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

Recent studies on local stress fields in proximity of crack and notch tips have shown that Strain Energy Density (SED), averaged in a circular control volume surrounding the point of stress singularities, represents a reliable engineering approach for assessing the brittle fracture of several brittle materials. It is worthy of notice that the application of SED criterion and the reliability of its results are strictly related to the proper determination of fracture parameters, i.e. the critical value of deformation energy Wc and the radius Rc of the control volume. This work presents an experimental methodology for their determination by means of notched specimens for different polyurethane densities, ranging from 100 to 651 kg/m3. Then, once obtained these critical parameters, the failure load in different types of notches and cracked specimens under mode I have been predicted. Moreover, for cracked specimens under mixed mode and mode II, the authors propose a personal approach that confirms PUR foams can be treated as brittle materials

Local strain energy density for the fracture assessment of polyurethane specimens weakened by notches of different shape INTRODUCTION n the last years, Polyurethane (PUR) has widely gained interest in industrial applications because of its high versatility.In fact, as other polymers like polyethylene (PE), PUR materials can be manufactured in a wide range of densities, obviously determining different applications.In fact at low densities (30 -200 kg/m 3 ), being rigid foams having a close cell cellular structure, they are employed as high-resilience seating, rigid foam insulation panels, microcellular foam seals and gaskets, high durable elastomeric wheels and tires and as automotive suspension bushings [1].Whereas, at higher I densities (> 200 kg/m 3 ) they show a porous solid structure, and they are used for fixtures and gauges, master and copy models, draw die moulds, hard parts for electronic instruments [1].Their mechanical properties have been extensively investigated in the past, showing a relationship, based on the geometry of cellular structure and the relative density, with solid materials used for manufacturing [2,3], and revealing a crushable behaviour in compression, characterized by the capability to absorb considerable amount of energy due to plateau and densification regions.Finally Marsavina [4] reports that they behave as brittle materials under tensile loading, being characterized by a linear elastic behaviour up to fracture.It is worth to notice that usually in the industrial applications components are designed with notches or they are affected by manufacturing induced defects, which are widely reported to reduce both tensile and fatigue strength [5][6][7].Several criteria have been proposed in literature for facture assessment of cracked and notched components [8][9][10][11][12][13][14][15][16], but among all a strain energy based (SED) approach has revealed to be the most robust in the assessment of brittle fracture resistance of several materials [17][18][19][20].The criterion states that brittle fracture failures occur when the strain energy density averaged in a circular control volume of radius R c , which surrounds a crack or notch tip, reaches a critical value W c dependent on the material.The robustness of this criterion has been proved by several works on different notches geometries and on different loading conditions [21][22][23][24][25][26][27].The main purpose of this work is to evaluate the effectiveness of SED criterion on PUR foams, aiming to experimentally evaluate the main parameters of this method, i.e.R c and W c .

MATERIALS
olyurethane materials of four different densities (100, 145, 300 and 708 kg/m 3 ) manufactured by Necumer GmbH -Germany, under commercial designation Necuron 100, 160, 301 and 651, were experimentally investigated.At low densities 100 and 145 kg/m 3 the materials have a rigid closed cellular structure, while the PUR materials of higher densities show a porous solid structure (300 and 708 kg/m 3 ).A QUANTA™ FEG 250 SEM was used to investigate the microstructures of the materials at different magnifications.The cell diameter and wall thickness were determined by statistical analysis, together with the density of PUR materials obtained experimentally according with ASTM D1622-08.The elastic properties Young modulus and Poisson ratio were determined by Impulse Excitation Technique (ASTM E-1876-01).Tensile strength was determined on dog bone specimens according with a gage length of 50 mm and a cross section in the calibrated zone with 10 mm width and 4 mm thickness, according to EN ISO 527, and described in the research published by Marsavina et al. [28].The mode I and II fracture toughness were determined on asymmetric semi-circular bend (ASCB) specimens.A detailed description of these tests is presented in [29,30].The experimentally values of elastic, mechanical and fracture toughness properties are presented in Tab. 1.

