Experimental and numerical investigation on compressive fatigue strength of lattice structures of AlSi7Mg manufactured by SLM

International Journal of Fatigue 128 (2019) 105181

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Experimental and numerical investigation on compressive fatigue strength of lattice structures of AlSi7Mg manufactured by SLM L. Boniotti, S. Beretta, L. Patriarca, L. Rigoni, S. Foletti

T

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Politecnico di Milano, Department of Mechanical Engineering, Via La Masa 1, I-20156 Milan, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: Additive manufacturing Lattice structures Fatigue strength Ratcheting

Micro-lattice materials represent nowadays a great opportunity for developing new ultra-lightweight materials. Design flexibility and multi-functional properties make them attractive for several applications in automotive, medical, space, aerospace and process industries. Predict fatigue resistance of these micro-structures is a key issue. In this work three different unit cells printed by a Selective Laser Melting process with an AlSi7Mg powder were analysed. Static properties and fatigue strength in compression have been studied by means of different experimental and numerical techniques: experimental tests, Digital Image Correlation, Micro-Computed Tomography and Finite Element analysis. Total fatigue life has been divided in three stages: ratcheting, damage initiation and propagation in one or few struts leading to final failure. The experimental results show that the initial ratcheting strain rate is significantly correlated to the life of each sample. The damage maps show that, even at fatigue limit, crack propagation in one or few struts is expected to occur. The effect of defects and local irregularities on the fatigue strength has been studied by means of finite elements models based on the asmanufactured geometry of the samples and on the definition of an equivalent stress amplitude based on a multiaxial high cycle fatigue criterion. A correlation between broken struts and maximum values of local equivalent fatigue stress amplitude has been demonstrated by post-test tomographies.

1. Introduction Nowadays, material engineering is focused on the new topic of miniaturize space-filling system by means of new materials considering the importance of environmental responsibility, smartness and multifunctionality [1]. In this scenario, micro-lattice materials are a great opportunity because of Additive Manufacturing (AM) which enables us to build new materials and to fabricate complex geometric shapes. Indeed, by means of layer-upon-layer techniques, such as Selective Laser Melting (SLM) processes, it is now possible to print materials with lightweight optimization, design flexibility. Micro-lattice materials represent a prime example of this concept. Microlattices can be described as a periodical arrangement of a regular cellular structure, named unit cell, that is repeated to create a continuum as it happens in natural materials like bones, corals and trees [2]. Lattice materials are made of micro-struts arranged together in such a manner as to results in a volume fraction much lower than the air voids and in a relative density ρ∗, defined as the ratio between the density of the lattice and the density of a traditional solid material with the same total volume, that is lower than 1 [3,4]. Accordingly, microlattice materials are made very attractive for many applications that

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requires good stiffness to weight ratio that can also take advantage of their multi-functional properties [5]. Their possibility of maximising surface areas is useful in medical industry for the characterization and production of scaffolds optimized for tissue and bone replacement in order to encourage osseointegration [6,7]. The capability to conduct heat, to propagate energy and to distribute an impact shock make lattice material suitable for automotive, aerospace and process industries [8–11]. From the space to the medical application, it is clearly important to know the mechanical behaviour and the mechanical properties of this micro-structure, both for the production of both non-critical and structural and primary components. Several applications, such as medical implants or space hardwares, have to satisfy specific requirements such as fatigue resistance in different loading conditions [12]. In the literature there are many studies about the fatigue resistance of SLM parts and it was demonstrated that there is a large scatter between these fatigue results and those of traditional materials [13]. This is due to the presence of defects and irregularities related to the AM process which make the 3D printed parts weaker than the parts made of wrought materials [14,15], while resistance could be potentially better than traditional processes if AM defects were not present [16–18].

