Microsoft Word - numero_41_art_34.docx A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 252 Focused on Crack Tip Fields On the use of length scale parameters to assess the static strength of notched 3D-printed PLA Adnan A. Ahmed, Luca Susmel Department of Civil and Structural Engineering, the University of Sheffield, Mappin Street, Sheffield S1 3JD, UK AAAhmed1@sheffield.ac.uk l.susmel@sheffield.ac.uk, http://orcid.org/0000-0001-7753-9176 ABSTRACT. This paper aims to investigate the accuracy of the Theory of Critical Distances (TCD) in estimating static strength of notched additively manufactured PLA as both notch sharpness and infill angle vary. The TCD takes as its starting point the assumption that the extent of damage under static loading can be assessed successfully by using two different material parameters, i.e. (i) a critical distance whose length is closely related to the material microstructural features and an inherent (i.e., a defect free) material strength. Plain and notched specimens of 3D-printed PLA were manufactured horizontally by making the deposition angle vary in the range 0-90. Using the TCD in the form of the Point Method, failures were predicted by directly post-processing the linear-elastic stress fields estimated through the well-known analytical solutions due to Glinka and Newport. Independently of the notch sharpness, the estimates being obtained were found to be highly accurate, falling within an error interval of about 20%. This result fully supports the idea that the TCD can successfully be used in situations of practical interest to design against static loading notched components of additively manufactured PLA by directly post-processing the results from simple linear-elastic Finite Element (FE) models. KEYWORDS. Additive Manufacturing; PLA; Theory of Critical Distances; Static Failure; Notches. Citation: Ahmed, A.A., Susmel L., On the use of length scale parameters to assess the static strength of notched 3D-printed PLA, Frattura ed Integrità Strutturale, 41 (2017) 252-259. Received: 28.02.2017 Accepted: 03.05.2017 Published: 01.07.2017 Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION s suggested by ASTM Committee F42 (www.astm.org/COMMITTEE/F42.htm), additive manufacturing is “the process of joining materials to make objects from 3D-model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies”. In more detail, additive manufacturing is an “additive” process that can be more effective than A A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 253 conventional “subtractive” technologies, with this holding true especially in the presence of complex shapes that would be very difficult to be manufactured by using traditional techniques. If attention is focused specifically on plastics, examination of the state of the art suggests that acrylonitrile butadiene styrene (ABS) and polylactide (PLA) are the two materials that are commonly employed along with low-cost 3D printers, where polymers can be additively manufactured by melting/extruding powders, wires and flat sheets. As mentioned earlier, the unique feature of additive manufacturing is that parts with complex geometries can be fabricated very easily, with the obtained objects being characterized by a remarkable level of accuracy in terms of both shape and dimensions. This makes it evident that 3D-printed objects can contain very complex geometrical features that are likely to act as local stress concentrators. Therefore, accurate and simple design methods are needed to allow structural engineers to effectively assess the static strength of 3D-printed materials by taking explicitly into account the presence of stress raisers of all kinds. As far static assessment of notched components is concerned, examination of the state of the art suggests [1-9] that the Theory of Critical Distances (TCD) is the most powerful candidate to be used to design additively manufactured components for the following reasons: (i) the TCD assesses the detrimental effect of notches independently from their shape and sharpness; (ii) it models explicitly the material morphology through ad hoc critical lengths; (iii) the required local stress fields can be estimated without adopting complex non-linear constitutive laws; (iv) the TCD can be applied along with the results from linear-elastic FE models, where the same solid models can be used also to inform the manufacturing process. In this challenging scenario, this paper aims to investigate