Variability, heterogeneity, and anisotropy in the quasi‐static

SAVE OFFLINE

Received: 11 December 2015

Revised: 7 October 2016

Accepted: 20 October 2016

DOI 10.1111/str.12219

F U L L PA P E R

Variability, heterogeneity, and anisotropy in the quasi‐static response of laser sintered PA12 components M. Faes1 | Y. Wang2 | P. Lava3 | D. Moens1 1

Department of Mechanical Engineering— Technology Campus De Nayer, KU Leuven, BE‐ 2860 St.‐Katelijne‐Waver, Belgium

2

Department of Materials Engineering— Technology Campus Ghent, KU Leuven, BE‐9000 Ghent, Belgium

3

MatchID Metrology Beyond Colors, BE‐9820 Merelbeke, Belgium Correspondence Matthias Faes, Department of Mechanical Engineering—Technology Campus De Nayer, KU Leuven, BE‐2860 St.‐Katelijne‐Waver, Belgium. Email: [email protected]

Abstract Additive manufacturing (AM) receives an increasing industrial interest thanks to its advantages in the economic production of highly complex and small‐series components. Especially laser sintering (LS) is in this context of particular interest for the production of plastic components, as it is generally deemed the most robust AM technology for polymer parts and therefore is expected to enable AM for functional components in the near future. However, to date, designers are often confronted with a severe lack of knowledge on the possible mechanical behavior of AM components. More specifically, the unit‐to‐unit variability, heterogeneity (within‐part variation), and anisotropy of the mechanical properties very often prove to be substantial and therefore require more elaborated studies in order to take these effects into account in the engineering of reliable components. Moreover, typical experimental results that are used for the determination of the elastic stiffness tensor are subject to variability, caused by the influence of the difference in thermal history between produced parts. This work therefore focuses first on the identification and quantification of the variability and heterogeneity in the quasi‐static response of laser sintering‐polyamide 12 (LS‐PA12) components. Second, also the anisotropy in this quasi‐static response is studied. For the first part, uniaxial tensile tests are performed and the variability on the quasi‐static properties is quantified by means of statistical analysis. Also, the elastic stiffness tensor is identified based on these tests. Next, the heterogeneity in the tested specimens is investigated by means of digital image correlation. Finally, in order to study the anisotropy in the quasi‐static properties, the Virtual Fields Method is applied to determine the variability in the elastic stiffness tensor of the LS‐PA12 material. A variability with a coefficient of variance of up to 6.5% on Young’s modulus was measured. It was also found that the production planning has an important influence on the homogeneity of the mechanical properties of the produced parts. Finally, the Virtual Fields Method showed that, contrary to most literature on the topic, the elastic properties of LS‐PA12 material is best described using an isotropic material model. K E Y WO R D S

additive manufacturing, digital image correlation, laser sintering, mechanical testing, virtual fields method

1 | IN T RO D U C T IO N 1.1 | Laser sintering of polymers Thanks to the large freedom in design, low time‐to‐market, and efficient usage of materials and resources, additive Strain 2017;53:e12219. https://doi.org/10.1111/str.12219

manufacturing (AM) receives an increasing industrial interest, especially in sectors where weight reduction plays an important role. This is for example the case for automotive and aerospace applications, where weight reduction is directly translated to a reduction in fuel consumption. Also for medical, patient‐oriented products, such as in the biomedical

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© 2016 John Wiley & Sons Ltd

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sector, a large impact is expected. The laser sintering (LS) process is in this context generally deemed to be the most robust AM process for plastic materials, with a realistic possibility of producing full functional components in the near future.[1,2] LS builds up complex parts based on a digital 3D model through a layer‐by‐layer approach. A fine layer, with a thickness between 60 and 150 μm and of polymeric powder, is deposited using a recoating blade or counter‐rotating rollers. This polymeric material is preheated to just below its melting point (for a semi‐crystalline polymer) or glass transition temperature (for an amorphous polymer) and selectively irradiated by a laser source, inducing a sintering process that connects the deposited particles and binds them to previously sintered layers. After completion of the build, the material is cooled down so that crystallization can occur and loose powder is removed.[3] Due to the layer‐by‐layer nature of this process, complexity can be added to the design without increasing the cost of the production. This enables the economic production of weight‐optimized, tailored small series.[4] A typical layout of a LS machine is shown in Figure 1. To date, different commercial machines exist, which differ in, for example, the type of laser, powder feed mechanism, or means of preheating.[5,6] In practice, polyamide 12 (PA12), which is a thermoplastic engineering polymer, is the most commonly used material for the production of components using LS.[1] 1.2 | Effect of process parameters on the quasi‐static properties It is generally accepted that the laser energy density (ED) is the most important factor that influences the quasi‐static mechanical properties such as Young’s Modulus (E), ultimate tensile strength (σr), or elongation at break (ϵr) of LS‐PA12 components, as indicated by numerous authors.[7–16] This parameter in fact consists of the power of the laser source (P), the scan speed of the laser (S), and the spacing between adjacent tracks (x):
 P J (1) ED ¼ Sx mm2

FIGURE 1 Typical layout of laser sintering machine (published with kind permission of Kruth et al.[5])

