Uncertainty analysis in composite material properties characterization using digital image correlation and finite element model updating

Accepted Manuscript Uncertainty Analysis in Composite Material Properties Characterization Using Digital Image Correlation and Finite Element Model Updating Tiren He, Liu Liu, Andrew Makeev PII: DOI: Reference:

S0263-8223(17)32048-2 https://doi.org/10.1016/j.compstruct.2017.10.009 COST 8984

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

5 July 2017 26 September 2017 2 October 2017

Please cite this article as: He, T., Liu, L., Makeev, A., Uncertainty Analysis in Composite Material Properties Characterization Using Digital Image Correlation and Finite Element Model Updating, Composite Structures (2017), doi: https://doi.org/10.1016/j.compstruct.2017.10.009

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Uncertainty Analysis in Composite Material Properties Characterization Using Digital Image Correlation and Finite Element Model Updating

1

Uncertainty Analysis in Composite Material Properties Characterization Using Digital Image Correlation and Finite Element Model Updating Tiren Hea Liu Liua∗ Andrew Makeevb a

School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, Texas

b

Abstract

This work presents an uncertainty analysis on composite material constitutive parameters, which are extracted using digital image correlation (DIC) and finite-element-model-updating (FEMU). The uncertainty is induced by the measurement system noise in DIC technique and the approximation error in the displacements and strains smoothing algorithm. The covariance matrix of the extracted material constitutive parameter has been given explicitly. Six material constitutive parameters were identified from a customized short-shear experiment simultaneously using an estimated optimal reconstruction mesh size as an illustration. Sensitivity of measurement noise and reconstruction parameter on extracted material properties has been investigated. The effects of region of interest (ROI) and DIC image number on uncertainties of extracted material properties have been addressed. It is suggested that there exist an appropriate ROI and the number of images, from which reliable material parameters can be identified, but much more data used in identification process always lead to smaller standard deviation and COV. It is observed that the material constants used to characterize the in-plane shear stress-strain behavior show strong robustness to measurement noise. However, the identified longitudinal Young's modulus is more sensitive to the measurement noise. Another key finding is that the reconstruction parameter in the global finite-element based approximation approach is critical for reliable material properties identification. Its value has to stay close to optimum for guaranteeing reliable identification of material properties. Key words: Uncertainty; Digital Image Correlation; Finite-element-model-updating;

∗

Corresponding Author: Telephone, (86)10 68918435; Email, [email protected] 2

Reconstruction parameter, Global finite-element based approximation

3

1. Introduction Composite materials with highly-anisotropic mechanical properties are becoming more essential for advanced structural designs. Growing worldwide demand has resulted in the rapid development of a

wide range

of composite

materials,

including

glass-,

carbon-reinforced polymer-matrix and multifunctional composites [1-3]. However, the lack of accurate material properties required for understanding of deformation and failure mechanisms causes significant delays in qualification of composite materials for structural applications, and results in extremely conservative designs. Also, the accuracy of finite element analysis (FEA) prediction strongly depends on the properties of materials evaluated experimentally. Improved three dimensional stress-strain constitutive relations are needed to more accurately characterize the mechanical response of advanced composites [4-9]. To enable efficient measurement of three-dimensional constitutive parameters of composite material behaviors, including nonlinear stress-strain relations, it is desirable to have a simple, yet effective method that can be used to extract multiple mechanical properties in constitutive relationships from a single test. Optical full-field measurement techniques such as reflection photoelasticity, moiré interferometer, grid method and digital image correlation (DIC), which can be used to acquire displacements and strains over the entire surface of a specimen, are found very promising for the experimental stress/strain analysis of materials and structures [10-14]. This information can change the philosophy of the experimental method-from extracting a single material property to identification of multiple material properties in stress-strain relations from a single experiment. The experimental method will reduce the amount of testing required to qualify new composite structural components. Better knowledge of material structural characteristics will lead to increased reliability. Thus full-field measurements have the capability of greatly improving efficiency for constitutive parameter determination in advanced composites. Among them, a DIC technique is easy to use, involves simple optics, less sensitive to vibration, no heavy surface preparation, and it can be applied on any class of material [15]. Although the optical full-field measurement techniques have found profound application in various domains, accuracy is a primary issue [16]. Errors in optical full-field displacement measurement, such as DIC could arise due to many sources such as illumination variations, quality of acquisition system, and image noise in the measurement system, or it could be due to the error associated with the implementation of correlation algorithm [17-20]. Then strains 4

can be computed from numerical differentiation of the estimated displacements. Unfortunately, the numerical differentiation can greatly amplify the noises contained in the displacements [17, 18]. Thus, extensive research works concentrated on eliminating the noisy data of the displacement. For instance, a global finite element (FE) based reconstruction approach can be applied to generate a smooth displacement field and strain field [21]. A thin-plate spline (TPS) smooth method is another well-known global smoothing approach [22]. Radial basis functions (RBFs) can also be applied to improve the accuracy of strain field estimation [21,23,24]. However all post-processing algorithms induce the approximation error inevitably. In order to use the full-field deformation measurement technique for material characterization, identification techniques have been developed to extract multiple constitutive parameters of material behaviors. For instance, the Virtual Field Method [25], which is based on the principle of virtual work written with a particular virtual field, can be used for identification of parameters in linear or nonlinear constitutive relations for composites [26-30]. Finite-element-model-updating (FEMU) method can also be used to extract material parameters in a constitutive model. It allows the result of a numerical simulation to match the experimental field in the sense of a given norm. Based on various cost functions, such as constitutive equation gap or equilibrium gap, constitutive parameters can be identified as the minimum of the cost function by updating finite element model iteratively [11]. The FEMU methodology is a priori suited for a wide range of applications, featuring complex geometrical and loading configurations [31-35]. In previous studies, four orthotropic planar elasticity parameters have been identified from uniaxial test of an open-hole specimen using the FEMU method [31]. Elastic properties of monolithic ceramic were identified using a high-order polynomial displacement field in the FEMU method [35]. Even though researchers have started exploring the mechanical characterization of composites using the optical full-field measurement technique, a thorough investigation needs to be done to study the influence of various errors in optical full-field deformation measurement, such as DIC on material property estimation [36, 37]. It is a key issue to evaluate quantitatively the uncertainty of identified constitutive parameters of material behaviors induced by the noise due to the deformation measurement system and the approximation error induced by the post-processing smoothing algorithm. The effect of region of interest (ROI), number of images, noise and smoothing algorithm parameter is of utmost importance and not many discussions are available in the literature [36].

