- Email: [email protected]
Constitutive parameter identification of 3D printing material based on the virtual fields method
Measurement 59 (2015) 38–43
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Constitutive parameter identification of 3D printing material based on the virtual fields method Xianglu Dai, Huimin Xie ⇑ AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 20 June 2014 Received in revised form 2 September 2014 Accepted 16 September 2014 Available online 28 September 2014 Keywords: 3D printing material Integrated deformation carriers Constitutive parameter identification Virtual fields method
a b s t r a c t In recent years, 3D printing technology has grown rapidly, and also has shown the great potential to be utilized in different fields. The identification of the constitutive parameters of materials fabricated by 3D printing is very important for product designing and technique selection. In this paper, a constitutive parameter identification method for 3D printing materials combining the integrated deformation carriers with the virtual fields method (VFM) is presented. The experimental process consists of three steps: fabricating the specimen with integrated deformation carriers by 3D printing; measuring the deformation fields by a full-field optical method; identifying the constitutive parameters by VFM. In the first step, the design method of the integrated deformation carriers is described in detail. Serving as a practice of the above process, a bending specimen with integrated deformation carriers was manufactured by the stereolithography technique, and the orthotropic constitutive parameters of this specimen at different temperatures were identified. The successful experimental results verify the feasibility of the proposed method, and show its advantages on aspects of high efficiency and easy processing as well. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the field of manufacture, there are three generic methodologies which include: subtractive methodology, formative methodology and additive methodology, for the production of prototypes or manufactured parts [1]. The first two methodologies are widely used, but show some drawbacks, e.g. difficulty in fabricating a part with a composite shape in the subtractive methodology; high cost when the output is small due to the expensive mold used in the formative methodology. The additive methodology is a relatively new approach to the manufacture of prototypes and end-use parts, and it used to be referred to as rapid prototyping (RP) [2,3], which is generally being called 3D printing now.
⇑ Corresponding author. E-mail address: [email protected] (H. Xie). http://dx.doi.org/10.1016/j.measurement.2014.09.033 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
The 3D printing includes a group of techniques that enable it to quickly fabricate a scale model of a physical product or an end-use part according to the threedimensional computer aided design (CAD) data, and there is no high requirement of complex implement and skilled model maker. In a 3D printing process, CAD data are reformatted into a special file in which the model is sliced to be many layers; then this file is transferred into a selected 3D printing system; afterwards this system will reproduce the model layer-by-layer; and finally a physical model of the original CAD data can be exported. Up to now, different methods have been developed to achieve these CAD data, such as magnetic resonance imaging (MRI), computed tomography (CT) scanning as well as point cloud data generated by engineering scanning or digitizing systems [3]. Since its emergence in 1980s, 3D printing has been witnessed rapid development: various forming techniques have been proposed [3,4], more and more materials and structures can be manufactured [5,6], and the application
X. Dai, H. Xie / Measurement 59 (2015) 38–43
fields are growing fast [7,8]. Recently, increasingly 3D printing parts have been used as end-use parts, and hence it is vital for the designers to characterize the mechanical properties of the materials produced by 3D printing system in order to optimize the material specification in their design process. However, as limited relevant literatures are available [2] and there are growing requirement to understand the mechanical behavior of 3D printing materials, the relative research is in urgent need. In this paper, our work focuses on the study of the constitutive parameter identification of 3D printing materials, and a constitutive parameters identification method combining the integrated deformation carriers with the VFM [9] is presented. The operation of this method includes three main steps: firstly, fabricate a specimen with integrated deformation carriers by 3D printing, and the integrated deformation carriers can be the lattices or speckles according to the selected optical method; then add the load on the specimen step-by-step and obtain the full-field displacement by an optical method, such as digital image correlation (DIC) [10–14], geometric phase analysis (GPA) [15–17], grid method [18] or moiré method [19]; finally, the VFM is employed to identify the desired constitutive parameters by using the data of the full-field displacement. The integrated deformation carriers could be fabricated on an interested area of specimen as needed, and then is used in the deformation measurement, so no more extra sensor is required (e.g. strain gauge or extensometer). The VFM [9,20,21] is a high efficient method which can identify many constitutive parameters in one test simultaneously, and thus it is time-saving and low cost; besides, the VFM is effective for both isotropic and anisotropic materials. In order to demonstrate the superiority of the presented method, the orthotropic constitutive parameters at different temperatures of a stereolithography [22] specimen with integrated lattices were identified. Besides, the influence on the identified parameters caused by missing data near the edge of calculated area of DIC method was analyzed by the finite element analysis (FEA). 2. The design of 3D printing specimen with integrated deformation carriers In general, deformation carriers are essential components for optical methods, for example, speckles for DIC, lattices or grid for GPA and grid method, grating (largearea lattices or grid) for moiré method. The deformation carriers are individually prepared after specimen is manufactured, and some conditions should be satisfied for obtaining the matched deformation carriers. For example: the material of deformation carriers should be compatible with the material of specimen (no reaction, easy bonding, small reinforced effect, low thermal mismatch). But it may be very difficult to meet these requirements. In order to tackle the difficulty, the deformation carrier and specimen can be generated with an integrated designing and fabrication. Benefiting from the flexibility of the 3D printing, the specimen and deformation carriers can be manufactured simultaneously; besides, the type, position, size and area of these carriers can be adjusted as needed.