whether the linear-elastic TCD is successful in performing the static assessment of 3D-printed notched PLA subjected to in service static loading. To conclude, it is worth observing that the present investigation was based on PLA since this material is a biodegradable, absorbable and biocompatible thermoplastic aliphatic polyester that is widely used to manufacture biomedical components having complex shape [10]. Further, thanks to its specific features, PLA is one of those polymers that can be additively manufactured at a relatively low-cost by employing commercial 3D-printers. STATIC ASSESSMENT ACCORDING TO THE THEORY OF CRITICAL DISTANCES ccording to the TCD, the static breakage of notched materials subjected to Mode I loading is avoided as long as a critical distance based effective stress, σeff, is lower than the material inherent strength, σ0 [1], i.e.:   0eff (1) An important feature of the TCD is that it assesses static strength by directly post-processing the linear-elastic stress fields damaging the material in the vicinity of the notch being designed, with this holding true independently from the ductility level of the material under investigation [1-4, 6]. This can be done successfully provided that material inherent strength σ0 is determined consistently [3, 4]. The procedure being recommended to be followed to determine σ0 experimentally will be reviewed in what follows. These considerations suggest that the TCD is bi-parametrical design method where the critical distance and the inherent material strength are the two material parameters being used. Under static loading, the TCD’s length scale parameter is recommended to be estimated according to this well-known relationship [1, 11]:          2 0 1 IcKL (2) where KIc is the plane strain fracture toughness. As soon as material length L is known, the required effective stress, σeff, can then be calculated directly according to either the Point Method (PM) or the Line Method (LM) as follows [1, 11]:          0, 2 eff y L r Point Method (PM) (3) A A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 254      2 0 1 0, 2 L eff y r dr L Line Method (LM) (4) Figure 1: Definition of the local systems of coordinates (a) and effective stress, eff, calculated according to the Point Method (b) and the Line Method (c). Figure 2: Manufacturing direction and orientation of the deposition filaments. The adopted symbols as well as the meaning of the effective stress determined according to definitions (3) and (4) are explained in Fig. 1. As to the different formalisations of the TCD, it is worth mentioning here that this powerful theory can be applied also in the form of the Area Method (AM) [1, 12] as well as of the Volume Method (VM) [1, 13]. In more nom nom  r x y 1 1 eff eff r r Point Method Line Method (a) (b) (c) L/2 2L yp x p 45° 45° p Filaments Reference Manufacturing Axis Build-Plate A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 255 detail, the AM postulates that the effective stress has to be determined by averaging σ1 over a semi-circular area centred at the notch tip and having radius equal to L [1]. In a similar way, the VM calculates σeff by averaging the linear-elastic maximum principal stress over a hemisphere centred at the apex of the stress raiser being assessed and having radius equal to 1.54L [12]. In the present investigation, owing to its simplicity, the accuracy of the TCD in estimating static strength of notched additively manufacture PLA will be assessed by applying this theory solely in the form of the PM. The definitions for σeff calculated according to Eqs (3) and (4) make it clear that inherent material strength σ0 plays a role of primary importance when the TCD is used to assess static strength of notched components. As far as brittle materials are concerned, much experimental evidence suggests that these materials have inherent strength that is always very close to the ultimate tensile strength, σUTS [1-3]. In contrast, when the final breakage is preceded by large scale plastic deformations, σ0 is seen to be somewhat larger than σUTS [1, 4, 6]. Further, σ0 takes on a value that is higher than σUTS also when the plain parent material fails by different mechanisms to those leading to the final breakage in the presence of stress raisers [1]. These considerations should make it evident that the only way to determine σ0 accurately is by running bespoke experiments involving notches whose presence results in different stress distributions in the vicinity of the tested geometrical features [1-6]. Figure 3: Experimental stress vs. strain curves generated by testing plain samples manufactured by adopting different values for manufacturing angle θp. 