3D PHOTOELASTICITY AND DIGITAL VOLUME
CORRELATION APPLIED TO 3D MECHANICAL STUDIES
A. Germaneau, P. Doumalin and J.-C. Dupré
Laboratoire de Mécanique des Solides, CNRS, UMR 6610, Université de Poitiers
SP2MI Bd Marie et Pierre Curie, Téléport 2, 86960 Futuroscope Chasseneuil, France
[email protected]
ABSTRACT
In this article, we show two methods based on similar optical properties allowing the analysis of 3D mechanical problems from
transparent model materials: 3D Scattered Light Photoelasticity (SLP) and Optical Scanning Tomography (OST) coupled to
Digital Volume Correlation (DVC), the 3D extension of Digital Image Correlation (DIC). Both techniques, easier to implement,
do not require any complex device and are based on analysis of light scattered by randomly distributed marks. Nevertheless,
each technique uses a particular scattered light phenomenon which involves fields of application and limitations. We present a
first comparison of these methods on a compression test performed with a spherical model. The obtained results show that
DVC seems to be better adapted for large strains than photoelasticity. However, 3D scattered light photoelasticty may give
better results for small strain values where DVC can be limited by its strain measurement uncertainty (0.1%).
Introduction
Generally in mechanics, 3D problems can be analyzed from simulations by finite elements and the results can be compared
with data measured on the surface of the specimen. However, in the case of structures with complex geometries or specific
loads, it can be necessary to determine experimentally the 3D strain field to validate more precisely the numerical approach in
the whole volume. For that, it is necessary to achieve a 3D view of the structure and one of the most encountered techniques
is X-ray Computed Tomography (CT). This method is often used to inspect the materials with a native heterogeneous
microstructure which gives a contrast due to differences of density of components [1-4]. Nevertheless, this technique requires
a heavy device, like a medical or laboratory tomograph or a synchrotron radiation device (X-ray microtomography) [5] and its
implementation can be also difficult when the natural local contrast is insufficient, for example for homogeneous materials. In
this case, some markers must be included in the material during its fabrication. This process is not always possible and is
reserved to some types of material as the ones obtained by powder metallurgy route [6]. In this work, we present two
techniques developed in our laboratory in order to study the elastic mechanical response of structures: Scattered Light
Photoelasticity (SLP) [7-10] and Optical Scanning Tomography (OST) coupled to Digital Volume Correlation (DVC) [11-14].
Each method gives different data. On one hand, SLP enables to obtain secondary principal stress (or strain) difference on a
slice inside the specimen. On the other hand, DVC which is the 3D extension of the Digital Image Correlation (DIC) [15-18]
gives 3D displacement in the whole specimen. The aim of this work is to determine the limitations, advantages and drawbacks
of both methods and allows us to choose the well adapted technique at each study. In this paper, we present the principle of
both techniques and we define the procedure to perform a 3D analysis of a model structure. A comparison of both techniques
is presented on a mechanical testing involving 3D mechanical effect: a located compression test with a spherical model.
3D Scattered Light Photoelasticity by Optical Slicing
Since the 1930s, the experimental study of stresses in 3D models has been usually performed by employing the frozen stress
technique coupled with a mechanical slicing and a two dimensional analysis of each slice in a classical polariscope [19-20].
Nevertheless, to avoid difficulties of this technique which requires a lot of time and several models for a whole study, nondestructive methods based on the polarization of the scattered or transmitted light in a birefringent material, have been
developed [21-25]. A few years ago, we have developed a simple and fast method using a numerical image processing of the
scattered light. The principle of our technique, using the polarization of the scattered light and the birefringence phenomenon
of transparent materials, consists in isolating a slice of the studied photoelastic model between two plane laser beams (Fig. 1)
[7-10]. The optical setup is constituted by a laser source, a separator device giving two beams, a convergent lens and a
cylindrical lens. In the direction perpendicular to the two illuminated sections, we record by CCD camera the scattered light
corresponding to a speckle pattern due to interferences of light beams of each section. The possibility of interferences
depends on the birefringence of the isolated slice.
Figure 1. Optical setup
The global scattered light intensity can be expressed by a coherent part (addition in amplitude) providing the speckle pattern
and an incoherent part providing the background (addition in intensity) and due to the fluorescence phenomenon of the
specimen and a part of no polarized scattered light.