5

It is the aim of the present study to investigate the combination of full field deformation measurements in conjunction with FEMU method to assess a set of reliable composite material constitutive parameters. Therefore a detailed methodology is presented to identify multiple constitutive parameters of composite material behaviors including tensile, compressive, and nonlinear shear stress–strain relations in one of principal material planes simultaneously. A custom short-beam shear (SBS) test method combined with the full-field non-contact 3D DIC technique and the FEMU has been employed for this purpose [38-40]. A global finite-element based algorithm is applied to reconstruct displacements and strains to reduce measurement “noise” [41]. The uncertainty of the extracted constitutive parameters of material behaviors have been evaluated explicitly. The effects of the ROI, the number of images, the noise induced in the measurement system and the reconstruction parameter in the global FE smoothing algorithm on the identified material constitutive constants have been investigated thoroughly. The material, the specimen geometry and the short beam shear test configuration are given in Section 2. Displacement fields are measured using a 3D-DIC system, which is described in Section 2. The identification method based on the FEMU is provided in Section 3. The global FE based smoothing algorithm used to reconstruct the displacements and strains is introduced in Section 4.1 and 4.2 respectively. Section 4.3 explains the uncertainty analysis on the extracted constitutive parameters of composite materials. Last, Section 5 deals with the reconstructed results and the corresponding uncertainties induced by optical system noise and smoothing algorithm. The influence of the ROI, the number of images, the noise in the measurement system and the approximation error induced by the smoothing algorithm on the uncertainty of the identified constitutive parameter has been discussed extensively. The concluding remarks are summarized in Section 6. 2. Mechanical experiment In a short beam shear (SBS) test, a customized specimen with a short span and uniform rectangular cross section is loaded under three point bending at room temperature. A schematic of the SBS specimen and experimental configuration is shown in Figure 1a. The geometry of the specimen is given in Fig. 1b. The SBS specimen is cut from a 34-ply 6.35 mm thick unidirectional IM7/8552 tape panel which had been cured at 350 oF using the manufacturer's specification for cure time [42]. It is machined at 44.5 mm length and 6.35 mm width. The support length l is 30.5 mm. The loading nose diameter is specified to be 101.6 6

mm to obtain shear delamination failure in the specimen. A standard lower-support diameter is 3.2 mm [43]. By loading the SBS specimens in the 1-3 (interlaminar) principal material plane, multiple constitutive parameters can be characterized. Matrix-dominated nonlinear shear stress-strain response up to failure can be obtained from a SBS test [39, 40]. The experimental setup comprises a 3D DIC system (Correlated Solutions, Inc.) for full-field

deformation

measurement

and

a

computer-controlled

MTS

Landmarks

servo-hydraulic machine (MTS 370.25) of 250 kN capacity. The SBS specimens were placed in the load frame and subject to monotonic load at a 1.27 mm/min crosshead displacement rate till failure. The 3D DIC setup consists of two CCD Cameras (IPX-16M3-L-1) having a spatial resolution of 4872 x 3248 pixels2, coupled with Sigma lenses (Sigma 105 mm f/2.8~C) of 105 mm focal length. The object distance is 180 mm. Random speckle patterns over specimen surface are obtained by spraying acrylic paints prior to the testing. Aperture is adjusted to achieve good field of view. One image per second is acquired using Vic-Snap software (Correlated Solutions, Inc.) for material characterization. Thus a total number of 48 images are acquired for material constitutive constant extraction consequently. In order to investigate the effect of ROI, two different regions of interest on the specimen surface are taken for material properties characterization. 2-mm wide gauge region between the supports and the loading nose on both the left and right side of the specimen is denoted as ROI-1 (Figure 2). Full-field region away from the supports and the loading nose is denoted as ROI-2 for comparison. The subset, ROI on the SBS specimen, speckle pattern (zoomed view), longitudinal and transverse displacements extracted from the ROI-2 on the right side through VIC-3D are shown in Figure 2. The displacements were extracted using VIC-3D software [15, 44]. The resolution of the image was 0.0115 mm/pixel. The average speckle size is 10 pixels. A subset size of 29 x 29 pixels2 along with a step size of 1 pixel is chosen for performing the DIC post-processing. 3. Finite Element Model Updating The identification of material constitutive parameters is be carried out by a finite-element-model-updating method, which is the most generic and intuitive method. The finite-element-model updating method is based on over-determined full-field displacement measurements and allows characterizing complex constitutive laws. The principle consists of updating material parameters in finite element models iteratively to minimize the objective function, which is assessed by the sum of the squared differences between experimental 7

determined and FEM-computed strains [11, 39]. The flowchart (Fig. 3) has been used for material characterization. A cost function Q(
) can be expressed in terms of unweighted least-squared difference between the experimentally determined and numerically calculated strains: 
 =  
 = ∑    
 − 







(1)

where  represents a set of unknown material parameters and n- the number of data used 

for evaluation of the squared differences.   and 

are the FEM-calculated strain

components and DIC-extracted strain components respectively. The following constitutive model was used for specimens loaded in the 1-3 principal material plane [39, 45]: 

" #$$    = !!− %$&  ! #$$ 0

%

− #&$ 

&&

#&&

0

0

, - + 0 + -  + ) ()*

(2)

where nonlinear shear stress-strain relations are characterized by the Ramberg-Osgood equation with three material parameters .13 , 213 , 313  4$&

 = 5 + $&

$

4 : 78$& 9 $& $&

(3)

After substituting Eqn. (2) and (3) into the cost function, it is related to the

FEM-calculated stresses -  , the DIC-extracted strains   and the set of unknown

material constants . To minimize the cost function, its partial derivative with respect to the material parameter is set to zero. <=> <>?


