Stress Relaxation and Creep of a Polymer-Aluminum Composite Produced through Selective Laser Sintering

This article discusses the rheological properties (stress relaxation and creep) of polymer-aluminum composite specimens fabricated through the selective laser sintering (SLS) from a commercially available powder called Alumide. The rheological data predicted using the Maxwell–Wiechert and the Kelvin–Voigt models for stress relaxation and creep, respectively, were in agreement with the experimental results. The elastic moduli and dynamic viscosities were determined with high accuracy for both models. The findings of this study can be useful to designers and users of SLS prints made from the material tested.


Introduction
Three-dimensional printing or additive manufacturing, originally known as Rapid Prototyping, is being increasingly used in various applications, including manufacturing, design, architecture and medicine [1][2][3]. 3D Printing has become a common process to fabricate not only models, patterns or prototypes, but also finished and semi-finished products. The latest developments in additive manufacturing, especially its application to medical devices and pharmaceuticals, have been discussed, for instance, in [4,5]. The articles analyze the current and future areas of use of 3D printing in manufacturing as well as recent advances in materials and designs.
The most important factors affecting the development of additive manufacturing are: − The ability to customize products according to individual needs and requirements. − The ability to design and fabricate elements complex in shape, which are impossible or very difficult to produce using conventional processes. − The ability to combine many components into one single product to save assembly costs.
Additive manufacturing technologies are suitable to print geometrically intricate internal structures, e.g., objects within objects, thin-walled products, or objects with a sponge, cellular or honeycomb structure retaining sufficient strength. Fused deposition modelling (FDM) is the most popular 3D printing process for quick and easy fabrication [6,7]. In medicine, for example, FDM technology has been used for hip joint reconstruction. Pre-surgical modelling involves simulating loads and calculating unit stresses, and developing a custom implant design [8].
In additive manufacturing, the layer-by-layer deposition considerably affects the material mechanical properties [9][10][11][12]. This is crucial in the manufacture of consumer products, particularly those subjected to external loads, where high material strength is required.
were designed in 3D CAD and saved in the * stl. format using the triangulation parameters in the export option: resolution (adjusted), deviation (0.016 mm tolerance), and angle (50 tolerance). Then, the specimen models were virtually oriented on the build tray of the printer, as illustrated in Figure  1. A layer thickness of 0.12 mm was used to fabricate the cylindrical specimens. The process parameters were selected in accordance with the recommendations of the manufacturer of the Alumide powder. After printing was completed, the specimens and the powder remains were removed from the build tray, and the specimens were prepared for the stress relaxation and creep tests.
The static compressive strength tests were carried out using an Inspekt mini 3 kN universal testing machine produced by Hegewald and Peschke MPT GmbH (Nossen, Germany), equipped with flat compression platens. After the measurement data were acquired, the test parameters were set using Labmaster software incorporated in the Inspekt mini. The specimens, one by one, were placed centrally on the lower platen in the vertical position. Then, the upper platen mounted in the crosshead grip was moved down to be in contact with the flat surface of the specimen.
The first stage of the stress relaxation test involved applying a strain of 5%, with the rate of the compression platen displacement v being 0.5 mm/s. In the second stage of the test, the crosshead motion was stopped to apply a constant strain of 5% for a predefined period of time, i.e., 7200 s. There was a decrease in the compressive load and, consequently, a decrease in compressive stresses (stress relaxation), which was illustrated in the form of a curve. In the third stage of the test, the platen returned to the initial (zero) position, which coincided with the unloading of the specimen. An example curve illustrating the whole stress relaxation test is shown in Figure 2a.
The first stage of the creep test consisted in applying a load of 300 N, which corresponded to a stress of approximately 3.82 MPa, with the rate of the compression platen displacement v being 0.5 mm/s. After the crosshead motion was stopped, a load of 300 N (a stress of 3.82 MPa) was maintained constant for 7200 s. The gradient of the strain-creep curve was reported to increase slightly. The third stage of the test involved unloading the specimen. The upper platen returned to the initial (zero) position and the test was stopped. An example creep curve is shown in Figure 2b. A layer thickness of 0.12 mm was used to fabricate the cylindrical specimens. The process parameters were selected in accordance with the recommendations of the manufacturer of the Alumide powder. After printing was completed, the specimens and the powder remains were removed from the build tray, and the specimens were prepared for the stress relaxation and creep tests.
The static compressive strength tests were carried out using an Inspekt mini 3 kN universal testing machine produced by Hegewald and Peschke MPT GmbH (Nossen, Germany), equipped with flat compression platens. After the measurement data were acquired, the test parameters were set using Labmaster software incorporated in the Inspekt mini. The specimens, one by one, were placed centrally on the lower platen in the vertical position. Then, the upper platen mounted in the crosshead grip was moved down to be in contact with the flat surface of the specimen.
