In-situ mechanical properties identification of 3D particulate composites using the Virtual Fields Method

Accepted Manuscript In-situ Mechanical Properties Identification of 3D Particulate Composites using the Virtual Fields Method B. Rahmani, E. Ghossein, I. Villemure, M. Levesque PII: DOI: Reference:

S0020-7683(14)00192-9 http://dx.doi.org/10.1016/j.ijsolstr.2014.05.006 SAS 8383

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International Journal of Solids and Structures

Received Date: Revised Date:

4 March 2014 2 May 2014

Please cite this article as: Rahmani, B., Ghossein, E., Villemure, I., Levesque, M., In-situ Mechanical Properties Identification of 3D Particulate Composites using the Virtual Fields Method, International Journal of Solids and Structures (2014), doi: http://dx.doi.org/10.1016/j.ijsolstr.2014.05.006

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In-situ Mechanical Properties Identification of 3D Particulate Composites using the Virtual Fields Method B. Rahmani

a,b

, E. Ghossein

a,b

, I. Villemure

b,c

, M. Levesquea,b∗

´ CREPEC, Laboratory of Multi-Scale Mechanics, Ecole Polytechnique de Montr´eal, P.O. Box 6079, QC, Canada H4T 1J4 b ´ Department of Mechanical Engineering, Ecole Polytechnique de Montr´eal, Montr´eal, P.O. Box 6079, QC, Canada H4T 1J4 c Research Center, Sainte-Justine University Hospital, 3175 Cote-Ste-Catherine Rd., Montr´eal, QC, Canada H3T 1C5 a

Abstract This paper presents an identification procedure based on the Virtual Fields Method (VFM) for identifying in-situ mechanical properties of composite materials constitutive phases from 3D full-field measurements. The new procedure, called the Regularized Virtual Fields Method (RVFM), improves the accuracy of the VFM thanks to the imposition of mechanical constraints derived from an appropriate homogenization model. The developed algorithms were validated through virtual experiments on particulate composites. The robustness of both the VFM and the RVFM was assessed in the presence of noisy strain data for various microstructures. A study was also carried out to investigate the influence of the size of region of interests on the reliability of the identified parameters. Accordingly, the optimum size of region of interest ´ Corresponding author at: Department of mechanical engineering, Ecole Polytechnique de Montr´eal, P.O. Box 6079, Station Centre-ville, Montr´eal, Qu´ebec, Canada H3C 3A7, Tel.: +1 514 340 4711 ext. 4857 Fax: +1 514 340 4176. Email address: [email protected] (M. L´evesque) ∗

Preprint submitted to International Journal of Solids and Structures

May 29, 2014

26

and morphological characteristics.

27

Key words: Mechanical properties identification, Virtual Fields Method,

28

Particulate composites, Full-field measurements

29

1. Introduction

30

Recent studies reveal that in-situ mechanical properties of composites

31

constituting phases can be significantly different than those obtained from

32

conventional testing on bulk samples (Gregory and Spearing, 2005; Hardiman

33

et al., 2012). Such differences can be attributed to different curing kinetics,

34

different chemical reactions, etc. Reliable in-situ mechanical properties are

35

the key for accurate predictions of damage evolution from micromechanical

36

modelling (Koyanagi et al., 2009; Wright et al., 2010).

37

Recent advances in imaging technologies, like micro-computed tomography

38

(µCT) (Bay et al., 1999; Martyniuk et al., 2013), enable the non-destructive

39

observation of internal deformation mechanisms in composites. When cou-

40

pled with a loading system and efficient Digital Volume Correlation (DVC)

41

algorithms (Mortazavi et al., 2014; Bay et al., 1999; Franck et al., 2007;

42

Réthoré et al., 2011; Bornert et al., 2004), such technologies pave the way

43

for the accurate in-situ characterization of local constituents through inverse

44

identification methods.

45

Several inverse mechanical properties identification methods have been re-

46

ported in the literature (Avril et al., 2008) for local properties identifica-

47

tion, such as the Equilibrium Gap Method (EGM) (Claire et al., 2002), the

48

equation error method (Gockenbach et al., 2008) and the constitutive com-

49

patibility method (Moussawi et al., 2013). The most extensively exploited

2

50

approach, however, is the Finite Element Model Updating (FEMU) method

51

(Okada et al., 1999; Kajberg and Lindkvist, 2004; Gras et al., 2013). It

52

updates iteratively the material parameters input into a representative fi-

53

nite element model to achieve the best possible match between numerically

54

predicted fields and their experimentally measured counterparts. The main

55

drawback of this method is its potentially prohibitive computational cost,

56

especially when dealing with three-dimensional (3D) problems.

57

As an alternative strategy, the Virtual Fields Method (VFM) (Pierron and

58

Grédiac, 2012) enables the direct (i.e. non-iterative) identification of param-

59

eters. This method, introduced first by Grédiac (Grédiac, 1989), was orig-

60

inally developed to identify the elastic properties of materials. It has been

61

successfully applied to identify bending and in-plane properties of composites

62

(Grédiac et al., 2002a; Avril and Pierron, 2007; Grédiac, 1996; Grédiac et al.,

63

2003; Pierron et al., 2007; Grédiac et al., 2002b). Efforts have also been

64

made to identify the orthotropic stiffness of laminated composites (Pierron

65

et al., 2000). The method is known to be less sensitive to measurement un-

66

certainties and noise, when compared to other identification techniques such

67

as FEMU and EGM (Avril and Pierron, 2007).

68

Different sources of errors such as important noise and artifact created by

69

current µCT scanners, as well as the systematic errors related to DVC can

70

induce uncertainties to the full field measurements. None of the above men-

71

tioned identification methods, including the VFM, can guarantee solution

72

uniqueness (nor providing information about multiple solutions), and varia-

73

tions in the measured data due to the presence of noise may cause changes

74

in the identified parameters.

3

75

Regularization constraints have been applied in different identification

76

methods (Florentin and Lubineau, 2010; Oberai et al., 2004) so as to stabi-

77

lize the identification problem. An improved VFM strategy, the Regularized

78

Virtual Fields Method (RVFM), has been recently proposed by the authors

79

(Rahmani et al., 2013) to determine the in-situ properties of fiber composites

80

from bi-dimensional (2D) full-field measurements. In this approach, regular-

81

ization constraints based on a micromechanical homogenization model were

82

exploited so as to regularize solving the related system of equations. The

83

RVFM demonstrated to be quite robust against noise effects, particularly for

84

composites including high strain heterogeneity. Most of the studies devoted

85

to VFM relied on 2D kinematic fields. Considering the recent advances in 3D

86

imaging techniques and the need for local characterization of 3D composites,

87

it would be of interest to evaluate, at least theoretically, the robustness of

88

such an approach for 3D problems.

