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Young Stress Analyst Competition 16th International Conference on Experimental Mechanics University of Cambridge, UK 2014
Young Stress Analyst Competition
16th International Conference on Experimental
Mechanics
University of Cambridge, UK
2014
CONTENTS Young Stress Analyst Competition Finalists Dislocation mediated hardening and relaxation in nanocrystalline palladium films revealed by on-chip HRTEM in-situ nano mechanical testing
Behnam Amin-Ahmadi, University of Antwerp (Belgium) .......................................... 1 Measuring dynamic inter-particle force transmission in opaque granular materials
Ryan Hurley, California Institute for Technology (USA) ............................................ 6 Validation of a modal frequency response model using pulsed laser DIC and image decomposition
Christopher Sebastian, University of Liverpool (UK) ................................................. 10 Dynamic response of hierarchical materials
Ramathasan Thevamaran, California Institute for Technology (USA) ......................... 13 Optical metrology with Hilbert-Huang fringe pattern analysis for experimental mechanics Maciej Trusiak, Warsaw University of Technology (Poland) ......................................... 17 P-adaptivity in digital image correlation Lukas Wittevrongel, University of Leuven (Belgium) ................................................... 24 First Reserve: Application of integrated digital image correlation to dynamical blade/casing interactions: towards estimation of contact force Romain Mandard, Ecole Centrale de Lille (France) ..................................................... 27 Second Reserve: Automatic optical crack tracking for double cantilever beam specimens Brett Krull, University of Illinois at Urbana-Champaign (USA) ...................................... 35
Dislocation mediated hardening and relaxation in nanocrystalline palladium films revealed by on-chip HRTEM in-situ nano mechanical testing
B. Amin-Ahmadi
Electron Microscopy for Materials Science (EMAT), University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium
PhD supervisors: Dr. H. Idrissi, Prof. Dr. D. Schryvers
Objectives:
Nanocrystalline (nc) metals present excellent mechanical properties such as strength and fatigue resistance but often show low ductility. Furthermore, nc systems show moderate to high rate sensitivity at room temperature which might help restoring the ductility but can have disadvantage of creep/relaxation effects in applications. Thin metallic films constitute ideal candidates for looking at the mechanics of nc systems as they can be easily produced with nano-grained structures frequently involving only one grain over the thickness (for high resolution transmission electron microscopy (HRTEM) observation) and sharp textures. Time dependent relaxation/creep mechanisms are amplified in nc materials compared to traditional microcrystalline systems which can be due to a change in deformation mechanisms. Different methods have been used to characterize the rate sensitive creep or relaxation behaviour at the nanoscale, involving nanoindentation, bending and direct tensile testing. The main shortcomings involve the difficulty to impose very small strain rates typical of real applications and to perform in-situ relaxation tests in order to characterize the deformation mechanisms. Also, the sensitivity to drift is increased at the nanoscale, which hampers imposing a constant load during long periods of time. In the present project a novel technique for stress and strain evolution measurement called "on-chip testing" and allowing in-situ HRTEM observations has been used for the first time for creep/relaxation experiments of nc Pd free standing beams.
Results and discussion
The concept of the on-chip method is to use the internal stress present in a long beam, ‘the actuator’ (30 nm-thick Si3N4), to deform another material attached to it, ‘the specimen’ (90 nm-thick ~2 µm wide Pd ribbon) by selectively back etching the Si substrate and SiO2 sacrificial layer yielding the actuator to contract and deform the Pd beams and enable direct in-plane TEM observation (Figure 1a). The stress and strain in the deformed Pd specimens are provided by the measurement of the specimen elongation. A complete stress-strain curve is generated by varying the actuator versus specimen length ratio to impose different
1
deformations. The relaxation experiment is performed by measuring the deformation at different time intervals. Note that both stress and strain vary with time. The stress and strain evolution during relaxation has been measured under pure uniaxial tension (Figure 1b). The fracture strain is around 3% and the yield stress ~500±60 MPa close to the internal stress measured after deposition. The mean activation volume was also measured for different plastic strains shown in the inset of Figure 1b.
Figure 1. On-chip testing method for mechanical characterization. (a) Schematic view of the structures allowing in-situ TEM observation. Notches have been milled at every two beams using focused ion
beam (FIB). (b) Stress-strain evolution of different Pd beams under uniaxial tension. The linear elastic regime has been measured by nanoindentation as equal to 120 GPa. The initial strain level εpr before
relaxation is indicated on the figure. The inset presents the mean activation volume evolution as a function of the initial strain εpr present in the different beams before relaxation. The activation volume
decreases when the initial strain level increases. (the on-chip method was developed at Université catholique de Louvain and the samples were prepared by Marie-Stéphane Colla [1])
The microstructure and texture of the as-deposited Pd films evaporated at 1 Å/s were characterized using both cross-sectional and plan-view thin foils prepared by FIB (Figure 2). The structure is columnar with 2 or 3 grains over the thickness and in-plane grain diameter of ~30 nm. The microstructure involves Σ3 60° {111} coherent twin boundaries (TBs) in ~ 25% of the grains. Automated Crystallographic Orientation Mapping in TEM (ACOM-TEM) shows that a fibre texture exists with the [110] direction oriented perpendicular to the film, with no in-plane preferential orientation.
2
Figure 2. a) ACOM-TEM orientation mapping of as deposited Pd film. Corresponding inverse pole figure along different directions are shown. b) Grain size distribution of (a). (c) Bright field
micrograph obtained on cross-sectional as-deposited Pd film (d) HRTEM image obtained in as-deposited films showing a Σ3 60° {111} perfectly coherent TBs.
HRTEM was performed in a TECNAI G2 (FEG, 200 kV) microscope in Antwerp. As soon as possible after the release of the films in Louvain-la-Neuve, the samples are introduced in the TEM for a first observation (t=0). The HRTEM observations were made in the notched area of the Pd beams to ensure the observation of a plastically deforming or relaxing zone and to provide landmarks for the follow-up of exactly the same well-oriented grains. Successive measures are performed by leaving the sample in the holder over a period of several days (up to more than a month) and returning to the microscope at specific intervals. The dislocation density was measured by counting extra half planes on inverse fast Fourier transforms generated from HRTEM images using masks applied on each g vector. The measurement of the dislocation density in a given grain at different times shows relaxation of the Pd beam with time.
During relaxation, the dislocation density decreases and reaches a steady-state regime (Figure 3a). Interestingly, Lomer-Cottrell dislocations have been detected. This kind of sessile dislocations are very effective to induce hardening and are typical of fcc metals with medium to high stacking fault energy (SFE), like Pd. More surprisingly, the Lomer-Cottrell locks disappear with time (Fig. 3b-d). Recently, a similar formation and breaking of Lomer-Cottrell junctions has been observed in Pt thin films with 10 nm grain size [2]. Numerical simulations predict that a stress of ~750 MPa is needed to destroy the junction in Pd [3], which falls in the range attained in this study.
3
Figure 3. (a) Average dislocation density (red solid squares) evolution of the Pd film versus time. The dislocation density in as-deposited Pd films is 4±0.7×1016 m-2. On the graph, t=0 represents the
dislocation density measured 4 hours after the release of the films. Dislocation density first increases when deformation is applied and then decreases upon relaxation. Numbers beside unfilled circles
indicate the corresponding grain size (nm). (b) to (d) Filtered HRTEM images showing the continuous evolution of the dislocation positions with time. Note the formation and the destruction of Lomer-
Cottrell dislocations indicated by L-C in (c). The perfect dislocations are indicated by "T" symbols.
Twins constitute also strong barriers to dislocation motion. Initially, in the as-deposited film, most of the Σ3 60° {111} TBs are perfectly coherent (Figure 2d). However, this coherency is lost during deformation due to the interaction of lattice deformation dislocations with the TBs (Figure 4a). The loss of coherency of these TBs keeps increasing upon relaxation as indicated by the progressive increase of the TB1 and TB2 thicknesses (Figure 4b and 4c). The TBs thickness term is defined as the distance separating the two last non-distorted twinning planes. Figure 4d shows the evolution of the TB thickness ratio with time, revealing an initial TB thickness increase and a saturation after 10 days which is in agreement with the dislocation density changes with time.
4
.
Figure 4. (a) to (c) HRTEM images showing Σ3 {111} TBs at t=0, 3 and 36 days, respectively. Note the increase of the TBs thickness from (a) to (b) in the filtered images at the upper right insets. No
significant change of the TBs thickness from (b) to (c) is observed. (d) Evolution of the TBs thickness ratio (T measured / T initial) of the TB1 and TB2 in (a).
Conclusions
As a summary, large creep rates are unexpectedly observed at room temperature. Despite the small 30 nm grain size, the relaxation mechanism is found to be mediated by the stress driven thermally activated nucleation and propagation of dislocations. The dislocations interact with the growth nanotwins present in the grains, leading to a loss of coherency of the twin boundaries. The density of stored dislocations first markedly increases with applied deformation, then decreases with time to drive additional deformation while no grain boundary mechanisms are observed. The impact of this fast relaxation can constitute a key issue in the development of a variety of micro- and nanotechnologies, such as Pd membranes used in hydrogen applications.