By noting A the complex amplitude of the coherent light, emerging of the specimen (A* its conjugate value) and I1F and I2F the
background intensities due to both beams, the global intensity can be expressed as:
I = A. A * + I1F + I 2 F
(1)
By introducing the intensity of each plane beam (I1 and I2), formula (1) can be decomposed by a part corresponding to the
background intensities (I1F and I2F) and a part due to the laser speckle intensities (I1g and I2g):
I = I 1 + I 2 + 2 I 1 g I 2 g γ cos(ψ 1 + ψ 2 + η)
I 1 = I 1F + I 1 g
(2)
I 2 = I 2F + I 2g
where ψ1 and ψ2 are the random phases of the speckle fields, η is a function of the optical characteristics of the slice and γ is
the correlation factor of both speckle fields given by:
γ 2 = 1 − sin 2 2α sin 2
ϕ
2
(3)
The birefringence state is assumed to be constant along the studied slice and is described by a 2D optical index tensor which
is the projection of the 3D tensor in the middle plane of the slice. We note α the orientation of the secondary principal axis of
this tensor and n1 and n2 its principal values. Then, the angular birefringence ϕ can be expressed by:
ϕ=
2πe
(n1 − n2 )
λ
(4)
where e is the thickness of the slice and λ is the wavelength of the light.
The difference (n1 - n2) can be relied with the secondary principal stress (σ1 and σ2) difference by using Maxwell relations:
(n1 − n2 ) = C (σ1 − σ 2 )
(5)
or with the secondary principal strain (ε1 and ε2) difference with the Neumann laws:
(n1 − n2 ) = K (ε1 − ε 2 )
(6)
where C and K are the photoelastic constants of the material determined experimentally.
Equation (3) is similar to the relationship of the light intensity obtained in a plane polariscope and so the scattered light
analysis gives equivalent fringes to the ones obtained in the case of a 2D photoelasticity study (isochromatic and isoclinic
fringes).
The photoelastic fringes cannot be observed directly with the intensity I because the scattered light intensity is a speckle field
and I1g, I2g, ψ1 and ψ2 are random values. So, a statistical analysis has been performed on the speckle pattern in order to
determine the correlation factor and so the fringes [8]. By considering the properties of speckle and by calculating the spatial
averages (noted . ) and the variances (vari) of the intensity fields described in equation (2), the correlation factor is
determined thus:
k 2γ2 =
var 2 − var12 − var22
2 I1 I 2
(7)
where k is a factor related to the scattered light properties of the material. Its value is determined experimentally and is
approximately equal to 1/10.
To calculate experimentally the correlation factor, we have to record three images corresponding to the speckle pattern for one
plane alone (I1), the second plane alone (I2) and both planes together (I).
Digital Volume Correlation and Optical Scanning Tomography
DVC principle
The DVC method is the extension in 3D cases [11,26-28] of DIC techniques usually used since the 1980s to measure plane or
3D displacements of surfaces [15-18,29,30]. The DVC has been firstly developed on volume images generated by X-ray CT
[26-28] and has been recently coupled with optical scanning tomography (OST) technique [12-14].
By DVC, the displacement field between two mechanical states (reference state and deformed state) of a studied sample is
measured on a 3D virtual grid defined in the reference volume. The displacement of each point of this grid is calculated by
intercorrelation of the grey levels of the neighborhood D surrounding the considered point in both states. D is composed of
several voxels and corresponds to a subset of the volume. By noting X and x the 3D coordinates (in voxels) of a same point in
the reference state and the deformed state, both configurations are linked by a 3D material transformation φ: x = φ (X). For a
subset D centered at the point X0 in the reference state, φ is approximated by its expansion at the first order corresponding to a
rigid body motion combined with a homogeneous deformation:
φ( X ) = X + U ( X ) ≈ X + U ( X 0 ) +
∂U
( X 0 ).( X − X 0 )
∂X
(8)
The displacement U(X0) of the subset center gives the intensity (u,v,w) of the rigid body translation. The local displacement
gradient expression
by 9 parameters:
∂U
( X 0 ) includes the rigid body rotation and the local stretch of the subset volume and is characterized
∂X
∂u ∂u ∂u ∂v ∂v ∂v ∂w ∂w ∂w
. Generally for small local strains, the local gradient component can
, , , , , ,
,
,
∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z
be neglected and so the approximation can be simply assimilated to a translation. The best parameters characterizing the
approximation (3 or 12 scalars) are those which minimize a correlation coefficient C which measures the degree of similarity of
grey level distributions in D and its transformed one by φ. We choose a formulation of C insensitive to small contrast and
brightness fluctuations which can appear in images: a normalized cross correlation formulation based on grey level gaps in
respect to the average on the subset.
( f ( X ) − f D ).( g (φ( X )) − g D )
X ∈D
C =1−
( f ( X ) − f D )2 .
X ∈D
( g (φ( X )) − g D ) 2
(9)
X ∈D
where X refers to voxels in D, f and g are respectively the grey levels in the undeformed and deformed images, f D and g D
are their averages over D and φ(D).
A trilinear interpolation of the grey levels in the deformed image is used in order to calculate the grey level variations between
two adjacent voxels. In this way, it is possible to achieve the position of the subset D in fractions of voxels (subvoxel precision).
