Optimization of compressive property for the development of triply periodic minimal surface lattice structure on polylactic acid polymeric material

Abstract

The present study deals with the process optimization of printing parameters for fabricating gyroid TPMS (triply periodic minimal surface) lattice structure incorporated compression samples on the polylactic Acid polymeric material for obtaining the maximum compressive strength. The design of experiments is followed for the process parameter optimization. The experiment was carried out by varying three printing process parameters and four levels such as printing speed (10 mm/sec, 20 mm/sec, 30 mm/sec, and 40 mm/sec), layer height (0.10 mm, 0.15 mm, 0.20 mm, and 0.25 mm), and nozzle temperature (190°C, 200°C, 210°C, and 220°C). The L₁₆ orthogonal array is employed for the experimental procedure and the Taguchi optimization technique is utilized for the optimization of the printing process parameters for obtaining maximum compressive strength for the fabricated gyroid TPMS lattice structure incorporated compression samples. The experimental results confirm that printing speed and layer height have major influence of 57.28% and 30.92% on the compressive properties of the fabricated samples. Based on the regression analysis results, it can be concluded that the proposed mathematical model has observed an error percentage of 2.1% and a good fit has been observed with the experimental results. The macroscopic view of the fractured samples depicts that the sample fabricated at a nominal printing speed of 20 mm/sec and layer height of 0.10 mm has obtained the highest compressive strength and lower buckling during compression test. The optimized combination of printing process parameters for obtaining maximum compressive strength is 20 mm/sec, 0.10 mm, and 210°C.

Keywords

Fused deposition modeling triply periodic minimal surface lattice structure 3D printing gyroid optimization

Introduction

Additive manufacturing can be briefly defined as the successful methodology to develop the most complex structures by successive layer deposition. Additive manufacturing plays a vital role in the field of free-form designing, complex geometry development, and reduction in material usage.¹ Additive manufacturing is widely classified into a broad spectrum of seven categories according to the ASTM F2792.² The fused deposition modeling (FDM) is the most preferred AM technology in which the feedstock material is provided in the form of a spool containing thermoplastic filament. This filament is fed into the heating unit where it gets heated up. The molten filament gets deposited in a sequence of successive layers on the pre-heated silicon bed to develop a 3D model. The FDM technology has a similar methodology like fused filament fabrication (FFF) technology and it was patterned by Stratasys. The widely accepted technology for the additive manufacturing of polymers is FFF.^3,4 Figure 1 depicts the process of the FDM technique and the variable elements in the FDM process.

The FFF technology has a broad spectrum of applications in the field of aerospace, medicine, construction, civil, etc.⁵ The FFF technology is significant because of its versatility, cost-effectiveness, ease to handle, wide range of thermoplastic material acceptance, design freedom, generation of complex geometries, etc. The widespread thermoplastic 3D printing filaments for the FDM-based 3D printer are PLA (polylactic acid), ABS (acrylonitrile butadiene styrene), PETG (polyethylene terephthalate glycol), nylon, and composite filaments like Wood-PLA,⁶ ceramic-PLA,⁷ and PLA-metal.⁸ For the current research work, the PLA 3D printing filament is considered because the composite filaments lead to the development of slight irregularity in the surface of the gyroid triply periodic minimal surface (TPMS) structure.

As it is understood, the FFF technology deals with the layer-by-layer deposition of the material. So, there is a probability of development of weak interactions between the successive layers of material which in turn affects the quality of the fabricated model. So, it is necessary to optimize the printing parameters for obtaining better results using the FFF technology.

