Strain analysis ecalais@purdue.edu

Plates vs. continuum

•  Most plates are rigid at the <1 mm/yr level => until know we have studied a purely discontinuous approach where plates are individual rigid entities.

•  But: –  Some plates are affected by GIA deformation –  Some plates are affected by broad tectonic deformation –  Some plate boundaries are broad

•  Need for a “continuous” approach = strain

Gordon and Stein, 1991

Strain - 1 D case •  Solid rod in a 1-dimensional coordinate system (0,x). Let us take two points A (x) and B (x+dx), very close to each other.

•  Apply a traction P in the x-direction at the end of the rod. The rod will elongate, points A and B will be displaced.

•  Assume infinitesimal displacements. The amount of displacement depends on the position of the point along the rod: the closer to the place where the force is applied, the larger the displacement will be.

•  Therefore, B will be displaced slightly more than A. The displacement at A is noted u(x), the displacement at B is noted u(x+δx)

Solid rod in a 1-dimensional coordinate system (0,x). Let us consider two points A (x) and B (x+dx), very close to each other.

Strain - 1D case •  Because the displacement δx is infinitesimal, we can write:

and:

•  Strain is defined as [change of length] / [original length]. Therefore:

•  Hence, infinitesimal strain is the spatial gradient of the displacement

€

u(x + dx) = u(x) +∂u∂xdx

€

ex =ΔLL

=A 'B'−ABAB

=dx+

∂u∂xdx

%

& '

(

) * − dx

dx=∂u∂x

€

" A " B = dx + u(x) +∂u∂x

dx$

% &

'

( ) − u(x) = dx +

∂u∂x

dx

2D case: velocity gradient tensor •  Assuming:

–  Local 2-dimensional Cartesian frame –  Infinitesimal displacements

•  Velocity (as a function of position X) can be expanded in a series:

•  With the two components:

•  Therefore, one can write:

€

v(X + δX) = v(X) +∂v∂X

δX

€

vx (X + δX) = vx (X) +∂vx∂x

δx +∂vx∂y

δy

vy (X + δX) = vy (X) +∂vy∂x

δx +∂vy∂y

δy

$

% & &

' & &

€

v(X + δX) = v(X) +∇V .δX

€

∇V =

∂vx∂x

∂vx∂y

∂vy∂x

∂vy∂y

$

%

& & & &

'

(

) ) ) )

with = velocity gradient tensor

Velocity gradient tensor •  Given 2 sites (i and j) separated by (small) distance δX, equation:

•  Becomes:

•  We can write it for 3 sites as:

•  Which can be solved using least-squares:

€

v(X + δX)v j

= v(X)vi −∇V .δX

€

vi − v j = vij =∇V .δX

€

Vij =

vx12vy12vx13vy13

"

#

$ $ $ $ $

%

&

' ' ' ' '

, A =

x12 y12 0 00 0 x12 y12x13 y13 0 00 0 x13 y13

"

#

$ $ $ $

%

&

' ' ' '

F =

∂vx ∂x∂vx ∂y∂vy ∂x∂vy ∂y

"

#

$ $ $ $

%

&

' ' ' '

€

Vij = A.F

Observables = Velocity differences

Model matrix = Position differences

(assuming point 1 is reference)

with:

Unknown = Velocity gradients

€

F = ATCV−1A( )

−1

CF−1

ATCV

−1Vij

Velocity gradient tensor: covariance

•  Observables are velocity differences.The covariance associated with vij can then be found using:

•  One usually knows the covariance matrix Cind of the individual site velocities -- the variance propagation law then gives:

•  CVij is then propagated in the least squares estimation (see previous slide) to get the covariance matrix of the unknowns CF. €

CVij= A.Cind .A

T

€

vxijvyij

"

# $

%

& ' =

−1 0 1 00 −1 0 1"

# $

%

& '

A

vxivyivxjvyi

"

#

$ $ $ $

%

&

' ' ' '

Velocity gradient tensor: covariance of unknowns

•  Given that:

•  One can write the variance on the east (for instance) component of the velocity difference as:

•  Therefore, if the covariance terms are large enough, velocities can be non-significant while relative velocities (and velocity gradient tensor) are… €

σ eij2 =σ ei

2 +σ e j2 − 2σ ei e j

€

CVij= A.Cind .A

T

Strain and rotation rates •  Displacements actually combine strain and a rigid-body rotation. •  Tensor theory states that any second-rank tensor can be

decomposed into a symmetric and an antisymmetric tensor. One can therefore write:

€

∇V =12∇V +∇VT[ ] +

12∇V −∇VT[ ]

⇔∇V =

∂vx∂x

12∂vx∂y

+∂vy∂x

&

' (

)

* +

12∂vx∂y

+∂vy∂x

&

' (

)

* +

∂vy∂y

,

-

.

