2 1 Introduction The study of cracks is part of the theories describing the degradation of materials and failure of structures [1]. Crack initiation, ...

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Controlling Stress Intensity Factor Histories with Digital Images Xavier Fayolle

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⋆

François Hild

Received: date / Accepted: date

Abstract It is shown that stress intensity factors (SIF) can be prescribed in mechan-

ical tests by using digital image correlation. A priori resolutions are validated with a posteriori results. A cascade controller is implemented to prescribe SIF histories. A

proof of concept is shown when testing a cracked elastomer. A tensile experiment is nally analyzed with a center cracked sheet made of commercially pure titanium. The linearity error is shown to be less than one order of magnitude higher for the closedloop system when compared to the SIF resolution. However, it is still small compared to the SIF ranges investigated herein.

Keywords Digital image correlation

·

fracture mechanics

measurement resolution

Laboratoire de Mécanique et Technologie (LMT-Cachan) ENS Cachan / CNRS / UPMC / PRES UniverSud Paris 61 Avenue du Président Wilson, F-94235 Cachan Cedex, France ⋆

Corresponding author: [email protected]

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full eld measurements

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2 1 Introduction

The study of cracks is part of the theories describing the degradation of materials and failure of structures [1]. Crack initiation, propagation and arrest are concepts that are important to master to design and monitor actual structures such as airplanes, cars, trains, or power plants [2]. In many cases, elastic approaches are still used to assess the criticality of cracks. One way of mechanically characterizing cracks is to consider stress intensity factors (SIFs) that describe stress singularities in the vicinity of the crack tip, or equivalently, displacement discontinuities across crack faces [3]. It is the latter property that will be used hereafter in conjunction with full-eld measurements.

Performing experiments on cracked samples or structures is a dicult task since crack propagation may become unstable. This is particularly true when load-controlled experiments are carried out since the SIF is generally increasing with the crack size. Conversely, when displacement controlled, the tests are usually easier to perform since crack propagation is likely to be stable. However, propagation may stop. The reason is that the parameters that control crack propagation are not necessarily the applied load or displacement, but crack opening displacements (CODs) [4, 5], SIFs [6], energy release rates [7] or

J -integral

[810].

Except for CODs for which displacement transducers (so-called crack gauges) are used on large structures (e.g., bridges, buildings) or COD (clip) gauges on smaller ones, the other quantities are only indirectly evaluated when using point data, provided the crack length is known or monitored. Load shedding techniques whereby the applied load is gradually decreased as the crack propagates [11] have been proposed to keep the nonlinearities conned in the vicinity of the crack tip. However, the crack tip position needs to be determined to relate the applied load to the SIF level. Conversely,

3 the sample geometry itself can be adapted to partially compensate for the load vs. SIF increase (e.g., edge-cracked tapered plate [12]). Yet another route consists of a linearly coupled load and displacement control loop that does not require the exact crack tip position to be determined [13]. One key parameter to tune is the so-called mixing ratio between displacement and remote load.

An alternative route is provided by full-eld measurement techniques, in particular Digital Image Correlation (DIC [14, 15]). Besides the qualitative aspect of detecting a crack, quantitative estimates of mechanically signicant properties have been reported such as opening loads in fatigue experiments [16, 17] and/or SIFs [1822] by least squares t using closed-form solutions [3]. However, all these analyses were performed a posteriori and not during the experiment itself. Pictures have already been used to drive experiments. For instance, by tracking points, average strains are controlled [23]. This is also possible by resorting to DIC [24]. In both cases, experiments could be performed in the presence of localized strains. The advantage of DIC is that it is, by construction, a full-eld measurement technique that was validated to analyze cracked media. In particular, its resolution may reach small values (typically 1 cpixel or even lower), which was shown to be compatible with the uncertainty level of strain gauges [24].