A significant number of research studies have been attempted in order to optimize the printing parameters (i.e., infill density, layer height, infill pattern, nozzle temperature, raster angle, etc.) for fabricating best-quality products. Anitha et al. studied the impact of various printing parameters like layer height, layer width, and printing speed on the surface roughness. The optimum surface roughness properties were observed at 0.3556 mm of layer height, 0.537 mm of road width, and 200 mm/sec of printing speed.⁹ Alafaghani et al. made an experimental study to determine the impact of the printing parameters such as infill density, nozzle temperature, build orientation, infill pattern, printing speed, and layer height. It was found that infill density, printing speed, and infill pattern has only a minor impact on the tensile properties of the fabricated parts.¹⁰ Rahman et al. determined six important factors which will have a greater influence on the fabricated parts. The widely suitable values for the six factors namely nozzle temperature, number of loops, printing speed, print bed temperature, infill density, layer height are 220°C, 1, 55 mm/sec, 110°C, 15%, 0.2 mm.¹¹ Vishal et al. has employed a study on the wear characteristics of the 3D printed silicon PLA composite.¹²

Following these researches, the various effects of process parameters like nozzle temperature, infill patterns, percentage of filler concentration in the matrix, bed temperature affect the production of the 3D-printed parts using various polymers like ABS, PLA/flax composite, PLA, and carbon fiber–reinforced composite have been examined earlier.^13–17 It is factual that the strength of the molded components is always slightly higher than the additively manufactured components.¹⁸

In recent years, the incorporation of lattice structures in the field of automobile and aerospace industries is widely developing. The incorporation of the lattice structure helps in attaining lesser weight, reduction in material usage, and increases the rate of energy absorption which in turn results in attaining a cost-effective, lesser fabrication time and it also helps in improving the mechanical properties of the fabricated product and also helps in solving the prevailing problems. A lattice structure is defined as a periodic pattern that is developed by the construction of an array of unit cells sequentially and it belongs to a category of cellular solids and the further classification of the lattice structure is shown in the Figure 2.¹⁹

The lattice structures have a broad spectrum of applications in various fields. The lattice structure has been incorporated in the fairings, space telescopes, payload adaptors, etc. The incorporation of the 3D lattice structures in polymeric materials has turned into an active area for research in the current era. The incorporation of various lattice structures and their implications are listed in Table 1.^20–26

Table 1.

Applications of incorporated lattice structures in various fields.

Sl. No	Author	Lattice type	Process	Industrial	Parts	Year	References
1	Maloney et al	Hollow lattice	Electroplating 3D print	Aerospace	Heat exchanger	2012	20
2	Stevenson et al	Diamond	Electron Beam Melting	Biomedical	Hip implant	2015	21
3	Heinl et al	Diamond and Hatched	Electron Beam Melting	Biomedical	Bone implant	2018	22
4	Moon et al	Kagome, pyramidal and hexagonal	Polyjet	Automotive	Wings	2014	23
5	Kumar et al	Hexagonal	Electron Beam Melting	Automotive	Heat exchanger	2009	24
6	Challapalli and ju	Octet-truss	Polyjet	Biomedical	Implant	2014	25
7	Chen et al	Inclined cubic at angle of 45°	Selective laser melting	Biomedical	Orthopedic implant	2016	26
8	Sabarinathan et al	Hexagonal structure	Fused deposition modeling	Structural		Present work

For this current research work, the TPMS lattice structures which belong to the 3D lattice structures category have been considered. The TPMS lattice structures are generally surfaces which have zero mean curvature and it also occupies a smaller area when compared to the conventional lattice structure. The reddish-orange region in Figure 3 depicts the curvatures in the fabricated gyroid TPMS lattice structure incorporated compression sample by surface analysis.

Wang et al. has fabricated glassy polymeric co-continuous composite having TPMS lattice structures.²⁷ The TPMS lattice structure helps in providing a higher surface-to-volume ratio when compared to the conventional strut-based lattice structures. The TPMS lattice structures consist of various structures like Schwarz Primitive, Diamond, Neovius, Schoen I-graph, wrapped package-graph (IWP), Fischer-Koch S, Schoen Gyroid, and Schwarz crossed layers of parallels (CLP).²⁸ The most commonly preferred TPMS lattice structure to ensure robust mechanical performance is the gyroid TPMS lattice structure. Maskery et al. evaluated the energy absorption of the gyroid structure made up of Al-Si10-Mg.²⁹ Jung et al. also have proposed that the gyroid TPMS lattice is capable of enhancing the mechanical properties.³⁰

The present research work deals with the optimization of process parameters for the fabrication of gyroid TPMS lattice structure incorporated with the PLA polymeric material. The polymeric samples for the current research work are fabricated by the FDM/FFF technique. This study has a vital impact on the process optimization of printing parameters for obtaining significant mechanical properties like compressive strength. The incorporation of the Taguchi L16 orthogonal array helps in the optimization of printing process parameters. The analysis of variance technique further assisted in the determination of the percentage contribution of all individual parameters. And it also helped in analyzing the variations between the predicted and experimental results.