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Generally, a higher ED leads to parts containing less pores, which makes them stronger and stiffer.[10,16,17] In this context, Zarringhalam et al. found that when more energy is applied to the PA12 powder, a higher degree of melting of the particles occurs.[18] However, when the applied energy reaches a material‐dependent limit value, the polymeric material degrades due to excessive heat accumulation, and consequently, the mechanical properties deteriorate.[10] Next to ED, also, the preheating temperature of the powder bed is found to show an important impact on the quasi‐static properties of these parts, because it influences the conditions under which the sintering of the polymeric particles occurs and thus the resulting degree of porosity in the component. Moreover, when the preheating temperature is too low (i.e., below the crystallization temperature of the material), unwanted crystallization occurs, leading to internal stresses in the material and possibly warping of the part, as was found by numerous authors.[6,13,14] In terms of process planning, the location of the component in the build platform also exhibits a significant effect on Young’s modulus of sintered PA12 material, due to inhomogeneity in the powder bed preheating temperature.[6,19] Also, the cooling rate of the material after completion of the build influences the morphology of LS‐PA12 parts and consequently, the mechanical properties. Zarringhalam et al. found that slower cooling after completion of the build leads to a higher degree of crystallinity in the resulting part, leading to stronger and slightly stiffer parts.[20] Finally, Jain et al. found that the delay time between the production of two subsequent layers significantly influences the tensile strength of PA12 parts produced via LS. They state that longer delay times yield lower tensile strength due to improper binding of subsequent layers.[21] It can be understood that the aforementioned production parameters directly influence the thermal history of the material during production by either impacting the temperature of the previous layer during scanning of the current layer (preheating temperature and its distribution throughout the powder bed, delay time between layers), the amount of thermal energy that is put into the current layer (energy density), or the time during which the sintered material is kept above its glass transition or crystallization temperature (cooling rate). Different researchers showed that in this way, the morphology of the parts in terms of porosity content as well as crystallinity is strongly influenced by these parameters. Consequentially, the mechanical properties of the resulting parts are also affected. Due to the large complexity of the effect of aforementioned production parameter variations and environmental variables during the LS process, produced components typically show a large variability in their mechanical response, with experimentally obtained Young’s moduli in literature reported ranging from 900 to 2300 MPa for PA12. For the tensile strength of parts produced with this material, a similar observation is made, with values ranging from 19.5[14] to 43 MPa,[22] even though samples were produced using similar machines.

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1.3 | Constitutive mechanical model of LS‐PA12 parts Research on the quasi‐static mechanical response of LS‐ PA12 parts shows that the mechanical response is typically governed by either isotropic, transverse isotropic, or orthotropic constitutive equations on a macroscopic scale, with the principal axes of the model oriented parallel to the three principal axes of the machine, as is shown by numerous authors.[19,22–26] In this regard, Ajoku et al. found orthotropic properties in terms of Young’s modulus, ranging from 1817 to 2047 MPa for the different axes of the model, using a 3D‐Systems Vanguard LS machine.[25] Jollivet et al. found similar values for an orthotropic model using an EOS Formiga LS machine.[22] On the other hand, Amado‐Becker et al. employed an ultrasonic technique to determine the elastic constitutive model of LS‐PA12 samples, identifying the samples as isotropic.[26] Cooke et al. found transverse isotropy in the Young’s modulus of LS‐PA12 samples, produced with a Sinterstation HiQ platform using a statistically relevant population of 144 test samples.[23] Finally, Wegner and Witt found that the degree of anisotropy is a function of the applied laser energy density during the process, showing transverse isotropy at lower laser energy density values and isotropy at higher values.[19] 1.4 | Objective and paper outline So far in literature, most studies studying mechanical properties of LS‐PA12 parts start from tensile specimens built according to the three principal axes of the machine. Although these samples explicitly incorporate the effect of the orientation of the intra‐ and inter‐layer bonding characteristics with respect to the loading and therefore are essential to build up the constitutive model, at the same time, the effect of the building orientation on the thermal history of the component throughout its building process in general stays unaccounted for. Indeed, these different samples often exhibit a significantly different thermal history due to differences in delay time between subsequent layers preheating temperature and post‐build cooling time, all of which are predominantly influenced by the build layout (number, orientation, and relative position of the samples in one build). In general, it is expected that this difference in thermal history can cause a high variability in the obtained results, which in some cases, might even surpasses the effect of the building orientation, causing a severe error when interpreting the data based on the constitutive model of the LS‐PA12 material. Based on these observations, it is concluded that there still remains a lack of knowledge on the possible mechanical material behavior of LS‐PA12 components. More specifically, the variability and anisotropy in the mechanical behavior of LS‐PA12 parts as observed in literature requires more elaborated studies in order to unravel the causes of the significant spread in the reported data. Furthermore, very little is currently known on the in‐component variability (heterogeneity) of the mechanical material properties due to

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the production process. It is expected that the same production parameters that affect the inter‐variability (variation between different specimens), most probably also impose a significant spread on the intra‐variability of the mechanical properties (variability within a specimen). Knowledge on these aspects is needed for the reliable design of functional components for realistic engineering applications and for the further optimization of the LS process. This work therefore focuses on the identification and quantification of the variability and anisotropy in the quasi‐ static response of LS‐PA12 components. Also, the heterogeneity in this quasi‐static response is studied. First, uniaxial tensile tests are performed and the variability on the quasi‐ static properties is quantified by means of statistical analysis. Also, the elastic stiffness tensor is identified based on these tests, according to the standard methodology in literature. Moreover, the heterogeneity in the tested specimens is investigated by means of digital image correlation (DIC), a contactless full‐field strain measurement technique. Finally, in order to study the anisotropy in the quasi‐static properties, it is proposed to use the virtual fields method (VFM) to determine the variability in the elastic stiffness tensor of the LS‐PA12 material. In this context, a test geometry is defined, which induces a biaxial strain state upon uniaxial loading. An overview of the employed methodology is given in paragraph 2. The results are listed and discussed in paragraph 3. Finally, paragraph 4 gives the conclusions of the paper and a vision for future research.

