Local strain energy density for the fracture assessment of polyurethane specimens weakened by notches of different shape

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1 Loal strain energy density for the frature assessment of polyurethane speimens weakened by nothes of different shape M. Peron, S.M.J. Razavi,F. Berto, J. Torgersen Department of Mehanial and Industrial Engineering, Norwegian University of Siene and Tehnology (NTNU), Rihard Birkelands vei 2b, 7491, Trondheim, Norway. L. Marsavina Department of Mehanis and Strength of Materials, Univeritatea Politehnia Timisoara, Timisoara, Romania ABSTRACT. Reent studies on loal stress fields in proximity of rak and noth tips have shown that Strain Energy Density (SED), averaged in a irular ontrol volume surrounding the point of stress singularities, represents a reliable engineering approah for assessing the brittle frature of several brittle materials. It is worthy of notie that the appliation of SED riterion and the reliability of its results are stritly related to the proper determination of frature parameters, i.e. the ritial value of deformation energy W and the radius R of the ontrol volume. This work presents an experimental methodology for their determination by means of nothed speimens for different polyurethane densities, ranging from 100 to 651 kg/m 3. Then, one obtained these ritial parameters, the failure load in different types of nothes and raked speimens under mode I have been predited. Moreover, for raked speimens under mixed mode and mode II, the authors propose a personal approah that onfirms PUR foams an be treated as brittle materials Citation: Peron, M., Razavi, S.M.J., Berto, F. Torgersen, J., Marsavina, L., Loal strain energy density for the frature assessment of polyurethane speimens weakened by nothes of different shape, Frattura ed Integrità Strutturale, 42 (2017) Reeived: Aepted: Published: Copyright: 2017 This is an open aess artile under the terms of the CC-BY 4.0, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original author and soure are redited. KEYWORDS. Strain Energy Density; PUR foams; Tensile frature; Critial radius; Frature parameters. INTRODUCTION I n the last years, Polyurethane (PUR) has widely gained interest in industrial appliations beause of its high versatility. In fat, as other polymers like polyethylene (PE), PUR materials an be manufatured in a wide range of densities, obviously determining different appliations. In fat at low densities ( kg/m 3 ), being rigid foams having a lose ell ellular struture, they are employed as high-resiliene seating, rigid foam insulation panels, miroellular foam seals and gaskets, high durable elastomeri wheels and tires and as automotive suspension bushings [1]. Whereas, at higher 214

