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Composites: Part B 66 (2014) 388–399

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Composites: Part B

journal homepage: www.elsevier .com/locate /composi tesb

A DIC-based study of in-plane mechanical response and fractureof orthotropic carbon fiber reinforced composite

http://dx.doi.org/10.1016/j.compositesb.2014.05.0221359-8368/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 (803) 777 2502; fax: +1 (803) 777 0106.E-mail address: [email protected] (A. Kidane).

Behrad Koohbor, Silas Mallon, Addis Kidane ⇑, Michael A. SuttonDepartment of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 February 2014Received in revised form 9 May 2014Accepted 16 May 2014Available online 5 June 2014

Keywords:A. Carbon fiberB. FractureB. Mechanical propertiesD. Mechanical testingDigital image correlation (DIC)

The in-plane elastic properties and quasi-static fracture response of an orthotropically woven carbonfiber reinforced composite are investigated using 3D digital image correlation. The elastic propertiesare determined as a function of fiber orientation, and the effect of fiber angles on the crack extensiondirection is investigated. The modified maximum hoop stress criterion is used to predict the crack exten-sion angle of the examined material. The predicted angles are in very good agreement with those foundexperimentally. Optical observations and local strain data from the quasi-static fracture studies revealthat crack initiation occurs well before the maximum far-field load is reached.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The in-plane elastic properties and quasi-static fractureresponse of an orthotropically woven, carbon fiber reinforced com-posite are investigated using 3D digital image correlation. Of par-ticular interest in this study are graphite/epoxy composites,which are especially suitable for many applications because oftheir exceptional thermo-mechanical attributes [1]. While thismaterial may possess many desirable characteristics, a completeset of mechanical properties, including fracture parameters, is req-uisite for proper engineering use.

Failure and damage as a result of fracture is a major concernwith the use of any composite material provoking substantialresearch in the area [2]. Crack propagation along the laminar inter-faces, including the effect of strain rate, has been most thoroughlystudied [3–6]. In-plane fracture in composite has been also docu-mented in the literature [7–11]. Most early experimental workrelied on conventional material testing techniques to obtainmechanical and fracture properties of composites, point measure-ments, load data and specimen geometry. These experimentaltechniques become more robust with the advent of the full fieldmeasurement technique, digital image correlation (DIC). Mostrecently Pollock et al. [12] utilized such full-field measurementsto extract elastic properties of woven glass/epoxy composites

through tensile loading of specimens extracted at different anglesrelative to the primary fiber direction.

The advent of DIC has also proven beneficial to the study of frac-ture mechanics, as seen in pioneering work by McNeill et al. [13],in which DIC data is used to extract stress intensity factors. Patakyet al. [14] utilized full-field measurements in conjunction with aleast squares algorithm to investigate mixed mode stress intensityfactors in an anisotropic, single crystal stainless steel. Such workdemonstrates the applicability and accuracy that full-field mea-surements offer to fracture studies.

However, the examination of criterion that predicts the directionof crack propagation in woven composites was missing from previ-ous work. Early work by Erdogan and Sih [15] that considered crackpropagation plane loading of a brittle material was one of the first todescribe such a methodology, suggesting that the crack will initiatealong a path perpendicular to the maximum tensile direction at thecrack tip. Kidane et al. [16,17] extended the method by employingthe minimum energy density criteria to study crack propagationin functionally graded materials subjected to thermo-mechanicalloading. Saouma et al. [18] extended the maximum circumferentialtensile stress theory presented by Erdogan and Sih [15] to predictthe direction of crack propagation in a homogeneous, anisotropicspecimen subjected to mixed mode loading, documenting the sub-stantial influence that material anisotropy has on the direction ofcrack propagation. Most recently, Cahill et al. [9] presented theuse of the extended finite element method (XFEM) in conjunctionwith a modified maximum hoop stress criterion to describe thedirection of crack propagation in a unidirectional fiber reinforced

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1

2 xy

= 90o

= 75o

= 30o

= 45o

= 60o

= 15o

= 0o

Fig. 2. Fiber orientation of the tensile specimens extracted from the wovencomposite sheet. The definition for the fiber orientation angle of the tensilespecimens, h, relative to the original sheet axes is shown.

B. Koohbor et al. / Composites: Part B 66 (2014) 388–399 389

composite that has a high degree of anisotropy. Although this modelappears applicable in unidirectional composites that exhibitextreme degrees of anisotropy, the use of this criterion has not beendocumented in woven composites, which can often exhibit nearlyorthotropic properties.