2 | M E T H O D O L O GY 2.1 | Uniaxial tensile testing In order to analyze the variability in the quasi‐static properties of LS‐PA12 components, a statistically significant population of 30 samples per orientation is built. Uniaxial tensile dogbone samples (ASTM D638—type III) are produced on a EOS P395 LS‐Machine in a blend of virgin and reused PA2200 PA12 material with a melt volume ratio of 24 cm3/10 min. Optimized commercial production parameters are employed. These samples are aligned with the three major building orientations of the machine (X, Y, Z) as indicated in Figure 2, with Z being the layering direction, resulting in a total test population of 90 samples. The different orientations are further denoted as Upright for the specimen, the main axis of which lies parallel to the Z‐axis, Edge for the specimen built on its edge along the X‐axis, and Flat for the specimen lying flat along the Y‐axis. Uniaxial tests are performed using an INSTRON 5900R testing machine, equipped with a 50‐kN load cell, at a strain rate of 1 mm/min. The length between the grips of the machine is kept constant for all tests at 100 mm, while also ensuring that the length of the specimen in the clamp is constant for all tests. An Instron 2620 50/5 mm extensometer with a gage length of 50 mm and maximum travel of 5 mm

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lenses and the corresponding load levels are registered. Thus, 255 grey levels can be measured. Deformation fields are calculated by considering a pixel and its neighborhood (subset) in the undeformed image f and tracking this subset in the deformed image. A zero‐normalized sum of squared differences matching criterion r is minimized to determine s, the displacement vector of a subset in image f with respect to image g:   2  32  ′ ′ f xi ; yj −f m g xi ; yj −gm 5 − r¼ ∑ ∑ 4 Δf Δg i¼−M j¼−M M

with

M

(3)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M  

2 Δf ¼ ∑ ∑ f xi ; yj −f m i¼−M j¼−M

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  i2 M M h  Δg ¼ ∑ ∑ g x′i ; y′j −gm

(4)

i¼−M j¼−M

FIGURE 2

Orientations along which the uniaxial samples were built

is used to measure the longitudinal strain averaged over the strain gage length. Young’s moduli (E) for each building orientation are calculated by determining the slope of the stress–strain relationship of the samples of the different orientations between 0.0005 and 0.0025 strain by means of linear regression. An orthotropic constitutive model is built under the assumption that the measured elastic mechanical properties in these directions correspond to their respective terms in the stiffness tensor: 0

σ 11

1

2

Q11

B C 6 @ σ 22 A ¼ 4 Q12 0 σ 66

Q12 Q22 0

0

30

ϵ11

1

7B C 0 5@ ϵ22 A Q66 ϵ66

(2)

where Qij are the in‐plane constitutive parameters of the assumed orthotropic material model. The tensile strength (σr) and elongation at break (ϵr) are considered as well. In order to study statistically the significance between the different considered orientations, pairwise t‐tests are employed. Additionally, subset‐based stereo DIC is employed both to determine Poisson’s ratio and to measure the homogeneity of the induced strain fields in the component. DIC is a contactless full‐field strain measurement technique, capable of measuring the occurring strain fields and crack propagation in loaded samples. Before the test, a random pattern is applied to the surface of the specimen using a spray painting technique. During the tensile test, a picture of the specimen is taken each 50 ms using two 8‐bit 2‐MP cameras, equipped with 50‐mm

where fm and gm are the ensemble averages of respectively the reference and deformed subset in f and g. The summation of i and j from −M to M indicates that the sum over the entire subset is considered. x and y are coordinates in the reference picture. x′ and y′ are the corresponding coordinates in the deformed subset, which are related to x and y by the shape functions that are employed. Specifically, affine shape functions were employed to account for deformation of the subset between images f and g. The zero‐normalized sum of squared differences criterion was specifically chosen for its robustness with respect to changes both in picture illumination and contrast, which is extremely relevant for stereo‐DIC measurements. Sub‐pixel accuracy is obtained by interpolating the pictures using bicubic splines.[27–30] A strain window of 15 px was used for the calculation of the transversal, longitudinal, and shear strain fields (ϵxx , ϵyy , ϵxy), resulting in a virtual strain gage of 91 px. The corresponding used step‐ and subset sizes are respectively 7 and 25 px. Missing data on the edges, originating from the employed correlation algorithm, is compensated for during the determination of the displacement fields. Data up to the edges are needed in view of a correct identification of material parameters by the VFM. This compensation method relies on the determined subset shape functions and does not involve an extrapolation step.[27] Ten images of the unloaded component are taken and averaged in order to determine the noise of the setup, expressed as a percentage of the bit‐depth of the cameras. The DIC measurements are performed using MatchID software[31]The parameters that are used for the DIC calculations are listed in Table 1. Poisson’s ratio was computed as vij ¼

ϵj ϵi

(5)

In order to obtain ϵj and ϵi, the strain fields ϵxx and ϵyy are averaged over the gage length of the tensile specimen. No

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TABLE 1

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Adopted parameters for the DIC measurements

Parameter

Value

Dynamic range

8 bit

Matching Criterion

ZNSSD

Interpolation

Bicubic splines

Shape function

Affine

Noise

0.4593%

Prefiltering

Gaussian, kernel 5

Step size

7 px

Subset size

25 px

Strain window

15 px

Virtual Strain Gage

91 px 1.213·10−04

Strain Resolution

Note. DIC = digital image correlation; ZNSSD = zeronormalized sum of squared differences.

separate test for the determination of Shear moduli Gij was performed. Huber’s equation was used to approximate the shear moduli Gij through the measured Young’s moduli and Poisson’s ratios. Huber[32] showed that Gij can be approximated by pffiffiffiffiffiffiffiffiffiffiffi E ii E jj  Gi;j ¼ pffiffiffiffiffiffiffiffiffi 2 1 þ vij vji

(6)

where Eii are the measured Young’s moduli and νij are Poisson’s ratios. The accuracy of this estimation largely depends on the independency of Gij with respect to E and ν. For a perfect orthotropic material, this dependency is zero, and conversely, the result of this equation will be inaccurate. The accuracy of this estimation will be cross‐checked by cross‐comparison of the results with the results of the VFM (Section 2.3). Finally, the degree of orthotropy of the quasi‐ static elastic properties of the LS‐PA12 material is defined as Oij ¼

E ii −1 E jj

(7)

In order to strengthen the conclusion on possible anisotropy, a pairwise t‐test is performed to check if the mean values of the measured Young’s moduli per orientation differ significantly. Additionally, the density of the components after production is determined by means of the Archimedes method using ethanol as fluid, as it is expected that this influences the mechanical properties of the specimens.[33] Ethanol was specifically used because the density of PA12 is lower than the density of water. The density of the component is calculated as ρ ¼ m*

