epoxy composites

Composites Science and Technology 88 (2013) 16–25

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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech

Use of diffuse approximation on DIC for early damage detection in 3D carbon/epoxy composites Pierre Feissel a,⇑, Julien Schneider b, Zoheir Aboura a, Pierre Villon a a b

Laboratoire Roberval de Mécanique, UTC, BP 20529, rue Personne de Roberval, 60205 Compiegne, France SNECMA Villaroche - Rond point René Ravaud, 77550 Moissy-Cramayel, France

a r t i c l e

i n f o

Article history: Received 16 October 2012 Received in revised form 19 June 2013 Accepted 18 August 2013 Available online 3 September 2013 Keywords: A. Structural composites B. Non-linear behaviour C. Anelasticity

a b s t r a c t This work is part of a global experimental approach aiming at studying the damage and rupture of 3D carbon/epoxy composites and based on multiinstrumented experiments. It focuses on the use of digital image correlation (DIC) in order to detect the development of local non-linearities corresponding to damage (with no other non-linearities) and is limited to small deformation. To that purpose, the strain field is required, hence derived from the measured displacement field, through a diffuse approximation algorithm. This approach is detailed and studied in terms of filtering and some criteria are proposed to choose the filtering parameters. A pragmatic approach based on the evolution of the linear approximation of the strain is proposed in order to detect local non-linearities leading to a local damage indicator. The effect of the filtering on the damage detection is then discussed. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The 3D composite reinforcements grow up as a new generation of composite materials and complement the traditional 2D reinforcements. The main advantage of these new architectures consists in overcoming a major drawback of composite laminates: delamination [1,2]. Various studies have shown the contribution of this type of reinforcements in terms of improvements concerning the interlaminar and off plan properties [3–7]. With these improved properties come modified in plan properties as well as new damage processes. The control of the damage process in these materials is therefore a key issue in order to develop engineer structures. This implies the identification, monitoring and understanding of the phenomena of damage, in particular to nurture any further development of model, aiming at being predictive, reliable and robust. The present study lies in a global research work, concerning the characterization of the damage and energy dissipation in such composites, in order to understand how the damage tolerance can be significantly increased. The global approach aims at highlighting the scenarios of failure of these materials under different types of loading (static, fatigue, impact, etc.). As performed on standard composites in the past [8–10], it is based on a rich multi-instrumentation to detect and track damage. This multiinstrumentation correlates measurements from conventional

⇑ Corresponding author. Tel.: +33 3 44 23 46 04. E-mail address: [email protected] (P. Feissel). 0266-3538/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2013.08.027

techniques such as extensometer gauges, post-mortem microscopic observations to newer techniques such as acoustic emission, infrared thermography, RX, videomicroscopy in situ observations and digital image correlation (DIC). The latter takes an important place in modern instrumentation [11,12]. It allows, when used with care, to detect local strain gradient in relation to the microstructure of the material [13,14]. Within this framework, the present paper focuses on the use of digital image correlation (DIC) for the detection and localization of the first sites of damage at very low strain levels (103). Due to its strong heterogeneity, the microstructure actually affects the location of the sites of damage. A keypoint to any further work is to fully understand the time and place for the advent of damage accurately. Hence the proposed approach, as detailed in Section 4, yields a local damage indicator from the DIC measurements, dedicated to small deformation and damage, as it is the case with 3D carbon/epoxy composites. As an evolution of the idea from [15], the proposed approach is based on the qualitative evolution of the local strain to loading ratio. Since DIC yields the displacement field, the strain field is to be derived from it. Such an operation is very sensitive to experimental noise, especially for low strain level [16] and we have to pay special attention to the control of noise when differentiating the measured displacement. Actually, at such strain levels, DIC usually suffers from low signal to noise ratio (SNR) and some specific approach is proposed in the following to deal with this issue. Other approaches such as the grid method [17,18] or the meshfree random grid method (MRG) [19,20], based on the tracking of well-defined patterns on the specimen, allow to improve the SNR at these strain levels. Nonetheless, it is chosen

