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Experimental and theoretical modeling of behavior of 3D-printed polymers under collision with a rigid rod
Accepted Manuscript Title: Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision with a Rigid Rod Author: Kamran Kardel Hamid Ghaednia Andres L. Carrano Dan B. Marghitu PII: DOI: Reference:
S2214-8604(17)30030-1 http://dx.doi.org/doi:10.1016/j.addma.2017.01.004 ADDMA 148
To appear in: Received date: Revised date: Accepted date:
18-12-2015 10-1-2017 23-1-2017
Please cite this article as: Kamran Kardel, Hamid Ghaednia, Andres L. Carrano, Dan B. Marghitu, Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision with a Rigid Rod, (2017), http://dx.doi.org/10.1016/j.addma.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision with a Rigid Rod
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Kamran Kardela,∗, Hamid Ghaedniab , Andres L. Carranoc , Dan B. Marghitub a Department
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of Manufacturing Engineering, 1100 IT Drive, Georgia Southern University, Statesboro, GA 30458-7991 b Department of Mechanical Engineering,1418 Wiggins Hall, Auburn University, Auburn AL 36849-5341 c Department of Industrial and Systems Engineering,3301 Shelby Center, Auburn University, Auburn AL 36849-5346
Abstract
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The behavior of five different 3D-printed polymers has been analyzed both theoretically and experimentally under low-speed collision conditions. The impact of a rigid rod with a flat specimen fabricated of 3D-printed materials was analyzed. An experimental setup has been designed in order to capture the motion of the rod during the impact using a high-speed camera. Image processing algorithms were developed to estimate the velocity before and after the impact as well as the coefficient of restitution. Also, permanent deformations after the impact were scanned with an optical profilometer. In this work, a theoretical formulation for the contact force during the impact is proposed. The impact was divided into two phases, compression and restitution, in which materials considered elastic-plastic in the first and fully elastic in the second one. The experimental results are used to measure the damping coefficient. Results show a good correlation between the proposed formulation for the contact force and the behavior of materials.
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Keywords: contact force, permanent deformation, coefficient of restitution, 3D-printed polymers, additive manufacturing
1. Introduction
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Additive manufacturing (AM) or 3D-printing enables the design and manufacture of complex geometries with polymers, ceramics, and metals which are not feasible with traditional manufacturing methods. Recent advances in AM has facilitated reductions in tooling needs, assembly and lead time, as well as increased efficiencies in the use of materials [1, 2]. Many relevant industry sectors such as automotive, aerospace, biosystems, and healthcare have already benefited by the use of AM [3, 4]. However, AM remains to be widely adopted across manufacturing industries [5]. Some factors limiting widespread incorporation of these technologies in industry sectors may ∗ Corresponding
author Email addresses: [email protected] (Kamran Kardel), [email protected] (Hamid Ghaednia), [email protected] (Andres L. Carrano), [email protected] (Dan B. Marghitu) Preprint submitted to Journal of LATEX Templates January 9, 2017
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be related to the lack of comprehensive information on mechanical properties of the rapidly evolving material set available to industry. Although there are extensive studies available on the mechanical properties of 3D-printed materials ([6]-[7]), new materials have appeared since and other mechanical properties have gained significance. This is particularly true for polymers and for the understudied field of mechanical impact. Understanding the behavior of new materials under dynamic contact as well as the history of the contact force during collision can provide insights leading towards more efficient mechanical designs. There are many research efforts on testing 3D-printed plastics in terms of mechanical properties available in archival literature [6, 8, 9, 10, 11]. The tensile strength of 3Dprinted polylactic acid (PLA) bars used in bone fixation and reconstruction has been tested [12]. Test specimens were fabricated from both low and high molecular weight PLA powders with chloroform used as a binder. The maximum tensile strength was found to be 15.94 ± 1.50 MPa for high molecular weight PLA and 17.40 ± 0.71 MPa for low molecular weight. More recently, research on the influence of print density on microstructure and mechanical properties of nylon parts in High Speed Sintering (HSS) showed that the mechanical properties are related to crystallinity [13]. Results showed that crystallinity decreased as print density increased. in turn, stiffness and tensile strength increased with crystallinity. In other effort, Acrylonitrile butadiene styrene (ABS) parts fabricated by Fused Deposition Modeling (FDM) were also studied for their tensile and compressive strengths [14]. It was found that for FDM parts made with a 0.003 inch overlap between roads, the tensile strength ranged between 65-72% of the strength of identical parts made with injection molded ABS. In another study, the mechanical properties, surface roughness, geometric and dimensional accuracy, and material costs of various types of materials processed with different types of AM technologies, including stereolithography, polymer extrusion, powder bed fusion, sheet lamination, and vat photopolymerization were tested and compared [15]. Other archival literature on fatigue behavior of laser-sintered nylon (PA12) showed that PA12 specimens under rotating-bending conditions will experience fatigue failure [16]. It was suggested that, rotating-bending at frequencies of up to 50 Hz is a valid way of determining the fatigue behavior of laser-sintered PA12. Parts made with stereolithography (SLA) process were investigated to elucidate the influence of layer thickness on mechanical properties of produced parts [17]. The results showed that, when the layer thickness decreases, the strength of parts made by SLA increases. Another effort pursued the effects of electroplating on polymer parts made with SLA [18]. A series of tests were conducted on samples coated with copper and nickel with varying thicknesses. Physical testing confirmed that a thicker coating (120 µm) increased impact energy and Young’s modulus but had a minimal effect on ductility of parts, in which a thinner coating (20 µm) resulted in a smoother surface finish. Also, a computer aided characterization approach has been used to evaluate the effective mechanical properties of porous biodegradable polyester tissue scaffold [19]. An approach for design and 3D-printing of porous tissue scaffold, the computational algorithm for finite element implementation, and the characterization of the effective mechanical properties of porous poly-ε-caprolactone scaffolds manufactured by polymer extrusion was provided. There are also previous research efforts on the mechanical properties of additively manufactured metals available in literature. These include experimental testing and simulation for fracture criteria, fatigue, tensile strength, among others [20, 21, 22, 23, 2
