Homogenization simulation of material extruded lattice structures

Abstract

The paper investigates the mechanical behavior of structural elements of a material extruded component using a multiscale analysis based on a homogenization method. The development and validation of a homogenization model start with designing a tailor-made lattice structure. The material model is described using elastoplastic properties and Hill's yield criterion. The numerical validation of the homogenized model and its comparison with the full detail is also described.

1. Introduction

Additive manufacturing (AM) is a set of processes to realize parts layer-by-layer, adding matter where necessary while reducing the amount of used material and promoting design innovation [1]. In material extrusion (MEX) processing, a thermoplastic feedstock, usually available as spooled filament, is fed into the hot-end portion of an extrusion head, heated, melted, and finally forced through an extrusion nozzle [2]. The resulting semi-molten polymer is deposited to realize a cross-section of the model. The stacking of consecutive cross-sections allows the creation of the final part [3]. Some specific problems arise with this deposition technique, such as deformation, shrinkage, raster bonding, and formation of voids, mainly due to the lack of adhesion between layers and rasters. Due to these problems, the strength of the MEX printed parts is lower than injection molded components [4] and affected by the variation of process parameters [5], where orientation and solidity are the most important [6].

Intricate structures, high-performance products, and functionally graded additive manufacturing raise challenges to evolving the print path usually generated with 3D slicing software and the need for innovative design chains to be industrially employed [7]. An example is the nonplanar deposition sequence based on the zigzag geometry. This path is generated by overlapping the nozzle movement along the Z-axis in positive and negative directions with the deposition trajectory in the XY plane [8]. However, generated slicing paths do not allow the direct evaluation of the mechanical part performance. Mechanical behavior and prediction are crucial topics in AM, and MEX in particular, because of the anisotropy induced by the infill pattern and printing direction and heterogeneity caused by pores. The material shrinkage in injection molding is compensated by high pressures, achieving compact and continuous parts. On the contrary, in MEX parts, every raster solidifies or shrinks before complete fusion, generating a porous mesostructured with a periodic arrangement of rasters and pores, which size, shape, quantity, and distribution depend on the geometry and the building parameters [9]. For this reason, the process-geometry-property relationship should be accurately investigated. The performance of MEX end-use components may arise only after a deep knowledge of the relation between the structural behavior and microstructural features. Critical material properties such as the elastic modulus are influenced by the tool path, infill density, and the geometrical non-uniformity induced by the printer's mechanical inaccuracy [10]. With the product complexity raised, the dimensional accuracy and structural integrity can become unpredictable, generating lightly optimized components with uncertain dimensions and material performance. Control and prediction of the material and structural defects are decisive in successfully producing complex geometries and mission-critical components using the full ME potentials [11]. In this context, the possibility of having a reliable 3D model for numerical simulation is crucial since the peculiar structural characteristics of the MEX parts are not included in the CAD model. Although detailed models are considered, there is a corresponding increase in the simulation complexity, directly affecting the time needed.

The paper investigates the effective mechanical behavior of structural elements of a MEX component with a multiscale analysis. Lattice structures unit cell describes the complex internal structure of MEX parts. Thus, the design of the lattice structure is presented together with the development of a robust framework to automatically generate the lattice structure from truss-based unit cells. A homogenization model is then validated using a material with elastoplastic properties and Hill's yield criterion. The numerical results of the homogenized model are compared with those of the full detail to verify the model's performance and reliability.

The paper is organized as follows. A State of the Art of the current AM design chain is reported in Section 2, highlighting the importance of predicting the properties and performance of the realized structures. Section 3 describes the CAD modeling of the lattice structure used for reproducing the typical MEX microstructure with a tailor-made spatial deposition. The implemented multiscale analysis based on the homogenization method is reported in Section 4. Finally, in Section 5, numerical results from the simulation chain are illustrated and discussed before reaching Conclusions.