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P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

here to use DIC because of its versatility and availability within the current work. The dealing of the noise can be introduced at the DIC level, for example by using some a priori mechanical information when seeking the displacement fields from the images [21], but this usually means a mechanical model of the experiments exists, even though some recent developments have shown the abilitity of mechanical regularisation at the DIC level with less reliable models [22]. Nonetheless, when using some standard DIC tools, the problem of differentiating noisy data remains and the displacement field is to be considered on a regular data grid. The most basic approach to derive the strain from the displacement field is to use finite difference where it is possible to choose the space step or the degree of approximation to control the filtering and the accuracy [23], but the effect on the filtering is often limited. It is also very common to apply approaches coming from signal and image processing such as kernel smoothing methods [24], where one difficulty is to reconstruct the data up to the edges of the measurement area. Alternatively, it appears to be very fruitful to choose reconstruction strategies based on approximation methods, because they lie on a theoretical framework. They can offer a way to introduce some mechanical information and guarantee the reconstruction to be optimal to a given criterion [25]; hence they can be seen as a way to construct optimal smoothing kernels. In approximation methods, the measurements are projected, usually through a least-squares formulation, on a function basis, which will be differentiated afterward, the two steps can be performed with two different basis [26]. In such approaches, the two keypoints in order to control the filtering of the noise are (i) the choice of the function basis [27–29] and (ii) the choice of local [30] or global [31] least-squares. In a previous study [32], it has been shown that good results could be reached when either the function basis (e.g. finite element functions) or the least-squares (e.g. diffuse approximation) were based on local spans. The method based on diffuse approximation developed in [32] appeared to be effective and will be used in the following. A keypoint of the use of the strain reconstruction methods is the choice of the filtering parameters in order to filter the noise while keeping the mechanical information. In the present paper, the filtering method is applied to a tensile test on a 3D composite in order to detect and localize early damage and a specific damage indicator is proposed. First the diffuse approximation (DA) is recalled and its theoretical behavior detailed. Then, the choice of the filtering parameters of the DA is discussed on real test data and a pragmatic criterion is proposed. Finally, the damage indicator is introduced and applied to the above mentioned example of a tensile test and the effect of the filtering on the results is discussed.

in uex. The filtering of the systematic error implies some mechanical a priori knowledge, as proposed in [21]. Here the filtering is not based on any mechanical considerations but will yield satisfactory results since the systematic error remains small on the studied cases. The output of the method is a reconstructed displacement and a reconstructed strain. The chosen strain is the infinitesimal strain tensor e, considering applications for carbon/epoxy composites with small deformation hypothesis. Furthermore, the tensorial strain components in the (eX, eY) basis are represented in a vectorial format, using the classical convention [33]:

2

3 ð2Þ

The aim of the reconstruction is to yield a strain field as close as possible to the one associated with uex. The reconstruction operator ~ as being linear (see Section 2.2), we can split the reconstruction of u the sum of the reconstruction of the exact field uex and the one of the perturbation alone. The reconstructed field uap(x), at any x, can therefore be written as:

uap ðxÞ ¼ uex ðxÞ þ duk ðxÞ þ dub ðxÞ

ð3Þ

where duk(x) is the approximation error due to the filtering of the exact field and dub(x) is the random error due to the noise. The same splitting can be applied to the reconstructed strain field eap. 2.2. Formulation The proposed approach is based on the use of local weighted least-squares [30]. Here, the local regression tool is the diffuse approximation (DA) [34]. The diffuse approximation was first proposed as an alternative to the finite element method for the solving of partial differential equations and has since been applied to various fields, such as optimization [35], mesh generation [36] or field transfer [37]. The parameter controlling the filtering is the span of the influence zone of each data point. One keypoint of the method is that it yields both a continuous displacement field and its derivatives (in a diffuse manner) at once, as explained in the following. For each component of ~ , the reconstructed field the displacement field, for example u is sought at any point x (x, y) of X, as the solution of the following minimization problem:

minJ x ðaðxÞÞ;

with

aðxÞ

J x ðaðxÞÞ ¼ 2. Strain reconstruction: Use of the diffuse approximation

1 X ~ ðxi ÞÞ2 wðx; xi Þðpðxi  xÞfaðxÞg  u 2 x 2VðxÞ

ð4Þ

i

where

2.1. Framework and notations In order to help describing the scenario of rupture from the DIC data, we aim at deducing the strain field from the displacement field. Before detailing the proposed method, its framework is first exposed. The input of the method is a set of displacement data on a regular grid of data points xi, on a zone denoted X. The dis~; v ~ Þ in the placements are expressed through their components ðu (eX, eY) basis associated with the directions of the CCD captor. Let the measurements be written as follows:

~ ðxi Þ ¼ u ~ðxi ÞeX þ v~ ðxi ÞeY ¼ uex ðxi Þ þ duðxi Þ; u

2

3

@u eXX @x 7 6 @v 7 7 e¼6 4 eYY 5 ¼ 6 @y 5 4 1 @u @v eXY ð þ Þ 2 @y @x

8i 2 ½1; N

ð1Þ

where du represents the perturbation on the measurements and uex is the exact mechanical field. du is assumed to be a random error and the measurement systematic error will be considered included

 x is constant with respect to the minimization;  the unknown {a} is a vector of coefficients and, afterwards, is defined at any x by (4), as: a⁄ = ArgminJx(a);  p(x) is the line vector of the monomials of the approximation basis, which is not necessarily polynomial. Here, a polynomial basis of degree 2 is chosen, since it appeared in [38] it was a good compromise between filtering and approximation errors;  V(x) represents the bounded neighborhood of x collecting the data points xi taken into account in the reconstruction at point x, ensuring the locality of the reconstruction;  w(x, xi) is a weighting function that equals zero outside V(x) and can be any positive function defined on V(x). Based on the fact the data grid is rectangular and periodic, the weighting function is chosen as:

P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

    jxi  xj jyi  yj wref wðx; xi Þ ¼ wref Rx Ry

ð5Þ

where wref is a dimensionless function such that it equals 1 at 0 and 0 at 1, while its first derivative is zeroing at 0 and 1 (this ensures the continuity of the diffuse approximation field and the one of its first derivative); it is chosen here as a cubic spline. Rx and Ry are chosen as functions of x but not of xi and are equal to one another. They are denoted R, which hence corresponds to the span of the weighting functions. These choices are not mandatory but allows a better numerical efficiency while keeping some generality. When not specified, R is given in terms of data points of the displacement data grid and, when specified, is given in pixels of the original picture. (4) can be written in a matricial manner as:

2

  T   1 min Pfag  U x W Pfag  U x with; aðxÞ 2

pðx1  xÞ

6 P¼4

... pðxN  xÞ

3 7 5

ð6Þ

i2VðxÞ

where the matrix W is a diagonal matrix collecting the weighting e x g the vector collecting the data on functions Wii = w(x; xi) and f U V(x). The solution {a(x)} of (6) is given as: 

fag ¼ ½MðxÞfU x g with;

1

½MðxÞ ¼ ½P T WP PT W

ð7Þ

With a polynomial basis of degree at least 1, the terms a2(x) and a3(x) represent the first order derivatives at point x in a diffuse manner:

uap ðxÞ ¼ a1 ðxÞ;

duap duap ðxÞ ¼ a2 ðxÞ and ðxÞ ¼ a3 ðxÞ dx dy

ð8Þ

where du stands for the diffuse derivative of u and is an approximadx tion of the exact derivative. This reconstruction can be performed for both components u and v of the measured displacement field ~ ðxi Þ and the strain field deduced from their diffuse first derivatives: u

"

ex U eðxÞ ¼ ½Me  ex V

2

# with;

6 ½Me  ¼ 4

M2 ðxÞ 0 M 3 ðxÞ 2

0

3

M 3 ðxÞ 7 5

ð9Þ

M2 ðxÞ 2



where Mi(x) is the ith line of matrix M(x) and V x is the displacements in the Y direction on Vx. The diffuse field is continuous and its value at any x is deduced from the solving of (6). Practically, we will choose the points where it has to be evaluated, depending on the application. In the current study, it is chosen to reconstruct the diffuse field at every data point xi, i 2 [1, N]. For large enough values of R (in practice R > 2), the problem (6) is well-posed at any data point. Hence it is possible to perform the reconstruction up to the edges of the domain X, even if the error will increase near the edges for a given R. Once the degree of the polynomial basis is chosen, the parameter to be tuned to control the filtering and the approximation error is the span R of the weighting functions. As R increases, the random error on the strain field will decrease whereas the approximation error will increase. There is hence a compromise to be found between both errors. 2.3. Reconstruction errors and criteria for R The minimization problem (4) of the quadratic criterion Jx leads to the solving of a linear system. This means, as suggested above, that we can study individually the filtering of the noise and the approximation error, along with their evolution with respect to the span R, based on (1) and (9).

