Bending behavior of additively manufactured lattice structures: numerical characterization and experimental validation
Nina Korshunova, Gianluca Alaimo, Seyyed Bahram Hosseini, Massimo Carraturo, Alessandro Reali, Jarkko Niiranen, Ferdinando Auricchio, Ernst Rank, Stefan Kollmannsberger
BBending behavior of additively manufactured lattice structures: numerical characterization and experimental validation
N. Korshunova ∗ , G. Alaimo , S. B. Hosseini , M. Carraturo , A. Reali , J.Niiranen , F. Auricchio , E. Rank , and S. Kollmannsberger Chair of Computational Modeling and Simulation, Technische Universit¨at M¨unchen, Germany Department of Civil Engineering and Architecture, University of Pavia, Italy Institute for Advanced Study, Technische Universit¨at M¨unchen, Germany Department of Civil Engineering, Aalto University, Finland
Abstract
Selective Laser Melting (SLM) technology has undergone significant development in the past yearsproviding unique flexibility for the fabrication of complex metamaterials such as octet-truss lattices.However, the microstructure of the final parts can exhibit significant variations due to the highcomplexity of the manufacturing process. Consequently, the mechanical behavior of these latticesis strongly dependent on the process-induced defects, raising the importance on the incorporationof as-manufactured geometries into the computational structural analysis. This, in turn, challengesthe traditional mesh-conforming methods making the computational costs prohibitively large. In thepresent work, an immersed image-to-analysis framework is applied to efficiently evaluate the bendingbehavior of AM lattices. To this end, we employ the Finite Cell Method (FCM) to perform a three-dimensional numerical analysis of the three-point bending test of a lattice structure and comparethe as-designed to as-manufactured effective properties. Furthermore, we undertake a comprehensivestudy on the applicability of dimensionally reduced beam models to the prediction of the bendingbehavior of lattice beams and validate classical and strain gradient beam theories applied in combi-nation with the FCM. The numerical findings suggest that the SLM octet-truss lattices exhibit sizeeffects, thus, requiring a flexible framework to incorporate high-order continuum theories.
Keywords: additive manufacturing, metamaterials, octet-truss lattice, Finite Cell Method, computedtomography, beam theories, strain gradient elasticity, Finite Element Method
Contents ∗ [email protected] , Corresponding authorPreprint submitted to Materials & Design
January 25, 2021 a r X i v : . [ c s . C E ] J a n Experimental setup 85 Numerical investigations 11
Mechanical metamaterials have received much attention in the past decades [29, 31]. One of themost common examples are octet-truss lattices. These regular, periodic structures are attractive formany industries due to the possibility of largely decoupling the effective stiffness and strength fromrelative density [5, 26, 32, 38]. One further advantage of the octet-truss lattices is the possibilityto relate their mechanical properties to the truss topology and geometry (see e.g. [5, 18, 26, 33]).Although this relation facilitates their design for specific applications, some geometrical constraintspush traditional manufacturing techniques of octet-truss lattices to their boundaries.Recent developments in additive manufacturing have provided a unique possibility to produce suchmetamaterials at very small scales. Yet, the design freedom comes at the cost of process complexity.The process-induced features, even defects, often occur in the produced structures, especially metallattices, thus altering the mechanical behavior of final parts [6, 8, 9, 22, 23]. Therefore, to achieve a re-liable prediction of the effective properties of these imperfect structures, as-manufactured geometriesshould be incorporated into computer-aided engineering (CAE). One of the common ways to acquirethe as-manufactured AM geometry is to perform a Computed Tomography (CT) scan [6, 8, 35]. Thescanned images provide extensive information about the microstructure of 3D printed componentsup to a scan resolution in the order of few microns. Thus, the CT-based analysis could lead to abetter prediction of the mechanical behavior of 3D printed structures.In the present work, we focus on the effective bending behavior of octet-truss lattices. The mostcommon numerical approaches for its prediction are three-dimensional (3D) Finite Element Analy-ses (FEA) and the application of one-dimensional (1D) beam theories. These techniques representthe two engineering extremes: one provides the most realistic solution, while the other delivers afast and quick approximation. 3D and 1D numerical analyses are commonly used in different ar-eas of engineering. Each of them faces major challenges when applied to additively manufacturedmetamaterials.To make CT images suitable for a traditional mesh-conforming three-dimensional analysis, ge-ometry reconstruction and mesh generation are required [21, 23, 34]. These steps tend to becomeespecially laborious when metamaterials are considered. Consequently, the numerical studies are of-ten conducted only on some specific regions of the lattices, for example on the periodic representativevolumes, or by modifying the idealized CAD models [4, 20–22]. To overcome these long and tedioussteps, a class of immersed domain methods has been developed. Immersed domain methods sepa-rate the geometrical representation from the applied discretization, thus, eliminating the necessityof