FULL- FIELD STRAIN RATE MEASUREMENT
BY WHITE-LIGHT SPECKLE IMAGE CORRELATION
Giovanni B. Broggiato1, Leobaldo Casarotto2, Zaccaria Del Prete1, Domenico Maccarrone1
1
Mechanics and Aeronautics Department – University of Rome “La Sapienza”
Via Eudossiana, 18 – 00184 Rome (Italy)
2
IPROM, Institut für Produktionsmeßtechnik – Technische Universität Braunschweig
Schleinitzstraße 20 – D-38106 Braunschweig (Germany)
[email protected]
[email protected]
[email protected]
[email protected]
ABSTRACT
This paper shows a numerical procedure able to process sequences of digital images and to return a full-field evaluation of the
strain rate. The processing procedure is based on a nonlinear least squares fitting performed globally, on the whole image,
and simultaneously on several images. The use of a highly optimized code allows to analyse long sequences in few minutes.
The calculations results are presented as movies built by blending the colour maps of the measured strain field with the
specimen pictures used in the correlation procedure.
Our application is focused to study the plastic behaviour of metals and, in particular, to highlight any transient phenomena that
may occur during yielding and strain hardening phases on the thin sheets used in sheet metal forming. A typical example for
such phenomena is the Portevin-Le Châtelier effect, a repetitive yielding of alloys in regime of plastic deformation.
Introduction
The strain localization in a material is associated to a non uniform field of the strain rate and to a consequent concentration of
the stress. The strain rate takes part to many constitutive equations and is one of the most important quantities determining the
material behaviour. A not uniform strain rate distribution may lead to the establishing of instability phenomena such as the
shear banding [1] and is often encountered during superplastic deformations [2]; furthermore, mechanical processes usually
induce a concentration of the strain rate, both in the case of cutting [3] and of material working as, for example, during the
wire-drawing [4].
Despite the importance of the strain rate to understand the dynamics of material deformation and failure, there are relatively
few attempts to investigate its distribution field, most of the experimental approaches performing only a global measurement.
While strain rate distribution measurement is well established in areas like fluidodynamics, medicine [5, 6] or geology [7], in the
field of the mechanics we have found, at the best of our efforts, only two attempts, both very recent and based on optical
techniques. In the first work, by Cordero and Labbé [8], an ESPI is used to observe the strain rate in an aluminium specimen
during a tensile test and to ascertain that the necking is preceded by a localization of the strain rate. The described set-up
consists of a double optical configuration in order to obtain both the displacement components in the axial and in the
transversal direction, which however can not be acquired together at the same instant. As a consequence, the tensors of the
strain and of the strain rate at the object surface can not be determined and the displacement fields in the two directions must
be considered separately. In the second work, Gilat et al. [9] performed a full field measurement of the strain and the strain
rate in a Hopkinson split bar test. The specimen was imaged with a photogrammetrical set-up based on two high-speed
cameras and the displacements on its surface were calculated by means of a conventional program for digital image
correlation. The strain rate was therefore obtained with a post-processing of the numerical results. Considered that a not
uniform strain rate distribution is often associated to fast phenomena, this last contribution indicates also that the recent
availability of high-speed cameras at affordable prices can disclose new advancements in this investigation field.
In this paper we present a novel method for full-field measurement of the strain-rate by means of a dedicated algorithm for
digital image correlation of dense sequences of white-light speckle images. The basic idea is to compare not just two images
at a time, but to extend the correlation procedure to an interval of pictures, where the path of a point is considered to be