References
[1] M. Coulombier, G. Guisbiers, M. S. Colla, R. Vayrette, J. P. Raskin, T. Pardoen, Rev Sci Instrum. 2012 (83) 105004.
[2] L. Wang, X. Han, P. Liu, Y. Yue, Z. Zhang, E. Ma, Phys. Rev. Lett. 2010 (105) 135501.
[3] D. Rodney, R. Phillips, Phys. Rev. Lett. 1999 (82) 1704.
5
Measuring Dynamic Inter-Particle Force Transmission in Opaque
Granular Materials
Ryan Hurley
Division of Engineering & Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
1 Introduction
The macroscopic behavior of granular materials, in applications ranging from geophysics to de-
fense, is governed by the particle-scale: inter-particle forces and particle kinematics. Researchers
have recognized this connection and have spent decades studying links between macroscopic and
microscopic quantities in an effort to better understand and model granular media. For instance,
researchers have studied relationships between inter-particle forces and macroscopic strength [1],
wave propagation [2], and dynamic friction [3]. Researchers have also developed experimental inter-
particle force measurement techniques, most notably those employing photoelasticty [4]. However,
such techniques remain limited to transparent birefringent materials, are applicable only to disks
in quasi-static settings, and do not resolve individual inter-particle forces in dynamic experiments.
This summary describes a new experimental technique, described graphically in Fig. 1, for
inter-particle force measurement in arbitrarily shaped opaque granular particles in both quasi-
static and dynamic settings. The objective of developing this method is to provide a new tool
for quantitatively studying the role of particle-scale response in macroscopic dynamic behavior,
particularly in opaque materials with arbitrary shape. The method can be applied to infer inter-
particle forces when the number of unknown force components exceeds the number of momentum
balance equations. An experimental example is presented for validation.
AVERAGEINTRA-PARTICLE
STRAINS
INVERSE PROBLEM FOR FORCES
INTER-PARTICLEFORCES
MULTI-OBJECTIVE OPTIMIZATION
DYNAMICEXPERIMENT
IMAGING
CONTACT TOPOLOGY
NpcX
c=1
f c = mpacm
NpcX
c=1
xc ⇥ f c ⇡ mp(xcmp ⇥ acm
p )
| {z }Kmf=bm
NpcX
c=1
xc ⌦ f c ⇡ Vp� + mp(xcm ⌦ acm)
| {z }Ksf=bs
f = arg minf
(||Ksf � bs||2 + �||Kmf � bm||2)
Figure 1: Method for inferring inter-particle forces.
6
2 The Method
As visually described in Fig. 1 the method described in this summary has two main steps: (1)
measurement of average intra-particle strains, contact points, and particle material constitutive law
parameters; (2) solution of an inverse problem to infer inter-particle forces.
In the examples presented here and any 2D experiment, measurement of average intra-particle
strains can be achieved with digital image correlation (DIC) and measurement of contact points
and particle edges can be achieved with Hough transforms. A constitutive law for the particle
material can be assumed and the relevant parameters can be measured using standard laboratory
stress-strain tests.
The formulation of the optimization problem for inferring inter-particle forces from measure-
ments involves momentum balance equations and relationships linking average intra-particle stresses
to contact forces. The momentum balance equations are written for each particle as
Nc∑
c=1
f c = mpacmp (1)
Nc∑
c=1
xc × f c = mp(xcmp × acmp ) +
∫
Vp
ρp(x× a)dv (2)
where Nc is particle p’s number of contact points, xc is the location of contact point c, Vp, ρg, mp,
xcmp , and acmp are the volume, density, mass, location of center of mass, and acceleration of center of
mass of the particles, and x and a are the position and acceleration relative to those of the center
of mass within the particle.
A relationship between intra-particle stresses and contact forces can be derived as
Nc∑
c=1
xc ⊗ f c = Vpσp +mp(xcm ⊗ acm) +
∫
Vp
x⊗ ρgadv (3)
where ⊗ represents a tensor (dyadic) product and σp is the volume-averaged stress in particle
p. The volume-averaged stress can be calculated from the average intra-particle strain using an
assumed constitutive law, for instance linear elasticity.
The numerical procedure used in this method is a multi-objective optimization scheme incor-
porating Eqs. (1)-(3). Eqs. (1) and (2) can be combined in the matrix form Kmf = bm where bm
contains the right hand sides of the equations, and f contains all unknown force components. Eq.
(3) can be written as Ksf = bs. The multi-objective optimization problem to solve is then
f = arg minf
(||Ksf − bs||2 + λ||Kmf − bm||2) (4)
where λ is a weight selected as the knee point of the optimal trade-off curve [5]. In practice, as in
the examples in this paper, the integral terms in Eqs. (2) and (3) can be neglected when solving
2
7
Eq. (4) with negligible effect. It is noted that the governing equations hold for any particle shape.
3 Example and Validation
The method outlined in this summary was validated in a simple example using a comparison with
a finite element (FE) simulation. A small 0.4m x 0.6m table was constructed with an air chamber
beneath the top surface fed by a 680m3/h fan (Fig. 2a). Air escaped from the tabletop through
1.6mm diameter holes drilled on a grid with 19mm spacing on the table top (Fig. 2b).
Six 44.45mm diameter x 6.35mm thick disks were cut from 60A durometer polyurethane. The
disks’ Young’s modulus was measured at E = 5.85 MPa through compression tests up to 13% strain
and Poisson’s ratio was estimated as ν ≈ 0.5. A concentric 38.1mm diameter x 1.19mm divot was
cut from the bottom of each disk (Fig. 2c), allowing the disks to float on a frictionless bed of air.
A speckle pattern was painted on the top of each disk to facilitate DIC measurements (Fig. 2b).
TABLETOP
LIGHTCAMERA
RIGID IMPACTOR v0
x
yRIGID
BLOCK
RIGIDBLOCK19mm
19mm
(a)
(b)
(c) (d)
12
56
34
Figure 2: (a) Experiment setup. (b) Table top and particle speckle. (c) Divot. (d) Initial conditions.
Two stiff wooden blocks were fastened to the tabletop. A stiff wooden impactor compressed
the disks between these blocks at an impact velocity of vx = −1.141m/s and vy = −0.66m/s (Fig.
2d). The 0.02 second impact event was captured with a Phantom v310 high-speed camera at 5000
frames per second. At each frame, intra-particle strains (measured with DIC software Vic2d) and
contact topology were measured and Eq. (4) was solved. Linear elasticity was assumed for the
grains to obtain the average intra-particle stresses.
A plane-stress FE simulation with the same initial conditions was performed in Abaqus. Fig. 3
shows the excellent agreement between experimental and FE strain fields and forces for a single time
step, as well as for the entire evolution of dynamic forces at major contact points. Given the poten-
tial uncertainty in initial conditions provided to the FE simulation, this agreement demonstrates
the remarkable accuracy of the method to capture the entire time-history of force evolution.
3
8
F (N) F (N)
F (N)F (N)
F (N) F (N)
Time (s)Time (s)
CONTACT 5 CONTACT 6
Time (s) Time (s)
Time (s)
CONTACT 1
CONTACT 3 CONTACT 4
CONTACT 2
Time (s)
EXPERIMENTFORCE MAGNITUDE FORCE MAGNITUDE
FE
(a)
t = 8ms
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
0 0.005 0.01 0.015 0.020
10
20
30
40
Time (s)
Force
(N)
FEMEq.(16)
Printed using Abaqus/CAE on: Thu Nov 21 22:21:29 Pacific Standard Time 2013
t = 8msEXPERIMENT FE
✏xx ✏xx
✏ij �0.04 ✏ij � 0.04✏ij = 0
(b)
FE FE
FE
FEFE
FEEq. 4
Eq. 4 Eq. 4
Eq. 4
Eq. 4Eq. 4
(c)
Figure 3: (a) DIC and FE strain field comparison. (b) Experimentally inferred and FE measuredforce comparison. (c) Time-history force comparison. See Fig. 2 for contact point numbers.
4 Conclusion
This study has illustrated a new experimental technique for measuring inter-particle forces in opaque
granular materials. The technique provides particle-level information in dynamic settings that ex-
isting methods like photoelasticity cannot. The formulation allows force inference in arbitrarily
shaped grains and an experiment on such shapes will be the focus of future work. As the devel-
opment of 3D imaging techniques progresses, the method offers the potential promise of allowing
inter-particle force measurement in opaque 3D grains such as those in real geologic materials.
References
[1] L. Rothenburg and R. J. Bathurst. Analytical study of induced anistropy in idealized granular materials.
Geotechnique, 4(1):601–614, 1989.
