“Micromechanics of additively manufactured structures:...

MEMORIAS DEL XXIV CONGRESO INTERNACIONAL ANUAL DE LA SOMIM 19 al 21 DE SEPTIEMBRE DE 2018 CAMPECHE, CAMPECHE, MÉXICO

Tema A2a Materiales: Manufactura aditiva

“Micromechanics of additively manufactured structures: computational and laboratory experiments”

Enrique Cuan-Urquizoa,b,* Atul Bhaskarb

a Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Epigmenio González #500 Fracc. San Pablo, 76130 Querétaro, Qro., México bComputational Engineering and Design Group, Faculty of Engineering and the Environment, Boldrewood Innotavion Campus, University of

Southampton, Burgess Road, Southampton SO16 7QF, United Kingdom

*Autor contacto. Dirección de correo electrónico: [email protected]

R E S U M E N

La relación entre la estructura y las propiedades mecánicas del material formado por manufactura aditiva de extrusión fue estudiada computacional y experimentalmente. Para predecir las propiedades mecánicas bajo cargas de tensión, se crearon modelos computacionales usando elementos finitos del tipo tetraedro. Las propiedades del material usadas para alimentar los modelos de elemento finito fueron obtenidas de la caracterización de filamentos simples bajo tensión. La predicción fue comparada contra datos obtenidos de pruebas de tensión de probetas fabricadas con manufactura de extrusión. Las probetas fueron manufacturadas con porosidad controlada, usando código G adecuado a parámetros estructurales, por medio de un programa desarrollado en MATLAB. Este programa permite tener control absoluto en las trayectorias que sigue el extrusor y por ende, de la estructura de la muestra. Finalmente, ambos resultados fueron comparados con un modelo teórico básico basado en Regla de mezclas. Las predicciones resultaron con un error menor al 10%.

Palabras Clave: Manufactura aditiva, modelación por deposición fundida, caracterización mecánica, tensión, relación estructura-propiedades.

A B S T R A C T

The relationship between the structure and the mechanical properties of the structured arrangement of material formed in the process of fused deposition modelling was studied computationally and experimentally. Computational models using tetrahedral finite elements were developed to predict the mechanical properties under tensile loading. Material properties data to feed the finite element simulations were obtained from tensile tests on single extruded filaments. Prediction of the apparent Young’s modulus was compared with measurements obtained from tensile tests of samples fabricated with fused deposition modelling. Samples with controlled structure were fabricated using customised G-codes generated with a developed MATLAB script. The script permitted full control on the extruder trajectories, hence tailoring the structured formed. Finally both, computational and laboratory experiments were compared with a basic theoretical prediction based on the rule of mixtures. Predictions resulted to be with less than 10% error.

Keywords: Additive manufacturing, Fused deposition modelling, Mechanical characterization, Tensile test, Structure-property relationship.

1. Introducción

Fused filament fabrication (FFF), commonly known as Fused deposition modelling (FDM) is probably the most

popular additive manufacturing (AM) technology. Applications of parts manufactured with FDM can be found in different areas, from robotics and vehicles [1], [2] to biomedical applications in dentistry [3], tissue engineering scaffolds [4], and prosthetics [5], [6]. FDM-machines

ISSN 2448-5551 MM 28 Derechos Reservados © 2018, SOMIM


consist of an extruder that moves linearly in two axes, and the third axis is adjusted with the distance between the extruder and a platform. Material is extruded and deposited in semi-molten state on the platform to form 2D profiles of the part to be fabricated. A shift of the extruder from the platform (or the platform from the extruder) allows the next layer to be deposited. The fabricated part is then constituted of the stack of 2D profiles which at the same time are formed by a structured distribution of extruded filaments (see fig. 1). The nature of the process will invariably produce anisotropic parts [7]. The literature found shows that different attempts to study the mechanical properties of FDM-parts have been carried out mainly for parts having a 100% of infill density, hence quasi-homogeneous solid parts [7]–[10]. These studies have been done mainly by experimental approaches. As the samples are quasi-solid, in the majority of them, the data measured converge to the properties of the parent material, making challenging to understand the influence of the structure on the properties. Some other works make use of analytical or semi-analytical models to predict the resulting mechanical properties. Analytical attempts to predict the effective mechanical properties can be classified in two main groups: classic laminate theory (CLT) approaches [11]–[13] and micromechanics approaches [14]–[17]. In [11]–[13], the stiffness matrix was fed using experimentally measured elastic properties. These properties were obtained from tensile tests on unidirectional samples. CLT has appeared to be a good approximation to predict the elastic properties. However, they are limited to parts fabricated with 100% infill, no separation between the extruded filaments was considered. To obtain the complete stiffness matrix, the Young’s moduli of a single layer in the longitudinal and transverse directions of the filaments are needed. The stiffness in the transverse direction is only obtained when the filaments within the layer interact, i.e. when they are bonded in the plane. The prediction of the mechanical properties using CLT fails when filaments are extruded with a gap between each other as the stiffness in the transverse direction vanished [18]. Micromechanics approaches involve taking into account the deformation mechanism of the extruded filaments, and incorporate their mechanics to predict the macro properties. Huang and Singamneni [16] studied the sensitivity of the effective stiffness and strength on unidireccional samples fabricated with filaments at various angles. These authors proposed an analytical model based on the plane stress response of layers of filaments bonded in a plane. Because the filaments are bonded within the layer, in [16] they considered the filaments cross-sections to be square with elliptical fillets in their analysis. Croccolo et al. [17] proposed an analytical model based on the structural modelling of the rasters as beams. The model in [17] showed excellent results in predicting the stiffness of the samples, but was also restricted to 100% density in them.