2. Design chain in additive manufacturing

One of the obstacles slowing the full adoption of AM in the industry is the unstructured knowledge of Design for AM due to multiple production methods linked to AM [12], with their benefits and drawbacks, and the limited attention on AM design research in industries and universities [13]. A typical design process chain to generate G-Code for AM is a multi-step procedure containing the main printing parameters, such as the infill pattern, layer thickness, and temperature profile [14]. The process can be completed by reacquiring the produced part [15] and machining the near-net-shaped component [16]. The improvement of AM is difficult because of the knowledge gap between operating conditions and build quality [17]. The design of an AM component is strictly dependent on the ability to simulate the AM process, compute the resulting material properties, and evaluate its effects on product performance [18]. Numerical simulation is essential to study an AM part, seeing the impact of the process-induced microstructure on the final structural performance. Some approaches analyze the fabrication directly [19,20], computing residual stresses and the deformed shape of the part [21]. Other techniques used multi-level modeling to investigate the effects of different fabrication features and defects on the AM structures, using various scales analyzing the same phenomena [22,23]. The situation becomes more complex when lattice structures are utilized. Lattice structures are frequently designed by using CAD modeling or AM-specific software. Within the AM-specific software, a library of unit cell topologies generates different lattice structures or infill patterns. Many manually generated structures are used because they are simplistic and can be broken down into basic geometric shapes. Their simplicity and simple parameterization make these structures the most used [24]. Lattice structure topology continues to receive significant interest with the application of AM in microstructure fabrication. Stochastic lattice structures can be random, such as in a metal foam, or non-stochastic lattice by pattern a repeated unit cell. The relationships between the unit cell topology and the effective mechanical properties in a non-stochastic lattice structure are crucial [25]. The slight change in the number and size of unit cells in a porous structure, associated with low infill densities, usually causes a significant computation difference in the design and printing [26]. A multiscale and multi-physics approach is highly desirable to face all the complex interactions of product, material, and process [27]. Multiscale approaches can face the complex nonlinear response of materials and the resulting structure. Among the multiscale methods used in designing lattice structures, the homogenization-based approach is one of the most used. The reason is connected to evaluating the effective properties of local microstructures computed with homogenization theory, reducing the computational effort during the analysis procedure [28].

3. CAD modeling of the lattice structure

AM allows the fabrication of tailor-made lattice parts with increased complexity and functionality. The proposed methodology to design and analyze the part made with the lattice structure is shown in Fig. 1, following three different paths.

Fig. 1.

Design chain.

The most straightforward path “Fabricate” was associated with product realization and consisted of slicing the CAD model with the lattice structure to manufacture it directly. The slicing software managed the part infill density, and no lattice structure was modified in the path. The numerical analysis with the Finite Element Method (FEM) was not carried out, not allowing the prediction of part properties and performance. This path was typically used for aesthetical components or assembly. The intermediate path “Analyze and Fabricate” was more complex and involved the analysis of the part performance. The CAD model with lattice structure meshed and then evaluated after selecting the infill density and wall parameters and adding the material properties. The STL model of the mesh was then sliced to fabricate the component at the end of the FEM Analysis. The prototype's performance was calculated, considering a high computation time for the numerical simulations. The last path, “Homogenize and Fabricate” considered the generation of a lattice structure starting from a Representative Volume Element (RVE) reproducing the infill pattern, later substituting it with equivalent homogenized solid cells with appropriate mechanical properties after performing the numerical simulation on it. As a result, the simulation time was reduced, preserving the accuracy of the computational results of the component performance. One of the evident advantages of using this last path was the possibility of being independent of the slicing limitation regarding the selection of unit cell and lattice size. This last path was implemented using the design environment Grasshopper integrated with CAD software Rhinoceros (Robert McNeel & Associates, Seattle, WA, USA). A graphical summary of the final Grasshopper data flow is presented in Fig. 2. Using this visual programming language, the flow created the lattice structures with tailor-made spatial deposition using the following steps:

• A.Define the volume for making the structure for a selected solid.
• B.Generate an initial unit cell using a truss-based lattice structure.
• C.Repeat the unit cell with different spacing along with the X, Y, and Z axes.
• D.Cut surfaces to confine the final lattice structure into the selected solid.
• E.Extract Lattice
• F.Repeat the design cycle with other volumes, eventually changing the unit cell and spacing.