ρeth −0:0012 þ 0:0012 0:99983*ðm−m′Þ

component, and m’ is the mass of the component submerged in ethanol. The measurement was corrected for air buoyancy by taking the density of air under standard condition (0.0012 g/cm3) into account. The factor 0.99983 is employed to correct the measurement for the buoyancy that is caused by immersing the measuring pan into the ethanol. Finally, the relative density is subsequently calculated by dividing by the density of fully dense PA12 (1.03 g/cm3).[33] It should be noted that only closed porosity is measured using this technique, as no coating was used. However, an initial study where the density measurements were performed using both coated and uncoated samples showed that the samples contain for 99.5% closed porosity. The influence of open pores is therefore neglected in this study. Finally, scanning electron microscopy (SEM) was performed to study the fracture surfaces of the tested specimens. Specifically, a Tescan Vega3 LMU SEM was used in combination with a low vacuum secondary electron detector. To prevent loading up of the uncoated polymer samples, images were taken at 10 keV, with a nitrogen chamber pressure of 15 Pa. 2.2 | Statistical analysis of the variability Because it is the objective of this paper to identify the variability on the mechanical properties, statistical analysis is required on the measured Young’s moduli. In the context of identifying the distribution that best describes the variability that is present in these data, the Anderson–Darling (AD) distance dAD between the data and some hypothesized distributions is computed.[34] In this regard, it is tested how the statistical distribution of Young’s modulus can be explained using a normal, log‐normal, Weibull, or gamma distribution. The AD distance is a quadratic‐type distance between the empirical distribution function Fn(x) (i.e., the measurement data) and the hypothesized distribution function Fθ(x), where more weight is given to the tails of the distribution as compared with the Kolmogorov–Smirnov or Cramé–Von Mises test. This is relevant for identification of distributions for use in reliability estimation. dAD is specifically computed as ∞

½F n ðxÞ−F θ ðxÞ2 −∞ F θ ðxÞð1−F θ ðxÞdF θ ðxÞ

dAD ¼ n ∫

(9)

Because Fn(x) is an empirical distribution (and thus a step function), the above integral is approximated using a finite sum over the parts in x. Numerically, dAD is computed as n

2i−1 ½ lnðF θ ðxi ÞÞ þ lnð1−F θ ðxnþ1−i ÞÞ (10) i¼1 n

dAD ¼ −n−∑ (8)

where ρ is the density of the produced component, ρeth is the density of ethanol (0.78800 kg/m3), m is the dry mass of the

where { x1 , x2 , … , xn} are the ordered n sample data points, being the results in terms of Young’s modulus of the performed tensile tests.

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The hypothesized distribution exhibiting the lowest AD distance is selected to be the best fitting distribution. Additional to dAD, a decision limit p is calculated. The candidate distribution will only be accepted if the corresponding p‐value is larger than the chosen threshold of α = . 05, because this expresses the confidence in the fitting of the distribution.

where σ is the stress tensor, b are volume forces working on Ω, and a is a certain distribution of acceleration through Ω. Finally, ϵ* is the virtual strain tensor, and u* is the virtual displacement vector. This equation is valid for any continuous, kinematic admissible (KA) virtual displacement field. In a static experiment, accelerations are negligible and volume forces such as gravity are usually constant throughout the test. Equation 8 therefore reduces to

2.3 | Determination of the constitutive parameters using the virtual fields method

∫ σ : ϵ dΩ ¼ ∫ T:u dΓ

In order to study the anisotropy in the elastic stiffness tensor of LS‐PA12 material based on a single tensile test, the VFM is applied and compared with the results obtained from uniaxial testing. For this purpose, a test geometry was defined, which induces a biaxial stress state upon uniaxial loading. This geometry is shown in Figure 3. The width of this part is 52 mm, and the thickness is 6 mm. The rectangular filleted hole in the middle measures 21.2 by 21.2 mm with a radius of 5 mm for the fillets. By uniaxial loading of this component, all constitutive parameters for plane‐stress are activated and can thus be measured through the VFM. For each orientation (Edge, Flat, and Upright), four samples are built and tested using DIC. The constitutive model parameters for each of these orientations are determined using the VFM, and it is checked whether they differ statistically. The VFM, as introduced by Grédiac and Pierron, is an experimental‐numerical technique to determine the elastic stiffness tensor Q, based on a measurement of the strain tensor and the traction T in a loaded component, regarded over the length Lf.[35–39] The VFM is based on the principle of virtual work (Equation 11), which for a body with a volume Ω and surface area Γ can be written as −∫ σ : ϵ dΩ þ ∫ T:u dΓ þ ∫ b:u dΩ ¼ ∫ ρ:a:u dΩ ∀u KA Ω

Γ

Ω

Ω

(11)

ET AL.

Γ

Ω

∀u KA

(12)

Finally, assuming plane‐stress in a homogeneous orthotropic material, Equation 9 yields Q11 ∫ ϵ11 :ϵ11 dS þ Q22 ∫ ϵ22 :ϵ22 dS Sv Sv   þ Q12 ∫ ϵ11 :ϵ22 þ ϵ22 :ϵ11 dS þ Q66 ∫ ϵ66 :ϵ66 dS Sv

¼ ∫ T:u dL

Sv

∀u KA

(13)

Lf

where Qij are the in‐plane constitutive parameters of the orthotropic material model, as presented in Equation 2, Sv is the surface of the component over which the strain fields are measured, and Lf is the length over which the traction T works on the component. The measurement of ϵ and u is discrete due to the employed DIC technique. Therefore, the integrals in the above equations are approximated using discrete sums over all the pixels of the cameras covering the part under consideration. By choosing the number of independent virtual fields equal to the number of unknown Qij, a set of equations is generated, which yields the constitutive model of the material. Piecewise parameterization of the virtual fields is employed by representing the measured surface as a 9 × 9 array. The employed fields are determined in order to minimize noise on the measurement data.[39] The engineering constants of

Biaxial testing geometry with a spray‐paint speckle pattern (left) and\r\npiecewise parametrization of the virtual fields depicted on a transversal strain\r\nfield (right). The total width of the specimen measures 52 mm

FIGURE 3

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the material (Ei, νij , and Gij ) are finally calculated from the thus obtained Qij using the following set of equations: Q11 ¼

Q22 ¼

Q11 ¼

E1

Eii [MPa]

νij [/]

Upright

1660

0.394

St. Dev.

