Creep Properties of a Viscoelastic 3D Printed Sierpinski Carpet-Based Fractal

Abstract

In this paper, the phenomenon of creep compliance and the creep Poisson’s ratio of a 3D-printed Sierpinski carpet-based fractal and its bulk material (flexible resin Resione F69) was experimentally investigated, as well as the quantification of the change in the viscoelastic parameters of the material due to the fractal structure. The samples were manufactured via a vat photopolymerization method. The fractal structure of the samples was based on the Sierpinski carpet at the fourth iteration. In order to evaluate the response of both the fractal and the bulk material under the creep phenomenon, 1 h-duration tensile creep tests at three constant temperatures (20, 30 and 40 °C) and three constant stresses (0.1, 0.2 and 0.3 MPa) were conducted. A digital image correlation (DIC) technique was implemented for strain measurement in axial and transverse directions. From the results obtained, the linear viscoelastic behavior regime of the fractal and the bulk material was identified. The linear viscoelastic parameters of both fractal and bulk materials were then estimated by fitting the creep Burgers model to the experimental data to determine the effect of the fractal geometry on the viscoelastic properties of the samples. Overall, it was found that the reduction in stiffness induced by the fractal porosity caused a more viscous behavior of the material and a reduction in its creep Poisson’s ratio, which means an increase in the compliance of the material.

1. Introduction

The recent progress in additive manufacturing (AM) techniques and technology has strongly benefited the construction of complex-shaped elements for lightweight and higher performance machines, improving the time of manufacturing and reducing costs [1,2]. AM has also enabled the manufacturing of many different lattice structures (metamaterials) in the macro- and micro- scales for a wide range of applications [3], exhibiting a new variety of physical properties which facilitate topology optimization of engineering elements [4,5]. In the area of fractals, AM has also been involved in recent decades. Fractals have been 3D-printed for art and decoration purposes [6], the construction of tunable properties devices [7,8], the construction of modern telecommunication electronic devices [9,10,11], and shock energy absorption elements [12]. Furthermore, in the field of the fluid dynamics, 3D-printed fractals have been reported to study the behavior of the permeability on fractal porous media [13]. In the same way, 3D-printed fractals have been investigated to characterize their mechanical (elasticity) behavior [14] and acoustic properties [15].

In most cases, 3D-printed lattice and fractal metamaterials are studied in terms of their elastic behavior, but the viscoelastic response has rarely been studied. According to the state-of-the-art of fractals’ mechanical properties analyses [16,17], most of the reported work of fractals is limited to numerical analysis or simulations only. Therefore, experimental evaluations are needed. In a recent work, Pascual-Francisco et al. [18] reported a pioneering study on the compressive creep behavior of various Bézier-based lattice structures manufactured by photopolymerization. The authors proposed a new methodology to determine the apparent viscoelastic properties of 3D-printed lattices and demonstrated that the degree of viscoelasticity of a lattice manufactured (3D-printed) with a flexible resin is easily altered by the type of their Bézier unit. In this sense, the viscoelasticity properties of 3D-printed lattices and fractals should be strongly recommended for having a better understanding of their mechanical and/or rheological performance either in the short or long term. i.e., quantifying the viscous and elastic parameters of such materials will allow us to know their behavior in the long term and under different stresses. In addition, such viscoelastic parameters could be useful for further numerical or analytical stress or strain analysis.

Thus, in order to expand the understanding of viscoelastic properties of 3D-printed fractals, the phenomenon of tensile creep and the creep Poisson’s ratio of a fractal was investigated in this work by using digital image correlation equipment as a non-contact strain measurement instrument. More specifically, the main purpose of this research was to quantify the change in the viscoelastic parameters of 3D-printed material when its macroscale porosity has a fractal structure. Particularly, thin rectangle polymeric samples with porosities based on the Sierpinski carpet were studied. The samples were manufactured via stereolithography (SLA), which is one of the most popular AM processes for rapid prototyping with polymers.