Fig. 2.

Graphical summary of the Grasshopper data flow.

A generic truss was a rectangular prism with base dimensions A × B and a fillet radius of r. The horizontal dimension A was a function of the extrusion nozzle diameter, whereas the vertical dimension B was equal to the layer thickness set during slicing. The truss elements of the unit cell were oriented with angles of α, ±β ∈ [α, γ], and γ (see Fig. 3). The complementary pitch p_i,j in the XY plane and normal pitch q_h,k along the Z-axis were used for each single truss element, using uniform or non-uniform spacing and overlapping by defining the main angle i = [α, ±β, γ], the complementary angle j, and the normal angle k. For the angle i equal to 0°, the complementary angle j was 90°, while for the 30° angle, the complementary angle was 60°. The normal angle k was always equal to 90° along the Z-axis.

Fig. 3.

Unit cell configuration.

The coding of a lattice structure reported the section dimensions, fillet radius, main angles, pitches, and overlapping in each direction (Table 1).

Table 1.

Coding of the unit cell configuration.

Parameter	Symbol	Value
Section dimensions	S	AxB [mm2]
Fillet radius	r	r [mm]
Angles	α, β, γ	0°–90°
Complementary pitch in XY plane	pi,j	i = [α, ±β, γ], 0–4 × A [mm]
Normal pitch in XY plane	pi,k	i = [α, ±β, γ], 0–4 × B [mm]
Vertical pitch along the Z-axis	qh,k	h = α, k = [ ±β, γ], 0–4 × A [mm]
Element shift along the X, Y or Z-axis	sh	n × pi,j/m × qh,k

Some examples are reported in Fig. 4 using the same rectangular section and varying only the horizontal and vertical pitches. These configurations were associated with different infill densities, starting from the less dense configuration A to the fully dense configuration D. The limitations in designing lattice structures were linked to the minimum wall thickness, surface quality, and support structure because these properties directly depended on the selected AM method [29].

Fig. 4.

Several configuration variants.

4. Multiscale modeling

The accuracy in predicting the mechanical behavior of a part with FEM using 3D cell elements depends on the element size. The FE approach was computationally demanding for the tiny size of the cell structure. Adopting FE cells of dimensions comparable to the MEX beads made it impractical to perform analyses reasonably on the entire structure because of the many mesh nodes and elements. The microstructure of a MEX part typically consists of an array of beads, which shape and packing generated voids of various shapes and sizes [30]. Multiscale material modeling aimed to predict material behavior at the macro level using the information at more minor scales [31]. The data was passed between scales through homogenization. Multiscale modeling also relied on computational homogenization of a statistically Representative Volume Element (RVE). An RVE was the smallest periodic volume with a detailed statistical description of the material properties used in a macroscale model. The homogenization process began by designing the RVE geometry and specifying its material properties. The meshed RVE was subjected to several load cases and computed its response.

The material characterization problem could be split into two distinct but related sub-problems. The first sub-problem involved the transition from macroscopic to mesoscopic scale, minimizing the distance between a reference response and its numerical counterpart. The second sub-problem was related on the transition from mesoscopic to microscopic one, with the aim of determining of both geometrical and elastic properties of material [32]. A homogenization scheme was applied to calculate macro-constitutive performance at a material point. The RVE was designed from the microstructure of the MEX paths for numerical homogenization using the previously described truss structure. Mesoscale RVE consisted of the deposited beads and layers, simulating the presence of voids created from fabrication. Homogenization detected the effective constitutive matrix and layer of the MEX‐made parts using the microstructural and mesostructured information data [33].