.

.

.

/

0

1 1 1 1

+0 1

2∂vx∂y

−∂vy∂x

&

' (

)

* +

12∂vy∂x

−∂vx∂y

&

' (

)

* + 0

,

-

.

.

.

.

/

0

1 1 1 1

Symmetric tensor Antisymmetric tensor

Strain and rotation rates

•  In other words:

•  With: €

∇V =exx exyexy eyy

#

$ %

&

' ( +

0 ω

−ω 0#

$ %

&

' (

E strain rate tensor

W rigid rotation

€

E =12∇V +∇VT[ ]

W =12∇V −∇VT[ ]

exx =∂vx∂x,eyy =

∂vy∂y,exy = eyx =

12∂vx∂y

+∂vy∂x

%

& '

(

) *

ω =12∂vx∂y

−∂vy∂x

%

& '

(

) *

Estimating strain rate •  Given 2 sites (i and j), one can write the velocity difference (w.r.t. site i) as a function of the

site position (w.r.t. site I) as:

•  For 3 sites as, this can be written as:

with:

•  Which can be solved using least-squares:

€

vij =∇V .δX⇔ vij = (E +W ).δX

⇔vxij = exxδx + exyδy +ωδyvyij = exyδx + eyyδy −ωδx' ( )

€

Vij =

vx12vy12vx13vy13

"

#

$ $ $ $ $

%

&

' ' ' ' '

, A =

x12 y12 0 y120 x12 y12 −x12x13 y13 0 y130 y13 y13 −x13

"

#

$ $ $ $

%

&

' ' ' '

S =

exxexyeyyω

"

#

$ $ $ $

%

&

' ' ' '

€

Vij = A.S

Observables = velocity differences

Model matrix = Position differences

€

S = ATCV−1A( )

−1

CS−1

ATCV

−1Vij

Estimating strain: A better approach

•  First, estimate velocity gradient components (and covariance) using least squares (see previous slides).

•  Then, recall that:

•  Therefore, strain and rotation rate components can be found using the following transformation matrix:

•  Propagate velocity gradient covariance to strain and rotation rate covariance using:

€

exx =∂vx∂x,eyy =

∂vy∂y,exy = eyx =

12∂vx∂y

+∂vy∂x

#

$ %

&

' ( ,ω =

12∂vx∂y

−∂vy∂x

#

$ %

&

' (

€

exxexyeyyω

#

$

% % % %

&

'

( ( ( (

=

1 0 0 00 0.5 0.5 00 0 0 10 0.5 −0.5 0

*

+

, , , ,

-

.

/ / / /

A

∂vx∂x∂vx∂y∂vy∂x∂vy∂y

#

$

% % % % % % % %

&

'

( ( ( ( ( ( ( (

€

CE ,W = AC∇V AT

Representing strain •  Tensor = independent of

the coordinate system = retains its properties independently from the ref. system.

•  Find reference system where: –  Shear strain (exy) is zero –  Two other components are

maximal •  Equivalent to diagonalize E:

–  Eigenvectors = principal axes of strain rate tensor

–  Eigenvalues = principal strain rates

Principal strains

•  Principal values of E: (= principal strains)

•  Principal directions: –  Use rotation matrix A: –  Expand AEAT and write that shear component = 0 to find:

€

det(E − λI) = 0

⇒ detexx − λ exyexy eyy − λ

%

& '

(

) * = 0

⇒ exx − λ( ) eyy − λ( ) − exyexy = 0

⇒ λ =exx + eyy2

±exx − eyy2

+

, -

.

