Ramberg used matrices to present the problem but he displayed solutions with algebra only.

In a companion paper, Ramberg (1975b) considered a three-dimensional specific case

combining a pure shear deformation with a single simple shear :

kx

DPS = 0

0

0

ky

0

1 0 γ

0

0 and DSS = 0 1 0

0 0 1

kz

DSS corresponds to a simple shear deformation in the XZ plane along the X-direction.

The simultaneous combination of these two strain components is obtained from heavy and

lengthy equations and remains suitable for this specific case only. However, it has been

shown very useful when studying the strain pattern within spreading nappes (Merle 1989).

Following this pioneer work, two- and three-dimensional specific cases have been solved

and analysed in detail. Theoretical studies of natural examples including strain variation in

nappes and thrust sheets (Coward and Kim, 1981; Sanderson, 1982; Coward and Potts, 1983;

Merle, 1986), deformation in anisotropic rocks (Weijermars, 1992), strain within

transpression and transtension zones (Fossen and Tikoff , 1993), thrust-wrench zones (Merle

and Gapais, 1997), basaltic dykes (Coward, 1980) and lava tubes (Merle, 2000) have shown

that the strain pattern approach is very useful to get a better insight into the kinematics of

geological processes. These cases are still relatively simple and may be managed with a set of

equations and not with matrices.

In two dimensions, for a deformation at constant volume with:

k 0

1 γ

DPS =

and DSS =

,

0 1/ k

0 1

Merle (1986) has given the following solution:

k

D =

0

γ (k −1 k)

2ln k

1 k

Tikoff and Fossen (1993) have extended this solution including volume change,

k1

DPS =

0

0

and k1 .k2 ≠ 1 ,

k2

which yields:

A Universal Method to Compute the Finite Strain Ellipsoid…

k

D = 1

0

5

γ (k1 − k2 )

ln(k1 k 2 )

k2

Tikoff and Fossen (1993) have also proposed a three-dimensional solution, but again this

applies to a particular case. The solution is algebraic and results from lengthy equations quite

difficult to handle. The case presented by Tikoff and Fossen corresponds to a 3x3 upper

triangular matrix of the form:

kX

D=0

0

ΓXY

kY

0

ΓXZ

ΓYZ

kZ

The complete solution may be calculated from a “simple” system with 9 equations and 9

variables and gives:

k1

D = 0

0

γ XY (k1 − k 2 )

γ XZ (k1 − k 3 )

ln(k1 / k 2 )

ln(k1 / k 3 )

k2

0

+

γ YZ γ XY (k 3 − k1 )

γ YZ γ XY (k1 − k 2 )

+

ln(k1 / k 2 ) ln(k 2 / k 3 ) ln(k 2 / k 3 ) ln(k1 / k 3 )

γ YZ (k 2 − k 3 )

ln(k 2 / k 3 )

k3

As an example of the valuable help such a matrix can bring to numerical simulations,

Merle (2000) has successfully used it to calculate the strain pattern within lava tubes.

However, deviating from this particular case, there is no analytical solutions to the equations.

Soto (1997) has followed this approach by extending it to all possible cases with change

of axes. The solution is universal in the mathematical sense as it allows the calculation of all

types of deformation. However, it also requires handling up to 5-pages long endless

equations.

To circumvent the problem, an approximate solution has often been used to study natural

examples. It consists in multiplying successively small increments of each strain component.

Merle and Gapais (1997) have used this "step by step" method to analyze the strain pattern

within thrust-wrench zones. However, changing the order of matrix multiplication, even for

very small increments, makes the final results slightly different, which reveals that the “stepby-step” method is convenient but not rigorous. In addition, this method requires a repetition

of calculations that only a software tool can perform.

THE INCREMENTAL DEFORMATION MATRIX

The importance of incremental stages has already been stressed out in previous studies

(e.g. Ramsay, 1967; Ramberg, 1975; Merle and Gapais, 1997). The final deformation

represented by the matrix D results from the succession of infinitesimal steps. However, the

same finite deformation may be obtained from multiple different combinations of incremental

6

Cécile Buisson and Olivier Merle

matrices. A given deformation history leads to a single finite strain ellipsoid but the reverse is

not true and many deformation histories may be deduced from a given finite strain ellipsoid.

Although the order of multiplication of strain components is essential in the resulting

finite strain, working with infinitesimal time intervals makes it possible to break this

constraint. The incremental matrix A is the sum of each individual incremental matrix. A is

the general incremental matrix representing an infinitesimal time t of deformation, which

simultaneously combines the incremental pure shear matrix APS and the six incremental

simple shear matrices AXZ, AZX, AYZ, AZY, AXY, AYX. This means that:

A = ∑ Aij = APS + AXZ + ...+ AYX

Each Aij does not correspond to any geological reality in itself. This is a mathematical

artifice but defining the incremental matrix A is a rigorous way to simultaneously combine

several strain components of the finite deformation.