This solution is researched with an automatic first gradient minimization procedure from an estimation in entire voxels which is
obtained by a direct systematic calculus [12,14].
DVC gives a discrete displacement field, the displacements of the centers of the subsets. In order to determine, at each point
X0 of coordinates (X0,Y0,Z0), the gradient of the transformation F defined by:
F=
∂x
∂U
=I+
∂X
∂X
(10)
we choose 3 vectors (dX1,dX2,dX3) from the 6 nearer neighboring points of X0 in the initial state. These 3 vectors describe a 3D
cross centered at X0 and define a parallelepipedic volume where we can consider F homogeneous. As the initial grid of points
is regular, the vectors (dX1,dX2,dX3) are respectively in the directions x,y,z, of the orthonormed basis. Then, we can express the
components of the gradient ∂U by finite differences [12]:
∂X
∂α α( X 0 + l 0 , Y0 , Z 0 ) − α( X 0 − l 0 , Y0 , Z 0 )
=
2.l 0
∂x
∂α α( X 0 , Y0 + l 0 , Z 0 ) − α( X 0 , Y0 − l 0 , Z 0 )
=
2.l 0
∂y
∂α α( X 0 , Y0 , Z 0 + l 0 ) − α( X 0 , Y0 , Z 0 − l 0 )
=
2.l 0
∂z
(11)
where α = u,v,w and l0 is the step of the 3D uniform grid. In this case, the gauge length is equal to 2l0.
Then the Green-Lagrange strain tensor E can be calculated by:
E=
1 T
( F .F − I )
2
(12)
Volume image by OST
The DVC method can be applied to any type of 3D images if the grey level of each voxel represents a data which follows the
material movement. Volume images are currently generated by X-ray CT. We propose a new device to analyze transparent
materials. The developed method is based on the phenomenon of scattered light created by randomly distributed particles,
added in the specimen during its elaboration. At each step of loading, a 3D image is obtained by scanning the specimen with a
plane laser beam in the z direction and with a motorized translation stage (with a smallest increment equal to 0.625 µm)
controlled by an integrated linear-scale encoder. At each position of the beam, we record an x-y 2D image of the illuminated
section where a random pattern due to scatterers appears. The volume is constituted by the succession of these 2D images.
The experimental setup is similar to the one used for the SLP (Fig. 1) but in this case, only one plane laser beam is employed.
In order to have cubic voxels, it is necessary to have the same spatial resolution in the z direction and in the x and y directions
of a slice. For that, the step between two successive slices (in z direction) has to be chosen from the magnification of the CCD
camera (x-y plane). To have a fill factor of voxels equal to 100%, the thickness of the plane laser beam must be equal to the
step between two successive slices. This spatial resolution depends to the specimen size and the added particles.
Experiments
Specimen manufacture
Specimens are made in epoxy resin which has mechanical properties well adapted to studies of model structures (linear
elasticity). Furthermore, this material is usually employed for photoelastic studies and easy to cast. To involve a scattered light
phenomenon, we must include particles. The blending of particles is an easy process, just added during the cast.
Nevertheless, both methods employ different properties of the scattered light which depend to the particles and the light
source. SLP technique uses polarization properties of the scattered light. For that, the size of the added particles must be of
the same order as the wavelength of the light source (0.5 µm, Rayleigh's law). The obtained random pattern is a light speckle
field due to interferences of the light scattered by each particle. For this study, a small quantity (0.05%) of silica powder with a
size of few microns is added for 3D photoelastic analysis. To observe scattered light properties, a polarizer is placed in front of
the CCD camera lens and we record the evolution of the light intensity according to the orientation of the polarizer. Fig. 2-a
presents the normalized intensity in a specimen containing Silica powder. The sinusoidal variation of the intensity shows that
the light is polarized as expected. For using DVC by OST, the speckle laser field must be eliminated, only lighted particles
must be observed. On this configuration, it has been shown [13,14] that a better measurement uncertainty is obtained. For
that, out of Rayleigh laws conditions, large particles like polyamide powder with a size from 150 to 200 µm are blended during
the material elaboration at a small quantity (0.5%). Fig. 2-b shows that these particles involve scattered light which is not
polarized. In this case, the observed pattern corresponds to the illuminated particles and not to interference.