Materials and methods

The FDM-based 3D printer is capable of only using thermoplastic 3D printing filaments because of its ideal physical and mechanical properties. The most widely available 3D printing thermoplastic filaments are PLA. Nowadays, PLA has become the widely used material in the FDM process. The main reason for adopting PLA is because of its biocompatibility. PLA is the most reliable biopolymer. The monomers of PLA are derived from renewable resources. For this current research work, the PLA 3D printing filament is considered. The PLA filament was purchased from Overture, the United States as shown in Figure 4(a). The appearance of the PLA 3D printing filament is white. The fundamental physical and mechanical properties of the 3D printing PLA filament are listed in Table 2. Figure 4(b) shows the FDM-based 3D printer during the fabrication of the gyroid TPMS lattice structure incorporated PLA compression samples.

Table 2.

Physical and mechanical properties of polylactic acid filaments.³¹

S. No	Property	Nominal range
1	Density (g/cm³)	1.24–1.26
2	Poisson’s ratio	0.33
3	Relative viscosity	4
4	Elongation at break (%)	0.5–9.2
5	Izod impact (J/m)	118
6	Tensile modulus (GPa)	2.02–4.15
7	Tensile strength (MPa)	37
8	Flexural modulus (GPa)	2.39–4.93
9	Flexural strength (MPa)	48–110
10	Melting point (°C)	150–175
11	Specific gravity (g/cc)	1.24
12	Glass transition temperature (°C)	55–60

Design modeling of the polymeric specimens

In order to fabricate the required polymeric samples, the most crucial step is the design modeling of the samples. The design modeling is achieved by using the suitable CAD (Computer-Aided Designing) software. In today’s world, a broad spectrum of CAD software’s available. For the current research work, the CATIA V5R20, CAD software is employed. The CAD model of the gyroid TPMS lattice structured compression sample is generated using CATIA V5R20, a CAD software. The dimensions of the gyroid TPMS lattice structured compression sample are 15 mm × 15 mm × 15 mm. Figure 5 depicts the 2D sketch and 3D CAD model of the gyroid TPMS structure incorporated PLA compression sample.

Sample fabrication

The gyroid TPMS lattice structured compression samples were designed according to the dimensions mentioned by Yubo Tao et al.³² The fabrication of the samples involves three crucial steps. The first step in the development of the 3D CAD model using CAD software like CATIA V5R20. After the generation of the 3D CAD model, the second step involves the conversion of the 3D CAD model into an STL file (Stereolithography). The third step involves the slicing of the STL files into G-codes (Geometric-Codes) by using suitable slicers. For this current research work, the ULTIMAKER CURA is used to slice the STL files. After the conversion of the STL files into G-codes, the G-codes are then saved in the storage device. The storage device is then inserted into the FDM printer for 3D printing. For this current research work, the FDM (Fused Deposition Modeling) technique is employed for the fabrication of the gyroid TPMS lattice structure incorporated PLA compression samples. Figure 6 depicts the entire process involved in the fabrication of the sample.

The Pratham 3.0, an FDM-based 3D printer was employed for fabricating the gyroid TPMS lattice structured compression samples. The compression samples were printed in the vertical direction. Some of the printing parameters were maintained constant during the fabrication of the samples. The constant printing parameters are tabulated in Table 3. The secondary pre-treatment processes are not employed to ensure net-zero deviations in the experimental results. For each testing parameter, five test samples are fabricated. As a result, a bunch of 80 samples has been fabricated for 16 trials. These 80 samples are required for optimizing the printing parameters.

Table 3.

