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Procedia Materials Science 3 ( 2014 ) 1342 – 1352

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2211-8128 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineeringdoi: 10.1016/j.mspro.2014.06.217

ScienceDirect

20th European Conference on Fracture (ECF20)

Evaluation of mixed mode fracture for PUR foams

Liviu Marsavinaa*, Dan Mihai Constantinescub, Emanoil Linula, Tudor Voiconia, Dragos Alexandru Apostolb, Tomasz Sadowskic

aPolitehnica University of Timisoara, Blvd. M. Viteazu, No. 1, Timisoara 300222, Romania bPolitehnica University of Bucharest, Splaiul Independentei, No. 313, Bucharest 060042, Romania

cLublin University of Technology, Nadbystrzycka 48 str., Lublin 20-618, Poland

Abstract

Polyurethane foams crush in compression and have a brittle fracture in tension, so their failure could be evaluated based on Linear Elastic Fracture Mechanics. Fracture toughness in mixed mode loading is of particular interest because foam cracking weakens the structure's capacity for carrying loads. Four fracture criteria (Maximum circumferential tensile stress, Minimum strain energy density, Maximum energy release rate, Equivalent stress intensity factor) were considered for evaluation of mixed mode fracture of three closed cell rigid polyurethane foams with densities: 100, 145 and 300 kg/m3. Mixed mode fracture tests were performed using asymmetric semi-circular specimen. The equivalent stress intensity factor criterion looks to give the better prediction of mixed mode fracture. Also the effect of cell orientation and the crack propagation angle were investigated. © 2014 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering.

Keywords: mixed mode fracture; polyurethane foams; fracture toughness; asymmetric semi-circular bend specimen

1. Introduction

The properties of foams are influenced by the properties of solid material (polymers, metals, ceramics), by the cellular structure topology (open or closed cells) and relative density *

s, with * density of cellular material and s

* Corresponding author. Tel.: +40-256-403577; fax: +40-256-403523.

E-mail address: [email protected]

© 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering

1343 Liviu Marsavina et al. / Procedia Materials Science 3 ( 2014 ) 1342 – 1352

the density of the solid material, Ashby (2005). The main characteristics of PUR foams are lightweight, high porosity and good energy absorption capacity, Gibson and Ashby (1997). Foam materials crush in compression, but show a brittle fracture in tension, Marsavina (2010). So, it is important to know the fracture properties. Sandwich structures with foam cores are usually loaded in bending, and crack in the foam core can initiate in mode I, mode II or mixed mode. The knowledge of the fracture properties of foam core becomes an important requirement for the design of such structures. Up to now most results are related to fracture toughness in mode I. A linear correlation between Mode I fracture toughness KIc and relative foam density s was observed by Danielsson (1996) on PVC Divinycell foams, and by Viana and Carlsson (2002) on Diab H foams. Brittle fracture without yielding and produced in Mode I was observed in experiments. Kabir et al. (2006) used the procedure described by ASTM D5045 (1996) for determining the fracture toughness of PVC and PUR foams. They investigated the effect of density, effect of specimen size, effect of loading rate and effect of cell orientation. Density has a significant effect on fracture toughness, which increases more than 7 times when the foam density increases 3.5 times. Burman (1998) presented fracture toughness results for two commercial foams Rohacell WF51 (density 52 kg/m3) and Divinycell H100 (density 100 kg/m3). The mode I fracture toughness KIc was obtained on Single Edge Notch Bending specimens and has values 0.08 MPa m0.5 for WF51, respectively 0.21 MPa m0.5 for H100. Burman (1998) also determined the Mode II fracture toughness using an End-Notch Flexure (ENF) specimen with values of 0.13 MPa m0.5 for WF51, respectively 0.21 MPa m0.5 for H100. Static and dynamic evaluation of mode I fracture toughness of PUR foams was presented by Marsavina and Sadowski (2008), Marsavina et al. (2013), and the increase of fracture toughness with density was also highlighted, Marsavina et al. (2014). Poapongsakorn and Carlsson (2013) presented the influence of loading configuration and cell size on fracture toughness of closed-cell PVC foams. Their results show that the fracture toughness obtained using four point bend specimens is significantly higher than that measured on three point bend specimens.