geometry reconstruction and simplifying the mesh generation process. In the present work, theFinite Cell Method (FCM) is employed to perform numerical analysis directly on CT scan images ofas-manufactured octet-truss lattices [7, 28].Concerning 1D analyses, the conventional continuum beam theories are not necessarily applicableto the evaluation of the metamaterial or effective bending behavior. They strongly rely on the as-sumption of the separation of scales, i.e., the microstructural characteristic length should be muchsmaller than the size of the representative volume element. Nevertheless, it has been determined ex-perimentally and numerically that these components cannot be described by conventional continuummodels, such as e.g. Euler-Bernoulli or Timoshenko beam theories, when the size of the periodiccell approaches the typical wavelength of the variation of the macroscopic mechanical fields. Suchdeviations are normally referred to as size effects. These effects can arise at different scales. Whenlattice or foam-like structures are considered, size effects can occur at the scale of millimeters [27]. If2his scale is comparable to the component dimension, size effects are crucial for the evaluation of thepart behavior. In metamaterials, size effects become especially pronounced when the correspondingstructures are loaded in shear or bending [37]. As an example, when lattice beams are considered, therelative bending rigidity increases significantly when the size of the representative cell of the latticeapproaches the thickness of the beam structure. This occurs if the beam structure is composed ofvery few layers of lattice cells in the thickness direction [13, 14]. In such scenarios, the strain gradientextensions of the classical continuum models are proven to be accurate in predicting the mechanicalbehavior of size-dependent lattice structures. These beam theories are especially relevant when addi-tively manufactured lattices are analyzed as the produced scales are rather small. However, as theyrequire the effective Young’s and shear moduli as input parameters, to the knowledge of the authorsof this paper they have not been validated for the as-manufactured octet-truss lattices.With this in mind, we aim to demonstrate and experimentally validate the proposed CT-basednumerical framework which allows us to accurately evaluate the bending behavior of as-manufacturedoctet-truss lattice structures. To this end, the framework provides an efficient tool to compare the as-designed to as-manufactured properties under loading. Additionally, we investigate and validate theaccuracy of the classical and the strain gradient beam theories by comparing their bending propertiesto the direct 3D numerical analysis of the as-manufactured and as-designed octet-truss lattice beams.The present article is organized as follows. In section 2 we start with a brief description of theFinite Cell Method for the numerical analysis of as-manufactured AM lattices. Then, section 3recalls the fundamentals of the classical and the strain gradient beam theories. In section 4, detailson the manufacturing process and the experimental setup of the bending tests are given. Further, thenumerical findings are discussed in section 5. This section starts by comparing the as-designed to as-manufactured bending behavior and validating the results of the three-dimensional numerical analysis(see section 5.1). The next section 5.2 validates the strain gradient beam theories and provides adiscussion on their applicability to octet-truss lattices produced by additive manufacturing. Finally,our conclusions are drawn in section 6.
The main idea beneath the Finite Cell Method is illustrated in Fig. 1.Ω (a)
Physical domain Ω ΩΩ e \ Ω (b) Extended domain Ω e α ≈ α =1 (c) FCM mesh (in bold) and indicatorfunction
Fig. 1:
The idea of the Finite Cell Method.First, an arbitrary complex shape defined on a physical domain Ω is immersed in a simplifiedbox-like domain Ω e . Due to its simplicity, Ω e can be trivially discretized with a structured grid ofcuboids, further referred to as finite cells . These elements provide the support for shape functionswhich are chosen to be integrated Legendre polynomials of order p .Second, the original boundary value problem must be recovered on the actual, physical domain.To achieve such a result, an indicator function α ( x ) is introduced into the problem formulation. It isdefined to be equal to one on all points of the physical domain Ω and to a small positive value in the3omain Ω e \ Ω. Then, the modified linear elastic weak form of the problem can be written as follows:Find u i ( x j ) ∈ H u (Ω e ) satisfying (cid:90) Ω e α ( x i ) C ijkl ∂u k ∂x l ∂δv i ∂x j d Ω e + β D (cid:90) Γ D u i δv i d Γ D == (cid:90) Ω e α ( x i ) b i δv i d Ω e + (cid:90) Γ N ˆ t i δv i d Γ N + β D (cid:90) Γ D ˆ u i δv i d Γ D (1)with H u (Ω e ) being the first-order Sobolev space, ˆ u indicating a prescribed displacement on thedomain boundary Γ D , and ˆ t is prescribed traction on boundary Γ N . In the present work, Dirichletboundary conditions are enforced using the penalty method with the penalty parameter β D .As the geometries under consideration stem from CT images, the spatial scalar function α ( x )can be conveniently related to the acquired Hounsfield scale. Since the analyzed parts are metalliclattices, the contrast between material and void in the scan is commonly very high.Therefore, the threshold value of Hounsfield units HU thres used to identify the metal and voidregions in the CT scan images can directly