[2] E. Somfai, J.-N. Roux, J. H. Snoeijer, M. van Hecke, and W. van Saarloos. Elastic wave propagation in
confined granular systems. Phys. Rev. E, 72:021301, Aug 2005.
[3] F. da Cruz, S. Emam, M. Prochnow, J.-N. Roux, and F. Chevoir. Rheophysics of dense granular materials:
Discrete simulation of plane shear flows. Phys. Rev. E, 72(2):021309, 2005.
[4] T. S. Majmudar and R. P. Behringer. Contact force measurements and stress-induced anisotropy in
granular materials. Nature, 435(1079):1079–1082, 2005.
[5] K. Deb and S. Gupta. Understanding knee points in bicriteria problems and their implications as preferred
solution principles. Eng. Opt., 43(11):1175–1204, 2011.
4
9
Validation of a modal frequency response modelusing Pulsed Laser DIC and image decomposition
Christopher SebastianSchool of Engineering, University of Liverpool
Introduction
A novel method of capturing the vibration of a com-ponent at resonance and comparing the results tothose from a simulation using image decompositionis presented. The motivation for this research isthe desire to validate computational models usingfull-field methods of stress and strain analysis suchas Digital Image Correlation (DIC). However, theacquisition of images from a vibrating componenttypically requires the use of expensive high-speedcameras. Instead, a method is described in whicha pulsed-laser is used in combination with standardcameras to capture the motion of a vibrating panel.There is limited literature on the use of pulsed lasersfor DIC, and only for 2D measurements [1].
The out-of-plane (z-direction) displacement wasmeasured using a pulsed-laser DIC (PL-DIC) systemwhile exciting the panel at resonance. The resultswere compared to those obtained from a modal fre-quency response simulation using Finite Element(FE) analysis. There are many challenges in compar-ing full-field experimental data to simulation data,including alignment of coordinate systems, scaling,and interpolation of data points.
To overcome these problems, image decompositionwas used to compress each of the large data setsinto feature vectors. The comparison was then madebetween the feature vectors rather than the originaldata sets. Previously, Wang et al. used image decom-position to compare mode shapes from experiment tosimulation, but they only examined the shapes andnot actual displacements [2]. In this example, themeasured and predicted displacement from the simu-lation were compared using image decomposition. Aconfidence interval was defined by the uncertainty inthe experimental data used to determine how wellthe data sets agreed.
Experimental Setup and Results
The design of the experimental setup was based inpart on the results obtained from the modal analysis(detailed in the next section). The modal analysiswas used to identify a suitable excitation point andfrequency to use for the experiment. The panel usedin these experiments was a 800mm wide by 400mmhigh aerospace component that had been milled froma single block of 7075 aluminum. The panel was flat
Pulsed Laser
FunctionGenerator
Amplifier
TimingBox
Shaker
DIC Cameras trigger signal
Computer
Figure 1: Diagram (top) showing the connections of thesystem and the triggering of the laser and cam-eras; picture of the front of the experimentalsetup (bot. left) and back showing the attach-ment of the panel to the shaker (bot. right).
on one side, which was speckled and imaged withthe PL-DIC system. The back side of the panelcontained some stiffening ribs and bosses which canbe seen in figure 1. The panel was suspended fromthe top corners by string and attached to a shakerusing a stinger.
A function generator was used to drive the shakerat 15 Hz, which excited the first torsional mode of
Page 1 of 3
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vibration of the panel. Images of the vibrating panelwere captured using a commercially available DICsystem (Q-400, Dantec Dynamics, Ulm, Germany)based on a pair of 2MP firewire cameras. A laserwith a 4 nanosecond pulse duration and wavelengthof 532 nm was used to freeze the motion of the panel.In addition to providing the driving frequency tothe shaker, the function generator also provided asynchronization signal to the DIC trigger hardware.A diagram of the experimental setup is shown infigure 1.
The DIC software was used to phase shift theimage acquisition relative to the driving frequency,permitting the capture of the entire cycle of vibrationof the panel. A total of 20 images were capturedin this fashion at 18 degree phase increments. Thepulsed laser was triggered at the same time as thecameras and provided the illumination of the vibrat-ing panel, effectively acting as strobe with a veryshort duration. An example of one of the measure-ment results is shown in figure 2. In order to measurethe amplitude of displacement of the excitation point,a laser Doppler vibrometer was aimed at the frontside of the panel where the stinger was attached.
Simulation Setup and Results
A modal analysis was performed first to aid in thedesign of the experiment, specifically to determinethe attachment of the stinger and the excitationfrequency. The nodes where the stinger was attachedwere constrained in the X, Y, and Z directions, andfrom rotation about the Z-axis. The analysis wasperformed using a commercial FE package (AltairHyperMesh and Optistruct, MI USA).
After the experiment was performed, the displace-ment measured by the laser vibrometer at the excita-tion point was used for a modal frequency responseanalysis. An enforced displacement of 1.3mm wasapplied to the nodes where the stinger was attached,and the response calculated at a frequency of 15 Hz.The components of displacement were output foreach element at this frequency. An example of theout-of-plane displacement for one of the phase stepsis shown in figure 2.
Image Decomposition
The z-component of displacement from the experi-ment and the simulation were compared using imagedecomposition. A total of 11 sets of data were com-pared, covering a phase range of 90-270 degrees in 18
degree increments. A rectangular section of data wascropped from both the experimental and simulationresults as shown in figure 2. The area was chosen inorder to crop out the discontinuities resulting fromholes or missing patches of data. 50 Tchebichef mo-ments were used to decompose the data the croppeddata sets. To check that 50 moments provided anaccurate representation of the data, a reconstructionof the original image was performed using the mo-ments and the MATLAB command corrcoef used tocompare the reconstructed to the original image [3].
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0
1 0
2 0
3 0
4 0
5 0
6 0
z - d i s p l a c e m e n t R B M R
p i x e l s
pixe
ls
Dis
plac
emen
t (u
m)
- 4
- 3
- 2
- 1
0
1
2
3
Figure 2: Out-of-plane (z-direction) displacement resultsin mm from the simulation (top) and the ex-periment (bottom). The dashed line indicatesthe area of comparison.
Figure 3 shows the results of plotting the momentsagainst each other, with the DIC on the x-axis andthe FE data on the y-axis using the procedure pro-posed by the authors [3]. If the data sets were inperfect agreement all of the moments would lie alonga line with a gradient of one. However, deviationsbetween the sets of moments will result in the in agradient other than one, as well as scatter. A lin-ear regression line has been fitted to the data andhas a gradient of 0.85. In this case a gradient of lessthan one indicates that the simulation is conservativein its prediction relative to the experimental data.The dashed lines indicate the expanded uncertaintyof the experimental measurement, obtained from aprior calibration. In this case the bands are given by±266µm and all of the moments fall within the areadefined by the bands.
Figure 3 tracks the difference found by subtractingthe simulation from the experiment for the first fivemoments over the measured phase range of 90-270degrees. The y-value of the plot covers a range of
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−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
DIC data Tchebichef moments
FE
A d
ata
Tch
eb
ich
ef
mo
me
nts
gradie
nt = 0
.85
15
2
34
Figure 3: Comparison of the moments for the 270 degreephase step. The first five moments are labeled.The dashed lines define the uncertainty fromthe experiment.
±266µm, which is the same as the expanded uncer-tainty of the measurement system. It can be seenthat none of the points fall outside of the bounds,which indicates good agreement between the experi-ment and simulation results over the measured phaserange.
90 180 270
−0.2
−0.1
0
0.1
0.2
Phase (degrees)
DIC
−F
EA
(m
m) 1
4
3
2
5
Figure 4: The difference between the first five experimen-tal and simulation moments over the measured180 degree phase range.
Figure 5 plots the change in the gradient of the fitline (as shown in figure 3) for each of the phase steps.Ideally the value of the gradient would be unity forall of the phase steps, indicating perfect agreementbetween the simulation and experimental data sets.In reality, the gradient will have a value other thanone, indicating a bias towards either the experimentor the simulation. In this case the gradient remainsbetween 0.8-1.0 for the majority of the range, exceptfor the inflection point when the panel was passingthrough the un-deformed state. At this point themoments are all very near to zero, so the gradienttakes extreme values.
90 180 270
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Phase (degrees)
Gra
die
nt of fit lin
e
Figure 5: Plot of the gradient of the fit line (see figure3) applied to the moments at each phase step.
Conclusions
A novel methodology for validating a direct frequencyresponse model using data obtained from Pulsed-Laser DIC and image decomposition has been pre-sented. A comparison was made between the twosets of data at an excitation frequency of 15Hz overa 180 degree phase of the vibration cycle. A confi-dence interval was defined based on the uncertaintyfrom the calibration of the experimental setup. Thedifference between the experimental and simulationdata sets was less than the confidence interval, so themodel was deemed to be an accurate representationof the experiment.