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24, 25, 26, 27]. Also the behavior of parts produced by Selective Laser Melting (SLM) has been compared to conventionally produced products for, traction, compression and flexural properties [28]. Additively manufactured AlSi10Mg using SLM was studied in terms of microstructure, high cycle fatigue, and fracture behavior [29]. For approximating flexural properties of bone, open cellular structures fabricated in Ti6Al4V using Electron Beam Melting (EBM) have been proposed for tissue scaffold and low stiffness implants [30]. In another study, the effect of build orientation and heat treatment on the mechanical properties of SLM Ti6Al4V lattice structures was investigated [31]. Different binding mechanisms were also investigated in powder bed fusion and SLM with metal powder to compare the mechanical properties and also improve understanding of these processes [7]. With respect to the dynamic behavior of 3D-printed materials, simulation of impact can be divided into two phases, compression and restitution[32]. The contact force between the objects during each of these phases is the most important factor needed to simulate the impact. The contact force itself can be divided into three phases, fully elastic, elastic-plastic and fully plastic phases. The fully elastic and fully plastic phases can be solved analytically. Hertzian theory[33] has been commonly used for the fully elastic phase. For the fully plastic phase there are few alternative approaches that can be found in available literature [34, 35]. The archival research available on the dynamic behavior of the 3D-printed materials during collision is very limited and not current, given the increased use of AM parts in industry and the evolution of the materials. Unfortunately, to date, the elastic-plastic phase is not amenable to a closed-form analytical solution. Arguably, for the collision problems the elastic-plastic phase plays the most important role. Many efforts have aimed at predicting the elastic-plastic contact forces for homogeneous metal materials, including Johnson [34], Jackson et al. [36], Kogut and Etsion [37], Ghaednia et al. [38], among many other models. A comprehensive comparison between different contact models for collision problems can also be found in literature [39]. Although, these contact models have been developed for quasi static contact; they have often been used in collision problems, such as those found in Stronge [40, 41, 42, 43, 44], Marghitu et al. [45] and Jackson et al. [46]. Conventionally, the coefficient of restitution is defined as the ratio between the rebound and initial velocities (i.e. Newton’s definition) or as the ratio between the restitution and compression impulses (i.e. Poisson’s definition) [32], which is used to account for the energy loss during the impact. The coefficient of restitution is easy to measure experimentally but lacks information on the history of the contact force during impact. Recently, Ghaednia et al. [47, 48] showed that permanent deformations after the impact can be used to achieve a better understanding of the history of the contact force during impact. In this paper, we focus on the dynamic behavior of traditional polymer materials (i.e. PLA, ABS,and acrylic) used in additive manufacturing and analyze them during collision with a rigid body. This includes both the motion of objects before and after the impact and permanent plastic deformations on the 3D-printed materials. We account for both the dynamic and static behavior of the material in the contact force. Due to ease of implementation, the Stronge formulation [44] for the contact force is used for the static phase of the force. For the dynamic behavior of the force, a nonlinear simple damper [48] has been incorporated to the Stronge’s formulation. 3
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Five different 3D printed materials have been tested during low velocity collision. A rounded end stainless steel rod was used as the rigid colliding object. Drop tests were conducted for eleven different initial velocities. Finally, experimental data for permanent deformations and the coefficient of restitution were collected and used to find the closest contact force formulation for each material.
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2. Theoretical Modeling
Figure 1 shows schematic of the normal impact of a rod with a flat. The contact force is applied on the tip of the rod in the Y-direction. The equation of motion for the rod is:
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where δ is the displacement of the tip of the rod, which is equal to the deformation on the flat, m is the mass of the rod and g is the gravitational acceleration. The equation of motion can be solved numerically to simulate the motion of the rod during the impact. The contact force Fn can be written as: ˙ Fn = Fs (δ) + Fd (δ)
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For the Fs the Stronge’s formulation in [44] is used. The static contact force is divided into two main phases, compression and restitution. The compression phase starts when the collision begins and continues until the relative normal velocity of the objects is zero. At this instant, the restitution phase starts and continues until the displacement of the tip, δ, reaches the permanent deformation, δr . The quasi-static contact force formulation, typically used for metals [47, 45, 38, 48, 49, 50], was found to be unacceptable in this experiment. The reason is that, because of the texture and material properties of 3D printed materials, the dynamic damping has a significant effect on the contact force during collision.
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where Fs and Fd are accounting for the static and dynamic behavior of the material, respectively. For Fd during the compression phase we propose the simple nonlinear damper equation: ˙ ˙ = b δ˙ 1.5 δ . (3) Fd (δ) ˙ |δ|
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The compression phase is divided into three sub-phases, fully elastic, elastic-plastic and fully plastic phase. For the fully elastic phase Hertzian[33] theory is used as follows: Fs =
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where E ∗ is the effective modulus of elasticity and R∗ is the effective radius of curvature, which can be calculated as: E ∗−1 =
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where νr and νf , Rr and Rf , and Er and Ef are the Poisson’s ratio, radius of curvature and modulus of elasticity of the rod and the flat, respectively. For our case, Rf = ∞ and Er = ∞, therefore R∗ = Rr and E ∗ = Ef /(1 − ν 2 ) The elastic phase continues until δ = δy , where δy is the deformation in which the yield starts: 2 3πνy Sy ∗ δy = R , (6) 4E ∗
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Figure 1: Schematic of the normal impact.
where the Sy is the yield strength of the flat. The elastic-plastic phase starts at δ = δy . The contact force is calculated from [44] as: 2 a a Fs = Fy 1 + 0.67νy−1 ln , (7) ay ay
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Where a/ay and Fy can be calculated as: 0.5 2 2δ 3πνy Sy a = −1 , Fy = νy Sy R∗2 π ay δy 4E ∗
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The Fully plastic phase starts at δ = δp = 84δy , and δp is the instant in which the fully plastic phase starts. The fully plastic contact force is: 2.8Fy 2δ Fs = −1 (9) νy δy 5
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2.2. Restitution phase
and Rr =
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The permanent deformation has been measured from the experimental results for each material presented in the following section. The general empirical equation for the permanent deformation is proposed as: δr = λ1 δm
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The restitution starts at δ˙ = 0, at this instant δ = δm and Fs = Fm . The restitution phase continues until δ = δr , and δr is the permanent deformation. The contact force for the restitution phase follows the Hertzian theory and can be calculated as:
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where λ1 and λ2 have been found from the experiments for each material. For the ˙ during the restitution phase we propose the following formula dynamic force, Fd (δ) ˙ = b |δ| ˙ 1.5 Fd (δ)
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δ − δr δm − δr
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which satisfies the boundary conditions during the restitution phase at both maximum and permanent deformations. Equation 13 presents a new idea which combines the conventional damper with the more recent approach to collision problems, with considering the effect of permanent deformations. The main problem with the conventional damper model happens when the damper results in a negative force at the end of collision, which is physically wrong. This problem has been solved in this study with adding the proposed coefficient to the conventional damper in Eq. 13.
