Modeling of Stress Distribution and Fracture in ABS, PLA, and Alumina-Filled PLA Filaments and FDM-Printed Specimens

Abstract

In the presented work, a probabilistic approach to fracture analysis of commercially available polyacrylonitrile butadiene styrene (ABS) and polylactide (PLA) filaments and fused deposition modeling (FDM)-printed tensile samples produced from them, as well as PLA filled with alumina (Al₂O₃) specimens, is discussed. The results of the analysis reveal the presence of symmetry between the stress distribution in the plastic–elastic and fracture stress regions. The value of the transition point from elastoplastic stress distribution to fracture distribution was established. It was shown that the probability density of the destruction of samples in a bound state obeys the Cauchy distribution. The resulting model well describes the stress distributions of tensile specimens, with a minimum deviation of 0.021% and a maximum deviation of 0.048% for the ceramic–polylactide (60Al₂O₃/40PLA) filament.

1. Introduction

Alumina (Al₂O₃) and the composites based on it are used in a wide variety of applications due to its outstanding corrosion and wear resistance, high electrical resistivity, good biocompatibility, and high hardness [1,2,3,4]. However, the widespread use of ceramic materials is limited by production technology. Usually these materials are machined using diamond tools, which is difficult and takes a lot of time. Therefore, modern fabrication methods are needed. An active desire of the industry to accelerate the process of creating new devices for the consumer market has led to the development of technologies reducing the design and manufacturing process. Traditional methods of materials research and development offer few ways to achieve both simultaneously. One such way may be through the use of data-driven techniques (materials informatics) [5] and enhanced characterization methods [6], while providing a new way to understand material performance and design. The key to the application of materials informatics is the creation of structured and machine-accessible data sets [7]. However, this process is resource-intensive, so software development will play an important role here. Nguyen et al. [8] presented a collection and analysis system for materials-to-device processes that support the capturing, curation, correlation, and coordination of digital material device data in real time and in trust prior to full data archiving and publication. Such systems should help shorten the cycle from discovery of new materials to production of new devices based on them. It should be noted, however, that these developments are only in the early stages. Another alternative technology that is already well established for creating three-dimensional objects at minimal cost is additive manufacturing (AM). A standard has been adopted that classifies terms used in AM technology by specific applications [9]. Fused deposition modeling (FDM) and fused filament fabrication (FFF) are the most widely used 3D printing technologies due to their availability and ease of use. A wide range of thermoplastics for layer-by-layer printing has been developed, among which polyacrylonitrile butadiene styrene (ABS) and polylactide (PLA) are the most common. These thermoplastics are attractive due to their relatively low melting point and shrinkage after printing (especially PLA), good strength properties, and low cost. In addition, these materials are used as a matrix to create composite filaments for FDM printing filled with various materials, such as wood, ceramics, metal, etc. The main advantage of these composites in the FDM method is their increased strength as compared with base thermoplastics. The rapid development of FDM requires the adaptation of existing design and engineering methods to the new technology and new materials. Fracture modeling is an integral part of the design of any constructions. The purpose of this simulation is to optimize the project of the final model based on the loads experienced by the various structural elements. The mathematical basis (numerical methods) to carry out engineering calculations is the various variations of the mesh method [10] and the finite volume method [11], as well as the physical basis is continuum mechanics [12]. A good example of the application of elasticity theory to the study of fracture of samples produced by FDM technology from ABS plastic is given in [13]. In the study of composite materials [14], the Weibull distribution was used to estimate the size distribution of fiber inclusions, which is often used in the study of various materials, including composite materials [15,16]. In [17], it was shown that the particle size distribution of Al₂O₃ particles in PLA does not follow the Weibull distribution and can be different in samples fabricated by the FDM technique from the same filament. In most cases, the uniparametric Weibull distribution is used by researchers to describe distributions of various quantities (stresses, nonmetallic or metallic inclusions, etc.) in materials of different nature, but no attention is paid to investigating the practical distribution of the quantity under investigation. Moreover, such processes as material failure proceed in certain stages, which depend on many factors (applied stresses, deformations, presence of inclusions, etc.) [18], and their description by one type of distribution does not fully reflect the picture of material behavior. Therefore, for the theoretical description of the processes that occur in real materials at failure, under the action of external stresses, the actual question is the description of the distribution of the various quantities affecting the failure of the material.