2.3.1. Random error From (9), the random error on the strain can be expressed as:

"

ex dU deb ðxÞ ¼ ½M e ðxÞ ex dV

# ð10Þ

and assuming the covariance matrix of the perturbations on the displacements is known, and denoted C, the covariance matrix of the reconstructed strain can be deduced as:

COVðdeb Þ ¼ M e CMTe

ð11Þ

A standard assumption is that the perturbation on the measurements is a white gaussian noise sample with known standard deviation c. In this particular case where C = c2Id, the covariance matrix of the reconstructed strain writes:

COVðdeb Þ ¼ c2 Me M Te

ð12Þ

This means we can characterize the level of random error on the strain from the level of noise on the measurements. It is illustrated on a numerical example based on [39]. A set of synthetic speckle images are made available by the authors. Here, 1024  1024 pixels (px) images created by TexGen are used with a 4 px random noise on the 256 gray levels and correspond to rigid body motions. The displacement is reconstructed from the images using CorreliQ4 [40] and a 10 px meshsize. The covariance matrix of the measurement C is estimated from the autocorrelation of the displacement field. Fig. 1 presents the standard deviation of eXX as a function of the span R, for 0.25 px rigid body motions. The red line corresponds to the standard deviation directly estimated on the reconstructed strain field. The blue line is the estimation based on (12), assuming C is diagonal, and the green line corresponds to the standard deviation from (11), taking into account the full estimated C matrix. (11) is validated on the example and it can be noted that the white noise hypothesis leads to an overestimated random error, yet with the same order of magnitude. Furthermore, the dash-dot line corresponds to (11) for a 0 px rigid body motion and the results are very similar. 2.3.2. Approximation error Concerning the approximation error, which corresponds to the systematic error of the reconstruction operator, it can be shown that it is related to the N + 1 Taylor remainder, where N is the degree of the polynomial basis in (4). For the sake of simplicity, this is illustrated for a degree 2 DA basis and on a scalar function of one variable, but it can be generalized to any degree or dimension. Based on its Taylor expansion, the exact field can be rewritten:

−4

4.5

x 10

Eq. (12) Eq. (11), displacement 0.25px estimation from field, displacement 0.25px Eq. (11), no displacement

4 3.5

standard deviation

18

3 2.5 2 1.5 1 0.5 0

4

6

8

10

12

14

16

span R Fig. 1. Standard deviation on eXX as a function of the span R.

P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

2

uex ðxi Þ ¼ pðxi 

uex ðxÞ 6 T 6 duex ðxÞ xÞ 4 dx d2 uex dx2

ðxÞ

3 7 7 þ rðxi Þ 5

ð13Þ

with r the remainder of the Taylor expansion. Hence, from the decomposition (13) of the measurement field, the minimization (4), with exact measurements, can be reformulated as:

0 min a

2

a1  uex ðxÞ

3

12

7 C B 6 1X T 6 a  duex ðxÞ 7 C wðx; xi ÞB 5  rðxi ÞA @pðxi  xÞ 4 2 2dx 2 i2VðxÞ d uex a3  dx2 ðxÞ

ð14Þ

2

3 a1  uex ðxÞ 6 a2  duex ðxÞ 7 The vector dak such that: dak ¼ 4 dx 5, is made up of the 2 a3  ddxu2ex ðxÞ approximation errors on the field and its derivatives and is the solution of the problem (14), with r(xi), xi 2 V(x) as data. This means the approximation error corresponds to the diffuse approximation of the Taylor remainder at point x. Estimating the approximation error requires therefore the estimation of higher order derivatives in order to estimate the Taylor remainder. Since the goal of the method is to reconstruct some reasonable first order derivatives, the estimation of higher order derivatives would require additional effort, that would better be dedicated to the first order derivative. The estimate of the approximation error would then not be appropriate anymore [30]. Such error estimators are still open questions and further work is needed to address this issue. Yet, Eq. (14) can be useful for an a prioripragmatic study of the filtering of representative numerical fields, in order to help in the choice of filtering parameter prior to the study of experimental results. 2.3.3. Choice of the span R Since the increase of R implies opposing effects on both errors, a trade-off is to be found. The problems in estimating the approximation error makes it difficult to use a criterion of balance that would lead a global error close to its minimal [32]. Hence [32] proposed a signal to noise ratio criterion that is interesting in cases with localized large strains but is not satisfactory in the case studied here. In the following section, we propose a pragmatic criterion based on virtual gauges that can help when choosing the filtering parameter R. 3. Filtering experimental data – Validation and choice criterion The following examples are based on a uniaxial tensile test on a 3D carbon/epoxy composite obtained by RTM process. The uniaxial tensile tests are performed on 6 samples of 30 mm width and 200 mm length between grips. The specimens are designed with an optimized dog-bone shape in order to avoid the grip effects and delamination (the latter is also limited by the 3D architecture of the fabric). Each specimen is instrumented with two Acoustic Emission sensors positioned near the grip. These two sensors are used to filter the signals in order to avoid fixtures noise and to locate the onset of damage in the useful area. The useful area of the specimen is painted with a white spray followed by a black Speckle pattern. As in [41], the speckle size is characterized by the autocorrelation radius defined as the radius corresponding to half height of the autocorrelation function of the reference image. This radius is equal to 2.3 px in the studied cases. The monitoring of the full-field displacement during the loading process, is performed using a high performance SensiCam QEPCO camera incorporating a black and white CCD sensor coded on 12 bits with a matrix of 1376  1040 pixels. The sensor is

19

maintained at 12 °C allowing it to achieve a low noise level. The framerate corresponds to 0.5 images/s and the exposure time is 25 ms. The tests are performed with presrcibed load leading to a strain rate of about 103 s1. The images taken during the test are then treated by DIC, with the software CorreliQ4 [40] yielding the displacement fields on the measurement zone. The DIC software is based on a global approach using a finite element formulation of the optical correlation problem, hence yielding the whole displacement field at once. Its controlling is mainly based, for the end-user, on the meshsize, denoted here as the variable h, replacing the zone of interest (ZOI) of standard local approaches. As for the span R for the diffuse approximation, the increase of the meshsize h would enhance the filtering of the noise but would increase the approximation error due to the filtering of the mechanical field. These two effects are coupled but their evolution with the meshsize h does not behave the same way as their evolution with the span R in the diffuse approximation step (the balance is not the same). Therefore, a question to address is to find a good value for the couple (h, R) in order to get results as accurate as possible. Here, this question is addressed from the experimental data point of view, trying to discuss basic ideas for the choice of the parameters from the experimental information alone. Two criteria will be studied, firstly a qualitative one based on the whole heterogeneous strain field reconstructed on the measurement zone and, then, a more quantitative one based on the standard deviation of the strain under a large enough virtual gauge centered on the zone. 3.1. Combining DIC with the filtering DA The strain field reconstructed for various spans R and meshsize h are compared for a given load during the tensile test. The strain field presented on Fig. 2 is the eXX field, for various values of h and R (both given in pixels of the original digital image). The subfigures with a value of h of 8 pixels show the effect of the span R on the filtering and the approximation. It seems that an intermediate span R leads to a good compromise. The use of a larger value of h (in particular h = 32 pixels) seems to yield a larger approximation error without significant improvement of the filtering. In this example, a value of h of 8 pixels and a span R between 32 and 48 pixels seem appropriate. Hence, it seems that the balance between approximation and random error is in favor of larger spans R for the diffuse approximation and smaller values of the meshsize h of the DIC tool. Nonetheless, the meshsize of the DIC tool needs to be large enough so that the correlation algorithm yields converged results. This evaluation and therefore the choice of the couple (h, R) remain strongly user dependant and lies on visual observations, which is not fully satisfactory since it lacks a systematic criterion. The following section aims at proposing a more quantitative criterion. 3.2. Virtual gauges and standard deviation Usually, virtual gauges are used to evaluate the mean strain on a zone as if a gauge was used, thus filtering the noise by meaning it. Nonetheless, the standard deviation under virtual gauges is precious information. This standard deviation can actually be explained from two causes:  the spatial heterogeneity of the mechanical strain field, either due to the loading or the material;  the random error on the strain. If the virtual gauge is large enough and assuming the ergodicity of the random error, the standard deviation of the latter under the virtual gauge is a statistical estimator of its true standard