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In order to conduct experiments, square-shape part with four anchor points in corR ners were designed in Solidworks with dimensions 76×76×15 mm. The equipment R used to fabricate the parts included BigRep ONE for Industrial PLA and Ultra PLA R R
with FDM technology, Makerbot for PLA with FDM technology, Stratasys Objet30 R R
using Verowhite acrylic with Poly-jet technology , and Cube using ABS with FDM technology. All samples were printed with 100% infill rate and same printing orientation patterns and parameters, except for Verowhite Poly-jet, to reduce the effect of printing direction on the dynamic strength of the materials. The experiments have been designed in order to measure permanent deformation of the plastic surface after each impact and motion of the rod. A rounded end stainless steel rod with length L = 304 mm, diameter D = 8.8 mm, and m = 0.567 kg, also with modulus of elasticity of Es = 212 GPa and yield strength of Sys = 750 MPa was used. Flat surfaces were 3D-printed and then finished with abrasive papers to the smoothest 6
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Trade Mark Industrial Ultra VeroWhite Makerbot Cube Tough
E (GPa) 3.2 2.3 2.0 3.4 1.8
Sy (MPa) 37 31 60 48 28
ν 0.360 0.360 0.375 0.360 0.350
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Table 1: Modulus of elasticity and yield strength of materials (by manufacturer specification)
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possible surface (Ra between 0.2 to 0.4 µm). The material properties of 3D-printed materials are listed in Table. 1. The schematic of the experimental setup has been shown in Fig. 2. The 3D-printed part was fixed on a massive metal flat using 4 bolts, which itself is fixed to the table with 4 clamps. The 3D printed part has been fixed to a massive table in order to reduce the effect of stress wave reflections. One should consider that the dynamic of impact depends on the relative velocity between the objects, therefore the results are useful for the case in which the rod was stationary as well as the case with both objects have motion. A magnet device is used to drop the rod vertically from different heights, H, between 0.025 m to 0.521 m. A four-head flexible microscopy illumination lights with a 1000 W projector have been used to capture clear videos with minimal noise during the impact. Each impact has been analyzed with two techniques: optical profilometry and image processing. To capture motion of the rod before, during, and after the impact, a highspeed camera capable of recording up to 200 000 frames per second (fps) has been used. The profile of each deformation has been measured with an optical profilometer after each impact. The maximum depth in the deformed region has been defined as permanent deformation.
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Figure 2: Schematic of the normal impact.
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3.1. Surface measurement 3D-printed parts and rod samples were polished so that the average roughness of the 3D-printed surfaces ranged between 0.2 to 0.4 µm. After each impact test, parts were scanned by optical profilometer to get 3D images and 2D profiles from deformed regions (Figure 3). The average of three cross sections of the deepest points in deformed region has been used to measure the permanent deformation. Distance between scanned points in X direction and profiles in Y direction was 3 µm. Also, The Z resolution of the instrument was 20 nm. μm
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Figure 3: 3D view and 2D profile of permanent deformation of one impact
In this study, for each 3D-printed part, 36 experiments for 3 specimens were performed for 12 different initial velocities from 0.5 m/s to 4.0 m/s. The coefficient of restitution and the permanent deformation are calculated for each initial velocity by averaging data from three experiments. 3.2. Image processings The motion of the rod and the radius of curvature of the tip of the rod has been calculated using image processing techniques. The radius of the curvature is needed in the contact force formulation while the motion of the rod is required to calculate the coefficient of restitution. The following subsections are explaining the process in detail. 3.2.1. Radius of curvature of the rod The radius of curvature of the tip of the rod is an important factor in our contact force formulation. Therefore to measure this parameter accurately image processing 8
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technique have been used. A 12 mega pixel image from the tip of the rod has been used for the measurements. Figure 4(a) shows the picture from the tip of the rod. The boundary of the rod has been identified and an eight-order polynomial has been fitted on the boundary, see Fig. 4(b). The radius of the curvature of the rod has been derived from the polynomial as, R = 0.0097 m.
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Figure 4: The image processing procedure to calculate the radius of curvature of the tip of the rod. (a) The original picture. (b) Processed image
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3.2.2. Motion of the rod The velocity of he rod during the tests is required for the model. Each impact collision has been recorded using a high speed camera at 10 000 frames per second. Every part of the rod, except the tip, has been painted in black. Two white adhesive strips have been attached to the rod for use in image processing (Fig. 5(a)). In the image processing each frame has been threshold to pure white and pure black, see Fig. 5(b). In order to track the motion of the rod, the positions of the three edges from the white strips was calculated. To find the position of the edges, the first derivative of the vertical direction was used. Each pixel has a gray value, g(x, y), which is an 8-bit number with x and y as the horizontal and vertical axis respectively. Let the frame size be m × n where m and n are the number of rows and columns, respectively, and the χ(y) be a coefficient that we defined as: n X ∂g(x, y) χ(y) = (14) ∂y x=1
χ(y) is the sum of vertical first derivation of g(x, y) for all pixels in each row. Figure. 5(c) shows χ(y) for the same frame as Fig. 5(a). The three large peaks show the position of the edges, which are shown in Fig. 5(b) with three horizontal lines. The average position of these three peaks have been used to calculate the position of the rod. Figure. 6 shows the vertical position of the rod during an impact test. The minimum point shows the instant the impact happens. To find the velocity before and after the impact two lines have been fitted on the data after and before the collision (Fig. 7). The slope of the lines show the velocities. For this specific experiment the initial vertical velocity is, vi = −3.258 m/s, the final vertical velocity is, vf = 1.794 m/s and coefficient of restitution, e = 0.550. 9
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Figure 5: The image processing procedure of the rod during one of the impacts. (a) The original frame. (b) Pure black and pure white image. (c) χ(y) for (b), sum of the first order vertical derivation.
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Figure 6: Vertical position of the rod during a test for industrial PLA (initial vertical velocity, vi = −3.258 m/s).