2. Creep Theoretical Considerations

When a viscoelastic material or element is subjected to a constant stress (either in tensile or compressive ways) during a certain period of time, it tends to “flow” and deform continuously. This continuous deformation is a consequence of the phenomenon of creep. To evaluate creep in bulk polymers, two main standard methods (ASTM-D2990 and ISO 899–1:2003 [19,20]) are commonly followed. A creep test consists of applying a constant stress to a specimen and measuring its deformation during a certain period of time. The response under creep varies depending on the type of material, micro- and macro-structure, stress and temperature. Hence, creep tests can be performed by applying a constant stress over time, which is often achieved by applying a dead weight for tensioning or compressing the specimen. In a complete creep test, in which the fracture of the material is desired, the behavior of the transient strain can be described in three stages. In the first stage (primary creep), the strain increases abruptly and continues rising at a decreased rate. The second or secondary creep stage increases the strain at an almost constant rate. It is usually the most prolonged stage in a creep test. Finally, the third stage or tertiary creep involves an increasing strain rate and ends in the fracture of the material [21]. Typically, the main creep behavior of materials may be obtained by performing short-term experiments as long as both the first and secondary stages are manifested [22]. The viscoelastic properties are represented by the elastic and viscous constants, which can be obtained by fitting the experimental creep strains to an appropriate viscoelastic model, namely, the Burger’s model, Maxwell´s model, Kelvin–Voight´s model, the standard linear solid (SLS) model, etc. [21,23]. It has been reported that the Burger´s model (also known as the 4-element model) correlates appropriately for common engineering plastics and elastomers [21,22,23,24]. As a reference, the Burgers model deploying the primary and secondary creep stages is presented in Figure 1, where ${\epsilon }_1$ and ${\epsilon }_2$ represent the strain of the spring and the dashpot, respectively, in the Maxwell unit, and ${\epsilon }_3$ represents the strain of the Kelvin–Voight unit. The creep strain function obtained with this model can be represented by Equation (1).

$$\epsilon \left(t\right)=\frac{{\sigma }_0}{R_1}+\frac{{\sigma }_0}{{\eta }_1}t+\frac{{\sigma }_0}{R_2}\left(1-e^{-\frac{R_2}{{\eta }_2}t}\right)$$

(1)

where $\epsilon \left(t\right)$ is the strain as function of time, ${\sigma }_0$ is the constant stress applied, $R_1$ and $R_2$ are the elastic constants, and ${\eta }_1$ and ${\eta }_2$ the viscous constants. In the Burger’s model, $R_1$ represents the elastic response of the spring connected in a series and is associated with the instantaneous strain, $R_2$ represents the maximum strain before the initiation of the second stage of creep, ${\eta }_1$ is associated with the slope of the creep strain in the secondary creep (higher values of ${\eta }_1$ means smaller slope) and ${\eta }_2$ is associated with the strain rate from the instantaneous strain to the initiation of the secondary creep. The methodology for obtaining the viscoelastic parameters of elastomers thorough the Burgers model is detailed in [22]. In summary, viscoelastic constants can be calculated from the model fitted to the experimental data as follows:

$$\begin{gathered} R_1=\frac{{\sigma }_o}{\overline{0A}} \\ {R}_2=\frac{{\sigma }_o}{\overline{AA´}} \\ {\eta }_1=\frac{{\sigma }_o}{\tan \beta } \\ {\eta }_2=\frac{{\sigma }_o}{\tan \phi -\tan \beta } \end{gathered}$$

(2)

In addition, the viscoelastic creep behavior of a material can be represented by its creep compliance $D\left(t\right)$, which is a viscoelastic property expressing the ratio of the creep strain and the constant stress applied; see Equation (3). Considering this property, a material may be considered as linearly viscoelastic when its creep compliance is independent of the level of stresses. In other words, the creep compliance magnitude must be the same for different stresses under the same temperature [21].

$$D\left(t\right)=\frac{\epsilon \left(t\right)}{{\sigma }_0},$$

(3)

3. Materials and Methods

3.1. Samples Preparation

Two different rectangle flat samples (1 ± 0.1 mm thickness) were manufactured via SLA and studied for the viscoelasticity testing: (i) bulk material, and (ii) fractal. SLA is a vat-photopolymerization 3D-printing technique in which the object becomes solid in a vat with liquid resin via an ultra-violet (UV) laser, curing the material selectively and forming the desired model layer by layer [17]. The geometry of the samples is shown in Figure 2. This configuration was chosen because the square Sierpinski carpet was easy to configure in rectangular samples. The porous fractal samples consisted of five fractal patterns as can be observed in Figure 2b. Each fractal pattern consisted of a four-iteration Sierpinski carpet, as can be seen in Figure 2c. The Sierpinski carpet is a fractal base on a square subdivided recursively into smaller squares. The basic Sierpinski carpet is a fractal subset of [0, 1]² defined in a similar way to the classical Cantor set, except that one removes the middle square out of a 3 × 3 block [25,26]. The dimensions of the samples and the fractal structure are reported in

Table 1. Dimensions of the samples for creep testing.