Nomenclature

a crack length E Young's modulus G energy release rate KI mode I stress intensity factor KII mode II stress intensity factor KIc mode I fracture toughness KIIc mode II fracture toughness Keff effective stress intensity factor Keq equivalent stress intensity factor Me mode mixity r polar radius R ASCB specimen radius S strain energy density factor S1, S2 support distances t ASCB specimen thickness W strain energy density YI, YII non-dimensional stress intensity factors

ratio between mode I and mode II fracture toughness shear modulus polar angle c crack propagation angle rr radial stress

circumferential stress r tangential stress

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Only few studies present the mixed mode fracture of polymeric foams, and only for PVC foams. Hallsttröm and Grenestedt (1997) investigated mixed mode fracture of cracks and wedge shaped notches in expanded PVC foams. Different types of specimens made of Divinycell H100 were investigated and the non singular T-stress was considered in formulation of fracture criteria. It was concluded that for predominantly mode II the use of T-stress improved the facture predictions. Three different densities of PVC foams were investigated using a Compact Tensile Specimen with Arcan fixtures to produce mixed mode conditions, Noury (1998). The ratio between mode II and mode I fracture toughness KIIc/KIc was found to be between 0.4 and 0.65 depending on foam density. For mixed mode loading the Richard fracture criterion gives better predictions of fracture limit and crack initiation angle.

Present study assessed the theoretical fracture criteria for PUR foam materials under mixed mode loading, using an asymmetric semi-circular bend (ASCB) specimen. Also the influence of density and cell orientation is investigated.

2. Review of mixed mode fracture criteria

2.1. Introductory remarks

For the in-plane mixed mode a fracture criterion should provide:

The angle of crack initiation c, A critical combination of stress intensity factors (KI and KII) and fracture toughness (KIc) in the form:

0,, IcIII KKKF (1)

The singular stress field around a crack under mixed mode loading can be written in polar coordinates (r, ) as:

.2

cos32

cos2

3sin2

sin241

;2

3sin32

sin32

3cos2

cos3241

;2

3sin32

sin52

3cos2

cos5241

IIIr

III

IIIrr

KKr

KKr

KKr

(2)

2.2. Maximum tensile stress (MTS) criterion

According to Erdogan and Sih (1963) criterion, based on stress from eq. (2), the crack growth starts radially from the crack tip at an angle = c perpendicular to the maximum tensile circumferential tensile stress ,max. The crack propagation becomes unstable when ,max reaches a critical value cr, which is a material parameter. The crack propagation angle can be found from:

0c

and 02

2

c

(3)

leading to:

01cos3sin cIIcI KK . (4)

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Solving eq. (4) the crack propagation angle results:

22

222

983

arccosIII

IIIIIIc KK

KKKK, (5)

and the fracture criterion from eq. (1) could be expressed:

IccIIc

Ic KKK sin

23

2cos

2cos 2 . (6)

2.3. Minimum strain energy density (SED) criterion

Sih (1974) postulated that the fracture occurs in the direction where the strain energy density is minimum, at a critical distance r0. For mixed mode loading the strain energy density W, can be expressed:

0rSW , (7)

where S represents the strain energy density factor:

,1cos3cos1cos11

1cos2sin2coscos116

1

2

2

II

IIII

K

KKKS (8)

with the shear modulus, for plane strain, for plane stress, and is the Poisson's ratio. According to this criterion the crack propagates when:

0c

dS and 02

2

c

S (9)

respectively, the fracture initiates when S reaches a critical value Scr:

2

81

Iccr KS . (10)

