Microsoft Word - numero_34_art_43 L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 387 Focussed on Crack Paths On the crack path under mixed mode loading on PUR foams L. Marsavina, E. Linul, T. Voiconi University Politehnica Timisoara, Romania liviu.marsavina@upt.ro, tudor_2015@yahoo.com, linul_emanoil@yahoo.com D. M. Constantinescu, D. A. Apostol University Politehnica Bucharest, Romania dan.constantinescu@upb.ro, apostolda@yahoo.com ABSTRACT. In this paper are presented the crack initiation angles obtained in polyurethane (PUR) foams under mixed mode loading. Closed cell rigid PUR foams having three different densities 100, 145, and 300 kg/m3 were investigated. Experiments were performed using Asymmetric Semi-Circular Bend and Single Edge Crack specimens. The obtained crack initiation values were compared with four fracture criteria to introduce: Maximum Tensile Stress, Strain Energy Density, Maximum Energy Release Rate and Equivalent Stress Intensity Factor, and a good agreement was observed. This allow to conclude that the theoretical fracture criteria developed for solid material could be used with success to predict the crack propagation angles in cellular materials like PUR foams. KEYWORDS. PUR foams; Crack path; Mixed mode. INTRODUCTION he main characteristics of polyurethane (PUR) foams are lightweight, high porosity and good energy absorption capacity, [1,2]. One of the main application of foam materials is their utilization as cores in sandwich structures. Of particular interest is the fracture toughness in mixed modes because foam cracking weaknesses the capacity of carrying load of the sandwich structure. There are several studies to determine the mode I fracture toughness and to investigate the influence of density, loading speed and loading direction [3-11]. A linear correlation between Mode I fracture toughness KIc and relative density of the foam was observed by Danielsson [7] on PVC Divinycell foams, Viana and Carlsson on Diab H foams [8]. Brittle fracture without yielding in mode I loading was observed by Kabir et al. [9] for PVC and PUR foams applying the procedure described by ASTM D5045. 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 [10] 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. He also determined the Mode II fracture toughness using End-Notch Flexure (ENF) specimen with values of 0.13 MPam0.5 for WF51, respectively 0.21 MPam0.5 for H100. Poapongsakorn and Carlsson [11] also presented fracture toughness results for PVC foams determined on Three (TPB) and Four Point Bending (FPB), investigating the cell size T L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 388 and loading rate. Their results show that the value of fracture toughness obtained in FPB loading configuration is higher with a factor of 2 comparing with those obtained on TPB. Only few studies present the mixed mode fracture of polymeric foams, and only for PVC foams. Hallsttröm and Grenestedt [12] 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, [13]. 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. Marsavina el. al. [14, 15] and Linul et al. [16] presented results mixed mode fracture toughness of PUR foams. They highlighted that the foam density is the major factor influencing fracture toughness, loading speed and loading direction having minor influence. The anisotropy of the foam was explained trough the cellular topology of foams. However, the crack initiation angle is less investigated [13, 15, 16]. Present study assessed the theoretical fracture criteria for crack initiation angle in PUR foam materials under mixed mode loading, using an asymmetric semi-circular bend and single edge crack specimens. ANALYTICAL MODELS FOR CRACK PROPAGATION he fracture initiation for a crack in-plane mixed mode conditions is described by:  The angle of crack initiation c,  A critical combination of stress intensity factors (KI and KII) and fracture toughness (KIc) in the form:   , , 0I II IcF K K K (1) The four most used fracture criterion are summarized below. Maximum Tensile Stress criterion (MTS) Erdogan and Sih [17] criterion, based on maximum tensile stress, consider that crack initiation 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 at a radius r:     ,max 2 Ic cr K r (2) The equation for crack initiation angle and the relationship between Stress Intensity Factors (SIF's) KI, KII and fracture toughness KIc for MTS criterion are presented in Tab. 1. Condition Mathematical formulation Crack initiation angle           2 2 2 2 2 3 8 arccos 9 II I I II c I II K K K K K K (3)  , ,I II IcF K K K         2 3cos cos sin 2 2 2 c c I II c IcK K K (4) Table 1: The MTS criterion. Minimum Strain Energy Density criterion (SED) Sih [18] postulated that the fracture occurs in the direction where the strain energy density S is minimum, at a critical distance r0:     2 1 8 cr IcS K (5) T L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 389 with  the shear modulus,  for plane strain, for plane stress, and  is the Poisson's ratio. Based on SED criterion the crack initiation angle and the relation between KI, KII and KIc are presented in Tab. 3. Condition Mathematical formulation Crack initiation angle                            2 2 2 sin 2 cos 1 2 cos ( 2 cos 1) 2sin sin 6 cos 1 0 I I II II K K K K (6)  , ,I II IcF K K K                                       2 2 2 1 1 cos cos 2 sin 2 cos 1 2 1 1 1 cos 1 cos 3 cos 1 c c I c c I II c c c II Ic K K K K K (7) Table 2: The SED criterion. Table 