Despite the huge amount of attempts to characterise the mechanical properties of parts fabricated using FDM, the structure-property relationship is still an open issue for various loading scenarios. This is attributed mainly to the fact that most of the works deal with quasi-solid samples, complicating the relation between structure and properties. Few works on these lines can be found [14], [19]. One of the few works that have treated such structured arrangement of filaments as a lattice material are the works of Cuan-Urquizo et al. [14], [15]. Lattice materials, in this case a collection of beams and rods in periodic arrangement, have effective or apparent properties as function of the architecture of the lattice and the parent material of which they are comprised of. When lattice elements, are of dimensions significantly smaller than those of the macro dimension, the material can be treated as homogeneous with apparent or effective properties. In order to relate such effective properties to the structure, the property is plotted as a function of the relative density�̅�, i.e. the ratio between the density of the cellular material and that of the material that is composed of. In the light of this gap in the knowledge, the present work is focused on the understanding of the elastic mechanical properties of FDM-components for low densities, i.e. lattice arrangement of extruded filaments. The lattice arrangement produced using FDM, is represented with the schematic shown in Fig 2. Note that each extruded filament is represented as cylinders, and the stack of such is sometimes called woodpile arrangement, as it resembles piles of wooden logs. This paper is arranged as follows: the description and details of the materials, experiments and analysis done are included in Section 2, results of such are presented in Section 3, and finally concluding remarks are mentioned in Section 4.

Figure 1 – SEM of the arrangement of filaments within a FDM-

component.

2. Materials and Methods

2.1. G-code generation

A conventional process to manufacture a part with FDM machines consists of the following steps. First, a CAD



model of the part to be printed is created and exported as a .stl file, as required by most of the 3D printers software. The software then slices the model, into 2D profiles of the part and generates the nozzle trajectories accordingly these trajectories are expound in the G-code. A G-code is a set of instruction lines containing manufacturing parameters and the coordinates for the extruder trajectory. Conventional G-code software restricts the control on the density of the part to be printed. Hence, in order to control the porosity of the samples, a MATLAB script was generated to produce such instruction lines. The input to the script were structural parameters such as: number of filaments, orientation, and distance between filaments. Some of the principal variables used as input for the script are depicted in fig. 2. Note that, the relative density, i.e. the volume fraction, was calculated from �̅� = 𝑉𝑠 𝑉𝑐⁄ = 𝜋𝑟 2𝜆⁄ , where Vs and Vc are the volume occupied by the fibres and the total volume including voids, respectively [14]. The output of such script is a G-code with all the set of instructions ready for the printer. Examples of the lattices formed using FDM and the G-codes generated are shown in Fig. 3.

Figure 2. Schematic of the lattice material obtained using FDM.

Different unit cells are shown for different relative densities.

Figure 3. Images of different porosities obtained by controlling the

G-code, (a) λx= λy= 1 mm so �̅� = 𝟎. 𝟑𝟗, (b) λx= 1 mm, λy= 2 mm so

�̅� = 𝟎. 𝟐𝟗 and (c) λx= λy= 2 mm so �̅� = 𝟎. 𝟏𝟗

2.2. Fused deposition modelling of slender structures

A set of 11 samples were fabricated using the G-codes described in the previous subsection, following ASTM-D638, and a BFB 3D Touch® 3D printer with polylactic acid (PLA) and a nozzle extruder of 0.5 mm diameter. Samples were oriented to lie parallel to the printing platform, as shown in fig. 4. While the separation between filaments was controlled within the G-code, the rest of the manufacturing parameters were kept constant in order to focus the analysis on the structure. These parameters are summarised in Table 1.

Figure 4 - Screen-shot of the (a) tensile sample manufacturing

simulation to ensure the correct trajectory of the nozzle generated,

and a photograph of the (b) physical tensile sample obtained right

after the printing process finished. Note the raft layers indicated,

these layers are printed to ensure the adhesion to the bed as well

as to bring support for the actual printed part.