Constant printing process parameter for the fabrication of gyroid triply periodic minimal surface incorporated polylactic acid compression samples using fused deposition modeling printer.

Sl. No	Printing parameters	Unit	Value
1	Bed temperature	°C	60
2	Wall thickness	mm	0.8 mm
3	Nozzle diameter	mm	0.4
4	Filament diameter	mm	1.75
5	Infill pattern	—	Line
6	Build orientation	—	X axis
7	Build adhesion	—	Skirt

Selection of process parameters for sample preparation

The main reason for adopting an orthogonal array technique is to reduce the maximum number of experiments without disturbing the impact of every individual parameter. The L₁₆ orthogonal array is developed by implementing various printing parameters such as printing speed, layer height, and nozzle temperature. Table 4 depicts the printing parameters and their levels concerning the fabrication of the gyroid TPMS lattice structures. The gyroid TPMS incorporated PLA compression sample is depicted in Figure 7.

Table 4.

Parameters and levels for the 3D printing of fused deposition modeling process.

Factors	Parameters	Units	Levels
A	Printing speed	mm/sec	10	20	30	40
B	Layer height	mm	0.10	0.15	0.20	0.25
C	Nozzle temperature	°C	190	200	210	220

Mechanical testing of the prepared sample

The compression test for the gyroid TPMS lattice structure modeled compression samples was tested in a Universal Testing Machine (Make: Bluestar, India). The machine is capable of handling a maximum loading capacity of about 50 kN. For conducting the compression test, the cube-shaped compression samples were fabricated. The experiment was carried out as per the ASTM D 695 standard with a loading speed of 2 mm/min. The compression samples were designed with dimensions of 15 mm × 15 mm × 15 mm. During the compression testing, the gyroid TPMS lattice structured compression samples fabricated at different process parameters are placed in between the two plates, and the corresponding compressive loads are applied for respective samples. For each test condition, five compression samples were experimentally tested and the average value is used for the optimization of the compressive strength property. Figure 8 depicts the compression test conducted for the gyroid TPMS lattice structure incorporated compression samples.

Taguchi experimental design and analysis

The Taguchi experimental technique is employed for optimizing the results by varying the process parameters. The experimental technique is employed for optimizing the compressive stress to reach maximum value.

The analysis of variance is performed in order to determine the significant printing process parameter for fabricating the gyroid TPMS lattice structured compression samples using the widely known polymeric material (i.e., PLA). Depending on the process parameters and experimental results, a suitable mathematical model is developed. The percentage contribution of every individual parameter was evaluated by utilizing the regression equation. The percentage error was determined by plotting the experimental results against predicted results. The maximum compressive strength, including the resilience parameter to obtain the maximum strength, is the required output without disturbing the quality of the fabricated specimens. The experimental results, such as the compressive strength are converted into an S/N ratio for the Taguchi experimental analysis.

The maximum compressive strength is considered as the output response parameter for optimizing the incorporation of the gyroid TPMS lattice structure in the PLA polymeric material using the FDM technique. For optimizing the compressive strength, the “larger the best” condition is adopted. The expression for the respective response is displayed in equation (1).

\frac{S}{N} = - 10 * l o g_{10} (\frac{1}{y^{2}})

(1)

where Y denotes the response such as compressive strength of the gyroid TPMS lattice structure incorporated in the FDM printed polymeric PLA sample.

The statistical analysis is developed using the Minitab 2020 statistical software tool. The analysis of variance technique is mostly preferred to determine the most influencing process parameter concerning various experimental conditions. In the existing 3D printing process, the confidence and significance levels were set at 95% and 5% in order to determine the impact of every individual parameter on the development of lattice structure in the fabricated components. The most influential factors were selected from the experimental results which include printing speed, layer height, and nozzle temperature. On the other hand, the regression equation and model for best fit were predicted with the help of independent factors such as printing speed (10 mm/sec, 20 mm/sec, 30 mm/sec, and 40 mm/sec), layer height (0.10 mm, 0.15 mm, 0.20 mm, and 0.25 mm), and nozzle temperature (190°C, 200°C, 210°C, and 220°C) and dependent variables such as compressive strength.