108

0.0034

CoV

6.5%

0.86%

Edge

1620

0.387

St. Dev.

91

0.0008

CoV

5.61%

0.21%

Flat

1680

0.409

St. Dev.

98

0.0067

CoV

5.83%

1.63%

(14)

ν2 E 1− 12E1 2

E2

(15)

ν2 E 1− 12E1 2

E 1 ν12 1−

Variability in the mechanical properties of the laser sintering samples, per orientation. The * indicates that the quantity was calculated from Eii and νij

TABLE 3

(16)

ν212 E 2 E1

Q66 ¼ G12

Gij ½MPa

O12 ½%

587

2.46

588

−3.57

600

1.2

(17)

3 | R E S ULT S A N D D I S C U S S I O N 3.1 | Uniaxial test results Table 2 shows the density (ρ), tensile strength (σr), and elongation at break (ϵr) of the 30 tested samples. The tensile strength for the Upright, Edge, and Flat orientations is measured to be respectively 42.4, 45.0, and 46.0 MPa with a respective coefficient of variance (CoV) of 1.6%, 0.66%, and 1.31%. Elongation at break for these samples measures respectively 8.7%, 23.0%, and 23.0%, with a respective CoV of 10.5%, 10.5%, and 9.91%. The density (ρ) for the Edge, Upright, and Flat samples is measured to be 0.966, 0.971, and 0.982 kg/m3, leading to relative densities of 93.8%, 95.3%, and 94.3%. Table 3 shows the measured constitutive properties as obtained from the uniaxial tensile testing. Young’s modulus for the Edge, Upright, and Flat orientations is measured to be 1620, 1660, and 1680 MPa, respectively, with a respective CoV of 5.61%, 6.5%, and 5.83%. Poisson’s ratio for Edge, Upright, and Flat orientations is found to be 0.394, 0.387, and 0.409, respectively, with a maximum CoV of 1.63% for the Flat direction. The shear modulus is calculated using Equation 3 and found to be 588.58, 586.48, and

600.54 MPa, respectively. The variability on the shear modulus is omitted as this value is calculated from the measured Young’s modulus and Poisson’s ratio. Finally, the degree of orthotropy between the different orientations is relatively low, with a maximum of 3.57% between the Edge and Flat direction. First, it can be concluded that a substantial variance of up to 6.5% is detected in Young’s modulus when the 30 replicas per orientation are considered. This variability stems from the fact that the tensile bars were distributed throughout the build platform of the machine. The influence of the location in the build platform on the mechanical properties was also noted by Goodridge et al.[6] and Cooke et al.[23] and is attributed to inhomogeneity in the preheating of the polymer throughout the build platform. Figure 4 shows in this context the measured Young’s modulus of the Upright specimens, as a function of the vertical (x) location in the build platform for this build, averaged over the five replicas that were placed at the same vertical location. As can be noted, a decreasing trend towards the sides is clearly visible, which is in correspondence with literature. Also, the standard deviation is considerably lower in the middle of the build platform, as compared with the sides. The same trend is visible when the measurements are plotted over the longitudinal (y) direction of the build platform.

TABLE 2 Density, tensile strength, and elongation at break for LS‐PA12 including the variability

ρ [kg/m3]

p (%)

Upright

0.982

95.3

42.4

8.7

St. Dev.

0.002

1.2

2.4

COV

0.18%

1.60% 93.8

σr [MPa]

Edge

0.966

St. Dev.

0.003

0.3

COV

0.32%

0.66%

Flat

0.971

St. Dev. COV

94.3

45.0

ϵr [%]

10.5% 23.0 2.4 10.5%

46.0

23.0

0.004

0.6

0.9

0.40%

1.31%

9.91%

Influence of the distance to the center of the build in the x direction on Young’s modulus

FIGURE 4

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This rather large variability is also visible on the elongation at break of these samples, whereas it is not as pronounced for the tensile strength. It is noteworthy that the density of the measured samples does not show a significant variability either. It can thus be concluded that it is not the density of the samples that is the morphological factor determining the measured variability in Young’s modulus and elongation at break. Concerning Young’s modulus, this can be explained by the fact that the effect of the location in the build masks the effect of a lower porosity. As concerns elongation at break, not only the number of pores in the specimen but also the shape and orientation of these pores w.r.t. the loading direction show an influence on the resulting elongation at break, as they act as stress concentrators in the part. Second, the observations clearly show that Young’s modulus for the different orientations is similar, as also indicated by the relatively low degrees of orthotropy that were obtained. Pairwise t‐testing between the Young’s moduli of the different orientations with a confidence level of 0.05 indeed shows that no statistically significant difference exists. Therefore, following these results, the elastic mechanical response of LS‐PA12 can be assumed to have isotropic constitutive parameters, confirming the findings of Amado‐ Becker et al.[26] and Wegner and Witt.[19] Due to the fact that the shear moduli were approximated using Huber’s law (Equation 6), no check of Lamé’s constraints is made. The explanation for the isotropy as compared with non‐ isotropic models that were found by other authors in the past can be found in the fact that the test specimens were produced using a state‐of‐the‐art LS machine with highly optimized parameter settings, as opposed to older machines that have been used in the referenced publications. These improvements include, for example, optimized scanning patterns, laser beam settings, improvements to the preheating homogeneity of the powder bed, and also continuing improvement in the sintering characteristics of the material itself. It is also interesting to note that the variability, measured in Young’s modulus exceeds the effect of the build orientation in the part. On the contrary, for the tensile strength and elongation at break, no isotropy is present. Considering tensile strength, the strength in the three directions is statistically different as shown by pairwise t‐testing. For the difference between Upright‐Edge, Edge‐Flat, and Upright‐Flat, the corresponding p‐value is for all combinations less than .001. The fact that the Upright specimens show a lower tensile strength, due to inter‐layer fracture, as is further explained in Section 3.2. As concerns the difference between Edge and Flat specimens, it is hypothesized that this is caused by the difference in cross‐sectional area of one layer. A bigger cross‐sectional area leads to a higher scanning time, which gives the material more time to “flow” and better fill the pores in the structure. Hence, a higher strength is obtained. This claim is strengthened by the higher density of the Flat specimens as compare with the Edge specimens.