Acknowledgements
Effort sponsored by the Air Force Office of ScientificResearch, Air Force Material Command, USAF, un-der grant number FA8655-11-1-3083. The U.S. Gov-ernment is authorized to reproduce and distributereprints for Governmental purpose notwithstandingany copyright notation thereon.
References
[1] Schmidt, Tyson and Galanulis, Exp. Tech.,27(4), 22-26.
[2] Wang, Mottershead, Ihle, Siebert, and Schubach,J. Sound & Vibration, 330(8), 1599-1620.
[3] Sebastian, Hack and Patterson, J Strain Analy-sis, 48(1), 36-47.
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Dynamic Response of Hierarchical Materials Ramathasan Thevamaran, California Institute of Technology, Pasadena, CA 91125.
The development of new engineered materials for better energy and impact absorption is critical for many societal needs—from protection against sports/accident related head injuries to protecting electronics/mechanical systems from impacts and vibrations. The many challenges faced in the development of engineered materials to improve energy absorption and impact mitigation has led researchers to address several fundamental questions: How do the microstructural features in different length scales respond to the applied loading and contribute to specific energy absorption? How do the deformation mechanisms depend on the rate at which the material is deformed? Given the likelihood of repeated impact events, how can we design material systems that offer recovery/healing to withstand fatigue? Can we engineer the microstructure of the material such that the dynamic mechanical properties are optimally tailored for the desired applications?
My research focuses on understanding the dynamic behavior of hierarchical materials with fibrous morphology. I use vertically aligned carbon nanotube (VACNT) foams as model materials—to understand the relation between structural organization of fibers across different length scales, and the bulk functional properties. The macro-scale VACNT foams studied have constituent CNT fibers organized hierarchically in multiple length-scales[1-2]: the entangled individual multi-walled carbon nanotubes (MWCNTs) form a forest-like system in the micro-scale and the bundles of these CNTs are aligned vertically in the meso-scale [Fig.1]. In previous studies, these materials have shown distinct mechanical characteristics in different loading regimes. Long duration stress relaxation experiments performed on VACNT foams have shown viscoelastic response[3]. Rate-independent stress-strain response was found in quasistatic compression experiments[4]. Small-amplitude vibration experiments have shown frequency independent response in linear dynamic regime[5]. Their mechanical behavior is influenced by different parameters such as bulk density, intrinsic density gradient along the height, surface roughness and CNT fiber orientation and alignment. Tuning these parameters offer versatility for engineering complex materials with desired properties.
The main objectives of my research are: (i) the development of a new experimental testing platform to measure and observe in-situ, the dynamic response of complex materials; (ii) the fabrication of CNT-based hierarchical materials with controlled microstructure, using chemical vapor deposition (CVD) synthesis techniques and standard lithographic approaches (iii) understanding the influence of microstructure on the bulk mechanical response of these materials and (iv) the development of numerical models to describe their dynamic response.
Experimental techniques.
The dynamic testing setup I developed[6] has four main components [Fig.2]: (i) an impact generator and low-friction striker guide to deliver controlled striker impacts to the samples at velocities between 0.5-10 ms-1; (ii) an impact force sensor to measure the transient force history during impact; (iii) a geometric moiré interferometer to measure the bulk dynamic deformation of the sample with micro-scale resolution; and (iv) a high-speed microscopic camera for in-situ visualization and characterization of complex deformations at different length scales. The experimental setup can be used to study dynamic indentation, rate-effects, and dynamic compression studies of various structured materials.
Fig.1: Hierarchical morphology of VACNT foams: the bundles of CNTs are aligned vertically in a meso-scale and the individual multi-walled carbon nanotubes (MWCNTs) are entangled to form a forest like system in the micro and nano scales.
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Fig.2: A schematic of the dynamic testing setup showing impact generator, force sensor and the optical system for geometric moiré interferometer.
Quasistatic compression experiments were performed on Instron E3000 commercial testing system to understand the dynamic phenomena in comparison to quasistatic mechanical response. Optical characterization techniques such as scanning electron microscopy (SEM), transmission electron microscopy (TEM) were used to study the morphological changes and deformation mechanisms due to the dynamic and quasistatic tests.
Material Synthesis.
The VACNT foams were synthesized using floating catalyst thermal CVD techniques. The number of walls of constituent MWCNTs and the VACNT foamʼs bulk density are tunable using different hydrogen concentrations during synthesis[2].
To understand the effect of geometry and periodic structural organizations on the bulk mechanical response, we engineered the microstructure of the VACNT foams by introducing an additional hierarchy in the meso-scale[7]. This was achieved by patterning 1D periodic arrays of lines and 2D periodic arrays of circle, orthogonal lines and concentric rings on the silicon substrate using photolithographic techniques; and then synthesizing VACNT arrays on these defined patterns into arrays of pillars, walls, orthogonal walls and concentric cylinders [Fig.3]. Varying the structural dimensions of these meso-structures such as pillar diameter, wall thickness, and periodicity of these structures allows us to tailor the bulk mechanical properties and the fundamental deformation mechanisms.
Significant outcomes.
VACNT foams exhibit a nonlinear dynamic stress-strain response with a loading-unloading hysteresis loop [Fig.4]. They dissipate energy through hysteresis, and they can be engineered to dissipate more than 200 times the energy dissipated by the commercial polymeric foams with comparable densities. The VACNT foams can deform up to 90% of its initial height and exhibit exceptional resilience to impact by recovering more than 85% of the deformation upon unloading[8].
Complex rate-effects are present on dynamic stress-strain response—the loading path was independent of strain-rate whereas the rate-effects were present during unloading (The dynamic unloading modulus normalized by the quasistatic unloading modulus increases with the strain-rate; inset of Fig.4)[8]. This loading response that is similar to quasistatic loading response transitions into shock formation when impacted at velocities higher than a critical velocity. The critical velocity was a function of the bulk density and increased with increasing bulk density.
Fig.3: VACNT structures with engineered microstructure: periodic arrays of walls, pillars and concentric cylinders.
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Mechanical properties such as unloading modulus, compressive strength, and energy dissipation also increase with the increasing bulk density.
Fig.4: Dynamic response of VACNT foams at increasing impact velocities. The inset shows the rate-effects on the unloading modulus.
Fig.5: (a) Deformation micrographs from high-speed microscopic imaging showing progressive buckling as the striker compresses the sample; the deformation is fully recovered upon unloading. (b) SEM image of permanent collective buckles. (c) TEM image of wrinkles on MWCNT walls.
The in-situ high-speed microscopy performed during dynamic deformation of VACNT foams revealed strain localization and characteristic deformations at different length scales—collective sequential buckling of bundles at the meso-scale and foam-like super compression at the macro-scale [Fig.5(a)]. Post-impact SEM characterization showed bending and buckling of individual nanotubes in micro-scale [Fig.5(b)]. TEM analysis of individual MWCNTs that underwent permanent deformation revealed wrinkles on the walls of the MWCNTs that is a characteristic of snap-through buckling [Fig.5(c)].
Through engineering the microstructure, we have shown that the structural properties of different geometries can be exploited to significantly improve the bulk mechanical properties [Fig.6]. For example, the 2D periodic array of concentric cylinders are 2 orders of magnitude lighter, but stiffer than the continuous VACNT foams; increasing pillar diameter keeping the bulk density in the same order increases the energy dissipation. We showed that different structural organizations could be used to create materials with broad range of tailored properties[7,9].
These studies provide insights into fundamental deformation mechanisms and energy dissipative characteristics of hierarchical materials. This understanding can guide the design of novel engineered materials for efficient energy absorption and impact mitigation. Further experimental and numerical studies are in progress to understand and predict the mechanical response of hierarchical, fibrous materials in different loading regimes.
0 0.1 0.2 0.3 0.4 0.5 0.60
5
10
15
20
Strain
Stre
ss (M
Pa)
1.45 ms−1
2.14 ms−1
3.03 ms−1
4.04 ms−1
0 0.1 0.2 0.3 0.40
5
10
15
20
StrainSt
ress
(MPa
)
Cycle 1 (0.01s−1)Cycle 2 (0.01s−1)Cycle 3 (0.01s−1)Cycle 4 (0.01s−1)2.50 ms−1(2345 s−1)3.27 ms−1(3067 s−1)3.91 ms−1(3667 s−1)4.54 ms−1(4258 s−1)
Loading
Unloading
(a)
2 40
0.4
0.8
Strain−rate (103 s−1)
Edy
n/Es
(b)
0 0.1 0.2 0.3
1
2
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4
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Stre
ss (M
Pa)
Striker
VACNT specimen
Sensor
1
1 32
Propagating buckling
Striker
VACNT specimen
Sensor
1 32
Propagating buckling
Strain
2
5
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4
5
321
20nm
(a)
(b)
(c)
20!m
Wrinkles
Fig.6: Dynamic response of VACNT foams with engineered microstructures.
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References.