Homogenization was suitable as a multiscale method due to the periodic nature of infills and lattice structures [34]. The definition required the cell pattern, spacing, and orientation. The material behavior of a single cell was detailed, given the mesoscale model. Topology optimization was also a crucial development to perform multi‐scale optimization of lattice structures [35]. Many problems require explicit solutions at multiple scales. However, an iterative solution process at each scale was highly computationally intensive. The intrinsic role of different scales in the mechanics of materials was well recognized. The proper understanding of the behavior, evolution, and mechanical response of materials at the micro-scale was critical.

At micro level, the initial position Y in the reference state Υ₀ and the deformation position y in the current state Υ were linked by the motion Eq. (1) as follows

$$y=\phi \left(Y\right)=Y+\omega \left(Y\right)$$ (1)

where ω was the micro displacement. The deformation gradient at micro level was

$$F={\nabla }_Y\phi \left(X;Y\right)$$ (2)

where X was the material point at macro level and ∇_Y the gradient operator at micro level. The self-equilibrium equation for the unit cell in Υ₀ at micro level was

$${\nabla }_Y\bullet P=0$$ (3)

with P the micro-scale 1st Piola-Kirchhoff (PK). The governing equation at micro level could be completed by introducing a constitutive model as a function deformation gradient F and other internal state variables accounting for inelastic material properties. The traction component of Piola vector, derived from Eq. (3),

$$T^{\left(N\right)}=P\bullet N$$ (4)

satisfying the condition

$$T^{\left[J\right]}+T^{\left[-J\right]}=0$$ (5)

with N the outward unit normal vector of the matching surface and J the basis vector of the Y_J-axis.

At macro level, the spatial position X in initial configuration χ₀ and the spatial position x in the deformation configuration χ were linked by the motion Eq. (6) as follows

$$x=\tilde{\phi }\left(X\right)$$ (6)

The deformation gradient at macro level was

$$\tilde{F}={\nabla }_X\tilde{\phi }\left(X\right)$$ (7)

with ∇_X the gradient operator. $\tilde{F}$ in Eq. (7) was the volume average of the corresponding deformation gradient at micro level over the unit cell, expressed by Eq. (8) as

$$\tilde{F}=\frac{1}{\left|{\mathrm{{\rm Y}}}_o\right|}\int FdY=\tilde{H}+1$$ (8)

with |Υ₀| the initial volume of the unit cell. The displacement gradient at macro level was

$$\tilde{H}={\nabla }_X\tilde{u}\left(X\right)$$ (9)

with $\tilde{u}$ in Eq. (9) the displacement field. 1st PK stress at macro level was the volume average of the corresponding stress at micro level over the unit cell

$$\tilde{P}=\frac{1}{\left|{\mathrm{{\rm Y}}}_o\right|}\int PdY$$ (10)

satisfying the following equilibrium at macro level

$${\nabla }_X\bullet \tilde{P}+\tilde{b}=0$$ (11)

with $\tilde{b}$ the body force.

The BVP of Eqs. (1), (2), (3), (4), (5), (6) at micro level was solved for $\omega$, $F$, $P$, whereas the BVP of Eqs. (7), (8), (9), (10), (11) at macro level was solved for $\tilde{u}$, $\tilde{F}$, $\tilde{P}$. The two-scale BVP consisted of BVPs at micro at macro level.

An elastoplastic formulation with small strains at room temperature was employed in the problem formulation for the constitutive material model. The strain tensor $\epsilon$ was

$$\epsilon ={\epsilon }_e+{\epsilon }_p$$ (12)

where ${\epsilon }_{\mathrm{e}}$ and ${\epsilon }_{\mathrm{p}}$ were the elastic and plastic tensors, respectively. The elastic part followed Hooke's law, whereas a yield criterion, a hardening rule, and a plastic flow rule defined the plastic part. The Hill yield function was defined as

$$f\left(\sigma ,\alpha \right)={\sigma }_{Hill}\left(\sigma \right)-\bar{\sigma }\left(\alpha \right)$$ (13)

where ${\sigma }_{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{l}}$ was the Hill stress, and was the hardness function [36]. The Hill stress was

$${\sigma }_{Hill}\left(\sigma \right)=\sqrt{F{\left({\sigma }_{xx}-{\sigma }_{yy}\right)}^2-G{\left({\sigma }_{zz}-{\sigma }_{zz}\right)}^2-H{\left({\sigma }_{xx}-{\sigma }_{zz}\right)}^2+2{N{\tau }^2}_{xy}+2{L{\tau }^2}_{yz}+2{M{\tau }^2}_{xz}}$$ (14)