The purpose of this study is to investigate the empirical probability density functions of stress distributions derived from tensile diagrams [17] and to determine the distribution types corresponding to the empirical functions of fracture probability densities of filaments and printed samples fabricated by FDM printing from ABS, PLA, and manufactured ceramic–polylactide (60Al₂O₃/40PLA) composites.

2. Materials and Methods

The ceramic–polymer (60 vol.% of Al₂O₃–40 vol.% PLA) mixture for filament extrusion was made in a PM 100 (Retsch, Haan, Germany) planetary mill using alumina (Plasmotherm Ltd., Moscow, Russia with d₅₀ = 40 µm) and PLA (eSun Ltd., Shenzhen China d₅₀ = 35 μm) powders. The obtained suspension was dried and sieved using a vacuum desiccator VO 400 (Memmert, Schwabach, Germany) and a sieve with a mesh size of 63 μm, respectively. Tensile test specimens according to Type 1B geometry [19] were printed from manufactured composite and commercial thermoplactic filaments (Bestfilament, Moscow, Russia) using a BlackWidow 3D printer (Tevo 3D, Zhanjiang, China) and tested on an Electropuls E10000 universal electrodynamic testing machine (Instron, Norwood, MA, USA). Details of the ceramic–polymer’s suspension processing, printing, and mechanical testing parameters are reported in a previous publication [13]. Fracture probability theoretical and tensile analysis is based on the analysis of fracture probability density behavior of filaments and FDM-printed samples made from ABS, PLA, and a manufactured alumina–polylactide composite. The analysis was performed numerically using software written in R language. The empirical probability density of fracture was calculated from the experimental data obtained [17] and on the basis of the theory stated in [20,21,22,23].

$${\widehat{f}}_h\left(\sigma \right)=\frac{1}{n}\sum _{i=1}^n\frac{1}{h{d}_{ik}}K(\frac{t-{\sigma }_i}{h{d}_{ik}})$$

(1)

where n is the number of points in the stretching diagram; K is the “kernel” function; k is an arbitrary positive number; d_ik is the distance from σ_i to k-nearest point in the stretching diagram data, consisting of n−1 remaining points; h is the smoothing parameter; and t is the local point where the state density function is defined.

Based on Equation (1), the probability density function of fracture probability of the filament and the specimens obtained by FDM printing technology is continuous and defined for all stresses from 0 to σ_n (where σ_n is the fracture stress). Based on the condition of continuity of function, ${\widehat{f}}_h(\sigma )$, thedensity of fracture probability of the filament and samples obtained by FDM printing technology can be decomposed into Fourier series:

$$\rho \left(\gamma \right)=\sum _{i=1}^n{\widehat{f}}_h({\sigma }_i)·\exp (-2\pi i\left(\gamma ,{\sigma }_i\right))$$

(2)

where $\rho \left(\gamma \right)$ is the spectral density of tensile stress distribution of the filament and samples obtained by FDM printing technology; ${\widehat{f}}_h(\sigma )$ is the probability density of failure of the filament and samples obtained by FDM printing technology calculated by Equation (1).

From Equation (2), it follows that all values of the spectral stress distribution density belong to the space of complex numbers, and according to the Euler equation, the behavior of the spectral function is determined by the argument, given ${\widehat{f}}_h\left({\sigma }_i\right)$ and ${\sigma }_i$:

$$\gamma =Re\left(\rho \left(\gamma \right)\right)/Im\left(\rho \left(\gamma \right)\right)$$

(3)

where γ is the argument of the spectral probability density function (2); Re(ρ(γ)) is the real part of the spectral probability density function; and Im(ρ(γ)) is the imaginary part of the spectral probability density function.

The behavior of argument Equation (3) and the modulus of the spectral density function were examined for symmetry using the following algorithm:

$${\phi }_i\left(\sigma \right)={\gamma }_{i}at\frac{d\gamma }{d\sigma }=0$$

(4)

$${\phi }_i\left(\sigma \right)+{\phi }_i\left({\sigma }_T-\sigma \right)=0$$

(5)

where ${\phi }_i\left(\sigma \right)$ is the value of argument Equation (3) of the spectral function corresponding to the maximum or minimum of the spectral function; ${\sigma }_T$ is the stress value at which the spectral probability density function exhibits symmetry properties.