4. Results
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Figure 7 shows the experimental results for the coefficient of restitution as a function of initial velocity for different 3D-printed materials. The initial velocity for all experiments ranged from 0.5 to 4 m/s. Industrial PLA and VeroWhite Polyjet showed the largest and Ultra PLA shows smallest coefficients of restitution for most initial velocities. The coefficients of restitution appear to be decreasing with increase in initial velocity. Also it can be seen that the coefficient of restitution is converging to a constant value for higher initial velocities. Figure 8 represents the permanent deformation after the impact as a function of the initial velocity. The permanent deformation increases with an increase of the initial ve10
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locity. Cube ABS and Ultra PLA show larger permanent deformations with significant differences compared to other materials. It could also be predicted from the material properties that the Cube ABS with the smallest yield strength deforms more than the other materials. Makerbot PLA and Industrial PLA show shallower permanent deformations. Permanent deformation for VeroWhite Polyjet were as measured as 10 times smaller than the average roughness (Ra ) of the surface, therefore the deformations for this material have been assumed negligible.
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Using numerical simulations and the experimental results, an empirical equation, Eq. (12), has been fitted to describe the permanent deformation of each of the materials. First the experimental results for the permanent deformation, Fig. 8, have been used to find λ1 and λ2 in Eq. (12). The results for each of the materials is presented in Table 2. Second the damping coefficient in Eq. (3) for each of the materials has been found, (Table 2). To calculate the damping coefficient, b, for each of the materials, the experimental results of coefficient of restitution for different initial velocities were used. For each material, the nonlinear differential equation of motion for a range of damping coefficients were solved using the Runge-Kutta numerical integration method. Then, the numerical results were compared with the experimental values of the COR for all of the initial velocities. The average error between the experiments and the theory was calculated for each value of the damping coefficient, b, of the corresponding material. The damping coefficient, b, with the least errorwas chosen for each of the materials. This gives only one damping coefficient for each material; however, it should be considered that this damping coefficient takes both the value and the curvature of the COR as a function of initial velocity. It should be notes that, the damping coefficient cannot be solved for each of the experimental results, because the nonlinear differential equations are irreversible in time. 11
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Figure 8: Permanent deformation vs. the of initial impact velocity
Table 2: Damping coefficient, b, (Eq. (3)), λ1 and λ2 (Eq. (12))
b (kg/s) 365 550 496 535 410
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λ2 1.219 0.817 0.786 0.514
The results in Tab. 2 along with Eqs. (1-12) can be used to simulate the impact for each of these materials. Figure 9 shows the comparison between the results from Table 2 with the experimental data for coefficient of restitution for each of the materials. As it can be seen, the proposed model matches the experimental results for all of the materials. The mean absolute deviation between the experiments and the model is less than 5%. The theoretical model show good match for“Makerbot PLA”, “Vero White Polyjet” and “Industrial PLA”. For “Ultra PLA” the experiments show some noise, which is reasonable for collision tests. For “Cube ABS” material it can be seen that the theorithical values do not fully matche with the experimental results. One reason for this is that for vi = 3.9 m/s detachment and delamination of the printed layers was observed. This phenomenon can require significant amounts of energy, and therefore decreases the COR. Figure 10 shows the comparison between the model and the experimental data for the permanent deformation. The models show a good agreement with average error less than 5% compare to the experiments. Results for the VeroWhite polyjet material are
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λ1 0.000393 0.008766 0.004606 0.049270
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Material Industrial PLA Ultra PLA Polyjet Verowhite Makerbot PLA Cube Tough ABS
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Figure 9: Verification of the formulation for coefficient of restitution of each of the materials. In each figure the dots show the experimental results and the lines show theoretical predictions.
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not presented because for the initial velocity range in our experiments the permanent deformations were negligible. The reason for significant difference observed between polyjet and FDM parts appear to be mostly because of the difference between technologies for printing polymers of Polyjet system and extrusion-based FDM systems tested in this study. In Polyjet, an acrylic photopolymer is projected over the previous layer and its curing depends on ultraviolet (UV) radiation. When the layer has been cured, the tray recesses by an amount equivalent to a layer height (28 µm in this case) along the Z axis, and this procedure is repeating until the part is accomplished [51, 52]. But in extrusion based systems, the part is made by extruding small roads of molten materials to form layers as the material (150-200 µm thickness) becomes hard immediately after extrusion 13
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from the nozzle. Based on the nature of the processes it can be concluded that, polyjet based materials would have much stronger bonding between layers than extrusion based materials which have thicker layers with no UV curing process, and then very stronger structure under impact in the range of this experiments initial velocities [53]. This could make a difference between the elastic and plastic phases, which is almost totally elastic for Polyjet printed parts in the range of initial velocities of this study. All printers used in this study are FDM-based, except for Polyjet 3D printer, and were setup with same printing pattern, speed, and nozzle temperature. Thus, it can be said that methods are the same and different behaviors mostly comes from the material properties. Ultra PLA
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Figure 10: Verification of the formulation for permanent deformation of each of the materials. In each figure the dots show the experimental results and the lines show theoretical predictions.
Conventionally only the experimental results for coefficient of restitution have been used to develop contact models for collision problems. This only requires the impulse to be correct. However, to match the model to the experiments for both the permanent deformation and the coefficient of restitution, the history of the contact force and the impulse during the impact should follow the material behavior. This increases the accuracy of the developed model, significantly. Finally, the material behavior during impact can be simulated using the proposed formulation. Figure 11 shows the contact force as a function of deformation for all of the materials. The initial velocity of the impact is vi = 0.68 (m/s) for all materials. The force starts with a positive value at δ = 0 because of the existence of a damper in the model. With the increase of the deformation the contact force increases until reaches the maximum value. At this instant the velocity is zero and the restitution phase starts. During the restitution phase the contact force decrease until the deformation reaches the 14
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permanent deformation, δ = δr . The impact ends at this instant and the contact force is zero.
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Industrial PLA Ultra PLA Makerbot PLA VeroWhite Polyjet Cube ABS
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The experimental and theoretical methods used in this study could be applied for other polymers. The constants, k, b, λ1 , λ2 , can be found in the presented formulation for new polymers with a few simple impact experiments. The presented model can be adapted for other materials to be used in simulations.
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Figure 11: Simulated contact force during the impact vs. deformation of the flat
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Although mechanical properties of 3D-printed materials have been widely studied and verified in many applications, there are no significant studies of the permanent deformation of 3D-printed materials. Five different 3D-printed materials have been analyzed during the impact. The contact force during the impact for each of the materials have been modeled. A damping force has been added to the Stronge’s formulation for the impact. Behavior of each of the materials during the impact has been analyzed experimentally for both the coefficient of restitution and the permanent deformations. Experimental results have been used to find the needed coefficients in the formulation for each of the materials. There are many parameters for 3D-printed materials that can affect the permanent deformations and coefficient of restitution such as the porosity, printing pattern, nozzle temperature, and layer thickness that can be studied in future work.