3: The Gmax criterion. Condition Mathematical formulation Crack initiation angle                                                                                            2 2 2 2 22 2 2 2 2 32 1 6 sin cos 8 cos sin 10 sin cos 1 cos 3 1 4 sin cos (3 cos 1) 8 sin cos 9 5 cos 1 cos 3 1 1 ln 1 1 I I II II I I II II K K K K K K K K                                                                                       1 2 2 2 2 2 22 1 1 1 11 1 3cos 1 8 sin cos 9 5 cos 0 cos 3 I I II IIK K K K (9)  , ,I II IcF K K K                                   2 2 2 2 2 2 2 1 1 4 3 cos 1 1 3 cos 8 sin cos 9 5 cos c c cc c I I II c c c II IcK K K K K (10) L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 390 Maximum Energy Release Rate criterion (Gmax) Hussain et al. [19] postulate that the crack propagation initiates when the Energy Release Rate G reaches a critical value GIc and expressed the energy release rate G in terms of stress intensity factors of initial crack:     2 ' Ic c Ic K G G E (8) with E’=E Young's modulus for plane stress, and E'=E/(1-2) for plane strain. The crack initiation angle could be found by solving Eq. (9) from Tab. 3, while the functional relationship between KI, KII and KIc is represented by eq. (10). Equivalent Stress Intensity Factor criterion (ESIF) Richard [20, 21] proposed a generalized fracture criterion, based on the equivalent stress intensity factor Keq, which is defined as:     221 4 2 2 I eq I II K K K K (11) 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 (12), which represents a correlation with a considerable number of experiments. Condition Mathematical formulation Crack initiation angle                   2 0 0155.5 83.4II IIc I II I II K K K K K K (12)  , ,I II IcF K K K    221 4 2 2 I I II Ic K K K K (13) Table 4: The ESIF criterion. MATERIALS AND METHODS Materials rack propagation studies were performed on rigid PUR foams having a closed cell microstructure. Three foam densities, manufactured by NECUMER GmbH (Germany), were investigated 100 kg/m3 (NECURON 100), 145 kg/m3 (NECURON 160) and 300 kg/m3 (NECURON 301). A statistical analysis of the foams microstructure is presented in [14]. The main properties provided by the manufacturer are listed in Tab. 5, [22]. According to the manufacturer the main applications of these foams are test models, draw dies, large volume models, back filling of moulds and patterns, and substructure for hard styling clay. Property Units of measure 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 Table 5: Material properties according to manufacturer [22] (approximate values). C L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 391 Determination of density The density determination of the considered foams was performed according with ASTM D 1622-03 [23]. A Sartorius GD 503 Class Balance was used for the mass determination, and the dimensions of the specimens (84 x 35 x 10 mm) were measured using a digital caliper Mitutoyo Digimatic. Fracture toughness determination and mixed mode loading tests The fracture toughness KIc and KIIc of PUR foams were determined using two types of specimens: Asymmetric Semicircular Bend (ASCB) and Single Edge Crack (SEC), Fig. 1. The choice of using these specimen types was that with the same shape is possible to produce from pure mode I loading to pure mode II loading, and intermediate mixed modes only by changing the position S2 of one support for ASCB specimens or by changing the loading direction  for SEC specimens. The ASCB 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 (S1≠S2), only by changing the position of one support. The considered geometry of the specimen has: R=40 mm, a=20 mm, t=10 mm, S1 = S2=30 mm (for symmetric loading and mode I fracture toughness determination), 12 mm, 8 mm, 6 mm, 4 mm (for mixed mode loading) and S2=2.66 mm (for mode II fracture toughness determination). The dimensions of the SEC specimens were W = 75 mm, t = 8 mm and a = 33.75 mm and the loading angle was  = 00 for mode I tests, 300, 450, 600 for mixed mode loading, respectively  = 900 for mode II tests. Tests were carried out at room temperature (20 0C) on a 5 kN Zwick Proline testing machine, using displacement control with 2 mm/min loading speed, Fig. 2. Typical load - displacement curves are shown in Fig. 3: a. the effect of applied mixed mode on ASCB specimens for the foam with density 145 kg/m3, and b. the influence of density on SEC specimens loaded in mode I. All tested specimens show a linear behaviour and a brittle fracture, which was confirmed by the fact that no plastic deformations remain after testing. The determination of fracture toughness is presented in more detail in [14, 15] for ASCB specimens and in [16] for SEC specimens. The experimental results for density and fracture toughness of investigated foams are presented in Tab. 6. It should be mentioned that the fracture toughness results are determined for specimens obtained in the rise direction (in plane). A discussion regarding the foams anisotropy is presented elsewhere, [14]. a. ASCB specimen b. SEC specimen Figure 1: Specimens used from fracture toughness and mixed mode loading tests. It could be observed that fracture toughness KIc and KIIc increases with density. The maximum relative differences between fracture toughness values obtained by ASCB and SEC specimens was 28% for foam with density 100 kg/m3 for mode II, while minimum relative difference was obtained for the same density for KIc 1.1%. Also, it could be