smooth accordingly to a chosen spatio-temporal function. This approach has two benefits: at first, a set of images are
compared simultaneously with the further constrain of a required regularity of movements, which leads to a more realistic
determination of the displacements field; secondly, the velocity of a point is found as the temporal derivative of the smooth
path determined in the correlation process. Once the velocity field is known, the strain rate tensor can be calculated at each
point according to the relations given in the next section.
Tensorial definition of the strain rate
Before introducing the strain rate tensor, a short overview on the other used tensors is given here [10].
The local deformation of a body can be described by means of two of its particles that move from the initial positions p0 and q0
to the new positions p1 and q1. These displacements are shown in Fig. 1 with the vectors up and uq, while their difference du
represents the relative displacement experienced by the points. The couple of points defines also the vector dx0, which is
transformed into dx1 by the local deformation. The vector can also be written as dx0= dS n, where dS is the original distance
between p0 and q0 (a scalar quantity) and the unit vector n indicates the direction of dx0, so that a unit relative displacement is
defined as du/dS.
q1
q0
uq
dx1
du
dx0 = dS n
n
p0
up
p1
Figure 1. The relation between initial and final positions of two points.
With the decomposition of vector du and unit vector n in rectangular coordinates and for dS → 0, the following relation can be
written:
du
=Jn
dS
where
J km =
∂u k
dx 0 m
(1)
(2)
The jacobian matrix J is known as the displacement gradient matrix and is actually a tensor that can be separated in its
symmetric and antisymmetric parts:
J = E+Ω
with
E=
1
T
1
(
J + J T ) and Ω = 2 (J − J )
2
(3.a)
(3.b)
where E is the strain tensor and Ω the rotation tensor, which are associated respectively to the strain and to a rigid rotation of
the region.
To model the displacement, the strain and the strain rate fields computed through the correlation of speckle images, it is
particularly appropriate to describe the local deformation in a Lagrangian framework [11], where the initial configuration is used
as reference (see Fig. 2).
As an example, on the left of Fig. 2, in the undeformed space x0–y0, a four node square element is given; on the right, in the
sequence of spaces xj–yj, a succession of deformed configurations of the same element are plotted. Introducing also a time
axis, we may say that the undeformed configuration is the snap shot of a deformable element at time t0 and the deformed ones
are the same element at time instants tj–1, tj, tj+1 and so on.
y0
yj–1
s
undeformed
referential
space (t = t0)
p
s
r
q
yj
r
s
q
p
x0
r
xj–1
p
yj+1
r
s
q
xj
q
p
deformed
spaces
(t = tj–1, tj, tj+1,)
xj+1
Figure 2. A four node plane element in the initial (undeformed) and in a set of deformed configurations.
By knowing (on each configuration) the four couples of node coordinates, a bilinear function f that maps the undeformed space
into each deformed ones can be univocally determined. This function, called mapping function, is defined as follows:
 f x ( x0 , y 0 ) = a x 0 + a x 1 x0 + a x 2 y 0 + a x 3 x0 y 0
 x1 
  = f ( x0 , y 0 ) = 
 y1 
 f y ( x0 , y 0 ) = a y 0 + a y1 x 0 + a y 2 y 0 + a y 3 x 0 y 0
(4)
The use of quadrilateral elements and bilinear shape functions will be explained in the next section.
We may also think that, along the three deformed configuration (j–1, j, j+1), where the time lapses in the interval t= [tj–1 …
tj+1], the node coordinates depend also on the time through quadratic functions, so that the mapping function will be now
written as:
 f x ( x0 , y0 , t ) = a x 0 + a x1 x0 + a x 2 y0 + a x 3 t + L + a x11 x0 y0 t 2
 x1 
  = f ( x0 , y 0 , t ) = 
2
 y1 
 f y ( x0 , y0 , t ) = a y 0 + a y1 x0 + a y 2 y0 + a y 3 t + L + a y11 x0 y0 t
(5)
Where the 12 unknown parameters (ax0 … ax11 and ay0 … ay11) can be evaluated solving a square set of equations built by
evaluating fx and fy at the four nodes p, q, r, s at the three instants tj–1, tj, tj+1.
The spatial gradient F of the mapping function f in the referential space is called deformation gradient tensor:
 ∂f x
 ∂x
F= 0
 ∂f y
 ∂x0
∂f x 
∂y 0 