39
In the measurement, the deformation carriers should match the selected optical method, specimen size and the deformation range. In addition, the machining accuracy of 3D printing process should not be neglected since it may limit the critical size of the deformation carriers. From our experience, two issues should be considered in the designing of integrated deformation carriers: firstly, the type of deformation carriers should be determined according to the selected optical method; second, the parameters of the deformation carriers (e.g. critical size and spatial area) should be optimized based on the sensitivity of the selected optical method. There are several suggestions that should be noted: 1. The original surface of the general 3D printing specimen may be rough, and it could serve as natural speckles, but some local region may lose the correlation in the DIC calculation. Therefore, the speckles are better to be directly fabricated on the measured surface. Besides, in some cases the lattices structure could serve as the speckles [23] which will be demonstrated in Section 4. 2. The critical size of the deformation carriers should be larger than the machining accuracy, otherwise the fabricated deformation carriers may be merged and cannot be recognized. 3. The height or depth of the deformation carriers should be less than 1/10 of the thickness of the specimen. It has been proved that when the carriers depth does not exceed 1/10 of the specimen thickness, the influence of the carriers on the mechanical properties of the specimen is very small [24]. 3. The principle of VFM VFM, proposed by Grédiac [9], is a constitutive parameters identification method based on the principle of virtual work and full-field heterogeneous deformation data. Compared with other available identification methods based on full-field deformation data, such as finite element model updating method [25], constitutive equation gap method [26], equilibrium gap method [27] and reciprocity gap method [28], VFM is non-iterative, noise tolerable and easy processing [29]. In addition, VFM has low requirement for the boundary condition of load since the virtual field can be changed flexibly to eliminate the disturbance of some unknown loads. In this paper, the main principle of VFM is introduced briefly, and the detailed description can be found in Refs. [20,30]. Suppose a solid with any shape is subjected to a prescribed loading and displacement, as shown in Fig. 1. Our target is to identify the constitutive parameters of this material from the deformation fields and the corresponding load that is measured by using load sensor. If ignoring the body forces, only the load condition over Sf and displacement condition over Su need to be processed. Generally, the load over Sf could be expressed as F (P, n), and over Su, where the displacement could be prescribed u ¼ u P and n indicate any point of Sf and the vector perpendicular to Sf at point P, respectively. Then, the principle of virtual work can be expressed as [30]:
40
X. Dai, H. Xie / Measurement 59 (2015) 38–43
4. Experiment
Fig. 1. Solid of any shape with the load condition over Sf and displacement condition over Su.
Z
r : e dV ¼
V
Z
FðP; nÞ u ðPÞdS
ð1Þ
Sf
where r is the stress field, e is the strain field, u is the displacement field, and V is the volume of the solid. The superscript ⁄ means virtual field, and the virtual strain fields are derived out form the virtual displacement fields. This equation is valid for any kinematically admissible (KA) virtual field which is u ¼ 0 over Su. Generally, the linear elastic constitutive equation can be written as:
rij ¼ Q ijmn emn
ð2Þ
In order to demonstrate the feasibility and to show the advantages of the presented method, the constitutive parameters of a stereolithography specimen were identified as an example. The stereolithography is a mature 3D printing technology which can be dated back to the mid1980s, as far as we know, the 1.2 lm machining accuracy of stereolithography technique has been achieved [6]. Stereolithography part is widely used as the master to produce mold for injection molding due to its relatively fine surface roughness; besides, the stereolithography product shows the trend of being used as the end-use part, and may work at high temperature in some cases, so it is necessary to study the mechanical behavior at different temperatures. As shown in Fig. 2, the stereolithography generally uses a laser beam to selectively cure the liquid resin layer-by-layer based on the CAD data. When a layer has been traced, the platform descends a specific distance which equals to the thickness of a single layer, and then the laser beam traces the next layer until the whole part is reproduced. In this study, a three-bending test specimen with integrated deformation carriers is fabricated by the stereolithography technique; then the DIC method is employed to get the displacement fields in both x and y directions at different temperatures (see Fig. 3); finally the displacement data are imported to the CamFit1.5 to identify the constitutive parameters.