 −  = ∑ @( A 



B

>9 ?

=0

(4)

The partial derivative of the DIC-extracted strains with respect to the material parameter is defined as the sensitivity matrix HIJ in the identification procedure and it is related to

FEM-calculated stress component -  . The entrance of it has been listed in Table 1. Thus,

(

data σ iFEM , εiexp

)

can be used in solving an ordinary least squares regression problem to

generate the average stress-strain response. A set of constitutive parameters of material behaviors can be extracted from solving the least squares problem of Eqn. (4). For instance, the material properties to characterize the axial normal stress-strain behavior can be extracted 8

through: M

 K  = HA  JL HA  J HA  JL H J

(5)

It can be seen that the uncertainties of the extracted material constitutive parameters can only be tracked to the uncertainties of the experimental determined strain field. Three material constants .13 , 213 , 313  in the log-linear shear stress-strain constitutive relation can also be updated in the numerical FE model and determined using OLS regression method.

To extract the material constitutive parameters, a three-dimensional finite element model was developed using ABAQUS. The model involves geometric nonlinearity, material nonlinearity, and contact interactions. Since symmetric boundary conditions are applied in the width and the length direction, only one quarter of the specimen was modeled. Multiple loading steps are defined in the analysis to match the loading steps at which the DIC measurements were taken. Eqn. (2) and Eqn. (3) are implemented in the orthotropic elastic model using the user defined subroutine UMAT available in ABAQUS. The approximate mechanical properties of IM7/8552 unidirectional tape composites in the current investigation have been measured in the previous work [45]. The experimental results showed that the in-plane shear stress-strain response in 1-2 principal material plane is similar to the interlaminar shear stress-strain response in 1-3 principal material plane. In addition, the shear modulus in 2-3 principal material plane is close to a transverse isotropic material #

NN approximation . = O% . Therefore the constitutive model for IM7/8552 unidirectional  N&

tape is assumed to obey transverse isotropy [39, 45]. Linear shear behavior is utilized in the 2-3 principal plane since the shear strain,  , is negligible in the specimen sections

An initial set of linear material constitutive parameters used by UMAT were obtained from the manufacturer's specification [42]. It has been shown that the identification procedure is quite robust to the initial approximation of the material constitutive parameters due to the "geometric" stress distribution [39]. The FEM-calculated stress and DIC-extracted strain at the nodes are used in the least-squares regression method. 4. Uncertainty Analysis 4.1. Displacement field reconstruction

The noisy displacements given through DIC analysis at the N data points in the ROI are 9

denoted as: PQRS T , U  = P T , U  + VPT , U  W QRS T , U  = W  T , U  + VWT , U 

(6)

P  and W  is are the unknown exact displacement components along x and y direction. VP

and VW represent the measurement error. The sources of uncertainty in the DIC system are

numerous due to complex measurement chains. However several studied showed that a significant part of the measurement error is composed of additive and uncorrelated white

noise [21, 46]. Thus the covariance matrix of the measurement error VP and VW are given as: cov[\ = cov[] = ^ H_J

(7)

where H_J is the identify matrix and ^ is the variance of the white noise. The full-field displacements can be reconstructed using a global finite-element (FE) based approximation method over the ROI zone [21]. This approach is chosen as the basis functions have only low interactions between each other and it limits reconstruction oscillations as the precision increases [21, 47]. The reconstruction parameter is the mesh size of the global FE based approximation method. A 4-node rectangular element was used in this study. The reference element is shown in Fig. 4. Each node has 2 DOF values (displacements along x and y axis). The displacements in the element can be given by [48]: PT, U = ∑c  ϕ T′, U′b WT, U = ∑c  ϕ T′, U′d

(8)

where the shape function ϕ T′, U′ is written using the local Cartesian system T′, U′ defined by:

T′ = T − T U′ = U − U

(9)

b and d are the node displacements with e representing the number of a local node. The

shape function is given as [48]:

ϕ = T′U′ /fg ϕc = f − T′U′/fg

ϕ = f − T′g − U′/fg ϕ = T′g − U′/fg

(10)

The displacement field can be written as: 10

PT, U = HiT, UJj WT, U = HiT, U Jk

(11)

where HiT, UJ is the elementary matrix of the shape functions. j and k are the

columns of the nodal displacements. Denoting HlJ as the assembled matrix made up of all the matrices HiT , U J stacked, e ∈ H1, nJ. HlJ collects the shape functions evaluated at the data points. It is given as:

\QRS = HlJj ]QRS = HlJk

(12)

\QRS  and ]QRS  are the columns made up of all the values of PQRS T , U  and

W QRS T , U , e ∈ H1, nJ. 3 is the number of the nodes in the global FE based approximation,

where n ≫ 3. Based on the least-squares minimization, j and k are the solutions of the following minimization:

L

minj ∥ HlJj − \QRS  ∥N ⟺ minj HlJj − \QRS  HlJj − \QRS  L

mink ∥ HlJk − ]QRS  ∥N ⟺ mink HlJk − ]QRS  HlJk − ]QRS 

(13)

The minimization problem (13) leads to two linear systems to be solved, yielding j

and k as:

j = HuJM HlJL \QRS  k = HuJM HlJL ]QRS 

(14)

where HuJ = HlJL HlJ, which is a symmetric and sparse matrix, with nonzero components located close to the diagonal [21].