/ 0

2

+ exy2

€

e12 = (exx − eyy )sinθ cosθ + exy (cos2θ − sin2θ) = 0

⇒ tan(2θ) =2exy

exx − eyy

€

A =cosθ −sinθsinθ cosθ$

% &

'

( )

€

sinθ cosθ =12sin 2θ( )

cos2θ − sin2θ = cos 2θ( )

Principal strains

•  Principal strains: –  One maximal, one minimal –  By convention, extension is

taken positive

•  Principal angle (direction of e1):

€

e1,e2 =exx + eyy2

±exx − eyy2

#

$ %

&

' (

2

+ exy2

€

tan(2θ) =2exy

exx − eyy

Back transformation •  Rotate principal strain rate tensor by angle -θ using rotation matrix A:

•  Recall that tensor rotation is given by:

•  Which gives:

€

exx = e1 cos2θ + e2 sin

2θ =e1 + e22

+e1 − e22

cos 2θ( )

eyy = e1 sin2θ + e2 cos

2θ =e1 + e22

−e1 − e22

cos 2θ( )

exy = −e1 sinθ cosθ + e2 sinθ cosθ = −e1 − e22

sin 2θ( )

$

%

& &

'

& &

€

sinθ cosθ =12sin 2θ( )

cos2θ − sin2θ = cos 2θ( )

€

A =cosθ sinθ−sinθ cosθ$

% &

'

( )

€

AEpAT =

e1 cos2θ + e2 sin

2θ −e1 sinθ cosθ + e2 sinθ cosθ−e1 sinθ cosθ + e2 sinθ cosθ e1 sin

2θ + e2 cos2θ

$

% &

'

( )

•  Shear strain (from previous transformation) is maximal when sin(2θ)=1, giving:

•  Recall the expression for e12 (which we set to zero to find the principal angle θ):

•  Angle at which shear strain is maximal is obtained by differentiating w.r.t. θ:

Shear strain

€

exy = −e1 − e22

sin 2θ( )

⇒ exy,max =e1 − e22

=exx − eyy2

%

& '

(

) *

2

+ exy2

€

e12 = (exx − eyy )sinθ cosθ + exy (cos2θ − sin2θ)

=12(exx − eyy )sin 2θ( ) + exy cos 2θ( )

€

e12 =max⇒ de12dθ

= 0

12(eyy − exx )cos 2θS( ) − exy sin 2θS( ) = 0

⇒ tan 2θS( ) =eyy − exx2exy

Shear strain

•  Maximum shear strain:

•  Maximum shear angle: €

exy,max =exx − eyy2

#

$ %

&

' (

2

+ exy2

€

tan 2θS( ) =eyy − exx2exy

€

θS = θ ± 45o

Example 1: pure extension •  Strain rate tensor:

–  exx = 0.00 ppb/yr –  exy = -0.00 ppb/yr –  eyy = 177.08 ppb/yr

•  Principal strains: –  e1 = 177.08 ppb/yr (most

extensional) –  e2 = 0.00 ppb/yr (most

compressional) –  theta = 90.00 (e1, CW from

north) •  Rotation:

–  omega = 0.00 deg/My

Example 2: simple shear

•  Strain rate tensor: –  exx = 0.00 ppb/yr –  exy = 88.54 ppb/yr –  eyy = -0.00 ppb/yr

•  Principal strains: –  e1 = 88.54 ppb/yr (most

extensional) –  e2 = -88.54 ppb/yr (most

compressional) –  theta = -45.00 (e1, CW

from north) •  Rotation:

–  omega = 0.09 deg/My

•  As any second rank tensor, strain rate tensor has 3 invariants = quantities that remain unchanged regardless of the reference system

•  First invariant = tensor trace

•  Second invariant = sometimes used to represent strain “magnitude”

•  Third invariant = determinant, same as second invariant in case of 2x2 symetric tensor

Strain invariants

€

IE = tr E( ) = exx + eyy = e1 + e2

IIE =12tr E 2( )− tr(EE)( ) = exxeyy − exy2 = e1e2

IIIE = det E( ) = exxeyy − exy2 = e1e2

Example: current strain rates in Asia

•  The 3x10-9 yr-1 line coincides with the 95% significance level •  A large part of Asia shows strain rates that are not significant at the 95% confidence level

and are lower than 3x10-9 yr-1

In practice •  Discretize space •  Using polygons with vertices

corresponding to data points •  For instance: Delaunay

triangulation: –  Circumcircle of every triangle

does not contain any other point of the triangulation

–  Minimize “sliver” triangles •  Using arbitrary polygons:

requires interpolation •  Calculate strain within each

polygon

WARNING: assumes homogeneous and

continuous strain within each polygon…

For instance

GPS velocities and principal strain rates in Central Asia

WARNING! •  Result depends on the size

of the elements used to discretize the domain…

•  Example: –  Generate a random 2D

velocity field (e.g. case of non significant residual velocities showing only noise)

–  Triangulate with smaller elements in the middle of the domain

–  Calculate strain –  Result will be apparently

larger strain rates where elements are smaller…