be used to define the indicator function as follows: α ( x ) = (cid:40) HU ≥ HU thres ε, ε << HU < HU thres (2)Finally, as the indicator function makes the domain integrands in Eq. (1) discontinuous over theboundaries of the physical domain, a special integration rule should be applied. For this purpose,multiple techniques have been proposed (see e.g. [1, 17]). However, the most efficient integrationrule for CT-based geometrical models is a voxel-based pre-integration introduced in [36]. In thiscase, the shape of an object is fully described by a grid of voxels with a constant Hounsfield scale.Such an underlying structure allows to further decompose every finite cell into a number of voxels m x × m y × m z . Then, the standard ( p + 1) quadrature rule can be applied to every voxel resulting in m x ( p + 1) × m y ( p + 1) × m z ( p + 1) integration points for one finite cell. Using these integration points,the integrands in Eq. (1) can be efficiently pre-computed for every voxel in an offline phase. Then,the resulting matrices are scaled in an online stage with the indicator function α ( x ) as in Eq. (2).Thus, this integration method provides an accurate and efficient technique to accurately computethe discontinuous integrands for CT-based geometries.Although the Finite Cell Method in combination with a voxel-based pre-integration techniqueprovides a powerful tool to perform numerical analysis directly on CT images, the size of the computedsystems remains large. Large linear systems occur because as-manufactured structures include aconsiderable number of small-scale features, which are significant for the overall behavior of theparts. As an example, the largest CT scan considered further in this paper has a resolution of2096 × ×
128 voxels, while the smallest significant geometrical variations have a size of 3 − et al. in [11, 12]. Although one can expect that 3D numerical simulations on the as-manufactured AM lattices providean accurate and realistic solution of the complex mechanical behavior, often a fast prediction isimportant for an early analysis stage. One of the approaches to obtain a quick solution is to usebeam theories.When slender beams with a small thickness-to-length ratio are considered, an Euler-Bernoullimodel can be used to evaluate bending rigidity, while the Timoshenko beam theory is more appro-priate when shear effects are not negligible. Both Euler-Bernoulli and Timoshenko beam models4ely on the determination of the effective Young’s modulus E ∗ , the moment of inertia I ∗ , and theeffective shear modulus G ∗ for the latter model. These three quantities are not straightforward toobtain when lattice structures are considered. The two most common ways to determine them areto perform experiments or to use a first-order numerical homogenization. The former will be con-sidered in this paper for the determination of the as-manufactured effective Young’s modulus viaa tensile test, while the latter is used for the determination of as-designed effective quantities andas-manufactured effective shear modulus G ∗ . For a detailed description of the first-order CT-basedhomogenization employed in this article, interested readers are referred to [16]. However, when thesize effects in the material characterization of lattice structures under bending play an importantrole in the macroscopic response, the classical beam theories might deliver incorrect results and mustbe further enhanced, e.g., by means of high-order models such as the strain gradient beam theorydescribed in the following. In the present work, three-point bending of the AM lattice beams is investigated. The structuredeforms in the xz -plane (see the 2D sketch of the problem in Fig. 2). xz FL Fig. 2:
A 2D sketch of a three-point bending setup.The boundary conditions for this test: w ( x = 0) = 0 , M ( x = 0) = 0 , w (cid:48) (cid:18) x = L (cid:19) = 0 , Q (cid:18) x = L (cid:19) = F x − coordinate runs along the central (neutral) axis of the beam and x = 0 is the coordinateof a fixed support, w is the deflection of a central axis of the beam, F is the applied force at themiddle of the beam span with respect to which the beam problem is symmetric, M is the standardbending moment and Q is the shear force in the beam. Given the previously defined bending problem, the classical Euler-Bernoulli solution delivers themaximum deflection at x = L : w EB = F L E ∗ I (4)where L is the length of the beam, E ∗ is the effective Young’s modulus. In the present work, weperform the homogenization such that the beam becomes a solid block made of homogeneous materialwith E ∗ . Hence, I is defined as a standard moment of inertia of a cross-section having the outerdimensions of the original structure.To account for shear deformations for higher thickness-to-length ratios, the solution of classical5imoshenko beam theory for three-point bending can be formulated as follows: w T = F L E ∗ I + F L G ∗ A (5)where G ∗ is the effective shear modulus and A is the cross-sectional effective area. Then, the maincharacteristic of the bending behavior is the bending stiffness or bending rigidity. It defines theresistance of the specimens to bending deformations and is determined as follows: D = Fw (6)where F is the applied load and w the determined displacement.With the help of the classical beam theories solutions, this quantity can be determined analyti-cally when all other parameters are known. The classical Euler-Bernoulli bending rigidity for theconsidered problem can be written as follows: D EB = Fw EB = 48 E ∗ IL = 4 E ∗ bh L (7)where b