Figure 2. (a) Evolution of the normalized intensity In of the scattered light with orientation θ of a polarizer of epoxy resin
containing silica powder and (b) of epoxy resin containing polyamide powder
Determination of photoelastic constants
For 3D photoelastic analysis, the determination of the differences of secondary principal stresses or strains needs the
knowledge of photoelastic constants C or K. These constants can easily be estimated from a mechanical test where the two
principal stresses (or strains) are known. We have evaluated those by performing a tensile test on specimens placed in a
classical 2D circular polariscope (Fig. 3). During the test, we record the evolution of the intensity, the true stress assumed
homogeneous is obtained from data of a loading cell and the strain is measured by a marks tracking technique [31]. Fig. 4
presents the evolution of the intensity according to principal stress and strain difference and thus we plot the fringe order. So,
with the formulas (4-6), we calculate the constants C and K. As a result, for our specimen in epoxy resin containing Silica
powder, the constants C and K are respectively equal to 43 ±0.5 Bw and 0.0995 ±1e-4.
Figure 3. Experimental setup for the determination of photoelastic constants
Figure 4. (a) Evolution of the intensity according to the principal stress difference and (b) according to the principal strain
difference; (c) Fringe order N according to the principal stress difference and according to the principal strain difference (d).
Mechanical testing
In order to compare the fields of applications and the performances of both methods, we have performed a located
compression test with a spherical model which involves 3D mechanical effects within the sample. As both methods do not use
3
the same optical properties, we have made two similar parallelepiped specimens (with dimensions 50x35x22 mm ): one for
SLP and one for the DVC by OST. In both cases, there are the same boundary conditions and the loading imposed with the
spherical model is equal to 150 daN. The measured Young moduli of both models are equal (2700 ± 10 MPa).
For OST, we have chosen a spatial resolution equal to 60 µm (according to the size of specimen and particles). DVC has been
3
processed with a correlation subset size of 31x31x31 vovels and the step of the 3D grid is set at 20 voxels. To have the same
gauge length with SLP, the thickness of the slice is fixed to 4 mm and the specimen is scanned with a step of 2 mm.
Results and discussion
The most significant differences between both methods are linked to the measurement process and data obtained. DVC gives
all the components of the displacement field after several hours of processing (Fig. 5-a) whereas we observe fringes in real
time with SLP (Fig. 5-b). Nevertheless, these fringes represent isovalues of principal stress or strain difference and a fringe
analysis process is necessary to obtain their numerical values [10]. However, this method is a powerful tool to visualize and
analyze quickly the stress state within a structure.
To compare more precisely obtained values, it is necessary to take same data in both cases. So, it is easier to calculate
principal strain difference from displacement field obtained than the inverse procedure. Then, we have calculated the principal
secondary strain difference from the photoelastic fringes (equation 5) on two straight lines A-B and A-C for successive
positions according to the thickness of the specimen (Fig. 6). For that, we have determined the positions at maxima and
minima of fringes with a measurement uncertainty less than 0.05 mm. From 3D displacements obtained by DVC in each point
of a 3D grid, we calculate (formula 12) the full strain tensor (Fig. 5-c) and we deduce the principal secondary strain difference
on the same lines A-B and A-C (Fig. 6). Fig. 7 presents profiles in the middle slice of the specimen.
Fig. 6 and Fig. 7 show that DVC seems to be better adapted for large strain measurement than SLP which is limited near the
contact zone. In the hatched area of the Fig. 6-b and 6-d, there are uncalculated points because of a too large number of
fringes. Nevertheless photoelasticty gives better results for small strains where DVC is limited by its strain measurement
uncertainty (0.1% [12-14]). Let us note that we have also an undetermined zone with DVC near the contact area (hatched
zone on Fig. 6-a and 6-c) due to the shadow of the spherical model.
y
x
Figure 5. (a) Displacement component Uy measured by OST; (b) Image of isochromatic fringes obtained by SLP on slice in the
middle of the specimen; (c) Strain tensor component εyy calculated from 3D displacements measured by OST.
Figure 6. Evolution of the principal strain difference ε1 - ε2: (a) along the line A-B from data given by OST and (b) from data
obtained by SLP; (c) Along the line A-C from data given by OST and (d) from data obtained by SLP.
Figure 7. Evolution of the principal strain difference ε1 - ε coming from a slice in the middle of the specimen (z=11 mm): (a)
along the line A-B; (b) Along the line A-C.
Conclusion
In this paper, we present two methods enabling to analyze 3D mechanical phenomena. These techniques use a model
structure made in a transparent material. We show an application of the methods on a mechanical testing with a loading by a
spherical model. Both methods are complementary. On one hand, DVC by OST is better adapted for large strain measurement
than SLP. On the other hand, SLP with a best accuracy is well adapted for small strain problems. Furthermore, this technique
allows us to visualize immediately the 3D stress state by analyzing photoelastic fringes. However, we obtained principal
secondary stress or strain difference. DVC by OST gives us 3D displacements on 3D grid but needs a large time acquisition
and processing. To conclude, both methods are effective for the analysis of 3D problems and be very helpful for example for
the validation of simulations and that is why they are used for mechanical problems in aeronautics.
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