LO (mm)	WO (mm)	T (mm)	A (mm)	B (mm)	C (mm)	D (mm)
100 ± 0.1	15 ± 0.1	1 ± 0.1	15 ± 0.1	5 ± 0.1	1.66 ± 0.1	0.55 ± 0.1

. The models of the samples were designed with the software Fusion 360 (Autodesk Inc., San Rafael, CA, USA); they were then converted to .stl files to facilitate the 3D printing. The specimens were manufactured with the printer 3D Mars 3 Ultra 4K (Elegoo, Shenzhen, China), which works with a curing UV laser of 40 W and a 405 nm wavelength. The polymer used was a flexible resin Resione F69 (Dongguan Godsaid Technology Co., Ltd., Dongguan, China), which is one of the most common 3D-printing resins and is widely used for the fabrication of tires, shoe samples, seals, gaskets, transmission belts, figure toys, etc. The bulk and fractal samples (as well as the dog bone specimens for the tensile test) were printed at a layer thickness of 100 μm and a printing angle of 46°, using fine conical supports. The samples were then dried for a few minutes. The unwanted resin in the samples was eliminated by summering them in a bath with isopropyl alcohol. Afterward, the supports were removed manually and the specimens were dried again and post-cured for 15 min at 60 °C with an UV light of 405 nm.

3.2. Mechanical Properties of the Constituent Material

The constituent material used in this paper (flexible resin Resione F69) was characterized in terms of its tensile strength, as specified in the procedures of the standard ASTM D638 [27]. For this, five dog bone (type V) specimens were manufactured using the same parameters used for printing the bulk and fractal samples for the viscoelasticity testing (same printer and same printing conditions). The tensile tests were conducted using a Shimadzu machine, model AG-IC (Figure 3a) (Shimadzu, Kyoto, Japan), with a load capacity of 100 kN, at a test speed of 10 mm/min. The load-displacement diagrams of the tests were exported from the machine’s software and then processed for further analysis, producing the characteristic stress–strain diagram shown in Figure 3b. The main mechanical properties obtained are summarized in

Table 2. Elastic mechanical properties of the constituent material.

Tensile Strength (MPa)	Percent Elongation at Break (%)	Young’s Modulus (MPa)	Nominal Strain at Break
4.87 ± 0.6	175 ± 9	23.53 ± 0.6	182 ± 8

.

3.3. Tensile Creep Tests

The tensile creep behavior of the bulk and fractal samples was evaluated in a creep test set-up instrumented with DIC equipment for measuring the strain (axial and transverse directions) with time. The experimental setup shown in Figure 4 was employed for conducting the tensile creep experiments. It has two main systems: (i) the load application; and (ii) the strain measurement system. The load application system consists of an electromechanical linear actuator and a temperature control chamber. The electromechanical actuator permits the load application by displacing the platform in a downward direction until the weight stack (with a predefined load) is fully hanging onto the specimen. The temperature control chamber enables controlling temperatures up to 120 °C through a PID controller, which maintained the temperature within ±1 °C close to the gauge length. In the case of the strain measurement system, a DIC equipment Dantec Dynamics System Q-450 was employed (Q-450: Dantec Dynamics, Skovlunde, Denmark). This system uses a CCD camera Phantom SpeedSense 9070 (Zeiss Makro-Planar 50 mm f/2 ZF.2 lens) (SpeedSense 9070: Phantom, Wayne, NJ, USA) with an image resolution of 1280 × 800 pixels. The hardware is connected to the software Istra 4D (Q-450: Dantec Dynamics, Skovlunde, Denmark) for the camera configuration, image processing strain computing, and graphics deployment. DIC is an optical hole-field measurement technique based on the comparison of digital images of an object at different states of deformation [28]. A speckle pattern (usually made with spray paint) has to be first generated over the surface of the object studied. Such a speckle pattern has to be such that the contrast is visible (i.e., if the object is black, the paint has to be white, and if the object is a light color, the paint has to be black). Through tracking algorithms, each point over the surface is identified in the deformed image, according to its light intensity. Then, numerical methods are employed to compute the deformation of the set of points, generating displacement or train maps of the object surface studied.

The test conditions considered in this work are reported in

Table 3. Tensile creep test conditions.

Stress (MPa)	Temperature (°C)	Rate of Image Acquisition (fps)	Test Duration (s)
0.1, 0.2, 0.3	20, 30, 40	1	3600

. Three repeatability tests were run for each condition for both the fractal and the bulk material, using new samples. The load application was synchronized with the image acquisition at a frame rate of 1 frame per second (fps). The first image was taken just before the linear actuator was activated. Then, once the dead load was applied by the displacing down of the platform of the linear actuator, the DIC system recorded and saved the images until the test ended. Afterwards, the DIC software computed the strains produced.