Results and discussion

The compression strengths of all the 16 different sets of samples which were fabricated with varying the printing parameters such as printing speed, layer height, and nozzle temperature are evaluated. Many researchers concluded that these printing process parameters will have a significant impact on the mechanical strength of the polymeric samples.^33–36 As a result, these printing process parameters were considered for studying its impact on the compressive strength of the gyroid TPMS lattice structure incorporated PLA polymeric samples. For each set of combination, five samples underwent a compressive test, and the respective compressive strengths of the fabricated samples are reported in Table 5. From the compression test experiment, the output response such as compressive strength was considered, and the impact of each printing process parameter and the optimal combinations of the printing process parameters were discussed in the below sections.

Table 5.

L₁₆ orthogonal array for the gyroid triply periodic minimal surface incorporated polylactic acid compression samples.

Sl. No	Printing speed (mm/sec)	Layer height (mm)	Nozzle temperature (°C)	Compressive stress (Mpa)	S/N ratio
1	10	0.10	190	27.44	28.7677
2	10	0.15	200	26.29	28.3958
3	10	0.20	210	25.03	27.9692
4	10	0.25	220	21.35	26.5880
5	20	0.10	200	35.41	30.9825
6	20	0.15	190	26.69	28.5270
7	20	0.20	220	32.21	30.1598
8	20	0.25	210	29.24	29.3195
9	30	0.10	210	31.01	29.8300
10	30	0.15	220	25.08	27.9866
11	30	0.20	190	19.54	25.8185
12	30	0.25	200	20.05	26.0423
13	40	0.10	220	24.87	27.9135
14	40	0.15	210	20.24	26.1242
15	40	0.20	200	16.24	24.2117
16	40	0.25	190	13.27	22.4574

General analysis of the signal-to-noise ratio for compressive strength

Table 5 depicts the L₁₆ orthogonal array and output responses for the gyroid TPMS lattice structure in the fabricated PLA sample. For each set of experimental runs, five samples are fabricated and tested for each corresponding set of parameters, and the average value is considered for the experimental and optimization process. For the S/N ratio, the maximum compressive strength is required for the fabricated gyroid TPMS lattice structure on the PLA polymeric material. So, the “larger the best” condition is adopted for the compressive strength analysis.

Figure 9 describes the signal-to-noise ratio plot for the compressive strength of the gyroid TPMS lattice structure of the PLA samples by varying the process parameters such as printing speed, layer height, and nozzle temperature.

Figure 9(a) depicts the main effect plot of tested PLA gyroid TPMS lattice-structured compression samples with varying printing parameters such as printing speed with various levels of 10 mm/sec, 20 mm/sec, 30 mm/sec, and 40 mm/sec. The results highlight that the maximum compressive strength of 29.75 MPa was observed at 20 mm/sec printing speed. Beyond 20 mm/sec, there is a significant decrease in the compressive strength. The higher printing speed of 40 mm/sec depicts the lowest compressive strength of 25.18 MPa. When the sample incorporated with the gyroid TPMS lattice structure gets fabricated at a higher printing speed, the air gaps will be developed in the gyroid TPMS lattice structure due to the improper supply of material. As a result of it, there will be slight deviations in the fabricated gyroid TPMS lattice structure. These imperfections in the gyroid TPMS lattice structure act as the main source for the development of the failure in the fabricated polymeric sample. It can be concluded that gyroid TPMS lattice structure incorporated compression samples fabricated at higher printing speeds (especially greater than 20 mm/sec) will be observed with lesser compressive strength when compared to the other samples fabricated at different printing speeds due to the development of the air gaps in the gyroid TPMS lattice structure. Therefore, it can be concluded that a nominal printing speed of 20 mm/sec develops the maximum compressive strength when compared to other conditions. A similar observation was made by Dhinakaran Veeman et al., and they concluded that the nominal printing speed of about 20 mm/sec develops the maximum compressive strength in the PLA/almond shell composites.³⁷