2 densities (> 200 kg/m 3 ) they show a porous solid struture, and they are used for fixtures and gauges, master and opy models, draw die moulds, hard parts for eletroni instruments [1]. Their mehanial properties have been extensively investigated in the past, showing a relationship, based on the geometry of ellular struture and the relative density, with solid materials used for manufaturing [2,3], and revealing a rushable behaviour in ompression, haraterized by the apability to absorb onsiderable amount of energy due to plateau and densifiation regions. Finally Marsavina [4] reports that they behave as brittle materials under tensile loading, being haraterized by a linear elasti behaviour up to frature. It is worth to notie that usually in the industrial appliations omponents are designed with nothes or they are affeted by manufaturing indued defets, whih are widely reported to redue both tensile and fatigue strength [5-7]. Several riteria have been proposed in literature for fature assessment of raked and nothed omponents [8-16], but among all a strain energy based (SED) approah has revealed to be the most robust in the assessment of brittle frature resistane of several materials [17-20]. The riterion states that brittle frature failures our when the strain energy density averaged in a irular ontrol volume of radius R, whih surrounds a rak or noth tip, reahes a ritial value W dependent on the material. The robustness of this riterion has been proved by several works on different nothes geometries and on different loading onditions [21-27]. The main purpose of this work is to evaluate the effetiveness of SED riterion on PUR foams, aiming to experimentally evaluate the main parameters of this method, i.e. R and W. MATERIALS P olyurethane materials of four different densities (100, 145, 300 and 708 kg/m 3 ) manufatured by Neumer GmbH Germany, under ommerial designation Neuron 100, 160, 301 and 651, were experimentally investigated. At low densities 100 and 145 kg/m 3 the materials have a rigid losed ellular struture, while the PUR materials of higher densities show a porous solid struture (300 and 708 kg/m 3 ). A QUANTA FEG 250 SEM was used to investigate the mirostrutures of the materials at different magnifiations. The ell diameter and wall thikness were determined by statistial analysis, together with the density of PUR materials obtained experimentally aording with ASTM D The elasti properties Young modulus and Poisson ratio were determined by Impulse Exitation Tehnique (ASTM E ). Tensile strength was determined on dog bone speimens aording with a gage length of 50 mm and a ross setion in the alibrated zone with 10 mm width and 4 mm thikness, aording to EN ISO 527, and desribed in the researh published by Marsavina et al. [28]. PUR Density Young s modulus [MPa] 30.18± ± ± ±15.00 Poisson s ratio [-] Tensile strength [MPa] 1.16± ± ± ±0.32 Mode I frature toughness [MPa m 0.5 ] 0.087± ± ± ±0.027 Mode II frature toughness [MPa m 0.5 ] 0.050± ± ± ±0.047 Table 1: Elasti, mehanial and frature properties of PUR materials, by varying the density. The mode I and II frature toughness were determined on asymmetri semi-irular bend (ASCB) speimens. A detailed desription of these tests is presented in [29,30]. The experimentally values of elasti, mehanial and frature toughness properties are presented in Tab. 1. EXPERIMENTAL INVESTIGATION Tensile test D ifferent nothed speimens were tested under tensile load. Nothed speimens with geometries presented in Fig. 1.a,b, having lateral V, rounded U and irular holes of different diameters D were tested in tensile. The U nothed speimens, with blunt urvature radius (R = 4.25 mm), were tested for eah density, respetively holed 215

3 plates with different diameters were tested only for the highest density (708 kg/m 3 ), geometries and maximum load presented in Tab. 2. Tests were performed at room temperature, on a Zwik/Roell Z005 testing mahine with 5 kn maximum load, using a loading rate of 2 mm/min. Four tests were performed for eah noth geometry. The speimens dimensions and the average maximum load are listed in Tab. 2. The obtained load-displaement urves show a linear behavior without plastiity, the failure ours suddenly and the behavior is brittle. Geometrial parameters [mm] PUR Density [kg/m Noth 3 ] Shapes l W b D R Average Maximum load F max [N] V U O Table 2: Geometrial parameters and average maximum load of nothed omponents. Noth shape: O Length l = 100 mm Width W = 25 mm Hole diameter [mm] Average maximum load [N] Table 3: The average maximum load from testing of speimens with hole on tensile. Bending of asymmetri semi-irular bend (ASCB) ASCB speimens with vertial rak were onsidered, Fig. 1.d. The rak tip was introdued using a razor blade. Different types of applied mixed mode are easily obtained only by hanging one of the supports position (S 2) and keeping onstant the other support (S 1 ). The load is applied on the symmetry axis of the speimen using three point bending grips. Stress intensity fators (SIFs) solution for ASCB speimen [31]: Pmax Ki ay i( a/ RS, 1/ RS, 2 / R) i III, (1) 2Rt were obtained by finite element analysis (Lazzarin and Filippi, [32] and are plotted for a rak length a = 20 mm, speimen radius R = 40 mm, distane to fixed support S 1 = 30 mm, thikness t = 10 mm, resulting a/r = 0.5, S 1/R = It ould be observed that hanging the distane S 2 from 30 mm to 3 mm, the loading onditions hange from pure mode I to dominant mode II onditions. Moreover, using a polynomial interpolation the exat position of left support, leading to pure mode II loading ondition, was determined at distane S 2 = 2.66 mm. The reorded load displaement urves were linear (no signifiant non-linearity identified) and the frature ourred suddenly, indiating that the speimens failed in a brittle manner, Fig. 1.e. Tab. 4 presents the average frature load values F max obtained at eah loading onfiguration for the four onsidered materials. For all the tested speimens the thikness was equal to 10 mm. The mixed mode ratio was quantified using the mode mixity through the dimensionless parameter M e, proposed by Ayatollahi and Torabi [33]. M e (S 2 [mm]) Density [Kg/m 3 ] 1(30) 0.83 (12) (8) (6) (4) (2.66) Average Frature Load Value [N] Table 4: Average frature loads values for ASCB speimens. 216