ET AL.

For elongation at break, no statistically significant difference between Edge and Flat samples is noted (p > .5), whereas there exists a difference between Edge and Flat on the one hand and Upright on the other hand (p < .001). These results do not correlate with the porosity of the parts, as the Upright samples are found to be the most dense, indicating that the porosity in these parts is the lowest, whereas they show the lowest tensile strength and elongation at break. Also, this is attributed to the mode of fracture (inter‐layer fracture), as is further discussed in Section 3.2. The resulting isotropic constitutive model and corresponding variability is shown in Table 4. In order to obtain these, the results that are presented in Table 2 are averaged over the entire test population of 90 specimens, including the three tested orientations (i.e., Edge, Upright, and Flat). Thus, the isotropic Young’s modulus was measured to have a mean value of 1653 MPa with a CoV of 5.0%. Poisson’s ratio was measured to have a mean value of 0.397 with a CoV of 1.9%. As a comparison, Young’s modulus for injection‐moulded PA12 was measured to be 1701 MPa by Van Hooreweder et al.[33]. 3.2 | Heterogeneity in the quasi‐static properties In order to study the heterogeneity in the quasi‐static elastic response of LS‐PA12 components, DIC is employed during the uniaxial tensile tests. Typical stress–strain relationships for the tested samples are shown in Figures 5, 7 and 8, illustrating the DIC measurements corresponding to the respective location on the stress–strain curve of the test. Figure 5 shows the longitudinal strain fields in an Edge specimen during loading on a true stress–strain curve. The specimen failed at a random location within the gage area. The failure is initiated in the middle of the specimen, from where it propagates towards the sides, until the specimen breaks. The fracture is ductile in the middle of the specimen, where the crack is initiated, indicating a stable crack propagation. This also indicates adequate coalescence between adjacent powder particles. At a certain point during loading, the stored elastic energy in the part exceeds the fracture toughness of the part and the fracture propagates instable, leading to a brittle fracture. This is also illustrated in Figure 6, where SEM images of the fracture surface from both the middle of the cross section of the test specimen as at the side of the cross section are shown. The picture from the middle of the specimen clearly shows crazing effects and plasticity, indicating a stable fracture growth. At the side of the specimen, a clearly brittle fracture surface is present. TABLE 4

Isotropic constitutive model for LS‐PA12 including the

variability Nominal

St. Dev.

COV

E11 [MPa]

1653

114.89

5.0%

ν12 [/]

0.397

0.0075

1.9%

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ET AL.

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Stress–Strain relationship for an Edge specimen with the corresponding digital image correlation strain fields at several load steps

FIGURE 5

Scanning electron microscopy (SEM) images of the fracture surface in an Edge specimen: (a) in the middle of the cross‐sectional area (b) at the side of the cross‐sectional area

FIGURE 6

These images also show that a considerable number of pores are present in the LS‐PA12 components. This is also evident from the comparably low relative density (93.8%) of the part. Finally, these images show adequate coalescence between adjacent powder particles, as no unsintered or partially sintered particles are visible.

Stress–Strain relationship for a Flat specimen with the corresponding digital image correlation strain fields at several load steps

FIGURE 7

Figure 7 shows the longitudinal strain fields in a Flat specimen. The difference with the Edge specimen is that the crack is initiated at the side of the specimen and propagates through a part of the thickness of the part. At a certain point during loading, the elastic energy that is stored in the resulting part of the specimen exceeds the loading capacity

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of the undamaged part and the part fractures abruptly, leading to a wedged fracture. This behavior is consistent for all tested parts. Observation of the fracture surfaces using SEM revealed similar effects as for the Edge specimens. At the location where the fracture is initiated (i.e., the side of the specimen), ductile behavior (crazing and plasticity) is noted, whereas in the rest of the cross‐section, a brittle fracture surface is present. Figure 8 shows the longitudinal strain fields in an Upright specimen. Brittle failure occurs at a relatively low level of stress and strain as compared with the Edge and Flat specimens. The crack is perpendicular to the gage area of the sample, parallel to the layering direction. This can be explained by the fact that the failure in an Upright sample stems from the separation of two successive layers, due to inter‐layer porosity,[2] which also poses an explanation for the lack of correlation between the tensile strength and elongation at break with the overall density of the samples. Due to the fact that the pores are oriented between the layers, their influence as stress‐concentrator is maximal, leading to brittle fracture and hence a lower tensile strength and elongation at break. This claim is further strengthened by the fracture surface that is shown in Figure 9. This picture indicates brittle inter‐layer fracture (i.e., separation of subsequent layers) as the primary mode of failure in the Upright specimens. Also here, a considerable number of pores is visible in the fracture surface, which is correlates with the on average comparably low relative density of the parts (95.3%). Closer inspection of the DIC measurements in Figure 7 shows heterogeneities in the strain fields of loaded Upright specimens that cannot be explained through the loading condition. These samples show locally higher strains of 5–10% in a repetitive stripe pattern. These locally higher strains are attributed to inter‐layer porosity, as these pores between layers directly act as stress concentrators and thus lead to a local strain magnification. Further, micro‐XCT measurements however are necessary to look into detail into the local number of pores and their morphology. It can as well be noted that a local higher strain exists in the sample throughout the test, as indicated by the red band in the middle of

ET AL.