Table 1 – Manufacturing parameters

Parameter Value Units

Nozzle speed 860 mm/s

Extrusion speed 45 rpm

Extrusion temperature 190 °C

2.3. Finite element model of woodpile structured material

Although, theoretically one can control the distribution of filaments, in reality the arrangement of filaments is restricted to intrinsic physical phenomena, such as the overhang between adjacent filaments. This phenomena could lead to extra complexity in the attempt to characterise the structure-property relationship. To overcome that, FEA appear to be a reasonable option, as it allows the modelling of structures with high degree of complexity. The FE model was built



using tetrahedral elements (C3D10) in ABAQUS, and it is shown in fig. 5. The material properties used to fed the FE model were the Young’s modulus of the PLA obtained from the single filament characterization E= 2290 MPa as determined in [15] and a Poisson’s ratio of 0.3 [20].

Figure 5 - FEA model showing the loading direction and the mesh

with tetrahedral elements.

A series of simulations with different relative densities were performed and the apparent Young’s modulus ⟨𝐸⟩ was calculated. A unit load was applied and distributed on all the nodes at one end of rectangular cuboid sample of the lattice material. The FEA samples, shown in fig. 5, consists of 16 axial and 24 transverse filaments. The relative density �̅� was varied across the simulations by keeping the diameter of the filaments (2r) fixed and changing the separation between them λ. The total length L of the samples along the load direction was also kept constant. Each FE model contained about 800K nodes and 500K elements after ensuring convergence.

2.4. Tensile tests of additively manufactured slender

structures

The tensile tests on the FDM-samples were carried out using an INSTRON 5569 machine. A sample under tensile loading is shown in fig. 6, along with a close-up to show the orientation of the filaments with the respect to the loading direction. The machine was operated in displacement control mode, the top grip moved upward at a strain rate of 5 mm/min applying a load in the vertical axis.

Figure 6 - One of the samples during the tension test and the

macrograph of the porosity showing the filaments aligned to the

loading direction. The loading direction is indicated with the red

arrows.

3. Results

3.1. Apparent Young’s modulus from tensile testing

The force-elongation curves obtained were processed to stress-strain curves and are shown in fig. 7. The apparent stress 𝜎∞ was calculated by dividing the force by the apparent cross-sectional area of the samples (assuming the sample to be homogeneous) A = 98 mm2. The strain was obtained by dividing the elongation by the original length between the grips L = 100 mm. The measured apparent Young’s modulus was obtained from the slope of the linear part of the stress-strain curves as ⟨𝐸⟩𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑= 640.6±39.7 MPa.

Figure 7 – The stress-strain curves of the whole set of samples.

3.2. Apparent Young’s modulus from finite element

The unit load applied was divided by the total effective area (including voids) to obtain the apparent stress 𝜎∞ . The displacement components at the end were averaged and divided by the total length of the bars L to obtain the



apparent strain ε. The effective Young’s modulus was then calculated using⟨𝐸⟩ = 𝜎∞/휀. The process was repeated for several samples of varying relative density. Since the diameter of the filaments was not varied, the actual strain was the same for all the cases simulated. The difference between all the cases is in the total effective area. This, as mentioned, was varied by changing λ. After obtaining the apparent Young’s modulus for all the cases simulated, this was plotted as a function of the volume fraction �̅� in fig. 8 with black squares.

3.3. Reconciliation with theoretical model: rule of

mixtures

In [14], [21], an expression to predict the apparent Young’s modulus along the filaments direction of the woodpile arrangement shown in Fig 2. was presented as function of the volume fraction �̅� and also of the fractional cross-sectional area φ exposed to loading. Both expressions are presented here respectively, ⟨𝐸⟩ = (1/2)𝐸�̅� (1) ⟨𝐸⟩ = 𝐸𝜑 (2) where E is the Young’s modulus of the parent material. We proceed to compare expressions (1) and (2) with computational and laboratory experiments next to understand their applicability and limitations. Equation (1) shows a linear scaling relationship with the relative density and it is shown in fig. 8 with a solid straight line. There is an excellent agreement with theoretically predicted values of the modulus of elasticity using equation (1) and the FEA simulation results. The linear trend is expected for stretch dominated filament deformation lattices. Equations (1) and (2) assume that filaments running orthogonally to the loading direction do not offer stiffness to the overall response of the lattice. This was confirmed by an inspection of the FEA results shown in Fig. 9. Note that perpendicular filaments remain stress-free, and all the stress are located in the axial filaments. This aspect results in an apparent Poisson’s ratio ⟨ν⟩ of 0 for this lattice material. This phenomena is due to the fact that the filaments perpendicular to the loading direction remain stress-free. But at the same time keeping the separation between the filaments parallel to the load constant, results in no displacement in the perpendicular direction of the load, i.e. ⟨ν⟩ ≈ 0.