Assuming that the measurement noise is a white noise with a covariance defined by Eqn.

(7), the covariance matrices of j and k are evaluated as:

covj = covk = ^ HuJM

(15)

It can be seen that the uncertainty of the reconstructed displacement field depends on both the variance of the white noise in the DIC measurement system and the matrix HuJ. 4.2. Strain field reconstruction

The strain field can be derived by differentiating the shape functions. Three strain-field components can be written as:

11

y\

vww  = x { = H|Jj yz y]

v}}  = x { = HJk y~

(16)

y\ y] € w}  = ‚ + ƒ = HJj + H|Jk y~ yz

where H|J = „ yz … and HJ = „y~ …. Only infinitesimal strains are considered. However the yl

yl

derived strain field is discontinuous and not appropriate for the material constitutive parameter extraction. In order to ensure the continuity of the approximate strain field, another strain field has to be reconstructed using a similar global FE based algorithm. The component of the reconstructed strain field is given as: †‡ ww  = HlJˆ‰‰ 

(17)

where ˆww  is the column of nodal strain component. The nodal strain components are the solutions of the global least-squares problem defined as:

min‰‰ Š ∥ vww  − HlJˆww  ∥ ‹Œ = min‰‰ Š ∥ H|Jj − HlJˆww  ∥ ‹Ω

min  Š ∥ v}}  − HlJˆ}}  ∥ ‹Œ = min Š ∥ HJ k − HlJˆ}}  ∥ ‹Ω

(18)

min  ∥ € w}  − HlJ‘w}  ∥ ‹Œ = min  ∥ H|Jk + HJj − HlJ‘w}  ∥ ‹Ω

€‰  

€‰  

The shape function in Eqn. (10) allows the integration can be calculated analytically in the ROI, providing matrices that are similar to the stiffness matrices in finite element analysis. The minimization problems defined by Eqn. (18) lead to linear system to be solved and yield: ˆww  = Hu’ JM Hl’ JL H|’ JHuJM HlJL \QRS  ˆ}}  = Hu’ JM Hl’ JL H’ JHuJM HlJL ]QRS 

(19)

“w}  = Hu’ J”$ Hl’ JL H|’ JHuJM HlJL ]QRS  + H’ JHuJM HlJL \QRS 

where Hu’ J, Hl’ J, H|’ J and H’ J are calculated based on the integration at Gaussian points in Eqn. (18).

The covariance matrix of the nodal strain induced by the random error in the displacement measurement can be determined based on Eqn. (15) and (19). For instance, the covariance of the nodal normal strain component ˆww  can be derived: where H•’ J = H|’ JL Hl’ J.

covˆww  = ^  Hu’ JM H•’ JL HuJM H•’ JHu’ JM

(20)

The off-diagonal terms of the covariance matrix in Eqn. (20) are at least one order of

magnitude lower than the diagonal ones. In addition, the diagonal terms of covˆww  are

nearly equal. Thus the variance of the strain field ˆww  can be approximated by the 12

averaged diagonal terms of covˆww .

covˆww  ≅ —ˆ‰‰  ^ H_J

(21)

Therefore the average standard deviation of the strain field can be approximated by —ˆ‰‰  ^ ,

where the constant —ˆ‰‰  is called the coefficient of sensitivity to noise. It is a function of the reconstruction mesh size in the global finite-element approximation approach. The

average standard deviation of the strain field ˆww  is also proportional to the standard deviation of the white noise in the DIC measurement system.

4.3. Uncertainty analysis of the extracted material constitutive parameter

The presence of white noise in the original DIC images and the approximation error in the reconstruction algorithm has to be tracked along the identification processing to provide uncertainties on the identified material constitutive parameters. From Eqn. (5), the uncertainty of the identified material constitutive parameter depends only on the uncertainty of the experimental extracted strain evaluated by Eqn. (21). The covariance matrix of the identified material constant can be derived explicitly from the OLS regression method as: H˜K J = — ^ HA  JL HA  JM

(22)

The diagonal terms of the covariance matrix in Eqn. (22) give the variance for each identified parameter separately. The off-diagonal terms, which provide the uncertainty to characterize all couplings between identified constants, have not been taken into account since the off-diagonal terms in the covariance matrix of the strain field (Eqn. 20) have been neglected. It has been clarified that the uncertainty of the identified material constitutive parameter is related to the uncertainty of the experiment-extracted strain field and the FEM-calculated stress field. 5. Results and Discussion 5.1 Displacement measurement error

It has been demonstrated that the uncertainty of the extracted material property is proportional to the variance of the displacement measurement error in DIC technique, which is assumed to be additive uncorrelated white noise. In order to evaluate the variance of the displacement measurement error and verify the assumption for the current SBS configuration, 10 images of the SBS specimen were taken at the loading-free state. The displacements have 13

been extracted using VIC-3D software with the same experimental configuration and analysis parameters described in Sec.2. The results obtained from 10 images have been averaged to take into account the influences of light source fluctuation and other indeterminate sources. The displacements consist of the known zero displacements and the measurement errors in the experiment system. Thus the measured displacements can be taken as the displacement measurement errors. The measured displacements at the loading-free state have been shown in Fig. 5. It is noted that the displacements remain very low. The histograms of the data and a standard Gaussian distribution have been given for comparison in Fig. 6. It is noteworthy that its distribution of the measurement error is nearly Gaussian. Therefore it has been validated that the displacement measurement error in the experiment system is additive and uncorrelated white noise. Using the subset size of 29 x 29 pixels, the standard deviation of the measurement error ^™



respectively.