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5. Conclusion
6. Acknowledgment 7. References 345
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neering 132 (2015) 62–69. [53] K. Kardel, An analytical and experimental study on 3d-printed custom surfaces for benthic algal biofilms, Ph.D. thesis, Auburn University (2016).
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S2214-8604(17)30030-1 http://dx.doi.org/doi:10.1016/j.addma.2017.01.004 ADDMA 148
To appear in: Received date: Revised date: Accepted date:
18-12-2015 10-1-2017 23-1-2017
Please cite this article as: Kamran Kardel, Hamid Ghaednia, Andres L. Carrano, Dan B. Marghitu, Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision with a Rigid Rod, (2017), http://dx.doi.org/10.1016/j.addma.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision with a Rigid Rod
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Kamran Kardela,∗, Hamid Ghaedniab , Andres L. Carranoc , Dan B. Marghitub a Department
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of Manufacturing Engineering, 1100 IT Drive, Georgia Southern University, Statesboro, GA 30458-7991 b Department of Mechanical Engineering,1418 Wiggins Hall, Auburn University, Auburn AL 36849-5341 c Department of Industrial and Systems Engineering,3301 Shelby Center, Auburn University, Auburn AL 36849-5346
Abstract
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The behavior of five different 3D-printed polymers has been analyzed both theoretically and experimentally under low-speed collision conditions. The impact of a rigid rod with a flat specimen fabricated of 3D-printed materials was analyzed. An experimental setup has been designed in order to capture the motion of the rod during the impact using a high-speed camera. Image processing algorithms were developed to estimate the velocity before and after the impact as well as the coefficient of restitution. Also, permanent deformations after the impact were scanned with an optical profilometer. In this work, a theoretical formulation for the contact force during the impact is proposed. The impact was divided into two phases, compression and restitution, in which materials considered elastic-plastic in the first and fully elastic in the second one. The experimental results are used to measure the damping coefficient. Results show a good correlation between the proposed formulation for the contact force and the behavior of materials.
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Keywords: contact force, permanent deformation, coefficient of restitution, 3D-printed polymers, additive manufacturing
1. Introduction
5
Additive manufacturing (AM) or 3D-printing enables the design and manufacture of complex geometries with polymers, ceramics, and metals which are not feasible with traditional manufacturing methods. Recent advances in AM has facilitated reductions in tooling needs, assembly and lead time, as well as increased efficiencies in the use of materials [1, 2]. Many relevant industry sectors such as automotive, aerospace, biosystems, and healthcare have already benefited by the use of AM [3, 4]. However, AM remains to be widely adopted across manufacturing industries [5]. Some factors limiting widespread incorporation of these technologies in industry sectors may ∗ Corresponding
author Email addresses: [email protected] (Kamran Kardel), [email protected] (Hamid Ghaednia), [email protected] (Andres L. Carrano), [email protected] (Dan B. Marghitu) Preprint submitted to Journal of LATEX Templates January 9, 2017
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be related to the lack of comprehensive information on mechanical properties of the rapidly evolving material set available to industry. Although there are extensive studies available on the mechanical properties of 3D-printed materials ([6]-[7]), new materials have appeared since and other mechanical properties have gained significance. This is particularly true for polymers and for the understudied field of mechanical impact. Understanding the behavior of new materials under dynamic contact as well as the history of the contact force during collision can provide insights leading towards more efficient mechanical designs. There are many research efforts on testing 3D-printed plastics in terms of mechanical properties available in archival literature [6, 8, 9, 10, 11]. The tensile strength of 3Dprinted polylactic acid (PLA) bars used in bone fixation and reconstruction has been tested [12]. Test specimens were fabricated from both low and high molecular weight PLA powders with chloroform used as a binder. The maximum tensile strength was found to be 15.94 ± 1.50 MPa for high molecular weight PLA and 17.40 ± 0.71 MPa for low molecular weight. More recently, research on the influence of print density on microstructure and mechanical properties of nylon parts in High Speed Sintering (HSS) showed that the mechanical properties are related to crystallinity [13]. Results showed that crystallinity decreased as print density increased. in turn, stiffness and tensile strength increased with crystallinity. In other effort, Acrylonitrile butadiene styrene (ABS) parts fabricated by Fused Deposition Modeling (FDM) were also studied for their tensile and compressive strengths [14]. It was found that for FDM parts made with a 0.003 inch overlap between roads, the tensile strength ranged between 65-72% of the strength of identical parts made with injection molded ABS. In another study, the mechanical properties, surface roughness, geometric and dimensional accuracy, and material costs of various types of materials processed with different types of AM technologies, including stereolithography, polymer extrusion, powder bed fusion, sheet lamination, and vat photopolymerization were tested and compared [15]. Other archival literature on fatigue behavior of laser-sintered nylon (PA12) showed that PA12 specimens under rotating-bending conditions will experience fatigue failure [16]. It was suggested that, rotating-bending at frequencies of up to 50 Hz is a valid way of determining the fatigue behavior of laser-sintered PA12. Parts made with stereolithography (SLA) process were investigated to elucidate the influence of layer thickness on mechanical properties of produced parts [17]. The results showed that, when the layer thickness decreases, the strength of parts made by SLA increases. Another effort pursued the effects of electroplating on polymer parts made with SLA [18]. A series of tests were conducted on samples coated with copper and nickel with varying thicknesses. Physical testing confirmed that a thicker coating (120 µm) increased impact energy and Young’s modulus but had a minimal effect on ductility of parts, in which a thinner coating (20 µm) resulted in a smoother surface finish. Also, a computer aided characterization approach has been used to evaluate the effective mechanical properties of porous biodegradable polyester tissue scaffold [19]. An approach for design and 3D-printing of porous tissue scaffold, the computational algorithm for finite element implementation, and the characterization of the effective mechanical properties of porous poly-ε-caprolactone scaffolds manufactured by polymer extrusion was provided. There are also previous research efforts on the mechanical properties of additively manufactured metals available in literature. These include experimental testing and simulation for fracture criteria, fatigue, tensile strength, among others [20, 21, 22, 23, 2