89

The main objective of this research work was to extend the VFM and RVFM

90

to 3D so as to obtain bi-phasic composites constituents properties assumed to

91

be uniform within each phase. Virtual experiments were conducted in which

92

3D composites reinforced by particles of different aspect ratios and volume

93

fractions were simulated. The virtual composites were artificially deformed

94

and 3D strain fields were obtained over the voxels of the microstructures.

95

The performance of both 3D VFM and RVFM was assessed against noisy

96

strain fields and for different sizes of Region of Interests (ROIs).

97

This paper is organized as follows. Section 2 deals with the theoretical as-

98

pects of the virtual fields method and its extension to 3D. Section 3 introduces

99

the RVFM identification methodology developed for particulate composites.

4

100

Sections 4 and 5 are related to the application of both VFM and RVFM

101

identification methodologies to the virtual composites and the related re-

102

sults, respectively. The paper ends up with the concluding remarks of the

103

study in section 6.

104

105

2. Theoretical background

106

2.1. The Virtual Fields Method

107

The VFM relies on writing the global equilibrium of a body subjected

108

to a given load through the principle of virtual work (Pierron and Grédiac,

109

2012) as Z −

∗

Z

T · u∗ dS = 0 ∀u∗ ∈ KA

σ : ε dV + V

(1)

∂V

110

where V is the volume and ∂V is the boundary of the body, σ is the stress

111

tensor, u∗ and ε∗ are the virtual displacement and the corresponding strain

112

fields, respectively, T are the tractions acting on the boundaries and KA

113

stands for Kinematically Admissible conditions. For linear homogeneous ma-

114

terials, the constitutive equation can be expressed as: σi = Qij εj

(2)

115

where Qij are the stiffness components to be identified and εj are full-field

116

measured strain components coming from experimental tests. Accordingly,

117

the principle of virtual work stated in Eq. (1) can be written as: Z Z ∗ − Qij εj εi dV + Ti u∗i dS = 0 ∀u∗ ∈ KA V

∂V

5

(3)

118

In the case of isotropy and considering the matrix representation of the

119

constitutive equation (2), the principle of virtual work (Eq. (3)) can be

120

written as: Z   1 1 1 Q11 ε1 ε∗1 + ε2 ε∗2 + ε3 ε∗3 + ε4 ε∗4 + ε5 ε∗5 + ε6 ε∗6 dV + 2 2 2 V Z  1 ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Q12 ε2 ε1 + ε3 ε1 + ε1 ε2 + ε3 ε2 + ε1 ε3 + ε2 ε3 − ε4 ε4 − ε5 ε5 − ε6 ε6 dV 2 2 2 V Z = Ti u∗i dS ∀u∗ ∈ KA (4) Sf

121

The principle of virtual work is then extended based on a set of indepen-

122

dent KA virtual fields (test functions). If as many independent virtual fields

123

as there are unknown parameters are chosen, Eq. (4) leads to the following

124

linear system of equations: A·q =b

(5)

125

where A is a square matrix, b is a vector whose components are the virtual

126

work of the applied forces corresponding to each virtual field and q is a vector

127

containing the unknown parameters of Q. The virtual fields may be chosen

128

among an infinite number of possibilities, but must meet two conditions: i)

129

have C 0 continuity; ii) be chosen such that the resulting equations are linearly

130

independent in order to the linear system be invertible. The stability of the

131

linear system against noisy measured data depends strongly on the level of

132

independence of the chosen virtual fields.

133

2.2. The Regularized Virtual Fields Method

134

The linear system presented in Eq. (5) is solved directly (i.e. by matrix

135

inversion method) in the VFM. In the RVFM (Rahmani et al., 2013), how6

136

ever, the idea is to solve Eq. (5) in an optimization framework where, for

137

the purpose of regularization, a set of mechanically relevant constraints are

138

added. To this end, a least square objective function is defined as  T   R(q) = A · q − b · A · q − b

(6)

139

The regularization constraints are defined as the discrepancy between the

140

effective properties predicted by an appropriate homogenization model and

141

those obtained from experimental tests. Hence, the RVFM aims at minimiz-

142

ing R(q) subjected to the following constraints: ˜ l (q) − Λ ˆ l ) |≤ γ | (Λ

(l = 1, 2, ..., M )

(7)

143

˜ and Λ ˆ represent the effective mechanical properties obtained from where Λ

144

a homogenization model and from experimental data, respectively; M is the

145

number of constraints and γ is a very small positive definite quantity. The

146

constraints are evaluated at each iteration by updating the sought param-

147

eters. Assuming that the composite effective shear and bulk modulii are

148

known, and also depend on the matrix and particles properties, the imposed

149

constraints restrict the optimization algorithm to follow rational search di-

150

rections relying on the effective properties. This regularization can be in-

151

terpreted as error averaging of the measured data that reduces significantly

152

the noise effects, thus biasing the solution towards points very close to those

153

of noise-free fields. Indeed, adding appropriate physical information when

154

minimizing R(q) can improve the accuracy of the identified parameters. It

155

should be noted that this regularization can be efficient as long as an accurate

156

homogenization model is employed in the constraints. 7

157

2.3. Homogenization models

158

Homogenization methods deliver estimates for the effective properties of

159

composites using information related to their constitutive phases mechanical

160

properties and geometry. The Mori-Tanaka model (Mori and Tanaka, 1973),

161

the Lielens’ model (Lielens et al., 1998) and the third order approximation

162

(TOA) (Torquato, 1991) are examples of analytical homogenization models

163

that have been widely used in the literature. The two latter have been proven

164

to be more reliable for delivering accurate effective properties, particularly for

165

composites having high volume fractions and contrasts of properties between

166

phases (Ghossein and Levesque, 2012).

167

3. Application of the VFM and RVFM to particulate composites

168

This section details the application of both the VFM and the RVFM to

169

particulate composites. The methodology followed three steps: i) The gener-

170

ation of virtual particulate composites 3D microstructures of various volume

171

fractions and particles shapes; ii) The computation of a 3D strain fields re-

172

sulting from imposed boundary conditions. The strain fields were perturbed

173

by additive Gaussian white noise prior to composite properties characteri-

174

zation. These modified fields were considered as "measured" strain fields;

175

iii) The identification of the virtual composites constitutive phases proper-

176

ties using the resulting "measured" strain fields. The RVFM optimization

177

problem was solved by a derivative free optimization method (i.e., relies only

178

on objective function evaluation and does not require derivative information)

179

and the relative constraints values were evaluated by an appropriate homog-

180

enization model. The following subsections provide methodological details 8

181

for the above-mentioned steps.

182

3.1. 3D microstructure of the composites and mechanical properties

183

The two types of composites studied in this work were composed of an

184

isotropic matrix reinforced by randomly distributed spherical or ellipsoidal

185

particles, as shown in Fig. 1. The 3D microstructures were generated us-

186

ing the Molecular Dynamic (MD) method implemented by Ghossein and

187

Levesque (Ghossein and Levesque, 2012, 2013). Table 1 shows the refer-

188

ence mechanical properties considered for the constitutive phases, where sub-

189

scripts p and m refer to the particles and matrix, respectively. The chosen

190

mechanical properties are typical of E-glass-epoxy composites.

191

3.2. Stress and strain fields computation

192

The composites were deformed through applying overall strains in differ-

193

ent directions in order to generate 3D strain fields. Stress and strain fields

194

were computed using a technique based on Fast Fourier Transforms (FFT)

195

that was initially proposed by Moulinec et Suquet (Moulinec and Suquet,

196

1998). The advantage of this method stems from its rate of convergence and

197

the fact that it does not require meshing. The method was implemented in

198

detail in (Ghossein and Levesque, 2012) and only specific details related to

199

the current study are provided in the following sub-sections.

200

3.2.1. Discretization of the microstructures

201

Microstructures were discretized into 256 × 256 × 256 voxels. For each

202

voxel, the position of 9 points was verified. The stiffness tensor of the voxel,

9

203

denoted by C(x), was obtained as follows:    C(x) = Cp if 5 points or more belonged to a particle   C(x) = Cm

204

205

206

(8)

Otherwise

3.2.2. Computation of the stress and strain fields using FFT The stress and strain fields were obtained by solving the Lippman-Schwinger equation (Moulinec and Suquet, 1998) in Fourier space:      n+1 −1 n n ε (x) = F F ε (x) − Γ0 (ξ) : F C(x) : ε (x)

(9)

207

where F and F −1 refer respectively to the Fast Fourier Transform and its

208

inverse. Γ0 (ξ) denotes the Green operator and is expressed as follow: Γ0 (ξ) =

1 λ0 + µ0 ξi ξj ξk ξl (δ ξ ξ + δ ξ ξ + δ ξ ξ + δ ξ ξ ) − ki l j li k j kj l i lj k i 4µ0 kξk2 µ0 (λ0 + 2µ0 ) kξk4 (10)

209

where ξ denotes the frequencies in Fourier space. µ0 and λ0 represent respec-

210

tively the shear and Lamé modulus of the reference material. These moduli

211

were given by: √ µ0 = − µm µf

(11a)

2 √ λ0 = − κm κf − µ0 3

(11b)

212

Equation (9) was solved iteratively until the strain field convergence was

213

achieved. The algorithm was initialized with uniform strains applied in dif-

214

ferent directions:

  −e νe 0 0     0 ε (x) =  0 −e ν e 0   0 0 e 10

(12)

215

where νe is the effective Poisson’s ratio of the composite and was computed

216

using the methodology presented in Section 3.3. In this study, e was set to

217

−0.02. This led to effective stresses σx and σy ≈ 0 (≈ 10−7 in practice)

218

and an effective stress σz = −Tz , where Tz depended on the composites

219

microstructure. Finally, Gaussian white noise was added to the resulting

220

strain fields. The standard deviation of the additive noise was approximately

221

10% of the mean strain values. Fig. 2 depicts typical simulated εz fields,

222

perturbed with that noise level.

223

3.3. Determination of effective properties

224

3.3.1. Effective properties of a single microstructure

225

The regularization constraints presented in Eq. (7) require the composites

226

effective properties. The effective stiffness tensor of a single microstructure

227

b was deduced from the relation between the volume averaged stresses and C

228

strains: b : hε(x)i hσ(x)i = C

(13)

229

where h·i represents volume averaging. Six orthogonal deformation states

230

b For example, the first column were applied to obtain all the terms of C.

231

was obtained by applying a unit strain field in the first principal direction

232

(ε011 (x) = 1 and εjj , for j = 2 to 6, =0). The five other columns were

233

computed similarly.

234

235

The effective elastic bulk and shear moduli κ b and µ b, respectively, were then calculated as:

11

1b C :: J 3

(14a)

1 b C :: K 10

(14b)

κ b= µ b= 236

where J and K are the isotropic projector tensors.

237

3.3.2. Representive Volume Element determination

238

For each combination of contrasts and volume fractions, the size of the

239

Representative Volume Element (RVE) was determined using the method-

240

ology of Kanit et al. (Kanit et al., 2003). For each number of particles,

241

several random realizations were performed and the effective properties were

242

obtained for each generated microstructure. The number of realizations was

243

increased until the width of a 95% level confidence interval on the mean

244

effective property was smaller than a prescribed value (see (Ghossein and

245

Levesque, 2012) for more detail). The procedure was then repeated for an

246

increasing number of particles until the arithmetic mean of both effective

247

moduli converged.

248

After determining RVE size, 3D microstructure of all composites with 200

249

particles, which was larger than the RVE, were generated and the corre-

250

sponding strain/stress fields were simulated. These largest sizes (as shown

251

in Fig. 1) were considered as ROI of dimension 1 × 1 × 1.

252

3.4. Parameters identification with VFM and RVFM

253

Both VFM and RVFM identification methods were applied with the aim

254

of retrieving the reference elastic parameters of the composites constituents

255

initially used to generate the artificially "measured" strain fields.

12

256

3.4.1. Identification using the VFM

257

Considering that the whole material was not homogeneous, the VFM

258

relation presented in Eq. (4) was developed for a two-phase material. For

259

factorizing the sought parameters out of the volume integrals, the overall

260

volume of the composite (V ) was split into V − V 0 and V 0 , i.e. the matrix

261

and particles sub-volumes, respectively. Hence, after defining Q11 , Q12 and

262

Q011 , Q012 as the stiffness components over V − V 0 and V 0 , respectively, Eq.

263

(4) becomes:

Q11

Z 

ε1 ε∗1

+

ε2 ε∗2

+

ε3 ε∗3

1 ∗ 1 ∗ 1 ∗ + ε4 ε4 + ε5 ε5 + ε6 ε6 dV + 2 2 2

V −V 0

Z 

Q12

ε2 ε∗1 + ε3 ε∗1 + ε1 ε∗2 + ε3 ε∗2 + ε1 ε∗3 + ε2 ε∗3 −

1 ∗ 1 ∗ 1 ∗ ε4 ε − ε5 ε − ε6 ε dV + 2 4 2 5 2 6

V −V 0

Q011

Z  1 ∗ 1 ∗ 1 ∗ ∗ ∗ ∗ ε1 ε1 + ε2 ε2 + ε3 ε3 + ε4 ε4 + ε5 ε5 + ε6 ε6 dV + 2 2 2 V0

Q012

Z  V

ε2 ε∗1

+

ε3 ε∗1

+

ε1 ε∗2

+

ε3 ε∗2

+

ε1 ε∗3

0

+

1 ∗ 1 ∗ 1 ∗ − ε4 ε4 − ε5 ε5 − ε6 ε6 dV 2 2 2 Z = Ti u∗i dS ∀u∗ ∈ KA (15)

ε2 ε∗3

Sf 264

Four KA independent virtual fields had to be chosen for each microstruc-

265

ture. The virtual fields were chosen so as to prevent any rigid body motion,

266

while not perturbing the stress and strain fields generated at Section 3.2.2.

267

Fig. 3 shows the values these fields took on the boundaries of the studied

268

ROIs. The following different sets of virtual fields were tested for all mi-

269

crostructures:

270

Set 1: 13

271

272

 ∗(1)   u1 = x   ∗(1)

 ∗(2)   u1 = 0   ∗(2)

u2 = 0     u∗(1) = 0 3

(16a)

 ∗(3)   u =0   1 ∗(3) u2 = 0     u∗(3) = z 3

 ∗(4)  u1 = 0    ∗(4) u2 = 0 (16c)  3    u∗(4) = z 3 3

(16d)

 ∗(2)  u1 = 0    y3 ∗(2) (17a) u2 =  3    ∗(2) u3 = 0

(17b)

 ∗(4)  u =0    1 ∗(4) u2 = 0 (17c)  3    u∗(4) = z 3 3

(17d)

 ∗(2)  u =0    1 πy ∗(2) (18a) u2 = sin( )  4    u∗(2) = 0 3

(18b)

273

u2 = y     u∗(2) = 0 3

(16b)

274

275

276

277

Set 2:  x2 ∗(1)   u =   1 2 ∗(1) u2 = 0     ∗(1) u3 = 0  ∗(3)   u =0   1 ∗(3) u2 = 0     u∗(3) = z 3

278

279

280

281

Set 3:  πx ∗(1)  u1 = sin( )    4 ∗(1) u2 = 0     u∗(1) = 0 3

14

282

 ∗(3)   u =0   1 ∗(3) u2 = 0     u∗(3) = z 3

283

 ∗(4)  u1 = 0    ∗(4) u2 = 0 (18c)  3    u∗(4) = z 3 3

(18d)

 ∗(2)   u =0   1 ∗(2) (19a) u2 = exp(y)     u∗(2) = 0 3

(19b)

 ∗(4)  u1 = 0    ∗(4) u2 = 0 (19c)  3    u∗(4) = z 3 3

(19d)

284

285

286

287

288

Set 4:  ∗(1)    u1 = exp(x)  ∗(1)

u2 = 0     u∗(1) = 0 3  ∗(3)   u =0   1 ∗(3) u2 = 0     u∗(3) = z 3

289

290

The virtual fields of Set 2 and Set 3 shown to be sufficiently independent

291

when employed for the microstructures with ellipsoidal and spherical par-

292

ticles, respectively. The effects of virtual fields definition on the identified

293

parameters is presented in section 4.

294

Consider, for example, the virtual fields of Set 1. Since the tractions in x and

295

y directions were null, the first two components of vector b were zero. As-

296

suming identical dimensions of the ROI in all directions (Lx = Ly = Lz = L),

297

the other components of vector b were determined using the corresponding

15

298

virtual displacements ZL ZL b3 = 0

∗(3) T3 (x, y, 0) u3

ZL ZL dx dy +

∗(3)

T3 (x, y, L) u3

0

0

dx dy =

0

ZL ZL 0+ 0

F3 z dx dy = F3 L3 (20a)

0

299

ZL ZL b4 =

F3 0

z3 F3 L5 dx dy = 3 3

(20b)

0

300

Hence, the following linear system of equations was built up from Eq. (15)

301

and the virtual fields presented above: 

R

R

ε1 dV

 V −V 0  R   V −V 0 ε2 dV   R  ε dV  V −V 0 3  R

ε3 z 2 dV

V −V 0

R

(ε2 +ε3 )dV

V −V 0

V0

R

R

(ε1 +ε3 )dV

V −V 0

R

R

ε2 dV

R

(ε1 +ε2 )dV (ε1 z 2 +ε2 z 2 )dV

R

(ε1 +ε3 )dV

V0

R

ε3 dV

V0

V −V 0

 (ε2 +ε3 )dV

V0

V0

V −V 0

R

R

ε1 dV

(ε1 +ε2 )dV

V0

ε3 z 2 dV

V0

R

(ε1 z 2 +ε2 z 2 )dV

          Q11  0                       Q12 0   0  =  3   Q  F3 L      11             0   F3 L5   Q12

3

V0

(21) 302

The integrals were approximated by discrete sums over the voxel points. For

303

instance: Z ε3 dV '

p X

εi3 v i

(22)

i=1

V −V 0 304

where p is the number of data points over (V − V 0 ) and v i are their volumes.

305

Appendix A presents the system of equations derived from Sets 1, 2 and 4.

306

The linear system in Eq. (21) was solved through matrix inversion method

307

(for the VFM) as well as by using a constrained optimization procedure (for

308

the RVFM) in order to determine the stiffness components. 16

309

The elastic parameters of the constitutive phases were directly related to the

310

sought stiffness components by the following relations: Q12 Q11 (1 − 2νm )(1 + νm ) , Em = Q11 + Q12 (1 − νm ) 0 0 Q Q (1 − 2νp )(1 + νp ) νp = 0 12 0 , Ep = 11 Q11 + Q12 (1 − νp )

νm =

311

(23)

3.4.2. RVFM optimization problem

312

The relevant equations for each system were used to create a least square

313

objective functions based on Eq. (6). Hence, the RVFM consisted of solving

314

the following optimization problem  T   min R(q) = A · q − b · A · q − b Subjected to

(24) |κ eH (q) − κ bF F T |≤ γ1 |µ eH (q) − µ bF F T |≤ γ2

315

where superscript H refers to homogenization model, F F T to the properties

316

computed by the F F T method, and γ1 and γ2 were set to 1% of the corre-

317

sponding effective properties. Lielens and TOA homogenization models were

318

used for predicting the effective shear and bulk modulii, respectively, for the

319

microstructures with spherical particles. Lielens homogenization model was

320

also exploited for the microstructures with ellipsoidal particles.

321

The optimization problem of the RVFM was solved with the Mesh Adap-

322

tive Direct Search (MADS) optimization method (Audet and Dennis Jr.,

323

2006), which demonstrated to be quite successful in a previous study by the

324

current authors (Rahmani et al., 2013). MADS is a frame-based global opti-

325

mization algorithm for solving nonlinear problems without requiring deriva-

326

tive information. The method is known to be quite robust for optimization 17

327

problems with nonsmooth objective functions subjected to nonsmooth con-

328

straints. The VFM solutions were considered as initial guesses for the RVFM

329

algorithm. The constraints values were evaluated at each iteration by substi-

330

tuting the trial parameters into the related homogenization model, and their

331

feasibility was checked by the constraints. The stopping criterion considered

332

for all optimizations was 300 objective function evaluation.

333

334

3.5. Parameters identification from small ROIs

335

In practice, for the sake of image magnification requirements strain in-

336

formation of a small region as a representative of whole microstructure (as

337

illustrated in Fig. 4) is processed in DVC measurements for identification

338

purposes. The load distribution on the boundaries of such small volumes is

339

not fully determined. Indeed, if the ROI is not large enough to be a Represen-

340

tative Volume Element (RVE), then the overall stresses over its boundaries

341

will differ from those applied on the whole sample. This represents an im-

342

portant challenge since the local mechanical properties must link the internal

343

strains (known) and the stresses (unknown).

344

Assuming that virtual fields in the VFM are carefully chosen, inaccurate

345

values of tractions might affect the accuracy of the identified mechanical pa-

346

rameters. On the other hand, strain fields measured with DVC at higher

347

magnifications (smaller ROIs) can better capture heterogenous deformation

348

patterns and discontinuities created due to phases contrast of properties.

349

Therefore, an optimum size of ROI which satisfies the requirements of both

350

RVE size and image magnification must be determined. In this study, the

351

influence of ROI size on the identification procedure for both spheres and 18

352

ellipsoids microstructures was investigated. To this end, smaller sub regions

353

with different sizes of 0.65 × 0.65 × 0.65, 0.5 × 0.5 × 0.5, 0.3 × 0.3 × 0.3 and

354

0.1 × 0.1 × 0.1 were considered for the identifications. Each ROI was consid-

355

ered as an independent continuum model (as illustrated in Fig. 3) that was

356

in equilibrium through the tractions created on its boundaries.

357

4. Results and discussions

358

4.1. Identified parameters from the whole microstructure

359

Table 2 shows the obtained parameters resulting from different sets of vir-

360

tual fields in the VFM for the composites A and B. As it can be seen, Set 3

361

and Set 2 led to much more accurate results than the other sets for composites

362

A and B, respectively. This is because they constitute sufficiently indepen-

363

dent equations in the related systems. The same two sets were demonstrated

364

to be accurate for the composites with larger volume fractions (i.e., compos-

365

ites A0 and B 0 , respectively) and were used for the parameters identifications.

366

Tables 3a and 3b compare the elastic properties identified using the VFM

367

and RVFM for composites A and A0 , respectively. The corresponding relative

368

errors of the identified parameters resulting from both methodologies are also

369

reported. The first identification was carried out using exact strain fields (i.e.,

370

without any additive noise to the strain data). In this case, the VFM leads

371

to relatively accurate parameters, except for the particles Poisson’s ratio. It

372

can be seen that the relative error of the particles parameters identified by

373

the VFM increases in the presence of noise, while the matrix parameters are

374

only slightly influenced by noise effects. This is most probably due to the

375

fact that the signal/noise ratio in the stiffer phase, i.e., the particles, is much 19

376

lower than that in the matrix phase. Table 4 shows the signal/noise ratio

377

(i.e., the ratio of mean strain value to the standard deviation of noise) for

378

the constitutive phases of the two composites. For the composite with larger

379

volume fraction of particles (composite A0 ), however, the identified matrix

380

parameters are less accurate than those of composite A. This might arise

381

from strain concentrations occuring in the matrix phase embedded between

382

close particles aligned in the loading directions.

383

The corresponding results of the RVFM for both volume fractions, which

384

show low relative errors for both phases in the presence of noise, confirm the

385

robustness of this method. Thanks to the regularization effects imposed by

386

the homogenization models, the RVFM results in more accurate parameters

387

than the VFM in the presence of noise. The lower accuracies for the particles

388

Poisson’s ratio identified by the RVFM can be associated to lower sensitivity

389

of the effective properties to the variations in this parameter than the other

390

parameters. This is shown in Fig. 5, which presents variations in the effective

391

parameters with respect to the variations of constituent phases parameters

392

for composite A. Similar trends were observed for composite A0 that are not

393

reported here.

394

Note that the way of computing the integrals in the VFM using discrete sums,

395

as described in Eq. (22), may also induce biases. No convergence study of the

396

reconstructed properties as a function of grid size was performed. However,

397

the chosen grid size was sufficient since convergence studies were performed

398

on its influence on the effective properties, which involves integrals computed

399

with Eq. (22).

400

Tables 5a and 5b present the identified parameters for the composites with

20

401

ellipsoidal particles (composites B and B 0 ), with and without noise. For the

402

noise-free cases, the VFM results in solutions very close to the target val-

403

ues. However, the presence of noise degrades the accuracy of the method,

404

especially regarding the particles properties. The acquired results indicate

405

that the RVFM is less sensitive to noise effects and leads to more accurate

406

solutions. Trends similar to those shown in Fig. 5 were observed for the sen-

407

sitivity of the effective properties to the variations in the constituent phases

408

of composites B and B 0 (not reported here).

409

4.2. Influence of ROI sizes on the accuracy of identified parameters

410

The mechanical parameters were identified using both VFM and RVFM

411

from noisy strain data of smaller ROIs. For each ROI size, the constituent

412

properties were obtained from 6 different realizations. These realizations

413

were in fact ’windows’ extracted from ROI 1 × 1 × 1 described in the previ-

414

ous sections. It should be noted that realizations extracted from larger ROIs

415

(0.65 × 0.65 × 0.65) overlapped and as a result, were not fully independent.

416

Table 6 lists the typical number of represented particles in each ROIs, for

417

composites A0 and B 0 , along with the RVE size. Corresponding average rela-

418

tive errors with respect to the exact values were subsequently derived. Table

419

7 presents the obtained results along with two-tailed 95% confidence inter-

420

val (CI) on the average values for composite A0 . The average relative errors

421

for the overall stress on the boundary of ROIs with respect to that of the

422

whole composite have also been reported in the table. For the largest ROI

423

(i.e., 0.65 × 0.65 × 0.65), the overall stress is very close to that of the whole

424

sample. This is due to the fact that the chosen size of ROI is very close to

425

that of the RVE. For this reason, the average parameters are also relatively 21

426

close to those resulting from the ideal condition (Table 3b). Identification

427

using ROI 0.5 × 0.5 × 0.5 leads to satisfactory results, although the overall

428

stress is slightly less accurate than the largest ROI. For the smaller ROIs

429

(i.e., 0.3 × 0.3 × 0.3 and 0.1 × 0.1 × 0.1), however, the average error in the

430

overall stresses increases as the size of the ROI decreases. This is obviously

431

associated with the fact that the chosen volumes are not large enough to

432

be a RVE and, therefore, the estimated stresses are considerably different

433

from those of the whole model. This could be considered as a source of er-

434

ror in the small ROIs that induces more uncertainties to the identifications,

435

when compared with larger ROIs. This observation can be explained by the

436

fact that the VFM depends directly on the accuracy of the applied stresses.

437

Moreover, the RVFM does not improve the accuracy of the parameters, when

438

compared to the similar case in the larger ROI. In some cases, the RVFM

439

even degraded the accuracy of the particles parameters, which was due to

440

the biases created in optimization as a consequence of error in the overall

441

applied stress.

442

Table 8 lists the average relative errors of the identified parameters as well

443

as the overall stresses resulting from different sizes of ROIs for composite

444

B0 . Similarly to the previous case, identification using the two larger ROIs

445

resulted in satisfactory parameters, whereas the smaller ROIs led to inaccu-

446

rate mechanical parameters, especially regarding the particles phase. It is

447

worth mentioning that a similar trend, in terms of the accuracy of identified

448

parameters with respect to ROI sizes, was observed for 2D composites in

449

(Rahmani et al., 2013). According to the quality of the resulting parameters,

450

the ROI 0.65 × 0.65 × 0.65 could be considered as the smallest size for both

22

451

microstructures.

452

Hence, the optimum size of ROI for a given composite with spherical particles

453

can be estimated, a priori, using the following relation 1 Lopt =

NRV E πds

3

!

3

(25)

6c

454

where Lopt is the edge length of the optimum ROI, NRV E is the size of RVE

455

(number of particles in the RVE), ds is the diameter of spheres and c denotes

456

particles volume fraction. The following relation can also be defined for the

457

composites including ellipsoidal particles 1 !

Lopt =

NRV E πde1 de2 de3 3 6c

(26)

458

where de1 , de2 and de3 are the ellipsoids diameters in different directions.

459

The above defined relationships could be useful for guiding the experimen-

460

talists in defining the optimum size of their ROIs for accurate properties

461

identification of composites with any mechanical or morphological proper-

462

ties.

463

Finally, it should be noted that besides the remarkable advantages mentioned,

464

the proposed identification approach is very demanding experimentally as it

465

requires 3D full field measured data at the microscale and effective material

466

properties at the macroscale.

467

5. Conclusions

468

An identification approach based on the Virtual Fields Method (VFM)

469

has been proposed to determine in-situ mechanical properties of composites 23

470

constitutive phases in 3D. Moreover, a Regularized Virtual Fields Method

471

(RVFM) consisting of homogenization-based constraints was developed so as

472

to regularize the identification procedure and therefore enhance the accuracy

473

of the identified parameters. For performance evaluation, the algorithms

474

were applied to 3D noisy full field strain fields of artificial particulate com-

475

posites including different volume fractions and particles geometries. The

476

obtained results demonstrate the capabilities of the VFM to determine ap-

477

propriate parameters of 3D composites in the presence of noisy strain fields.

478

The RVFM, however, by taking advantage of regularization effects, leads to

479

more accurate results. Depending on the type of microstructure in terms of

480

the particles geometry, appropriate homogenization models were employed

481

so as to enhance the accuracy of the identified parameters.

482

Different ROIs were tested to investigate the influence of their size on the

483

corresponding overall tractions and consequently on the accuracy of the iden-

484

tified parameters. It was found that for ROIs smaller than RVE, neither of the

485

VFM and the RVFM resulted in appropriate mechanical parameters due to

486

the inadequacy of the overall tractions on the boundaries. A helpful compre-

487

hensive relationship has also been developed, by which the experimentalists

488

can efficiently determine the optimum ROI size to obtain properties within

489

an adequate range of accuracy. This study could also be very useful for es-

490

timating, a priori, the required magnification of 3D images for composites of

491

any mechanical and morphological characteristics.

24

492

493

Appendix A The linear system made from the virtual fields of Set 2: 

R

R

ε1 xdV

V −V 0

  R  2 V −V 0 ε2 y dV   R  ε dV  V −V 0 3  R

(ε2 x+ε3 x)dV

V −V 0

(ε1 y 2 +ε3 y 2 )dV

R

ε2 y 2 dV

R

R

(ε1 +ε2 )dV

V −V 0

ε3 dV

V0

(ε1 z 2 +ε2 z 2 )dV

V −V 0

R



         0  Q11               R         (ε1 y 2 +ε3 y 2 )dV   Q 0 12 V0   0  =  3 R  (ε1 +ε2 )dV   Q11  F3 L          V0         5  0   F3 L   R R

(ε2 x+ε3 x)dV

V0

V0

R

R

ε1 xdV

V0

V −V 0

ε3 z 2 dV

V −V 0

R

ε3 z 2 dV

V0

(ε1 z 2 +ε2 z 2 )dV

Q12

3

V0

(27) 494

The linear system derived from the virtual fields of Set 3: 

R

π ε 4 1

cos( πx )dV 4

V −V 0  R  πy π V −V 0 4 ε2 cos( 4 )dV   R  ε dV  V −V 0 3  R ε3 z 2 dV

V −V 0

R V −V 0

R V −V 0

π (ε 4 2

cos( πx )+ε3 cos( πx ))dV 4 4

π (ε 4 1

cos( πy )+ε3 cos( πy ))dV 4 4 R

V0

R V0

π ε 4 1

cos( πx )dV 4

π ε 4 2

cos( πy )dV 4

R

(ε1 +ε2 )dV

V −V 0

R

R

R V0

R V0

ε3 dV

V0

(ε1 z 2 +ε2 z 2 )dV

V −V 0

R

ε3 z 2 dV

π (ε 4 2

cos( πx )+ε3 cos( πx ))dV 4 4

   Q11            πy πy π  (ε cos( )+ε cos( ))dV 3 4 1 4 4  Q12    0  R  (ε1 +ε2 )dV Q11      V0       0  R (ε1 z 2 +ε2 z 2 )dV

V0

V0

=

       

0 0

       

   F3 L3         5  F3 L   3

25



(28)

Q12

495

The linear system made from the virtual fields of Set 4: 

R

ε1 exp(x)dV

V −V 0  R  V −V 0 ε2 exp(y)dV   R  ε dV  V −V 0 3  R ε3 z 2 dV

V −V 0

R

(ε2 exp(x)+ε3 exp(x))dV

V −V 0

R

R

ε1 exp(x)dV

V0

(ε1 exp(y)+ε3 exp(y))dV

V −V 0

R

R

R

ε2 exp(y)dV R

(ε1 +ε2 )dV

ε3 dV

V0

(ε1 z 2 +ε2 z 2 )dV

V −V 0

(ε2 exp(x)+ε3 exp(x))dV

V0

V0

V −V 0



     Q11         R     (ε1 exp(y)+ε3 exp(y))dV   Q 12 V0   0  R  (ε1 +ε2 )dV Q11      V0       0  R R

R V0

ε3 z 2 dV

=

       

0 0

       

   F3 L3          F3 L5   3

26

Q12

(ε1 z 2 +ε2 z 2 )dV

V0

(29)

496

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Okada, H., Fukui, Y., Kumazawa, N., 1999. An Inverse Analysis Determin-

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ing the Elastic-Plastic Stress-Strain Relationship Using Nonlinear Sensi-

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tivities. Computer Modeling and Simulation in Engineering 4, 176–185.

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Pierron, F., Grédiac, M., 2012. The Virtual Fields Method. Springer, New

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York.

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Pierron, F., Vert, G., Burguete, R., Avril, S., Rotinat, R., Wisnom, M., 2007.

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Identification of the orthotropic elastic stiffnesses of composites with the

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virtual fields method: sensitivity study and experimental validation. Strain

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43 (3), 250–259.

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Pierron, F., Zhavoronok, S., Grédiac, M., 2000. Identification of the through-

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thickness properties of thick laminated tubes using the virtual fields

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method. International Journal of Solids and Structures 37 (32), 4437–4453.

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Rahmani, B., Villemure, I., Lévesque, M., 27 Sep. 2013. Regularized virtual

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field method for mechanical properties identification of composite mate-

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ages. The Journal of Strain Analysis for Engineering Design 46 (7), 683–

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Torquato, S., 1991. Random heterogeneous media: microstructure and improved bounds on effective properties. Applied mechanics reviews 44, 37.

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Wright, P., Moffat, A., Sinclair, I., Spearing, S., 2010. High resolution to-

617

mographic imaging and modelling of notch tip damage in a laminated

618

composite. Composites Science and Technology 70 (10), 1444–1452.

32

(a)

(b)

(c)

(d)

Figure 1: 3D microstructures of the particulate composites: (a) composite A with spherical particles at a volume fraction of 20%; (b) composite A0 with spherical particles at a volume fraction of 50%, (c) composite B with ellipsoidal particles of aspect ratio 10 at a volume fraction of 10% and (d) composite B0 with ellipsoidal particles of aspect ratio 10 at a volume fraction of 20%.

33

z

z y

y

x

x

(a)

y

(b)

z

z y

x

(c)

x

(d)

Figure 2: Noisy εz for (a) composite A, (b) composite A0 , (c) composite B and (d) composite B0 , subjected to an overall uni-axial compressive stress in z

34

σz

uy=0 z

uz=0

σz

y

x

uz=0

ux=uy=uz=0

Figure 3: Free body diagram of a sample ROI

35

Tz T’z=?

T’z=?

z y

x

Tz

(a)

(b)

Figure 4: (a) Schematic representation of a complete sample being submitted to an external load, (b) Schematic region of interest for which strains would be computed from a DVC algorithm applied on images obtained from µCT.

36

Variations in kTOA (%)

40 35

Ep

30

Em

25

νp ν

m

20 15 10 5 0

2

4

6

8

10

12

14

Variation in constituent phases parameters (%)

(a)

Variations in µLielens (%)

12 Ep 10

Em

8

νp ν

m

6 4 2 0

2

4

6

8

10

12

14

Variation in constituent phases parameters (%)

(b)

Figure 5: Sensitivity of the effective parameters with respect to variations in the constituent phases properties for composite A; (a) effective bulk modulus predicted by TOA method (b) effective shear modulus predicted by Lielens model

37

Table 1: Reference elastic mechanical properties for the virtual composites Composite

volume fraction

A

20

A0

50

B

10

B0

20

particles

Ep (GPa)

νp

Em (GPa)

νm

Spherical

74

0.2

3.5

0.35

Ellipsoidal

74

0.2

3.5

0.35

Table 2: Identification results for composite A using different sets of virtual fields in the VFM for a noise level of 2% (ROI 1 × 1 × 1) Virtual fields

Ep (GP a)

νp

Em (GP a)

νm

Reference values

74

0.2

3.5

0.35

Set 1

3.88

0.499

6.22

0.531

Set 2

79.70

0.350

3.32

0.250

Set 3

74.75

0.227

3.51

0.343

Set 4

79.76

0.362

3.25

0.219

0.247

Composite A

Composite B Set 1

5.54

0.487

4.57

Set 2

72.10

0.212

3.57

0.343

Set 3

20.80

0.446

4.4

0.269

Set 4

1210

3.19

2.38

0.555

38

Table 3a: Identified parameters and corresponding relative errors () for composite A (ROI 1 × 1 × 1)

Method

Noise level

Ep (GP a) (%)

νp (%)

Em (GP a) (%)

νm (%)

Reference values

–

74

0.2

3.5

0.35

VFM

Exact data

73.71 (0.4%)

0.217 (8.5%)

3.53 (0.85%)

0.344 (1.7%)

VFM

10%

70.55 (4.7%)

0.124 (38%)

3.54 (1.1%)

0.360 (2.8%)

RVFM

10%

73.40 (0.8%)

0.213 (7.5%)

3.53 (0.85%)

0.345 (1.4%)

Table 3b: Identified parameters and corresponding relative errors () for composite A0 (ROI 1 × 1 × 1) Method

Noise level

Ep (GP a) (%)

νp (%)

Em (GP a) (%)

νm (%)

Reference values

–

74

0.2

3.5

0.35

VFM

Exact data

73.49 (0.7%)

0.194 (3%)

3.65 (4.3%)

0.339 (3.1%)

VFM

10%

71.61 (3.2%)

0.224 (12%)

3.82 (9.1%)

0.318 (9.1%)

RVFM

10%

74.01 (0%)

0.182 (9%)

3.47 (0.85%)

0.362 (3.3%)

Table 4: Signal/noise ratio for the constituent phases of different composites Composite

spheres

matrix

A

2.27

22.18

A0

4.36

34.45

B

3.30

18.95

B0

3.63

21.81

39

Table 5a: Identified parameters and corresponding relative errors () for composite B (ROI 1 × 1 × 1)

Method

Noise level

Ep (GP a) (%)

νp (%)

Em (GP a) (%)

νm (%)

Reference values

–

74

0.2

3.5

0.35

VFM

Exact data

72.27 (2.4%)

0.212 (6%)

3.56 (1.7%)

0.343 (2%)

VFM

10%

70.83 (4.3%)

0.221 (10.5%)

3.60 (2.8%)

0.341 (2.6%)

RVFM

10%

73.20 (1%)

0.207 (3.5%)

3.53 (0.85%)

0.346 (1.1%)

Table 5b: Identified parameters and corresponding relative errors () for composite B 0 (ROI 1 × 1 × 1)

Method

Noise level

Ep (GP a) (%)

νp (%)

Em (GP a) (%)

νm (%)

Reference values

–

74

0.2

3.5

0.35

VFM

Exact data

76.10 (2.8%)

0.189 (5.5%)

3.51 (0.3%)

0.339 (3.1%)

VFM

10%

78.95 (6.7%)

0.190 (5%)

3.39 (3.1%)

0.338 (3.4%)

RVFM

10%

75.81 (2.4%)

0.205 (2.5%)

3.52 (0.6%)

0.344 (1.7%)

Table 6: Number of particles in different ROIs ROI size

0.1

0.3

0.5

0.65

1

RVE

Number of particles

1

6

25

55

200

60

40

Table 7: Average relative error of the identified parameters for composite A0 from different ROIs (noise level=10%)

Method

Ep error (CI)

νp error (CI)

Em error (CI)

νm error (CI)

Stress error

ROI 0.65 × 0.65 × 0.65 VFM

1.7% (0.7 , 2.8)

12.9% (9.7 , 16)

8.1% (5.8 , 9.9)

8.7% (6.1 , 10)

RVFM

1.9% (1 , 2.8)

8.5 % (6.8 , 10)

1.8% (1.3 , 2.2)

2.5% (1.7 , 3.4)

1% ROI 0.5 × 0.5 × 0.5 VFM

1.6% (0.6 , 2.6)

15.9% (6.3 , 25)

7.7% (5.3 , 10)

8.9% (7.1 , 10.8)

RVFM

2.0% (0.9 , 3.1)

10.0% (7.5 , 12)

1.7% (1.1 , 2.2)

2.8% (2.1 , 3.6)

2.1% ROI 0.3 × 0.3 × 0.3 VFM

9.4% (4.1 , 14.8)

6.9% (4.1 , 9.6)

12.9% (2.6 , 23)

9.4% (5.8 , 12.9)

RVFM

11.1% (4.2 , 17.8)

10.2% (8.8 , 11.4)

3.4% (1.1 , 5.8)

2.8% (1.2 , 4.2)

8.83% ROI 0.1 × 0.1 × 0.1 VFM

19.4% (13.9 , 24.7)

7.0% (5.6 , 8.3)

16.1% (5.6 , 26)

9.0% (6.7 , 11.4)

RVFM

20.2% (15.6 , 24.7)

8.9% (4.2 , 13.6)

5.2% (3.6 , 6.7)

4.2% (1.4 , 6.9)

18.7%

41

Table 8: Average relative error of the identified parameters for composite B0 from different ROIs (noise level=10%) Method

Ep error (CI)

νp error (CI)

Em error (CI)

νm error (CI)

Stress error

ROI 0.65 × 0.65 × 0.65 VFM

3.9% (2.4 , 5.5)

6.5% (4.4 , 8.7)

3.3% (1.7 , 5.3)

2.9% (2.1 , 3.7)

RVFM

3.2% (0.8 , 5.7)

9.5% (8.0 , 11.1)

0.6% (0.2 , 1.0)

1.8% (1.3 , 2.2)

1.8% ROI 0.5 × 0.5 × 0.5 VFM

3.7% (2.1 , 5.1)

6.2% (4.8 , 7.6)

3.9% (2.3 , 5.4)

3.2% (2.9 , 3.5)

RVFM

3.1% (1.9 , 4.2)

7.7% (6.3 , 9.0)

1.7% (0.7 , 2.7)

2.3% (1.6 , 2.9)

2.3% ROI 0.3 × 0.3 × 0.3 VFM

5.5% (1.1 , 10)

7.3% (4.9 , 9.5)

6.8% (4.3 , 9.3)

3.6% (3.2 , 3.9)

RVFM

5.8% (1.4 , 10)

10.3% (6.9 , 14)

1.4% (0.6 , 2.1)

1.2% (0.3 , 2.1)

VFM

25% (21 , 29)

6.3% (3.8 , 8.9)

25.5% (19 , 32)

3.1% (2.4 , 3.8)

RVFM

26% (21 , 30)

13.5% (7.6 , 16)

17.6% (12 , 31)

2.7% (2.2 , 3.1)

6.5% ROI 0.1 × 0.1 × 0.1 25.3%

42

619

BIOGRAPHIES

620

621

Behzad Rahmani has been a PhD student in the Department of Mechanical

622

Engineering at Ecole Polytechnique de Montreal since 2010. He received his

623

Bachelor in Mechanical Engineering form the University of Tabriz in 2004

624

and completed his Master in Mechanical Engineering in the university of

625

Mazandaran in 2007. His research interests lie in the area of inverse methods

626

in engineering, optimization methods, composite materials and Finite Ele-

627

ment methods.

628

629

Isabelle Villemure’s background is structural engineering with a B.Eng.

630

from Polytechnique Montreal, Canada, a M.A.Sc. from UBC, Vancouver,

631

Canada, and a Ph.D.degree in biomedical engineering from the University

632

of Montreal, Canada. She subsequently continued her training as a post-

633

doctorate in bioengineering at the University of Calgary, Canada. She is

634

a Professor at Polytechnique Montreal in mechanical and biomedical engi-

635

neering, and a Researcher at the Sainte-Justine University Hospital Center,

636

Montreal. Her research aims at establishing how mechanical forces impact

637

bone growth and development to leverage this knowledge in the design of

638

novel orthopedic treatments for progressive skeletal deformities in children.

639

640

Martin Lévesque is an Associate Professor in the Department of Me-

641

chanical Engineering at Ecole Polytechnique de Montreal where he has been

642

a faculty member since 2005. He completed his Ph.D. at ENSAM in Paris in

643

2004. He is currently the holder of the Canada Research Chair in Multiscale

43

644

Modelling of Advanced Aerospace Materials. He is also responsible of the

645

Laboratory for Multi-scale Mechanics at Ecole Polytechnique de Montreal.

646

His research interests are focused on Solid mechanics, Composite materials,

647

Viscoelasticity, Fatigue and Structural analysis.

44

45