is the depth and h is the thickness of the homogenized rectangular cross-section.Analogously, the classical bending rigidity using the Timoshenko beam theory is defined as: D T = Fw T = D EB E ∗ IG ∗ AL = D EB E ∗ G ∗ (cid:18) hL (cid:19) (8)Eq. (8) shows that for a fixed length L the bending rigidity D T approaches D EB when thicknessapproaches zero, whereas for constant thickness-to-length ratios the Timoshenko and Euler-Bernoullirigidities stay apart. In the scope of the present work, we also consider strain gradient beam theories elaborated in [13, 25].In the following, the derivation for the Euler-Bernoulli beam is described in greater detail. Thestrain energy density for a 3D body following Mindlin’s strain gradient elasticity theory of form II isformulated as follows [24]: W II = 12 C ijkl ε ij ε kl + 12 A mijnkl ∂ m ε ij ∂ n ε kl (9)where C ijkl and A mijnkl stand for the linear and high-order elasticity tensors, ε ij is the engineeringstrain tensor, and ∂ m ε ij and ∂ n ε kl denote the partial strain gradient. Following the assumption ofweak non-locality for isotropic materials [19], the high-order elasticity tensor can be further simplified: A mijnkl = g δ mn C ijkl (10)where g is an intrinsic length scale parameter affecting the macroscopic behavior and δ mn is theKroenecker delta. Parameter g can be interpreted as a high-order material parameter for a specificmicrostructure.Using the principal of virtual work, the variation of the internal energy takes the form δ (cid:90) Ω W II d Ω = (cid:90) Ω (cid:0) C ijkl ε kl δε ij + g δ mn C ijkl ∂ l ε kl ∂ k δε ij (cid:1) d Ω (11)where δ indicated the variation. 6hen, the dimensional reduction to the strain gradient Euler-Bernoulli beam theory is performed.The displacement components u = ( u x , u y , u z ) obey the same relationships as for the classical beamtheory: u x = − z ∂w ( x ) ∂x , u y = 0 , u z = w ( x ) (12)where x is the coordinate along the main axis of the beam, z is the direction perpendicular to it, and y is the out-of-plane coordinate, as depicted in Fig. 2. This leaves the transverse deflection w as theonly unknown.Furthermore, the only non-zeros stress and strain components are σ xx and ε xx . With this back-ground, the formulation of a generalized moment R ( x ) can be introduced: R ( x ) = (cid:90) A ∂σ xx ( x, y, z ) ∂z dA (13)where A = A ( x ) is the cross-sectional area of the beam.Then, the variation of the internal energy in Eq. (11) with Eq. (12) and Eq. (13) simplifies to the1D energy expression over the main axis of the beam: δ (cid:90) Ω W II d Ω = (cid:90) L (cid:0) M + g R (cid:1) ∂ ( δw ) ∂x dx + (cid:90) L g ∂M∂x ∂ ( δw ) ∂x (14)Applying the Hamilton’s principle the strong formulation of the one-parameter strain gradient Euler-Bernoulli elasticity model can then be formulated:( M + g R − ( g M (cid:48) ) (cid:48) ) (cid:48)(cid:48) = f ∀ x ∈ (0 , L ) (15)where f is the externally applied force, g is an unknown high-order material parameter, and thehigh-order term (( g M (cid:48) ) (cid:48) ) (cid:48)(cid:48) is responsible for the description of boundary layer effects. As the macro-scopic behavior of the beam is of interest, the strong form of the governing equation for constanthomogenized parameters can be further simplified:( E ∗ I + E ∗ Ag ) w (cid:48)(cid:48)(cid:48)(cid:48) = f ∀ x ∈ (0 , L ) (16)The analytical solution of Eq. (16) under the absence of body load with the boundary conditionsdescribed in Eq. (3) takes the form: w EBgr = F L (cid:0) E ∗ I + E ∗ Ag (cid:1) (17)Eq. (17) compared to the solution of the classical Euler-Bernoulli theory in Eq. (4) introduces theintrinsic length scale parameter g which acts as a high-order material parameter depending on themicrostructure of the unit cell. This parameter characterizes the size-dependent beam behavior whenthe thinnest beams show a stiffening effect.The solution of the strain gradient Timoshenko beam theory can be derived in a similar mannertaking into account the respective assumptions: w Tgr = F L (cid:0) E ∗ I + E ∗ Ag (cid:1) + F L G ∗ A (18)Eq. (18) is also similar to the solution of the classical Timoshenko theory except for the presenceof the intrinsic material parameter g . The bending rigidities (with rectangular cross sections A = bh )7orresponding to these deflections can be shown to follow, respectively, the formulae: D EBgr = D EB (cid:18) (cid:16) gh (cid:17) (cid:19) D Tgr = D T (cid:18) (cid:16) gh (cid:17) (cid:19) (19)revealing the size effect for decreasing values of h with a fixed value of g .To sum up, both the classical and the strain-gradient theories could provide a quick estimate ofthe bending behavior of the considered beam-like lattice structures. In the following, the predictionsprovided by these theories will be compared to the full 3D numerical and experimental analysisperformed on the AM octet-truss beams. Furthermore, their accuracy and applicability will beevaluated with the help of experimental three-point bending tests. The experimental and numerical investigations are held on octet-truss lattices. A representative unitcell of such structures is depicted in Fig. 3. As the main focus of the present work is the investigationof lattice bending behavior, an octet-truss unit cell indicated in Fig. 3 is used to construct the fourbeam-like structures shown in Fig. 4. These beams have the same length of 128 mm (32 cells) and thesame width of 8 mm (2 cells) but different heights (thicknesses): 4, 8, 12, and 16 mm, respectively(1 , ,
3, and 4 unit cells). Thus, the constructed thickness-to-length ratios are 0 .