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 S tr es s [M Pa ] Strain [mm/mm] p=0 (a) 0 5 10 15 20 25 30 35 40 45 50 0 0.005 0.01 0.015 0.02 S tr es s [M Pa ] Strain [mm/mm] p=30 (b) 0 5 10 15 20 25 30 35 40 45 50 0 0.005 0.01 0.015 0.02 S tr es s [M Pa ] Strain [mm/mm] p=45 (c) 0 5 10 15 20 25 30 35 40 45 50 0 0.005 0.01 0.015 0.02 S tr es s [M Pa ] Strain [mm/mm] p=60 (d) 0 5 10 15 20 25 30 35 40 45 50 0 0.05 0.1 0.15 0.2 S tr es s [M Pa ] Strain [mm/mm] p=90 (e) A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 256 EXPERIMENTAL RESULTS lain and notched specimens were additively manufactured via 3D-printer Ultimaker 2 Extended+ by using as parent material New Verbatim filaments of white PLA with initial diameter of 2.85mm. The values of the adopted manufacturing parameters were as follows: nozzle size=0.4 mm, nozzle temperature=240ºC, build-plate temperature=60ºC, layer height=0.1 mm, shell thickness=0.4 mm, fill density=100%, and print speed=30 mm/s. According to Fig. 2, the flat samples being tested were manufactured horizontally on the build-plate by setting angle θp equal to 0º, 30º, 45º, 60º, and 90º. In particular, while the extruded filaments were deposited, layer upon layer, always at ±45º to the axis yp of the build-plate, the angle between the longitudinal axis of the specimens and axis yp was varied in the range 0º-90º (see Fig. 2). All samples had thickness, t, equal to 4mm, with the un-notched specimens having net width equal to 15 mm. The sharply U-notched specimens being manufactured had net width, wn, equal to 15.4 mm, gross width, wg, to 24.9 mm, and notch root radius, rn, equal to 0.5 mm. These dimensions returned a value for the net stress concentrator factor, Kt, equal to 4.76. The specimens containing the intermediate U-notches had the following average dimensions: wn=15.3 mm, wg=24.9 mm, and rn=1.0 mm (resulting in a Kt value equal to 3.51). Finally, the dimensions of the bluntly U-notched specimens were as follows: wn=15.2 mm, wg=25.1 mm, and rn=3.0 mm (Kt=2.22). For the notched specimens being investigated the corresponding values for the stress concentration factors were estimated using Peterson’s diagrams [14]. A number of specimens containing crack-like notches were also additively manufactured in order to determine the fracture toughness for t=4 mm. In particular, the relevant dimensions for these specimens were as follows: wn=16.4 mm, wg=24.9 mm, and rn≈0.01 mm. By setting the displacement rate equal to 2 mm/min, the plain specimens as well as the samples containing both U- notches and crack-like notches were tested under quasi-static tensile loading by using a Shimadzu universal machine. In the un-notched specimens, the local strains were measured during testing via an axial extensometer with gauge length equal to 50 mm. Three different specimens were tested for any geometry/manufacturing configuration that was investigated. (a) (b) (c) Figure 4: Influence of manufacturing angle θp on E (a), σUTS (b), and KC (c). 2000 2500 3000 3500 4000 0 10 20 30 40 50 60 70 80 90 E [ M Pa ] Manufacturing Angle, θp [] +2SD -2SD E=3296 MPa 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 σ U T S [M P a] Manufacturing Angle, θp [] +2SD -2SD σUTS=42.5 MPa 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 10 20 30 40 50 60 70 80 90 K C [M Pa ·m 1/ 2 ] Deposition Angle, θp [] +2SD -2SD Kc=3.4 MPa·m 1/2 P A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 257 The stress vs. strain diagrams reported in Fig. 3 show the mechanical behaviour as measured by testing the un-notched specimens. These charts make it evident that the stress vs. strain response of the additively manufactured material being tested was characterised by a mechanical behaviour that was predominantly linear up to the maximum stress value being recorded during testing. In terms of reference mechanical properties, the results generated by testing the plain samples returned the following average values: Young’s modulus, E, equal to 3296 MPa, 0.2% proof stress, σ0.2%, equal to 40.3 MPa and ultimate tensile strength, σUTS, equal to 42.5 MPa. The charts reported in Figs. 4a and 4b summarise the influence of manufacturing angle θp on both E and σUTS. These diagrams make it evident that the