The present study focuses on the development of a general androbust methodology for extraction of mechanical properties andfracture parameters for orthogonally woven composites usingfull-field measurements. To our knowledge, this is the first detailedstudy of woven composites where the elastic properties, stressintensity factor and crack direction are investigated as a functionof fiber orientation using full field measurement technique. Theelastic properties of the examined composite are first determinedexperimentally with the help of 3D DIC, based on which the elasticstress fields are extracted. Stress fields extracted from fractureexperiments, in conjunction with the maximum modified hoopstress criterion, are then utilized to predict the angle of crack prop-agation as a function of fiber orientation angle.

2. Experimental procedure

2.1. Material and specimen geometry

The material examined in the present work is a three-layerorthogonally woven carbon fiber-reinforced epoxy resin matrixcomposite with a nominal density of 1384 kg/m3. The plain weavestructure of the laminate consisted of two mutually orthogonaldirections (warp and weft) with an approximate volume fractionof 70% for the reinforcing carbon fibers. The weave structure canbe seen in a magnified image of the material surface in Fig. 1. Rect-angular 175 � 25 (mm2) coupon specimens were cut from sheets of0.8 mm thick material using a CNC water jet. Specimens wereextracted from the plate at fiber orientation angles ranging from0� to 90� in 15� increments. The schematic orientation of such spec-imens is given in Fig. 2.

2.2. Mechanical testing

To obtain mechanical properties, tensile tests were performed onrectangular coupon specimens. Loading was applied monotonicallyuntil failure using displacement control at a rate of 1.5 mm/min.Loading rate was chosen such that sufficient number of data pointsand images could be collected, at a sampling rate of 2 Hz, to charac-terize each specimen, while the rate is sufficiently low to be consid-ered quasi-static. Tests were conducted on an MTS 810 tensile loadframe fitted with hydraulic grips. Specimens were fastened into themachine with approximately 30 mm in length held by each of thehydraulic grips, with 6 mm thick aluminum tabs placed on either

1mm

Fig. 1. Magnified image of the as-manufactured structure of the carbon-fiberreinforced composite used in this work. Fiber bundles oriented along 0� and 90�directions within the surface weave geometry are evident. The yarn dimensionwithin the material is approximately 2 � 2 (mm2).

side of the specimen thickness within the grips to protect againstdamage from the serrated, hardened steel grips. Load and displace-ment data were acquired from the load cell and displacement sen-sors, while stereovision was simultaneously used to acquire pairsof images during the loading process from one side of the specimenthat had been painted to obtain a speckle pattern with random con-trast variations (see Fig. 3c).

2.3. Quasi-static fracture testing

Quasi-static experiments were performed on initially flawedspecimens using an MTS 810 tensile frame. The geometry of thefracture specimens is shown in Fig. 3. The 160 � 50 (mm2) single-edge notched specimens were extracted from 0.8 mm thick sheetsusing a procedure similar to the aforementioned procedure. Again,fiber orientation angles (h) were varied from 0� to 90� in 15� incre-ments. An initial 25 mm long notch was cut into each specimenusing the CNC water jet to produce an a/W ratio of 0.5. Prior to test-ing, the notch on each specimen was struck with a razor blade toeffectively produce a sharp crack; approximate crack length was26 mm in all experiments. To perform each fracture experiment,specimens were fastened into the hydraulic grips utilizing alumi-num tabs to protect the roughly 30 mm length portion of the spec-imen held in each grip. Loading was applied monotonically at aloading rate of 0.72 mm/min until failure. It is noted that the loadingrate for each fracture test is kept lower than the rate used for theunflawed sample test so that enough data and images are acquiredat 2 Hz sampling rate, as the test duration for fracture test is shorterthan the unflawed tensile test. Synchronized load, axial displace-ment and stereo images were collected at 2 Hz. Each experimentwas tested at least twice to ensure the results obtained duringmechanical and fracture tests are repeatable.

2.4. Digital image correlation (DIC) method

During the entirety of the present experimental study, full-fieldstrain and displacement fields were obtained with the use of stereo-vision in conjunction with 3D DIC. To facilitate use of the DICmethod, the surface of each specimen was first coated with a thinlayer of white paint. A fine, high contrast speckle pattern of blackpaint was then sprayed on the white surface. A typical speckle pat-tern applied on the specimen’s surface, along with its gray scale his-togram, is shown in Fig. 3. Specimen response during tensile andfracture experiments was observed using a pair of Point Grey Grass-hopper 5 megapixel cameras fitted with 35 mm lenses, operating ata resolution of 2448 � 2048 pixels. External lights sources wereused to promote uniform illumination of the specimen in the regionof interest.

Fig. 3. Illustration of fracture specimen (a), with a close-up details of the crack-tip vicinity (b). (x–y is the basic testing coordinate system, where x is parallel to the initialnotch axis and y is parallel to loading axis. 1–2 is the fibers coordinate system, where 1 is parallel to 0� fibers and 2 is parallel to the 90� fibers.) A typical speckle pattern (c)with its relative gray-scale histogram (d) is also shown.

