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The grid method for in-plane displacement and strain
measurement: a review and analysis
Michel Grediac, Frédéric Sur, Benoît Blaysat
To cite this version:
Michel Grediac, Frédéric Sur, Benoît Blaysat. The grid method for in-plane displacement and
strain measurement: a review and analysis. Strain, Wiley-Blackwell, 2016, 52 (3), pp.205-243.
�10.1111/str.12182�. �hal-01317145�
The grid method for in-plane displacement and
strain measurement: a review and analysis
Michel GREDIAC
1†
, Fr´ed´eric SUR
2
, Benoˆıt BLAYSAT
1
1
Clermont Universit´e, Universit´e Blaise Pascal, Institut Pascal, UMR CNRS 6602
BP 10448, 63000 Clermont-Ferrand, France
2
Laboratoire Lorrain de Recherche en Informatique et ses Applications, UMR CNRS 7503
Universit´e de Lorraine, CNRS, INRIA projet Magrit, Campus Scientifique, BP 239, 54506
Vandoeuvre-l`es-Nancy Cedex, FRANCE
†
corresponding author, tel: +33 4 73 28 80 77, fax: +33 4 73 28 80 27,
michel.grediac@univ-bpclermont.fr
Abstract: The grid method is a technique suitable for the measurement of in-plane dis-
placement and strain components on specimens undergoing a small deformation. It relies
on a regular marking of the surfaces under investigation. Various techniques are proposed
in the literature to retrieve these sought quantities from images of regular markings, but
recent advances show that techniques developed initially to process fringe patterns lead
to the best results. The grid method features a good compromise between measurement
resolution and spatial resolution, thus making it an efficient tool to characterize strain
gradients. Another advantage of this technique is the ability to establish closed-form
expressions between its main metrological characteristics, thus enabling to predict them
within certain limits. In this context, the objective of this paper is to give the state of
the art in the grid method, the information being currently spread out in the literature.
We propose first to recall various techniques which were used in the past to process grid
images, to focus progressively on the one which is the most used in recent examples: the
windowed Fourier transform. From a practical point of view, surfaces under investigation
must be marked with grids, so the techniques available to mark specimens with grids
are presented. Then we gather the information available in the recent literature to syn-
thesise the connection between three important characteristics of full-field measurement
techniques: the spatial resolution, the measurement resolution, and the measurement bias.
Some practical information is then offered to help the readers who discover this technique
to start using it. In particular, programmes used here to process the grid images are
offered to the readers on a dedicated website. We finally present some recent examples
available in the literature to highlight the effectiveness of the grid method for in-plane
displacement and strain measurement in real situations.
Keywords: displacement, full-field measurement, grid method, strain.
This is the author-manuscript version of
M. Gr´ediac, F. Sur, and B. Blaysat. The grid method for in-plane displacement
and strain measurement: a review and analysis. Strain, vol. 52, no. 3, p. 205–
243, Wiley, 2016.
DOI: 10.1111/str.12182
1
1 Introduction
Full-field measurement techniques are now widespread in the experimental mechanics com-
munity [1, 2]. This is mainly due to the combined effect of the decreasing cost of cameras,
their increasing performance, and the very nature of the quantities they provide: fields of
measurements such as displacements or strains. Indeed this wealth of data can be used
in different ways for a better characterisation of materials and structures. It offers for
instance the possibility to validate the output of numerical models obtained with finite el-
ement calculations, to observe phenomena whose location cannot be predicted in advance,
to help propose suitable constitutive models able to describe and predict these observed
phenomena, and even to identify their governing parameters. Some techniques based on
incoherent light such as moir´e [3, 4, 5, 6] have been known for a long time but they require
the use of two gratings: a reference and a deformed one. Others such as electronic speckle
interferometry [7], holographic interferometry [8] and moir´e interferometry [9] rely on co-
herent light, so they are somewhat complicated to implement in experimental mechanics
laboratories. Because of their ease of use, white-light techniques have become more pop-
ular in experimental mechanics. In particular, digital image correlation [10] is now widely
accepted, as reflected in the huge body of literature available on this technique or on its
use in experimental mechanics, see for instance the review proposed in [11]. Beside this
well-known technique based on the processing of randomly marked surfaces, a white-light
technique based on the processing of regular patterns, namely the grid method, is also
employed. This latter technique combines three main advantages:
• Like DIC, this is a non-interferometric technique;
• It can potentially take benefit of some efficient and well-established procedures de-
veloped for processing fringes obtained with interferometric techniques;
• It relies on reproducible patterns.