Equation (4) describes the search for the extremums of the dependence of the argument Equation (3) of the spectral function, and Equation (5) is the symmetry law of the dependence of the argument of the spectral density of states on stresses. For the analysis of the empirical fracture probability density function, the behavior of the theoretical probability density left and right of the stress ${\sigma }_T$ corresponding to Equation (5), the symmetry breaking in the vicinity of ${\sigma }_T,$ and the changes occurring in the theoretical distribution left and right of ${\sigma }_T$ are of primary interest [24]. The theoretical distribution closest to the empirical distribution was determined by the minimum value of the Akaike and the Bayes criterions. The theoretical distributions were as follows:

• Normal.

$$f\left(\sigma \right)=\frac{1}{sd·\sqrt{2\pi }}\exp (-\left(\frac{{(\sigma -\mu )}^2}{2·{sd}^2}\right))$$

(6)

where sd is the standard deviation of the stress; μ is the expectation of the stress distribution.

2.

logarithmically normal.

3.

logistic.

4.

Cauchy.

$$f\left(\sigma \right)=\frac{1}{\pi ·s·\left[1+{\left(\frac{\sigma -x_0}{s}\right)}^2\right]}$$

(7)

where s is the scale factor; x₀ is the shift factor.

The mode of the Cauchy distribution is x₀.

5.

Weibull.

$$f\left(\sigma \right)=\frac{a}{b}{(\frac{\sigma }{b})}^{(a-1)}\exp (-{(\frac{\sigma }{b})}^a)$$

(8)

where a is the Weibull distribution shape factor; b is the scale factor of the Weibull distribution.

The mode of the Weibull distribution is calculated using the equation:

$$mod=\frac{b·{(a-1)}^{\frac{1}{a}}}{a^{\frac{1}{a}}}$$

(9)

where mod is the mode of the Weibull distribution.

6.

Poisson.

7.

Exponential.

Calculation of specimen fracture probability densities based on the identified theoretical probability densities was carried out using the equation:

$$f_i\left(\sigma \right)=\sum _{k=1}^l{\omega }_k{f}_k\left(\sigma \right)-\sum _{m,j=1}^l{f}_m\left(\sigma \right)f_j\left(\sigma \right)$$

(10)

where $f_i\left(\sigma \right)$ is the probability density of failure of the i-th sample; k is the stress distribution number; ${\omega }_k$ is the weight of the k-th probability density in the sample fracture probability density; and $f_k\left(\sigma \right)$ isthe probability density of the distribution at the k-th stress distribution.

The first term of Equation (10) describes a weighted failure probability density function [25], where each of the probability densities corresponds to distributions to the left and right of ${\sigma }_T$. The second term of Equation (10) accounts for the mutual intersection of the probability densities of stresses in the material.

The determination of the total probability density of fracture for the bound state of the filament or specimens made by FDM printing technology from the same material was calculated using Equation [26]:

$$\overline{f\left(\sigma \right)}=\prod _{i=1}^N{f}_i\left(\sigma \right)$$

(11)

where f_i(σ) is the fracture probability density of the i-th sample; N is the number of tested samples; and $\overline{f\left(\sigma \right)}$ is the total probability density of fracture for the bound state of the filament or specimens made by FDM printing technology.

When carrying out a bound-state study, Equation (11) was applied to an empirical state density function (1) and to a theoretical one (10).

Each derived common probability density function (11) was assigned to a theoretical probability density function (6)–(9), and the divergence (“discrepancy”) of the common probability densities was found using Equation (3):

$$D_f=\frac{1}{n+1}\sqrt[2]{\sum _{i=1}^n{(f_i^T\left(\sigma \right)-f_i^P(\sigma \left)\right)}^2}$$

(12)

where $f_i^T(\sigma )$ is the mean theoretical probability density function of the static fracture stress distribution; $f_i^P(\sigma )$ is the practical mean probability density function of the static fracture stress distribution; and n is the number of points in the state density function.

3. Results and Discussion

In [17,26], an analysis of fracture probabilities based on static tensile diagrams of filament and printed samples was performed. This paper presents a further development of this theory based on a stress-dependent failure probability density analysis. For a series of static tensile tests, tensile diagrams of the filament and specimens obtained by FDM printing technology [17] were obtained. The empirical fracture probability densities calculated from Equation (1) are shown in Figure 1.