The first stage of the stress relaxation test involved applying a strain of 5%, with the rate of the compression platen displacement v being 0.5 mm/s. In the second stage of the test, the crosshead motion was stopped to apply a constant strain of 5% for a predefined period of time, i.e., 7200 s. There was a decrease in the compressive load and, consequently, a decrease in compressive stresses (stress relaxation), which was illustrated in the form of a curve. In the third stage of the test, the platen returned to the initial (zero) position, which coincided with the unloading of the specimen. An example curve illustrating the whole stress relaxation test is shown in Figure 2a.
The first stage of the creep test consisted in applying a load of 300 N, which corresponded to a stress of approximately 3.82 MPa, with the rate of the compression platen displacement v being 0.5 mm/s. After the crosshead motion was stopped, a load of 300 N (a stress of 3.82 MPa) was maintained constant for 7200 s. The gradient of the strain-creep curve was reported to increase slightly. The third stage of the test involved unloading the specimen. The upper platen returned to the initial (zero) position and the test was stopped. An example creep curve is shown in Figure 2b.  The curves in Figure 2a,b are divided into three zones. The first zone or segment 1 represents a rapid increase in load: − to achieve a displacement of 0.5 mm in stress relaxation tests; − to achieve a force of 300 N in creep tests.
The rapid increase in load applied to a specimen is represented by the unit step function In theory, the rapid increase in load occurs at an infinitely high rate. This, however, is not possible in reality. In the experiments, load was applied at a relatively high rate; hence the quasistep function. The second zone of the curve (Segment 2) is the proper stress relaxation curve ( Figure  3a) or the proper creep curve (Figure 3b). Only these curves will be analyzed in this article. The third zone (Segment 3) illustrates a decrease in load, a return of the upper platen to the output (zero) or the end of test position. The stress relaxation tests were performed by placing a specimen between two platens of the universal testing machine and compressing it, as depicted in Figure 3a. The specimens after the tests are shown in Figure 3b.    The curves in Figure 2a,b are divided into three zones. The first zone or segment 1 represents a rapid increase in load: − to achieve a displacement of 0.5 mm in stress relaxation tests; − to achieve a force of 300 N in creep tests.
The rapid increase in load applied to a specimen is represented by the unit step function ε(t) = ε 0 H(t). In theory, the rapid increase in load occurs at an infinitely high rate. This, however, is not possible in reality. In the experiments, load was applied at a relatively high rate; hence the quasi-step function. The second zone of the curve (Segment 2) is the proper stress relaxation curve (Figure 3a) or the proper creep curve (Figure 3b). Only these curves will be analyzed in this article. The third zone (Segment 3) illustrates a decrease in load, a return of the upper platen to the output (zero) or the end of test position. The curves in Figure 2a,b are divided into three zones. The first zone or segment 1 represents a rapid increase in load: − to achieve a displacement of 0.5 mm in stress relaxation tests; − to achieve a force of 300 N in creep tests.
The rapid increase in load applied to a specimen is represented by the unit step function In theory, the rapid increase in load occurs at an infinitely high rate. This, however, is not possible in reality. In the experiments, load was applied at a relatively high rate; hence the quasistep function. The second zone of the curve (Segment 2) is the proper stress relaxation curve ( Figure  3a) or the proper creep curve (Figure 3b). Only these curves will be analyzed in this article. The third zone (Segment 3) illustrates a decrease in load, a return of the upper platen to the output (zero) or the end of test position. The stress relaxation tests were performed by placing a specimen between two platens of the universal testing machine and compressing it, as depicted in Figure 3a. The specimens after the tests are shown in Figure 3b.   The stress relaxation tests were performed by placing a specimen between two platens of the universal testing machine and compressing it, as depicted in Figure 3a. The specimens after the tests are shown in Figure 3b. The material of the prints was analyzed using a Nikon Eclipse MA200 microscope equipped with NIS 4.40 AR elements imaging software. Top surfaces of the specimens were examined. The images of the material structure are shown in Figure 4. The material of the prints was analyzed using a Nikon Eclipse MA200 microscope equipped with NIS 4.40 AR elements imaging software. Top surfaces of the specimens were examined. The images of the material structure are shown in Figure 4. The microscopic images reveal that the build direction had no considerable effect on the material structure. It can be seen, however, that the material has voids, with this suggesting that not all the grains of the polyamide powder were fused and bonded. Some of the spherical grains were partially bonded. Aluminum grains are irregular, but they are distributed relatively uniformly in the material structure. The material tested is a typical polyamide-based composite. The microscopic images reveal that the build direction had no considerable effect on the material structure. It can be seen, however, that the material has voids, with this suggesting that not all the grains of the polyamide powder were fused and bonded. Some of the spherical grains were partially bonded. Aluminum grains are irregular, but they are distributed relatively uniformly in the material structure. The material tested is a typical polyamide-based composite.