Figure 8 – A comparison between the analytically predicted

apparent Young’s modulus ⟨𝑬⟩ (solid line) and the results obtained

via FEA (squares), showing linear scaling relationship with the

relative density �̅� . A single obtained from the mean of the

experimental measurements is also plotted along with the error

bars at one standard deviation.

A single point of data coming from the laboratory experiments is also shown in fig. 8. The difference is attributed to the fact that equation (1) assumes the lattice to be infinite. For experimental data, equation (2) shows better results, as it accounts for a finite number of filaments as shown next.

The value for the apparent Young’s modulus was calculated using equation (2). The fractional area was calculated as φ = Na/A, where Na is the total number of filaments parallel to the loading direction. Since full control on the lattice fabricated is achieved with the customized G-code the exact number of filaments parallel to the loading direction are known Na = 98; 7 layers of 14 filaments each. The area of a single filament is taken as a = 0.302 mm2 [15] and the Young’s modulus of the PLA E= 2290 MPa as mentioned in Section 2.3. The apparent Young’s modulus was then calculated as ⟨E⟩calculated = 692 MPa; this value is based on equation (2) in conjunction with the single filament properties measured.

Figure 9. Deformed and undeformed FE model. Note that the

filaments running perpendicular to the loading direction remain

stress-free.



A comparison of the predicted trend with experimental data is shown in fig. 10. The predicted trend is shown in solid black line while the mean of the experimental data is presented with dots showing the distribution of one standard deviation.

Figure 10 – Stress-strain curves of the elastic part of the tensile tests

and the predicted value using the rule of mixtures. The predicted

trend is shown in solid black line. The mean of the experimental data

is shown with dots. The error bars are shown at one standard

deviation from the experimental mean.

Note a systematic difference between the calculated trends of 10%, which is always above the directly measured trend for complete lattices. Figure 10 indicates a slightly softer lattice than predicted using single filament properties. There are two possible reasons for this. Firstly, the complete lattice shows sag of filaments overhung between orthogonally placed supporting filaments underneath. This slight curvature, also observed microscopically, as well as variability in the manufacturing conditions between those for a single filament and those for complete lattices, result in a softer response.

Figure 10 – A sample after being tested under three-point bending.

Another experimental validation of equation (1) was performed by submitting FDM-samples (ASTM D790) under three-point bending using a Texturometer Brookfield with load cell capacity of 50Kg (490 N). The separation between supports was set to L=68mm. Samples were fabricated for a

�̅� = 0.5, in order to reduce the influence of shear [14], It was demonstrated in [14] that beams composed of this lattice arrangement have higher sensibility to shear deformations in the transverse displacements for lower values of relative density. The apparent Young’s modules of the samples fabricated with a PLA with E=2290MPa is calculated as⟨𝐸⟩ =572.5𝑀𝑃𝑎. The maximum displacement was set to be 𝛿 =2.6mm, the average of force registered was F=104.44 N. The cross-section area of the samples was 25mm width and 6mm thick, hence a second moment of area of I=450mm4. The calculated force to produce a transverse displacement of 2.6 mm was obtained from the simply supported beam formula: 𝛿(48⟨𝐸⟩𝐼 𝐿3⁄ ) = 𝐹.Using this equation, the calculated force resulted as 102.29 N, showing an excellent prediction with 2% error. A sample after three-point bending is shown in fig. 11.

4. Conclusions

The apparent Young’s modulus of the woodpile lattice material along filament direction ⟨𝐸⟩ was measured and compared with a model that neglects deformation in filaments across the loading direction. The model utilises experimentally measured single filament properties. The comparison between the model and the FEA data showed excellent agreement for models with various relative densities. The experimental data and the calculated value showed minor differences attributed to manufacturing inconsistencies. The laboratory experiments were done on a set of FDM-manufactured samples. The apparent Young’s modulus ⟨𝐸⟩ was shown to be dominated by the filaments stiffness in tension or compression. The apparent Young’s modulus was predicted by means of the rule of mixtures, as function of the cross-sectional area exposed to loading. The effective property resulted to be stretch-dominated showing a linear dependence to the volume fraction ⟨𝐸⟩~�̅� . The woodpile lattice is a material arrangement that is directional dependent. The models and analysis presented in this manuscript are restricted to the loading direction exposed. The mechanical response when loaded at any other direction will be significantly different. Currently, research on these lines is underway.

Acknowledgements

This work has been conducted under the financial support of CONACYT during the PhD studies of the first author at University of Southampton, UK.

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