infill direction had little effect on the overall mechanical behaviour of the additively manufactured material being investigated. In particular, the experimental results expressed in terms of both E and σUTS in Figs. 4a and 4b, respectively, are seen to be all within two standard deviations, SD, of the mean. The results generated by testing specimens with crack-like notches are summarised instead in the KC (for t=4 mm) vs. θp diagram of Fig. 4c. The values for KC shown in this chart were estimated according to Linear Elastic Fracture Mechanics (LEFM), i.e. by using the following standard relationship [15]:     C fK a (5) In Eq. (5) α is the shape factor (estimated as recommended in Ref. [16]), σf is the nominal failure stress referred to the gross area, and a is the crack-like notch depth. Since, according to Fig. 4c, the experimental values for KC are within two standard deviations of the mean, it is possible to come to the conclusion that also the fracture resistance of the additively manufacture polymer under investigation was marginally affected by manufacturing angle θp. To conclude, it is worth observing that, due to such a high level of consistency, the notched specimens being tested were manufactured by setting angle θp solely equal to 0º, 30º, and 45º. Figure 5: Linear-elastic stress fields in the incipient failure condition and determination of critical distance L. VALIDATION BY EXPERIMENTAL DATA y considering notched specimens of polymethylmethacrylate (PMMA), in 2004 Taylor et al. [17] observed that the inherent strength, σ0, of this material takes on a value that is equal to about 2·σUTS. Accordingly, the standard formalisation of the TCD considered in the present paper can be used to perform the static assessment of PMMA as long the notches being assessed result in stress concentration factors that are larger than σ0/σUTS≈2. Having recalled these important aspects, in the preliminary investigating summarised in the present paper the accuracy of the TCD in estimating the static strength of notched additively manufactured PLA was checked by forming the following simplifying hypotheses: 0 50 100 150 200 250 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1 [MPa] Distance, x [mm] Linear-elastic stress fields Sharp Intermediate Blunt 0=85.0 MPa Error= +20% Error= -20% L/2=0.467 mm UTS=42.5 MPa B A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 258  according to Figs. 4b and 4c, the static strength and the fracture toughness of the additively manufactured material being tested were assumed to be independent from manufacturing angle θp;  as per PMMA [17], the inherent strength, σ0, of the additively manufactured material under investigation was taken invariably equal to 2·σUTS – i.e., σ0=2·σUTS=2·42.5 MPa=85 MPa;  the linear-elastic stress fields in the vicinity of the stress concentrators being tested were estimated by simply using the analytical solutions devised by Glinka and Newport [18]. Fig. 5 shows the linear-elastic stress fields determined in the incipient failure condition, where the centre of the adopted system of coordinates was taken coincident with the notch tip (Fig. 1a). It is worth observing here also that, for any considered notched geometry, the corresponding stress field was calculated by setting the nominal net stress equal to the average failure stress determined from the nine tests run by using the specimens manufactured with angle θp equal to 0º, 30º, and 45º (Fig. 2). As shown in Fig. 5, the critical distance value was estimated according to the PM via the point at which the linear-elastic stress-distance curve due to the sharp notch and the horizontal straight line modelling the plain material inherent strength intersected each other [1]. According to Fig. 5, this simple procedure returned a value for L of 0.934 mm. Fig. 5 makes it evident also that by setting L/2=0.467 mm and σ0=85 MPa, the PM was seen to be very accurate also in estimating the average static strength of the specimens containing both the intermediate and the blunt notches. In order to show the overall accuracy of the PM in performing the static assessment of notched additively manufactured PLA, the chart of Fig. 6 summarises the error associated with the individual experimental results, with the error being calculated as:         0 0 % 100 eff Error (6) According to the error diagram reported in Fig. 6, it is possible to conclude by observing that the TCD applied in the form of the PM was seen to