= 5.33 x 10-5 mm and ^š



= 4.8 x 10-5 mm have been determined

The standard deviation of the displacement measurement error was analyzed using different subset sizes in VIC-3D software. The results show the standard deviation of displacement measurement error decreases with increasing subset size in Fig. 7. The average confidence margin ^ , which is an estimation of the displacement measurement error provided

by VIC-3D software, always underestimates the standard deviation of the measured displacement. In order to evaluate the underestimation between ^™ the normalized deviation ^™



− ^/^™





and ^, the variation of

as a function of the subset size has been

illustrated in Fig. 7. It shows that the underestimation between ^™ 

increasing the subset size. The normalized deviation is 25% ^™



and ^ increases with

with the subset size = 29 x

29. It is proposed that the standard deviation of the displacement measurement error can be estimated by the average confidence margin given by VIC-3D software and the relationship is 

given as ^™

= 1.3^ for the current experiment system.

5.2 Reconstruction results

In order to verify displacement and strain field reconstructions, a comparison of fields given using synthetic data and reconstructed fields has been made. The significance of synthetic data is that the exact displacement field and the measurement errors are known, allowing approximation errors induced by the global FE reconstruction algorithm to be evaluated quantitatively. The synthetic data is obtained from the same three-dimensional solid 14

finite-element model developed in Sec. 3. The mesh was refined significantly to ensure that synthetic displacements and strains are very close to experimental ones. The linear material constitutive properties used in the model to generate synthetic data were given by the prepreg manufacturer as an approximation [42]. Reconstruction is performed based on the synthetic data without white noise initially. Thus, the error in the reconstructed field is only induced from the global FE approximation algorithm. For the sake of simplicity and illustration purpose, the reconstruction mesh used in the global finite-element approximation approach is square with a mesh size of 0.24 x 0.24 mm2. The displacement error induced by the global FE approximation approach at an arbitrary point is defined as: ›\ T , U  = œP†‡ − PžŸ  + W †‡ − WžŸ 

(23)

The strain error is defined as: žŸ 

žŸ 

žŸ 

†‡ †‡ †‡ › T , U  =   ww − ww  + }} − }}  + w} − w} 

(24)

The synthetic field, the reconstructed field and the approximation error map for displacements and strains are shown in Fig. 8 and 9. The comparisons indicate that reconstruction from the synthetic data without white noise based on the global FE approach induces approximation errors, but a good agreement between the exact displacement field and reconstructed displacement field has been achieved. The approximation error for the displacement reconstruction is fairly small. However larger strain approximation error is presented in the region with higher shear strain distribution. It indicates that the approximation error induced by the reconstructed algorithm is related to deformation and increases with the deformation. The normalized average displacement and strain approximation errors are defined as: ›̅\ = ›̅ =



¢ 

∑¢  „

$

£™¤¥¦ M™§¨©  NOš ¤¥¦ Mš §¨© N£ N … £™§¨© NO£š §¨©N

$

N N £ ’¤¥¦ M’ §¨© N O7’ ¤¥¦ M’ §¨© 9 O7«¤¥¦ M«§¨© 9 £ N ‰‰  ‰ ‰‰  ‰

∑¢ ª ¢ 

N

N

£ ’§¨©N O7’ §¨© 9 O7«§¨© 9 £ ‰‰  ‰

¬

(25)

where M is the total number of points in the region used for identification of material properties. Figure 10 shows the variations of the normalized average displacement errors and strain errors with the reconstruction mesh size. It can be noted that the error induced by the reconstruction algorithm solely decreases and saturate with decreasing mesh size for the case

15

without white noise. The normalized average strain error is remarkably higher than the displacement error (Fig. 10c), which suggests that constructing cost function using reconstructed displacement field can reduce the uncertainty of the extracted material constitutive parameter induced by the approximation error due to reconstruction algorithm. The strain error amplification scale increases significantly with decreasing reconstruction mesh size for the case without white noise. It indicates that the strain error amplification due to reconstruction algorithm is more remarkable with smaller mesh size. In order to investigate the effect of the displacement measurement noise on the reconstruction, a white noise with a standard deviation of 2.79 x 10-4 mm has been superposed to the synthetic data. Figure 11 shows the displacement fields given by the synthetic data superposed with the white noise and the reconstructed fields. The results show that the noise has been reduced through reconstruction algorithm. The variations of the normalized average displacement and strain errors with the reconstruction mesh size is given for the case with white noise in Fig. 10, where the total error is attributed to the displacement measurement error and the approximation error due to reconstruction. It has been observed that the variations are non-monotonic. The error induced by the white noise increases with decreasing mesh size, however the approximation error due to reconstruction algorithm decreases with decreasing mesh size. Therefore an optimized mesh size can be achieved with minimization of the total error. It is inferred that the optimized mesh size depends on both the displacement measurement error and the deformation field, and an optimized reconstruction mesh size can be determined by tuning the measurement error and the approximation error in the reconstruction field. It is worth noting that the normalized average strain error is still higher than the displacement error for the case with white noise. The strain error amplification scale still decreases with increasing the reconstruction mesh size (Fig. 10c), but the variation is not significant. Therefore it is beneficial to identify material constitutive parameters from the reconstructed displacement field. It is rational to assume that the standard deviation of the displacement measurement error remains the same with loading since it is mainly governed by the gray value noise of camera and pattern gradients with a given subset [49, 50]. However, the gradients of displacement and strain fields increase with loading. Thus the optimized reconstruction mesh size could vary for each image taken during loading. Fig. 12a shows the average strain variance as a function of the reconstruction mesh size for five different loadings using synthetic data. A white noise with a standard deviation of 2.79 x 10-4 mm has been superposed to the data. The 16

results show that the optimized reconstruction mesh size decreases with loading, which is attributed to increasing displacement and strain gradients. It suggests that as the displacement measurement error is reduced significantly, the optimized mesh size is governed by deformation and a small reconstruction mesh size is favorable for identification procedure. The variation of average strain variance as a function of applied loadings is presented in Fig. 12b. It is found that the average strain variance increases with loading significantly for the case without displacement measurement noise, which indicates that the approximation error induced by the reconstruction algorithm solely increases with deformation. The similar trend is observed for the case with ^ = 2.79 x 10-4 mm.