A comparative analysis of the probability densities of each filament and tensile specimen under static loading shows that the maximums of the probability densities corresponding to the modes of the stress distribution density have a wide scatter (Figure 1A). For specimens manufactured by FDM printing, the scatter of modes decreases, and the values of maximums of probability density functions are divided into three close groups: the first group with samples number 1 and 3, the second group (intermediate) with sample 2, and the third group with samples 4 and 5 (Figure 1B). The failure probability density behavior of ABS plastic filament has a smaller scatter of modes of distribution in comparison with 60Al₂O₃/40PLA and two main groups of scatters of maximum probability densities (first group samples 1 and 3 and second group samples 2, 4, and 5) (Figure 1C). In samples obtained by FDM printing technology, the scatter of maxima and modes is more uniform (Figure 1D). For PLA plastic filament, two groups with different mode positions of fracture probability densities are observed. The first group contains samples 1–3, and the second group contains samples 3–6 (Figure 1D). Such behavior is like the tensile diagrams of PLA filament specimens [17]. For the samples made by FDM printing technique from PLA plastic, the differences in the values of probability density function maxima and the positions of modes of the probability density functions are close to each other (Figure 1E).

Figure 2 shows the calculation of argument Equation (3) of spectral density (2) state for printed samples made from ABS, PLA, and 60Al₂O₃/40PLA using FDM technology.

For example,

Table 1. Positive and negative values of arguments Equation (3) of the spectral probability density (2) satisfying Equation (4) for printed 60Al₂O₃/40PLA samples.

Sample Number	Positive Value of the Argument	Negative Values of the Argument
1	313.1180	−313.1180
	221.7899	$-\infty$
	67.0347	−67.0347
	65.6173	−65.6173
	23.3037	−23.3037
	18.7718	−18.7718
	14.6878	−14.6878
	13.7451	−13.7451
	12.7882	−12.7882
	12.6956	−12.6956
	12.3003	−12.3003
	10.4754	−10.4754
2	164.6106	$-\infty$
	23.9434	−23.9434
	21.9585	−21.9585
	14.9844	−14.9844
	14.2770	−14.2770
	12.6253	−12.6253
	12.5044	−12.5044

presents the values of arguments (3) of fracture spectral density (2) satisfying Equation (4) for printed 60Al₂O₃/40PLA samples.

The numerical analysis (Table 1) and the dependencies presented in Figure 2 of the calculation of the argument Equation (3) of the spectral probability density function (2) of fracture (2) show the presence of symmetry in the behavior of the argument Equation (3) values corresponding to Equation (5). The symmetry breaking occurs in the vicinity of the point ${\phi }_i\left(\sigma \right)\to -\mathrm{\infty }$. The number of peaks and minima decreases for the printed 60Al₂O₃/40PLA samples with a fracture diagram corresponding to a brittle fracture mechanism. Similar results were obtained for the filaments and plastic tensile samples. The behavior of the argument Equation (3) of spectral function of fracture probability density function (2) presented in Table 1 is the same for all studied samples as filaments as well as printed specimens.

Table 2. Stress values corresponding to ${\phi }_i\left(\sigma \right)\to -\mathrm{\infty }$ calculated from Equation (4).

Sample Number	Material Type	Geometry	σT, MPa	$\overline{{\mathit{\sigma }}_{\mathit{T}}}$, MPa	sd, MPa
1	60Al2O3/40PLA	Filament	20.58	21.08	1.04
2			21.45
3			21.50
4			22.29
6			19.56
1		Type 1B	22.92	23.13	0.39
2			23.15
3			23.38
4			23.60
5			22.59
1	ABS	Filament	21.12	24.28	7.70
2			20.86
3			39.87
4			19.65
5			22.58
6			21.62
1		Type 1B	17.43	17.60	0.21
2			17.71
3			17.43
4			17.84
1	PLA	Filament	25.81	25.60	0.50
2			26.49
3			25.11
4			25.22
5			25.55
6			25.44
1		Type 1B	28.95	29.12	0.23
2			28.92
3			29.32
4			29.01
5			29.38

shows the stress values (${\sigma }_T$) at which the spectral probability density function exhibits symmetry properties corresponding to the argument Equation (3) of the spectral density of fracture probability tending to minus infinity.

The analysis of “transition stress” behavior shows that the general trend corresponds to the behavior of strength and yield strength of the filament and printed samples [17] but has a lower standard deviation for all materials and sample types considered in the presented work. Samples produced by FDM technology have higher “transition stress” than filaments, except for samples made of ABS plastic.