be capable of accurately estimating the static strength of notched additively manufactured PLA, with its usage returning predictions falling within an error interval of ±20%. Figure 6: Overall accuracy of the TCD applied in the form of the PM in estimating static strength of additively manufactured PLA. CONCLUSIONS ccording to the analyses summarised in the present paper, the following conclusions can be drawn:  the TCD is successful in predicting static failures in notched additively manufactured PLA;  the estimates obtained by applying the TCD in the form of the PM were seen to fall within an error interval of ±20%; -100 -80 -60 -40 -20 0 20 40 60 80 100 0 5 10 15 20 25 30 35 40 45 E rr o r [% ] Deposition Angle, θp [] Point Method (PM) Kt=4.76 Kt=3.51 Kt=2.22 Error= +20% Error= -20% A A. A. Ahmed et alii, Frattura ed Integrità Strutturale, 41 (2017) 252-259; DOI: 10.3221/IGF-ESIS.41.34 259  satisfactory results were obtained by forming the hypothesis that the mechanical behaviour of additively manufactured PLA follows a simple linear-elastic constitutive law;  more work has to be done in this area to systematically check the accuracy and reliability of the TCD when used to predict static failures in additively manufactured materials not only characterised by different elasto-plastic behaviours, but also failing by different mechanisms. REFERENCES [1] Taylor, D. The Theory of Critical Distances: A new perspective in fracture mechanics. Elsevier, Oxford, UK (2007). [2] Taylor, D. Predicting the fracture strength of ceramic materials using the theory of critical distances. Engng. Fract. Mech. 71 (2004) 2407-2416. [3] Susmel, L., Taylor, D. The theory of critical distances to predict static strength of notched brittle components subjected to mixed-mode loading. Eng Frac Mech, 75 3-4 (2008) 534-550. [4] Susmel, L., Taylor, D. On the use of the Theory of Critical Distances to predict static failures in ductile metallic materials containing different geometrical features. Engng. Fract. Mech. 75 (2008) 4410-4421. [5] Susmel, L., Taylor, D. The Theory of Critical Distances to estimate the static strength of notched samples of Al6082 loaded in combined tension and torsion. Part I: Material cracking behaviour. Engng. Fract. Mech. 77 (2010) 452–469. [6] Susmel L., Taylor D. The Theory of Critical Distances to estimate the static strength of notched samples of Al6082 loaded in combined tension and torsion. Part II: Multiaxial static assessment. Engng. Fract. Mech. 77 (2010) 470–478. [7] Susmel, L., Askes, H. Material length scales in fracture analysis: from Gradient Elasticity to the Theory of Critical Distances. Computational Technology Reviews 6 (2012) 63-80. [8] Susmel, L., Askes, H., Bennett, T., Taylor, D. Theory of Critical Distances vs. Gradient Mechanics in modelling the transition from the short- to long-crack regime at the fatigue limit. Fatigue Fract Engng Mater Struct. (2013). [9] Askes, H., Livieri, P., Susmel, L., Taylor, D., Tovo R. Intrinsic material length, Theory of Critical Distances and Gradient Mechanics: analogies and differences in processing linear-elastic crack tip stress fields. Fatigue Fract Engng Mater Struct. 36 (2013) 39-55. [10] Hamad, K., Kaseem, M., Yang, H.W., Deri, F., Ko, Y.G.. Properties and medical applications of polylactic acid: a review. eXPRESS Polymer Letters 9 (2015) 435–455. [11] Whitney, J.M., Nuismer, R.J. Stress fracture criteria for laminated composites containing stress concentrations. J. Composite Mater. 8 (1974) 253-265. [12] Sheppard, S.D. Field effects in fatigue crack initiation: long life fatigue strength. Transactions of ASME, Journal of Mechanical Design 113 (1991) 188-194. [13] Bellett, D., Taylor, D., Marco, S., Mazzeo, E., Guillois, J., Pircher, T. The fatigue behaviour of three-dimensional stress concentrations. Int. J. Fatigue 27 (2005) 207-221. [14] Pilkey, W.D., Pilkey, D.F. Peterson's Stress Concentration Factors. Wiley, USA, (2008). [15] Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, CRC Press, USA, (2005). [16] Tada, H., Paris, P.C., Irwin, G.R. Stress Analysis of Cracks Handbook, ASME Press, USA, (2000). [17] Taylor, D., Merlo, M., Pegley, R., Cavatorta, M.P. The effect of stress concentrations on the fracture strength of polymethylmethacrylate. Materials Science and Engineering A 382 (2004) 288-294. [18] Glinka, G., Newport, A. Universal features of elastic notch-tip stress fields. Int. J. 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