5.3 Uncertainty of the extracted material constitutive parameter

The ordinary least-squares regression presented in Section 3 is used to extract material stress-strain constitutive parameters for the unidirectional SBS specimen loaded in the 1-3 principal material plane. The initial numerically-calculated stress components extracted from the full-field region (ROI-2 in Fig. 2) of the three-dimensional solid finite element model are combined with the reconstructed strain components. The material constitutive relationship is generated by coupling FEM-calculated stresses and DIC-reconstructed strains and updated. This iterative procedure continues until the change in the constitutive model is negligible. The convergence criteria used as the threshold to stop the iteration is that the relative value of the parameter in the constitutive relationship updates less than 0.5%, and the variation of the normalized square root of the cost function in Eqn. (1) is less than 1%. Figure 13 shows the typical converged results for the ordinary linear least-squares regression between FEM-calculated stresses and DIC-reconstructed strains extracted from the full-field region (ROI-2). The strain reconstruction mesh size, which is determined by minimization of the normalized average strain error, varies for each acquired image. The data indicates a linear relationship between axial normal stress and strain can be identified for extraction of tensile and compressive axial Young's modulus (Fig. 14a and b). A three-parameter Ramberg-Osgood relationship between shear stress-strain is identified in Fig 14(c). It is noteworthy that the normalized average strain error depends on both the displacement measurement error and the mesh size in the reconstruction algorithm. The optimal mesh size is unknown since the gradients of displacement and strain fields are unknown in the experiments. In order to estimate an appropriate reconstruction mesh size, the 17

linear elastic material constitutive model is assumed with materials properties given by the prepreg manufacturer [42], and a numerical FEM simulation was performed to generate synthetic field. The synthetic field is considered as an approximation to experimental field, but it could underestimate the deformation due to material linearity. The standard deviation of the displacement measurement error ^­ = 2.79 x 10-5 mm, which is given by the average confidence margin in VIC-3D software for the current SBS configuration, is superposed to the

synthetic data. The appropriate reconstruction mesh size is determined further by minimization of the normalized average strain error defined by Eqn. (25) in ROI. The "optimal" reconstruction mesh size is estimated for each image and used in the experimental strain field reconstruction. The converged expectation, standard deviation and the coefficient of variation (COV) of each identified material constitutive parameter based on FEM-calculated stresses and DIC-reconstructed strains in the 2-mm wide gauge region (ROI-1) and the full-field region (ROI-2) have been listed in Table 2. The results demonstrate that the COV for the axial tensile, compressive Young's modulus and Poisson ratio related parameter are higher than that of the parameter used to characterize the in-plane shear Ramberg-Osgood relationship due to much higher shear strain distribution in the SBS specimen. The results also indicate that the converged expectation of material constitutive parameter identified from the smaller ROI (2 mm wide gauge section region) are consistent with those extracted from the full-field region, and the maximum deviation is no more than 5%. However it is worthy of note that the COV of the parameter extracted from larger ROI is smaller than that extracted from smaller ROI. It stands to reason that the variance of identified parameter is related to the number of the stress data used in the identification procedure (Eqn. 22). The number of the stress data in the full-field region is much more than that in the 2-mm wide gauge region, which leads to smaller variance of the identified material parameter and the COV as well. The comparison suggests that the expectation of material constitutive parameter can be determined from a smaller ROI [38-40], but the variance of the identified material parameter is higher than that extracted from larger ROI. The variation of expectation and COV of material parameter as a function of number of images used in the identification procedure is presented in Fig. 14. The material parameter has been normalized by the identified result used all images during the test. One image in Fig. 14 represents the last image taken before the specimen failure; 2 images represent the last two

18

images, and so on. The results demonstrate that the expectation of identified material parameter is insensitive to the number of images except Poisson ratio related parameter. The largest deviation is no more than 7%. The COV decreases with increasing the number of images and converges after the last 10 images used in the identification procedure. The comparison of identified material parameters from smaller and larger ROI (Table 2) and the results in Fig. 14 both indicate that there exist an optimal region of interest and number of images, from which reliable expectation of material constitutive parameter can be identified. But more data points used in the identification procedure can lead to smaller standard deviation and COV of identified material constitutive parameter. It has been derived explicitly that the uncertainty of identified material parameter is related to displacement measurement error and reconstruction mesh size used in the global FE based approximation algorithm (Eqn. 22). The variance of identified material parameter is proportional to the variance of displacement measurement error as the reconstruction mesh size remains unchanged. The effect of displacement measurement error on the identified material constitutive parameter has been investigated and the results are presented in Fig. 15

using synthetic data. A white noise with a standard deviation of ^ has been superposed to

characterize displacement measurement error. The left axis represents the identified material

parameter, which has been normalized by the nominal value identified with ^­ = 2.79 x 10-5. The right axis represents the normalized standard deviation of material parameter, which increases linearly with the standard deviation of the displacement measurement error. It is noteworthy that the material parameter used to characterize in-plane nonlinear shear stress-strain behavior shows strong robustness to displacement measurement error and can be readily identified. Very good agreement between the identified material parameter and the nominal value is achieved with ^ ≤ 10^­ (Figs. 16b, c and d). However, the axial Young's modulus extracted from the full-field ROI is sensitive to ^ -a reliable material constant cannot be identified with ^ ≥3^­ . This occurs due to increased sensitivity to noise when the

normal strain component ww is much smaller than the shear strain distribution in the SBS test.