Figure 9(b) displays the main effect plot for the process parameter, such as layer height with different levels of 0.10 mm, 0.15 mm, 0.20 mm, and 0.25 mm. From the experimental results, the maximum compressive strength of about 29.37 MPa was recorded at 0.10 mm. While exceeding this corresponding value, a significant decrease is observed in the compressive properties of the fabricated sample. The experimental results depict that the layer height of about 0.10 mm concludes the highest compressive strength. When the sample is fabricated at a layer height above 0.10 mm, the compressive strength decreases significantly. It is factual that as the layer height increases, the layer count decreases. This means that the fabricated component has only a lesser number of layers when fabricated at a lower layer height. As the number of layers decreases, the strength of the fabricated sample also decreases. In addition, the incorporated gyroid TPMS lattice has only a thin surface. Therefore, when the incorporated gyroid TPMS lattice structure is fabricated at higher layer heights, the uniform surface thickness cannot be ensured. As a result, the strength of the compression sample incorporated with gyroid lattice structure reduces when the layer height exceeds 0.10 mm. A similar result was observed by Sai et al., that the minimum layer height of about 0.10 mm has proved to be the best condition for obtaining maximum compressive stress by the ANFIS model and whale optimization algorithm.³⁸

Figure 9(c) depicts the main effect plot for the process parameter, such as nozzle temperature with different levels of 190, 200, 210, and 220°C. From the experimental results, the maximum compressive strength of about 28.31 MPa was recorded at 210°C. While exceeding this temperature, a significant decrease is observed in the compressive properties of the fabricated sample. The experimental results depict that the nozzle temperature of about 210°C shows the highest compressive strength. Beyond this nozzle temperature limit, the compressive strength of the fabricated samples gets reduced. The main reason behind the decrease in the compressive strength is due to the extra ductility obtained by the PLA polymeric material above 210°C. Alafaghani and Qattawi have also referenced a similar observation on the samples fabricated using the PLA polymeric material by the means of the FDM technique.³⁹

From the experimental results, it can be concluded that the optimum condition for obtaining higher compressive strength for incorporating gyroid TPMS lattice structure in the polymeric PLA material will be a combination of a printing speed of 20 mm/sec, a minimum layer height of 0.10 mm, and a nozzle temperature of 210°C.

From the deviations in the signal-to-noise ratio, the delta value and rank were given to every individual process parameter. Based on the respective results, the sequence of influencing factors for the printing of the gyroid TPMS lattice structured PLA polymeric compression sample was studied. Table 6 displays the response table of the signal-to-noise ratio value for the compressive strength property with respect to the 3D printed gyroid TPMS lattice structured PLA polymeric compression samples. For the compressive strength results of the FDM printed gyroid, TPMS lattice structured PLA polymeric compression samples, the most significantly influencing printing process parameters and sequence were printing speed > layer height > nozzle temperature. Among the various printing process parameters, the printing speed and layer height have a major influence on the compressive strength property of the additively fabricated gyroid TPMS lattice-structured PLA polymeric compression samples and the printing temperature contributes to the minor quantity of the lattice structure incorporation for the compressive properties.

Table 6.

Response table for the various printing process parameter with respect to compressive strength.

Level	Printing speed (mm/sec)	Layer height (mm)	Nozzle temperature (°C)
1	27.93	29.37	26.39
2	29.75	27.76	27.41
3	27.42	27.04	28.31
4	25.18	26.10	28.16
Delta	4.57	3.27	1.92
Rank	1	2	3

Analysis of variance (ANOVA) of the produced gyroid triply periodic minimal surface lattice structured compression samples

The analysis of variance is conducted to determine the most influencing process parameter and it helps in identifying the contribution of every individual parameter for the corresponding experiments. For the present research work, the compressive strength is only considered as the output response for the fabricated gyroid TPMS lattice structure. The inspection for the additive manufacturing of the gyroid TPMS lattice structure with various process parameters was conducted with a confidence level of about 95% and a significance level of 5%.

