*3.4. Comparison of the Load-Bearing Capacity*

To design the computational model, we destroyed the specimen A1 with a real thickness of *tr* = 0.80 mm (after evaluating stiffness data) to obtain the value of the load-bearing capacity. The maximal failure load of the FS *Fmax* = 276 N was measured in the experi-

ment. This force was important for estimating the value of the plastic strain *ε pb* = 0.0339, which was obtained using full non-linear model (with contact) after subjecting specimen A1 to the maximal failure load *Fmax*, see Table 9. The plastic strain field corresponding to the failure load force *Fmax* N with details on the critical area is shown in Figure 12. Plastic deformation occurred locally on the borders of the pattern empty area. A critical area with the highest plastic strain was near the edge of the plunger, where additional bending of FS acted. This is also the area of the breakage of the specimen during the initial experiment.

After calculating the value of plastic strain and locations, authors were able to compare the pattern types A2 and A3 with A1. A failure load was also calculated for specimens A2 and A3 with the real thickness of *tr* = 0.80 mm. The highest plastic strain on specimen A2 and A3 occurred in the area similar to that of the specimen A1. The load-bearing capacity values corresponding to all calculated failure loads are listed in Table 10. It should be noted that only one specimen was destroyed and values in Table 10 cannot be viewed as statistically correct; still, it gave us some approximate values of the load-bearing capacity of all geometries.

**Table 9.** Comparison of failure load *Fmax* from experiment and simulation.


**Figure 12.** The plastic strain field corresponding to the failure load *Fmax* = 276 N for specimen A1, real thickness *tr* = 0.80 mm.

**Table 10.** Calculated values of load-bearing capacity.


#### **4. Discussion**

In this paper, two ways of determining the stiffness of 3D printed flexible structures (FS) are presented. The first way was the construction of a novel testing device replacing

the traditional tensile and flexural testing techniques used for testing of internal structure properties [25,26]. The second way is a custom-developed validated computational model based on the Finite element method (FEM). It was also demonstrated that the computational model is suitable for the description of the FS behavior and, as such, could be used to evaluate the FS stiffness. The agreement between the initial experiment and simulation was achieved and the model was subsequently used for a parametric study with three different FS pattern types, each of them with three variations of pattern geometries and five thickness variations (Figure 11).

Even though all testing devices were 3D printed, they were stiffer than tested specimens. This influence was analyzed by finite element analysis of the fastening system. Maximal deformation of the fastening system (in the direction of force) was only *u* = 0.11 mm corresponds to load force *F* = 250 N. Deformation of the fastening system did not significantly affect performed experiments (compared to deformations of specimens). Therefore, the influence of the fastening system stiffness could be neglected. For measuring the specimens with higher thickness, a new robust fastening system is necessary to design.

The proposed experimental technique was found to be an important complementary tool in the design of the 3D printed FS with a three-point star pattern by non-linear computational modeling. Moreover, a computational model for analyzing the FS loadbearing capacity was presented, yielding results with satisfactory accuracy compared to the real, experimentally verified, failure load. Such a non-linear computational model can be used for prediction of the FS load-bearing capacity and for prevention of specimen breakage during testing. One should not forget the possibility of plastic strains occurring in small curvatures at higher load forces. It should be noted that comparing linear and non-linear computational models, the latter one describing laboratory experiment closely.

This paper presents two different computational models, both solving a static structural problem with elasto-plastic behavior defined by bilinear material model. In the first model for stiffness determination, we used a simplified non-linear model without a plunger. For precise evaluation of the load-bearing capacity, however, a non-linear model with a plunger had to be constructed. Although the material model provided satisfactory results, it could be substituted with an even better one, such as Chaboch material model, if even more accurate results were necessary. Nevertheless, the main advantage of the presented non-linear material model is its simplicity, needing only two input variables.

It should be noted that the thicknesses of the computational models were corrected to fit the real printed ones and, therefore, they were comparable with experiments. The deviation from the planned specimen thickness occurred during printing. This might have been caused by the low thickness of the shell to which a lower than required amount of the Nylon powder may have adhered. Measurement was performed on various places of the specimen using the vernier caliper. In some areas, the thickness was reduced by 0.18 mm, sometimes by 0.22 mm. The thickness of the computational model was reduced on average by 0.20 mm. To increase the accuracy of the thickness, it would be necessary to use a very accurate 3D scanner and to subsequently insert the acquired point cloud into the calculation.

Assuming the isotropic material behavior could be viewed as an incorrect premise since printed materials do not behave the same in all directions. The material model should be constructed for orthotropic or, even better, anisotropic behavior. Some results from the measurement suggest a possibility of creep and viscoelastic behavior under load.

During the experiment, strain and displacement were also measured by the DIC method; however, obtained data were not satisfying. The measurement was limited by the used template size, which had to be set to "large" to obtain some data. Another complication was the presence of a "holed" curved surface, which caused difficulties in detecting the pattern on the outer surface. All these problems meant that although it was possible to correctly assess the displacement, the strain/stress measurement could not be properly evaluated. In future experiments, therefore, the use of the DIC method should be

redesigned to prevent the mentioned problems. Successful use of this method on a curved surface is described in a paper by Halama et al. [27].