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P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

Fig. 2. Strain field eXX reconstructed for various couples (h, R).

deviation. In this paper, the virtual gauge covers 70% of the measurement zone and corresponds to 6500 data points, hence leading to an accurate enough estimation of the random error level. This will first allow us to validate the theoretical filtering of the noise by the DA method then to propose a pragmatic criterion for the choice of the couple (h, R).

curves, the latter being the result of two filterings (one through CorreliQ4 and one through the DA filtering). Nonetheless, the model of the noise seems appropriate, if the initial strain random error level is tuned to the experimental one. This study, before loading, allowed us to characterize the magnitude of the strain random error deb.

3.2.1. Filtering of the experimental noise During the tests, the measurement perturbations on the displacements are not exactly white noise samples, as illustrated in the example of Section 2.3.1. Even if the method is not currently able to filter the deterministic part of the perturbation (which would imply some mechanical considerations), we aim at verifying that a white noise hypothesis can still be reasonable and that the filtering on real measurements is as expected from a theoretical point of view. To that purpose, the DA filtering is applied on displacements coming from CorreliQ4 applied on two images, both taken before the loading of the specimen. There are hence no displacement and the signal can be considered as a typical perturbation. Fig. 3(a) shows the standard deviation under a virtual strain gauge as a function of the span R of the Diffuse Approximation (R is given in pixels of the original digital image). The standard deviation is drawn for three values of h and in the theoretical case of a white gaussian noise, the magnitude being scaled to the experimental one. We can note that the curves for the three values of h are in the continuity of each other, yielding a global curve very analogous to the theoretical one. For a given span R, the increase of the meshsize h increases the filtering of the noise. This could explain the discrepancy between the theoretical and experimental

3.2.2. A pragmatic criterion for the choice of the span R and the DIC meshsize h A first criterion is proposed in order to help choosing the value of the couple of parameters (h, R). As mentioned above, the standard deviation of the strain under the virtual gauge is due to two distinct reasons: the unfiltered perturbations and the spatial variation of the heterogeneous mechanical strain field. In Section 3.2.1, the evolution of the error due to the noise as a function of the span R of the Diffuse Approximation has been studied and its experimental magnitude characterized. The theoretical curve of the standard deviation as a function of R (scaled to the level of noise of the experiment and shifted to the same initial value) can be superimposed to the one of the standard deviation under the virtual gauge for a given load, as in Fig. 3. When the standard deviation under the virtual gauge reaches a level lower than the minimum level corresponding to the filtering of the noise (horizontal dashed line on Fig. 3), we can assure that the filtering is affecting the quality of the mechanical field. Therefore, around the horizontal line, there is a compromise to be found between filtering and loss of deterministic information. On our example, a value of h of 32 pixels is too large and the compromise of a value of h of 8 pixels and a span R of 32 pixels proposed in

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P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

3

−3 x 10

6

theorical filtering

theorical filtering

5

measurement h = 8

standard deviation on εXX

standard deviation on εXX

2.5

measurement h = 16

2

measurement h = 32 1.5 1

measurement h = 8 measurement h = 16

4

measurement h = 32 3 2 1

0.5 0

−3 x 10

0

20

40

60

80

100

120

140

160

0

0

20

40

60

80

100

span R, in pixels

span R, in pixels

(a) perturbation alone

(b) tensile test

120

140

160

Fig. 3. Standard deviation of eXX as a function of the span R.

4. Detecting local damages

where the ini group of snapshots corresponds to the very beginning of the test, where we can consider, from Acoustic Emission, no damage occurred yet. We can also define an incremental gap of the linear approximation of the strain as:

4.1. Proposed methodology

k Dekþ1;k ðxÞ ¼ ekþ1 lin ðxÞ  elin ðxÞ lin

Section 3.2.1 is confirmed here by this simple quantitative criterion. The latter is a first step toward a more objective choice of the filtering parameter.