This last feature can be an asset for future standardization of full-field measurement
techniques. The main drawback for the grid method is that the surface must be marked
with a pattern, which must be as regular as possible. However, even optimised random
patterns have been defined for DIC [12], thus depositing a controlled pattern is also an
issue with DIC.
In this context, the main objective of this paper is to focus on the technical charac-
teristics and the metrological performance of this contactless optical method. The paper
is organized as follows. Regularly marked surfaces have been employed for a long time
to measure displacement and strain components, but processing images of these regular
markings can potentially be made with different approaches which are briefly presented
in Section 2. We discuss then the different techniques available to mark surfaces with
grids in practice. This is the aim of Section 3. Section 4 is a reminder on the windowed
Fourier transform (WFT), which is the most commonly technique used to process grid
images. Providing the metrological performance is necessary for any measuring tool, so
Section 5 discusses in depth the performances and limits of the grid method, with a partic-
ular emphasis put on three metrological characteristics which are first defined, namely the
measurement resolution, the bias and the spatial resolution. Then we discuss in Section 6
some practical problems raised when using this technique, before finally presenting in
the last section some recent applications of the grid method for material characterisation
purposes.
2
2 The grid method: what are we talking about?
We address here the measurement of in-plane displacement and strain components by
processing images of flat specimens on which a regular marking such as a grid has been
deposited before in-plane deformation. The idea is quite old since it flows from intuition:
considering a regular marking enables us to easily distinguish features such as particular
points on deformed marked surfaces, and thus facilitates tracking them during deforma-
tion. The use of grids to build regular distributions of points by considering either the
intersection of the lines or the dot points located in between is broadly documented in the
literature of the 40’s-60’s, as reported in a survey paper published in the late 60’s [13]. Iso-
lated but regularly distributed points were also employed in the same spirit [14]. At that
time, pictures of deformed grids were processed by hand, by measuring the coordinates of
the points, and deducing the displacement by subtracting the coordinates between current
and reference configurations. Even though the progressive dissemination of affordable dig-
ital cameras and powerful computational resources together with recent advances in image
processing led finding the coordinates of the points automatically and more precisely than
by hand [15], this technique was mainly used for measuring large displacements and strains
like those occurring in composite or metal sheet forming [16, 17, 18, 19, 20, 21]. It is now
only scarcely mentioned in recent papers such as [22, 23, 24], and seems to fall progressively
into disuse. The main cause is the emergence and dissemination of digital image correla-
tion (DIC) in the experimental mechanics community [25, 10]. DIC can potentially process
images of regular markings [26] but this technique mainly relies on randomly marked sur-
faces, which undoubtedly constitutes an advantage from a practical point of view. This
has led this technique to be now widely accepted and employed in experimental mechan-
ics, as illustrated in [27]. Despite the versatility and the user-friendliness of DIC, this
technique suffers from some disadvantages pointed out by experts in the field, for instance
in case of non-homogeneous and small deformation [11]. On the contrary, interferometric
techniques exhibit a better measurement resolution, but they require the use of various
optical components such as mirrors, lasers or beam splitters, which make them difficult to
use in an industrial context. In addition, these techniques are often highly sensitive to vi-
brations. Trying to overcome the disadvantages of DIC without relying on interferometry
has led some researchers to still consider regular grids instead of random patterns, this
time not by tracking the intersecting points between the lines or the dot points located
in between, but by employing numerical tools developed for a similar objective: namely
processing fringes by extracting phase distributions from regular patterns.
It is clear that from a practical point of view, depositing regular patterns may appear
incongruous since this induces an additional problem compared to merely spraying a ran-
dom marking and applying DIC. This is all the truer as grids featuring a frequency of
some lines per millimeter are the most suitable ones for thoroughly analyzing phenomena
which commonly occur at the scale of usual tensile specimens for instance. However such
grids are presently not easily commercially available. This leads users to order specifically
printed grids, and then transfer them on the specimens to be tested, see [28] for instance,
which does not facilitate the dissemination this technique. Despite this relative drawback,
this technique has a big advantage: the possibility to reveal localized phenomena thanks to
its good compromise between spatial resolution and measurement resolution, as illustrated
in the examples shown in Section 7, which makes it an appealing tool.