The effect of the reconstruction mesh size on the identified material constitutive parameter has been investigated and presented in Fig. 16 using synthetic data, in which a white noise with a standard deviation ^° = 2.79 x 10-5 mm is superimposed. The identified

material parameter is normalized by the nominal value, which is identified with an optimal reconstruction mesh size. The results indicate that all identified material parameters are 19

sensitive to reconstruction mesh size. With increasing mesh size, all extracted material parameters deviate significantly away from the nominal values. The variance of identified material parameter decreases with increasing reconstruction mesh size; this is related to dependence on the standard deviation of the reconstructed strain, which decreases with increasing mesh size. It has been pointed out that no parameter can be extracted reliably if an inappropriate mesh size is used for reconstruction. Therefore an estimation of an appropriate reconstruction parameter, based on minimization of average reconstruction strain error using synthetic data or other complementary prior information is necessary for improving reliability of material constitutive parameter identification procedures. 6. Concluding Remarks

In the present work, an elaborate study is carried out to investigate the effects of intrinsic displacement measurement error in optical full-field measurement technique and approximation error due to smoothing algorithm on the uncertainty of extracted composite material constitutive parameters. As an illustration, six material parameters in the constitutive relation for a unidirectional composite specimen has been extracted simultaneously based on full-field 3D DIC measurements combined with FEMU method through a customerized short-shear-beam experiment. The displacement measurement error and the approximation error due to the reconstruction algorithm have been studied carefully, which induce the uncertainty of extracted material constitutive parameter. The covariance matrix of identified material constitutive parameter is derived explicitly. The effects of region of interest (ROI) and the number of image used in the material parameter extraction has been investigated. It is found that the displacement measurement error in the current experimental configuration is nearly Gaussian white noise. The standard deviation decreases with increasing the subset size used in the VIC-3D analysis. It can be estimated by the confidence margin provided by VIC-3D software through a calibration procedure. The measured noisy displacements and strains can be reconstructed through a global FE based approximation approach and the noise has been reduced significantly. The normalized average displacement error (Eqn. 25) is much smaller than the normalized average strain error - suggesting that using a reconstructed displacement field in the identification is beneficial to reduce the error effect on material constitutive parameter extraction. Large deformation always results in larger strain variance, which indicates that average strain error increases with deformation. An optimized reconstruction parameter exists and can be determined by tuning the measurement 20

error and the approximation error to minimize the average strain error. Six material constitutive parameters were identified from the SBS experiment simultaneously using an estimated optimal reconstruction mesh size. It is found that the expectation of constitutive parameter extracted from smaller ROI is consistent with that extracted from larger ROI, but the standard deviation is higher. The expectation of the material parameter is insensitive to the number of images used in the identification process except for the Poisson ratio related parameter. The COV decreases with increasing the number of images and converges after 10 images. It is suggested that there exist an appropriate region of interest and the number of images, from which reliable material constitutive parameters can be identified, but much more data used in identification process always lead to smaller standard deviation and COV. The effects of the variance of the displacement measurement error and the reconstruction mesh size on the uncertainty of extracted material constant have been studied using synthetic data. The results demonstrate that the material constitutive parameter used to characterize in-plane shear stress-strain behavior shows strong robustness to displacement measurement error. However, the identified axial Young's modulus is more sensitive to measurement error and a reliable material parameter could not be extracted with increasing standard deviation of displacement measurement error. Another key finding is that the reconstruction mesh size in the global FE based approximation approach is critical for reliable material parameter identification. If the reconstruction mesh size significantly deviates from an appropriate value, the linear least-square regression procedure cannot guarantee extracting accurate material parameters. Acknowledgement

This work is sponsored by the National Natural Science Foundation of China under Grant No. (11472043). Dr. L. Liu is supported under the Program for New Century Excellent Talents at Chinese Universities and the 111 Project (B16003). This support is gratefully acknowledged.

21

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Figure Captions Figure 1. (a) Short-beam shear (SBS) experimental setup combined with DIC full-field displacement measurement system to determine multiple constitutive parameters of composites from a single experiment. (b) Geometry of the SBS specimen. Figure 2. (a) Subset, ROIs and zoomed view of speckle pattern on the SBS specimen surface; (b) Longitudinal displacement u, (c) Transverse displacement v extracted from ROI-2 on the right side throughVIC-3D Figure 3. Schematic flowchart for the material constant identification procedure using a short-beam shear experiment combined with a general FEMU method. An ordinary linear least-squares regression optimization technique was applied in the FEMU. Figure 4. Rectangular element geometry and local node numbers with local Cartesian system used for the global finite-element reconstruction approach. Figure 5. (a) Measured averaged displacement field u obtained for the loading-free state; (b) Measured averaged displacement field v obtained for the loading-free state. Figure 6. Histogram of displacements at the loading-free stage for the current SBS configuration with a standard Gaussian distribution. (a) Displacements u along x-axis; (b) Displacements v along y-axis. Figure 7. Standard deviation of measured displacement error v.s. subset size Figure 8. Displacement fields from synthetic data without measurement noise and reconstructed results. (a) Displacement field u along x-axis, (b) Displacement field v along y-axis, (c) Displacement error map with the mesh size of 0.24 mm. Figure 9. Strain fields from synthetic data without measurement noise and reconstructed results. (a) Normal strain ww (b) Shear strain w} . (c) Strain error map with the mesh size of 0.24 mm. Figure 10. Normalized average error as a function of mesh size (a) Displacement error; (b) Strain error; (c) Error amplification scale, ›̅’ /›̅™ Figure 11. Displacement fields from synthetic data and reconstructed results. The displacement measurement error with the standard deviation of 2.79 x 10-4 mm is superposed to the synthetic data. (a) Displacement u along x-axis, (b) Displacement v along y-axis. Figure 12. (a) Average strain variance v.s. mesh size for 5 different loadings, (b) Average strain variance v.s. applied loadings Figure 13. Typical converged result for material constitutive relationship derived through least-square approximation coupling of reconstructed strain and FEM-calculated stress. (a) Axial normal tensile stress-strain data and average response; (b) Axial normal compressive stress-strain data and average response; (c) In-plane shear stress-strain data and average response. Figure 14. (a) Expectation and (b) COV of identified material constitutive parameter as a function of number of images in the identification procedure. Figure 15. Variation of normalized identified material parameter and normalized standard deviation as a function of standard deviation of the displacement measurement error (a) Axial normal tensile and compressive modulus; (b) In-plane shear modulus; (c) Value of k13 in Ramberg-Osgood relationship; (d) Value of 1/n13 in Ramberg-Osgood relationship.