observed that the mode II fracture toughness values are lower than the mode I ones for all foam densities and specimen type, with the exception of 300 kg/m3 foam density using ASCB specimens. L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 392 Figure 2: Experimental set - up with details of ASCB specimen in three point bending grips (top right), respectively with SEC specimen in Arcan grips (left and middle right). Property Units of measure NECURON 100 160 301 Density kg/m3 100.35±0.25* 145.53±0.22* 300.28±1.38* Mode I fracture toughness (ASCB) MPa·m0.5 0.087 0.131 0.372 Mode II fracture toughness (ASCB) MPa·m0.5 0.049 0.079 0.374 Mode I fracture toughness (SEC) MPa·m0.5 0.088 0.108 0.337 Mode II fracture toughness (SEC) MPa·m0.5 0.069 0.095 0.319 Table 6: Material properties determined experimentally (* standard deviation values). 0 50 100 150 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 L o a d [ N ] Displacement [mm] 2.66 4 6 8 12 30 ASCB Specimens 145 kg/m3 density 0 50 100 150 200 250 300 350 0.0 0.5 1.0 1.5 2.0 2.5 L o a d [ N ] Displacement [mm] 300 145 100 300 145 100 Density [kg/m3] a) b) Figure 3: Typical load - displacement curves: a) Influence of applied mixed mode for ASCB specimens; b) Influence of density for SEC specimens loaded in Mode I. L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 393 RESULTS AND DISCUSSIONS his paper is focused on the determination the crack initiation angle and the obtained crack paths under mixed mode loading of PUR foams. The obtained crack paths were digitized using a NIKON D5100 Digital Camera. Fig. 4 presents the crack paths for ASCB specimens for six loading configurations: a. mode I, b.-e. mixed modes, f. mode II. Except for the mode I case, when the crack propagates like a straight line, all other cases show curvilinear crack paths. Figure 4: Crack paths resulted from testing ASCB specimens. In Fig. 5 the crack paths for SEC specimens are shown. More straight crack trajectories resulted on testing the SEC specimens for all loading configurations. a. 0 0C b. 0 0C c. 45 0C d. 0 0C e. 0 0C Figure 5: Crack paths resulted from testing SEC specimens. Figure 6: Measurement of crack initiation angle. The crack initiation angle c was measured on each specimen using Sigma Scan Pro software as the angle between the tangent (red line) to the crack at the initiation point and the crack initial direction (blue line) in Fig. 6. Fig. 7 presents the mean values and the range (minimum to maximum values) of the crack initiation angle c measured on the specimens versus applied mixed mode loading Me = Arctg(KII/KI) side by side with the predicted crack propagation angles by the four 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 T L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 394 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 (especially to MTS and ESIF), developed for brittle solid materials. Noury et al. [13] also found, investigating PVC foams in mixed mode loading, that MTS criterion gives the closest crack initiation angles to experimental values. Comparing the two type of specimens the crack initiation angles obtained on SEC specimens are above those results for testing ASCB specimens, and closer to the theoretical predictions. 0 15 30 45 60 75 90 0 15 30 45 60 75 90  c [d eg re es ] Me [degrees] Density 100 kg/m3 0 15 30 45 60 75 90 0 15 30 45 60 75 90  c [d eg re es ] Me [degrees] Density 145 kg/m3 a. Density 100 kg/m3 b. Density 145 kg/m3 0 15 30 45 60 75 90 0 15 30 45 60 75 90  c [d eg re es ] Me [degrees] Density 300 kg/m3 c. Density 300 kg/m3 Legend Figure 7: Comparison between experimental and theoretical crack initiation angles. In order to investigate the effect of density, the mean value of crack initiation angle is plotted versus the support position S2 for ASCB specimens in Fig. 8.a, respectively versus loading angle  for SEC specimens, Fig. 8.b. The crack initiation angles are slightly different for the investigated densities. This is more pronounced for ASCB specimen, where maximum relative difference between densities 145 and 300 kg/m3 and mode II loading (S2=2.66 mm) is 22.3%. The maximum relative difference for SEC specimens was observed for loading angle = 300 is 7.2% and this could be considered in the range of experimental determination. CONCLUSIONS rack initiation angle on mixed mode loading for PUR foams of three different densities was investigated experimentally. Two specimens ASCB and SEC were considered, which allow to produce mixed mode loading only by changing some loading configurations. The obtained values were compared with four most used fracture C L. Marsavina et alii, Frattura ed Integrità Strutturale, 34 (2015) 387-396; DOI: 10.3221/IGF-ESIS.34.43 395 criteria MTS, SED, Gmax, respectively ESIF, and a good agreement was observed. This allows to conclude that the theoretical fracture criteria developed for a solid material could be used with success to predict the crack propagation angles in cellular materials like PUR foams. Crack paths presented in Figs. 4 and 5 show that crack initiates in mixed mode or mode II, but grows to regain the symmetry and to eliminate mode II. 0 10 20 30 40 50 60 70 80 90 2.66 4 6 8 12 30 c [ d eg re es ] Support position S2 [mm] ASCB specimens 100 145 300 Density 0 10 20 30 40 50 60 70 80 90 0 30 45 60 90  c [d eg re es ] Loading angle [degrees] SEC specimens 100 145 300 Density a. ASCB specimens b. 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