Combine Eq. (1) with Eq. (2), it can be derived out:
Z V
Q ijmn emn eij dV ¼
Z
4.1. Experimental preparation
FðP; nÞ u ðPÞdS
ð3Þ
Sf
Assuming that the constitutive material is homogeneous, the Q ijmn ’s do not depend on x, y and z. Therefore, the Eq. (3) can be written as:
Q ijmn
Z V
emn eij dV ¼
Z
FðP; nÞ u ðPÞdS
ð4Þ
Sf
According to Eq. (4), the problem turns to determining the Q ijmn ’s by any different virtual fields according to KA. Taking an orthotropic material as an example, the constitutive equation can be written as follows in the case of plane stress (x–y plane, Q xy ¼ Q yx due to the symmetry) [30]:
8 9 > < rxx > = > :
2
Q xx
9 38 > < exx > = 7 0 5 eyy > :c > ; Q ss xy
Q xy
Considering machining accuracy of the selected stereolithography system (SLA-250), a three-point bending test specimen with the deformation carriers was designed with the dimension size as shown in Fig. 3, and the specimen was solidified (resin, DSM’s SomosÒ GP Plus 14122) by alternately tracing in x and y directions at the adjacent layers perpendicularly. The lattices are formed by the array of square grooves with the edge length 0.25 mm and depth 0.20 mm, see Fig. 3(b) and (c). Although the designed lattices are regular, the natural roughness (surface roughness Ra25) makes the lattices appear to be short-range disorder,
0
ryy ¼ 6 4 Q yx Q yy > 0 0 rss ;
ð5Þ
Combining Eq. (4) with Eq. (5), we can get [30]:
Q xx
Z S
exx exx dS þ Q yy
þ Q ss
Z S
Z
cxy cxy dS ¼
S
eyy eyy dS þ Q xy
Z
Z S
ðeyy exx þ exx eyy ÞdS
FðP; nÞ u ðPÞdA
ð6Þ
Sf
Therefore, if four different (uncorrelated) virtual fields are offered, the Q ijmn ’s can be identified. Fortunately, a free VFM software called CamFit1.5 [30] is developed by Grédiac’s group, which is competent for the identification under the plane stress measurement.
Fig. 2. Schematic view of the stereolithography process.
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X. Dai, H. Xie / Measurement 59 (2015) 38–43
on the measured surface by an ion sputtering to increase the reflection efficiency. Besides, the mean intensity gradient method [31] is used to check the image quality of the designed lattices, and the values are around 10, which indicates the lattices are qualified as the speckles for DIC method. In order to identify the constitutive parameters at different temperatures, the test was performed with a selfdeveloped high temperature DIC test system, which includes an oven with a quartz window, a testing machine for loading and a DIC equipment for deformation measurement, as illustrated in Fig. 4. In the measurement, the specimen was heated up to 65 °C, then let the specimen slowly cool down to 60 °C, 50 °C, 40 °C, 30 °C and 21 °C (ambient temperature), the specimen was loaded at each temperature, and the images were recorded under different load values, respectively. Fig. 3. The dimension size of the three-point bending test specimen: (a) schematic diagram of the specimen; (b) morphology of the square grooves; (c) height map of the square grooves.
Fig. 4. The self-developed high temperature DIC test system.
so these lattices can serve as both grating and speckles. In this study, we regarded these lattices as speckles and used DIC method to obtain the full-field displacement. Gold film with thickness of several tens nanometers level was coated
4.2. Experimental results and analysis In the data processing, the region (marked with the dotted frame) on the left half part of specimen was calculated by CamFit1.5, and this region (1235 pixels 780 pixels, 1 pixel is equivalent to 0.025 mm) is illustrated by a white dash rectangle in Fig. 5. Since local stress concentration happens near the region of the loading point, it was kept out from the analysis region to eliminate its disturbance on the identified parameters [32]. As for each temperature, the fields of 6 load steps were used to identify the orthotropic elastic parameters through the vertical shear model, the mesh size was set as 40, and the orthotropy angle was 90° in CamFit1.5. The identified results, four elastic constants (see Eq. (5)), are given in Table 1. Obviously, the elastic constants reduces as the temperature rising, and the stereolithography specimen shows distinct anisotropy. When using DIC to calculate a displacement, the data near the edge of calculated area is difficult to be obtained, because only the center pixel can be located when matching the subset (see Fig. 5). Therefore, it is necessary to analyze its influence on the identified parameters caused by the missing data. In this study, we used the FEA method to simulate this three-point bending test, and then imported the obtained displacement fields (at the surface
Fig. 5. Displacement fields on the left half part of specimen (21 °C, 77N): (a) u field; (b)
v field.
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X. Dai, H. Xie / Measurement 59 (2015) 38–43 Table 1 Identified results of orthotropic elastic parameters under different temperature. Temperature (°C)
Q xx (MPa)
Q yy (MPa)
Q xy (MPa)
Q ss (MPa)
21 30 40 50 60
1243.19 572.37 245.26 55.63 25.28
1356.71 675.41 310.87 77.01 37.46
589.46 245.85 91.85 40.16 16.22
527.96 203.81 88.75 24.98 13.30
preset value and identified value, respectively. Obviously, it can be found that the relative error grows as D increasing; when D < 0.5 mm the relative errors of the identified results are less than 10%. Moreover, the Q xx is most sensitive to the change of D since the direct strains exx near the top and bottom boundaries are acute, and exx makes a major contribution to the identification of Q xx , so when the data near the top and bottom sides of the specimen are missing, the error of the obtained Q xx rises quickly. In this study, the subset of 29 pixels 29 pixels is set in the DIC process, so D = 0.35 mm (14 pixels near edge are missed, 1 pixel is equivalent to 0.025 mm) which is less than 0.5 mm, therefore the errors of the identified parameters are less than 10% (relative error evaluated by FEA: Q xx : 9.3%, Q yy : 4.1%, Q xy : 5.2%,Q ss : 0.1%). 5. Conclusion In this paper, a constitutive parameters identification method for 3D printing materials is presented, and during the operation of this method, integrated designing for manufacturing the specimen and the deformation carrier is utilized.