Scanning electron microscopy (SEM) image of the fracture surface in the Upright specimen

FIGURE 9

the specimen. Also, yielding was initiated at that exact location. A closer observation of the LS build that was used to produce the samples reveals the cause of this inhomogeneity. From Figure 10, it is clearly observed that the strain pattern in the DIC results is strongly influenced by the spatial distribution of the parts in the build topology. Indeed, the location where the strain reaches its peak value corresponds to those layers in the Upright sample where an Edge sample was built at the same time. An explanation for this is found in the increase in scanning time of these layers, which permits the sintered layer of material to cool further down as compared with layers in the Upright specimen where the Edge part is not present. This decrease in temperature in its turn disadvantages the sintering conditions for the next layer, leading to a lack of interconnection between the subsequent layers and thus a locally lower stiffness. Therefore, it can be concluded that other samples in the build significantly influence the local properties of LS‐PA12 material, which corresponds to the findings by Jain et al. They found that the delay time between two subsequent layers in a part influences the mechanical stiffness of that part, which is attributed to the binding process between the particles.[21]

Stress–Strain relationship for an Upright specimen with the corresponding digital image correlation strain fields at several load steps

FIGURE 8

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FIGURE 10 Heterogeneous strain profile in upright built laser sintering samples with a cross section of the placement of Upright and Edge specimens in the build volume

3.3 | Analysis of the anisotropy based on the VFM The transverse (ϵxx), longitudinal (ϵyy), and shear (ϵxy) strain fields in an Upright built biaxial sample at a load level of 3250 N are shown in Figure 11. As is evident from these plots, a biaxial strain state is induced in the loaded sample upon uniaxial loading. This strain state activates all in‐plane constitutive parameters, hence permitting the identification of the constitutive model using the VFM. It is assumed that the material follows an orthotropic constitutive model and that the strain fields are constant through the thickness of the specimen (plain stress). The first assumption is based on the scientific literature on the subject, the latter on the fact that the thickness of the part (6 mm) is small compared with the other dimensions. The noise level, resolution in strain, and displacement and the spatial strain resolution are listed in Table 5. As can be noted, the resolution in both displacement and strain is fine enough to yield accurate results in the VFM calculations. The results of the VFM calculations are shown in Table 6. The primary Young’s moduli (Eii) and shear moduli (Gij) are found to be slightly lower as compared with the results obtained by the uniaxial testing. Poisson’s coefficients however correspond well. This also proves that application of Equation 6 yields an accurate estimation of the shear modulus for LS‐PA12 material. This may be explained through the heterogeneity of the LS‐PA12 component, as also indicated in Figure 10. In comparison to conventional techniques, where Young’s modulus calculation is based on strain values obtained by dividing the elongation of the component by the gage length, VFM uses the full‐field DIC strain data.

FIGURE 11 Measured strain fields in the transverse (εxx), longitudinal (εyy), and shear (εxy) in the biaxial test sample at a load of 3250 N

Therefore, local heterogeneities impact the calculation of the constitutive parameters more significantly. On the other hand, also, the LS process itself can affect these properties. Due to the different geometry of the biaxial and uniaxial samples, the thermal history will differ as well. Moreover, the topology of both builds was different, further influencing this difference in thermal history and thus mechanical properties. The results in Table 6 also confirm the nearly isotropic constitutive parameters, with a maximum degree of orthotropy of 8.68% for the Edge direction. Also, Poisson’s ratio and the Shear moduli are consistent for all tested orientations, further strengthening this conclusion. With a maximum CoV of 5.37% for Young’s modulus, 2.43% for Poisson’s ratio, and 8.23% for the shear modulus, the variability on the results of the VFM test population is smaller then what was observed during uniaxial testing. This can be attributed to the rather small test population of four replicas that was used for the VFM tests as compared with the 30 replicas for the uniaxial testing. Moreover, the biaxial parts were all located in the center of the build, whereas some replicas of the uniaxial samples were located closer to the edges of the machine. Therefore, less effect of preheating temperature heterogeneities is to be expected for the biaxial specimens. When the measurements of Young’s moduli of the samples under the different orientations (Edge, Flat, and Upright) are analyzed with respect to the principal axes of the machine (X, Y, Z), the nearly isotropic assumption is affirmed, as can be seen in Table 7. This analysis shows Young’s moduli of respectively 1583, 1667, and 1647 MPa for the X, Y, and Z axis of the LS machine. A pairwise t‐test with a confidence level of 0.05 indeed shows that no statistically significant

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Noise, resolution, and spatial resolution of the digital image correlation measurement

TABLE 5

Parameter

Anderson–Darling distances of the hypothesized distributions for Young’s modulus

TABLE 8

Value

Noise

0.78% 1.043·10−02 mm

Displacement resolution

ET AL.

Isotropic

Edge

Flat

Upright

Distribution

AD

p

AD

p

AD

p

AD

p .36

Normal

.27

.68

.35

.37

.79

.03

.78

Strain resolution

1.025·10−04

Log‐normal

.33

.51

.36

.41

.78

.04

.79

.34

Virtual strain gage

99 px

Weibull

.58

.14

.85

.02

.91

.02

.82

.29

Gamma

.31

.25

.391

.25

.83

.03

.84

.23

Variability in the mechanical properties of the laser sintering samples, per orientation, as obtained by the virtual fields method

TABLE 6

E11[MPa]

E22 [MPa]

ν12 [/]

G12 [MPa]

Upright

1596

1626

0.41

545

St. Dev.

66.56

75.06

0.01

37.9

COV

5.37

4.62

2.43

8.23

Edge

1550

1697

0.41

545

St. Dev.

61.91

36.97

0.01

29.26

COV

4.11

2.27

2.43

5.46

Flat

1575

1667

1.42

544

St. Dev.

33.17

65.51

0.01

14.15

COV

2.11

3.92

2.38

22

O12 −1.84

−8.68

−0.55

4 | C O NC LU S I ON S

Young’s moduli as obtained by the virtual fields method, listed by the principal axis of the laser sintering machine

TABLE 7

X [MPa]

Y [MPa]

Z [MPa]

1626

/

1597

St. Dev.

66.6

/

66.6

Edge

1550

/

1697

Upright

noted that the p‐value of the calculated AD distances for the Flat orientation is lower than the limit of .05, which is due to multi‐modality in the data. Therefore, these results should be interpreted with care. The mean and standard deviation of the corresponding distributions can be found in Tables 3 and 4. Following these identified distributions, these data can be readily used to represent the process variability in any design calculation that requires Young’s modulus.

St. Dev.

63.8

/

59.09

Flat

1575

1667

/

St. Dev.

33.17

65.51

/

Combined

1583

1667

1647

St. Dev.

68.01

65.51

55.47

difference between the orientations exists. Combined into one constitutive model, this yields a Young’s modulus for the LS‐PA12 material of 1632 MPa with a standard deviation of 74.31 MPa and a Poisson’s ratio of 0.41. Computation of the shear modulus with these properties using Lamé’s equation yields a Shear modulus of 587 MPa, which further proves the nearly isotropic nature of the material.

3.4 | Identification of variability distributions in the uniaxial data Table 8 shows the calculated Anderson–Darling distances when respectively a normal, log‐normal, Weibull, and a gamma distribution are fitted to the Young’s modulus data. As is evident, the lowest AD distance is achieved with a normal distribution for both the isotropic Young’s modulus (as determined in Section 3.1) as well as the data considered for all different orientations separately. It should however be

This paper describes an experimental investigation to determine the variability, heterogeneity, and anisotropy in the quasi‐static mechanical properties of PA12 parts produced via LS. The parts were built along the principal axis of the machine. Specifically, the elastic constitutive model of the material, including Young’s modulus, Poisson’s coefficient, and the shear modulus were investigated using uniaxial tensile tests and digital image correlation. Additionally, the VFM was employed to determine the in‐plane constitutive parameters of LS‐PA12 material in a single test. The variability in Young’s modulus was investigated by performing 30 tensile tests for each axis of the machine, resulting in a population of 90 specimens. A rather large CoV was found in Young’s moduli (5%) and Poisson’s coefficient (1.9%) of all the produced samples combined. Regarding the different orientations, a CoV of 6.5%, 5.61%, and 5.83% for Young’s modulus of respectively the Upright, Edge, and Flat samples was found. Therefore, the use of deterministic numerical simulations in the design of a LS‐ PA12 component, while neglecting this variability on the quasi‐static properties, could result in a severe bias with respect to the actual mechanical behavior of the produced part. Using the Anderson–Darling methodology, a normal distribution for both the isotropic constitutive model and the data for all different orientations separately was identified. This enables the usage of non‐deterministic design calculations in the context of reliable design of LS‐PA12 components, for example, in the framework of Monte Carlo‐based techniques. Concerning the heterogeneity, the difference in fracture behavior between Edge, Flat, and Upright built specimens was investigated using DIC. No distinct difference in failure between the Edge and Flat orientations was found. The

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Upright sample on the other hand failed due to inter‐layer porosities. These specimens show a crack initiation at the side, stemming from the separation of subsequent layers. It was also found that the simultaneous production of parts, loaded in the layering direction, can affect the local stiffness properties of this part. This is possibly attributed to the increased scanning time of a layer, by introducing extra parts, which in its turn affects the local density of the specimen. This finding poses important implications for the optimal planning of LS‐PA12 build topologies in the context of producing mechanically reliable components for functional end‐use applications. Further work however is needed to fully understand the underlying phenomena. This research will combine X‐ray computer tomography with DIC. Literature on the subject of anisotropy in the constitutive elastic model of LS‐PA12 parts mainly shows orthotropic or transverse‐isotropic parameters. In this study, it was found that the quasi‐static elastic response of these parts is best described using an isotropic constitutive model, with a Young’s modulus of 1653 MPa and a Poisson’s ratio of 0.397. The explanation for this may be found in the fact that the test specimens were produced using a state‐of‐the‐art LS machine with highly optimized parameter settings, as opposed to older machines that have been used in the referenced publications. As a validation of these findings, the VFM was applied to a biaxial complex test geometry to calculate the in‐plane orthotropic constitutive parameters of the built specimens within a non‐dogbone configuration. A slight discrepancy between the uniaxial and biaxial tests was noted and attributed to the difference in thermal history between the different samples. Thermal monitoring however is needed to validate this assumption. The VFM measurements also indicate a nearly isotropic constitutive model with a Young’s modulus of 1632 MPa and a Poisson’s coefficient of 0.41. AC KNOWLEDGMENTS

The authors would like to acknowledge the Agency for Innovation by Science and Technology in Flanders (IWT) for the funding in the framework of the TETRA project 130211 AMPLIFY, Additive Manufacturing of Polymers: Innovating Functionality through reliability. The authors would also like to acknowledge Dr. T. Craeghs and M. Pavan of Materialise NV for the production of the test samples and interesting discussion of the results. Finally, the authors would also like to thank Prof. J. Ivens for the help in interpreting the results. REFERENCES [1] T. Wohlers, T. Caffrey, Wohlers Report 3D printing.pdf, Wohlers Associates, Fort Collins, 2013. [2] D. K. Leigh, D. L. Bourell, J. J. Beaman. Jr., Scan. Electron Microsc. 2011, 453, DOI: 10.1115/ISFA2012-7102 [3] H. L. Marcus, J. W. Barlow, J. J. Beaman, D. L. Bourell, JOM 1990, 42, 8, DOI: 10.1007/BF03220915

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How to cite this article: Faes M, Wang Y, Lava P, Moens D. Variability, heterogeneity, and anisotropy in the quasi‐static response of laser sintered PA12 components. Strain 2017;53:e12219. https://doi.org/ 10.1111/str.12219