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24, 25, 26, 27]. Also the behavior of parts produced by Selective Laser Melting (SLM) has been compared to conventionally produced products for, traction, compression and flexural properties [28]. Additively manufactured AlSi10Mg using SLM was studied in terms of microstructure, high cycle fatigue, and fracture behavior [29]. For approximating flexural properties of bone, open cellular structures fabricated in Ti6Al4V using Electron Beam Melting (EBM) have been proposed for tissue scaffold and low stiffness implants [30]. In another study, the effect of build orientation and heat treatment on the mechanical properties of SLM Ti6Al4V lattice structures was investigated [31]. Different binding mechanisms were also investigated in powder bed fusion and SLM with metal powder to compare the mechanical properties and also improve understanding of these processes [7]. With respect to the dynamic behavior of 3D-printed materials, simulation of impact can be divided into two phases, compression and restitution[32]. The contact force between the objects during each of these phases is the most important factor needed to simulate the impact. The contact force itself can be divided into three phases, fully elastic, elastic-plastic and fully plastic phases. The fully elastic and fully plastic phases can be solved analytically. Hertzian theory[33] has been commonly used for the fully elastic phase. For the fully plastic phase there are few alternative approaches that can be found in available literature [34, 35]. The archival research available on the dynamic behavior of the 3D-printed materials during collision is very limited and not current, given the increased use of AM parts in industry and the evolution of the materials. Unfortunately, to date, the elastic-plastic phase is not amenable to a closed-form analytical solution. Arguably, for the collision problems the elastic-plastic phase plays the most important role. Many efforts have aimed at predicting the elastic-plastic contact forces for homogeneous metal materials, including Johnson [34], Jackson et al. [36], Kogut and Etsion [37], Ghaednia et al. [38], among many other models. A comprehensive comparison between different contact models for collision problems can also be found in literature [39]. Although, these contact models have been developed for quasi static contact; they have often been used in collision problems, such as those found in Stronge [40, 41, 42, 43, 44], Marghitu et al. [45] and Jackson et al. [46]. Conventionally, the coefficient of restitution is defined as the ratio between the rebound and initial velocities (i.e. Newton’s definition) or as the ratio between the restitution and compression impulses (i.e. Poisson’s definition) [32], which is used to account for the energy loss during the impact. The coefficient of restitution is easy to measure experimentally but lacks information on the history of the contact force during impact. Recently, Ghaednia et al. [47, 48] showed that permanent deformations after the impact can be used to achieve a better understanding of the history of the contact force during impact. In this paper, we focus on the dynamic behavior of traditional polymer materials (i.e. PLA, ABS,and acrylic) used in additive manufacturing and analyze them during collision with a rigid body. This includes both the motion of objects before and after the impact and permanent plastic deformations on the 3D-printed materials. We account for both the dynamic and static behavior of the material in the contact force. Due to ease of implementation, the Stronge formulation [44] for the contact force is used for the static phase of the force. For the dynamic behavior of the force, a nonlinear simple damper [48] has been incorporated to the Stronge’s formulation. 3
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Five different 3D printed materials have been tested during low velocity collision. A rounded end stainless steel rod was used as the rigid colliding object. Drop tests were conducted for eleven different initial velocities. Finally, experimental data for permanent deformations and the coefficient of restitution were collected and used to find the closest contact force formulation for each material.
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2. Theoretical Modeling
Figure 1 shows schematic of the normal impact of a rod with a flat. The contact force is applied on the tip of the rod in the Y-direction. The equation of motion for the rod is:
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(1)
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where δ is the displacement of the tip of the rod, which is equal to the deformation on the flat, m is the mass of the rod and g is the gravitational acceleration. The equation of motion can be solved numerically to simulate the motion of the rod during the impact. The contact force Fn can be written as: ˙ Fn = Fs (δ) + Fd (δ)
(2)
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For the Fs the Stronge’s formulation in [44] is used. The static contact force is divided into two main phases, compression and restitution. The compression phase starts when the collision begins and continues until the relative normal velocity of the objects is zero. At this instant, the restitution phase starts and continues until the displacement of the tip, δ, reaches the permanent deformation, δr . The quasi-static contact force formulation, typically used for metals [47, 45, 38, 48, 49, 50], was found to be unacceptable in this experiment. The reason is that, because of the texture and material properties of 3D printed materials, the dynamic damping has a significant effect on the contact force during collision.
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where Fs and Fd are accounting for the static and dynamic behavior of the material, respectively. For Fd during the compression phase we propose the simple nonlinear damper equation: ˙ ˙ = b δ˙ 1.5 δ . (3) Fd (δ) ˙ |δ|
2.1. Compression phase
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The compression phase is divided into three sub-phases, fully elastic, elastic-plastic and fully plastic phase. For the fully elastic phase Hertzian[33] theory is used as follows: Fs =
4 ∗ ∗0.5 1.5 E R δ , 3
(4)
where E ∗ is the effective modulus of elasticity and R∗ is the effective radius of curvature, which can be calculated as: E ∗−1 =
(1 − νr2 ) (1 − νf2 ) 1 1 + , R∗−1 = + , Er Ef Rr Rf
(5)
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Fn
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where νr and νf , Rr and Rf , and Er and Ef are the Poisson’s ratio, radius of curvature and modulus of elasticity of the rod and the flat, respectively. For our case, Rf = ∞ and Er = ∞, therefore R∗ = Rr and E ∗ = Ef /(1 − ν 2 ) The elastic phase continues until δ = δy , where δy is the deformation in which the yield starts: 2 3πνy Sy ∗ δy = R , (6) 4E ∗
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Figure 1: Schematic of the normal impact.
where the Sy is the yield strength of the flat. The elastic-plastic phase starts at δ = δy . The contact force is calculated from [44] as: 2 a a Fs = Fy 1 + 0.67νy−1 ln , (7) ay ay
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Where a/ay and Fy can be calculated as: 0.5 2 2δ 3πνy Sy a = −1 , Fy = νy Sy R∗2 π ay δy 4E ∗
(8)
The Fully plastic phase starts at δ = δp = 84δy , and δp is the instant in which the fully plastic phase starts. The fully plastic contact force is: 2.8Fy 2δ Fs = −1 (9) νy δy 5
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2.2. Restitution phase
and Rr =
2
1
3.