The solution of the problem under consideration may be found by managing incremental

stages taking into account all strain components of the finite deformation. This allows the

calculation of the incremental deformation matrix A. In a recent paper (Provost et al., 2004),

it has been shown that the way to calculate the finite deformation matrix D from the

incremental deformation matrix A, and the opposite, is trivial.

According to these authors, D = exp(L∆t) and L∆t = ln(D) where L(t) is the velocity

gradient tensor and L∆t is the time integration of L(t). L(t) describes the deformation

process(es) acting at any moment of the deformation history. The matrix D represents a tensor

describing the resulting, observable final deformation. Following the notation used in this

paper, this means that:

A = ln D and D = e A

THE FINITE DEFORMATION MATRIX FOR ANY STRAIN

COMBINATION

It results from the above considerations that we can easily combine pure and simple

shears in all conceivable cases. There are, at most, seven separated matrices, one for pure

shear and six for simple shear:

k X

DPS = 0

0

1

DYX = γ YX

0

0

kY

0

1 γ XY

0

,

0 DXY = 0 1

0 0

kZ

1 0 γ XZ

0

,

0 DXZ = 0 1 0 ,

0 0 1

1

1 0 0

1

0 0

,

,

1 0 DYZ = 0 1 γ YZ DZX = 0

0 0 1

γ ZX

0 1

0 0

1 0

0 1

A Universal Method to Compute the Finite Strain Ellipsoid…

1 0

and DZY = 0 1

0 γ ZY

7

0

0

1

For incremental steps, we define εX, εY and εZ as the infinitesimal pure shear components

and γXZ, γZX, γYZ, γZY, γXY and γYX as the infinitesimal simple shear components. As an

example, the incremental pure shear matrix is given by :

εX

APS = ln(DPS ) = 0

0

0

εY

0

0

0

εZ

( )

Similarly: AXZ = ln(DXZ ) and in general: Aij = ln Dij

Then, we may combine these incremental matrices to obtain A, the incremental matrix at

each incremental stage: A =

∑A

ij

. Finally, we come back to the finite deformation matrix,

D (figure 1):

kX

D = e = ΓYX

ΓZX

A

ΓXY

kY

ΓZY

ΓXZ

ΓYZ

kZ

Figure 1. Separate strain components of the finite deformation are estimated from the field or models

(This gives at most 7 matrices). Then, incremental matrices related to each strain component are

calculated. Summing each individual incremental matrix allows to obtain A, the general incremental

matrix. From A, it is possible to come back to the finite deformation matrix D.

8

Cécile Buisson and Olivier Merle

APPLICATION OF THE METHOD TO LAVA DOMES

This universal method to calculate the finite deformation matrix has been used to analyze

the strain pattern within lava domes (Buisson, 2001; Buisson and Merle, 2004). Numerical

modeling was achieved with 3D Cartesian co-ordinates with X and Y lying along a horizontal

basal plane on which the spreading takes place. Z is the vertical axe and the origin is centered

on top of the feeding conduit, at the vent from which lava flows on the basal XY plane.

The whole deformation is far from simple. Experiments reveal a pure shear component

with concentric lengthening (Y direction), vertical shortening (Z direction) and lengthening or

shortening in the radial direction (X direction) according to the distance from the vent

(Buisson and Merle, 2002). Two simple shear components are simultaneously combined with

the pure shear one, which correspond to vertical and horizontal simple shearing acting in the

XZ plane : γXZ and γZX. This may be described with three matrices at the final stage of

deformation:

k X

DPS = 0

0

1 0 γ XZ

1

0

0 , DXZ = 0 1 0 and DZX = 0

0 0 1

γ ZX

kZ

0

kY

0

0 0

1 0

0 1

As explained above, deformation is not commutative and the order of multiplication of

these matrices is critical:

DPS D XZ DZX ≠ D XZ DPS DZX ≠ D XZ DZX DPS ≠ etc

The finite deformation matrix D, which expresses the simultaneous combination of these

three components, is of the form:

kX

D = 0

ΓZX

0

kY

0

ΓXZ

0

k Z

The matrix D is not a 3x3 upper triangular one and the solution provided by Tikoff and

Fossen (1993) cannot be used. According to the method shown above, it is simply needed to

calculate the incremental matrices related to each strain component:

εX

APS = ln(DPS ) = 0

0

and

0

εY

0

0 0 γ XZ

0

,

0 AXZ = ln(DXZ ) = 0 0 0

0 0 0

εZ

A Universal Method to Compute the Finite Strain Ellipsoid…

AZX

0

= ln(DZX ) = 0

γ ZX

9

0 0

0 0

0 0

Summing these incremental matrices, we obtain the incremental matrix A:

A = APS + AXZ + AZX

εx

= 0

γ ZX

0

εy

0

γ XZ

0

εz

and can come back to the finite deformation matrix D:

D = e A = e A PS +A XZ +A ZX

In the case of lava domes, the dome was divided into seven domains with specific values

of pure and simple shears parameters as observed from experiments. Then, we use the finite

deformation matrix D to define the total strain pattern in the three directions of space. This

numerical method has given an invaluable insight of the deformation within the structure,

which could not be obtained from analogue modeling. Attitude of flattening planes,

orientation of the stretching axis and shape of the strain ellipsoid throughout the structure has

been revealed for the first time (Buisson and Merle, 2004). Obviously, this has implication in

the understanding of lava domes emplacement and kinematics.

CONCLUSION

This new method to calculate the finite deformation matrix provides an analytical

solution to a problem which was still left unsolved for three dimensions despite many studies

on the topics (e.g. Ramberg, 1975; Tikoff and Fossen, 1993, Soto, 1997). Using a computer, it

is just routine to calculate the 3D deformation matrix for any stain combination. The method

is universal and discards the “step-by-step’ method that was applied for complex strain

combination for which rigorous solutions were missing. Likewise, it makes unnecessary the

use of endless equations proposed in previous algebraic solutions, as it provides a simple and

rapid method, understandable in a flash and practicable by everybody.

REFERENCES

Buisson, C. (2001). Cinématique et déformation dans les dômes de lave: Modélisations

analogique et numérique. PhD thesis, Univ. Blaise Pascal, Clermont-Ferrand, France;

276p.

Buisson, C. and Merle, O. (2002). Experiments on internal strain in lava dome cross-sections.

Bulletin of Volcanology , 64, 363-371.

10

Cécile Buisson and Olivier Merle

Buisson, C and Merle, O. (2004). Numerical simulation of strain within lava domes. Journal

of Structural Geology, 26, 847-853.

Coward, M.P. (1980). The analysis of flow profiles in a basaltic dyke using strained vesicles.

J. Geol. Soc. London, 137, 605-615.

Coward, M.P. and Kim, J. H. (1981). Strain within thrust sheets in: Thrust and Nappe

tectonics. Geological Society special publication, 9, 275-292.

Cowards, M. P. and Potts, G. J. (1983). Complex strain pattern developed at the frontal and

lateral tips to shear zones and thrust zones, J. Struct. Geol., 5, 383-399.

Flinn, D. (1979). The deformation history and the deformation ellipsoid, J. Struct. Geol.,

Vol.1, n°4, 299-307.

Jaeger, J.C. (1969). Elasticity, Fracture and Flow. Chapman and Hall Ltd, London.

Malvern, L.E. (1969). Introduction to the mechanics of a continuous medium. Prenctice- Hall,

Inc. Englewood cliffs, NJ; 713pp.

Means, W.D. (1976). Stress and strain. Springer Verlag, New York; 339p.

Merle, O. (1986). Pattern of stretch trajectories and strain rate within spreading-gliding

nappes. Tectonophysics, 124, 211-222.

Merle, O. (1989). Strain pattern within spreading nappes. Tectonophysics, 165, 57-71.

Merle, O. (2000). Numerical modelling of strain in lava tubes. Bulletin of Volcanology, 62,

53-58.

Merle, O. and Gapais, D. (1997). Strains within thrust-wrench zones. Journal of Structural

Geology, 19, 1011-1014.

Provost, A.; Buisson, C. and Merle, O., (2004). From progressive to finite deformation, and

back. Journal of Geophysical Research, 109, B2, pages.

Ramberg, H. (1975a) Particle paths, displacement and progressive strain applicable to rocks.

Tectonophysics, 28, 1-37.

Ramberg, H. (1975b) Superposition of homogeneous strain and progressive deformation in

rocks. Bull. Geol. Inst. Uppsala, N.S. 6, 35-67.

Ramberg, H (1981). Gravity, deformation and the Earth's crust. Academic Press, London;

452p.

Ramsay, J.G. (1967). Folding and fracturing of rocks. Mac Graw-Hill Book Co., New York;

568p.

Ramsay, J.G. and Huber, M. (1983). The technique of modern structural geology, volume 1 :

strain analysis. Academic press, London; 307p.

Sanderson, D. J. (1982). Models of strain variation in nappes and thrust sheets: a review.

Tectonophysis, 88, 201-233.

Soto, J.I. (1997). A general deformation matrix for three dimensions. Mathematical Geology,

29, 93-130.

Tikoff, B. and Fossen, H. (1993). Simultaneous pure and simple shear : the unifying

deformation matrix. Tectonophysics, 217, 267-283.

Weijermars R. (1992) Progressive deformation in anisotropic rocks. J. Struct. Geol, 14, 723742.

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