DIC is appealing when trying to control experiments, in particular when the input signals cannot be given by a strain gauge, an extensometer or a transducer (e.g., measuring a displacement or a load). DIC has already been used to control cracking experiments [25, 26]. Either post-processed DIC measurements [25] or integrated DIC [26] have been used to evaluate the SIF history. In the present paper, the former approach is used to control experiments in which the crack does not propagate. The goal is to assess the resolution of the closed-loop system and to prove the feasibility

4 on two very dierent materials (i.e., a very soft elastomer and a metallic material). To design the closed-loop software, the SIF resolution has to be assessed. This procedure is performed by rst evaluating the displacement resolution of the implemented DIC code. Thereafter, it is possible to assess the resolution of SIFs. The performance of the controller is analyzed when an edge-cracked plate made of a soft material (i.e., elastomer) is tested. Last, experimental results are analyzed when an SIF history is prescribed on a center cracked sheet made of commercially pure titanium. In both cases, the a priori resolutions are compared with linearity errors for the open-loop and closed loop congurations.

2 SIF Resolution

When designing a closed-loop system, one key quantity is the resolution of the measuring device of the controlled variable (i.e., the SIF in the present case). To evaluate the SIF resolution, the displacement resolution is needed. It is based upon the analysis of the noise sensitivity of DIC [27, 28]. The SIF resolution is then obtained by propagating the uncertainty induced by the least squares procedure to evaluate the mode I SIF from the measured displacements.

2.1 Displacement Resolution

In practice, the region of interest (ROI) is selected for the SIF gauge. Within the ROI, smaller correlation windows or zones of interest (ZOIs) are chosen. For each ZOI in the reference image, its corresponding position is sought in subsequent pictures. The pattern-matching algorithm is based on cross-correlation. The cross-correlation is a function giving the maximum likelihood between two ZOIs shifted by a constant

5 displacement. Its computation is performed via fast Fourier transforms [2932]. The location of this maximum gives the most likely displacement between the ZOI in the reference and deformed congurations. For a sub-pixel estimate, a parabolic interpolation is performed around the maximum pixel value and the displacement corresponds to the maximum of the interpolated correlation function. Then the ZOI in the deformed conguration is shifted by an amount equal to the evaluated displacement using the shift/modulation property in Fourier space. Since interpolation errors occur, these three steps need to be iterated until convergence is reached. In the present case, since very small displacement increments are sought, a constant displacement assumption for each ZOI is deemed to be sucient [24].

An estimate of the displacement resolution of the correlation algorithm is needed. The following procedure is applied to the actual picture of the experiment with the elastomeric material. The CCD digital camera used herein has a

1280 × 1024-pixel

denition and a 12-bit dynamic range. Telecentric lenses are used to minimize the (spurious) eects of out-of-plane motion. To have independent estimates of the displacements in the following analyses, the shift equal to the ZOI size

δ

between consecutive ZOIs is at least

ℓ. The noise level of the camera is of the order of σ = 20 gray lev-

els. By following the derivations of Ref. [33], the corresponding standard displacement resolution is expressed as

√ σu =

where

Gf

contrast),

2σp Gf ℓ

(1)

is the root mean square (RMS) value of the picture gradient (i.e., its mean

p

the physical size of one pixel.

6 These a priori results are compared with actual (i.e., a posteriori) measurements for which a set of 10 pictures is acquired for the same conguration. It was checked that the mean displacement was very small for all the analyzed cases. Figure 1 shows the standard displacement resolution ZOI size, the smaller the resolution

σu

σu .

as a function of the ZOI size

ℓ;

the larger the

However, if the ZOI size increases, the spatial

resolution is degraded, i.e., less independent measurement points are obtained. The ZOI size also needs to be adapted to the speckle size (here of the order of

≈2

pixels

when characterized by an average correlation radius [34], i.e., suciently small with respect to

ℓ).

Furthermore, as the ZOI size increases, so does the computation time. A

good compromise for the present study is given by a ZOI size of 32 pixels for which a standard displacement resolution of the order of

2 × 10−3

Last, when interpolating the displacement resolution

10−2

is found, to be compared to

when

σu

and

√

pixel is achieved.