∂f y 
∂y 0 
(6)
which embodies the stretch and the rotation of body evolution without loss of significance when large strain and finite rotations
are present. Through the right Cauchy-Green tensor C the strain state can be completely assessed; in fact, being C = FTF,
and λ1,2 its eigenvalues, the principal logarithmic strains will be:
ε 1,2 =
1
ln λ1, 2
2
Moreover F and C are joined to the strain tensor E by the relations:
E=
[
]
1 T
1
F F − I = [C − I ]
2
2
(7)
Dealing with the distribution of velocities, a clear distinction between the material (or referential) and the spatial description has
to be made in that the first one is suited to describe the motion of individual material particles, which is needed to define
displacements, strains and strain rate, while the last one only permits to observe the flow of particles through fixed points and
is especially useful in the fluid dynamics.
To describe the strain rate field the velocity gradient tensor L, defined as spatial gradient of the particle (or node, specifically)
velocities, will be used here. Because node velocities are just the time derivative of the mapping function, we may write:
 ∂2 f x
 ∂t ∂x
L= 2 0
 ∂ fy
 ∂t ∂x
0

∂2 f x 
∂t ∂y 0 
∂2 f y 
∂t ∂y 0 
(8)
The introduction of the velocity gradient L yields to the definition of two further tensors:
D=
1
1
(
L + LT ) and W = L − LT
2
2
(
)
(9)
that are referred to as the deformation rate tensor D and the spin tensor W, respectively.
In particular, we will use the deformation rate tensor D to evaluate the stretch rate along the direction defined by the unit vector
n as:
And the shear rate as:
ε&n = n D n
(10)
γ& = n1 D n2
(11)
where n1 and n2 are two orthogonal unit vectors.
Finally, notice that one can relate the strain rate tensor with the deformation rate tensor by means of the following equation:
E& = F T D F
(12)
In case of small deformations the tensor F is very close to the unity, which allows us to assume
E& ≅ D
(13)
In the following sections an algorithm for measurements of the strain rate tensor according to Eqs. (8 and 9) will be presented.
In addition, we may also say that for all the application shown within this paper the assumption of Eq. (13) is fully verified.
Image correlation procedure
Experimental techniques based on numerical correlation of white light speckle images have recently reached considerable
results in full-field strain measurement. These improvements are not only due to the continuous growth of digital cameras
resolution and computers capabilities but also to the introduction of new and more powerful algorithms that are particularly
suited for full-field analyses. In particular, the use of a “global approach” in image correlation, which constrains the computed
displacement field to be continuous across the entire images, allows a more reliable description of material behaviour.
In a global correlation approach, the field to be studied is divided into sub-images in the same way as the element mesh is built
for a finite element analysis. During the correlation computation, the elements on the deformed image cannot freely change
their shape to match the elements on the undeformed image, but it is required that their displacements accomplish the
congruency constraint caused by the shape functions that describe the deformation of each image element.
Considering the correlation between two images, the evaluation of the displacements may be faced as follows: the
undeformed picture is divided in a fixed and regular grid of square elements; the portion of the image that is framed by the four
corner nodes of each element is associated to them. On the deformed image a similar operation is performed, but here the
node grid is neither fixed nor regular. Consequently, to associate a square sub-image to each element of the deformed grid,
we have to remap the content of each of them into sub-images equal in size and shape to the undeformed ones. To perform
the remapping operation we have to resample the deformed image on a distorted pixel grid: a bilinear shape function is used
to locate the new sampling points and a bicubic interpolation scheme is adopted to estimate the gray level of the resampled
pixels.
The final goal of the correlation algorithm is to arrange the nodes of the deformed grid so that the sub-images coming from the
resampling operation look (in speckle distribution) as close as possible to the corresponding sub-images of the reference
image. This task can be mathematically posed as a nonlinear least squares minimization of the difference between the
element sub-images in the two configurations. Thus, the error e to be minimized may be written as:
e=
∑
sid ( X d ) − siu ( X u )
2
(14)
i