390 B. Koohbor et al. / Composites: Part B 66 (2014) 388–399

System parameters obtained from calibration for the stereo cam-era arrangements used in both the mechanical testing and the frac-ture testing are shown in Table 1. The camera 1 and camera 2coordinate systems, as well as the camera orientation angles a, b,and c are also depicted, with the Z-axis nominally along the opticalaxis of each camera. From the calibration data, stereo angles (bs) forthe mechanical and fracture testing were 21.8� and 36.5�, respec-tively. Images collected were analyzed using commercial softwareVic-3D (VIC-3D website: www.correlatedsolutions.com) with sub-set and step sizes of 25 and 3 pixels, respectively.

Table 1System parameters obtained from calibration of the stereo camera arrangements usedin the mechanical and the fracture testing.

Parameter Mechanical testing Fracture testing

Camera 1 Camera 2 Camera 1 Camera 2

Center (x) (pixels) 1238 1223 1237 1187Center (y) (pixels) 1038 1073 1022 1016Focal length (x) (pixels) 5174 5158 5198 5152Focal length (y) (pixels) 5174 5157 5198 5152Skew (�) 0.796 �0.913 0.524 1.105

Alpha (�) 0.354 �0.735Beta (�) 21.856 36.47Gamma (�) 0.364 �0.307Tx (mm) �146.8 �185.45Ty (mm) �2.169 0.181Tz (mm) 29.08 68.63

3. Data analysis

3.1. Elastic properties

Elastic stress-strain relation of an orthotropic lamina underplane-stress conditions can be written in matrix form as [12]:

exxeyyexy

264

375 ¼

s11ð¼ 1=E1Þ s12ð¼ �t12=E1Þ 0s21ð¼ �t12=E1Þ s22ð¼ 1=E2Þ 0

0 0 s66ð¼ 1=G12Þ

264

375

rxxryyrxy

264

375ð1Þ

where eij and rij are the components of the strain and stress tensors,respectively. In order to determine the in-plane mechanical proper-ties of the woven composite as a function of fiber orientation, off-axis tensile specimens were tested. The tensile load, however, wasapplied along the specimen axis, making an angle h with the princi-pal horizontal fiber orientation (see Fig. 2).

The stress components along the material axes 1 and 2 can bewritten in terms of the axial stress, rh, and the fiber orientationangle, h, as:

r11 ¼ rh cos2 h r22 ¼ rh sin2 h r12 ¼ �rh sin h cos h ð2Þ

Using the quadratic elasticity potential function and the stress com-ponents shown in Eq. (2), the stored energy, w, can be written as afunction of axial stress, in-plane elastic constants and the fiber ori-entation angle as [6,12]:

w ¼ 12r2h

1E1

cos4 hþ 1G12� 2t12

E1

� �sin2 h cos2 hþ 1

E2sin4 h

� �ð3Þ

where subscripts 1 and 2 in the present work denote the materialproperty at h = 0� and h = 90�, respectively. Using the definition ofaxial strain for a linear elastic material, eh can be written as:

http://www.correlatedsolutions.com

0

100

200

300

400

500

600

700

0 0.05 0.1 0.15 0.2

Stre

ss (

MP

a)

Strain

0153045607590

0

10

20

30

40

50

60

70

80

90

100

0 0.0005 0.001 0.0015 0.002

Stre

ss (

MP

a)

Strain

0

15

30

45

60

75

90

Fig. 4. Typical tensile stress–strain curves as a function of fiber orientation (a) and aclose-up view of the linear elastic region (b). A linear regression method was used


eh ¼@w@rh¼ rh

E1h2ðhÞ and Eh ¼

E1h2ðhÞ

ð4Þ

where

h2ðhÞ ¼ cos4 hþ d2d1

sin4 hþ d3d1

sin2 h cos2 h

d1 ¼1E1

d2 ¼1E2

d3 ¼1

G12� 2t12

E1

ð5Þ

By performing tensile experiments to obtain rh � eh curves in thelinear region for h = 0� and h = 90�, d1 and d2 can be determined,respectively. At this point, the only unknown parameter would bed3, which can be numerically assessed using a linear least squareoptimization process which involves the minimization of the fol-lowing metric [12]:

U ¼XMJ¼1

XNi¼1

eih�rih�� h

2ðh�; d1; d2; d3ÞE1

" #2

h2ðh�; d1; d2; d3Þ ¼ cos4 h� þd2d1

sin4 h� þ d3d1

sin2 h� cos2 h�ð6Þ

where M is the number of specimens tested at angle h⁄ and N denotesthe number of rh � eh data obtained for each specimen. From valuesof rh and eh measured at several load magnitudes for specimens withh⁄ = 15�, 30�, 45�, 60�, 75�, the value of the parameter d3 can beobtained from the optimization process. Additionally, the Poisson’sratio as a function of fiber angles, t(h), can be experimentallyobtained by finding the ratio of transverse to longitudinal strain com-ponents for each specimen. Once d3 and t(h) are obtained, shear mod-uli at different angles, G(h), of the material can be determined fromEq. (5). Note that the level of accuracy of the results depends on thenumber of specimens tested at each h⁄, as well as the number of datapoints obtained in the linear region of rh � eh curves. The proceduresthat are followed to obtain the mechanical properties can be summa-rized as follows:� Measure the linear portion of tensile r � e of a specimen loaded

along with fiber angle h = 0� and determine d1 according to Eq.(5).� Measure the linear portion of tensile r � e of a specimen loaded

along with fiber angle h = 90�, and determine d2 according to Eq.(5).� Determine the Poisson’s ratio from tensile experiments of spec-

imens with fiber angles are oriented ath = 0�, 15�, 30�, 45�, 60�, 75�, 90�.� Measure rh � eh values at several loads within the linear defor-

mation region from tensile experiments for specimens withfiber angles are oriented at h = 15�, 30�, 45�, 60�, 75�.� Since d1, d2 and rh � eh values at different angles are known use

Eq. (6) to determine the unknown d3 using a linear least squareoptimization process.� Once d3 is obtained, Eq. (5) can again be used to find the values

for shear moduli at different angles.

3.2. Crack-tip displacement fields

The in-plane Cartesian displacement components, u and v, inthe vicinity of stationary and propagating cracks in orthotropicmaterials can be written as [7,19]:

u ¼ KIffiffiffiffipp Ac66 �

l1a

s11 þ p2l3s12� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2 uþ 1p2

sin2 u

sþ cos u

vuut264

þ pl2l5aql6

s11 þpql4l5

l6s12

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 uþ 1

q2sin2 u

sþ cos u

vuut375 ffiffiffirp

ð7-aÞ

v ¼ KIffiffiffiffipp Ac66 �p

l1a

s12 þ p2l3s22� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2 uþ 1p2

sin2 u

s� cos u

vuut264

þ q pl2l5aql6

s12 þpql4l5

l6s22

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 uþ 1

q2sin2 u

s� cos u

vuut375 ffiffiffirp

ð7-bÞ

where KI is the stress intensity factor for Mode-I fracture, and sijdenotes the components of the material’s compliance matrix as pre-sented in Eq. (1). The results obtained from the tensile experimentswere used to determine the material’s compliance matrix. Quanti-ties r and u in Eq. (7) are the radial and angular coordinates of anarbitrary point. Expressions for p and q are defined as:

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 � a2

qr; q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 � a1

qrð8Þ

Where the quantity a1 is given as:

2a1 ¼ aþ a1 � 4b1b2; a2 ¼ aa1 ð9Þ

With

a ¼ c66c11ð1�M21Þ

a1 ¼c22

c66ð1�M22Þ2b1

¼ c12 þ c66c11ð1�M21Þ

2b2 ¼c12 þ c66

c66ð1�M22Þð10Þ

cij in Eq. (10) are the elements of the material’s stiffness matrix. Inthis case also, the cij matrix was obtained as the inverse of the mate-rial’s compliance matrix (sij), which was determined earlier basedon the elastic properties found from the tensile experiments. M1

to find the elastic moduli of the specimens in this region.

0

20

40

60

80

100

120

140

-0.001 0 0.001 0.002 0.003

Stre

ss (

MP

a)

Strain

Longitudinal strain

Transverse Strain

0

5

10

15

20

25

30

35

40

-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004

Stre

ss (

MP

a)

Strain

Longitudinal strain

Transverse Strain

(a)

(b)

Fig. 5. Longitudinal and transverse strain as a function of stress for tensilespecimens of (a) h = 0� and (b) h = 45�.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0 15 30 45 60 75 90

Measured data

Reported in Ref. [24]

(o)

Poi

sson

's r

atio

, v(

)

Fig. 6. Variation of apparent Poisson’s ratio, t(h), as a function of fiber orientationangle. The data points were measured in the present work, and the dotted lineshows the variation according to Ref. [24].

2.85

2.9

2.95

3

3.05

3.1

3.15

3.2

0 15 30 45 60 75 90

Shea

r M

odul

us, G

(G

Pa)

0

10

20

30

40

50

60

70

0 15 30 45 60 75 90

(a)

(b) Experimental Analytical (Eq. 4)

Reported in Ref. [24]

(o)

(o)

E(G

Pa)

Fig. 7. Variation of (a) apparent Shear Modulus and (b) apparent Elastic Modulus, asa function of fiber orientation angle h. The dotted line in (b) refers to the resultsreported in Ref. [24].

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2 2.5 3 3.5 4

Loa

d (N

)

Extension (mm)

0

15

30

45

60

75

90

Fig. 8. Far-field load–extension curves obtained for specimens with different fiberorientation.


and M2 are non-dimensional parameters, defined as the ratio of thecrack tip velocity to the longitudinal wave velocity, cl, and shearwave velocity, cs, of the materials as:

M1 ¼ccl

M2 ¼ccs; cl ¼

ffiffiffiffiffiffiffic11q

rcs ¼

ffiffiffiffiffiffiffic66q

rð11Þ

In the case of stationary and/or quasi-static crack growth, the cracktip velocity, c, is assumed to be zero. The parameters li(i = 1 . . . 6)and Ac66 in Eq. (7) are also written as:

(a)

1 mm 1 mm

(b)

Fig. 9. Crack bridging as a result of fiber pull-out and fiber cross-over at early stages of crack growth for specimens of (a) h = 45� and (b) h = 90�.


l1 ¼2b1p2

ða� p2Þð1�M21Þþ 2b� a l2 ¼

2b1q2

ða� q2Þð1�M21Þþ 2b1 � a

l3 ¼ 1�M22 �2b1

a� p3

l4 ¼ 1�M22 �2b1

a� q2 l5 ¼ �l3 �M22 l6 ¼ �l4 �M

22

Ac66 ¼l6

pql4l5 � p2l3l6ð12Þ

The displacement components at arbitrary radial and angularpositions in the crack tip vicinity were used to find the stressintensity factor, KI, based on an over-deterministic least squarealgorithm [20–22]. To avoid any possible error introduced by the

0

500

1000

1500

2000

2500

0 0.5 1 1.5

Loa

d (N

)

Exten

0

400

800

1200

1600

2000

0 0.5 1

Loa

d (N

)

Crack e

(a)

Crack initiates (Pinit)

Pmax = 1875 N

Pini

Fig. 10. (a) Crack tip location at different load magnitudes for the specimen with h = 45load magnitude for the same specimen. Here, Pinit refers to the magnitude of the load at

crack tip nonlinearity effects, data points were selected to be suf-ficiently far from the location of the crack tip. Regions with0:5 < rB

� �< 2, where B is the specimen thickness, were considered

as this criterion [17]. In addition, to increase the accuracy, a largenumber of data points (minimum 100) were used in the calcula-tion of KI for each specimen. The numerical analysis was per-formed using an in-house computer code written on theMATLAB™ platform.

3.3. Criterion for crack propagation direction

The modified maximum hoop stress criterion is used to predictthe direction of crack growth. The maximum hoop stress criterionstates that the crack will propagate perpendicular to the direction

2 2.5 3 3.5

sion (mm)

1.5 2 2.5 3

xtension (mm)

Maximum Load (Pmax)

(b)

t = 1606 N

�. The crack tip location is marked on each image. (b) Crack tip location vs. far-fieldwhich matrix cracking initiates.

0

0.005

0.01

0.015

0.02

0.025

0.03

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.4 0.8 1.2 1.6

Loa

d (N

)

Extension (mm)

1 mm

Point of Interest

Load Strain

0.012

0.014

0.016

0.018

0.6 0.65 0.7 0.75 0.8

yy

Extension (mm)

Crack growth

(a)

(b)

Ope

ning

str

ain,

yy

Fig. 11. (a) Load and strain history for the h = 45� specimen. The onset of crackinitiation is marked with the dashed line. The strain curve was extracted for a pointlocated along the line of the original crack, 1 mm ahead of the crack tip, asschematically shown in (b).

0

10

20

30

40

50

60

70

80

90

100

0 15 30 45 60 75 90

0 15 30 45 60 75 90K

I(M

Pa.

m0.

5 )

DIC dataLoad data - correctedLoad data - uncorrected

0

10

20

30

40

50

60

70

80

90

100

0

500

1000

1500

2000

2500

3000

3500

4000

4500

% d

iffe

renc

e

Loa

d (N

)

Corrected LoadUncorrected Load% difference

(a)

(b)

(o)

(o)

Fig. 12. (a) Variation of Mode-I stress intensity factor, KI, with fiber orientation. (b)The fracture loads (corrected and uncorrected) and their difference as a function offiber orientation.


in which the hoop/circumferential stress is maximum [23]. Themodified criterion, as proposed by Sauma et al. [18] takes intoaccount the angular variation of the fracture toughness of thematerial, making it effective for orthotropic materials consideredin this work.

The modified hoop stress for orthotropic material can beexpressed as [18]:

rmodhoop ¼rx sin2 uþ ry cos2 u� rxy sin 2u

E2E1

cos2ðu� hmaterialÞ þ sin2ðu� hmaterialÞð13Þ

where hmaterial denotes the fiber orientation angle with respect toperpendicular to the loading direction. E1 and E2 denote the mate-rial’s elastic moduli along its principal fiber directions, obtainedfrom the tensile experiments in the present work. Note that, thestress components rxx, ryy and rxy are functions of angular position,u, and the fiber orientation angle, hmaterial, [19].

4. Results and discussion

4.1. Elastic properties

Following the experimental procedure described earlier, theelastic properties of the examined materials were determined asa function of fiber orientation angles. Typical stress–strain curvesfor the composite as a function of fiber orientations angle areshown in Fig. 4. The elastic portion of the stress–strain relation

has been magnified and shown in Fig. 4. As clearly seen in Fig. 4,the stress–strain relations are consistent with previous studies[12] and show different trends for different fiber orientations. Forthe case of 0� and 90� fiber orientations, the failure was abruptwith small strain and high failure stress; no progressive failurewas observed. On the other hand, failure in the case of off-axis fiberorientation is progressive, leading to a relatively large strain beforecomplete failure. This is expected since the fibers are in the princi-pal loading direction and are the primary load carrying members.Since the matrix has little or no contribution to the tensile strengthfor 0� and 90� specimens, the stress strain results exhibit a mono-tonic linear response. In the case of off-axis specimens, the matrixtakes the load initially, as none of the fibers are aligned with theloading direction. As the loading process progresses, the fibersbegin sharing the load and a non-linear stress-strain response withprogressive damage occurs. A similar response was observed byPollock et al. [12] for an orthotropic woven glass–epoxy composite.

From the full-field displacement data obtained using 3D DIC,longitudinal and transverse strain components were also deter-mined for each tensile specimen. Fig. 5 depicts the longitudinaland transverse strain components for specimens of h = 0� andh = 45�. The ratio of transverse to longitudinal strains increases sig-nificantly as fiber orientation angle increases from 0� to 45�, indi-cating that the apparent Poisson’s ratio will vary substantially asa function of fiber orientation relative to loading axis. The experi-mentally determined values of the apparent Poisson’s ratio exhib-ited by the material are plotted with respect to the fiberorientation angle in Fig. 6. The values obtained here are in a goodagreement with literature data [24].

Once the elastic moduli and the apparent Poisson’s ratio of thecomposite specimens at different fiber orientation angles areobtained, the method described earlier was used to extract the

(a) (b)

Fig. 13. Stress contours for specimens of (a) h = 45� and (b) h = 90�. The original crack tip is located at (0, 0) and the stress values are in MPa. Dimensions are also in mm.


material’s shear modulus. Using a least-square approach and Eq.(6), the values for di’s in Eq. (5) are calculated as:,

d1 ¼ 0:02167 GPa�1 d2 ¼ 0:01733 GPa�1 d3 ¼ 0:31432 GPa�1

Then using d3 along with the other elastic constants found ear-lier, the material’s apparent in-plane shear modulus G(h) as a func-tion of fiber orientation angle h are obtained, as shown in Fig. 7a.

Having all the necessary parameters, the theoretical variation ofE(h) was calculated from Eq. (4) and the results were compared tothe experimental findings. Fig. 7b demonstrates the comparisonbetween the theoretical and experimental results. A numericalcomparison of the experimental and analytical results shows a rel-ative small difference, less than 3%, in all the cases, confirming thatthe theoretical formulation presented above has a high level ofaccuracy which was achieved in all of the results found in thiswork. Another point worth mentioning is that the material


properties of the specimens at 0� and 90� fiber orientation anglesare not the same, and as shown in Fig. 7 the elastic moduli arenot distributed symmetrically with the fiber orientation. A similarbehavior was reported in the work performed by Pollock et al. [12],with this phenomenon explained as being the result of a differentnumber of reinforcement fibers used along the 0� and 90�directions.