σu

2σ/Gf = 8.0 × 10−2

by

A/ℓ,

a value

A = 7.4 ×

(according to Equation (1)

ℓ are expressed in pixels). These two levels being close, the present results

validate the a priori estimate of the standard displacement resolution

σu . The low level

of the displacement resolution is obtained thanks to a mean contrast

Gf

of the order

of 355 gray levels / pixel.

2.2 SIF Resolution: A Priori Estimate

The displacement eld

x-axis (x < 0)

u

in an innite solid containing a semi-innite crack along the

is written as a combination of a double semi-innite series [3]. In the

following, only the dominant term associated with the mode I SIF is considered under a plane stress hypothesis (i.e., the considered samples have small thicknesses compared

7 with the other dimensions)

ux =

KI E

K uy = I E where

r, θ

√

r θ cos [3 − ν − (1 + ν) cos θ] 2π 2

√

r θ sin [3 − ν − (1 + ν) cos θ] 2π 2

denote the polar coordinates with respect to the crack tip,

coordinates with respect to the crack tip,

E

Young's modulus, and

(2)

x, y ν

the cartesian

Poisson's ratio.

This displacement eld is rewritten as

u = KI υK

where

υK

(3)

corresponds to the displacement eld of a unitary mode I SIF.

Let us now assume that the displacement is known at a number of discrete points (i.e., the center of each ZOI) with

n = 1, . . . , nx ny

denote the number of ZOIs in the are denoted by

um .

and

y

directions. The measured displacements

v1 , v2 ,

rotation

v3 (xn ))

and the mode I SIF eld

The corresponding amplitudes are determined by minimization of

the quadratic cost function

T

T ({a}) =

∑

∥um (xn ) −

n where

nx , ny

The identication basis consists of elds associated with rigid

body motions (i.e., translations

v4 (xn ) = υK (xn ).

x

close to the crack tip, where

xn

∑

ai υi (xn )∥2

(4)

i

{a} gathers the unknown translations u0x , u0y , rotation ω , and SIF KI . A linear

system is solved

[T ]{a} = {b}

(5)

with

Tij =

∑ n

υi (xn ) · υj (xn ) ,

bi =

∑ n

υi (xn ) · um (xn )

(6)

8 It was mentioned that displacements are corrupted by measurement noise,

ηu ,

which

is considered as white in the present case since each ZOI is independent of the other ones [34]. Consequently, the optimal SIF extractor is given by the least squares procedure [35]. Moreover, it was checked that the acquisition noise was a Gaussian distribution. It is of interest to investigate the eect of uncertainties on the resulting evaluation of

{a}.

By considering the linear system to be solved for

{a},

it is observed that no

systematic error (or bias) will result from the noise, and the covariance matrix

[C]

reads

[C] = [T ]−1 σu2 When the ROI is centered about the crack tip, the only o diagonal term of is associated with

υx · υK (xn ).

(7)

[T ]

A rst order estimate of the diagonal terms of the

covariance matrix is given by (no index summation)

Cii ≈ σu2 /Tii

(8)

which consists of neglecting any cross correlation between the various amplitudes (i.e.,

Cij ≪ Cii ,

when

i ̸= j ).

Further, the following approximation is used to assess

which corresponds to the variance

σi2

of amplitude

ai , the other parameters being xed,

∫ Rmax ∫ 2π 1 1 ∑ 2 2 [υix (r, θ) + υiy (r, θ)] rdrdθ ∥υi (xn )∥2 ≈ 2 nx ny πR max 0 0 n where

Rmax

Lx

(9)

is determined by assuming a surface equivalence of the ROI

2 πRmax = Lx Ly

where

Cii ,

and

Ly

(10)

denote the width and height of the ROI. The centers of the ZOIs

correspond to the collocation points of the approximation of the previous integral. The standard SIF resolution

σK σK

is normalized to become

√

nx ny Rmax ≈ Eσu

√

6π 19 − 10ν + 3ν 2

(11)

9 When the Poisson's ratio

ν

varies from

−1

to

sionless standard SIF resolution ranges from the order of unity (since (ranging from

√

nx = ny = 2

6π/19 ≈ 1).