Tensile test
ifferent notched specimens were tested under tensile load.Notched specimens with geometries presented in Fig. 1.a,b,c having lateral V, rounded U and circular holes of different diameters D were tested in tensile.The U notched specimens, with blunt curvature radius (R = 4.25 mm), were tested for each density, respectively holed P D 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 Zwick/Roell Z005 testing machine with 5 kN maximum load, using a loading rate of 2 mm/min.Four tests were performed for each notch geometry.The specimens' dimensions and the average maximum load are listed in Tab. 2. The obtained load-displacement curves show a linear behavior without plasticity, the failure occurs suddenly and the behavior is brittle.
Table 2: Geometrical parameters and average maximum load of notched components.
Table 3: The average maximum load from testing of specimens with hole on tensile.

Bending of asymmetric semi-circular bend (ASCB)
ASCB specimens with vertical crack were considered, Fig. 1.d.The crack tip was introduced using a razor blade.Different types of applied mixed mode are easily obtained only by changing one of the supports position (S2) and keeping constant the other support (S 1 ).The load is applied on the symmetry axis of the specimen using three point bending grips.Stress intensity factors (SIFs) solution for ASCB specimen [31]: were obtained by finite element analysis (Lazzarin and Filippi, [32] and are plotted for a crack length a = 20 mm, specimen radius R = 40 mm, distance to fixed support S1 = 30 mm, thickness t = 10 mm, resulting a/R = 0.5, S1/R = 0.75.It could be observed that changing the distance S2 from 30 mm to 3 mm, the loading conditions change from pure mode I to dominant mode II conditions.Moreover, using a polynomial interpolation the exact position of left support, leading to pure mode II loading condition, was determined at distance S2 = 2.66 mm.The recorded load-displacement curves were linear (no significant non-linearity identified) and the fracture occurred suddenly, indicating that the specimens failed in a brittle manner, Fig. 1.e.Tab. 4 presents the average fracture load values F max obtained at each loading configuration for the four considered materials.For all the tested specimens the thickness 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]. , independent of the notch opening angle and loading type.If the material exhibits an ideally brittle behaviour until fracture, the parameter Wc is calculated from the ultimate tensile strength σu: Under the situations when plain specimens exhibit a non-linear behaviour, whereas the notched specimens behave linear, Seweryn [35] recommended that the stress σu should be replaced by "the maximum normal stress existing at the edge at the moment preceding the cracking" determined on tensile specimens with blunt curvature radius, where semi-circular notches are recommended.In plane problems, the control volume becomes a circle or a circular sector with a radius R c in the case of cracks or pointed V-notches in mode I or mixed, I + II, mode loading (Fig. 2a and b).Under plane strain conditions, a useful expression for Rc has been provided considering the crack case [36,37]: If the critical value of the NSIF is determined by means of specimens with α ≠ 0, the critical radius can be estimated by means of the expression: When 2α = 0, K1C equals the fracture toughness KIC.For rounded V-notches, a crescent-shaped control volume bounded by two radii differently centered was introduced: a circular notch edge with radius q as the inner boundary and a circle arc with radius r 0 + R c as the outer boundary, (Fig. 2c).The length r 0 represents the distance between the origin of the polar coordinates (used to express the stress field) and the notch tip.The parameters r0 depends from q then it's function only from the geometry (the opening angle 2α); r0 is defined as 0 / ( 1) r qR q   where The radius of the control volume and the critical strain energy density depend only from the mechanical properties of the material as the Young's Modulus, the fracture toughness, the Poisson's ratio and the ultimate tensile strength σu or σt.

Determination SED parameters
he quasi ideally brittle behavior, for these foams, is exhibited for notched components, Fig. 1e, so the ultimate tensile strength σ u should be substituted with σ t , the maximum normal tension presents at the notch tip in the moment that proceed the crack, tension calculated in a notched specimen under tensile load, specimen with a bland curvature radius.This σt can be evaluated using U notched specimen with a curvature radius greater than 4 mm, a bland notch: is recommended to use a semi-circular notches, in this paper it has been choose to use a plate with symmetric U notch.A linear-elastic finite element analysis was carried out in ANSYS 14.5 software for all specimen geometries.Based on symmetry of loading and boundary conditions quarter of geometry was considered.The average maximum load was applied to the models for each notch geometry as uniaxial loads.The PLANE184 plane 8-node biquadratic elements with a suitably high mesh density in the area of the notch tip were employed, the analysis are under plane strain conditions.According with the procedure described above, it's possible to define the tension at the notch tip.In Tab. 4 is exhibit the tension σ t and the parameters of SED method, calculated through Eq. 2 and 3.