4.2. Stress intensity factor and crack propagation direction

Several typical far-field quasi-static loading histories for singleedge cracked specimens with different fiber orientations withrespect to the initial crack direction are shown in Fig. 8. A constantincrease in the load magnitude prior to fracture, followed by a sud-den drop, was observed for specimens with h = 0� and h = 90�, typ-ical of a brittle fracture response. By increasing the specimen fiberorientation angle from 0� to 45�, both the peak value and the load-ing slope were remarkably reduced, reaching minima for the spec-imen with h = 45�. The same trend was found when varying h from90� to 45� as well. The gradual crack growth observed for speci-mens with h approaching 45� is consistent with the failure modeobserved in unflawed samples in Section 4.1, indicating a changein failure mechanism for specimens with fiber orientation angledeviating from the loading axis. From a magnified image of atypical failure surface, shown in Fig. 9, it was observed that forspecimens with fibers aligned with the loading axis, the failure ispredominantly by fiber fracture. In the case of off-axis specimens,failure is associated with matrix cracking and fiber pullout. Fig. 9shows a typical failure mode during early stages of crack propaga-tion for two different cases with fiber angles h = 45� and h = 90�.

4.2.1. Stress intensity factorTo quantify the stress intensity factor, both the load and displace-

ment/strain fields from DIC measurements are considered. How-ever, the comparison made between the load data and the imagestaken during the loading process revealed that crack initiation takesplace well before the load magnitude reaches its peak value. A mag-nified view of the crack tip area at different load levels for a specimenhaving h = 45� is shown in Fig. 10a. The crack extension vs. far-fieldload was also plotted for this specimen in Fig. 10b. It is obvious fromthe images that the crack has already initiated some time before themaximum load is reached. To find the correct load magnitude asso-ciated with the material’s crack initiation, the history of the openingstrain component (eyy) was examined for a point located very nearto, and at an angleu = 0� relative to, the original notch tip. This strain

5 mm

(a) (

Fig. 14. Contours of the modified hoop stress for the specimens with (a) h = 60� and (b)along the angle of crack propagation.

component experiences a distinct jump in its value at the actualmoment of crack initiation, as depicted in Fig. 11. This type of strainresponse can be used to find the exact load magnitude at which thefracture initiates. After measuring the corrected crack initiation loadin this way for each specimen, the critical stress intensity factor as afunction of fiber orientation angle was calculated using load dataand specimen geometry. The stress intensity factor present in eachof the specimens was also calculated using the displacement fieldsobtained from DIC and the asymptotic displacement fields withthe over-deterministic approach explained earlier. The calculatedKI values obtained using the over-deterministic approach are com-pared to those found using specimen load and geometry inFig. 12a. As shown in Fig. 12a, the KI values determined using bothmethods exhibit similar trends and are in relatively good agreement.Noting that the KI value calculated from the corrected load is in bet-ter agreement with the value obtained from strain fields suggeststhe necessity for optical observation to determine the instant of frac-ture. By including such observations, the critical stress intensity fac-tor of such materials can be obtained. In cases where only far-fieldload is considered, the stress intensity factor of the material maybe substantially over-estimated (see uncorrected load data forinstant of crack propagation in Fig. 12a). Variation of the peak loadas obtained from the load cell (uncorrected fracture load) and thecorrected fracture load values at the instant of fracture are presentedin Fig. 12b for specimens with different fiber orientation angles. Aminimum of 16% difference was obtained between the correctedand uncorrected fracture load magnitudes.

4.2.2. Crack propagation directionIn order to predict the angle at which crack propagation takes

place, the modified hoop stress criterion described in Section 3.3is used. Using the full-field strain components obtained from DICalong with the elastic constants determined for the material, thefull-field stress components are calculated. These stress compo-nents are directly determined from Eq. (1) using known ei and sijvalues within the area of interest. Note that the elastic constantssij that are used in Eq. (1) are those obtained from mechanical test-ing, explained earlier in Section 3.1 and the strains ei are obtainedfrom DIC measurements. The distribution of the correspondingstress components rxx, ryy and rxy for two different specimensobtained through this process are depicted in Fig. 13. Using these

stress components, the modified hoop stress rmodhoop� �

field was

determined within the same region of interest on which the fullfield stress components were calculated. The distribution of mod-ified hoop stress in the crack tip neighborhood is shown in

5 mm

b)

h = 90�. Experimental results confirm that the maximum modified hoop stress lies


Fig. 14. The experimentally observed results showing the crackinitiation angle for all the specimens have been depicted inFig. 15. Inspection of Figs. 14 and 15 indicates that the crack pathfollows the direction in which the modified hoop stress is maxi-mized. To give more detailed insight into prediction of the crackpath, the material’s elastic constants, along with the stress formu-lae proposed for orthotropic materials [10,11,18], was used toanalytically determine the distribution of the modified hoop stressin the vicinity of the crack for specimens with different fiber orien-tation. In this case, the stress components, rx, ry and rxy in theneighborhood of the crack for Mode-I fracture are calculated as:

rx ¼KIffiffiffiffiffiffiffiffiffi2prp Re l1l2l1�l2

l2ðcosuþl2 sinuÞ

1=2�l1

ðcosuþl1 sinuÞ1=2

!" #

ð14-aÞ

Fig. 15. Experimentally observed direction of crack initiation for different specimens wi