0.5,

the approximation to the dimen-

≈ 0.77

≈ 1.13

to

(see Figure 2), i.e., of

Figure 2 shows that for small numbers of ZOIs

to 6), the proposed approximation is a good rst order es-

timate. It can be shown that when the number of ZOIs becomes very large, the present approximations lead to an underestimation of the order of 20% of the standard SIF resolution. Equation (11) also shows that the higher the Young's modulus Further, the larger increasing

Rmax

Rmax ,

σK .

the lower

E,

the larger

σK .

However, some care should be exercised since

may induce a ROI that no longer belongs to the

K -dominant region so

that the chosen kinematic basis is no longer relevant. This can be checked by assessing the identication error

√ η=

The closer increasing

η

to

σu ,

T ({a}) nx ny

(12)

the better the identication. It is also possible to decrease

σK

by

nx ny , which is accompanied by an increase in computation time. A nal way

is to decrease the displacement resolution

σu .

This is achieved by considering larger

ZOIs, again inducing a higher computation cost. This last discussion shows that the SIF resolution

σK

is the result of a trade-o

between the resolution level and the computation time. This is particularly important for a closed-loop system in which the reference signal is obtained by the analysis of digital images. By using the results of the previous section, the SIF resolution becomes

σK

√

√ nx ny Rmax 2σp ∝ E Gf ℓ

(13)

Equation (13) summarizes the dierent compromises that have to be considered when designing a controller with a DIC-based evaluation of SIFs. The left hand side term

10 σK

√

nx ny Rmax /E

analysis of

nx ny

Young's modulus

is related to the sought quantity itself, which is associated with

points in a ROI of characteristic radius

E.

The second parameter,

√

2σp/Gf ℓ,

Rmax

of a material with

is the displacement resolu-

tion, which is related to the analyzed texture (i.e., its mean contrast camera (of acquisition noise resolution

ℓ.

σ)

with a chosen magnication (i.e.,

p),

Gf )

by a given

and to the spatial

It is worth noting that the SIF resolution is valid when the crack tip is

centered in the ROI. If this hypothesis is not satised, there are additional correlations that need to be accounted for. However, for the sake of simplicity, they are not considered in the following analyses.

Last, the fact that only the very rst mode I displacement eld associated with the presence of a crack is chosen is related to the limited number of measurement points, themselves linked to speed of the controller. This choice implicitly assumes that the ROI belongs to the K-dominant area and that a pure mode I loading is achieved. These hypotheses may be checked a posteriori with a more complete analysis considering more terms of Williams' series [21, 17, 36] and more measurement points. Furthermore, the estimate of the stress intensity value may be biased as additional terms of Williams' series are needed to describe the measured displacement eld. The level of the identication error is one means of checking whether the considered basis is sucient when compared to the displacement resolution.

3 Controlling SIF Histories via DIC

The general setting of the controller is identical to the one that was used to prescribe strain histories to a sample [24], namely, a two-loop system is implemented to allow a slow outer loop to compute the current value of the SIF to be compared to the

11 prescribed level. One cycle consists of an image acquisition, image saving (on the hard disk) and the evaluation time of the SIF through DIC calculations. It requires a total time of 2.4 s. This is not a limitation per se as the use of GPUs may signicantly reduce this time [37]. The aim here is to show the feasibility of SIF control, more than `real time' issues. The inner loop consists of a displacement prescribed signal with a frequency of 20 Hz compatible with electromechanical (and servohydraulic) testing machines.

In the present application, the input signal is an SIF. In the following, the loading history is given by linear ramps loading and unloading the crack. To cancel out the steady state error between the input signal (i.e., a ramp) and the measured SIF (feedback), a double integral controller is implemented in a cascade system (Figure 3). However, a double integral controller, by itself, is unstable. A zero is added in the low

◦

frequency range to raise the phase by up to 45 , thereby yielding a sucient phase

◦

margin (from the 180

limit). Last, a unitary open loop gain is sought so that SIF

signals are directly controlled.