During the incorporation of the gyroid TPMS lattice structure in the PLA polymeric materials, the experimental process parameters such as printing speed, layer height, and nozzle temperature have a significant impact on the compressive strength. Similar observation was also by Sabarinathan et al., on the tensile behavior of the hexagonal lattice structure on the PLA polymeric material.³⁵ Table 7 depicts the Analysis of Variance (ANOVA) results of the output responses such as compressive strength. From the experimental results, the impact of each process parameter should be studied among the experimental output response parameters such as compressive strength. Figure 10 shows the percentage contribution of the printing process parameters. The widely influencing process parameter and the percentage contribution of every individual process parameter concerning the compressive strength were printing speed of 57.28%, layer height of 30.92%, and nozzle temperature of 9.89%.

Table 7.

ANOVA results for compressive strength property of the 3D printed gyroid triply periodic minimal surface structure incorporated polylactic acid compression samples.

Source	Compressive strength
Source	DF	Adj SS	Adj MS	Percentage contribution (%)
Printing speed (mm/sec)	3	302.08	100.692	57.28
Layer height (mm)	3	163.05	54.349	30.92
Nozzle temperature (°C)	3	52.07	17.356	9.89
Error	6	10.10	1.684	1.91
Total	15	527.29		100

Regression analysis

The regression equation was developed from the experimental results by varying the input process parameters such as printing speed, layer height, and nozzle temperature which in turn concerns the output responses such as compressive strength. By implementing the usage of MINITAB statistical software, the mathematical models are developed for the output responses such as compressive strength. By using the regression equation, the R square value was determined by plotting the predicting values against the experimental values of the response parameters such as compressive strength. Figure 11 depicts the R square value of the compressive strength for the incorporated gyroid TPMS lattice structure in the PLA fabricated compression samples and the residual plot for the compressive strength of the fabricated PLA samples. The results depict that almost every point in the probability plot matches with the normal line, which in turn satisfies the accuracy of the predicted model and the observed fit is good.

Compressive strength = 11.4 - 0.2609 speed - 0.549 height + 0.1431 temp

(2)

For the regression analysis of the compressive strength experiment, the regression equation is predicted based on the experimental results, and the regression equation is shown in equation (2). The experimental results depict that the R square value concerning the output responses such as compressive strength will be 0.863. A similar fit was also observed by Sabarinathan et al. on the sol-gel grain recovery process.⁴⁰

Confirmation experiment for the optimal combinations

The experimental results significantly depict the optimal combination for the output responses such as compressive strength for fabricating the gyroid TPMS lattice structured compression samples using the polymeric PLA material. Table 8 depicts the predicted and experimental results of the gyroid TPMS structure incorporated PLA compression samples fabricated at optimized conditions.

Table 8.

Predicted and experimental results of the gyroid triply periodic minimal surface structure incorporated polylactic acid compression sample fabricated at optimized conditions.

Response parameter	Optimal combination	Predicted value (MPa)	Experimental value (MPa)	Percentage error (%)
Compressive strength	Printing speed-20 mm/secLayer height-0.10 mmNozzle temperature- 210°C	31.823	31.154	2.1

The experimental and predicted values for the output response parameter such as compressive strength was 31.154 MPa and 31.823 MPa. The percentage of error was determined from the experimental and predicted results. For the compressive strength, the error percentage lies within the acceptable range of about 2.1%. Dhinakaran et al. observed similar kind of error value on the compressive behavior of wood particle reinforced PLA composite. And also, they concluded that the value within 5% limit are acceptable for the regression equation.³⁷

The mathematical model and the experimental results for the optimal combination of the fabricated gyroid TPMS lattice structure are determined. And the stress-strain curve for the optimized experimental condition is depicted in Figure 12. This stress-strain curve depicts the maximum compressive strength of about 31.154 MPa obtained for the gyroid TPMS lattice structure incorporated compression sample fabricated at the optimized conditions.