(δm δr )
(10)
(11)
The permanent deformation has been measured from the experimental results for each material presented in the following section. The general empirical equation for the permanent deformation is proposed as: δr = λ1 δm
δm δy
λ2 ,
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4 ∗ E Fm 3
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4 ∗ ∗0.5 E Rr (δ − δr )1.5 , 3
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The restitution starts at δ˙ = 0, at this instant δ = δm and Fs = Fm . The restitution phase continues until δ = δr , and δr is the permanent deformation. The contact force for the restitution phase follows the Hertzian theory and can be calculated as:
(12)
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where λ1 and λ2 have been found from the experiments for each material. For the ˙ during the restitution phase we propose the following formula dynamic force, Fd (δ) ˙ = b |δ| ˙ 1.5 Fd (δ)
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δ − δr δm − δr
˙ δ , ˙ |δ|
(13)
3. Experiments
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which satisfies the boundary conditions during the restitution phase at both maximum and permanent deformations. Equation 13 presents a new idea which combines the conventional damper with the more recent approach to collision problems, with considering the effect of permanent deformations. The main problem with the conventional damper model happens when the damper results in a negative force at the end of collision, which is physically wrong. This problem has been solved in this study with adding the proposed coefficient to the conventional damper in Eq. 13.
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In order to conduct experiments, square-shape part with four anchor points in corR ners were designed in Solidworks with dimensions 76×76×15 mm. The equipment R used to fabricate the parts included BigRep ONE for Industrial PLA and Ultra PLA R R
with FDM technology, Makerbot for PLA with FDM technology, Stratasys Objet30 R R
using Verowhite acrylic with Poly-jet technology , and Cube using ABS with FDM technology. All samples were printed with 100% infill rate and same printing orientation patterns and parameters, except for Verowhite Poly-jet, to reduce the effect of printing direction on the dynamic strength of the materials. The experiments have been designed in order to measure permanent deformation of the plastic surface after each impact and motion of the rod. A rounded end stainless steel rod with length L = 304 mm, diameter D = 8.8 mm, and m = 0.567 kg, also with modulus of elasticity of Es = 212 GPa and yield strength of Sys = 750 MPa was used. Flat surfaces were 3D-printed and then finished with abrasive papers to the smoothest 6
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Trade Mark Industrial Ultra VeroWhite Makerbot Cube Tough
E (GPa) 3.2 2.3 2.0 3.4 1.8
Sy (MPa) 37 31 60 48 28
ν 0.360 0.360 0.375 0.360 0.350
cr
Type PLA PLA Acrylic PLA ABS
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Table 1: Modulus of elasticity and yield strength of materials (by manufacturer specification)
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possible surface (Ra between 0.2 to 0.4 µm). The material properties of 3D-printed materials are listed in Table. 1. The schematic of the experimental setup has been shown in Fig. 2. The 3D-printed part was fixed on a massive metal flat using 4 bolts, which itself is fixed to the table with 4 clamps. The 3D printed part has been fixed to a massive table in order to reduce the effect of stress wave reflections. One should consider that the dynamic of impact depends on the relative velocity between the objects, therefore the results are useful for the case in which the rod was stationary as well as the case with both objects have motion. A magnet device is used to drop the rod vertically from different heights, H, between 0.025 m to 0.521 m. A four-head flexible microscopy illumination lights with a 1000 W projector have been used to capture clear videos with minimal noise during the impact. Each impact has been analyzed with two techniques: optical profilometry and image processing. To capture motion of the rod before, during, and after the impact, a highspeed camera capable of recording up to 200 000 frames per second (fps) has been used. The profile of each deformation has been measured with an optical profilometer after each impact. The maximum depth in the deformed region has been defined as permanent deformation.
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3.1. Surface measurement 3D-printed parts and rod samples were polished so that the average roughness of the 3D-printed surfaces ranged between 0.2 to 0.4 µm. After each impact test, parts were scanned by optical profilometer to get 3D images and 2D profiles from deformed regions (Figure 3). The average of three cross sections of the deepest points in deformed region has been used to measure the permanent deformation. Distance between scanned points in X direction and profiles in Y direction was 3 µm. Also, The Z resolution of the instrument was 20 nm. μm
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Figure 3: 3D view and 2D profile of permanent deformation of one impact
In this study, for each 3D-printed part, 36 experiments for 3 specimens were performed for 12 different initial velocities from 0.5 m/s to 4.0 m/s. The coefficient of restitution and the permanent deformation are calculated for each initial velocity by averaging data from three experiments. 3.2. Image processings The motion of the rod and the radius of curvature of the tip of the rod has been calculated using image processing techniques. The radius of the curvature is needed in the contact force formulation while the motion of the rod is required to calculate the coefficient of restitution. The following subsections are explaining the process in detail. 3.2.1. Radius of curvature of the rod The radius of curvature of the tip of the rod is an important factor in our contact force formulation. Therefore to measure this parameter accurately image processing 8
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technique have been used. A 12 mega pixel image from the tip of the rod has been used for the measurements. Figure 4(a) shows the picture from the tip of the rod. The boundary of the rod has been identified and an eight-order polynomial has been fitted on the boundary, see Fig. 4(b). The radius of the curvature of the rod has been derived from the polynomial as, R = 0.0097 m.
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3.2.2. Motion of the rod The velocity of he rod during the tests is required for the model. Each impact collision has been recorded using a high speed camera at 10 000 frames per second. Every part of the rod, except the tip, has been painted in black. Two white adhesive strips have been attached to the rod for use in image processing (Fig. 5(a)). In the image processing each frame has been threshold to pure white and pure black, see Fig. 5(b). In order to track the motion of the rod, the positions of the three edges from the white strips was calculated. To find the position of the edges, the first derivative of the vertical direction was used. Each pixel has a gray value, g(x, y), which is an 8-bit number with x and y as the horizontal and vertical axis respectively. Let the frame size be m × n where m and n are the number of rows and columns, respectively, and the χ(y) be a coefficient that we defined as: n X ∂g(x, y) χ(y) = (14) ∂y x=1
χ(y) is the sum of vertical first derivation of g(x, y) for all pixels in each row. Figure. 5(c) shows χ(y) for the same frame as Fig. 5(a). The three large peaks show the position of the edges, which are shown in Fig. 5(b) with three horizontal lines. The average position of these three peaks have been used to calculate the position of the rod. Figure. 6 shows the vertical position of the rod during an impact test. The minimum point shows the instant the impact happens. To find the velocity before and after the impact two lines have been fitted on the data after and before the collision (Fig. 7). The slope of the lines show the velocities. For this specific experiment the initial vertical velocity is, vi = −3.258 m/s, the final vertical velocity is, vf = 1.794 m/s and coefficient of restitution, e = 0.550. 9
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Figure 6: Vertical position of the rod during a test for industrial PLA (initial vertical velocity, vi = −3.258 m/s).