Contrary to what was performed previously (i.e., using a servohydraulic testing machine [24]), an electromechanical testing machine is chosen. The machine manufacturer gives a rise time of 50 ms for 10

µm displacement increments. To avoid accounting

for the frequency response of the testing machine itself the computer process is delayed by 300-ms hold time (i.e., only the steady state response is needed). The consequence is that the total cycle time of the outer loop is 2.7 s (rather than 2.4 s). The testing machine is modeled as a sti equation (i.e., no Laplace function). To check various aspects of the controller depicted in Figure 3 and to estimate the zero value of the con-

r

troller Simulink

[38] is used. Figure 4 shows the error (E) and reference (R) signals.

Apart from the changes of slope that induce high (unavoidable) errors, the system has

12 a behavior that is consistent with what was expected (e.g., zero steady state error for ramps). One important aspect of the controller is related to gain

G0 that links the prescribed

SIF (outer loop) with the displacement of the testing machine crosshead (inner loop). To compute

G0 ,

the total displacement associated with a sample of initial length

cracked at its mid-height is needed. The edge crack size is denoted by the sample by

b. The relative displacement ∆up

∆up = L0

where

Σ

is the remote stress, and

The applied stress

Σ

v2

Σ E

of the two

( 1+

4av2 L0

L0 -apart sections reads [39]

) (14)

KI

Y

(15)

is a dimensionless parameter dependent upon

two expressions, gain

G0

a/b.

by

√ KI = Y Σ πa where

a and the width of

a dimensionless parameter dependent upon

is related to the SIF

L0 ,

a/b

[39]. By combining these

becomes

G0 =

∆up L0 √ = KI Y E πa

( 1+

4av2 L0

) (16)

All the parameters dening the SIF controller have now been introduced. The fol-

TM

lowing section is a proof of concept of the implemented controller in a LabVIEW environment [40].

4 Proof of Concept

4.1 Studied Material

Being a feasibility study, the implemented controller is used to perform a test on a cracked elastomer. The displacement range was kept suciently low to avoid various

13 non linear aspects associated with elastomers [41]. To evaluate the elastic parameters of the studied material, a tensile test on a dogbone sample has been carried out. Digital images were acquired during the test and the same DIC code was used to evaluate the mean longitudinal and transverse strains. The strain resolution

σϵ

is determined by

following the same type of derivation as for the SIF resolution. The only dierence is given by the interpolating displacement eld, which is linear (i.e., three unknowns per direction)

σϵxx with

350

σu = √ Lx nx ny

√

12(nx − 1) , nx + 1

√ σϵyy

σu = √ Ly nx ny

12(ny − 1) ny + 1

(17)

Lx = δ(nx − 1) and Ly = δ(ny − 1). With the chosen parameters (i.e., Ly > Lx =

pixels,

ℓ = 32

pixels, and

nx = ny = 4)

the strain resolution is less than

5 × 10−6 .

Figure 5 shows the stress/strain response of the material. As explained above, the stress and strain levels remain very low. By using a least squares t of the measured data, a Young's modulus

E = 7.6

MPa and Poisson's ratio

ν = 0.4

are obtained.

Further, most of the uctuations observed in Figure 5 are due to the load cell (whose resolution is of the order of 1 N or 0.02 MPa) and not the DIC code since the strain range is ten thousand times the corresponding resolution.

4.2 A Posteriori Validation of Strain and SIF Resolutions

With the experimental conguration, the strain resolution is less than the SIF resolution is evaluated as one pixel is

√ σK = 2 × 10−6 MPa m,

p = 53 µm, Rmax = 275

pixels, and

ℓ = 32

5 × 10−6

and

since the physical size of

pixels. When performing the

strain and SIF analyses on the same set of pictures as those used in the displacement resolution analysis, it is found that the strain resolution is of the order of the SIF resolution

√ σK = 2.1 × 10−6 MPa m.

4 × 10−6

and

These two values are very close to the a

14 priori estimates proposed herein. Consequently, the a priori displacement, strain and

SIF resolutions are validated a posteriori.

4.3 Open-Loop Response

The response of the open-loop controller is shown in Figure 6. 137 pictures were acquired. Except for the changes of slope, the RMS dierence with a piecewise linear response is of the order of

√ 1.4 × 10−5 MPa m.