This section aims at proposing a methodology in order to detect local damages and material non-linearities from the strain fields obtained through DIC and diffuse approximation, in the framework of small deformation and assuming the material behavior corresponds mainly to elasticity with damage. The local non-linearities can be studied from the evolution of the strain field along time. Even with the DA filtering, comparing two strain states directly yields very noisy data that is difficult to interpret. This is due to the fact that such a comparison is comparable to a time derivative, hence a second order derivative of the measurements. This is even more a problem if the strain magnitude remains small as in the following example. That is why we propose, from the basic idea from [15], to define the linear approximation of the strain at each data point at several times of the test. Hence, the snapshots are gathered along time in Ng groups, each composed of Nk snapshots. Within each group collecting Nk snapshots, the linear approximation for any component e of the strain tensor at any data point is defined as the solution eklin of the following minimization: Nk X  k 2 min F i elin  eðx; F i Þ

eklin

for the kth group

ð15Þ

i

where, for i 2 {1, . . . , Nk}, Fi is the corresponding load. eklin represents the linear approximation field for a load of 1N or the slope of the strain-load response at each DIC data point. This linear approximation of the strain can be used to define a local damage indicator. When eklin remains constant along time (or k), it means everything is linear and thus no damage occurs. Once eklin evolves with respect to time (or k), some damage or local non-linearities occurs implying a strain redistribution. Hence we can define damage indicators from the discrepancy between the linear approximations at various stages of the test. The first indicator proposed is defined as the gap to the initial elastic strain: k ini Dek;ini lin ðxÞ ¼ elin ðxÞ  elin ðxÞ

ð16Þ

ð17Þ

kþ1;k lin

De > 0 implies a local damage and strain redistribution to the zones where Dekþ1;k < 0. Let us note that, since the local stress is unlin known, the indicator is based on the global load and the zones with negative Dekþ1;k are related to the stress field redistribution and not lin healing of the material. Hence, the damage indicator is limited to zones where it is positive and remains qualitative. Furthermore, the incremental gap should help emphasize the damage initiation and propagation. 4.2. Application to a 3D composite The indicators proposed in Section 4.1 are applied on a ±45° tensile test on a 3D carbon/epoxy composite with several loadings and unloadings as described in Fig. 4. The loadings and unloadings are part of the general protocol of the study, in order to distinguish between damage and permanent strain, as well as between traction and compression behavior. The experimental setup is the same as described in Section 3. One objective of this work is to detect damage initiation sites. The nature and architecture of this material generates the first damage, detected by acoustic emission, at very low strain levels (103). At these strain levels and without filtering, the strain fields are very noisy and therefore unusable to detect local non-linearities. DIC is performed on about 500 images coming from the test, a large number of pictures is thus available. The DIC is performed with CorreliQ4, with a meshsize h of 8 pixels and the diffuse approximation is used with a span R of 64 pixels. Despite a loss in spatial resolution, a larger value than the one in Section 3 is chosen, since the latter was optimal for the strain field, whereas, here, the discrepancies between strain fields at different times are studied, emphasizing the noise even more. The studied strain field is eXX, corresponding to the tensile direction. A tracking of the test by acoustic emission (AE) is also performed, with a filtering threshold at a 55 dB magnitude, because of the acoustic surrounding during the test. In previous treatment of the data, it was considered damage was detected when AE is over a 80 dB magnitude. The various groups of snapshots for the linear approximation of the strain are represented on Fig. 4. Six different groups along

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P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

Fig. 4. Loading history.

time are chosen for this study, each made up of about 20 snapshots. The strain-load curve on Fig. 4(b) illustrates that the carbon/epoxy 3D composite is stiff and that the loss of global stiffness due to damage remains quite small. The linear approximations of the eXX field are represented on Fig. 5 for a span R of 64 pixels for the different studied groups of snapshots. Considering Fig. 5(a), the mean of the linear approximation on the first three groups is ploted, and will be considered as the initial elastic strain field eini lin of the specimen, before damage occurs. This choice is motivated mainly by the acoustic emission activity, which is almost neglectable before group 4 as shown Fig. 5 (furthermore, if no damage occured during group 1, the unloading and loading until group 3 should not initiate damage). We can observe the heterogeneity of the strain field due to the mesostructure of the composite. The other maps are referred by their chronological numbers from Fig. 4.