Fig. 6. The schematic diagram of displacement field imported to CamFit1.5 (white dash rectangle, u field as example).
Fig. 7. The relative errors of identified results.
nodes) to the CamFit1.5 to analyze the influence. The Abaqus software was employed, and the orthotropic elastic material model with the parameters of Q xx ¼ 1000MPa, Q yy ¼ 2000MPa, Q xy ¼ 500MPa and Q ss ¼ 300MPawas used. The FEA model shared the same boundary condition of load and dimension size with specimen in our experiment, the mesh size was set as 0.2 mm. As shown in Fig. 6, the displacement fields with missing data near the edge (D = 0.2, 0.4, 1, 1.4, 1.8 mm) were imported to the CamFit1.5 respectively to simulate the actual identification. It is worth noting that before the displacement fields being imported to the CamFit1.5, these fields were flipped in the left-right direction. The identified results are given in Fig. 7, the relative error means ðQ idt Q set Þ=Q set , where Q set and Q idt are the
(1) This method shows obvious advantages as follows: integrated designing and manufacturing, flexible adjustment of deformation carriers (type and parameter), no reinforced effect on the specimen; in addition, the additional advantage of this method is one experiment for multi-parameter identification simultaneous, and less time consumption for experiment operation. (2) The feasibility and advantages of this method was demonstrated by an experiment combining stereolithography specimen, bending test, DIC and a VFM software. From this experiment, we can conclude that the integrated deformation carriers can be utilized to measure the deformation up to temperature which the measured material can withstand. (3) The influence on the identified parameters caused by missing data near the edge of calculated area is analyzed, and the results show this influence is less than 10% in our experiment. Form this study, we found the presented method has the potential to be widely used in 3D printing materials characterization. Acknowledgements The authors are grateful to the financial support from the National Basic Research Program of China (‘‘973’’ Project) (Grant Nos. 2010CB631005, 2011CB606105), the National Natural Science Foundation of China (Grant Nos. 11232008, 91216301, 11227801, 11172151), Tsinghua University Initiative Scientific Research Program. References [1] S.O. Onuh, K.K.B. Hon, Optimising build parameters for improved surface finish in stereolithography, Int. J. Mach. Tools Manuf. 38 (1998) 329–342.
X. Dai, H. Xie / Measurement 59 (2015) 38–43 [2] R. Hague, S. Mansour, N. Saleh, R. Harris, Materials analysis of stereolithography resins for use in rapid manufacturing, J. Mater. Sci. 39 (2004) 2457–2464. [3] S. Upcraft, R. Fletcher, The rapid prototyping technologies, Assembly Automat. 23 (2003) 318–330. [4] J.P. Kruth, M.C. Leu, T. Nakagawa, Progress in additive manufacturing and rapid prototyping, CIRP Ann.-Manuf. Technol. 47 (1998) 525– 540. [5] J.P. Kruth, Material incress manufacturing by rapid prototyping techniques, CIRP Ann.-Manuf. Technol. 40 (1991) 603–614. [6] X. Zhang, X.N. Jiang, C. Sun, Micro-stereolithography of polymeric and ceramic microstructures, Sens. Actuators, A 77 (1999) 149– 156. [7] F.P. Melchels, J. Feijen, D.W. Grijpma, A review on stereolithography and its applications in biomedical engineering, Biomaterials 31 (2010) 6121–6130. [8] C.X.F. Lam, X.M. Mo, S.H. Teoh, D.W. Hutmacher, Scaffold development using 3D printing with a starch-based polymer, Mater. Sci. Eng., C 20 (2002) 49–56. [9] M. Grédiac, Principe des travaux virtuels et identification/principle of virtual work and identification, Comptes Rendus del’Académie des Sciences, II/309:1–5. Gauthier-Villars (1989). (in French with Abridged English Version). [10] M. Sutton, W. Wolters, W. Peters, W. Ranson, S. McNeill, Determination of displacements using an improved digital correlation method, Image Vis. Comput. 1 (1983) 133–139. [11] B. Pan, K. Qian, H. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20 (2009) 062001. [12] Z. Hu, H. Xie, J. Lu, H. Wang, J. Zhu, Error evaluation technique for three-dimensional digital image correlation, Appl. Opt. 50 (2011) 6239–6247. [13] Z. Hu, H. Xie, J. Lu, T. Hua, J. Zhu, Study of the performance of different subpixel image correlation methods in 3D digital image correlation, Appl. Opt. 49 (2010) 4044–4051. [14] Z. Hu, H. Luo, Y. Du, H. Lu, Fluorescent stereo microscopy for 3D surface profilometry and deformation mapping, Opt. Express 21 (2013) 11808–11818. [15] M. Hÿch, L. Potez, Geometric phase analysis of high-resolution electron microscopy images of antiphase domains: example Cu3Au, Philos. Mag. A 76 (1997) 1119–1138. [16] X. Dai, H. Xie, Q. Wang, Geometric phase analysis based on the windowed Fourier transform for the deformation field measurement, Opt. Laser Technol. 58 (2014) 119–127.