4. Results and Analysis

4.1. Creep Strain and Creep Compliance

Figure 5a–f presents the mean creep strain results obtained from the three test repeats for the different bulk and fractal samples and test conditions. In order to quantify the variation in the measurements, the standard deviations of the three repeats were computed and are shown in each curve. The test time was suitable for obtaining the primary and secondary creep stages for all cases. Nonetheless, the fractal samples reached the tertiary creep stage and were broken, so the results of the creep strain for the fractal at 40 °C under 0.3 MPa stress are not presented in Figure 5f. This means that such a temperature and stress caused a rapid increase in the creep strain in the fractal, and ended in failure. Both bulk and fractal materials were found to exhibit creep strain curves with the shape of that from the Burgers model, in particular at 20 °C, under the different stresses tested, i.e., they exhibited the characteristic first and secondary creep stages of the Burgers model shown in Figure 1. However, interesting behavior of the bulk material was observed at 30 and 40 °C: the creep strain tended to decrease slightly in the second creep stage contrary to that exhibited at 20 °C and that stated in the Burgers model, i.e., apparently, no further creep strain occurred after the initial strain. A similar decrease was also seen for the fractal sample at 40 °C (see Figure 5f). In contrast to the bulk material, the temperature caused a significant increase in the creep strain for the fractal samples (due to the reduction in stiffness); the strain tended to increase abruptly with the temperature, in particular in the primary creep stage.

The mean creep compliance property for each sample under the different conditions was determined from the mean creep strain data from Figure 5a–f. The obtained creep compliance behaviors are shown in Figure 6a–f. According to those results, the creep compliance (under the three different stresses) behavior of both bulk and fractal materials almost overlap only at 20 °C, see Figure 6a,d. Thus, it can be inferred that both the bulk and porous fractal only exhibited a linear viscoelastic behavior at 20 °C under all the stresses applied (0.1, 0.2 and 0.3 MPa), i.e., the condition of linear viscoelasticity is achieved for such conditions. In the other cases, the temperature was able to promote significant alterations to the structure of the polymer generating non-linear viscoelasticity to both the bulk and fractal materials. Hence, a single mean creep compliance curve was generated for each material by averaging the results from the three stresses tested. The mean creep compliance curves obtained are shown in Figure 7a–f. The error shown in these plots corresponds to the standard deviation obtained from the mean creep compliance curves produced under the different stresses (0.1, 0.2 and 0.3 MPa).

4.2. Linear Viscoelastic Parameters

Considering that the porous fractal and the bulk material behave linearly viscoelastic only at 20 °C, the Burgers model was only fitted to those experimental creep strain results obtained at 20 °C under the three stresses to determine their viscoelastic constants. Thus, the creep curves of the three stresses and their respective fitted curves are shown in Figure 8. The elastic and viscous constants ($R_1$, $R_2$, ${\eta }_1$ and ${\eta }_2$) obtained from the fitted models with the corresponding coefficient of determination are presented in

Table 4. Viscoelastic (creep) constants of bulk and fractal samples as obtained from fitted Burgers models.

Material	Stress (MPa)	${\mathit{R}}^2$	${\mathit{R}}_1 \mathbf{\left(}\mathbf{MPa}\mathbf{\right)}$	${\mathit{R}}_2 \mathbf{\left(}\mathbf{MPa}\mathbf{\right)}$	${\mathit{\eta }}_1 \mathbf{(}\mathbf{MPa}. \mathbf{s}\mathbf{)}$	${\mathit{\eta }}_2 \mathbf{(}\mathbf{MPa}. \mathbf{s}\mathbf{)}$
Bulk	0.1	0.97	25	7.38 ± 0.85	81,851 ± 0.16	821.78 ± 11
	0.2	0.96
	0.3	0.97
Fractal	0.1	0.99	25	9.7 ± 0.51	38,197 ± 0.35	2001.33 ± 7.7
	0.2	0.99
	0.3	0.98

.

Regarding the viscoelastic parameter reported in Table 4, it is noteworthy that the elastic constant $R_1$ is similar to the Young’s modulus (23.53 ± 0.6 MPa) obtained experimentally by the tensile test. This is because, according to the Burgers model, $R_1$ is estimated dividing the applied stress by the instantaneous strain, which is the main elastic response; it is also due to the fact that the stresses considered in the creep tests remain in the linear elastic region of the stress–strain diagram of Figure 3. Furthermore, it can be observed that $R_1$ is the same for the bulk and fractal materials. However, considerable changes occurred to the other parameters: $R_2$, which represents the maximum strain before the initiation of the second stage of creep, increased by 31.5% in the fractal with respect to the bulk; ${\eta }_1$, which is associated with the slope of the creep strain in the secondary creep, was 53% higher in the bulk than in the fractal (according to Equation (1); higher values of ${\eta }_1$ mean a decreasing strain which is related to the densification in the bulk material); and ${\eta }_2$, which is associated with the strain rate from the instantaneous strain to the initiation of the secondary creep, was 143.5% higher in the fractal with respect to the bulk. These differences are obvious to some extent because the fractal, which is a porous material (in the macro-scale), flows or deforms more rapidly with respect to the bulk, meaning that the fractal is more compliant than the bulk.