Table 3. Types and parameters of theoretical distributions left and right of the ${\sigma }_T$ critical point.

Sample Number	Material Type	Geometry	Before σT	After σT	“b” by Weibull, before σT	“a” by Weibull, before σT	“b” by Weibull, after σT	“a” by Weibull, after σT
1	60Al2O3/40PLA	Filament	Weibull	Weibull	11.3516	1.6319	35.5636	5.3541
2			Weibull	Weibull	11.7840	1.6077	36.7638	5.4249
3			Weibull	Weibull	11.8617	1.6327	39.2420	4.9499
4			Weibull	Weibull	13.4186	2.0907	35.1329	6.2619
6			Weibull	Weibull	10.7935	1.6311	32.5863	5.6789
1		Type 1B	Weibull	Weibull	12.6561	1.6382	42.6263	4.8264
2			Normal 1	Weibull	11.5214	6.6790	42.8089	4.8630
3			Normal 1	Weibull	11.6288	6.7440	43.8685	4.7689
4			Weibull	Weibull	13.0004	1.6261	43.1739	4.9325
5			Weibull	Weibull	12.4487	1.6264	41.4362	4.9169
1	ABS	Filament	Weibull	Weibull	12.3382	1.9228	33.6510	6.1397
2			Normal 1	Weibull	10.3907	6.0190	34.7253	5.6850
3			Weibull	Normal1	21.9928	1.6238	40.8020	0.5130
4			Weibull	Weibull	10.8189	1.6197	33.4263	5.4899
5			Weibull	normal	14.4278	2.4710	32.3603	5.6225
6			Weibull	Weibull	12.5277	1.8802	34.5337	6.1107
1		Type 1B	Weibull	Weibull	9.6416	1.6487	31.8517	4.9390
2			Weibull	Weibull	9.7648	1.6314	32.3896	4.9339
3			Weibull	Weibull	9.6093	1.6296	32.5916	4.7928
4			Weibull	Weibull	9.8286	1.6270	33.0255	4.8547
1	PLA	Filament	Weibull	Weibull	14.2244	1.6256	45.2415	5.2436
2			Normal 1	Weibull	13.1847	7.6446	46.2874	5.2664
3			Normal 1	Weibull	12.5007	7.2434	44.4017	5.1744
4			Weibull	Weibull	13.9334	1.6389	43.0200	5.4649
5			Weibull	Weibull	14.0810	1.6252	43.9315	5.3986
6			Weibull	Weibull	14.0002	1.6184	43.6629	5.4121
1		Type 1B	Weibull	Weibull	15.9913	1.6399	53.8382	4.8269
2			Weibull	Weibull	15.9092	1.6178	53.5672	4.8524
3			Weibull	Weibull	16.1192	1.6151	54.5735	4.8206
4			Normal 1	Weibull	14.4431	8.3639	53.6236	4.8645
5			Weibull	Weibull	16.1923	1.6277	54.4008	4.8548

¹ For the normal (6) distribution, the values of the mathematical expectation and the standard deviation of the stress of the theoretical distribution set by the Akaike and Bayes criterion are given.

compares the theoretical and empirical distributions using Akaike and Bayes criteria for stress intervals before and after σ_T.

Analysis of the behavior of the types of the closest theoretical distributions and their parameters shows that a change in the distribution type occurs in six of the thirty-one samples studied, which does not allow us to state that σ_T is a phase transition point of the second kind. However, the main parameters of the theoretical probability densities of the stress distributions change significantly, indicating a change in the nature of the stretching process. The shape factor “a” (8) of the failure probability density functions estimated to the left of σ_T varies weakly with the material type and state of the samples, while the estimate of the shape factor a to the right of σ_T shows little sensitivity to the shape of the sample but is weakly sensitive to the material type. The scale factor “b” (8) when estimated to the right and left of ${\sigma }_T$ is sensitive to the shape and material type of the sample. Figure 3 shows the theoretical distribution probability densities derived from the analysis of the stress distribution to the left and right of σ_T and the practical fracture probability density of the 60Al₂O₃/40PLA filament specimens according to Equation (1).

Analysis of the results of the theoretical distributions shows that the mode of the first theoretical distribution is in the region of elastic stresses, while its symmetric distribution is in the region of stresses close to fracture stresses. To extract the main distribution describing the probability density of fracture according to Equation (10), a total probability density of fracture of the bound states for the filament and samples obtained by FDM printing was calculated. Figure 4 shows the results of the average probability density calculation for the filament and printed samples.