Although the measurement using the DIC method was not successful, it sparked a few ideas on how to improve the procedure. First, background should be added to the inner surface to create contrast with the outer pattern. Second, some post-manufacturing process should be applied to ensure that the outer surface does not remain as porous as it is after printing. Third, the measuring software should be set up in a better way for this analysis; in particular, a better template size should be chosen. In our experimental setup, unfortunately, only the displacement could be evaluated without a significant error. Fourth, cameras should be better positioned to ensure measurement of displacement in the part of the sphere furthest from the rim. The main reason for preparing a better DIC measurement is obtaining a strain field from which one could derive the limit value of plastic strain (currently, the limit value is obtained from the computational model).

#### **5. Conclusions**

In this paper, we present that using a non-linear computational model, the stiffness can be determined with satisfactory results. The difference between modeling results and experimental data is less than 15%, which means that our simplified non-linear computational model was suitable for evaluating the stiffness properties of FS. As far as the load-bearing capacity is concerned, a full computational non-linear model was calibrated for force *Fmax* = 276 N which gave us limit value of plastic strain. This being said, the simplified model had an overwhelming advantage in solving time—it was approximately 160% faster than the full model; therefore, the simplified model was more useful in the parametric study. For example, the computing time for a simplified non-linear model (Specimen A1, *tr* = 0.80 mm) was 35 min. (84 iterations) on a standard machine (workstation Intel i7-8700K, 12 cores, 16 GB RAM, SSD). Approximately 25 s. was needed for each iteration. If we used a full non-linear model, the time for each iteration would be multiplied by more than 2 (i.e., 53 s. per iteration, summary 103 iterations, total time 91 min.). It must be also emphasized that the full non-linear model was not ideally converging to a solution every time. Additional tuning of the contact pair was necessary and better boundary condition needed to be set (in this case, the displacement boundary condition converged better than the force boundary condition). However, this model is highly recommended for evaluating load-bearing capacity.

Out of the three virtually tested pattern types, the pattern A performed best, mainly thanks to its high load-bearing capacity. It offered a high stiffness, which helped prevent the specimen A from changing shape under the applied load (see Figure 9). This was also confirmed in the analysis calculating the failure load. Another advantage was that the stiffness of specimen A scaled almost linearly with thickness with thickness, making it easy to design a specific stiffness value. Specimen A2 was the stiffest of the patterns A. Also, it had the highest load-bearing capacity. These properties were caused by having smaller gaps than the specimen A1. On the other hand, Specimen A3 was, compared to the other two geometries compliant, which also meant a low failure force. It was thanks to wider gaps than specimen A1. Other patterns did not offer such mechanical properties (such as stiffness and load-bearing capacity).

Our paper studied a shell, considering 2D pattern only. However, if an application needs a compliant (or rather even more compliant) structure, studying 3D-spatial pattern structures would be beneficial. The 3D-spatial structure might offer additional advantages compared to 2D structures only. For example, considering the biomechanical use, they might provide better sweat drainage and airflow, increasing the patients' comfort. Finally, a compliant structure could also be ensured by choosing another material (one with a lower Young's modulus than PA12), such as Thermoplastic Polyurethane (TPU).

It is necessary to mention here the cost-saving character of this virtual model. Should all configurations from Table 4 be evaluated experimentally, many more shells would have to be printed, which would immensely increase the costs. Using the presented approach, only a specific configuration is printed and evaluated experimentally, which would save roughly 90% of the costs—and this calculation is based on an experiment with a low number of specimens. To support a standard statistical evaluation, the experiment should use at least 5 to 10 specimens for each proposed structure, which would rocket the costs sky-high and the savings thanks to the use of our software solution could be over 98%.

It should be noted that the authors will continue their research on this topic and will expand on tested (2D and 3D-spatial) specimens in the following years.

**Author Contributions:** Conceptualization, P.M. and J.P.; methodology, P.M. and V.R.; software, V.R. and P.M.; validation, V.R., M.S., P.M. and D.R.; formal analysis, V.R.; investigation, P.M. and V.R., R.H. and M.F.; resources, V.R., P.M. and R.H.; data curation, P.M. and V.R.; writing—original draft preparation, V.R.; writing—review and editing, M.S. and P.M.; visualization, V.R. and M.S.; supervision, P.M. and M.F.; project administration, P.M.; funding acquisition, R.H. and P.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by The Ministry of Education, Youth and Sports from the Specific Research Project SP2020/23, by The Technology Agency of the Czech Republic in the frame of the project TN01000024 National Competence Center-Cybernetics and Artificial Intelligence and by Structural Funds of the European Union within the project Innovative and additive manufacturing technology—new technological solutions for 3D printing of metals and composite materials, reg. no. CZ.02.1.01/0.0/0.0/17\_049/0008407.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable to this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