This value can be compared to the SIF

resolution given above. There is a factor 7 between the two results. To explain this dierence, there are three types of cause to be listed. First, the resolution analysis performed herein does not account for sub-pixel interpolation errors [34]. In the present case, it is expected that they are non negligible since the displacement resolution is found to be very small. Second, the linearity error is also due to the crosshead displacement, which itself has its own resolution and errors. Last, the choice of only 4 displacement elds induces identication and model errors of the SIF level. It is expected that the model error be very small because of the low level of SIF. Conversely, the identication error is expected to increase with the SIF level. This trend was experimentally observed when the change of

η

against the SIF was analyzed.

These results show that the resolution analysis provides a lower bound to the actual resolution of the open-loop system. The controller itself induces its own error, which will be assessed in the next section.

4.4 Closed-Loop Response

Figure 7 shows the input, output and error signals of the implemented feedback system. From these results it is concluded that such a system is able to drive an SIF history with

15 a very small error (Figure 7(b)) for SIF amplitudes ranging from 0.008 to

√ 0.04 MPa m

(Figure 7(a)). The errors are the highest at slope changes, as expected from the simulated system (Figure 4). Their absolute level is less than

√ 5 × 10−4 MPa m,

i.e., one

hundred times less the maximum SIF level, the SIF history was indeed controlled in a very satisfactory manner. 1437 pictures were acquired during this experiment.

On a more quantitative basis, the RMS error is equal to

√ 8×10−5 MPa m when the

whole history is analyzed. The latter includes the 11 slope changes of the SIF history (Figure 7(a)). When these changes are not considered, the RMS error decreases by a factor of 3 to reach

√ 2.7 × 10−5 MPa m.

This last value includes the resolution of the

DIC measurements, the SIF extraction and the performance of the controller itself. When compared with the a priori SIF resolution (σK obtained with an open-loop conguration (σK

√ = 2 × 10−6 MPa m),

√ = 10−5 MPa m)),

and that

it is concluded that

the controller contributes to about 60 % of the total error. This level is to be expected from a standard closed-loop system.

5 Test on Commercially Pure Titanium

Having shown that the designed controller is able to accurately prescribe SIF histories on a soft material, the nal step of the present study is to perform the same type of experiment on a stier material (i.e., with a Young's modulus in the 100-GPa range). In the present case, commercially pure titanium (i.e., T35) is chosen. This type of material has already been studied to estimate crack propagation parameters [36, 42]. The sample has a CCT geometry with a width of 30 mm, a height of 150 mm, and thickness of 0.3 mm. The initial crack size is equal to 6 mm. In the present case, it is believed that the plane stress solution is a good approximation.

16 Figure 8(a) shows the input and output signals for the four applied cycles when the SIF range varies between 6 and

√ 9 MPa m.

In the present case, 563 pictures are

acquired. The overall response of the controller is deemed satisfactory as no particular deviations are observed. Contrary to the previous case, the change of slope is not accompanied by a higher error (Figure 8(b)). This is due to the fact that the overall level of the latter is signicantly higher. The RMS error with respect to linearity is found to be equal to

√ 9.2 × 10−2 MPa m.

Being more than 60 times lower than the

minimum prescribed amplitude, the implemented system can be used to control cyclic tests with prescribed SIF histories

For the chosen (T35) grade, the Young's modulus is equal to 100 GPa and Poisson's ratio

ν = 0.33

[36]. With these elastic parameters, the a priori SIF resolution becomes

√ σK = 1.4 × 10−2 MPa m

since the physical size of one pixel is

(18)

p = 13.4 µm, Rmax = 400 pixels, and ℓ = 32 pixels.

For the open-loop system, an RMS error with respect to linearity is found to be equal to

√ √ 4.1×10−2 MPa m (to be compared to 9.2×10−2 MPa m for the closed-loop system). In the present case, there is a factor 3 dierence between the a priori SIF resolution and the RMS linearity error of the open-loop response, and a factor 7 dierence between the SIF resolution and the RMS error of the SIF controller. With the chosen SIF amplitudes whose level is close to SIF threshold in high cycle fatigue [36], it can be concluded that the controller can be used to monitor crack opening during fatigue cycles with SIF amplitudes greater than a few

√ MPa m.