As a last remark, it is worth mentioning that the grid method is basically suitable for
in-plane measurements, but it has been recently shown that it could be extended to the
simultaneous measurement of out-of-plane displacement [29].
3
The grid method would not exist without techniques able to deposit grids on the
surface of specimens, so let us now examine the procedures which can be employed to
deposit grids on the surface of flat specimens.
3 Marking surfaces with grids
As recalled above, grids have been deposited on flat specimens for a long time. In [13,
30, 31], comprehensive lists of techniques available for obtaining grids were given some
decades ago. Some of these techniques are still valid with current printing techniques, but
the pitches reached and the quality of the obtained grids are generally not available. We
thus offer below a state of the art on the techniques employed to mark specimens with
grids, by going from the finest to the largest scales.
The finest grid that has been considered for in-plane deformation measurement seems
to be the atomic lattice of the substrate itself observed with high resolution electron
microscopes (HREM), see Ref. [32, 33, 34], or atomic force microscopes (AFM) [35, 36, 37].
With such natural markings, the grid pitch can be as small as some angstr¨oms.
At a higher scale, pitches of some tenths of µm to some µm can be obtained by
various techniques such as nanoimprint lithography [38]. With the so-called electron beam
moir´e technique, the lines of the reference grid are ”drawn” on the substrate by using the
electron beam of a scanning electron microscope (SEM) [39, 40, 41, 42], or milled using a
focused ion beam [43, 44, 45, 46]. Microgrids can also be obtained by metal sputtering and
evaporation [26, 47, 48] or by appropriate etching of the specimen [49]. A simple technique
also consists in replicating the negative imprint of a mold grating, see [50, 51, 52, 53, 54] for
instance. In this case, the surface of the specimen is polished, cleaned and gel-coated with
a suitable epoxy resin beforehand. Optical techniques can also be employed. For instance,
photoresist coatings insulated through a mask lead to grid featuring similar pitch values as
in the preceding examples [55, 56]. An interferometric setup can also be used to generate
a regular marking, as described in [57].
Even at this scale, grids can be used to form moir´e fringes from which in-plane displace-
ment fields are deduced. The reference grid is typically the SEM monitor [58] in this case,
or a fringe pattern generated by a computer [37], while the deformed grid is that printed
on the specimen. Another technique is based on such microgrids: moir´e interferometry. In
this case, the grid deposited on the specimen acts as a diffraction grating and interference
fringes are obtained with two coherent laser beams [9, 59]. Microgrids can also be used to
improve the contrast of the natural surface marking, the images being then processed by
tracking the dot points [26] or by using digital image correlation [60]. Microgrids can also
be considered as regular patterns whose phases are modulated by the displacement, thus
justifying to employ techniques based on Fourier analysis to retrieve the displacement and
strain fields [32, 33, 34, 57, 61] for instance.
Grid frequencies lying between 250 and 2000 lines per inch (thus pitches lying between
102 and 13 µm, respectively) are reported in [62] by evaporation of aluminium through
a mask. In a more recent work, similar grids were obtained by depositing a thin layer
of metallic film onto the specimen surface, which was covered beforehand with a copper
gilder grid acting as a mask. A very thin layer of gold was then sputter-coated onto the
specimen surface. The zone underneath the mesh bars was not coated contrary to the
space inside each grid, thus generating regular arrays of spots. Pitches equal to 12.5 µm
are reported in [22], but other pitch values can potentially be obtained with the same
technique: from 12.5 to 500 µm in [63].
In the preceding cases, the imaging system which captures the grid images is often
4
![Figure 13: Strain field near a crack tip in aluminium specimen (see [203]). dimensions in pixels, 1 pixel represents 40 µm on the specimen. a- shear strain field. b- close-up of zone A. c- close-up of zone B](/figures/figure-13-strain-field-near-a-crack-tip-in-aluminium-3eyecwng.webp)