03, 0 .
06, 0 .
09, and0 .
13 respectively.
Fig. 3:
CAD model of the octet-truss unit cell [15].8 a) Beam 2 × × × × × × × × Fig. 4:
Investigated CAD models of the octet-truss beam structures.The specimens for experimental testing were printed in the laboratory 3DMetal@UniPV using aselective laser melting metal 3D printer Renishaw AM400. For the production of the specimens,stainless steel powder SS 316L-0407 was used. According to the material data sheet of the pro-ducer [30], the considered setup leads to a bulk material with Young’s modulus 190 GPa ± ◦ C in the chamberNabertherm LH120/12 for 2 hours are shown in Fig. 5.
Fig. 5:
Printed specimens after heat treatment.Prior to performing any experimental test, the four bending specimens were subjected to a com-puted tomography to acquire the as-manufactured geometries. The CT scans were performed witha Phoenix V CT scanner with a resolution of 61 µ m.Then, to validate the numerical frameworks proposed in sections 2 and 3, three main quantitieswere determined experimentally. These are the porosity of the printed lattices structures, the effectiveYoung’s modulus, and the bending rigidity. Porosity of the printed structure
The overall porosity of the lattice structures is measured for two reasons. The first motivation isto compare the experimentally determined porosity value to the as-designed CAD-based ones, thus,providing the first estimate on the geometrical variations of the as-manufactured geometries withrespect to the original CAD models shown in Fig. 4. The second reason is to experimentally verifythe porosity values determined from the acquired CT scan of every beam. The porosity values are9etermined by evaluating the mass of the specimen m. Then, considering the printed density ρ indicated in [30] the overall porosity can be calculated as: φ = 1 − mρV (20)where V is the measured volume of the bounding box of the specimen. Together with the mea-sured porosity values, the measurement uncertainty is computed based on the accuracy of the usedinstrumentation. Effective Young’s modulus
The second quantity of interest is the effective Young’s modulus of the octet-truss lattice. This valueis important for the investigation of the applicability of the beam models as described in section 3.The as-manufactured effective Young’s modulus E ∗ is determined via a tensile test of the samplelattice specimens. The experiment is performed in the material mechanics laboratory with the helpof the MTS Insight System. For the elongation measurements, a video extensometer is used (seeexperimental setup in Fig. 6). The effective Young’s Modulus is then computed according to ASTME111 standard [2]. The determined value is E ∗ = 12 533 ±
751 MPa together with the correspondingmeasurement error.
Fig. 6:
Experimental setup of a tensile experiment on an octet-truss lattice structure [15].
Bending rigidity
The final experimentally determined value is the bending rigidity of the octet-truss lattice beamsas defined in Eq. (6). This quantity describes the characteristic overall (global) resistance of thestructure against the bending deformation. The values of bending rigidity of the four 3D printedstructures of Fig. 5 is experimentally measured by a three-point bending test under quasi-staticconditions and displacement-controlled velocity (see Fig. 7). The span ( L ) between the supports is120 mm, while the applied point load ( F ) is transferred in the middle of the span of the beam. Duringthe experiment, the imposed displacement and the corresponding force are recorded. The bendingrigidities of the beams are then computed by using Eq. (6). All tests are performed in both elasticand plastic regime. However, for the aim of this work only the elastic characteristics are considered.Experimental results will be discussed together with the numerical values in the following sections.10 a) Beam 2 × × × × × × × × Fig. 7:
Bending of beam specimens.