The σ(α) in the yield function is a hardness function using the cumulative plastic strain α as the hardness parameter and was called the reference yield stress. F, G, H, L, N, and M were data-fitted coefficients [37]. The goal at the macro level was to study the behavior of the entire cell set, realized with various geometric parameters, forming a dense or sparse material. Each heterogeneous cell was replaced with an equivalent homogenized solid with appropriate mechanical properties. Average quantities per RVE for macroscopic strain <ε> and stress <σ> were assigned to this cell, where <> was the average operator per volume, ε the local strain value, directly dependent on the displacements field u, and σ the local stress value. Homogenization with periodic boundary conditions computed effective mechanical properties [38].

The modeling capabilities of the homogenization theory allowed the evaluation of the equivalent material properties, defined as the homogenized material constants, and the actual stress field on a micro-scale for a composite material with linearly elastic periodic microstructures. The former process was called Homogenization, whereas the latter was Localization [39]. The solution of the macro-scale boundary value problem (BVP) within the framework of the nonlinear homogenization was implemented by using a two-scale computation scheme for elastic-plastic deformations of composites which completely coupled the micro- and macro-scale BVPs [40]. A unit cell was assigned to each integration point in the macro-scale FE model, and the solutions of the macro-scale BVP, in terms of macroscopic stress and strain, were the volume average of the corresponding solutions of the micro-scale BVP. The macro-scale analysis shared the same structure as a nonlinear FE analysis. A Newton-Raphson iterative procedure was performed on each time interval with the discretized two-scale variational forms. The suggested scheme relied on numerical material testing (NMT) of the RVE, like the computational homogenization method for linear problems. A series of NMTs on the RVE was performed to obtain the nonlinear macro-scale material behavior. Utilizing the measured data in the NMTs, the material parameters in the assumed constitutive model are identified with an appropriate parameter identification method. Once the macro-scale material behavior was correctly fitted with the identified parameters, the macro-scale analysis was performed. The macro-scale deformation history at any point in the macro-structure was applied to the RVE to evaluate the micro-scale response.

The computational steps for the resolution of the two scale BVP were the following:

• 1.Chhose a constitutive material model at macro level.
• 2.
  Perform NMTs on a unit cell model with FE analysis

  • -
    Impose the displacement gradient $\tilde{H}$ at macro level and the displacement vector $q^{\left[J\right]}$ of the external points.
  • -
    Make the expanded BVP at micro level by setting ${\omega }^{\left[J\right]}$-${\omega }^{\left[-J\right]}=q^{\left[J\right]}$
  • -
    Compute 1st PK stress at each incremental step n for all loading patterns by solving the expanded BVP at micro level.
• 3.
  Find material parameters at macro level

  • -
    Adopting the NMT data, compute the macro-scale 2nd PK stress.
  • -
    Construct a function using the material parameters p.
  • -
    Identify the material parameters p at macro level.
• 4.
  Perform the FE analysis at macro level

  • -
    Compute the BVP at macro level with the constitutive model built with material parameters just computed.

Further details and a more in-depth description of the mathematical formulation can be found in Refs. [[39], [40], [41], [42]]. The approach was based on a bottom-up scheme computing the effects of the processing conditions on the microstructure and coupling meso with the macroscopic simulations to predict the mechanical behavior of the whole MEX part [43]. The resulting behavior was evaluated with appropriate mechanical tests numerically validated. This approach reduced the computational effort by neglecting the local deformation of every single cell. The values of the homogenized properties of the material were calculated by performing numerical/virtual tests without carrying out expensive experimental campaigns. After an appropriate validation process, the results obtained with these methods predicted the macroscopic behavior of the structures without the need to model their microstructure.