Application SED method on specimens with different type of notch, mode I
Through the SED parameters determined previously, is possible to apply the SED method on the notched specimens tested in the previous paragraphs.In the same way followed to determine the σ t tension, the SED method were applied through linear elastic finite element analysis, using plane elements (PLANE 184) and creating the control volume around the notch tip.The results are reported in Fig. 3.a.All the specimens are in mode I loads configuration.For the majority of the results, the scatter band is contained between + 10 % and -22 %, a reasonable dispersion in engineering field.
Application SED method on specimens with different type of notch, mode I ASCB specimens were tested under pure mode I, pure mode II and mixed mode I+II.The first approach is to use the SED parameters defined for mode I (Tab.5) in the case 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 contained between ± 10 %, Graph 1b.It has been noticed that the strain energy density increase from mode I to mode II.If it's possible to T define the SED parameters, then the hypothesis that the material has a brittle behavior is valid and in the crack case (the control volume is a sector centered at the notch tip, Fig. 2.b) the strain energy density can be express through eq. ( 3).
The authors proposed the following approach: the control volume remains the same in all load configurations and it's equal to the control volume defined under pure mode I: in this way it's possible to recalculate the value of the critical strain energy density in mixed mode I+II and in pure mode II.Under this hypothesis, the scatter band is contained between ± 10 %, as it seen Fig. 3.c.In Fig. 3 the error is calculated using the W c defined through the σ t tension; the W c can be redefined through the mean value of the strain energy density of each specimens.The new values of W c are listed in Table 5: the errors using these values of critical energy density are presented in Fig. 4

CONCLUSIONS
xcept some value, the relative errors is contained between +10 % and -22 %, a reasonable prediction in engineering field, Figs. 3 and 4. From these results it's possible to notice that the SED method works for these foams and the parameters (Rc and Wc) can be determined through experimental tests on tensile notched specimens.In a research by Negru et al. (2015) it's defined the inherent stress for the Theory of Critical Distance (TCD), an approach based on the same theory of SED method, they belong to the linear elastic mechanic fracture 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 compared with the inherent stress of TCD method, Tab. 6.Two different approach give values of the tensions very similar, with the same order of magnitude, this confirms that it's possible to apply the SED method on these foams and the tensions σt that valid the SED for each type of notch have the same order of magnitude and it's similar.The personal approach works but the hypothesis that the control 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 can be treated as a brittle materials and the SED approach can be applied, in fact the strain energy density defined through Eq. 3 differs less than ± 8% from the numerical strain energy density defined through the numerical investigations.It's possible to define a new values of critical energy density for each density that permit to decrease the errors, in fact more than 95 % of the results are contained between ± 15 %, so the new values of Wc fit better the results.The paper represents an entry level approach for the determination of the SED parameters for these foams, it's necessary further studies and tests.E

Figure 2 :
Figure 2: Control volume for sharp V notch (a), crack case (b) and rounded V notch under mode I loading.

Figure 3 :
Figure 3: Ratio between the predictions of maximum loads and experimental loads: a) notched specimens, b) ASCB specimens under mixed mode, c) personal approach for ASCB specimens under mixed mode, d) all notched specimens under mode I.

Figure 4 :
Figure 4: Ratio between the predictions of maximum loads and experimental loads using the new values of critical energy density.

Table 1 :
Elastic, mechanical and fracture properties of PUR materials, by varying the density.

Table 4 :
Average fracture loads values for ASCB specimens.

Table 5 :
Values of tension at the notch tip and respective SED parameters.

Table 6 :
New values of critical energy density that fit better the results.

Table 7 :
Comparison between stress of TCD method and SED method.