Macroscopic failure analysis of the tested samples

Figures 13 and 14 displays the fractured compression samples concerning various experimental process parameters. In all the observed cases, the failure mode of the fabricated samples started from the walls of the compression samples. The fabricated sample’s failure aroused upon imperfections and especially the deformation which took place in the internal part of the fabricated samples which consisted of the gyroid TPMS lattice structure. During the application of the compressive load, the fabricated PLA polymeric compression sample started to compress in the loading direction. At this stage, the internal gyroid TPMS lattice structure gets compressed along the loading direction, and then it begins to fail. The experimental parameters such as higher printing speed, higher layer height, and lower nozzle temperature influence the initiation of the crack in the incorporated gyroid TPMS lattice structure and walls of the fabricated samples. The samples fabricated at these parameters only have the ability to withstand lower compressive strength. On the contrary, the samples fabricated at a nominal printing speed of about 20 mm/sec, lower layer height, and a moderate nozzle temperature of about 210°C have more compressive strength.

These macroscopic fracture images clearly satisfy the following conclusions: First, in the case of printing speed optimization, higher compressive strength is observed in the samples fabricated at 20 mm/sec. When the sample is fabricated beyond 20 mm/sec, the compressive strength decreases gradually. This is because when printing speed increases, the generation of more voids in the fabricated sample takes place which in turn reduces the strength of the fabricated sample. When the sample fabricated at a higher printing speed is subjected to the loading conditions, the sample undergoes a significantly higher bulging and buckling. On the contrary, when the sample is fabricated at an optimum speed of 20 mm/sec, the sample has the ability to withstand higher compressive strength. This is because the optimum speed of 20 mm/sec ensures that there will be no generation of voids in the fabricated sample.

Second, in the case of layer height optimization, the sample fabricated with a layer height of 0.10 mm has observed a maximum compressive strength. When the sample is fabricated with higher layer heights, the samples start to fail at the earlier stages during the loading conditions. This is because as the layer height increases, the number of layers decreases. This decrease in the number of layers helps in initiating the crack propagation at the initial conditions of loading which in turn decreases the strength of the fabricated sample.

Conclusions

In this current research work, the commercially available biodegradable PLA polymeric material is utilized for fabricating the gyroid TPMS lattice structure incorporated compression samples by the FDM technique. The ANOVA optimization technique is employed for the optimization of the printing process parameters to fabricate the best gyroid TPMS lattice structure incorporated compression samples. The L₁₆ orthogonal array is developed by implementing various printing parameters such as printing speed, layer height, and nozzle temperature. The inferences observed from the experimental and optimization results are listed below, by varying the printing process parameters such as printing speed, layer height, and nozzle temperature.

The optimal combination of printing process parameters for fabricating PLA polymeric gyroid TPMS lattice structure incorporated compression sample are printing speed of about 20 mm/sec, layer height of about 0.10 mm, and nozzle temperature of 210°C. The influential printing process parameters are printing speed, layer height, and nozzle temperature and their sequence is printing speed > layer height > nozzle temperature. The contributions of the printing process parameters are 57.28% for printing speed, 30.92% for layer height, and 9.89% for nozzle temperature. From the regression analysis results, the percentage of error for the compressive strength is about 2.1%. From the overall results, the optimal combination of the printing process parameter for fabricating the gyroid TPMS lattice structure incorporated compression samples can be determined.

The TPMS gyroid lattice structure plays a significant role in the development of lightweight structures. These developed structures help in enhancing the rate of energy absorption and they can be used for lightweight structural applications such as household applications and damping materials in kitchen shelves and cupboards. On the other hand, the gyroid structures were extensively employed in the impact energy absorption, aerospace load withstanding, biomedicine and elastic cushioning applications. The gyroid TPMS structure is also employed in various devices like photovoltaic and electrochromic devices. The zero mean curvature in the gyroid TPMS structure helps in achieving the uniform load distributions under the loading conditions.

Limitations & future study

This study fails in a particular condition like when the loading direction during the compressive testing is perpendicular to the direction of the printing during the fabrication of the sample. In such cases, a sudden collapse takes place. This sudden collapse is attributed by the printing direction and stacking orientation of the successive layers of material.

The current research deals with the process optimization of the printing process parameters. In the future, the optimization of the design aspect with respect to the various commercially available polymeric materials will be carried out.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Chennai Instiute of Technology (CIT/CAR/2021/003).

ORCID iDs

Sabarinathan Palaniyappan

Dhinakaran Veeman

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