Mathematical Model
Materials able to respond elastically to a rapidly applied load and slowly increasing deformation can be described mathematically by combining two properties: elasticity and viscosity. This ability can be described linearly using the laws of a Hookean solid and a Newtonian liquid. The elastic and viscous behaviors of a viscoelastic solid material are mechanically interpreted using two simple single-parameter models: one for a spring and the other for a hydraulic damper.
The simple models are connected in series and/or in parallel to form a system-a mechanical model-describing certain behaviors of a real solid body such as creep and relaxation.
The general equation of the condition describing the two phenomena is: .
Multi-parameter models may be difficult to use as they require determining a greater number of parameters to calculate the material elasticity and viscosity constants. In this study, a five-parameter model will be considered and Equation (1) will be used to analyze: Since Equations (2) and (3) apply to two different phenomena, the stress, strain and relevant coefficients are distinguished by the superscripts r and c for stress relaxation and creep, respectively.
Equations (2) and (3) can be solved analytically. The integral Laplace transform is used for this purpose. The general form of this transform for the second and first-order derivatives can be written by the formulas: In this method, the problem is solved using the Heaviside function and the Dirac delta function: The Laplace transforms of these functions are given by: Solving this problem analytically will allow us to follow the calculation process and unify the coefficients of Equation (1).

Maxwell-Wiechert Stress Relaxation Model
The system of springs and dampers used for the five-parameter Maxwell-Wiechert model is illustrated in Figure 5.
Solving this problem analytically will allow us to follow the calculation process and unify the coefficients of Equation (1).

Maxwell-Wiechert Stress Relaxation Model
The system of springs and dampers used for the five-parameter Maxwell-Wiechert model is illustrated in Figure 5. Equation (2) can be written as: Solving this ordinary differential equation requires defining the initial conditions. In the experiment, the initial conditions were: The coefficients of Equation (11) The Laplace transform of the other terms of Equation (12) yields: The solution of Equation (13) can be written as: After algebraic calculations, we have Equation (14) is solved by using the inverse Laplace transform, which gives: Equation (2) can be written as: Solving this ordinary differential equation requires defining the initial conditions. In the experiment, the initial conditions were: σ r (0) = ε r 0 E r 0 and d dt σ r (0) = 0. The coefficients of Equation (11) are thus as follows: with H(t) being the Heaviside or unit step function. Transforming Equation (11) we get: The Laplace transform of the other terms of Equation (12) The solution of Equation (13) can be written as: After algebraic calculations, we have A r = ε r 0 E r 0 , B r = ε r 0 E r 1 , C r = ε r 0 E r 2 . Equation (14) is solved by using the inverse Laplace transform, which gives: or, in the general form: where: τ r 1 = Equation (3) was solved for creep in a similar way.

Kelvin-Voight Creep Model
The configuration of the springs and dampers used for modelling the material creep is illustrated in Figure 6. In the creep model, when σ c = σ c 0 , then the stress is constant. (16) where:

Kelvin-Voight Creep Model
The configuration of the springs and dampers used for modelling the material creep is illustrated in Figure 6. In the creep model, when σ σ = 0 c c , then the stress is constant.

For calculation purposes, Equation (3) is written as:
With the initial conditions being where: Figure 6. Kelvin-Voight creep model.

For calculation purposes, Equation (3) is written as:
With the initial conditions being ε c (0) = 0 and d dt ε c (0) = 0, the coefficients of Equation (17) are: For the case of simple creep loading, σ c (t) = σ c 0 H(t). Algebraically, the integral Laplace transform gives: where: The inverse Laplace transform of the strain-time function yields or, in the general form: where: σ c 0 -constant stress, n-number of basic models, i-consecutive number of the model, τ c i -delay of elasticity of the i-th Kelvin model expressed by: where: µ c i -viscosity of the i-th model and E c i -elastic modulus of the i-th model. Thus, is the transformed strain-time function.