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[17] X. Dai, H. Xie, H. Wang, C. Li, Z. Liu, L. Wu, The geometric phase analysis method based on the local high resolution discrete Fourier transform for deformation measurement, Meas. Sci. Technol. 25 (2014) 025402. [18] C. Badulescu, M. Grédiac, J.D. Mathias, D. Roux, A procedure for accurate one-dimensional strain measurement using the grid method, Exp. Mech. 49 (2009) 841–854. [19] D. Post, H. Bongtae, I. Peter, High Sensitivity moiré: Experimental Analysis for Mechanics and Materials, Springer, 1997. [20] M. Grédiac, E. Toussaint, F. Pierron, Special virtual fields for the direct determination of material parameters with the virtual fields method. 1––Principle and definition, Int. J. Solids Struct. 39 (2002) 2691–2705. [21] T. Guélon, E. Toussaint, J.B. Le Cam, N. Promma, M. Grédiac, A new characterisation method for rubber, Polym. Test. 28 (2009) 715–723. [22] P.F. Jacobs, Fundamentals of stereolithography, in: Proceedings of the Solid Freeform Fabrication Symposium, 1992. [23] H. Jin, W. Lu, J. Korellis, Micro-scale deformation measurement using the digital image correlation technique and scanning electron microscope imaging, J. Strain Anal. Eng. Des. 43 (2008) 719–728. [24] M. Tang, H. Xie, J. Zhu, X. Li, Y. Li, Study of moiré grating fabrication on metal samples using nanoimprint lithography, Opt. Express 20 (2012) 2942–2955. [25] K.T. Kavanagh, R.W. Clough, Finite element applications in the characterization of elastic solids, Int. J. Solids Struct. 7 (1971) 11–23. [26] G. Geymonat, F. Hild, S. Pagano, Identification of elastic parameters by displacement field measurement, C.R. Mec. 330 (2002) 403–408. [27] D. Claire, F. Hild, S. Roux, Identification of a damage law by using full-field displacement measurements, Int. J. Damage Mech. 16 (2007) 179–197. [28] M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM J. Appl. Math. 50 (1990) 1635–1644. [29] S. Avril, M. Grédiac, F. Pierron, Sensitivity of the virtual fields method to noisy data, Comput. Mech. 34 (2004) 439–452. [30] F. Pierron, M. Grédiac, The Virtual Fields Method: Extracting Constitutive Mechanical Parameters From Full-Field Deformation Measurements, Springer, 2012. [31] B. Pan, Z. Lu, H. Xie, Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation, Opt. Lasers Eng. 48 (2010) 469–477. [32] B. Guo, H. Wang, H. Xie, P. Chen, Elastic constants characterization on graphite at 500 °C by the virtual fields method, Theor. Appl. Mech. Lett. 4 (2014) 021010.
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Constitutive parameter identification of 3D printing material based on the virtual fields method Xianglu Dai, Huimin Xie ⇑ AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 20 June 2014 Received in revised form 2 September 2014 Accepted 16 September 2014 Available online 28 September 2014 Keywords: 3D printing material Integrated deformation carriers Constitutive parameter identification Virtual fields method
a b s t r a c t In recent years, 3D printing technology has grown rapidly, and also has shown the great potential to be utilized in different fields. The identification of the constitutive parameters of materials fabricated by 3D printing is very important for product designing and technique selection. In this paper, a constitutive parameter identification method for 3D printing materials combining the integrated deformation carriers with the virtual fields method (VFM) is presented. The experimental process consists of three steps: fabricating the specimen with integrated deformation carriers by 3D printing; measuring the deformation fields by a full-field optical method; identifying the constitutive parameters by VFM. In the first step, the design method of the integrated deformation carriers is described in detail. Serving as a practice of the above process, a bending specimen with integrated deformation carriers was manufactured by the stereolithography technique, and the orthotropic constitutive parameters of this specimen at different temperatures were identified. The successful experimental results verify the feasibility of the proposed method, and show its advantages on aspects of high efficiency and easy processing as well. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the field of manufacture, there are three generic methodologies which include: subtractive methodology, formative methodology and additive methodology, for the production of prototypes or manufactured parts [1]. The first two methodologies are widely used, but show some drawbacks, e.g. difficulty in fabricating a part with a composite shape in the subtractive methodology; high cost when the output is small due to the expensive mold used in the formative methodology. The additive methodology is a relatively new approach to the manufacture of prototypes and end-use parts, and it used to be referred to as rapid prototyping (RP) [2,3], which is generally being called 3D printing now.
⇑ Corresponding author. E-mail address: [email protected] (H. Xie). http://dx.doi.org/10.1016/j.measurement.2014.09.033 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
The 3D printing includes a group of techniques that enable it to quickly fabricate a scale model of a physical product or an end-use part according to the threedimensional computer aided design (CAD) data, and there is no high requirement of complex implement and skilled model maker. In a 3D printing process, CAD data are reformatted into a special file in which the model is sliced to be many layers; then this file is transferred into a selected 3D printing system; afterwards this system will reproduce the model layer-by-layer; and finally a physical model of the original CAD data can be exported. Up to now, different methods have been developed to achieve these CAD data, such as magnetic resonance imaging (MRI), computed tomography (CT) scanning as well as point cloud data generated by engineering scanning or digitizing systems [3]. Since its emergence in 1980s, 3D printing has been witnessed rapid development: various forming techniques have been proposed [3,4], more and more materials and structures can be manufactured [5,6], and the application
X. Dai, H. Xie / Measurement 59 (2015) 38–43
fields are growing fast [7,8]. Recently, increasingly 3D printing parts have been used as end-use parts, and hence it is vital for the designers to characterize the mechanical properties of the materials produced by 3D printing system in order to optimize the material specification in their design process. However, as limited relevant literatures are available [2] and there are growing requirement to understand the mechanical behavior of 3D printing materials, the relative research is in urgent need. In this paper, our work focuses on the study of the constitutive parameter identification of 3D printing materials, and a constitutive parameters identification method combining the integrated deformation carriers with the VFM [9] is presented. The operation of this method includes three main steps: firstly, fabricate a specimen with integrated deformation carriers by 3D printing, and the integrated deformation carriers can be the lattices or speckles according to the selected optical method; then add the load on the specimen step-by-step and obtain the full-field displacement by an optical method, such as digital image correlation (DIC) [10–14], geometric phase analysis (GPA) [15–17], grid method [18] or moiré method [19]; finally, the VFM is employed to identify the desired constitutive parameters by using the data of the full-field displacement. The integrated deformation carriers could be fabricated on an interested area of specimen as needed, and then is used in the deformation measurement, so no more extra sensor is required (e.g. strain gauge or extensometer). The VFM [9,20,21] is a high efficient method which can identify many constitutive parameters in one test simultaneously, and thus it is time-saving and low cost; besides, the VFM is effective for both isotropic and anisotropic materials. In order to demonstrate the superiority of the presented method, the orthotropic constitutive parameters at different temperatures of a stereolithography [22] specimen with integrated lattices were identified. Besides, the influence on the identified parameters caused by missing data near the edge of calculated area of DIC method was analyzed by the finite element analysis (FEA). 2. The design of 3D printing specimen with integrated deformation carriers In general, deformation carriers are essential components for optical methods, for example, speckles for DIC, lattices or grid for GPA and grid method, grating (largearea lattices or grid) for moiré method. The deformation carriers are individually prepared after specimen is manufactured, and some conditions should be satisfied for obtaining the matched deformation carriers. For example: the material of deformation carriers should be compatible with the material of specimen (no reaction, easy bonding, small reinforced effect, low thermal mismatch). But it may be very difficult to meet these requirements. In order to tackle the difficulty, the deformation carrier and specimen can be generated with an integrated designing and fabrication. Benefiting from the flexibility of the 3D printing, the specimen and deformation carriers can be manufactured simultaneously; besides, the type, position, size and area of these carriers can be adjusted as needed.
39
In the measurement, the deformation carriers should match the selected optical method, specimen size and the deformation range. In addition, the machining accuracy of 3D printing process should not be neglected since it may limit the critical size of the deformation carriers. From our experience, two issues should be considered in the designing of integrated deformation carriers: firstly, the type of deformation carriers should be determined according to the selected optical method; second, the parameters of the deformation carriers (e.g. critical size and spatial area) should be optimized based on the sensitivity of the selected optical method. There are several suggestions that should be noted: 1. The original surface of the general 3D printing specimen may be rough, and it could serve as natural speckles, but some local region may lose the correlation in the DIC calculation. Therefore, the speckles are better to be directly fabricated on the measured surface. Besides, in some cases the lattices structure could serve as the speckles [23] which will be demonstrated in Section 4. 2. The critical size of the deformation carriers should be larger than the machining accuracy, otherwise the fabricated deformation carriers may be merged and cannot be recognized. 3. The height or depth of the deformation carriers should be less than 1/10 of the thickness of the specimen. It has been proved that when the carriers depth does not exceed 1/10 of the specimen thickness, the influence of the carriers on the mechanical properties of the specimen is very small [24]. 3. The principle of VFM VFM, proposed by Grédiac [9], is a constitutive parameters identification method based on the principle of virtual work and full-field heterogeneous deformation data. Compared with other available identification methods based on full-field deformation data, such as finite element model updating method [25], constitutive equation gap method [26], equilibrium gap method [27] and reciprocity gap method [28], VFM is non-iterative, noise tolerable and easy processing [29]. In addition, VFM has low requirement for the boundary condition of load since the virtual field can be changed flexibly to eliminate the disturbance of some unknown loads. In this paper, the main principle of VFM is introduced briefly, and the detailed description can be found in Refs. [20,30]. Suppose a solid with any shape is subjected to a prescribed loading and displacement, as shown in Fig. 1. Our target is to identify the constitutive parameters of this material from the deformation fields and the corresponding load that is measured by using load sensor. If ignoring the body forces, only the load condition over Sf and displacement condition over Su need to be processed. Generally, the load over Sf could be expressed as F (P, n), and over Su, where the displacement could be prescribed u ¼ u P and n indicate any point of Sf and the vector perpendicular to Sf at point P, respectively. Then, the principle of virtual work can be expressed as [30]:
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X. Dai, H. Xie / Measurement 59 (2015) 38–43
4. Experiment
Fig. 1. Solid of any shape with the load condition over Sf and displacement condition over Su.