Figure 9 shows the change of the measured load with respect to the prescribed SIF. There is a linear relationship between these two quantities, thereby validating the small scale yielding hypothesis. When measuring the RMS error to linearity, it is equal

17 to

√ 9.5 × 10−2 MPa m

when expressed in terms of SIF, or 1.7 N in terms of load. The

former is very close to the RMS error observed for the controller itself. The latter is of the order of what is expected from the used load cell (see Figure 5). This last result validates the whole procedure developed herein.

6 Summary

It was shown that it is possible to control in a mechanical test a stress intensity factor (SIF) history by resorting to digital images. In the present case, a local approach to digital image correlation is considered to measure the displacement eld. The latter is then post-processed to evaluate the rigid body motions of the crack tip and the mode I stress intensity factor (SIF). Since this procedure requires almost 3 seconds per control cycle, a two-loop system is implemented. The inner loop is displacement-controlled and the outer loop is SIF-controlled.

Resolution analyses were performed to design the controller. A priori estimates of displacement, strain and SIF resolutions were validated a posteriori by analyzing a series of ten pictures for which no displacements were prescribed. The SIF resolution was then compared with the linearity error of the open-loop and closed-loop system. There is almost an order of magnitude dierence when the former is compared with the latter ones. This result is to be expected as it includes not only measurement uncertainties but also other aspects of the controller. With the two studied cases, it was shown that the SIF range is signicantly larger (i.e., almost two orders of magnitude) compared with the linearity error of the controller. It is concluded that the SIF histories prescribed herein are suciently accurate to monitor gradual crack openings and closures in cyclic tests.

18 Having validated the implemented controller, the next step is to consider crack propagation. This step requires an accurate estimation of the crack tip position, which was not discussed herein. The same procedure as that proposed by Hamam et al. [17], which is based on canceling out the amplitude of the rst supersingular displacement eld, will be considered. As already mentioned, additional displacement elds accounting for, say the T-stress or other subsingular contributions [43, 36], may also be added. They will yield a more robust estimate of the stress intensity factor. These issues being solved, gradual propagation (e.g., in high cycle fatigue) will enable the controller to be used in a `real time' conguration since the crack tip motion per cycle is expected to be very small so that the determination of its position will be needed only at large time increments.

Acknowledgments

This work was partially supported by a grant from Région Île-de-France (plateforme francilienne d'expérimentation mécanique de troisième génération). The authors thank Dr. Roux for useful discussions during the project.

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23 List of Figures

1

Standard displacement resolution

σu

as a function of the ZOI size

ℓ.

The symbols correspond to the analysis with the reference picture of the SIF-controlled test on the studied elastomer. The solid line is the interpolation given by Equation (??) . . . . . . . . . . . . . . . . . . . .

2

Dimensionless standard SIF resolution tion of the Poisson's ratio

ν

σK

√

nx ny Rmax /Eσu

for dierent numbers

nx ny

24

as a func-

of ZOIs. The

symbols correspond to the value of the Poisson's ratio of the studied elastomer, and the solid line to the approximation given by Equation (??) 25

3

Block Diagram of the two-loop controller. Points R, E and O correspond to the signals shown in Figure 4 . . . . . . . . . . . . . . . . . . . . . . .

4

Simulated response of the controller (see Figure 3). (a) Reference (R) SIF history, (b) error signal (E = O

5

26

−

R)

. . . . . . . . . . . . . . . . .

27

Stress / strain response of a tensile test on the studied elastomer. The stress uctuations are due to the standard resolution of the load cell (i.e., of the order of 1 N or 0.02 MPa)

6

. . . . . . . . . . . . . . . . . . .

