The finite strain ellipsoid for any strain combinations .pdf

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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

Cécile Buisson and Olivier Merle
Laboratoire Magmas et Volcans, CNR-IRD-UBP
5 rue Kessler, 63 038 Clermont-Ferrand

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 + …)


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

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
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.




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…


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 =  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 :


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



kZ 

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

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,
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


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 :


DPS =  0
 0


1 0 γ 

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:

D =

γ (k −1 k)
2ln k 
1 k 

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

DPS = 

 and k1 .k2 ≠ 1 ,

which yields:

A Universal Method to Compute the Finite Strain Ellipsoid…

D = 1



γ (k1 − k2 )

ln(k1 k 2 )

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:

 0



kZ 

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

 k1

D = 0



γ XY (k1 − k 2 )

γ XZ (k1 − k 3 )

ln(k1 / k 2 )

ln(k1 / k 3 )



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 )



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
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 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


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

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


DYX = γ YX
 0


1 γ XY

0  DXY = 0 1
0 0
kZ 

1 0 γ XZ 

0 DXZ = 0 1 0 ,
0 0 1 

1 0 0 
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




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 :

APS = ln(DPS ) =  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 =



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

D (figure 1):
 kX

D = e = ΓYX



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.


Cécile Buisson and Olivier Merle

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

k X

DPS =  0
 0

1 0 γ XZ 

0  , DXZ = 0 1 0  and DZX =  0
0 0 1 
γ ZX
kZ 


0 0

1 0
0 1

As explained above, deformation is not commutative and the order of multiplication of
these matrices is critical:

The finite deformation matrix D, which expresses the simultaneous combination of these
three components, is of the form:

D =  0


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:


APS = ln(DPS ) =  0
 0



0 0 γ XZ 

0  AXZ = ln(DXZ ) = 0 0 0 
0 0 0 
εZ 

A Universal Method to Compute the Finite Strain Ellipsoid…



= ln(DZX ) =  0
γ ZX


0 0

0 0
0 0

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


 εx

= 0
γ ZX



γ XZ 

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.

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.

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analogique et numérique. PhD thesis, Univ. Blaise Pascal, Clermont-Ferrand, France;
Buisson, C. and Merle, O. (2002). Experiments on internal strain in lava dome cross-sections.
Bulletin of Volcanology , 64, 363-371.


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.
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tectonics. Geological Society special publication, 9, 275-292.
Cowards, M. P. and Potts, G. J. (1983). Complex strain pattern developed at the frontal and
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Jaeger, J.C. (1969). Elasticity, Fracture and Flow. Chapman and Hall Ltd, London.
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Geology, 19, 1011-1014.
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