4.3. Creep Poisson’s Ratio

The software implemented in this work also permits us to export the transverse strain. The measurement of the transverse strain during the same creep tests was useful to determine the so-called creep Poisson’s ratio of both the bulk and fractal materials under different stresses and temperatures. The creep Poisson’s ratio, ${\nu }_{creep}\left(t\right)$, which is analogous to the Poisson’s ratio for an elastic material, is the ratio between the transverse creep strain and the longitudinal creep strain, as expressed by Equation (4).

$${\nu }_{creep}\left(t\right)=\frac{{\epsilon }_x\left(t\right)}{{\epsilon }_y\left(t\right)},$$

(4)

where ${\epsilon }_x\left(t\right)$ and ${\epsilon }_y\left(t\right)$ are the transverse and axial creep strains, respectively.

Thus, the creep Poisson’s ratio of bulk, ${\nu }_b$, and fractal, ${\nu }_f$, was calculated from the creep strain results exhibiting linear viscoelastic behavior (results obtained at 20 °C). In Figure 9, the creep Poisson’s ratios of the bulk and fractal materials are presented. The error represents the standard deviation obtained from the three test tresses. For both, an almost constant value was obtained. Averaging the data for each material, the creep Poisson’s ratio of the bulk and fractal was 0.47 ± 0.03 and 0.34 ± 0.04, respectively. The creep Poisson’s ratio of the porous fractal was lower than the bulk material, which can be related to the larger axial creep strain produced in the fractal than in the bulk.

In order to summarize the effect of the fractal structure on the viscoelastic properties of the studied material, the relative parameters were estimated by normalizing their magnitudes with respect to those of the bulk material. The relative viscoelastic constants and the creep Poisson’s ratio (mean value) are shown in Figure 10. It can be observed that the relative parameters $‹R_2›/R_2$ and $‹{\eta }_2›/{\eta }_2$ increased whereas $‹{\eta }_1›/{\eta }_1$ and the relative creep Poisson’s ratio decreased. Thus, the fractal porosity in the material makes the material behave more viscously than elastically, which is ascribed to a reduction in its stiffness. The parameter associated with the slope of the strain (rate of flow) in the secondary stage of creep is ${\eta }_1$; thus, the higher the ${\eta }_1$ the smaller the slope, i.e., for smaller values of ${\eta }_1$ this means that the material flows more rapidly (higher slope of creep). Therefore, the decrease in $‹{\eta }_1›/{\eta }_1$ means that the material is more viscous.

5. Conclusions

In this work, the phenomenon of creep in additive manufactured fractal and bulk materials was investigated. Creep tests at three different temperatures (20, 30 and 40 °C) and three stresses (0.1, 0.2 and 0.3 MPa) were conducted. It was found that both fractal and bulk samples are linearly viscoelastic up to 0.3 MPa at 20 °C. For these conditions, the viscoelastic parameters of both materials were determined by employing the creep Burgers model, which is defined by two elastic and two viscous parameters. Accordingly, these viscoelastic parameters fully characterize the materials under creep and could be used to investigate their creep curves for any stress up to 0.3 MPa at 20 °C.

Based on the relative parameters obtained, it was observed that the fractal material is more viscous than the bulk, mainly because of the reductions in its parameter ${\eta }_1$, which causes an increase in the slope of the creep curve in the secondary stage of creep. Additionally, the creep Poisson’s ratio of both samples was investigated. It was found that this parameter decreased in the fractal, which means that the axial strain is higher in the fractal than in the fractal samples. This behavior is ascribed to the reduction in stiffness in the fractal. Overall, based on the results obtained, it can be concluded that the Sierpinski carpet fractal geometry on flat sheets increases the compliance of polymeric materials. In addition, the viscoelastic parameters obtained in this research paper can be useful for predicting the creep strains in bulk and fractal (Sierpinski carpet) materials made of the resin Resione F69. Furthermore, regarding the Poisson’s ratios estimated, this mechanical property can also be implemented for strain and stress analysis in such materials.