5. Simulation chain and results

The multiscale material modeling was applied to compute the mechanical response of a MEX flat bone specimen, the dimensions following the ISO 527-1:2019 “Plastics - Determination of tensile properties – Part 1: General principles”. The initial section and the reference length of the sample were 4 × 10 mm² and 80 mm, respectively. The other dimensions are reported in Fig. 5.

Fig. 5.

Tensile test dimensions (mm).

The investigated material was a natural unfilled polyetheretherketone (PEEK) supplied by Intamsys Technology Co. Ltd. (Shanghai, China). This high-performance engineering polymer was chosen for its outstanding resistance to harsh chemicals, excellent mechanical strength, and dimensional stability. PEEK can also maintain stiffness in application with temperatures continually up to 170 °C. Aerospace, oil and gas, food and beverage processing, and semiconductor are some of the main fields of application. The main mechanical properties declared by the supplier are reported in Table 2.

Table 2.

PEEK material properties.

Properties	Value	units
Density	1.30	g/cm3
Young's modulus	3.74	GPa
Tensile strength	99.90	MPa
Elongation at break	9.10	%

The CAD specimen geometry consisted of a sparse core and an externally fully dense structure. The core was lightened by alternating stacked roads at various angles and spacing, creating the structure lay-up during deposition (see Fig. 6).

Fig. 6.

Stress states of the RVE.

The main steps of the simulation chain were:

• •Identify the Repetitive Unit Cell (RUC) representing the RVE of the core.
• •Define the nonlinear mechanical properties assigned to the RVE at the mesoscale level.
• •Characterize the RVE and create the homogenized cell.
• •Perform nonlinear FEM analyses of homogenized and full-detail specimens.
• •Compare results and validate the homogenization process.

The RVE was constructed with a parametric CAD. Its translation along the three principal axes was necessary to build the trabecular core structure, leaving the remaining portion fully dense. The solid geometry of the RVE was imported into the Ansys Workbench (Ansys Inc., Canonsburg, PA, USA), and the mechanical properties of the bulk material were assigned to it. The material was modeled as anisotropic elastoplastic with the Hill plasticity criterion. The Hill yield function extends the von Mises yield criterion for considering the anisotropy of an elastoplastic material [34]. The characterization process of the homogenized anisotropic elastoplastic behavior was carried out with Multiscale. Sim (Cybernet Systems Co. Ltd., Tokyo, Japan). After generating the mesh and imposing the boundary conditions, numerical material tests were carried out, and comparable properties were computed using Ansys Mechanical. One linear plus six nonlinear analyses were performed, establishing on the RVE three uniaxial tensile states along with X, Y, and Z directions and three pure shear states in the three coordinated planes XY, XZ, and YZ, considering the occurrence of large deflections. Additional details on the simulation parameters are: (1) A high-order 3D, 10-node element SOLID187 having three degrees of freedom (translations) at each node was used. (2) The maximum element size was 0.1 mm (3) A sensitivity analysis conducted for element size less than 0.1 mm gave no significant improvements in results while increasing solution times. (4) The RUC mesh consisted of more than 170,000 FE elements. (5) Cyclicity constraints were adopted to impose the periodicity of the boundary conditions of the RUC. (6) A maximum deformation of 6% was set in all directions. (7) The maximum number of sub-steps to reach this total deformation was 25. (8) Sparse Matrix Direct Solver was used to solve the Full Newton-Raphson algorithm (NROPT). (9) The tangent stiffness matrix was automatically updated at each step of the full NROPT.

Fig. 7 reports stress fields of tensions and shears of the RVE at a deformation of 6%. The relationship between homogenized stress-strain <σ>−<ε> and local stress-strain σ-ε of the mesh nodes of the RVE was identified by fitting the coefficients of each test state and characterizing the anisotropic behavior. The homogenization process was carried out by identifying the principal direction, reaching the plastic deformation, and using it as the master curve. In contrast, the stress-strain curves in the other two main directions were a function of the master curve suitably scaled according to the identified Hill parameters.