28

Analysis of the controller as an open-loop system. Change of the SIF as a function of the picture number when two loading/unloading cycles are prescribed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

29

Analysis of the controller as a closed-loop system. (a) Comparison of the input and output SIF signals of the controller when a single edge cracked sample made of elastomer is tested. (b) Error signal in the controller . .

30

24 8

(a) Comparison of the input and output SIF signals of the controller when a CCT sample made of T35 is tested. (b) Corresponding error signal in the controller . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

31

Measured load vs. SIF for the CCT sample made of T35. The crosses are experimental points and the solid line corresponds to a linear interpolation to be expected when the small scale yield hypothesis applies . .

32

Standard displacement resolution (pixel)

25 −2

10

DIC Equation (1)

−3

10

−4

10

1

10 Fig. 1

2

10 ZOI size (pixels)

3

10

Standard displacement resolution σu as a function of the ZOI size ℓ. The symbols

correspond to the analysis with the reference picture of the SIF-controlled test on the studied elastomer. The solid line is the interpolation given by Equation (1)

Dimensionless standard SIF resolution

26

Fig. 2

1.5 1.4 1.3

4 16 36 Equation (11)

1.2 1.1 1 0.9 0.8 0.7 −1

−0.5 0 Poison’s ratio, ν

0.5

√

Dimensionless standard SIF resolution σK nx ny Rmax /Eσu as a function of the Pois-

son's ratio ν for dierent numbers nx ny of ZOIs. The symbols correspond to the value of the Poisson's ratio of the studied elastomer, and the solid line to the approximation given by Equation (11)

27 CORRELISIF R

reference SIF signal

TESTING MACHINE + SAMPLE

E

+-

(z-0.3) (z-1)2

+ +

G0

O

CAMERA + IMAGE PROCESSING + SIF CALCULATION

inner loop 1

Fig. 3

Block Diagram of the two-loop controller. Points R, E and O correspond to the signals

shown in Figure 4

28 2

SIF (a.u.)

1.5

1

0.5

0 0

200

400 600 Picture number

800

1000

(a) 0.02 0.015

Error (a.u.)

0.01 0.005 0 −0.005 −0.01 −0.015 −0.02

0

200

400 600 Picture number

800

1000

(b) Fig. 4

Simulated response of the controller (see Figure 3). (a) Reference (R) SIF history, (b)

error signal (E = O − R)

29

0.4

Stress (MPa)

0.3 0.2 0.1 0 −0.02

Fig. 5

0

0.02 Strain

0.04

0.06

Stress / strain response of a tensile test on the studied elastomer. The stress uctua-

tions are due to the standard resolution of the load cell (i.e., of the order of 1 N or 0.02 MPa)

30 0.012

SIF (MPa.m

1/2

)

0.01 0.008 0.006 0.004 0.002 0 0

Fig. 6

50 100 Picture number

150

Analysis of the controller as an open-loop system. Change of the SIF as a function of

the picture number when two loading/unloading cycles are prescribed

31 0.04

output input

0.035 SIF (MPa.m1/2)

0.03 0.025 0.02 0.015 0.01 0.005 0 0

500 1000 Picture number

1500

(a) −4

x 10

Error (MPa.m

1/2

)

5

0

−5 0

500 1000 Picture number

1500

(b) Fig. 7

Analysis of the controller as a closed-loop system. (a) Comparison of the input and

output SIF signals of the controller when a single edge cracked sample made of elastomer is tested. (b) Error signal in the controller

32

10

output input

SIF (MPa.m

1/2

)

8 6 4 2 0 0

100

200 300 400 Picture number

500

600

500

600

(a)

0.3

Error (MPa.m1/2)

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 0

100

200 300 400 Picture number (b)

Fig. 8

(a) Comparison of the input and output SIF signals of the controller when a CCT

sample made of T35 is tested. (b) Corresponding error signal in the controller

33

200

controller interpolation

Load (N)

150

100

50

0 0

Fig. 9

2

4 6 1/2 SIF (MPa.m )

8

10

Measured load vs. SIF for the CCT sample made of T35. The crosses are experimental

points and the solid line corresponds to a linear interpolation to be expected when the small scale yield hypothesis applies