where Xd and Xu are the sets of node coordinates of the deformed and undeformed grid and sid and siu are respectively the i-th
sub-image taken out from the deformed and undeformed pictures. Considering a sequence of images, the deformed image will
T
be the generic j-th image and the set of its sub-images will be identified as: Sj = { s1j, …, sij, …} ; similarly the undeformed set
T
will be: S0 = { s10, …, si0, …} . Thus the aim of the fitting problem is to evaluate the coordinate set Xj that satisfies the following
equality:
S j (X j ) = S 0 ( X 0 )
(15)
The least squares solution of the previous equation can be computed by iteratively solving its linearized form, obtained through
a series expansion (stopped at the first term) of its left member, so that it can be written:
S j ( X ′) + ∑
k
∂S j
∂x k
(x
kj
− x k′ ) = S 0 ( X 0 )
(16)
X =X′
T
T
where X’ = { x’1, …, x’k, …} is a first guess of the unknown set of node coordinates Xj = { x1j, …, xkj, …} .
Moving Sj on the right, Eq. (16) yields to the following linear set of equations:
J ∆X = d
(17)
where J is the jacobian matrix whose elements are the derivatives ∂sij/∂xk of the deformed element sub-images relevant to
their node coordinates and evaluated at first guess node locations X’; ∆X is the increments vector that tells us how much the
previous guess should be increased to have a new estimation of Xj (when the ∆X norm is sufficiently small, we can stop to
iterate); d is the vector that stores the differences between the undeformed and deformed element sub-images. The norm of d
is the error e of Eq. (14).
The set of equations (17) is overdetermined because its coefficient matrix J has much more rows then columns. It can be
solved in three steps as follows:
A = JTJ ,
b = JTd ,
∆x = A−1b
(18)
The number of iterations needed to get a good estimation of Xj is typically less than ten. Nevertheless, this procedure is quite
time consuming because at each iteration we have to compute again the jacobian matrix that, though sparse, is a huge matrix.
A good strategy to cut down the computational load is to linearize Eq. (15) by working on its right term. If we now built the
series expansion of S0 we will get an equation quite similar to Eq. (16):
S j ( X ′) = S 0 ( X 0 ) + ∑
k
∂S 0
∂x k
(xk′ − xk 0 )
(19)
X =X0
that yields to a set of linear equations whose coefficient matrix can keep constant through the entire procedure. In fact, at the
end of each iteration the increment vector ∆X is not applied to the undeformed grid (as it should be, if the same procedure
shown above would have been used), but it is mapped onto the deformed grid to update its node coordinates. The mapping
rule between the undeformed and the deformed grid is simply the inverse of the transformation used to warp the quadrangular
elements of the deformed configuration to square sub-images.
As a consequence, now the steps required to get a second guess of the deformed node coordinates will be:
1) computation of the jacobian matrix J,
T
2) computation of the square and symmetric matrix A = J J,
3) factorization of A using Cholesky decomposition,
4) computation of the vector b = JT d,
5) computation of vector ∆X applying a back-substitution algorithm to A factors,
6) mapping ∆X from the undeformed to the deformed configuration,
7) updating of deformed grid coordinates;
Even if now a further step (the mapping operation) has been added, in the succeeding iteration only steps 4 to 7 have to be
repeated again and, above all, we need not to repeat the steps 1 to 3 for any of the successive correlation with the other
images of the sequence.
The evident advantage of this procedure in the correlation of two images is even more significant when, for the estimation of
the strain rate field, a multi-image correlation is attempted.
Suppose we want to process our images five by five (from index j-2 to j+2 across the generic j-th image) in order to make all
of them to be compliant with Eq. (15) and to make also the node coordinates xkj on the five images to be not free parameters
but to fulfill a linear regression on quadratic curve. That is to say:
S j −2 (X j −2 ) = S j −1 (X j −1 ) = S j (X j ) = S j +1 (X j +1 ) = S j + 2 (X j + 2 ) = S 0 ( X 0 )
(20)
xkj = a 0 kj + a1kj j + a 2 kj j 2
(21)
where xkj are the components of vector Xj. Notice that in Eq. (21) the image index j plays the role of the time, thus the
regression coefficient a1kj is, ultimately, the time derivative of the x coordinate of the k-th node at the time of the j-th image.
These values will be used to compute the components of the velocity gradient tensor L of Eq. (8).
Coming back to the multi-image fitting problem, we may now build the series expansion of the last right term of Eq. (20) that
yields to the following set of five sets of equations:

∂S0
 S j −2 (X ′j −2 ) = S0 ( X 0 ) + ∑
k ∂x k


M
M


∂S
 S j +2 (X ′j +2 ) = S0 ( X 0 ) + ∑ 0

k ∂x k
(x′
j −2 k
− xk 0 )
X =X0
M
(x′
j +2 k
(22)
− xk 0 )
X =X0
that can be rewritten in a more compact form as:
J [∆X j −2
∆X j −1
∆X j
∆X j +1
∆X j + 2 ] = [d j −2
d j −1
dj
d j +1
d j + 2 ] (23)
where ∆Xj-2 … ∆Xj+2 are the increments of node coordinates and the vectors dj-2 … dj+2 are the difference among the element
sub-images of the five configurations, Sj-2 … Sj+2, and the undeformed initial sub images S0.
This mathematical expression has not the meaning of single set of equations, but still contains five separate sets that can be
solved disjointedly. Its iterative solution satisfies Eq. (20) (in the least squares sense) but not yet Eq. (21). To introduce the
constrain due to the wish of smoothing the node displacements by a linear regression on five time steps by a quadratic curve,
a weighting matrix W is appended to the right term of Eq. (23) to couple the five sets of equations each other. Doing so, we
obtain the following expression:
J [∆X j − 2 L ∆X j + 2 ] = [d j −2 L d j + 2 ] W
(24)
where W is equal to:
 31 9
 9 13
1 
W=
 − 3 12
35 
− 5 6
 3 − 5
−3 −5 3 
12 6 − 5

17 12 − 3

12 13 9 
− 3 9 31 
(25)
The origin of the W coefficients is in linear regression theory. In fact, if we want to fit a parabola y= a0 + a1t + a2t2 on five
equispaced samples y1 … y5 at time instants t = {-2, -1, 0, 1, 2}, we should solve the following overdetermined set:
 y1 
1 − 2 4
y 
1 − 1 1 a

  0   2 
1 0 0  a1  =  y 3 

   
1 1 1 a 2   y 4 
1 2 4
 y5 
(26)
By calling m the coefficient matrix, a the vector of the three unknown coefficients and y the vector of the five known samples,
to compute the values y’ of the fitting parabola at the sampling locations, we have, at first, to evaluate a=(mTm)-1mTy, then
premultiply it by m:
y ′ = m a = m (m T m ) m T y = W y
−1
(27)
Where the product m(mTm)-1mT is precisely the weighting matrix W.
The matrix W couples the sets of Eq. (23) because their right terms are now a linear combination of the original dj-2 … dj+2
vectors and also because the solutions ∆Xj-2 … ∆Xj+2, modifying all five deformed grids, cause the need of updating all d
vectors at each iteration.
The last issue is the computation of node speeds, that is to say the a1 coefficient of the fitting parabolas. From the previous
simple regression example, we can obtain (by expanding the (mTm)-1mT product) that:
dy
1
= a1 = [ − 2 − 1 0 1 2] y
dt t = 0
10
(28)
This rule, applied to node coordinates, yields to:
dX j
dt
t =t j
 X Tj−2 
 T 
 X j −1 
1
[ − 2 − 1 0 1 2]  X Tj 
=
10
 T 
 X j +1 
 X Tj+ 2 