ry ¼KIffiffiffiffiffiffiffiffiffi2prp Re 1l1�l2

l1ðcosuþl2 sinuÞ

1=2�l2


!" #

ð14-bÞ

rxy ¼KIffiffiffiffiffiffiffiffiffi2prp Re l1l2

l1�l21

ðcosuþl1 sinuÞ1=2�

1


!" #

ð14-cÞ

where l1 and l2 are the complex roots of the general character-istic equation:

s11l4 � 2s16l3 þ ð2s12 þ s66Þl2 � 2s26lþ s22 ¼ 0 ð15Þ

With sij denoting the components of the material’s compliancematrix (see Eq. (1)). For an orthotropic case similar to that consid-ered in this work, Eq. (15) reduces to the following form [9]:

th (a) h = 0�, (b) h = 15�, (c) h = 30�, (d) h = 45�, (e) h = 60�, (f) h = 75� and (g) h = 90�.

0

0.2

0.4

0.6

0.8

1

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90φ (o)

0 15 30

45 90

Normalized hoop stress

Fig. 16. Variation of normalized hoop stress, rmodhoop=ðK1=ffiffiffiffiffiffiffiffiffi2prp

Þ, as a function angularposition from the crack tip for some selected specimens.

0

10

20

30

40

50

0 15 30 45 60 75 90

Cra

ck in

itia

tion

ang

le (

o )

analytical prediction

experimentally obtained

Fiber Orientation (o)

Fig. 17. Comparison of analytical predictions and experimental results for crackinitiation angle as a function of fiber orientation.


s11l4 þ ð2s12 þ s66Þl2 þ s22 ¼ 0 ð16Þ

In which the constants, sij, are experimentally determined. Onceall the stress components are analytically determined in the cracktip vicinity, Eq. (13) can be utilized to obtain the distribution of themodified hoop stress within the region of interest. The variation ofnormalized modified maximum hoop stress as a function of angu-lar position around the crack tip is shown for specimens with dif-ferent fiber orientation angles in Fig. 16. The crack initiation anglewas theoretically found as the angle at which the magnitude of themodified hoop stress reaches a maximum. Comparing the resultsfound in this stage, with those obtained experimentally (Fig. 15),indicates very good agreement between both sets of results. Com-parison of predicted and measured crack growth initiation anglefor different fiber angles is shown in Fig. 17. A very slight discrep-ancy is observed between the analytical and experimental results,which might be associated with (a) slight errors in the experimen-tally measured angles or (b) the original fibers might have not beenperfectly straight within the composite sheet.

5. Conclusions

A detailed and robust digital image correlation-based metrologyis used to determine the elastic and fracture properties of wovencomposites. In-plane mechanical parameters are quantified andthe fracture response of an orthotropically woven carbon fiber

reinforced composite are determined using DIC measurements.The stress intensity factor at the onset of crack extension is calcu-lated based on both the far-field load and also the displacementfields obtained from 3D-DIC. A detailed study of the material’s frac-ture response reveals that the far-field load and specimen geome-try are insufficient to extract accurate estimates for fractureparameters such as the stress intensity factor. Rather, the experi-ments should be accompanied with either (a) optical observationsto detect the onset of crack extension and/or (b) a correspondingmeasurement of local strain/displacement fields at a location veryclose to the initial crack tip to detect the abrupt changes in thesecomponents that occurs when crack propagation initiates.

Finally, the modified maximum hoop stress is utilized to analyt-ically predict the angle of crack initiation. The predicted angle ofcrack initiation is shown to be in very good agreement with themeasured crack initiation angle, indicating that this criterion isexpected to be reliable for use in both experimental as well asnumerical studies conducted on similar material systems.

Acknowledgement

The financial support of NASA through EPSCOR under Grant No.21-NE-USC_Kidane-RGP, the College of Engineering and Comput-ing and the Department of Mechanical Engineering at the Univer-sity of South Carolina is gratefully acknowledged.

References

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A DIC-based study of in-plane mechanical response and fracture of orthotropic carbon fiber reinforced composite1 Introduction2 Experimental procedure2.1 Material and specimen geometry2.2 Mechanical testing2.3 Quasi-static fracture testing2.4 Digital image correlation (DIC) method

3 Data analysis3.1 Elastic properties3.2 Crack-tip displacement fields3.3 Criterion for crack propagation direction

4 Results and discussion4.1 Elastic properties4.2 Stress intensity factor and crack propagation direction4.2.1 Stress intensity factor4.2.2 Crack propagation direction

5 ConclusionsAcknowledgementReferences

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