The gaps to the initial elastic strain field, Dek;ini lin , are drawn on Fig. 6 for a span R of 64 pixels. Fig. 6(a)–(c), gives an idea of both the effect of the noise and the validity of the fact no damage occurs before group 4, since it compares each of the three first groups with the initial one. It can be noted that the obtained gaps are not purely due to noise (in particular for group 2) but their magnitudes remain lower than on the following maps and we will consider they define the threshold for the detection of damage. From group 4, the strain increases (due to local damage) on the upper part of the zone in a first time, inducing an unloading of the central zone. Even if the magnitude of the gap for group 4 remains low, this scenario is confirmed on group 5. In a second time, it seems the damage propagates to the whole zone following alternative ±45° strips damaging and unloading. The AE data ploted on Fig. 4 confirms the development of damage and the cross-validation between AE and DIC allows to decrease the AE threshold for damage

Fig. 5. Linear approximation of the strain field eXX for the various groups.

P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

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Fig. 6. Discrepancy to the elastic strain: Dek;ini lin (16).

Fig. 7. Incremental discrepancy field: Dekþ1;k (17). lin

Fig. 8. Discrepancy between group 5 and the initial one: De5;ini – effect of the span R – initial group is made of groups 1–3. lin

onset detection. This tendencie is confirmed by Fig. 7 where the incremental discrepancy field, Dekþ1;k , is ploted. The incremental lin information helps emphasize the scenario of rupture, where the damage bands appear more clearly from groups 5 and 6.

4.3. Discussing the number of snapshots per group The previous study was performed with a span R of 64 pixels, that is quite large with respect to the heterogeneity of the mechan-

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P. Feissel et al. / Composites Science and Technology 88 (2013) 16–25

Fig. 9. Discrepancy between group 5 and group 3: De5;3 lin – effect of the span R.

ical field. We could wish to diminish its value to improve the spatial resolution, but the effect of the noise is to be controlled. This Section focuses on the effect of the span R and the number of snapshots in the groups on the quality of the proposed damage indicators. Fig. 8 presents the effect of the span R on the discrepancy field De5;ini lin between the strain coming from group 5 and the initial one. On this example, it is obvious that a span R of 64 pixels is too large and some local information is lost. Thanks to the large number of snapshots (about 60) used for the estimate of eini lin , a good filtering of the noise is achieved and we can afford to take a smaller span R to get more detailed local non-linearities. Nonetheless, Fig. 9 shows the same reconstruction with the same color range for each span R from a reference strain coming from group 3 alone. It can be seen that small spans lead to strains dominated by the noise. A span R of 64 pixels, if too large, allows at least to find out where the main zones of damage appears, in such an example. This last example suggests that by taking more snapshots we can afford to reduce the span R and therefore reduce the approximation error. To that purpose, an evolution of the diffuse approximation filtering tool has been developed in order to filter the displacement both in time and space [42]. 5. Conclusion The aim of the work presented in this paper is to propose a tool to contribute to the study and the understanding of the scenario of rupture in particular for 3D carbon/epoxy composites where the stiffness of the fibers implies low strain levels and where the elastic strain field is heterogenous. A damage indicator has been defined based on the linear approximation of the strain and its evolution. It was shown that this indicator can help detect the spatial and temporal localization of the damage sites. Furthermore, the coupling of this indicator with the acoustic emission data allowed to improve the detection of damage at early stages of loading. Due to the low level of strain field, its computation from the displacement field has to take into account the perturbations on the measurements and a dedicated filtering tool was proposed. This tool is based on the diffuse approximation and appears to be suitable to the strain reconstruction from DIC. The parameter to control the filtering, on this study, is the span R of the diffuse approximation that has to be chosen as a compromise between the random error and the approximation one. A pragmatic criterion based on the standard deviation under a virtual gauge has also been proposed that can help the user in the choice of the filtering parameter. It appeared on the treated examples based on an averaging of the strain on several snapshots that an increase of the number of snapshots for the averaging improves the spatial resolution of the results. Hence, the extension of the diffuse approximation to both space and time is currently under progress. Furthermore, another

interesting point with the diffuse approximation is that any mechanical field can be introduced in the function basis, which can help regularize the measurement through any a priori information, for example in the case of crack propagation.

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