Z
r : e dV ¼
V
Z
FðP; nÞ u ðPÞdS
ð1Þ
Sf
where r is the stress field, e is the strain field, u is the displacement field, and V is the volume of the solid. The superscript ⁄ means virtual field, and the virtual strain fields are derived out form the virtual displacement fields. This equation is valid for any kinematically admissible (KA) virtual field which is u ¼ 0 over Su. Generally, the linear elastic constitutive equation can be written as:
rij ¼ Q ijmn emn
ð2Þ
In order to demonstrate the feasibility and to show the advantages of the presented method, the constitutive parameters of a stereolithography specimen were identified as an example. The stereolithography is a mature 3D printing technology which can be dated back to the mid1980s, as far as we know, the 1.2 lm machining accuracy of stereolithography technique has been achieved [6]. Stereolithography part is widely used as the master to produce mold for injection molding due to its relatively fine surface roughness; besides, the stereolithography product shows the trend of being used as the end-use part, and may work at high temperature in some cases, so it is necessary to study the mechanical behavior at different temperatures. As shown in Fig. 2, the stereolithography generally uses a laser beam to selectively cure the liquid resin layer-by-layer based on the CAD data. When a layer has been traced, the platform descends a specific distance which equals to the thickness of a single layer, and then the laser beam traces the next layer until the whole part is reproduced. In this study, a three-bending test specimen with integrated deformation carriers is fabricated by the stereolithography technique; then the DIC method is employed to get the displacement fields in both x and y directions at different temperatures (see Fig. 3); finally the displacement data are imported to the CamFit1.5 to identify the constitutive parameters.
Combine Eq. (1) with Eq. (2), it can be derived out:
Z V
Q ijmn emn eij dV ¼
Z
4.1. Experimental preparation
FðP; nÞ u ðPÞdS
ð3Þ
Sf
Assuming that the constitutive material is homogeneous, the Q ijmn ’s do not depend on x, y and z. Therefore, the Eq. (3) can be written as:
Q ijmn
Z V
emn eij dV ¼
Z
FðP; nÞ u ðPÞdS
ð4Þ
Sf
According to Eq. (4), the problem turns to determining the Q ijmn ’s by any different virtual fields according to KA. Taking an orthotropic material as an example, the constitutive equation can be written as follows in the case of plane stress (x–y plane, Q xy ¼ Q yx due to the symmetry) [30]:
8 9 > < rxx > = > :
2
Q xx
9 38 > < exx > = 7 0 5 eyy > :c > ; Q ss xy
Q xy
Considering machining accuracy of the selected stereolithography system (SLA-250), a three-point bending test specimen with the deformation carriers was designed with the dimension size as shown in Fig. 3, and the specimen was solidified (resin, DSM’s SomosÒ GP Plus 14122) by alternately tracing in x and y directions at the adjacent layers perpendicularly. The lattices are formed by the array of square grooves with the edge length 0.25 mm and depth 0.20 mm, see Fig. 3(b) and (c). Although the designed lattices are regular, the natural roughness (surface roughness Ra25) makes the lattices appear to be short-range disorder,
0
ryy ¼ 6 4 Q yx Q yy > 0 0 rss ;
ð5Þ
Combining Eq. (4) with Eq. (5), we can get [30]:
Q xx
Z S
exx exx dS þ Q yy
þ Q ss
Z S
Z
cxy cxy dS ¼
S
eyy eyy dS þ Q xy
Z
Z S
ðeyy exx þ exx eyy ÞdS
FðP; nÞ u ðPÞdA
ð6Þ
Sf
Therefore, if four different (uncorrelated) virtual fields are offered, the Q ijmn ’s can be identified. Fortunately, a free VFM software called CamFit1.5 [30] is developed by Grédiac’s group, which is competent for the identification under the plane stress measurement.
Fig. 2. Schematic view of the stereolithography process.
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X. Dai, H. Xie / Measurement 59 (2015) 38–43
on the measured surface by an ion sputtering to increase the reflection efficiency. Besides, the mean intensity gradient method [31] is used to check the image quality of the designed lattices, and the values are around 10, which indicates the lattices are qualified as the speckles for DIC method. In order to identify the constitutive parameters at different temperatures, the test was performed with a selfdeveloped high temperature DIC test system, which includes an oven with a quartz window, a testing machine for loading and a DIC equipment for deformation measurement, as illustrated in Fig. 4. In the measurement, the specimen was heated up to 65 °C, then let the specimen slowly cool down to 60 °C, 50 °C, 40 °C, 30 °C and 21 °C (ambient temperature), the specimen was loaded at each temperature, and the images were recorded under different load values, respectively. Fig. 3. The dimension size of the three-point bending test specimen: (a) schematic diagram of the specimen; (b) morphology of the square grooves; (c) height map of the square grooves.