[1] Cao A, Dickrell PL, Sawyer WG, et al. (2005) Super-compressible foamlike carbon nanotube films. Science 310:1307–10. doi: 10.1126/science.1118957
[2] Raney JR, Misra A, Daraio C (2011) Tailoring the microstructure and mechanical properties of arrays of aligned multiwall carbon nanotubes by utilizing different hydrogen concentrations during synthesis. Carbon N Y 49:3631–3638. doi: 10.1016/j.carbon.2011.04.066
[3] Lattanzi L, Raney JR, De Nardo L, et al. (2012) Nonlinear viscoelasticity of freestanding and polymer-anchored vertically aligned carbon nanotube foams. J Appl Phys 111:074314. doi: 10.1063/1.3699184
[4] Raney J, Fraternali F, Daraio C (2013) Rate-independent dissipation and loading direction effects in compressed carbon nanotube arrays. Nanotechnology. doi: 10.1088/0957-4484/24/25/255707
[5] Teo EHT, Yung WKP, Chua DHC, Tay BK (2007) A Carbon Nanomattress: A New Nanosystem with Intrinsic, Tunable, Damping Properties. Adv Mater 19:2941–2945. doi: 10.1002/adma.200700351
[6] Thevamaran R, Daraio C (2014) An Experimental technique for the dynamic characterization of soft complex materials, Experimental Mechanics (in press)
[7] Lattanzi L, De Nardo L, Raney JR, Daraio C (2014) Geometry-Induced Mechanical Properties of Carbon Nanotube Foams. Adv Eng Mater n/a–n/a. doi: 10.1002/adem.201300524
[8] Thevamaran R, Daraio C (2014), Rate-effects and shock formation in vertically aligned carbon nanotube foams (in preparation)
[9] Lattanzi L, Thevamaran R, Daraio C, Impact response of vertically aligned carbon nanotube foams with engineered microstructure (in preparation)
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Optical metrology with Hilbert-Huang fringe pattern analysis for experimental mechanics
M. Trusiak1 and K. Patorski1
1 Institute of Micromechanics and Photonics, Warsaw University of Technology, Poland
1. Introduction
Optical whole-field measurement methods with simultaneous acquisition and parallel
processing of experimental data are well suited to evaluate mechanical properties of macro
and microscale objects in static and dynamic regimes. They enable non-invasive, contact-
less, fast, automatic and very accurate investigations of, e.g., in-plane displacements/strain
fields, vibration testing and 3D shape evaluation. The measurand is encoded in the output
fringe pattern phase (fringe period and orientation) or amplitude distribution (fringe
contrast). Quantitative analysis is performed using computer-aided automatic fringe pattern
analysis (AFPA) methods [1,2]. The accuracy and calculation speed of the most attractive,
from the experimental point of view, single shot techniques depend mainly on the
algorithmic solutions applied.
In this contribution we advocate several novel algorithms developed by us for efficient fringe
filtering and phase/amplitude demodulation based on the notion of the Hilbert-Huang
transform (HHT). Their robustness and effectiveness is exemplified by processing fringe
patterns obtained using two powerful experimental mechanics techniques, i.e., time-average
interferometry and 2D grating (moiré) interferometry.
2. Hilbert-Huang fringe pattern analysis
Hilbert-Huang processing comprises empirical mode decomposition (EMD) algorithm used to
prepare a signal for subsequent Hilbert spectral analysis [3]. EMD is an adaptive and data-
driven algorithm proposed as an alternative to linear integral transform methods. Unlike the
Fourier-based or wavelet approaches EMD is not bounded by fixed basis functions and
uncertainty in time-frequency localization. EMD adaptively dissects a rather small,
meaningful number of so-called intrinsic mode functions (IMFs) from the analyzed signal
using sifting process. Appropriately managing a set of IMFs one obtains a very powerful
signal processing tool. For efficient image analysis the bidimensional EMD (BEMD) was
proposed [4]. The practical impact of BEMD is limited by the calculation time – envelope
spline interpolation required during sifting process is the most expensive part of the
algorithm (several hours per decomposition). To overcome this limitation FABEMD (Fast
Adaptive BEMD) approach was proposed [5] with fast envelope estimation using nonlinear
order-statistics-based filtering followed by smoothing operation (several minutes per
decomposition).
17
Our group has successfully employed the FABEMD method for studying the moiré
phenomenon [6,7], low quality fringe pattern adaptive enhancement [8-11] and optical
sectioning in structured illumination microscopy [12]. For further considerable computation
time reduction the EFEMD (enhanced fast EMD) solution was reported by us in [9]. Utilizing
mathematical morphology aided sifting process we are able to complete 512x512 pixels
image full decomposition under one second. Combined with automatic selective
reconstruction (ASR) the EFEMD algorithm constitutes a state-of-the-art fringe pattern
adaptive filtering approach called the ASR-EFEMD [9].
Due to the local zero-mean value every extracted BIMF is suited for further Hilbert
transformation (HT) enabling phase and amplitude demodulation. In our work we employ
Hilbert spiral transform (HS) [13] as a very efficient 2D extension of the original one-
dimensional HT (much better results than the partial Hilbert transform ones [14]).
3. Results and discussion
To exemplify the versatility and robustness of the developed Hilbert-Huang transform
algorithms for experimental mechanics studies we present and discuss results obtained
using 1D digital speckle pattern interferometry and 2D grating (moiré) interferometry for in-
plane displacement and strain analysis, and time-averaged interferometry for silicon
microelement vibration testing.
3.1 1D digital speckle pattern interferometry
To corroborate the potential of the ASR-EFEMD method [9] for enhancing complex, low
quality experimental data we have processed real DSPI fringes, Fig. 1, obtained in the studies
of in-plane displacements of a stiff epoxy polymer cantilever beam [15]. Proposed technique
is robust to considerable fringe spatial frequency variations over the whole correlogram,
strong speckle noise and fringe pattern modulation and background changes.
(a) (b) Fig.1 Experimental (263x622 pixels) DSPI fringe pattern [15] (a); enhanced and normalized correlogram using the ASR-
EFEMD method [9] (b).
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3.2 2D grating (moiré) interferometry
Proposed novel contribution to the analysis of grid patterns encoding the information about
in-plane displacement fields in mutually orthogonal directions comprises three main
procedures: (1) crossed fringe pattern is resolved into two fringe families using novel
orthogonal empirical mode decomposition (OEMD) approach, (2) separated fringe sets are
enhanced and normalized using the ASR-EFEMD technique aided by mutual information
detrending, and (3) the Hilbert spiral transform is employed for the fringe phase
demodulation (see [16] for details). Proposed algorithm diagram is outlined in Fig. 2.
Fig. 2 Flow chart of the proposed decomposition method exemplifying additive superimposition of simulated noisy fringe
sets. The OEMD method is used for resolving two orthogonal fringe families, the ASR-EFEMD technique is applied for
extracted single fringe pattern filtering and normalization, and the Hilbert spiral transform is employed for phase
demodulation.
Experimental data analysis is conducted for low quality crossed interferogram, Fig. 3(a),
obtained using the 2D moiré interferometry setup [17-19] in the stretched fabric test [17].
Continuous u(x,y) and v(x,y) in-plane displacement fields were calculated following the
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processing path presented in Fig. 3. The method can be applied to multiplicative and additive
type crossed gratings with both sinusoidal and binary component structures [16]. Adaptive
filtering enables fast processing and analysis of complex and defected patterns.
(a) (b) (c) (d)
(e) (f) (g)
(h) (i) (j)
(k) (l) (m)
Fig.3 (a) Experimental crossed interferogram (310x410 pixels image corresponding to 4,5x6 mm2 area of the sample under
test; stretched fabric sample with transferred reflective diffraction grating 1200 lines/mm; carrier fringes introduced by
tilting illumination beams [17]), (b) BIMF containing vertical fringes extracted from (a) using the OEMD algorithm, (c)
vertical fringes enhanced and normalized using modified ASR-EFEMD, (d) wrapped phase fringes obtained using the Hilbert
spiral transform, (e) unwrapped phase maps corresponding to the u(x,y) displacement field obtained by proposed approach
and (f) the 5-frame temporal phase shift method; (g) plot illustrating cross-sections along 50th row of the u(x,y)
displacement field obtained using Hilbert-Huang transform (red line) and TPS (blue line) methods (c) (h-m) analogous
processing results for the horizontal fringe set (cross-section along 200th row).