Results and Discussion
Figure 7a-c show the stress relaxation test results and the corresponding best fit curves obtained for the three types of specimen differing in the build direction (X, Y and Z, respectively). The experimental stress relaxation curve is a black continuous line while the best fit stress relaxation curve approximated by Equation (15) is a broken line in red, blue and green. Each figure includes values calculated by the program using a certain number of iterations. These are the parameters E r 0 , E r 1 , E r 2 , τ r 1 and τ r 2 describing the Maxwell-Wiechert model. There are also approximation errors with the goodness of fit data including the reduced chi-squared χ 2 and the R-squared R 2 . 19) or, in the general form: where: σ 0 c -constant stress, n-number of basic models, i-consecutive number of the model, τ c idelay of elasticity of the i-th Kelvin model expressed by: where: μ c i -viscosity of the i-th model and c i E -elastic modulus of the i-th model.
is the transformed strain-time function.

Results and Discussion
Figure 7a-c show the stress relaxation test results and the corresponding best fit curves obtained for the three types of specimen differing in the build direction (X, Y and Z, respectively). The experimental stress relaxation curve is a black continuous line while the best fit stress relaxation curve approximated by Equation (15)    Each experimental stress relaxation curve was approximated by Equation (15) From the discussion of Equation (15), it is clear that, for t = 0, the stress is: If we assume that then the stress for t = 0 is: where: r s E -equivalent modulus of elasticity. However, when t → ∞ , the limit of Equation (15) is: The equivalent elastic moduli and stresses obtained for the different types of specimen at t = 0 and t → ∞ in the stress relaxation tests are given in Table 2.  Figure 7. The results of the stress relaxation tests and the best fit curves: 1-experimental stress relaxation curves, 2-stress relaxation curves approximated by Equation (15): (a) for the specimens built in the X direction, (b) for the specimens built in the Y direction, (c) for the specimens built in the Z direction.
Each experimental stress relaxation curve was approximated by Equation (15); this required determining the parameters E r 0 , E r 1 , E r 2 , τ r 1 and τ r 2 . The Levenberg-Marquardt algorithm, which is an iterative procedure, was used for curve fitting. The results are given in Table 1. From the discussion of Equation (15), it is clear that, for t = 0, the stress is: If we assume that then the stress for t = 0 is: where: E r s -equivalent modulus of elasticity. However, when t → ∞, the limit of Equation (15) is: The equivalent elastic moduli and stresses obtained for the different types of specimen at t = 0 and t → ∞ in the stress relaxation tests are given in Table 2. The relaxation times τ r 1 and τ r 2 are the ratios of elastic moduli to dynamic viscosities: where: µ r 1 and µ r 2 -dynamic viscosities. The formulae in Equation (27) and the data in Table 1 were used to calculate the coefficients µ r 1 and µ r 2 for each specimen type. The values, rounded to the nearest integers, are provided in Table 3. From the experimental data, it is evident that the polymer-aluminum composite Alumide (all build directions considered) shows slightly lower anisotropy than pure polyamide [19]. The composite material also exhibits lower stress relaxation and creep than pure polyamide P12 additively manufactured through SLS.
The creep test data and the best fit curves are shown in Figure 8. Each experimental creep curve was approximated by Equation (22); this involved determining the parameters ε c 0 , ε c 1 , ε c 2 , τ c 1 and τ c 2 . The curve fitting was performed using the Levenberg-Marquardt algorithm. The results are provided in Table 4.  From the experimental data, it is evident that the polymer-aluminum composite Alumide (all build directions considered) shows slightly lower anisotropy than pure polyamide [19]. The composite material also exhibits lower stress relaxation and creep than pure polyamide P12 additively manufactured through SLS.   The elastic moduli E c 0 , E c 1 and E c 2 , and viscosities µ c 1 and µ c 2 for the Kelvin-Voight model, illustrated in Figure 4, were calculated from the experimental data using Equation (21). It is important to note that, in the case of creep, the times τ c 1 and τ c 2 are the retardation times or delays of elasticity of the constituent models. The calculation results obtained for this model are given in Table 5. Table 5. Elastic moduli and dynamic viscosities obtained for the Kelvin-Voight model ( Figure 8).  From the discussion of Equation (19), it is clear that, at t = 0, the strain is: