Assessment of different forecasting strategies for the breaking load resistance in notched specimens 3d-printed in PLA using additive manufacturing (FFF) technology

The introduction of fused deposition modelling in the industrial sector to produce functional components in thermoplastic materials, such as PLA, requires knowledge of the performance limits of such elements during the design phase. Predicting the breaking load resistance of notched specimens is essential to evaluate the mechanical performance of components manufactured using this technology. This study compares different methodologies based on the critical distance criterion (TCD) for predicting the static breaking strength of printed notched specimens made with Fused Filament Fabrication (FFF) technology using PLA material. Specimens with different topologies of V-notches were printed according to a standardized configuration. Various analysis techniques were applied to determine the breaking strength of each specimen. By comparing the results obtained with experimental data to assess their accuracy and reliability, it has been demonstrated that these methodologies, coupled with the concept of equivalent elastic material (EMC), can be useful for predicting static breaking load.


Introduction
The Nowadays, additive manufacturing is a widely used technology, even in the industrial sector, allowing for the easy production of highly complex geometries, as it is well-known.Fused deposition modeling (material extrusion) is undoubtedly the most utilized additive technology, especially for polymer materials [1].In the industrial world, there is an increasing tendency to design structural components capable of withstanding significant static and dynamic loads.In various fields, such as aerospace, there have been considerations to replace aluminum in many significant applications [2][3].This new trend is also due to the knowledge gained from scientific research in the mechanical characterization of these materials and the standardization of processes and technological controls that are gradually defining standard parameters for successful mechanical performance of finished components, both statically and dynamically [4][5][6].
In this context, there is still a lack of knowledge regarding the behavior of notched components produced through additive manufacturing.In other words, understanding the behavior of these components under specific manufacturing conditions, where notches do not represent material defects, is essential.As is well-known, the presence of notches has a direct effect on the material's strength.Therefore, this article focuses on the fracture behavior of PLA (polylactic acid) produced through additive manufacturing (3D printing), in the presence of V-notches with fillets.Different fillet radii IOP Publishing doi:10.1088/1757-899X/1306/1/012019 2 and various resisting specimen thicknesses will be analyzed, while using a specific parametric setting previously optimized by the authors [7].
The presence of a notch requires the use and/or development of specific approaches for predicting the fracture strength of these materials.While most of these methodologies were originally developed for brittle materials [8], the need to analyze the fracture strength of ductile materials arises from the fact that most structural engineering components are typically made from such materials.Numerous criteria have been presented in the literature, formulated through various approaches, including global criteria [9][10], energy-based criteria using the Strain Energy Density (SED) [11][12][13][14], cohesive zone methods [15][16][17][18], those based on fracture mechanics advance [19], and methods based on the Critical Distance [20][21][22][23][24][25].The latter will be the focus of this study.The analysis of the effect of the notch on 3D printed PLA specimens will be based on the Theory of Critical Distances (TCD), a set of methodologies widely applied in recent years, especially for brittle or quasi-brittle materials, characterized by the use of a material length parameter.This methodology, first developed by Nember [26] and Peterson [27], has subsequently been elaborated in various forms by other authors such as Taylor [28,29], who independently formulated the theory in slightly different ways for predicting fracture in brittle materials [30] and composites [31].Only in recent years has attention shifted towards applying this methodology to polymeric plastic materials [32], leveraging specific calibration activities (experimental and/or numerical).The works of Cicero, Susmel, and Torabi [33,24] are examples of this.This article, instead, aims to demonstrate that by leveraging the concept of Equivalent Elastic Material (EMC) introduced by Torabi [34], it is possible to derive all the necessary parameters for setting up the TCD methods in all their forms from the results of a simple tensile test of the material.Subsequently, a theoretical prediction of the method can be made using finite element analysis.
In conclusion, the results of two different TCD methodologies will be compared to identify the best prediction solution.Section 1 will present the methods at a theoretical level, along with the procedure for correction using the concept of Equivalent Elastic Material.Section 2 will gather experimentally obtained results, while Section 3 will present the main predictive results.Finally, in Section 4, the conclusions will be drawn.

Theory of Critical Distance (TCD)Some text.
The origin of TCD can be traced back to the works of Neuber and Peterson [26,27] in 1958, where it was used for predicting fractures, both static and fatigue, under the assumption of linear-elastic behavior of the material.Only in recent decades has this theory been extensively developed for the analysis of different types of materials, fracture processes, and loading conditions.In general, this theory states that the fracture of a notched component can be avoided as long as the effective stress   (calculated based on material-related parameters) remains below a value of intrinsic strength  0 (stress related to the type of applied load and the material itself), as stated in equation (1).
In other words, the entire linear-elastic stress field around the notch tip is represented by an appropriate effective stress   , while the material's strength is represented by a suitably chosen stress value related to the type of loading or material in use.
The effective stress can be calculated using different methodologies [29], all of which incorporate a length parameter known as the critical distance .In static fracture analyses, this critical distance is calculated as expressed in equation ( 2): where   represents the fracture resistance of the material, and  0 (as mentioned above) is the intrinsic strength of the material.We can immediately state that only in particular situations where the behavior of the involved material is linearly elastic (both micro and macroscopically),  0 coincides with the ultimate tensile strength of the material   , and   can be replaced by the material's fracture toughness   [31].In these rare cases, the application of the TCD method does not require any correction.However, in cases like the one analyzed by the authors, where the material, PLA, exhibits a clear plastic phase, a proper evaluation of these parameters is necessary for the method to function correctly.Regardless of the values of   and  0 identified as suitable for the analyzed situation, the TCD method can be formalized with different methodologies [20], the main ones of which, being rapid and efficient, are compatible with industrial requirements.In particular, among these methodologies, we find the Point Methodology (PM) and the Linear Method (LM).These will be analyzed and detailed below:

The Point Method (PM)
. This represents the simplest implementation of the TCD.This methodology suggests that the effective stress   is evaluated at a point located at a distance of /2 along the bisector of the notch, as expressed in equation ( 3) and depicted in figure 1.

Line Method (LM)
. This methodology suggests that the effective stress is evaluated as the average stress calculated over a length equal to 2L, along the bisector of the notch, as described by equation ( 4) based on figure 2. In most cases, the illustrated methodologies are developed with the support of finite element analysis, which allows replicating the loading condition and evaluating the stress distribution around the notch.The choice of which stress to consider will mainly depend on the type of loading in place.In our case, we will always refer to the maximum principal stress.However, it is not always straightforward, and in some cases, multi-axial phenomena may need to be taken into account by considering von Mises equivalent stress or Tresca stress.
As mentioned earlier, regardless of the chosen TCD method, accurate predictions rely on an appropriate selection of   and  0 parameters.For materials that do not exhibit linear-elastic behavior until fracture,  0 does not coincide with   .In scientific literature,  0 is usually "calibrated" through a dedicated experimental program on specimens with different sizes and/or notch types.In this case, the critical distance  is defined as the value obtained from the intersection of two Stress-Distance curves (obtained through finite element modeling) of specimens with different notches.The calibration process represents a fundamental challenge in the application of TCD methodologies and is a clear obstacle to their widespread implementation in industrial practice [24].
For this reason, the authors have chosen to use an equivalent elastic material approach, which will be extensively described in the next paragraph, for determining  0 value.They will only later use experimental results to evaluate the effectiveness of the method used.

The Equivalent Material Concept (EMC)
As mentioned earlier, the TCD method is based on the strict assumption of having a linear-elastic behavior of the material.It is essential to respect this concept when applying the method.If we approximate a ductile material to have a brittle behavior, we would underestimate the energy required to cause its fracture.As shown in Figure 3a), the area under the real Stress-Strain curve of a ductile material is significantly larger than the line that approximates its behavior as that of a linear-elastic material.In other words, if we were to take  0 equal to the   of the tensile test of a ductile material, we would lose a non-negligible energy contribution.
For this reason, the concept of equivalent material, introduced by Torabi [35], is used by the authors in this research activity to find a material with a fictitious linear-elastic behavior that stores the same energy contribution as the ductile material.In practical terms, this means evaluating the curve of linear-elastic fracture of the equivalent fictitious material, having the same elastic modulus and the area under the real curve resulting from a simple tensile test of the material under study, as shown in Figure 3b).From this curve, it is easy to obtain the value of  0 , which corresponds to the ultimate tensile strength of the equivalent fictitious material  _ .Regarding the value of   , in this research activity, it has been replaced by the fracture toughness of the considered material,   [36].

Specimens Fabrication
The fabrication of the specimens followed, whenever possible, the ASTM D638 standard [37], both for the smooth and notched sample production.All specimens were fabricated using an Ultimaker 3 Extended FDM printer with a 0.4 mm diameter nozzle, and green Ultimaker PLA filament with a diameter of 2.85 mm.
As known in the literature, the correct selection of process parameters is crucial for successful printing, especially to optimize the material's strength and load-carrying capacity.For this reason, all samples were printed using a single layer height of 0.2 mm, 100% infill created with a line pattern and a direction of +/-45°, printed at a speed of 35 mm/s.The printing temperature was set at 215°C, while the build plate temperature was maintained at 60°C.For more details on the parameter settings, reference [4,38] can be consulted, where the same authors conduct research to optimize printing parameters.All specimens were printed in a horizontal position to optimize their mechanical properties.
Six dog bone samples were made to perform standard tensile tests on PLA, choosing Type I samples in accordance with the reference standard.Based on the geometries of the standard specimens, different notch geometries were created, as shown in Figure 4. Various notch sets were built by varying the notch thickness (H) and the fillet radius (ρ), while keeping the opening angle 2α=90° constant.A total of 9 different sets of samples, each containing 6 specimens, were fabricated, resulting in a total of 54 samples.The geometric parameters were varied as shown in the table in Figure 4.For clarity, the latter will be referred to as Hxry, where x indicates the thickness of the sample and y the fillet radius.The different types of carvings will allow us to assess the influence of varying geometry on the strength of the specimens and the method's ability to predict their fracture.
For all samples, after removing any burrs following ASTM D638 standards, the angles and surfaces were smoothed using fine abrasive paper.Following this procedure, all surfaces were defect-free, and the samples were ready for testing procedures.
All tensile tests, both for standard specimens and notched ones, were performed in accordance with ASTM D638 [37], ASTM E132 [39], and ASTM D5045 [40] standards.Specifically, an Instron 3382 machine was used for force control during the tests.An Instron W-6280 extensometer was employed to determine deformation.Following the mentioned standards, the machine's speed of advancement was set to 2 mm/min, and data were acquired at a sampling frequency of 5 Hz.

Standard Tensile Test
From a standard tensile test, by monitoring the evolution of the specimen's area, it is possible to extrapolate the average results of the true Stress-Strain curve.Specifically, a value of the elastic modulus  equal to 3400  and an ultimate tensile strength   = 62 MPa at  = 0,023 /  elongation can be calculated.Consequently, it will also be possible to evaluate the value of the critical real energy density  = 0,789 /∛.The values of the Poisson's ratio  = 0,36 and the material's fracture toughness   = 126,491 MPa √ 2 were obtained from the literature [38].With this, all the necessary values for calibrating and implementing the TCD method are now available.

Test di trazione su campioni intagliati
Figure 5 shows the graphs of the average values, for each set, of the tensile tests on notched specimens.The figures are grouped by specimens with a constant H section and variable fillet radius.As expected, it can be observed that the presence of the presence of notch not only reduces the material's strength but also makes its behavior much more brittle.Figure 5 presents the average results in terms of the maximum force recorded at the material's fracture.

Application of the concept of equivalent material and determination of the critical length
As extensively described in subsection 2.2, based on the data obtained from the standard tensile test on smooth dog bone specimens, it is now straightforward to derive the value of the ultimate tensile strength of the equivalent fictitious material,  _ .As shown in Figure 4b) and from the knowledge of the  value, defined as the area under the real Stress-Strain curve, as shown in Figure 4a), it is possible to calculate the value of  _ from equation ( 5), assuming the same slope of the elastic-linear curve ( which is equivalent to the elastic modulus E).
Considering the known value of   =   for the material, it is possible to derive, using equation (2), the critical length  = 0,477 .4.2.Application of the method using FEA: Workflow.
Figure 6 shows the workflow describing the procedure used to predict fracture force using the critical distance methods through the support to a finite element model (FEM).The process begins by accurately modeling the notched specimen geometries and their corresponding loading/boundary conditions within the finite element code.In this specific case, a shell representation of the specimens was used in Ansys Workbench software, assuming plane stress conditions, which are justified by the small thickness of the specimens and the typical stress distribution around the carving [24,34].The various sets of specimens were then loaded with a unit force applied to the gripping area in the machine (red area in figure 6 a)).The same specimen was fixed on the opposite side by restraining the displacement in the force direction (blue area).
Since TCD, different from other methodologies, is highly influenced by element size [41], an appropriate mesh pattern was applied around the carving to extract stress values at locations of interest for the various TDC methodologies using a simple linear-elastic analysis (example of the maximum stress results are shown in Figure 6b).3) and ( 4) from which the values of the effective stress σ_eff can be derived for each of the proposed methods.
In this application, the maximum principal stress values were considered, as there were no multiaxial phenomena present, and the authors believe that this value represents the stress state of the specimen well.Thanks to the results of the numerical analysis, it is possible to extrapolate, for each experimental set, the trend of the maximum principal stress  _ as a function of the distance r from the apex of the notch along its bisector.As visible in Figure 6c), from equations ( 3) and (4), it is then possible to obtain the values of the effective stress   for each of the proposed methods.Obtaining the values of the predicted maximum allowable force   is now possible through the use of a simple ratio expressed by equation ( 6):

Results and discussions
Table 1 presents the predictions obtained through the proposed TCD methodologies calibrated using the assumption of leveraging the concept of equivalent elastic material.From these results, it is possible to observe that: the predictions made using the PM method are highly accurate, with a maximum deviation from the experimental mean of -11.9% obtained for specimens with a cross-section of 10 mm and a notch fillet radius of 0.5 mm.The average estimation error for this method is 6.5%, and there is no apparent tendency to overestimate or underestimate the experimental data.The PM method's predictions are suitable for both large and small thicknesses and for wide and severe fillet radii.On the other hand, when using the LM method, the percentage error increases in almost all geometric notch conditions but remains below 20%.This method exhibits a clear tendency to overestimate the fracture load, consistent with other reported cases in the literature [24].

Conclusions
In this article, two simple methodologies (Point Method and Linear Method) derived from the Theory of Critical Distance (TCD) for predicting fracture load resistance were tested and compared with experimental results.PLA specimens with V-notch manufactured through additive manufacturing, having different cross-sections and notch fillet radii, were subjected to tensile tests to evaluate the accuracy and reliability of each method.A significant finding was that the use of the concept of equivalent elastic material (EMC) can be effectively beneficial for estimating the intrinsic strength  0 , and consequently, the effective stress   .Thanks to the excellent results obtained from this research, it can be asserted that these methodologies, initially developed for application on brittle materials, can be applied to situations involving components with a considerable degree of plasticity before fracture when corrected using the EMC approach.
Under this assumption, the combination of the TCD methods and the concept of equivalent material leads to remarkably accurate predictions, especially when using the PM method, with deviations from correct predictions limited to a maximum of -11%.In conclusion, we can affirm that the combination of the TCD method and the equivalent material concept can be a valuable tool for designers.By employing simple linear finite element analyses, they can accurately predict the behavior of notched components with substantial plasticity before fracture, such as those made from

Figure 1 .
Figure 1.Example of Stress-distance curve from the notch tip and definition of the PM method.

Figure 2 .
Figure 2. Stress-distance curve from the notch tip and definition of the LM method.

Figure 3 .
Figure 3. a) Real Stress-Strain curve of a ductile material in a tensile test.b) Linear-elastic fracture curve of the fictitious material with the same elastic modulus and the same area under the real curve.

Figura 4 .
Figura 4. Dog bone sample on the left with notched geometry and definition of characteristic geometric parameters for each set of samples.

Figure 5 .
Figure 5. Graphs of the average values of the tensile tests on notch specimens grouped by specimens with a constant H section and variable fillet radius.a) section equal to 10 mm, b) section equal to 13 mm, c) section equal to 15 mm.

Figure 6 .
Figure 6.Workflow describing the procedure used to predict fracture using the critical distance methods.a) Finite element code schematization, application of unit load, and necessary constraints.b) Example of stress distribution around the notch.c) Equations (3) and (4) from which the values of the effective stress σ_eff can be derived for each of the proposed methods.

Table 1 .
Predictions obtained through the proposed TCD methodologies calibrated using EMC.