25

Figure 16. Variation of normalized identified material parameter and normalized standard deviation as a function of reconstruction mesh size (a) Axial normal tensile and compressive modulus; (b) In-plane shear modulus; (c) Value of k13 in the Ramberg-Osgood relationship; (d) Value of 1/n13 in the Ramberg-Osgood relationship.

26

Figure 1. (a) Short-beam shear (SBS) experimental setup combined with DIC full-field displacement measurement system to determine multiple constitutive parameters of composites from a single experiment. (b) Geometry of the SBS specimen.

27

Figure 2. (a) Subset, ROIs and zoomed view of speckle pattern on the SBS specimen surface; (b) Longitudinal displacement u, (c) Transverse displacement v extracted from ROI-2 on the right size through VIC-3D software.

28

Figure 3. Schematic flowchart for the material constant identification procedure using a SBS experiment combined with general FEMU method. An ordinary linear least-square regression optimization technique was applied in the FEMU.

29

Figure 4. Rectangular element geometry and local node numbers with local Cartesian system used for the global finite-element reconstruction approach.

30

Figure 5. (a) Measured averaged displacement field u obtained for the loading-free state; (b) Measured averaged displacement field v obtained for the loading-free state.

31

Figure 6. Histogram of displacements at the loading-free stage for the current SBS configuration with a standard Gaussian distribution. (a) Displacements u along x-axis; (b) Displacements v along y-axis.

32

Figure 7. Standard deviation of measured displacement error v.s. subset size

33

Figure 8. Displacement fields from synthetic data without measurement noise and reconstructed results. (a) Displacement field u along x-axis, (b) Displacement field v along y-axis, (c) Displacement error map with the mesh size of 0.24 mm.

34

Figure 9. Strain fields from synthetic data without measurement noise and reconstructed results. (a) Normal strain ww (b) Shear strain w} . (c) Strain error map with the mesh size of 0.24 mm.

35

Figure 10. Normalized average error as a function of mesh size (a) Displacement error; (b) Strain error; (c) Error amplification scale, ›̅/›̅\

36

Figure 11. Displacement fields from synthetic data and reconstructed results. The displacement measurement error with the standard deviation of 2.79 x 10-4 mm is superposed to the synthetic data. (a) Displacement u along x-axis, (b) Displacement v along y-axis.

37

Figure 12. (a) Average strain variance v.s. mesh size for 5 different loadings, (b) Average strain variance v.s. applied loadings

38

Figure 13. Typical converged result for material constitutive relationship derived through least-square approximation coupling of reconstructed strain and FEM-calculated stress. (a) Axial normal tensile stress-strain data and average response; (b) Axial normal compressive stress-strain data and average response; (c) In-plane shear stress-strain data and average response.

39

Figure 14. (a) Expectation and (b) COV of identified material constitutive parameter as a function of number of images in the identification procedure

40

Figure 15. Variation of normalized identified material parameter and normalized standard deviation as a function of standard deviation of the displacement measurement error (a) Axial normal tensile and compressive modulus; (b) In-plane shear modulus; (c) Value of k13 in Ramberg-Osgood relationship; (d) Value of 1/n13 in Ramberg-Osgood relationship.

41

Figure 16. Variation of normalized identified material parameter and normalized standard deviation as a function of reconstruction mesh size (a) Axial normal tensile and compressive modulus; (b) In-plane shear modulus; (c) Value of k13 in the Ramberg-Osgood relationship; (d) Value of 1/n13 in the Ramberg-Osgood relationship.

42

Table 1. The entrance of the sensitivity matrix HIJ in the identification procedure
̅

(1/E11)

 ² ²
̅

 -

 ² ²
̅


̅

(v31/E33)

 −-

0

0


̅

(1/G13)


̅c (1/k13)


̅±(1/n13 )

0

0

0



 ) ) ´ µ´ µ 3 2

³#¢ )

$ M9 :$&

7

$

 :$&

 ) ) ´ µ ¶3 ´ µ 2 2

Table 2. The standard uncertainty of each identified material constitutive properties for the unidirectional SBS specimens loaded in 1-3 principal material plane ROI-1 (2-mm wide gauge section)

Material

Standard Deviation

Coefficient of

(SD)

Variance (COV)

1/174.9

1/(174.9×11.1)

9.0%

1/GPa

1/135.3

1/(135.3×13.2)

v31/E33

0.022/8890

1/G13, 1/MPa

ROI-2 (Full-field) Standard

Coefficient of

Deviation (SD)

Variance (COV)

1/181.5

1/(181.5×29.9)

3.35%

7.58%

1/134.3

1/(134.3×39.1)

2.56%

(0.0218×0.014) /8890

13.6%

0.019/8788

(0.019×0.047) /8788

4.68%

1/4441.1

1/(4441.1×112.3)

0.89%

1/4448.5

1/(4448.5×234)

0.43%

1/k13, 1/MPa

1/210.7

1/(210.7×49.8)

2.01%

1/213.8

1/(213.8×99)

1.01%

n13

1/4.57

1/(4.57×42)

2.38%

1/4.45

1/(4.45×84.8)

1.18%

constants

1/E11T, 1/GPa

Mean

Mean

1/E11C,

43