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3.3 Time-averaged interferometry
Characteristic features of the time-averaged vibration testing are the experimental setup
simplicity, independence of the vibration frequency value and full-field analysis. In the case
of harmonic vibration the interferogram modulation changes are described by the zero-
order Bessel function J0 of the first kind. Its argument encodes the vibration amplitude
distribution. Once the pattern modulation envelope is determined utilizing the Hilbert spiral
transform, the vibration amplitude information is extracted. Most comprehensive approach
uses heterodyning with phase modulation (complicated setup) for the temporal phase
shifting analysis of Bessel fringes. In some cases of harmonic vibration modes, however, the
vibration amplitude sign change localizations can be predicted. Tracing dark Bessel fringe
centers representing equidistant contours of equal amplitude serves the purpose. This can
be very difficult for considerable vibration amplitudes when high order Bessel fringes
become very dark.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i) Fig. 4 (a) Time-averaged frame of a square (1mm x 1mm) silicon micro-membrane vibrating at 172 kHz (single resonant
mode excitation) and the results of two frame Hilbert transform processing: (b) modulation distribution and (c) dark fringe
center contour lines; (d) time-averaged frame for a circular (1 mm diameter) silicon micro-membrane vibrating at 365 kHz
(single resonant mode excitation) and the processing results: (e) modulation distribution and (f) dark fringe center contour
lines; (g) time-averaged frame for a circular silicon micro-membrane vibrating at 930 kHz (complex resonant mode
excitation) and the processing results: (h) modulation distribution and (i) dark fringe center contour lines.
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Time-averaged fringes in interferometric vibration testing of MEMS (microelectromechanical
systems) are additive moirés unaffected by the carrier fringe displacements. These features
and Hilbert transform vulnerability to additive trend are utilized for visualization of centers
of dark Bessel fringes [20]. Two frames with shifted carrier fringes are subtracted for
background and noise correction. Next two normalized images are calculated with slightly
different bias levels and subtracted. The method proposed is very fast (computation time
below 0.25 second) and robust - it does not require precise phase stepping between two
time-averaged frames, strictly cosinusoidal carrier fringes and linear recording. Figure 4
shows exemplifying results of the proposed two-frame Hilbert spiral transform Bessel fringe
contouring and amplitude demodulation.
4. Conclusions
Several novel algorithms recently developed by our group for efficient fringe filtering and
phase/amplitude demodulation based on the notion of the Hilbert-Huang transform were
presented. Their robustness, efficiency and accuracy were corroborated by processing fringe
patterns obtained using various experimental optical techniques. We advocate their use for
optical testing in experimental mechanics.
References
1. D.W. Robinson, and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics Publishing, 1993).
2. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
3. N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and non-stationary time series analysis,” Proc. Roy. Soc. Lond. A 454, 903-995 (1998).
4. J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vision Comput. 21(12), 1019-1026 (2003).
5. S.M.A. Bhuiyan, R.R.Adhami, and J F Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process., ID728356(164), 1-18 (2008).
6. K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19(27), 26065-26078 (2011).
7. M. Trusiak, and K. Patorski, “Space domain interpetation of incoherent moiré superimpositions using FABEMD, ” Proc. SPIE 8697, 18th Czech-Polish-Slovak Optical Conference on Wave and Quantum Aspects of Contemporary Optics, 869704 (December 18, 2012) .
8. M. Trusiak, K. Patorski, and M. Wielgus, "Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform," Opt. Express 20(21), 23463-23479 (2012)
9. M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52(1), 230–240 (2014)
10. K. Patorski, M. Trusiak, and M. Wielgus, “Fast Adaptive Processing of Low Quality Fringe Patterns by Automated Selective Reconstruction and Enhanced Fast Empirical Mode Decomposition,” Fringe 2013, 185-190 (2014).
11. M. Trusiak, and K. Patorski, “Optical fringe pattern processing using empirical mode decomposition based algorithms,” The European Conference on Lasers and Electro-Optics (CLEO), CH_P_24 (2013).
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12. K. Patorski, M. Trusiak, and T. Tkaczyk, "Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing," Opt. Express 22(8), 9517-9527 (2014)
13. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18(8), 1862–1870 (2001).
14. M. Wielgus, and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50(28), 5513-5523 (2011).
15. K. Patorski, and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng 1997;36(7):2010–5.
16. M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
17. K. Pokorski, and K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt. 51(35), 8433-8439 (2012).
18. D. Post, B. Han, and P. Ifju, High Sensitivity Moirè: Experimental Analysis for Mechanics and Materials Science and Technology (Springer, 1994).
19. K. Patorski, Handbook of the Moirè Fringe Technique (Elsevier, 1993).
20. K. Patorski, and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21(14), 16863-16881 (2013).
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p 1/3
P-Adaptivity in Digital Image Correlation
L. Wittevrongel1, D. Debruyne
1, S.V. Lomov
2 and P. Lava
1
1 Department of Materials Engineering, University of Leuven Campus Ghent, BE 2 Structural Materials, Department of Materials Engineering, University of Leuven, BE
1. INTRODUCTION
The DIC technique has been widely used in experimental mechanics due to its low experimental cost such as simple
setup, simple specimen preparation and low requirements in measurement environment. The traditional approach is a
subset based approach, having some important drawbacks, keeping it from being used in some specific domains. The
lack of inter-subset continuity increases sensitivity of the local approach against noise, leading to noisy measured
displacements. These measurements should be smoothed out prior to differentiation to minimize their effects on the
strain. The downside of this method is that the choice of subset, step and strain window influences the spatial resolution
and thus makes data very user dependent. For this reason measuring a high gradient strain field with small amplitudes is
difficult using subset based DIC. To avoid the non-continuous displacement field, several global approaches were
presented. The current implementations of these global approaches all use a fixed degree of freedoms (DOF), ranging
from lower (Q4 element) to higher order (24 node elements) meshes. The use of fixed DOF leads to very user/approach
dependent results, as natural smoothing is performed. Here, to circumvent this dependency a self-adapting global DIC
procedure is proposed. The main topic discussed is the algorithm itself.
2. CONCEPT
The self-adapting procedure is based on conservation of optical flow. By minimizing the sum of squared differences (1)
with respect to the displacement parameters used in the displacement description (2), a linear system (3) is obtained.
→
Where f(x) and g(x) are respectively the reference and deformed image, d the displacement described by the shape
functions Φi (x) and displacement parameters diα. The choice of shape functions defines the number of DOF’s, and thus
the order/polynomial degree of the elements used (e.g. Q4, Q8 elements). Other than in the current approaches, the DOF
are not fixed in the self-adaptive approach. By using error/convergence estimators, the algorithm adapts/updates the
mesh where needed (e.g. regions with highly heterogeneous deformation) in analogy with adaptive finite element
procedures. By using hierarchical shape functions, updating the mesh can be done very efficiently without changing the
geometry and number of elements. Adding extra DOF’s only leads to expanding the linear system with extra parameters
and coefficients regarding the newly introduced DOF. For example, updating a mesh from n to n’ DOF (adding m DOF)
leads to following linear system:
with:
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p 2/3
This way of updating is analogous to p-refinement in FEA, as each region gets extra DOF (higher order functions) when
updated. By using the p-refinement, the self-adaptive procedure independently determines which DOF is needed to
describe the deformed surface based on the error/convergence estimators. As a consequence, homogenous deformation
will require lower order shape functions, while - in contrast - heterogeneous regions will require higher order shape
functions. The estimators are based on displacement and strain norms, much like the ones used in p-methods in FEA.
The new method is called p-DIC. A simple flowchart of the newly proposed method is shown in Fig 1.
Fig 1: Flowchart of the p–DIC algorithm.
3. ADAPTIVITY
The concept of the new method, p-DIC, is shown on an numerical deformed image. The displacement imposed is a
horizontal sinusoidal deformation field. The frequency of the sine wave increases from left to right, resulting in an
image with a variation in needed spatial resolution. The correlation is performed with several sizes of meshes. As the
frequency rises from left to right, also the order of elements should rise from left to right. Changing element size, will
influence the element order in a similar way. Larger elements need higher orders than small elements. The order
distribution for 3 different sizes of mesh is shown in fig 2.
Fig 2: Distribution element orders for changing spatial resolution and element size.
For traditional DIC, the correlation parameters (subset, step, strain window) will heavily change the data. The p-DIC
method is, due to the self-adapting feature, less influenced by the user input (mesh size) and thus maintain accuracy.
The spatial resolution is, in comparison to the local method, not limited by initial user settings.
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p 3/3
4. APPLICATION
As application, a tensile test is numerically simulated. A holed aluminum specimen is used, producing a heterogeneous
strain field. The images are correlated using an arbitrary first order mesh with element size 100 pixels. Because of the
adaptive procedure, element orders will be increased if needed. In fig 3, the reference image, deformed images and final
element order distribution is shown.
Fig 3: Reference image, deformed image and element order distribution
Regions around the hole will produce more heterogeneous deformations, resulting in higher order elements. Correlating
the same images with different mesh sizes barely influence final results, proving the concept of p-DIC. The error of
displacement for the subset method and p-DIC method is shown in fig 4.
Fig 4: Error distribution horizontal and vertical displacement for subset and p-DIC method.
