Theoretical Feasibility Research on 3D Printing Technology Applied to Statics Model Experiment

Scale test also known as model experiment is an effective tool
to simulate and investigate phenomena, which can’t be established...


Introduction
Scale test also known as model experiment is an effective tool to simulate and investigate phenomena, which can't be established directly [1][2]. Based on dimensional analysis and similarity theory, the similarity criteria can be formulated mathematically to obtain equivalent relationships between prototype and model [1][2][3].
However, traditional modeling has troubles in making complex structure as a whole in only one step. And then the indispensable post-treatments will change the state of stress, resulting in experimental errors, even at the cost of rocket high expenditure and large human labor amount. This step undermines the benefit from cheap substitution materials and reduces the efficiency of the whole optimization process.
Contrary to traditional modeling, 3D printing can detours all the above drawbacks through directly making complex geometries in one step, such as models, assembly fixtures and production molds.
However, the accuracy and the reliability of statics model experiments strictly depend on theoretically mathematical equations [1][2]. In detail, in form of the π theorem, it enables an object or a system to be represented by a dimensionless similarity function of dimensionless parameters (π values) [1][2][3][4]. That is, objects or systems are considered completely similar for the same dimensionless parameters (π values) [1][2][3][4]. Whereas, traditional statics model experiments with strict dimensionless function are hardly to be established through 3D printing items. In other words, 3D printing statics model is restricted by printing material and scale or size of the model. Even though 3D printing materials satisfy the requirements for balancing system, too large or too small model is difficult to be built and directly used in static experiments for poor practicability and operability [1][2][6][7], such as limited 3D printing device and test equipment. Thus, it is necessary to establish a method to make 3D printed models customized for actual testing purpose equivalent to the prototype in detailed structure and static indexes as stress, strain and displacement [6][7][8]. In fact, some studies have been carried out to solve above problems.
Murphy proposed a strategy to relax the similarity constraints by intentionally casting the prediction equation into the more complex form [39]. Kristin L.Wood et al. provided an improved similarity method that utilizes a geometrically simple specimen pair, in order to design the prototype with more freedom [40].
To make it feasible, this paper reduces the internal constraints of the statics structural system. That is, the dimensionless similarity function is theoretically simplified based on reducing constrains of weight and Poisson's ratio, so as to loosen the limits to 3D printing model materials. The method is interpreted through popular cases in detail,when different material and Poisson's ratio encountered between 3D printed model and prototype. The validated pathway will methodologically benefit and enhance the accuracy of statics researches through 3D printing model experiments.

Theoretical simplification
Compensation method without gravity  Figure 1. The x axis forwards in the same direction as the acceleration of gravity.
The similarity ratio Cx is defined as the ratio of the correspondent physical quantities of the prototype to the corresponding model, with the subscript x represents the homologous physical quantity.
ratios, as followsEq. (7) can be replaced by similarity: Then the first similarity criterion equation can be obtained by Eq. (6) and Eq. (9), as follows: Similarly, the other similarity criterion equations can be obtained based on Eq.
(2) to Eq. (5), as follows: Eq. (14)  However, both the above conditions are very hard to be satisfied.
Obviously, the strict dimensionless equation becomes an obstacle.
That is, Eq. (16) needs to be simplified to release the limits.
, , , , 0 f π π π π π = (16) First, the structure weight is regarded as an external force or , , 0 f π π π = (18) Polylactic acid (PLA), widely used 3D printing material, is selected for the model to investigate a prototype in Q345R (16Mn), one of the most common engineering materials. The parameters of the two materials are summarized in Table 2 [41]. The calculation  Table 3: Calculation procedure of similarity ratios.
Taking gravity as force, similarity ratio of gravity in above case is equal to similarity ratio of concentrated force, or. Actually, the ratio of prototype weight to model weight is. This means the model weight is reduced and an additional weight calls for compensation, with G the weight of prototype, evenly loaded of course.  '  '  2  3  5  2  3  5  2  3  5  2  3  5   1  1  1  1  1  1  , ,  ,  ,  , ,

Method for inequivalent Poisson's ratio
where correction coefficient is used to revise dimensionless parameters of 3D printing model. According to Eq. (18)  ( ) ' 2,3, 4,5 Therefore, prediction can be provided in Figure 2, of the 3D printing model experiment to substitute for the prototype one. In Figure 2, the correction coefficient must be given between  According to Figure 2, two computer simulation experiments are required, and the 3D print model experiment needs a proportional conversion. Although the correction coefficient can be calculated, the above calculation process is still very troublesome. Therefore, the analysis of Fig. 3 simplifies the simulation process. 3D printing model C is established on the basic of simplified statics system. 3D printing model C can equivalently study prototype A by means of adjusting correction coefficient. In addition to same Poisson's ratio, prototype is established depending on same similarity ratio as above process. Thus, 3D printing model and prototype possess same dimensionless parameters or dimensionless similarity Eq. (19). At this stage, Except for Poisson's ratio, Prototype A and prototype is identical. Then the correction coefficient between 3D printing model C and prototype A can be found by simulating prototype A and prototype. In other words, the correction coefficient is available by regression or trail of the prototype Poisson's ratio with the aid of computer. Furthermore, the impact can also be studied of different 3D printing materials on static prediction results through the establishment of curve. If so, the appropriate 3D printing material can be chosen for specific statics experiment.

Case analysis of correction coefficient
For simplified statics system, methods of compensation and correction ensure equivalence of modelexperiments, assisted by additional weight available through case study. Similarly, method for inequivalent Poisson's ratio is herein expound in detail by means of a specific case analysis.
The truss beam is selected in the large column as the object of computer simulation, shown in Figure 4 [42]. Obviously, any simple statics structure here can be used to illustrate the point of the paper.
Geometric parameters and load of prototype are summarized in Table 4. Properties of prototype material subjected to working conditions are summarized in Table 5. The point of measurement is recommended to select the one where stress and strain are easy to converge. Therefore, the point is the best candidate near the axis of left-and-right symmetry on truss beam. Then the simulated results of the deformation, stress and strain of the selected points are summarized in Table 6.    ' ' Obviously, the value of l and q is equal to ' l and ' q , individually.
So, the Eq. (21) can be transformed into Eq. (22), Combining with Eq. (22) and Table6, the results of correction coefficient can be summarized in Table 7     After correction coefficient is obtained, prediction process in

Conclusions
This work simplifies statics system on the basis of theoretical analysis. To guarantee equivalence of 3D printing model experiment, weight compensation and correction are proposed of dimensionless parameters through treating weight as external applied load. Correction coefficient is calculated by computer simulation for correction of dimensionless parameters. Essentially, the correction coefficient is a modification to the stress, strain and displacement, resulted from different Poisson's ratio between 3D printing material and prototype material. After relevant case analyses, the method is validated and illustrated. This proves the theoretical feasibility of 3D printing to be applied to static model experiment under similarity theory with different module size and materials. Obviously, the research method is strictly applicable in prediction for the elastic behavior. It can be also used to qualitatively analyze the mechanical behavior of breaking stage through 3D printing scale test. Generally speaking, the methodology will benefit analogical statics investigations in the age of 3D printing.