Analysis of the behavior of the average probability density function of the filament samples shows that when generalizing the probability density functions (Equation (11)), bimodality in the behavior of the density function is observed (Figure 4A—cyan line: first mode indicated by the red vertical line and second mode indicated by the green vertical line), whereas the average probability density function for the printed samples has no bimodality (Figure 4A—pink line: mode indicated by the blue vertical line). At the same time, for printed 60Al₂O₃/40PLA samples, the shifted mode of fracture probability density increases, and the maximum value of the average total density of fracture probability of linked states for these specimens is reduced in comparison with the maximum average total density of fracture probability of linked states of the 60Al₂O₃/40PLA filament.

Figure 5 shows the average overall probability density of fracture of bound states (11) calculated for the three groups of samples 4 and 6, 1 and 5, and 2 and 3.

The results of the calculation of the total probability density function of bound states failure calculated for two samples and compared with theoretical probability density function for Cauchy distribution are presented in Figure 5. The figure shows that the theoretical function well describes the maximum of average density of states for samples 2 and 3 and 4 and 6 (Figure 5C,D) and has a rather big discrepancy for samples 1 and 5 (Figure 5A). Further analysis of the approximation of the theoretical probability density function to the practical one showed that bimodality in the probability density function of the stress distribution is introduced by sample No. 5; so, it is excluded from subsequent studies.

Figure 6 shows the results of calculations of the overall average probability density of fracture of the bound states of the samples selected based on the probability density behavior analysis.

Table 4. Divergence of theoretical and practical mean probability densities of filament and printed samples.

Material	Geometry	Discrepancy with Cauchy Distribution	Discrepancy with Weibull Distribution
60Al2O3/40PLA	Filament	0.033	0.039
ABS		0.042	0.048
PLA		0.024	0.0285
60Al2O3/40PLA	Type 1B	0.020	0.023
ABS		0.027	0.033
PLA		0.015	0.018

shows the results of calculation of divergence (12) for each filament and samples made by FDM printing technology.

Analysis of obtained values shows that the largest divergence between theory and practice is observed for the average probability density function of ABS plastic and 60Al₂O₃/40PLA, in spite of the good visual similarity of curves in Figure 6D. Comparison of the divergence of Cauchy and Weibull distributions with the total probability density function of bound state failure obtained from (1) and (11) by practice shows that the Cauchy distribution has less divergence value with practice than the Weibull distribution.

Table 5. Comparison of mean values of conditional stress of fracture obtained by geometrical analysis of fracture diagrams [17] with the stress corresponding to the maximum mean probability density of fracture.

Material	Geometry	$\overline{{\mathit{\sigma }}_{\mathit{p}}}$, MPa	sd σp, MPa	x0 (7), MPa	s (7), MPa	b (8), MPa	a (8), MPa	Mod (9), MPa
60Al2O3/40PLA	Filament	38.77	2.02	39.13	2.996	39.38	11.332	39.060
ABS		36.30	0.58	37.48	2.270	37.22	14.163	37.028
PLA		47.37	1.35	46.78	4.480	48.25	9.256	47.658
60Al2O3/40PLA	Type 1B	40.88	4.66	40.09	6.520	42.63	5.559	41.136
ABS		30.11	0.45	31.85	4.270	32.13	6.428	31.296
PLA		52.84	0.36	54.69	8.350	56.30	5.745	54.457

presents a comparison of the stress values corresponding to the maximum mean total probability density of failure of the bonded states in static tension with the values of the conditional limit of failure obtained on the basis of the geometrical analysis of static fracture diagrams [13].

Analysis of the deviations of the mean stress of fracture of the filament and printed samples from the stress values corresponding to the Cauchy distribution probability density mode (7) shows that the maximum deviation is 1.5%. The divergence of the Weibull distribution mode values from the mean value of fracture stress shows that the maximum deviation of the stress value corresponding to the Weibull distribution mode (9) from the mean fracture stress is 3.9% and is observed for printed ABS samples. The results of this analysis show that the Cauchy distribution well describes the overall probability density of the stress distribution of the bound states in the tested filament and printed specimens. It was also found that the mode of Cauchy distribution was close to the fracture stress of the tested specimens. Considering Equation (7), the theoretical representation of the fracture probability density is as follows:

$$f\left(\sigma \right)=\frac{1}{\pi ·s·\left[1+{(\frac{\sigma -{\sigma }_p}{s})}^2\right]}$$

(13)

where σ is external stresses; σ_p is material fracture stress; and s is the material-type-dependent coefficient.