Fig. 7.

Stress states of the RVE.

Fig. 8 shows the stress-strain curve of every single state. The reference state was the σ_ZZ because of its low elastic modulus value. The other curves were scaled at the exact value of the plastic strain, allowing the computation of Hill's coefficient. The resulting average stress-strain curve of the core is shown in Fig. 9.

Fig. 8.

Stress-strain curve of every single state of the RVE.

Fig. 9.

Stress-strain curve of the homogenized cell.

Once the homogenization of the mechanical properties of the RVE, the specimen analysis was performed. The model was initially simplified. The core structure with the original stacked cell was substituted with homogenized cells with a solid structure with the material properties still computed (Fig. 10).

Fig. 10.

Homogenized model vs. original model.

The skin remained unmodified. The fixed constraint was applied at the left surface of the specimen, and a displacement of 10 mm was imposed at the left surface along the X-axis. The von Mises yield criterion was used for the mechanical evaluation of the homogenized and fully detailed geometries. Nonlinear analysis was carried out, considering the effects of material nonlinearity for small displacements. Since the sample was subjected only to tension, Hill and von Mises's solutions were perfectly equivalent. The number of elements in the full detail model was more than 3.4 million, using a mesh discretization of 0.1 mm. The ratio between the number of mesh elements of the full and homogenized models was 77.8%. Consequently, the solution time of the homogenized model was 3.1% of the full model, equal to 288 h, with an evident computational timesaving. In particular, the total homogenized computational time was 6.5 h to calculate the unit cell plus 2.5 h for the homogenized model. Fig. 11, Fig. 12 show the stress and strain fields, pointing out a high numerical agreement between models. A slight variation was observed in the region of width change because the homogenized model simplified the structure transition between core and skin, with possible stress concentration at the interlayer and intralayer. Fig. 13 reports the comparison between the stress-strain curves of the homogenized and full detail models, extracted showing the good agreement between results of the two models.

Fig. 11.

Equivalent von Mises stresses.

Fig. 12.

Equivalent total strain.

Fig. 13.

Stress-strain curve of homogenized model vs. original model.

This paper numerically.

6. Conclusions

This paper numerically investigated the effective mechanical behavior of a MEX part. The approach considered the generation of a lattice structure starting from an RVE reproducing the infill pattern, later substituting it with equivalent homogenized solid cells with appropriate mechanical properties after performing the numerical simulation. The structure consisted of a trabecular core and tough skin. A homogenization material model was used, revealing that simplifying the geometry drastically decreased the computation time without affecting the numerical results. The comparison between the full-detail model and the homogenized model indicated that model reduction did not affect the quality and computation of the numerical results but only the total computation time.

Further research will be addressed to propose and study more lattice unit cells tailor-made for the MEX process, evaluating other deformation mechanisms (compression shear) and their application on industrial components.

Author contribution statement

Roberto SPINA, Prof: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper.

Maria Grazia Guerra: Contributed reagents, materials, analysis tools or data; Wrote the paper.

Silvia Dirosa: Performed the experiments; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Giulio Morandina: Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Funding statement

Prof Roberto SPINA was supported by Programma Operativo Nazionale Ricerca e Competitività [B36G18001430005], Ministero dell’Istruzione, dell’Università e della Ricerca [232/2016].

Data availability statement

Data will be made available on request.

Declaration of interest's statement

The authors declare no competing interests.

List of Acronym

AM

Additive Manufacturing

BVP

Boundary Value Problem

CAD

Computer-Aided Design

EMF

Enhanced Metafile

FE

Finite Element

FEM

Finite Element Method

MEX

Material Extrusion

NMT

Numerical Material Testing

NROPT

Newton-Raphson algorithm

PEEK

Polyetheretherketone

RUC

Repetitive Unit Cell

RVE

Representative Volume Element

STL

Stereolithography