In this section, the results of the numerical investigations on the octet-truss lattices are discussed indetail.First, the behavior of the octet-truss lattice structures undergoing a bending load case is analyzednumerically in section 5.1. In this section, the as-manufactured and as-designed octet-truss beamsare compared geometrically and the differences are quantified by means of the macroscopic porositydefined in Eq. (20). Then, the direct numerical simulation of the three-point bending test is performedon both CAD and CT geometries. The achieved numerical results are finally compared to theexperimental values.Second, in section 5.2, the applicability of the beam theories described in section 3 is investigated.Both, the classical and the strain-gradient Euler-Bernoulli and Timoshenko beam theories are appliedto analyze the behavior of both as-designed and as-manufactured octet-truss lattice beams.
Geometrical comparison
To highlight the macroscopic differences between the as-manufactured geometry extracted from CTscan images and the as-designed geometric model, zoomed views on both geometries are depictedin Fig. 8. From a thorough comparison of the two geometric models (see Fig. 9), the followinggeometrical features of as-manufactured geometry can be observed compared to the as-designedones: • larger truss thickness; 11 partially melted material powder particles in overhanging surfaces opposite to the build direc-tion; • excess material collection in the nodes.These features are well-known side effects of the SLM printing process. It is also established inliterature [3, 6, 15], that these geometrical features have a strong influence also on the numericalresults, and thus as-designed models lead to a quite inaccurate prediction of the mechanical behaviorof lattice structures. Fig. 8:
Zoom on the geometrical features of the as-manufactured (left) and as-designed (right) bend-ing specimen (build direction marked with the black arrow).12 ig. 9:
Comparison of as-manufactured and as-built octet-truss bending specimen 2 × × × × × × ± × × ± × × ± × × ± Tab. 1:
Porosity comparison of the beam specimens.
Direct numerical simulations of three-point bending test
In order to further support the above observations, we carry out a numerical simulation of thethree-point bending test described in section 4. Numerical experiments are performed for each oneof the four specimens on both as-designed (CAD) and as-manufactured (CT) geometrical models.In both cases, the same boundary conditions and load cases are applied as in the experimentalsetup. The simulation of the as-designed geometry is carried out by using Comsol ™ with quadratictetrahedral Finite Elements, whereas as-manufactured geometry is simulated using the high-orderFinite Cell Method as described in section 2 with finite cells of polynomial degree p = 3 containing2 × × × × (a) Overview of the Finite Cell mesh with a representativedisplacement distribution (b)
Zoom on the Finite Cells in the cornerof the beam (c)
Zoom on one periodic cell with FiniteCells (in black) (d)
Zoom on one periodic cell with FiniteCells (in black) and voxels (in blue)
Fig. 10:
Finite Cell mesh with 51 × ×
32 cells for 2 × ×
32 beam specimen.14 a) Displacement field in mm in the load direction (b)
Von Mises stress in MPa Distribution
Fig. 11:
Displacement and von Mises stress distributions for as-printed beam 2 × ×
32 utilizingthe Finite Cell Method.The numerical bending rigidities are, then, computed by using Eq. (6) and their values are com-pared to the experimental ones in Fig. 12.The qualitative comparison of these results shows that the as-designed and as-manufactured ge-ometries follow the same tendency of a higher rigidity value for thicker beams. Nevertheless, quan-titatively the relative errors in the bending rigidity value are always above 40%. This gap is largelydriven by the geometrical difference between the as-manufactured and as-designed geometries. Asthe CT-based and experimental porosity values shown in Tab. 1 are lower than the designed ones,the as-designed bending rigidity should agree with this trend. According to the results in Fig. 12the as-manufactured bending rigidity is larger than the designed one, thus, supporting the describedtendency. Furthermore, the numerical simulation on the printed geometry via computed tomographyprovides an excellent agreement with the experimental tests, with a relative error always below 4%. × × × × × × × × . . . B e nd i n g r i g i d i t y D ,[ N mm ] Experimental values Numerical as-built bending test Numerical as-designed bending test
Fig. 12:
Comparison of bending rigidity obtained by numerical bending tests on the original as-designed geometry and on the as-manufactured geometry obtained from CT-scan data.15 .2 Experimental validation of strain gradient beam theory for octet-truss lattices
Since in a three-point bending it is often desired to predict the mechanical behavior by dimensionallyreduced beam models, we investigate more carefully the applicability of the beam models describedin section 3 to octet-truss lattice structures.The beam models rely on the identification of effective quantities, such as Young’s modulus E ∗ and shear modulus G ∗ . As briefly mentioned in section 3, there are two ways to obtain the neces-sary quantities. For the as-designed geometries, only the first-order homogenization can be applied,as there is no possibility to perform experimental tests on it, while for the as-manufactured struc-tures, the effective Young’s and shear modulus can be measured experimentally. In the scope of thiswork, only the as-manufactured Young’s modulus of octet-truss lattices is experimentally evaluated,whereas the effective as-manufactured shear modulus is determined by means of the first-order ho-mogenization technique mentioned in section 3. Tab. 2 summarizes the effective quantities used inthe following. Effective quantity As-designed As-manufactured E ∗ , MPa 7 356 12 533 ± † G ∗ , MPa 2 742 5 651 Tab. 2:
Effective mechanical quantities of the octet-truss specimens. † Experimental measure
Fig. 13 shows the normalized bending rigidity
D/D EB with respect to the beam height h (see Eq. (7)).The normalization is performed with respect to the Euler-Bernoulli bending rigidity D EB solutionas follows: DD EB = Dw EB F = w EB w (21)where w EB is the classical Euler-Bernoulli solution for three-point bending as in Eq. (4), w is theexperimentally recorded maximum deflection, and D is the compared bending rigidity.As the as-manufactured and as-designed geometries have different effective properties, the bendingrigidities are normalized with the Euler-Bernoulli solutions using the respective quantities from Tab. 2and, thus, they are plotted separately in Fig. 13a and 13b.16
10 15 200.91.01.11.21.31.41.5 Height h, [mm] N o r m a li ze db e nd i n g r i g i d i t y D / D E B ,[ - ] Experimental bending testNumerical as-manufactured bending testStrain gradient Euler-Bernoulli beam theoryStrain gradient Timoshenko beam theoryClassical Timoshenko beam theoryClassical Euler-Bernoulli beam theory (a)
Experimental and CT-based values N o r m a li ze db e nd i n g r i g i d i t y D / D E B ,[ - ] Numerical as-designed bending testStrain gradient Euler-Bernoulli beam theoryStrain gradient Timoshenko beam theoryClassical Timoshenko beam theoryClassical Euler-Bernoulli beam theory (b)
CAD-based values
Fig. 13:
Normalized bending rigidities of the octet-truss lattice beams with respect to the beamheight.In both plots of Fig. 13, the dashed lines indicate the results predicted by the classical beamtheories, while the solid lines stand - the strain-gradient beam theories. The blue dots correspond tothe experimental bending rigidity, whereas the crosses indicate the results of the numerical bendingsimulation computed on the as-manufactured specimen from Fig. 12. Both values are normalizedwith the analytical Euler-Bernoulli solution using the as-manufactured effective Young’s modulusfrom Tab. 2. The brown dots in Fig. 13b indicate the CAD-based results of the numerical bendingtest and again the results are normalized with the Euler-Bernoulli solution with the as-designedeffective Young’s modulus from Tab. 2. Since as-designed geometry allows for further reduction ofthe considered thickness-to-length ratios, an extra point is added at the height of 2 . . Classical beam theory using as-manufactured and as-designed geometry
As the normalization is performed with respect to the corresponding classical Euler-Bernoullisolution, the dashed black lines remain at the value 1 for both as-manufactured and as-designedgeometries. If the octet-truss lattice beams were to follow this behavior, all bending rigidities wouldlay on a straight line. However, neither as-manufactured nor as-designed values seem to comply withthe assumptions of the Euler-Bernoulli theory. Thus, the classical Euler-Bernoulli theory cannot beapplied to the characterization of the bending behavior of the considered octet-truss lattices.The classical Timoshenko beam theory indicated with the green dashed line converges to the Euler-Bernoulli theory with the decreasing beam height. These states correspond to extremely slenderbeams, thus, making shear effects of minor importance. The as-manufactured geometry results asshown in Fig. 13a propose that only the thickest specimen with 2 × × .
13 follows the Timoshenko theory. However, the rest of the points do not followthis curve. The as-designed bending behavior as depicted in Fig. 13b shows a similar trend, wherefor the thickest specimens the points lay on the curve. Although the Timoshenko beam theory seemsto provide a better solution compared to Euler-Bernoulli, none of them can capture the observed17ending behavior well.
Strain gradient beam theory using as-manufactured geometry
Fig. 13a indicates the presence of a stiffening effect. When the height of the beam is close to thecharacteristic size of the unit cell, the size effects affect the macroscopic bending behavior of thecomponents and cause stiffer behavior in comparison to a standard prediction of the classical beamtheories. This size-dependent bending phenomenon is precisely captured by the strain gradient beamtheories on the as-manufactured geometries.The strain gradient beam theories as described in section 3 introduce an additional material param-eter g . This high-order parameter is unknown a priori and can only be determined by a calibration ofthe solid lines to the obtained numerical and experimental solutions (or by other generalized homoge-nization procedures [10]). As mentioned in [13], this intrinsic length parameter behaves as a materialparameter and it is independent of loading, problem type, or the beam model. This quantity onlydepends on the underlying geometry. Thus, it must be the same for both strain gradient Timoshenkoand Euler-Bernoulli theories. The value of the high-order material parameter g is determined as0 .