(29)
where tj is the time of the j-th image.
The entries of vector dXj/dt = {v1j, …, vkj, …}, which are the node velocities, will then be mapped by a bilinear shape function
within each element to allow the computation of their spatial gradient for the evaluation of the velocity gradient tensor L as
shown in a previous section.
Finally notice that the solution of Eq. (24) is obtained by applying five times the same back-substitution subroutine used in the
correlation of two images. Also, since the jacobian matrix J has been computed for the undeformed configuration, its
factorization remains unchanged for the entire processing of the whole sequence. Thus it should be computed only once.
On a PC unit equipped with a Intel Core Duo T2400 1.83 GHz and 1 GB Ram the presented procedure requires about 0.8
seconds to process each time step from a sequence of images with 512 × 1024 pixels where a grid of 96 elements with 80 ×
80 pixels is defined.
Applications
The algorithm was first employed for the investigation of the Portevin-Le Châtelier (PLC) effect, a phenomenon of repetitive
yielding that may occur during the plastic deformation of metal alloys. Depending on the experimental conditions, different
patterns of correlation exist among the regions where the deformation intermittently concentrates. These regions, called PLC
bands, and their correlation patterns are the object of our investigation because they are a bridge between macroscopically
measurable deformations and the mechanisms in the atomic lattice that originate the effect. In our experimental setup, a
tensile specimen is continuously imaged by means of two cameras, a line-scan camera for a real-time survey of the whole
gauge length and a high-speed camera that frames a small region and is triggered by the first one. A more detailed description
of the phenomenon and of the experimental procedure is given in [12], nonetheless we want to draw the attention to the fact
that these plastic instabilities originate from a strain-rate sensitivity of the material. Indeed, yieldings appear in the form of
shear bands and are intrinsically regions where the strain rate concentrate. Although this concentration is the object of the
mathematical models for the description of the phenomenon [13], no measurements of the strain rate distribution have been
previously performed. This task poses in fact great technical difficulties because of the short duration of a single PLC band,
underneath 10ms, and can be undertaken only with high-speed cameras. Until now, the strain rate inside a PLC band could
only be measured with small strain gauges [14] or estimated from some boundary conditions [13], but we propose here a fullfield technique based on picture sequences taken with the high-speed camera at a rate of 1000 fps. Since the camera frames
a small portion of the specimen, an 8 × 4 mm² area with a spatial resolution of 8 µm/pixel, it needs to be triggered in order to
capture pictures only in case of band emergence in that area. This full-field measure permits the tensorial calculations
therewith the determination of the shear rate, which is of relevant importance considered that the bands originate from a
shearing mechanism. Advantage of this approach is that the shear rate values can be directly employed in the constitutive
equations that relate the deformation to the dislocations dynamics without passing through the calculation of resolved shear on
the base of strain values as it is currently done [15].
Figure 3. The distribution of equivalent strain and of displacements in horizontal and vertical direction in a type C PLC band.
The last scheme indicates the relative movements between the two parts of the specimens.
Figure 4. Time evolution of longitudinal strain rate on a grid node close to a PLC band.
The novel algorithm permits to measure and visualize the distribution of strain and displacements inside PLC bands. In Fig. 3
an example of these distributions is given for a type C band; the three fields refer to the same time instant and show a highly
concentrated strain and a relative displacement between the two specimen parts that is oriented along the band direction.
Figure 5. Evolution of the shear rate in a type C PLC band. The framed area is 8 × 4 mm², the time interval between the
-1
frames is 1 ms and the colour scale is from 0 to 1.8 s .
PLC bands are also a short-lived phenomenon as shown in Fig. 4. In fact, the evolution of the shear strain rate in occasion of
this band can be seen in Fig. 5, where a shear rate avalanche quickly crosses the specimen within a time of about 5 ms.
Before this fast movement, the band was announced by the anomalous orientation of the shear rate in Fig. 6. The frame in this
figure precedes by 4 ms the sequence in Fig. 5 and refers to an instant in which the shear rate, although approximately null, is
not uniformly oriented. Indeed, the direction of the maximal shear rate is generally about 45°, as in any uniform tensile state,
but it is vertical inside the band. This pattern remains during the whole band emergence.
Figure 6. The direction of the maximal shear rate an instant before the appearing of the band in Fig. 5.
The shear rate is oriented vertically inside the band, by 45° outside.
Conclusions
In this research we developed and applied a new method of full field strain measurement based on white light speckle image
correlation. Starting from a tensorial definition of the strain rate, the image correlation procedure we propose here has the
advantage of requiring fewer steps to calculate successive guesses of the deformed node coordinates and allows for the
estimation of the strain rate field with a multi-image correlation attempt. This optimized computation technique requires less
than 1 second on a commercial personal computer to process each time step for a sequence of images with 512 × 1024
pixels.
The algorithm was tested for the investigation of the Portevin Le Chatelier effect and demonstrated to be fully satisfactory in
detecting the characteristics of the PLC bands. Moreover, with an image acquisition rate of 1kHz we succeeded in measuring
the evolution of the shear rate in a type C PLC band. We believe the method holds the promises to reach a quantitative
evaluation of this and of other more common effects either on metal sheets and on other material types and shapes.
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