4 Figure 1: Nothed speimens on tensile, (a) lateral rounded V noth, (b) lateral U nothes, () irular hole and (d) ASCB speimen; (e) typial load-displaement urves on tensile and bending speimens. THEORETICAL BACKGROUND B erto and Lazzarin [27], and later Radaj and Vormwald [34], presented omprehensive overview of the volumebased strain energy density riterion. Below, only a reminder of the main onepts of the SED regarding brittle frature of nothed omponents is reviewed. SED riterion assumes that failure ours when the mean value W of deformation energy in a loal finite volume around the noth tip (ontrol volume) reahes a ritial value W ; the failure ours when W W, independent of the noth opening angle and loading type. If the material exhibits an ideally brittle behaviour until frature, the parameter W is alulated from the ultimate tensile strength σ u: W 2 /2E (2) u Under the situations when plain speimens exhibit a non-linear behaviour, whereas the nothed speimens behave linear, Seweryn [35] reommended that the stress σ u should be replaed by the maximum normal stress existing at the edge at the moment preeding the raking determined on tensile speimens with blunt urvature radius, where semi-irular nothes are reommended. In plane problems, the ontrol volume beomes a irle or a irular setor with a radius R in the ase of raks or pointed V-nothes in mode I or mixed, I + II, mode loading (Fig. 2a and b). Under plane strain onditions, a useful expression for R has been provided onsidering the rak ase [36, 37]: R v v K I 4 t 2 (3) If the ritial value of the NSIF is determined by means of speimens with α 0, the ritial radius an be estimated by means of the expression: R K 1 I 4 1 EW (4) When 2α = 0, K 1C equals the frature toughness K IC. For rounded V-nothes, a resent-shaped ontrol volume bounded by two radii differently entered was introdued: a irular noth edge with radius q as the inner boundary and a irle ar with radius r 0 + R as the outer boundary, (Fig. 2). The length r 0 represents the distane between the origin of the polar oordinates (used to express the stress field) and the noth tip. The parameters r 0 depends from q then it s funtion only from the geometry (the opening angle 2α); r 0 is defined as r 0 qr /( q 1) where q (2 2 ) /. 217

5 Figure 2: Control volume for sharp V noth (a), rak ase (b) and rounded V noth under mode I loading. The radius of the ontrol volume and the ritial strain energy density depend only from the mehanial properties of the material as the Young s Modulus, the frature toughness, the Poisson s ratio and the ultimate tensile strength σ u or σ t. NUMERICAL INVESTIGATIONS Determination SED parameters T he quasi ideally brittle behavior, for these foams, is exhibited for nothed omponents, Fig. 1e, so the ultimate tensile strength σ u should be substituted with σ t, the maximum normal tension presents at the noth tip in the moment that proeed the rak, tension alulated in a nothed speimen under tensile load, speimen with a bland urvature radius. This σ t an be evaluated using U nothed speimen with a urvature radius greater than 4 mm, a bland noth: is reommended to use a semi-irular nothes, in this paper it has been hoose to use a plate with symmetri U noth. A linear-elasti finite element analysis was arried out in ANSYS 14.5 software for all speimen geometries. Based on symmetry of loading and boundary onditions quarter of geometry was onsidered. The average maximum load was applied to the models for eah noth geometry as uniaxial loads. The PLANE184 plane 8-node biquadrati elements with a suitably high mesh density in the area of the noth tip were employed, the analysis are under plane strain onditions. Aording with the proedure desribed above, it s possible to define the tension at the noth tip. In Tab. 4 is exhibit the tension σ t and the parameters of SED method, alulated through Eq. 2 and 3. Density [Kg/m 3 ] σ t [MPa] R [mm] W [MJ/m 3 ] Table 5: Values of tension at the noth tip and respetive SED parameters. Appliation SED method on speimens with different type of noth, mode I Through the SED parameters determined previously, is possible to apply the SED method on the nothed speimens tested in the previous paragraphs. In the same way followed to determine the σ t tension, the SED method were applied through linear elasti finite element analysis, using plane elements (PLANE 184) and reating the ontrol volume around the noth tip. The results are reported in Fig. 3.a. All the speimens are in mode I loads onfiguration. For the majority of the results, the satter band is ontained between + 10 % and 22 %, a reasonable dispersion in engineering field. Appliation SED method on speimens with different type of noth, mode I ASCB speimens were tested under pure mode I, pure mode II and mixed mode I+II. The first approah is to use the SED parameters defined for mode I (Tab. 5) in the ase of the mode II and mixed mode: for the higher densities, in mixed mode and in mode II the error is greater than 35 %, while for the lowest densities the error is ontained between ± 10 %, Graph 1b. It has been notied that the strain energy density inrease from mode I to mode II. If it s possible to 218