349 [mm] for the as-manufactured octet-truss lattice (see Tab. 3). This intrinsic length parametercharacterizes the size effects in the octet-truss lattice structures via both Euler-Bernoulli and Tim-oshenko strain gradient beam theories. Its order is close to the smallest strut size diameter of theunit cell of 0 . .
13 seems to be away from it. This can suggest that for the last configuration the strain-gradientTimoshenko theory is more appropriate. However, the measurement error bars on the experimentaldata indicate that both theories could be applicable for this setup and the last point can as well layon the black solid line. Furthermore, the CT-based porosity value for the thickest beam is furtheraway from the experimental one. Thus, it could lead to uncertainty in the computed bending rigidity.To further clarify this let us look at the as-designed results.As-designed g , [mm] As-manufactured g , [mm]Octet-truss beam 0.244 0.387 Tab. 3:
Comparison of as-designed and as-manufactured high-order intrinsic length parameter ofthe octet-truss specimen.
Strain gradient beam theory using as-designed geometry
As already pointed out, the effective quantities obtained on the as-designed model are far from theexperimentally determined bending rigidity and are depicted separately in Fig. 13b.Curiously, for the as-designed geometry, a weaker stiffening effect is observed. For the thickness-to-length ratio of 0 .
03 (i.e., for the thinnest beam), the CAD-based results show about 8.4% stiffeningcompared to the thickest observation, while the as-printed analysis indicated 9.5%.This is also reflected in the intrinsic high-order material parameter g . It is determined as g = 0 . h < omparison between as-manufactured and as-design results All in all, the overall stiffening tendency is similar to the one observed from the experimental andas-manufactured numerical analysis. But the as-manufactured values are about 50% higher thanthe designed ones as shown in Fig. 12. The as-manufactured computations always lie within theuncertainty range of the experimental measurements, whereas as-designed numerical results neverfall in this range. This rather large difference has been observed in similar studies conducted by thesame authors on tensile behaviors of octet-truss lattices [15].Moreover, when a closer study on the as-manufactured and as-designed geometries is undertaken,the stiffening trend differs. Firstly, we have observed that the considered octet-truss beams experiencesize effects, such that classical beam theories are not applicable to approximate the bending behavior,whereas strain gradient beam theories provide a much more accurate description. Secondly, the as-manufactured bending rigidities show a stronger stiffening effect than the designed ones, as alsoreflected in the intrinsic material parameter determined for both geometries. This observation wellcorrelates to all other material characteristics determined by the authors.
The numerical analysis of additively manufactured metamaterials can be prohibitively expensive andoften impossible at full scale. In the present work, we have shown and validated an efficient numericalframework to incorporate complex as-manufactured geometries in a direct image-to-analysis workflow.The achieved numerical results are fully supported by the experimental tests performed on the octet-truss lattices. These findings suggest that in both direct numerical simulations and beam theoriesthere is a strong need to incorporate as-manufactured geometries into the numerical analysis of AMproducts. In particular, the direct numerical simulation of CT-based as-manufactured geometriesdelivers results very close to the experimental measurements, whereas numerical analysis computedon the as-designed model fails to correctly predict the mechanical behavior of these metamaterials,presenting relative errors in the bending rigidity value always above 40%.Furthermore, we have demonstrated the applicability of classical and strain gradient beam theoriesto the prediction of the bending behavior of AM octet-truss lattices. This work has confirmed thatsize effects arise in these metamaterials, thus raising the importance of the high-order continuumtheories. Additionally, we validated the strain gradient beam theories in combination with the FiniteCell Method. In particular, a high-order intrinsic material parameter was determined directly fromthe numerical analysis of the as-manufactured geometries. As this material parameter is independentof the problem type, it can be used for the dimensionally reduced modeling of such octet-truss latticecomponents under different loadings and boundary conditions.To conclude, the proposed numerical framework provides an accurate and flexible tool to analyzethe behavior of as-manufactured metamaterials. Furthermore, these results represent an excellentinitial step toward the validation of the strain gradient continuum theories in the field of additivemanufacturing. In this line of research, we intend to incorporate the demonstrated technique intothe analysis of the statistically similar CT models of such mechanical metamaterials in the future.This step would allow expanding the capabilities of the proposed image-to-material-characterizationworkflow.
Acknowledgements
Data Availability
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