4. CONCLUSION
A new correlation algorithm is presented. The algorithm is based on global digital image correlation and adopts features
from the adaptive finite element. The region of interest is described by an adaptive element mesh. A p-refinement
scheme is implemented so that the elements in the mesh are capable of rising (automatically) in DOF when the error
estimators indicate them to do so. Using a numerical simulated test, a comparison of the traditional local and newly
presented p-DIC is performed. Results from the comparison indicate that the p-DIC method has a smaller distribution of
error compared to the local method. Besides the advantage in performance at optimal settings, another big advantage is
found. Because of the self-adapting mesh, the method becomes less user dependent. The spatial resolution is, in
comparison to the local method, not limited by initial user settings. Also measurements can go until the edges and an
error indication can be provided. In other work, an in depth validation of the method is performed. Future work is
mainly aimed at the further development of the error estimators as they are a key in the p-DIC procedure.
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Application of integrated Digital Image Correlation
to dynamical blade/casing interactions:
towards estimation of contact force
Romain Mandard
Laboratoire de Mecanique de LilleEcole Centrale de Lille
59650 Villeneuve d’Ascq, France
SNECMA (SAFRAN Group)Site de Villaroche
77550 Moissy-Cramayel, France
Introduction
The efficiency of aircraft engines is directly affected by the clearance between rotating blades and surroundingcasing. Minimizing the in-service clearance to a few tenths of millimeters has been shown to improve efficiency, butmay be critical to blade integrity in the event of contact. In order to achieve a compromise between compressorperformance and blade integrity, an abradable coating is deposited on the casing. In the event of blade-casingcontact, the blade tip abrades the coating, which has been designed to be easily worn. Experimental knowledgeof wear mechanisms, blade dynamics and contact force is paramount to the proper design of abradable materialsand the prediction of their lifetimes through numerical simulations. Different kinds of test rigs have been developedto study the interactions between blades and abradable coatings [1–5]. However, most of these studies do notinvestigate the couplings between blade vibrations and wear of abradable coating, which have been encounteredduring full-scale experiments [6]. In this phenomena, contact force and blade kinematics are key features but aretricky to measure. In order to study interactions of few milliseconds between flexible blades and abradable coatings,a specific test rig has been developed at ONERA 1 and presented in previous papers [7–9]. A first study [8], basedon the time-frequency decomposition of experimental data, has allowed to propose an experimental methodologyfor the estimation of blade/abradable-coating contact force [9]. One of the main conclusions was that the bladecan be modeled as a continuous Euler-Bernoulli bending beam. The boundary conditions were chosen accordingto the friction-induced vibrational modes identified with help of time-frequency analysis. The present work aimsto confirm the validity of this model and the associated boundary conditions by means of integrated Digital ImageCorrelation (iDIC). Moreover, the technique was implemented to measure the blade bending displacement andstrain fields during interactions. The results obtained with iDIC were compared with 1D-sensor data (displacementsensor and strain gauges).
1ONERA, the French Aerospace Lab, 59000 Lille, France.
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Experiment
The test rig consists of a rotating cylinder whose external surface is coated with an abradable AlSi-Polyester material,as shown in Figure 1a. A blade is fastened to the small rigid unit (Fig. 1b), which is moved (displacement DN )toward the abradable coating by means of a piezoelectric actuator. Blade geometry has been simplified comparedto the full-scale condition in order to focus on blade bending (Fig 1c). Blade characteristics have been providedin paper [9]. Blade/coating relative speed VT is created by spinning the coated cylinder; once the target speed isreached, an electrical pulse Ve is sent to the actuator to generate the displacement DN of the blade root. The dataacquisition system is then triggered to record:
� Images of the vibrating blade captured from the side by means of a high-speed camera, at 12 500 frames/s,with a resolution of 1024 x 1024 pixels;
� Blade bending displacement DT at 45 mm from the clamped end (by means of a laser displacement sensor);
� Blade longitudinal strain at 5 mm from the clamped end on both top and bottom faces (resp. εAxx and εBxx,by means of strain gauges).
Figure 1: Test rig instrumentation : a) overview b) incursion cell and c) speckled blade.
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An experiment was conducted at a tangential speed VT = 92 m/s and abradable temperature Ta = 150 ◦C. Theraw data recorded during the interaction are shown in Figure 2. The observation of blade bending displacementDT (blue line) and strain signals εAxx and εBxx (green lines) suggests that a series of blade impacts occurred duringthe 12-ms interaction.
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DN (
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) or
Ve (
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ε xx
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/m)
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DN
Ve
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εxx A
εxx B
t1
Figure 2: Raw signals obtained in the experiment.
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Measurement of blade displacement and strain fields
The bending displacement and strains fields ~d were estimated for the series of digital images captured during theexperiment, by means of a DIC algorithm, whose principle is the minimization of function Φ(~d):
Φ(~d) =
∫∫image
(g(~u) − f(~u− ~d)
)2dxdy (1)
where f(~u) and g(~u) are the gray levels of the reference and deformed blade, respectively. In the framework
of integrated DIC, the displacement field ~d is decomposed onto a basis of relevant functions which are a priorichosen [10]. In the present case, the function basis was chosen according to the previous time-frequency analysis [8].The blade bending kinematics is described as the superposition of Euler-Bernoulli bending modes, with a clamped-free boundary condition. A rigid-body displacement in direction ~x is added to the basis, in order to take intoaccount the displacement DN . Thus, the displacement field was searched as:
~d(x, y) =N∑i=1
TFi
(−y ∂Zi(x)
∂x~x+ Zi(x)~y
)+ TRB~x (2)
where Zi(x) are the Euler-Bernoulli beam mode shapes, N is the number of modes considered, TFi are the bending
parameters and TRB is the rigid-body parameter. The function basis and the optimization algorithm (Newton-Raphson) were prototyped in a lab-made Python code and then implemented in the EmCor software 2. TF
i andTRB were identified for each image of the experiment. Once parameters TF
i are identified, Equation 2 can bederivated to reconstruct the bending strain field εxx(x, y):
εxx(x, y) = −yN∑i=1
TFi
∂2Zi(x)
∂x2(3)
The strength of iDIC is that the strain field can be decomposed onto the modal basis as shown in Figure 3 (imagecorresponding to time t1 (cf. Fig. 2), for N = 2).
2DIC software developed at Laboratory of Mechanics of Lille, France.
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Abradable coating
BladeCONTACT FORCE ?
VTx
y
Small rigid unit
5 mm a)
εxx 1F
c)
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/m
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b)
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/m
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800400
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120016002000
εxx 2F
d)
μm
/m
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800400
0400800
120016002000
Figure 3: a) Image captured at time t1, b) total strain field εTOTxx identified for N = 2, c) contribution of the first
bending mode ε1Fxx d) contribution of the second bending mode ε2Fxx .
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In order to assess the relevance of the model and the iDIC algorithm, the temporal signals recorded with thelaser displacement sensor and the strain gauges were compared with iDIC results at the locations of the 1D sensors(Figure 4). Both displacement DT and bending strain εxx measured at 5 mm from the clamped end were correctlyidentified by iDIC, which confirmed the validity of the model.
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)
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a)
b)
iDIC
iDIC
Laser displacement sensor
ε xx
(μm
/m)
Time (ms)
blade/coating interaction
free oscillations
Figure 4: a) Displacement DT and b) strain εxx : comparison between iDIC and 1D sensors.
Figure 5 highlights the contributions of the first and second bending modes in the displacement DT and strainεxx during the interaction. The second mode has a negligible contribution in the displacement DT which supportsthe method employed in paper [9] to estimate the contact force. By contrast, the first mode is not sufficient torepresent the total strain, as shown in Figure 3 as well.
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free oscillations
εxx1F
εxx2F
εxxTOT
Time (ms)
DT1F
DT2F
DTTOT
a)
b)
Figure 5: Contribution of first and second bending modes (resp. 1F and 2F) within a) displacement DT and b)strain εxx.
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Conclusion
Integrated Digital Image Correlation was implemented in order to measure blade displacement and strain fieldduring a blade/abradable-coating interaction of few milliseconds. The results were compared with signals obtainedwith a displacement sensor and strain gauges, which allowed to validate the analytical model of the blade and thechosen boundary conditions. Thanks to this model, the blade/coating contact force could be estimated accordingto the methodology presented in paper [9]. Tangential and normal components of the contact force (resp. fT andfN ) are shown in Figure 6.
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DT (
mm
)
ε xx
(μm
/m)
Time (ms)
blade/coating interaction
free oscillations
(εxxA - εxx
B)
DT
0 2 4 6 8 10 12 14 16 18 20
0.0
0.2
0.4
0.6
0.8
1.0 fT / fTmax
fN / fNmax
a)
b)
Figure 6: a) Tangential and normal components of the contact force (normalized), b) displacement and strainrecorded during the experiment (1D sensors).