6 define the SED parameters, then the hypothesis that the material has a brittle behavior is valid and in the rak ase (the ontrol volume is a setor entered at the noth tip, Fig. 2.b) the strain energy density an be express through eq. (3). e K e K W (5) ER ER I 2 II 2(1 1) 2(1 2) The authors proposed the following approah: the ontrol volume remains the same in all load onfigurations and it s equal to the ontrol volume defined under pure mode I: in this way it s possible to realulate the value of the ritial strain energy density in mixed mode I+II and in pure mode II. Under this hypothesis, the satter band is ontained between ± 10 %, as it seen Fig. 3.. In Fig. 3 the error is alulated using the W defined through the σ t tension; the W an be redefined through the mean value of the strain energy density of eah speimens. The new values of W are listed in Table 5: the errors using these values of ritial energy density are presented in Fig. 4; exept Neuron 301, the satter band is ontained between ± 15 %. Density [Kg/m 3 ] R [mm] W [MJ/m 3 ] Table 6: New values of ritial energy density that fit better the results. Figure 3: Ratio between the preditions of maximum loads and experimental loads: a) nothed speimens, b) ASCB speimens under mixed mode, ) personal approah for ASCB speimens under mixed mode, d) all nothed speimens under mode I. 219

7 Density [Kg/m 3 ] σ t [MPa] σ 0,TCD [MPa] Table 7: Comparison between stress of TCD method and SED method. Figure 4: Ratio between the preditions of maximum loads and experimental loads using the new values of ritial energy density. CONCLUSIONS E xept some value, the relative errors is ontained between +10 % and 22 %, a reasonable predition in engineering field, Figs. 3 and 4. From these results it s possible to notie that the SED method works for these foams and the parameters (R and W ) an be determined through experimental tests on tensile nothed speimens. In a researh by Negru et al. (2015) it s defined the inherent stress for the Theory of Critial Distane (TCD), an approah based on the same theory of SED method, they belong to the linear elasti mehani frature theory (LEFM). The inherent stress in TCD method is equivalent to the failure stress and this tension is defined in a different way: the stress σ t defined in this paper is ompared with the inherent stress of TCD method, Tab. 6. Two different approah give values of the tensions very similar, with the same order of magnitude, this onfirms that it s possible to apply the SED method on these foams and the tensions σ t that valid the SED for eah type of noth have the same order of magnitude and it s similar. The personal approah works but the hypothesis that the ontrol volume remains the same it s only a personal view of the problem; this assumption it has been made only to prove that the PUR foams an be treated as a brittle materials and the SED approah an be applied, in fat the strain energy density defined through Eq. 3 differs less than ± 8% from the numerial strain energy density defined through the numerial investigations. It s possible to define a new values of ritial energy density for eah density that permit to derease the errors, in fat more than 95 % of the results are ontained between ± 15 %, so the new values of W fit better the results. The paper represents an entry level approah for the determination of the SED parameters for these foams, it s neessary further studies and tests. 220

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