2.4. Maximum energy release rate criterion (Gmax)

Hussain et al. (1974) investigates the infinitesimal kink of a crack at an angle , and expressed the energy release rate in terms of stress intensity factors of initial crack:

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22

'1

III KKE

G , (11)

where:

sin21cos

sin23cos

1

1

cos34

2

2

III

III

II

I

KK

KK

KK

, (12)

resulting:

22222

2 cos59cossin8cos311

1

cos31

'4

IIIIII KKKKE

G (13)

with E’=E Young's modulus for plane stress, and E'=E/(1- 2) for plane strain. The angle of crack propagation c is found by maximizing G( ):

0c

dG and 02

2

c

G. (14)

Crack extension occurs when G( c) reaches the critical value GIc:

EKG Ic

c

2

, (15)

resulting:

1cos59

cossin8cos311

1

cos314

22

2

22

2

2

Ic

IIc

Ic

IIIcc

Ic

Ic

c

c

c

KK

KKK

KK

c

(16)

A similar criterion was proposed by Chang et al. (2006) for generalized mixed mode loading (KI, KII and KIII).

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2.5. Equivalent stress intensity factor (ESIF) criterion

Richard (1985, 2005) proposed a generalized fracture criterion, based on the equivalent stress intensity factor Keq, which is defined similar to maximum principal stress eq, as:

IcIIII

eq KKKKK 22 421

2, (17)

with =KIc/KIIc. Crack starts to propagate when Keq reaches the fracture toughness of the material KIc. For the crack initiation angle Richard proposed an empirical expression:

2

00 4.835.155III

II

III

IIc KK

KKK

K, (18)

which represents a correlation with a considerable number of experiments.

3. Materials and method

Closed cell solid polyurethane (PUR) foams of three different densities (Necuron 100, 160 and 301), manufactured by NECUMER GbmH (Germany), were used in experiments. The main properties of the investigated foams are provided by the manufacturer and presented in Table 1. The microstructures of the investigated foams were obtained using QUANTATM FEG 250 SEM, and are presented at 500x magnification in Fig. 1.

Fig. 1. SEM microstructures for investigated PUR foams (magnification 500x): (a) density 100 kg/m3, (b) density 145 kg/m3, (c) density 300 kg/m3.

Table 1. Main properties of the investigated PUR foams.

Property Necuron

100 160 301

Density, [kg/m3] 100 145 300

Temperature resistance, [0C] 120 120 65

Compressive strength, [MPa] 2 3 5

Flexural strength, [MPa] 1.5 2.5 6

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Statistical analysis of the microstructure provide the mean and standard deviation of cell size on the rise direction (named ''in plane'') and flow direction (named ''out of plane'') and the cell thickness, Table 2.

Table 2. Main properties of the investigated PUR foams.

Cellular structure property Necuron

100 160 301

Cell length in plane, [ m] 104.5±9.4* 83.8±9.6* 68.5±33.9*

Cell length out of plane, [ m] 120.2±14.5* 88.1±11.2* 67.8±32.1*

Cell wall thickness, [ m] 2.9-5.8 5.1-13.1 3.8-21.8

Density (determined), [kg/m3] 100.35±0.25* 145.53±0.22* 300.28±1.38*

* standard deviation The density of the foams was determined according with ASTM D 1622-08, using cubic specimens of 15x15x15

mm, an electronic balance Sartorius LA230S for weighting and a digital caliper Mytotoyo for dimension determination. The mean values of density are also presented in Table 2.

The asymmetric semi-circular bend (ASCB) specimen, Fig. 2 was adapted to perform mixed mode fracture toughness tests. This semi-circular specimen with radius R, which contains an edge crack of length a oriented normal to the specimen edge, loaded with a three point bend fixture, was proved to give a wide range of mixed modes from pure mode I (S1=S2) to pure mode II, only by changing the position of one support, Ayatollahi et al. (2011), Negru et al. (2013). The considered geometry of the specimen has: R=40 mm, a=20 mm, t=10 mm, S1=30 mm and S2=30, 12, 8, 6, 4, 2.6 mm. The Stress Intensity Factors (SIFs) of the ASCB specimen are expressed in the form, Ayatollahi et al. (2011):

IIIiRSRSRaYatR

FK ii ,,,,2 21 , (19)

where the non-dimensional SIFs Yi (a/R, S1/R, S2/R) were determined by finite element analysis, and are plotted in Fig. 3.

Fig. 2. The geometry of ASCB specimen. Fig. 3. Variations of the non-dimensional SIFs YI, and YII.

The tests were performed on a Zwick/Roell of 5 kN testing machine at room temperature with a loading rate of 2 mm/min. Support rollers of diameter 20 mm were used for the bending fixture. For each position of support S2 four specimens were tested. Typical load - displacement curves for the foam with 300 kg/m3 and different S2 positions are shown in Fig. 4. Brittle fracture was observed for all tested specimens with an abrupt drop of load to zero after

1349 Liviu Marsavina et al. / Procedia Materials Science 3 ( 2014 ) 1342 – 1352

reaching the maximum load. The linear elastic behavior was confirmed during the tests when no cushioning and plastic deformations remain after finishing the test.

Fig. 4. Typical load – displacement curves for foam Fig. 5. Out of plane and in plane position of the ASCB specimens. with density 300 kg/m3 and different support distances S2.

4. Results and discussions

The mode I fracture toughness KIc was determined for the case of symmetric loading (S1 = S2 = 30 mm) on an ASCB specimen - mean values are given in Table 3 together with the mode I fracture toughness determined for the same foams using Single Edge Notched Bend (SENB) specimen according to ASTM D 5045-99, as presented by Marsavina et al. (2010, 2013), together with the mode II fracture toughness KIIc for S2 = 2.6 mm. From Table 3 it could be observed that both mode I and mode II fracture toughness increases with density. Similar results of Mode I fracture toughness were obtained on both types of specimens SENB and ASCB. Maximum relative difference between KIc was 8.2 % for 145 kg/m3 density and minimum value was 0.8 % for 300 kg/m3 density. Also the fracture toughness KIc and KIIc values determined out of plane (direction 3 in Fig. 5) are higher than those in plane results (direction 1 in Fig. 5), indicating anisotropic behavior of investigated foams. This is also confirmed in Fig. 6, which presents the corresponding values of effective stress intensity factors Keff=[(KIc)2+(KIIc)2]0.5 corresponding to fracture load versus the mode mixity Me=Arctg(KII/KI). It could be observed that for both densities (100 and 145 kg/m3) the out of plane effective SIF values are higher than those in plane. For the foam of density 300 kg/m3 the out of plane fracture toughness could not be determined because the specimen radius was higher than the PUR plate thickness 25 mm.

Table 3. Fracture toughness results for the investigated PUR foams.

Specimen Fracture toughness

Loading direction

Necuron

100 160 301

SENB KIc, [MPa m0.5] In plane 0.089±0.003* 0.121±0.003* 0.369±0.030*

KIc, [MPa m0.5] Out of plane 0.103±0.005* 0.126±0.002* 0.339±0.020*

ASCB

KIc, [MPa m0.5] In plane 0.087±0.003* 0.131±0.003* 0.372±0.014*

KIc, [MPa m0.5] Out of plane 0.106±0.007* 0.143±0.003* not available

KIIc, [MPa m0.5] In plane 0.050±0.002* 0.079±0.004* 0.374±0.013*

KIIc, [MPa m0.5] Out of plane 0.064±0.002* 0.090±0.006* not available

* standard deviation On ASCB specimens the mode II fracture toughness is lower than the mode I fracture toughness for low density

foams and almost equal for 300 kg/m3 density.

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(a) foam density 100 kg/m3 (b) foam density 145 kg/m3 Fig. 6. Effect of loading plane on fracture toughness.

Figs. 7 (a)-(c) present the experimental results together with the fracture limit curves predicted by the four fracture criteria considered. It can be observed that for low density foams (100 and 145 kg/m3 densities) the experimental results fall between the Gmax and ESIF criteria. While for the foam with 300 kg/m3 the experimental data are more scattered and close to SED and ESIF criteria. Based on these results it could be concluded that for rigid PUR foams the Richard’s ESIF criterion is most reliable to predict mixed mode fracture. This could be also explained by the fact that it takes into account the ratio between mode I and mode II fracture toughness =KIc/KIIc. The ratio decreases with increasing relative density. Noury et al. (1998) found same results investigating mixed mode fracture of PVC foams.

(a) foam density 100 kg/m3 (b) foam density 145 kg/m3 (c) foam density 300 kg/m3 Fig. 7. Effect of loading plane on fracture toughness.

Fig. 8 presents the crack path for four applied mixed modes Me (a. mode I, b. predominantly mode I, c. predominantly mode II, d. mode II). Except for the mode I case, when the crack propagates like a straight line, all other cases show curvilinear crack paths. Crack propagation angle was measured on each specimen, Fig. 9. Fig. 10 presents the mean values of the crack propagation angle c measured on the specimens versus applied mixed mode loading Me, side by side with the predicted crack propagation angles by theoretical criteria. It could be observed that for predominantly mode I loadings Me<450 the measured values are in good agreement with the predicted ones. For predominantly mode II loading (Me>450) the experimental crack propagation angles differ from the predicted values. It can be also observed that the foam with density 300 kg/m3, with a microstructure close to a porous solid gives closer propagation angles to theoretical predictions, developed for brittle solid materials.

1351 Liviu Marsavina et al. / Procedia Materials Science 3 ( 2014 ) 1342 – 1352

Fig. 8. Crack paths for different positions of S2 support.

Fig. 9. Measuring the crack propagation angle. Fig. 10. Crack propagation angle versus mode mixity.

5. Conclusions

The main conclusions of this study are:

The asymmetric semi-circular bend specimen was adopted to determine the fracture toughness of polyurethane rigid foams under mixed mode loading. The advantages of this specimen are the simple geometry, the use of classic bending fixtures for loading the specimens and the ability to produce full range of mixed modes, from pure mode I to pure mode II, only by changing the position of one support.

The density of foams is the most important parameter influencing the fracture toughness. The order of magnitude for mode I fracture toughness of PUR foams is between 0.087 MPa m0.5 for density of 100 kg/m3 to 0.372 MPa m0.5 for density of 300 kg/m3. These results are in agreement with those obtained on Single Edge Notched specimens (Table 2). The mode II fracture toughness ranges between 0.050 MPa m0.5 for density of 100 kg/m3 to 0.374 MPa m0.5 for density of 300 kg/m3.

The anisotropy of the foams is highlighted by the determination of fracture toughness on two planes: in plane and out of plane directions (Fig. 6).

Four theoretical fracture criteria were assessed to characterize the failure of rigid PUR foams. The experimental results proof that the equivalent stress intensity factor criterion of Richard is most suitable for this type of plastic

1352 Liviu Marsavina et al. / Procedia Materials Science 3 ( 2014 ) 1342 – 1352

foams (Fig. 9). This is the only criterion which takes into account the ratio between mode I and mode II fracture toughness, =KIc/KIIc. This parameter decreases with increasing the relative density of the foam. However, some previous experimental results have shown that the conventional fracture criteria fail to provide reliable estimates for experimental results if different test specimens are used, revealing that the T-stress plays an important role in mixed mode brittle fracture, Smith et al. (2006), Aliha and Ayatollahi (2010).

Experimental crack propagation angles are compared with theoretical estimates. Good agreement was obtained for predominantly mode I loading Me<450 (Fig. 10).

Acknowledgements

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project PN-II-ID-PCE-2011-3-0456, contract number 172/2011.

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