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References
[1] M. Cuny, S. Philippon, P. Chevrier, F. Garcin, Experimental measurement of dynamic forces generated duringshort-duration contacts: application to blade-casing interactions in aircraft engines, Exp Mech 54 (2014) 101-114
[2] N. Fois, J. Stringer, M.B. Marshall, Adhesive transfer in aero-engine abradable linings contact, Wear 304 (2013)202-210
[3] C. Padova, J. Barton, M.G. Dunn, S. Manwaring, Experimental results from controlled blade tip/shroud rubsat engine speed, J. Turbomach.-Trans. ASME 129 (2007)
[4] R. K. Schmid, New high temperature abradables for gas turbines, Ph.D. thesis, Swiss Federal Institute ofTechnology, Zurich, Swiss (1997)
[5] Laverty W.F., Rub energetics of compressor blade tip seals, Wear 75 (1982) 1-20
[6] A. Millecamps, J.-F. Brunel, P. Dufrenoy, F. Garcin, M. Nucci, Influence of thermal effects during blade-casingcontacts experiments, Proceedings of the ASME IDETC/CIE, USA (2009)
[7] S. Baız, J. Fabis, X. Boidin, Y. Desplanques, Experimental investigation of the blade/seal interaction, Proc.Inst. Mech. Eng. Part J.-J. Eng. Tribol. 227 (2013) 980-995
[8] R. Mandard, J.-F. Witz, Y. Desplanques, J. Fabis, J. Meriaux, Wavelet analysis of experimental blade vibrationsduring interaction with an abradable coating, J. Tribol.-Trans. ASME 136 (2014)
[9] R. Mandard, J.-F. Witz, X. Boidin, Y. Desplanques, J. Fabis, J. Meriaux, Interacting force estimation duringblade/seal rubs, Tribo. Int. (2014), In Press http://dx.doi.org/10.1016/j.triboint.2014.01.026i
[10] F. Hild, S. Roux, Digital image correlation: from displacement measurement to identification of elastic prop-erties - a review, Strain 42 (2006) 69-80
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Automatic Optical Crack Tracking for Double
Cantilever Beam Specimens
Brett P. Krull∗†
University of Illinois at Urbana-Champaign, USA
I. Introduction
Interlaminar fracture poses one of the greatest performance concerns for laminated composite materi-als, limiting reliability and leading to premature catastrophic failure.1–3 The DCB specimen provides aconvenient experimental platform to evaluate fracture toughness for a variety of composite materials.1–4 Aset of reliable and comprehensive guidelines for composite DCB sample fabrication, fracture testing, anddata analysis can be found in ASTM D5528.4 However, ASTM D5528 requires many tedious, manual, andpotentially subjective crack length measurements for each GIc calculation because crack length is opticallydetermined from hand-marked delineations along the side of each specimen.
Figure 1. a) Schematic of experimental setup showing top and side-mounted CCD cameras and lower light source. b)Photograph of DCB sample during fracture test. c) Diagram of DCB sample with labeled test variables.
Automated optical inspection known as “machine vision” is implemented for the purposes of qualitycontrol and inspection. A charge-coupled device (CCD) records an image which undergoes analysis todetermine dimension or quality characteristics. We’ve developed a method of automatically tracking andmeasuring cracks in fiber-reinforced composite DCB specimens. Striving for compatibility with ASTMD5528, our automatic tracking system requires only the addition of a CCD imaging system and light sourceto the typical test setup (Fig 1). Simple reconfiguration of commercially available software integrates themechanical testing equipment and image processing algorithms into a single graphical user interface (GUI)(Fig 2).
∗Department of Materials Science and Engineering, 1304 West Green Street, Urbana, IL 61801, USA.†Beckman Institute of Advanced Science and Technology, 405 North Mathews Avenue, Urbana, IL 61801, USA.
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Figure 2. GUI from integrated mechanical testing and automatic crack-tracking program.
II. Results and Discussion
Automatic measurements are acquired by placing a light under the DCB specimen and monitoring crackpropagation from an overhead CCD. The translucent E-glass fabric/epoxy matrix composite allows uniformillumination of the DCB sample. As the fracture test proceeds, delamination between middle plies of thesample creates a contrast difference between the intact (lighter) portion of the sample and the fractured(darker) portion due to scattering from the reflective crack plane. National Instruments LabVIEW softwareis used to construct a specialized subroutine to automatically measure the crack length from calibratedoptical images while simultaneously recording load-displacement data. Crack position is detected basedon the intact/fractured contrast differences via machine vision image processing functions. The programsearches within a defined region of interest (ROI) to locate the end of the sample and the leading edge ofthe crack to calculate the crack length:
a = a0 + af − am, (1)
where a is the total crack length, a0 is the precrack length, af is the available fracture length, and am is theoptical measurement length (Fig 3a).
An optical calibration factor (OCF) is initially determined as a mm per pixel ratio from the user-inputdistance between the pre-crack and the end of the DCB specimen (af , Fig 3a). The OCF requires insitu adjustment during fracture tests because crosshead displacement causes the distance between the DCBsample and the CCD to decrease. The samples appear larger and thus, the original OCF becomes invalid. Abaseline “reference” test is performed using an intact DCB specimen with a 50.8 mm length of black paperattached to the dorsal surface to simulate a constant crack length. Over 50 mm of crosshead displacement,the measured crack length shows an artificial increase of more than 10% (Fig 3b). As such, the OCF mustbe updated throughout the fracture test to ensure accurate crack length measurements.
A “linear-fit calibration” method is implemented for OCF adjustment using data acquired from thereference sample. The reference sample exhibits a nearly linear increase in crack length over the displacementrange investigated. A least squares linear regression is performed to fit an analytical calibration curve tothe data and calculate an OCF adjustment based on crosshead displacement. Measurements of the 50.8mm reference sample averaged 50.81 ± 0.15 mm (within 0.3%) for the linear-fit calibration method over theentire range of displacement. The error approximates the calibrated length of one pixel.
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Figure 3. a) Schematic of edge detection measurement system. The edge detection overlays lines for the ROI (green),edges detected (red), and center of the search region (blue). The pixel distance between the two detected edges (am) isused to calculate the crack length. b) Crack size as a function of crosshead displacement for 50.8 mm standard samplewith and without OCF adjustment.
A dual-camera setup allows direct comparison of automatic crack-tracking to ASTM D5528 (Fig 4). Theautomatic tracking successfully follows the edge of the crack front as the mid-ply delamination propagates(Fig 5a). The average error in crack length between the automatic crack tracking and ASTM D5528 manualmeasurements is 1.8 ± 0.6 mm. The familiar “thumbnail” shape of the crack front is the primary source ofdiscrepancy between the two methods. The automatic method gives crack lengths based on the apex of thecrack whereas ASTM D5528 returns lower crack lengths from the edge of the specimen.
Figure 4. Overhead (left) and side (right) images of fracturing DCB specimen at crack lengths 20 mm (a, b); 50 mm(c, d); and 80 mm (e, f). Scale bar = 10 mm.
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Calculations of mode I critical strain energy release rate (GIc) are performed according to modified beamtheory (MBT):2,4, 5
GIc =3Pδ
2b (a+ |∆|), (2)
where P is the applied load, δ is the crosshead displacement, b is the specimen width, a is the total cracklength, and |∆| is a correction factor2 to account for non-zero rotation at the delamination front.
Figure 5b shows the average GIc for specimens tested with both ASTM D5528 and automatic tracking.Despite the disparity in measured crack length, the two methods differ in GIc by less than 1%. Thus, theautomatic crack tracking technique and ASTM D5528 provide comparable data for mode I fracture toughnessevaluation of composite DCB specimens.
Figure 5. a) Crack lengths for representative DCB sample as measured by ASTM D5528 and automatic tracking. b)Comparison of average GIc from 4 samples according to both crack length measurements.
III. Conclusions
Automatic crack tracking in DCB specimens via edge detection software is simple, reliable, and accurate.Automatic tracking provides a more continuous crack length data set with superior spatial resolution of0.15 mm compared to 5 mm with the accepted standard. The experimental setup is highly adaptable andinexpensive to implement as it involves minimal equipment additions and is compatible with traditionalcomposite DCB specimen fabrication.
References
1De Moura, M.F.S.F., Campilho, R.D.S.G., Amaro, A.M., & Reis, P.N.B. Interlaminar and intralaminar fracture charac-terization of composites under mode I loading, Composite Structures 92(1) 144-149 (2010).
2Hashemi, S., Kinloch, A. & William, J. Corrections needed in double-cantilever beam tests for assessing the interlaminarfailure of fibre-composites, Journal of Materials Science Letters 2 125-129 (1989).
3Davidson, B.D. An Analytical Investigation of Delamination Front Curvature in Double Cantilever Beam Specimens,Journal of Composite Materials 24(11) 1124-1137 (1990).
4ASTM International. ASTM D5528: Standard Test Methods for Mode I Interlaminar Fracture Toughness of UnidirectionalFiber-Reinforced Polymer Matrix Composites (2007).
5De Morais, A., De Moura, M., Goncalves, J.P., & Camanho, P. Analysis of crack propagation in double cantilever beamtests of multidirectional laminates, Mechanics of Materials 35(7) 641-652 (2003).
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