With (11) in mind:

$$\overline{f\left(\sigma \right)}={(\frac{1}{\pi ·s·\left[1+{(\frac{\sigma -{\sigma }_p}{s})}^2\right]})}^N$$

(14)

where $\overline{f\left(\sigma \right)}$ is the overall probability density of destruction; N is number of samples, where N ≥ 2.

The analysis of the result obtained for the total probability density of fracture of the associated states and the number of specimens shows that with an increasing number of specimens, the total (11) fracture probability density decreases and, provided the material is unchanged, leads to a decreasing fracture probability (subject to the same stresses). The comparison of the behavior of the total failure probability density function with the probability density function of each sample, in particular 60Al₂O₃/40PLA samples, shows that the main maximum of the failure probability density function is described by the Cauchy distribution (13), and the previously determined stress probability densities (see Table 3 and Figure 3) have almost no influence on the total failure probability density function of the associated states, provided that the number of associated samples N ≥ 2. A generalization of the results of the stress distribution to the left and right of ${\sigma }_T$ (Table 3) with the stress distribution (13) highlighted in the analysis of the total probability density of fracture of the bonded states taking into account Equation (10) is given in Figure 7.

Table 6. Discrepancy between the theoretical description of the density of state according to Equation (10) and calculations based on Equation (1).

Sample Number	Material	Geometry	“Unconformity”
1	60Al2O3/40PLA	Filament	0.00031
2			0.00031
3			0.00021
4			0.00038
6			0.00048
1	60Al2O3/40PLA	Type 1B	0.00025
2			0.00031
3			0.00028
4			0.00024
5			0.00023
1	ABS	Filament	0.00033
2			0.00028
3			0.00034
4			0.00024
5			0.00039
6			0.00037
1	ABS	Type 1B	0.00032
2			0.00034
3			0.00039
4			0.00035
1	PLA	Filament	0.00033
2			0.00032
3			0.00031
4			0.00047
5			0.00024
6			0.00024
1	PLA	Type 1B	0.00025
2			0.00029
3			0.00028
4			0.00028
5			0.00026

presents the calculation of the “unconformity” (12) between the theoretical description obtained from Equation (10) and the probability density obtained from Equation (1) for 60Al₂O₃/40PLA filament samples.

The divergence values obtained show that the model presented in this paper well describes the fracture probability density of the 60Al₂O₃/40PLA filament samples. It is worth noting that similar results were obtained for all materials and geometry types considered in this paper.

4. Conclusions

The analysis of empirical stress probability densities by means of Fourier analysis has established that a region of stress symmetry is observed in the fracture of ABS, PLA, and 60Al₂O₃/40PLA filaments and tensile samples produced using FDM technology. It was found that in this region, the argument of the probability density function tends to minus infinity, this point being the “boundary” separating the distribution of the plastic region from the distribution of fracture stresses. Fracture probability density function results show that the Cauchy distribution is applicable to describe the fracture risk of ABS, PLA, or 60Al₂O₃/40PLA specimens. The comparison of the empirical distribution function with the theoretical Cauchy distribution shows a good agreement between theory and practice. When constructing the model, the mode of the Cauchy distribution corresponds to the fracture stress, and the scale parameter of this distribution depends on the material type. Provided specimens are related by some criterion (e.g., belonging to the same material), the overall probability density function becomes a degree function, where the degree is the number of specimens tested in tension. In the case of the Cauchy distribution, the probability density function decreases, indicating that the probability of failure also decreases, provided the distribution of stresses tested on the specimen is maintained. In this study, it was found that the contribution of the plastic stress distribution does not affect the final probability density of fracture when specimens are bonded. The results obtained from the study and description of the fracture probability density of the filaments and printed specimens made from ABS, PLA, and 60Al₂O₃/40PLA show that the stress distribution at failure is more closely described by the Cauchy distribution, rather than the Weibull distribution, as was previously thought. The main limitation of this work is the lack of strain analysis of the investigated specimens; the issue of strain analysis during the tensile testing of filament and printed specimens will be considered in future works.