Measurement of the mode II intralaminar fracture toughness and R-curve of polymer composites using a modified Iosipescu specimenand the size effect lawCatalanotti, G., & Xavier, J. (2015). Measurement of the mode II intralaminar fracture toughness and R-curve ofpolymer composites using a modified Iosipescu specimen and the size effect law. Engineering FractureMechanics, 138, 202-214. https://doi.org/10.1016/j.engfracmech.2015.03.005

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Download date:01. Sep. 2021

Measurement of the mode II intralaminar

fracture toughness and R-curve of polymer

composites using a modified Iosipescu

specimen and the size effect law

G. Catalanotti a,∗, J. Xavier baDEMec, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto

Frias, 4200-465, Porto, Portugal.

bOptics and Experimental Mechanics Laboratory, INEGI, Campus da FEUP, Rua

Dr. Roberto Frias 4200-465 Porto, Portugal.

Abstract

A modified Iosipescu specimen is proposed to measure the mode II intralaminarfracture toughness and the corresponding crack resistance curve of fibre reinforcedcomposites. Due to the impossibility of scaling the specimen, a modification of theclassical size effect method is proposed. The classical Iosipescu shear feature wasused and tests were coupled with digital image correlation to support the proposedapproach. Experiments were performed on IM7/8552 material system and the R-curve was obtained. The steady–state value of the fracture toughness of the ply isfound to be equal to R0ss = 34.4 kJ/m2.

Key words:

A. Polymer-matrix composites (PMCs), B. Fracture toughness, C. Analyticalmodelling, D. Mechanical testing

1 Introduction1

The intralaminar fracture toughness is a key parameter used to screen and to2

qualify material systems, and is an input parameter for strength analysis mod-3

els. Some of those, as recent progressive damage models, use softening laws4

∗ Corresponding authorEmail address: giuseppe.catalanotti@fe.up.pt (G. Catalanotti).

Preprint submitted to Eng Fract Mech 30 January 2015

that can be completely defined once the R-curve of the material is known [1–5

4]. Others, as Finite Fracture Mechanics (FFMs) models [5–8], require the6

fracture toughness (together with the strength of the laminate) as input pa-7

rameters.8

Despite the importance of knowing the intra-laminar fracture toughness of9

composite laminates, very few tests methods were developed in the recent10

past. Generally the R-curve is measured using Compact Tension or Compact11

Compression specimens [9], the last if the R-curve associated with the propa-12

gation of a kink band is required. Several difficulties arise in the execution of13

the experimental tests, in particular the measurement of the crack extension14

and the computation of the J-Integral around the crack tip [10]. Moreover, the15

orthotropy of the material complicates the tests and the data post-processing.16

Those difficulties drastically increase when measuring the R-curve in mode17

II. In this case no tests method exists and, to the authors’ best knowledge,18

no relevant work has been done using fibre-reinforced plastics. However, the19

measurement of the R-curve in mode II could be very valuable because it20

would enable the generalization of several analysis models developed for mode21

I propagation (e.g., FFMs and progressive damage models).22

The authors recently proposed the use of the size-effect methods developed23

by Bazant (mainly for rocks and ceramics) to measure the R-curve of fibre re-24

inforced composites [11,12]. It was demonstrated that size effect method can25

be effectively used to measure the R-curve of a laminate (or a ply), drasti-26

cally simplifying the experimental tests. Indeed, using the size effect law it is27

not required to measure the crack extension, a parameter whose experimental28

measurement is quite difficult when using heterogeneous materials, particu-29

larly under compression, under shear, and in the presence of bridging effects.30

In this paper the size effect method is applied to a modified Iosipescu speci-31

men. This experimental method is commonly used to measure the elastic and32

strength parameters of fibre–reinforced plastic in shear and its ability to ap-33

ply a uniform shear stress field in the gauge section of the specimen has been34

demonstrated [13]. The details of the method are given in [14]. Therefore, the35

Iosipescu test method was chosen to apply the shear stress to a cracked spec-36

imen. The main inconvenient of this method is that the experimental device37

is standardized and it is not possible to scale specimens of the same geometry38

to use the classical size effects method [15]. To overcome this limitation, a39

modification of the size effects method is proposed. Digital image correlation40

was coupled to the Iospiescu tests to provide the strain field at the centre of41

the specimens.42

2

2 Analytical Model43

2.1 Linear-elastic material44

Let x1 and x2 be the material axes of a two-dimensional orthotropic solid.45

The energy release rate in mode II for a crack propagating in the x1 direction46

reads:47

GII =1

EK2

II (1)

where E is the equivalent modulus and KII is the stress intensity factor in48

mode II. The equivalent modulus reads [16]:49

E =(

s11 s221 + ρ

2

)−1/2

λ−1/4 (2)

where slm are components of the compliance tensor computed in the x1-x250

coordinate system, and λ and ρ are two dimensionless elastic parameters given51

as [16]:52

λ =s11s22

ρ =2 s12 + s662√s11s22

(3)

The measurement of the ply fracture toughness is based on balanced cross ply53

laminates [9]. The plies that are perpendicular to the direction of the crack54

propagation (0◦ plies) have a fracture toughness that is two or three orders55

of magnitude higher than that of the plies parallel to the crack propagation56

(90◦ plies). This is due to the fact that the former plies experience a fibre57

dominated failure whilst the latter experience a matrix dominated failure.58

Neglecting the possible small interactions between these two different failure59

modes, an energy balance shows that the fracture toughness of the laminate60

along the x1 direction is simply one half of the fracture toughness of the 0◦61

ply for the same direction.62

If a balanced cross ply is used (λ = 1), equation (2) yields:63

E = E(

1 + ρ

2

)−1/2

(4)

where E is the Young’s modulus along a preferred direction of the material.64

3

Consider now a modified Iosipescu specimen as shown in Figure 1. The width65

of the specimen is 2W = 20mm while the length is 80mm. The specimen66

has a gauge section of width 2w, created with two symmetric notches, and67

two symmetric cracks on the gauge section of length a. Two dimensionless68

parameters are defined: (i) the ratio between the crack length and the width69

of the gauge section, α = a/w; and (ii) the ratio between the width of the70

gauge section and the width of the specimen, β = w/W . Figures 1(a) and 1(b)71

shows the specimen when 0 < β < 1 and β = 1, respectively.72

[Fig. 1 about here.]73

The stress intensity factor of equation (1) depends on the dimension of the74

specimen (a characteristic size, w), on the geometry (α and β), and on the75

remote applied load P or stress τ :76

KII = τ√wφ (α, ρ, β) =

P

2 t√wφ (α, ρ, β) (5)

where τ = P/ (2wt).77

Since β = w/W , the previous equation can be rewritten as:78

KII = τ√

β Wφ (α, ρ, β) (6)

Using (1), the energy release rate reads:79

GII =W

Eβ τ 2 φ2 (α, ρ, β) . (7)

Equation (7) yields the energy release rate for a given applied stress τ . At the80

peak load, when the maximum stress is τu, the energy release rate equals the81

mode II fracture toughness of the material:82

R (∆a) =W

Eβ τ 2u φ

2

(

α0 +∆a

β W, ρ, β

)

(8)

where ∆a is the crack increment and α0 = a0/w, where a0 is the initial crack83

length. The R-curve is a material parameter that does not depend on the84

shape or dimension of the specimen. In particular it does not depend on the85

parameter β. Therefore, taking the derivative of R with respect to β, it follows86

that ∂βR = 0. Deriving equation (8) with respect to β and imposing ∂βR = 087

yields:88

4

∂βR =W

E∂β(

β τ 2 φ2)

= 0. (9)

If τ = τ (β) (for a given W ) is known and under the hypothesis that α0 is89

a constant, equation (9) can be solved for β = β (∆a). Substituting β (∆a)90

in equation (8) yields the R-curve for 0 < β < 1 or, in other words, for91

0 < w < W .92

It should be noted that the procedure described enables the determination93

of part of the R-curve (in the range 0 < β < 1) but not the entire R-curve.94

This is due to the fact that it is not possible to extrapolate the region of95

the R-curve when w → +∞ because 0 < w < W . To obtain the entire R-96

curve the size effect law must be obtained with a characteristic length that97

ranges in the interval [0,+∞). In the impossibility of scaling the specimens,98

the size effect law can be obtained by mapping the part of the R-curve already99

defined. Consider a specimen with a constant β, for example β = 1 as shown100

in Figure 2.101

[Fig. 2 about here.]102

The crack driving force for this specimen reads:103

GII =1

Ew τ 2 ψ2 (10)

where ψ = φ|β=1. At the peak load the following system of equations is satis-104

fied:105

GII = R∂∆aGII ≥ ∂∆aR

(11)

where ∂∆aGII = ∂∆aR represents the condition for unstable crack propagation.106

Equation (11) represents the tangency condition between the crack driving107

force and the R-curve, as shown in Figure 3. Therefore, for a given w⋆, the108

value of τ ⋆ tangent to the R-curve can be found. The strategy used here is quite109

simple: for a given characteristic dimension w⋆ the stress τ for which there is110

just an intersection point (⋆ in Figure 3) is the stress τ ⋆. This is illustrated111

in Figure 3 in which, for a given point of the R-curve (indicated with ⋆), only112

two crack-driving force curves are defined (assuming a0 constant). The known113

portion of the R-curve (in red in Figure 3) extends between two points that114

are tangent to the driving forces curve identified with βmin and βmax.115

5

βmin and βmax are the limiting values of β. Since the numerical model described116

in the following section is defined for β ∈ [0.2, 1.0], βmin and βmax are taken117

as 0.2 and 1.0, respectively.118

[Fig. 3 about here.]119

By repeating this procedure several times, the calculation of several pairs120

(w⋆, τ ⋆) are obtained. By fitting these points it is possible to define the size121

effect law (see Table 1 for suggested expressions of the size effect laws), τ =122

τ (w), that is necessary to calculate the entire R-curve, R (∆a).123

[Table 1 about here.]124

To do that it should be noted that the R-curve is a material parameter and it125

does not depend on w. Therefore, ∂wR = 0. Substituting (10) in the first (11)126

and deriving with respect to w yields:127

∂wR =1

E∂w(

w τ 2u ψ2)

= 0. (12)

Knowing the size effect law, τ = τ (w), under the hypothesis that similar128

specimens are considered (α0 is constant) equation (12) can be solved for129

w = w (∆a). Substituting w = w (∆a) in the first (11) yields the entire R-130

curve. In this case the steady state value can be obtained simply taking the131

limit, for w → +∞, of R in the first of (11):132

Rss = limw→∞

R (∆a) =1

Ew τ 2 ψ2

0 (13)

where ψ0 = ψ (α = α0).133

2.2 Elasto-plastic materials134

The method proposed in the previous section is rigorous for an elastic body135

and it may be used when the plastic region is negligible when compared to the136

characteristic crack length. This is not the case when dealing with cross ply137

laminates loaded in shear where the material behaviour is non–linear. There-138

fore, the R-curve obtained using the method proposed in the previous section139

may substantially differ from the real one. However, this discrepancy should140

affect only the rising part of the R-curve whilst it should not influence its141

plateau. In fact, the steady–state value of the R-curve, Rss, is obtained for an142

infinitely large specimen (see equation (13)) for which LEFM is valid. Sup-143

pose, for example, that the material exhibits a Ramberg–Osgood relationship144

6

between stress and strain in shear:145

γ =τ

G+m

τ0G

(

τ

τ0

)n

(14)

where G, τ0, γ, m, and n, are the shear modulus, the yield stress, the shear146

strain, and the two constants of the Ramberg–Osgood equation, respectively.147

Under linear elastic conditions, the J-integral equals the energy release rate:148

Je =1

Ew τ 2 ψ2 (15)

For a fully plastic cracked body the J-integral reads [17]:149

Jp = mγ0 τ0w g (α) h (α, n)(

τ

τ0

)n+1

(16)

where γ0 is the yield strain of the material, and g and h are two correction150

factors. For an elasto-plastic cracked body the J-integral is normally computed151

superposing the elastic and the fully plastic solutions [18–20]:152

J =1

Ew τ 2 ψ2 (α + αcorr) +mγ0 τ0w g (α) h (α)

(

τ

τ0

)n+1

(17)

where αcorr is the correction for strain hardening. At the peak stress, τu, the153

elasto-plastic J-integral equals the material fracture toughness:154

R =1

Ew τ 2u ψ

2 (α + αcorr) +mγ0 τ0w g (α) h (α)(

τuτ0

)n+1

(18)

It is straightforward to calculate the steady state value of Rss as the limit of155

(18) for w → +∞. Taking into account that the correction factor αcorr = rp/w,156

with rp = rp (n, τ0) being Irwin’s correction, it follows that limw→+∞

αcorr = 0.157

Also, the limit of the second term in (18), Jp, is equal to zero. Without loss158

of generality, suppose that the size effect law can be fitted using a linear159

regression II fit (see Table 1):160

1

w τ 2u= A

1

w+ C (19)

The limit of Jp, when w → +∞, reads:161

7

limw→+∞

Jp = limw→+∞

mγ0 τ−n0 g (α) h (α, n)

w(

A+ Cw)

n+1

2

= 0 (20)

The previous limit is equal to zero because n > 1. Therefore the steady state162

value of the R-curve, Rss, computed as the limit, for w → +∞, of the elasto-163

plastic J-integral is exactly that computed in equation (13) using LEFM.164

3 Numerical Model165

The dimensionless functions φ and ψ are calculated for the problem under166

analysis, using the Finite Element Method (FEM). For this purpose a para-167

metric Finite Element model was created using Python [21] together with the168

Abaqus 6.8-3 Finite Element code [22].169

The characteristic distance w is taken as constant and equal to the unity,170

while the variables are: the shape parameters or, in other words, the crack171

length 0 < α < 1 and the notch length 0.2 < β < 1; and the dimensionless172

parameter ρ that takes into account the effect of the orthotropy of the material173

(0 ≤ ρ ≤ 20). Figure 4(a) shows the mesh of the finite element model while174

figures 4(b)–4(c) show the deformed shape overall the specimen and, in detail,175

in the crack.176

[Fig. 4 about here.]177

The finite element used is the 4-node quadratic, reduced integration element,178

CPS4R. The Virtual Crack Closure Technique (VCCT) [23] is used to compute179

the energy release rates, GI and GII . For this purpose, a Python script was180

written to be used into Abaqus. Both energy release rates are calculated and181

compared to ensure that the crack propagation occurs in mode II. In addition,182

the total energy release rate computed was compared with that obtained from183

the Abaqus built-in method to calculate the J-Integral. This redundant mea-184

sure was used to validate the Python script. Displacements are applied on the185

model and frictionless contact is assumed. The applied load P is calculated186

summing the forces at the reaction nodes.187

Using the results obtained in the FE analysis the correction factor φ is ap-188

proximated by the following polynomial:189

φ =α

1− α

∑

i

∑

j

∑

k

Φijk αi−1 ρj−1 βk−1 (21)

8

where Φijk is the element of the matrix Φ with indexes i, j, and k. The matrix190

Φ is shown in Table 2 .191

[Table 2 about here.]192

The fitting surface φ is shown, together with the numerical points, in Figure 5.193

[Fig. 5 about here.]194

The maximum error of equation (21) in fitting the numerical points is 5%,195

with an average error of 2%. Function ψ can trivially obtained posing β = 1196

in equation (21). However, in order to reduce the error a new fit was calculated.197

The correction factor ψ is computed using the following polynomial:198

ψ =α

1− α

∑

i

∑

j

Ψij αi−1 ρj−1 (22)

where Ψij is the element, with the indexes i and j of the matrix Ψ defined as:199

Ψ =

−0.03644 0.22363 −0.00437 2.21749E−5

5.23616 1.44575 −0.16601 0.00449

−7.39060 −6.89756 0.63003 −0.01629

1.55840 9.24044 −0.79486 0.02025

0.84958 −4.05275 0.33844 −0.00855

(23)

The average error obtained using the expression of equation (22) is 1% while200

the maximum error is less than 2%. Figure 6 shows the comparison between the201

fitting surface ψ and the numerical points obtained using the aforementioned202

finite element model.203

[Fig. 6 about here.]204

4 Experiments205

IM7/8552 16-ply cross-ply ([90/0]8s) laminate with a nominal thickness of 4206

mm is used here. The nominal ply thickness is 0.125 mm, and the details207

of the manufacturing process given in [11]. The pre-crack was machined in a208

CNC machine equipped with a 1 mm drill bit. The Young’s modulus (along209

the orthotropic axes of the material) and the dimensionless parameter ρ of the210

cross-ply are E = 90649MPa and ρ = 8.5, respectively.211

9

The tests were carried out on a universal testing machine with a displace-212

ment rate of 2mm/min. The load was measured with a load cell of 100kN.213

A Iosipescu shear fixture was used to transfer the vertical movement of the214

cross-head of the testing machine in predominant shear at the specimen cen-215

tral section [24]. In particular, this set-up was used to apply mode II loading216

to the modified Iosipescu specimen. A torque wrench was used to tighten the217

sliding wedges of the fixture to a torque of 1Nm. Specimens were centered218

within the fixed and movable parts of the fixture.219

The fracture tests were coupled with digital image correlation (DIC). The220

photo-mechanical set-up is shown in Figure 7. In this work, the ARAMIS DIC-221

2D system by GOM was used [25,26]. A 8-bit Charged Coupled Device (CCD)222

Baumer Optronic FWX20 camera coupled with an Opto-Engineering telecen-223

tric lens TC 23 36 was selected (Table 3). With this type of lens, the mag-224

nification of the optical system can be kept constant over a defined working-225

distance range. The measurement are therefore less sensitive to any parasitic226

out-of-plane movement of the specimen. The gauge section at the specimen227

centre was painted using an airbrush in order to guarantee the refired speckled228

surface for DIC measurements. The illumination and shutter time were then229

adjusted in order to enhance the contrast of the target image. Based on rigid-230

body translation tests carried out systematically before the fracture tests [26],231

an estimation of the displacement and strain resolution was obtained in the232

range of 2×10−2 pixel (0.36 µm) and 0.02-0.04%, respectively.233

[Table 3 about here.]234

[Fig. 7 about here.]235

Six different geometries were tested, indicated with labels from A to F, cor-236

responding to a variation of β from 0.5 to 1.0. Table 4 shows the relevant ge-237

ometrical parameters together with the experimental results obtained. Three238

specimens were tested for each geometry. The dimensionless parameter α0 was239

taken as α0 = 0.5.240

[Table 4 about here.]241

Figure 8 shows the specimen after testing. The specimens failed by mode II242

crack propagation.243

[Fig. 8 about here.]244

The digital image correlation is used to verify the validity of the tests per-245

formed. Figure 9 shows the contour plot of the shear strain, γ12. Any misalign-246

ment in the load condition are checked in this way to ensure a predominant247

mode II crack propagation.248

10

[Fig. 9 about here.]249

Fitting the experimental results shown in Table 4, the law τu (β) is obtained.250

It should be noted that τu (β) is not, strictly speaking, a size effect law because251

it is not obtained scaling all the specimen’s dimension. This law is valid for252

the specimen shown in Figure 1 (with the width W=20 mm). Nevertheless,253

as we showed in the previous section, this specific size effect law can be used254

to obtain part of the R-curve. Due to the fact that τu (β) is not a size effect255

law, no particular caution should be taken in choosing the fitting function to256

use. However, it should be ensured that the strength decreases monotonically257

with β. In other words, it is not necessary to use the fitting formulas suggested258

in Table 1 because some characteristic of those formula are not required (for259

example, to provide a finite value of the strength when the characteristic size260

tends to infinity). The best fitting of the experimental data reported in Table261

4 was obtained with the linear regression II fit:262

1

βτ 2u= Aβ

1

β+ Cβ (24)

The best fitting (R2=0.91) is obtained for Aβ = 6.666 × 10−5MPa−2 and263

Cβ = 4.084 × 10−5MPa−2. The experimental results and the fitted curve are264

shown in Figure 10.265

[Fig. 10 about here.]266

Having obtained the law τu (β), part of the R-curve can be calculated solving267

the systems of equations (8)–(9) as shown in Figure 11.268

[Fig. 11 about here.]269

To obtain the entire R-curve the actual size effect law should be estimated as270

explained in the previous section (see Figure 3).271

A very simple algorithm was written in Matlab for this purpose allowing the272

definition of the points shown in Figure 12. The numerical points obtained273

were fitted again (together with the experimental point for β = 1.0 because274

this point belongs to this size effect law), using the fitting formulas reported275

in Table 4. The best fitting (R2=0.99) was obtained with the linear regression276

II:277

1

wτ 2u= A

1

w+ C (25)

where A = 6.964× 10−5MPa−2 and C = 3.46× 10−6 MPa−2mm−1. The corre-278

11

sponding points and the fitted curve are shown in Figure 12.279

[Fig. 12 about here.]280

Using the size-effect law obtained is possible to extrapolate the entire R-curve281

solving the equations (12) and (11).282

[Fig. 13 about here.]283

The steady state value of the R-curve and the length of fracture process zone284

are calculated as [15]:285

Rss =ψ20

E

1

C= 17.92 kJ/m2

lfpz =ψ0

2 ψ0

A

C= 4.95 mm

(26)

where ψ0 = ψ|α=α0and ψ0 = (∂ψ/∂α)|α=α0

. As previously mentioned, the286

fracture toughness of the 0◦ ply is obtained as twice the fracture toughness of287

the cross ply laminate, R0 (∆a) = 2R (∆a) . Therefore the steady state value288

of the ply fracture toughness is R0ss = 34.4 kJ/m2.289

It is worth to find a simple analytical expression of the R-curve to use in290

strength prediction models. A formula generally used is [11,12,15]:291

R = Rss (1− (1−∆a/lfpz)κ) if 0 ≤ ∆a ≤ lfpz < 0

R = Rss if ∆a ≥ lfpz(27)

For the material system used here the best fit (R2=0.997) is κ = 2.36.292

5 Conclusion293

The main conclusions of this work can be summarized as follows:294

• A new cracked Iosipescu specimens was proposed to measure the intralami-295

nar fracture toughness and R-curve in mode II of fibre reinforced composites.296

• In the impossibility of scaling the specimen a different formulation of the297

classical size effect method was proposed. The new formulation enables the298

measurement of the part of the R-curve that can be calculated once that299

τu = τu (β) is known. Differently from the classical size effect method β is300

12

not a characteristic size but a dimensionless parameter that depends on the301

shape of the specimen.302

• The entire R-curve can be calculated using the classical size effect method303

after determining the size effect law τu = τu (w). This is calculated mapping304

the known portion of the R-curve.305

• The ply steady state value of the fracture toughness for the material system306

investigated was found to be R0ss = 34.4 kJ/m2 and the corresponding307

length of the fracture process zone is 4.95mm.308

The drawbacks of the proposed methods are (i) that the rising part of the309

R–curve may substantially differs from the real one for cross ply laminates310

because the plastic behaviour of the material is ignored, and (ii) the maximum311

crack propagation obtained is limited by the gauge section of the specimen,312

therefore, for material that exhibit very large length of fracture process zone,313

the size effect obtained using this methodology may not be reliable. In this314

case the specimen should be scaled and a different test method should be315

proposed.316

Acknowledgements317

The first and second author would like to acknowledge the support of the318

Portuguese Foundation for Science and Technology under the grants FCT-319

DFRH-SFRH-BPD-78104-2011 and the Ciencia2008 program, respectively.320

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15

List of Figures394

1 Specimen’s geometry (dimensions in mm). 17395

2 Geometry of the virtual specimen. 18396

3 R-curve and mapping procedure. 19397

4 Finite element model of the specimen (α = 0.5, β = 0.5, and398

ρ = 20). 20399

5 φ for different values of α, ρ, and β: fitting surface and400

numerical points. 21401

6 ψ for different values of α and ρ: fitting surface and numerical402

points. 22403

7 Photo–mechanical set-up. 23404

8 Specimens after testing. 24405

9 Field of the shear strains γ12 obtained using Digital Image406

Correlation. 25407

10 τu (β) law, fitting and experiments. 26408

11 Portion of the R-curve obtain solving (8) and (9). 27409

12 Size effect law. 28410

13 R-curve of the investigated laminate. 29411

16

(a) 0 < β < 1

(b) β = 1

Fig. 1. Specimen’s geometry (dimensions in mm).

17

Fig. 2. Geometry of the virtual specimen.

18

Fig. 3. R-curve and mapping procedure.

19

Z

Y

X

(a) mesh

(b) deformed shape and τ12 contour plot

crack tipscracks

(c) detail of the deformed shape

Fig. 4. Finite element model of the specimen (α = 0.5, β = 0.5, and ρ = 20).

20

00.1

0.20.3

0.40.5

0.60.7

0.80.9

0.20.3

0.40.5

0.60.7

0.80.9

1−1

0

1

2

3

4

α [–]β [–]

φ[–

]

ρ = 0

ρ = 20

Fig. 5. φ for different values of α, ρ, and β: fitting surface and numerical points.

21

00.1

0.20.3

0.40.5

0.60.7

0.80.9

0

5

10

15

200

0.5

1

1.5

2

2.5

3

3.5

4

α [–]ρ [–]

ψ[–

]

Fig. 6. ψ for different values of α and ρ: fitting surface and numerical points.

22

Fig. 7. Photo–mechanical set-up.

23

Fig. 8. Specimens after testing.

24

Fig. 9. Field of the shear strains γ12 obtained using Digital Image Correlation.

25

1 1.2 1.4 1.6 1.8 21

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9x 10

−4

1/(βτ

2 u)

[MP

a−

2]

1/β [–]

experimentsfitted curve

Fig. 10. τu (β) law, fitting and experiments.

26

0 0.5 1 1.5 20

2

4

6

8

10

12

∆a [mm]

R[k

J/m

2]

crack driving force curvesR−curve

Fig. 11. Portion of the R-curve obtain solving (8) and (9).

27

0.1 0.15 0.2 0.25 0.3 0.351

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

−5

1/w [mm−1]

1/(wτ u

)[M

Pa−

2m

m−

1]

mappingexperimentsfitted curve

Fig. 12. Size effect law.

28

0 2 4 6 8 100

5

10

15

20

25

30

∆a [mm]

R[k

J/m

2]

crack driving force curvesR−curve (experiments)R−curve (mapping)

Rss=17.2kJ/m2

lfpz = 4.95mm

Fig. 13. R-curve of the investigated laminate.

29

List of Tables412

1 Size effect law fits [15]. 31413

2 Elements of Φ matrix. 32414

3 Optical system components and measurement parameters. 33415

4 Experimental results. 34416

30

Table 1Size effect law fits [15].

Regressions fit Formula Fitting parameters Rss lfpz

Bilogarithmic ln σu = lnM√N + w

M , Nφ20

EM2 φ0

2 φ0

N

Linear regression I1

σ2u

= Aw + C A, Cφ20

E

1

A

φ0

2 φ0

C

A

Linear regression II1

wσ2u

= A1

w+ C A, C

φ20

E

1

C

φ0

2 φ0

A

C

31

Table 2Elements of Φ matrix.

k

(i, j) 1 2 3 4 5

(1,1) 1.794 -1.308 0.1716 -0.01097 0.0002523

(1,2) -10.43 13.21 -1.36 0.0792 -0.001765

(1,3) 25.35 -32.6 2.888 -0.1572 0.003473

(1,4) -16.95 38.9 -3.18 0.1614 -0.003451

(1,5) 0.1882 -17.92 1.474 -0.07248 0.001496

(2,1) -0.3254 6.094 -0.6752 0.04665 -0.001181

(2,2) 34.16 -36.52 1.377 -0.08447 0.002929

(2,3) -38.05 64.97 3.783 -0.2963 0.003905

(2,4) -5.217 -85.65 -5.751 0.5185 -0.009382

(2,5) 13.79 53.24 0.8606 -0.1592 0.003198

(3,1) -2.122 -11.88 1.238 -0.0982 0.002733

(3,2) -55.28 34.68 3.842 -0.1465 -0.001238

(3,3) -5.196 -2.537 -30.47 1.794 -0.02811

(3,4) 63.33 34.79 38.99 -2.568 0.04627

(3,5) -4.591 -64.79 -12.11 0.93 -0.01781

(4,1) -0.1686 12.14 -1.385 0.1204 -0.003441

(4,2) 60.16 -18.51 -5.287 0.1265 0.003617

(4,3) -3.963 -45.88 35.77 -1.841 0.024

(4,4) -5.052 14.51 -45.77 2.713 -0.04409

(4,5) -54.11 50.67 14.8 -1.009 0.01765

(5,1) 1.106 -5.201 0.6829 -0.06038 0.001703

(5,2) -29.87 8.16 1.231 0.04213 -0.003985

(5,3) 25.31 13.52 -11.51 0.4595 -0.00217

(5,4) -40.86 0.4896 15.21 -0.782 0.009515

(5,5) 47.24 -22.61 -4.832 0.2949 -0.004118

32

Table 3Optical system components and measurement parameters.

Camera-lens optical system

CCD camera Baumer Optronic FWX20

8 bit, 1624×1236 pixels

lens TC 23 36 Telecentric lens

Magnification: 0.243 ± 3 %

Field Of View: 29.31 × 22.1 mm

Working Distance: 103.5 ± 3 mm

Working F-number: 8

Image recording

Acquisition frequency 1 Hz

Exposure time 0.7 ms

Project parameter – Subset

Subset size 15×15 pixels2 (0.270×0.270 mm2)

Subset step 13×13 pixels2 (0.234×0.234 mm2)

Project parameter – Strain

Strain length 5×5 subsets2 (1.17×1.17 mm2)

Resolution

Displacement 2×10−2 pixel (0.36 µm)

Strain 0.02-0.04%

33

Table 4Experimental results.

specimen β [–] w [mm] τ [MPa] SD [MPa] IC 95% [MPa]

A 0.5 5 107.0 3.0 3.3

B 0.6 6 104.3 4.9 5.5

C 0.7 7 103.3 2.3 2.6

D 0.8 8 102.3 1.1 1.3

E 0.9 9 98.0 1.0 1.1

F 1.0 10 95.0 2.6 2.9

34