4. Results
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Figure 7 shows the experimental results for the coefficient of restitution as a function of initial velocity for different 3D-printed materials. The initial velocity for all experiments ranged from 0.5 to 4 m/s. Industrial PLA and VeroWhite Polyjet showed the largest and Ultra PLA shows smallest coefficients of restitution for most initial velocities. The coefficients of restitution appear to be decreasing with increase in initial velocity. Also it can be seen that the coefficient of restitution is converging to a constant value for higher initial velocities. Figure 8 represents the permanent deformation after the impact as a function of the initial velocity. The permanent deformation increases with an increase of the initial ve10
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locity. Cube ABS and Ultra PLA show larger permanent deformations with significant differences compared to other materials. It could also be predicted from the material properties that the Cube ABS with the smallest yield strength deforms more than the other materials. Makerbot PLA and Industrial PLA show shallower permanent deformations. Permanent deformation for VeroWhite Polyjet were as measured as 10 times smaller than the average roughness (Ra ) of the surface, therefore the deformations for this material have been assumed negligible.
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Using numerical simulations and the experimental results, an empirical equation, Eq. (12), has been fitted to describe the permanent deformation of each of the materials. First the experimental results for the permanent deformation, Fig. 8, have been used to find λ1 and λ2 in Eq. (12). The results for each of the materials is presented in Table 2. Second the damping coefficient in Eq. (3) for each of the materials has been found, (Table 2). To calculate the damping coefficient, b, for each of the materials, the experimental results of coefficient of restitution for different initial velocities were used. For each material, the nonlinear differential equation of motion for a range of damping coefficients were solved using the Runge-Kutta numerical integration method. Then, the numerical results were compared with the experimental values of the COR for all of the initial velocities. The average error between the experiments and the theory was calculated for each value of the damping coefficient, b, of the corresponding material. The damping coefficient, b, with the least errorwas chosen for each of the materials. This gives only one damping coefficient for each material; however, it should be considered that this damping coefficient takes both the value and the curvature of the COR as a function of initial velocity. It should be notes that, the damping coefficient cannot be solved for each of the experimental results, because the nonlinear differential equations are irreversible in time. 11
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Table 2: Damping coefficient, b, (Eq. (3)), λ1 and λ2 (Eq. (12))
b (kg/s) 365 550 496 535 410
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The results in Tab. 2 along with Eqs. (1-12) can be used to simulate the impact for each of these materials. Figure 9 shows the comparison between the results from Table 2 with the experimental data for coefficient of restitution for each of the materials. As it can be seen, the proposed model matches the experimental results for all of the materials. The mean absolute deviation between the experiments and the model is less than 5%. The theoretical model show good match for“Makerbot PLA”, “Vero White Polyjet” and “Industrial PLA”. For “Ultra PLA” the experiments show some noise, which is reasonable for collision tests. For “Cube ABS” material it can be seen that the theorithical values do not fully matche with the experimental results. One reason for this is that for vi = 3.9 m/s detachment and delamination of the printed layers was observed. This phenomenon can require significant amounts of energy, and therefore decreases the COR. Figure 10 shows the comparison between the model and the experimental data for the permanent deformation. The models show a good agreement with average error less than 5% compare to the experiments. Results for the VeroWhite polyjet material are
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λ1 0.000393 0.008766 0.004606 0.049270
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not presented because for the initial velocity range in our experiments the permanent deformations were negligible. The reason for significant difference observed between polyjet and FDM parts appear to be mostly because of the difference between technologies for printing polymers of Polyjet system and extrusion-based FDM systems tested in this study. In Polyjet, an acrylic photopolymer is projected over the previous layer and its curing depends on ultraviolet (UV) radiation. When the layer has been cured, the tray recesses by an amount equivalent to a layer height (28 µm in this case) along the Z axis, and this procedure is repeating until the part is accomplished [51, 52]. But in extrusion based systems, the part is made by extruding small roads of molten materials to form layers as the material (150-200 µm thickness) becomes hard immediately after extrusion 13
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from the nozzle. Based on the nature of the processes it can be concluded that, polyjet based materials would have much stronger bonding between layers than extrusion based materials which have thicker layers with no UV curing process, and then very stronger structure under impact in the range of this experiments initial velocities [53]. This could make a difference between the elastic and plastic phases, which is almost totally elastic for Polyjet printed parts in the range of initial velocities of this study. All printers used in this study are FDM-based, except for Polyjet 3D printer, and were setup with same printing pattern, speed, and nozzle temperature. Thus, it can be said that methods are the same and different behaviors mostly comes from the material properties. Ultra PLA
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Figure 10: Verification of the formulation for permanent deformation of each of the materials. In each figure the dots show the experimental results and the lines show theoretical predictions.
Conventionally only the experimental results for coefficient of restitution have been used to develop contact models for collision problems. This only requires the impulse to be correct. However, to match the model to the experiments for both the permanent deformation and the coefficient of restitution, the history of the contact force and the impulse during the impact should follow the material behavior. This increases the accuracy of the developed model, significantly. Finally, the material behavior during impact can be simulated using the proposed formulation. Figure 11 shows the contact force as a function of deformation for all of the materials. The initial velocity of the impact is vi = 0.68 (m/s) for all materials. The force starts with a positive value at δ = 0 because of the existence of a damper in the model. With the increase of the deformation the contact force increases until reaches the maximum value. At this instant the velocity is zero and the restitution phase starts. During the restitution phase the contact force decrease until the deformation reaches the 14
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permanent deformation, δ = δr . The impact ends at this instant and the contact force is zero.
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The experimental and theoretical methods used in this study could be applied for other polymers. The constants, k, b, λ1 , λ2 , can be found in the presented formulation for new polymers with a few simple impact experiments. The presented model can be adapted for other materials to be used in simulations.
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Figure 11: Simulated contact force during the impact vs. deformation of the flat
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Although mechanical properties of 3D-printed materials have been widely studied and verified in many applications, there are no significant studies of the permanent deformation of 3D-printed materials. Five different 3D-printed materials have been analyzed during the impact. The contact force during the impact for each of the materials have been modeled. A damping force has been added to the Stronge’s formulation for the impact. Behavior of each of the materials during the impact has been analyzed experimentally for both the coefficient of restitution and the permanent deformations. Experimental results have been used to find the needed coefficients in the formulation for each of the materials. There are many parameters for 3D-printed materials that can affect the permanent deformations and coefficient of restitution such as the porosity, printing pattern, nozzle temperature, and layer thickness that can be studied in future work.
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6. Acknowledgment 7. References 345
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