Fig. 4. The self-developed high temperature DIC test system.
so these lattices can serve as both grating and speckles. In this study, we regarded these lattices as speckles and used DIC method to obtain the full-field displacement. Gold film with thickness of several tens nanometers level was coated
4.2. Experimental results and analysis In the data processing, the region (marked with the dotted frame) on the left half part of specimen was calculated by CamFit1.5, and this region (1235 pixels 780 pixels, 1 pixel is equivalent to 0.025 mm) is illustrated by a white dash rectangle in Fig. 5. Since local stress concentration happens near the region of the loading point, it was kept out from the analysis region to eliminate its disturbance on the identified parameters [32]. As for each temperature, the fields of 6 load steps were used to identify the orthotropic elastic parameters through the vertical shear model, the mesh size was set as 40, and the orthotropy angle was 90° in CamFit1.5. The identified results, four elastic constants (see Eq. (5)), are given in Table 1. Obviously, the elastic constants reduces as the temperature rising, and the stereolithography specimen shows distinct anisotropy. When using DIC to calculate a displacement, the data near the edge of calculated area is difficult to be obtained, because only the center pixel can be located when matching the subset (see Fig. 5). Therefore, it is necessary to analyze its influence on the identified parameters caused by the missing data. In this study, we used the FEA method to simulate this three-point bending test, and then imported the obtained displacement fields (at the surface
Fig. 5. Displacement fields on the left half part of specimen (21 °C, 77N): (a) u field; (b)
v field.
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X. Dai, H. Xie / Measurement 59 (2015) 38–43 Table 1 Identified results of orthotropic elastic parameters under different temperature. Temperature (°C)
Q xx (MPa)
Q yy (MPa)
Q xy (MPa)
Q ss (MPa)
21 30 40 50 60
1243.19 572.37 245.26 55.63 25.28
1356.71 675.41 310.87 77.01 37.46
589.46 245.85 91.85 40.16 16.22
527.96 203.81 88.75 24.98 13.30
preset value and identified value, respectively. Obviously, it can be found that the relative error grows as D increasing; when D < 0.5 mm the relative errors of the identified results are less than 10%. Moreover, the Q xx is most sensitive to the change of D since the direct strains exx near the top and bottom boundaries are acute, and exx makes a major contribution to the identification of Q xx , so when the data near the top and bottom sides of the specimen are missing, the error of the obtained Q xx rises quickly. In this study, the subset of 29 pixels 29 pixels is set in the DIC process, so D = 0.35 mm (14 pixels near edge are missed, 1 pixel is equivalent to 0.025 mm) which is less than 0.5 mm, therefore the errors of the identified parameters are less than 10% (relative error evaluated by FEA: Q xx : 9.3%, Q yy : 4.1%, Q xy : 5.2%,Q ss : 0.1%). 5. Conclusion In this paper, a constitutive parameters identification method for 3D printing materials is presented, and during the operation of this method, integrated designing for manufacturing the specimen and the deformation carrier is utilized.
Fig. 6. The schematic diagram of displacement field imported to CamFit1.5 (white dash rectangle, u field as example).
Fig. 7. The relative errors of identified results.
nodes) to the CamFit1.5 to analyze the influence. The Abaqus software was employed, and the orthotropic elastic material model with the parameters of Q xx ¼ 1000MPa, Q yy ¼ 2000MPa, Q xy ¼ 500MPa and Q ss ¼ 300MPawas used. The FEA model shared the same boundary condition of load and dimension size with specimen in our experiment, the mesh size was set as 0.2 mm. As shown in Fig. 6, the displacement fields with missing data near the edge (D = 0.2, 0.4, 1, 1.4, 1.8 mm) were imported to the CamFit1.5 respectively to simulate the actual identification. It is worth noting that before the displacement fields being imported to the CamFit1.5, these fields were flipped in the left-right direction. The identified results are given in Fig. 7, the relative error means ðQ idt Q set Þ=Q set , where Q set and Q idt are the
(1) This method shows obvious advantages as follows: integrated designing and manufacturing, flexible adjustment of deformation carriers (type and parameter), no reinforced effect on the specimen; in addition, the additional advantage of this method is one experiment for multi-parameter identification simultaneous, and less time consumption for experiment operation. (2) The feasibility and advantages of this method was demonstrated by an experiment combining stereolithography specimen, bending test, DIC and a VFM software. From this experiment, we can conclude that the integrated deformation carriers can be utilized to measure the deformation up to temperature which the measured material can withstand. (3) The influence on the identified parameters caused by missing data near the edge of calculated area is analyzed, and the results show this influence is less than 10% in our experiment. Form this study, we found the presented method has the potential to be widely used in 3D printing materials characterization. Acknowledgements The authors are grateful to the financial support from the National Basic Research Program of China (‘‘973’’ Project) (Grant Nos. 2010CB631005, 2011CB606105), the National Natural Science Foundation of China (Grant Nos. 11232008, 91216301, 11227801, 11172151), Tsinghua University Initiative Scientific Research Program. References [1] S.O. Onuh, K.K.B. Hon, Optimising build parameters for improved surface finish in stereolithography, Int. J. Mach. Tools Manuf. 38 (1998) 329–342.
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