Corresponding author. E-mail address: [email protected] (S. Foletti).

https://doi.org/10.1016/j.ijfatigue.2019.06.041 Received 3 December 2018; Received in revised form 24 June 2019; Accepted 27 June 2019 Available online 05 July 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 128 (2019) 105181

L. Boniotti, et al.

Nomenclature

E Ni Nf Kc Ki Ks R2 Rm Rp02 s0 s−1 ΔF Δ ∊eff ∊a ∊m ∊̇m σa, eq σa∗, eq σlogN σI , σII , σIII σm∗

Abbreviations AM BCC CV DIC FCC LVDT SC SC-BCC SC8-FCC SLM μCT

Additive Manufacturing Body-Centered Cubic elementary lattice Coefficient of Variation Digital Image Correlation Face-Centered Cubic elementary lattice Linear Variable Displacement Transducer Simple Cubic elementary lattice Combined SC, BCC lattice structure Combined SC, FCC lattice structure Selective Laser Melting Micro-Computed Tomography

Symbols a ratio between the stiffness of a 4 × 4 × 12 and a 4 × 4 × 4 specimen A% elongation at failure

Elastic Modulus number of cycle at damage initiation number of cycle at failure stiffness of the first and the last row of cubic cells stiffness of a single inner row of cubic cells total stiffness of the specimen coefficient of determination in linear fitting ultimate tensile strength yield strength fatigue strength at 107 cycles at R = 0 fatigue strength at 107 cycles at R = −1 applied force range effective strain range strain amplitude mean strain ratcheting strain rate equivalent stress amplitude with mean stress correction equivalent stress amplitude scatter in fatigue life principal stresses equivalent mean stress

structures, [30–34]. Fatigue tests were complemented by CT-scans taken before and after cyclic tests, allowing us to describe the failure modes of the different cells. Finally, fatigue strength of lattice structures was studied by means of a multiaxial high cycle fatigue criterion starting from the stress state obtained with a finite element analysis (FEA) of as-manufactured geometries extracted from the pre-test tomographies. FEA and experimental results were compared in terms of failure location and fatigue strength showing good agreement.

In particular, in the case of micro-lattices by means of the MicroComputed Tomography ( μCT ) it is possible to evaluate how much the as − manufactured geometry of the printed specimens differs from the as − designed one. Moreover, by analysing these micro-lattices with an optical microscope, it is possible to evaluate the microstructure of the material and evaluate how it is affected by printing parameters and by the printing orientation in respect of the building platform, leading to surface defects [19–21]. Several works analysed these geometrical defects[22–25] and the surface roughness [26] that profoundly affects the mechanical properties of lattice materials. Indeed, it is possible to observe that the failures of micro-lattices start from the geometrical irregularities on the struts surface or adjacent to the strut junctions and not from internal pores [20,27]. As for fatigue properties of micro-lattice structures, most of the results in the literature refer to bending dominated cells, that are widely considered for applications in bio-mechanics. On the other hand, even if not so widely investigated from the point of view of fatigue, stretch dominated lattices have the best static properties for a given relative density [22,28,29]. In terms of mechanisms, three stage of the fatigue life were found considering the average deformation and the deformation amplitude during cyclic test: ratcheting (cumulative deformation), crack initiation (or damage of a few struts) and crack coalescence (damage diffusion over the cellular structure) [30–34]. In terms of properties, some papers [35,36] introduced S-N diagrams normalized compared to the static properties with fatigue limits that depend on the cell type. A similar concept was also adopted in [37,38], with an interpretation of the fatigue properties in terms of the local stresses at strut intersections calculated by beam’s theory (although the stress concentrations in diamond cells, considered in those papers, are not very high). In this work, the fatigue properties under compressive loads of three different micro-lattice structures printed from AlSi7Mg powder by SLM process were analysed by means of a local fatigue approach. The cells are two stretch dominated cells, namely a body-centered and a face-centered cubic structural units, and a bending dominated cell (a cubic structure rotated at 45°). Monotonic tests in compression on lattice structures were conducted for evaluating the non-uniform stress/strain distribution in the specimen height resulting from a non-perfect contact between the compression plates of the testing machine and the specimen. The fatigue behaviour of micro-lattice was thoroughly characterised in compression by analysing the deformation behaviour during the fatigue tests adopting a method proposed for metal foams and cellular

2. Experimental tests – Bulk material properties The mechanical properties of micro-lattice structures are affected by several factors, such as the printing parameters, the properties of the used aluminum alloy powder, and the surface roughness. In order to study the mechanical properties of the bulk material in the same printing condition, several monotonic tensile tests and three point bending fatigue tests have been performed. The specimens were printed with the same printing parameters used for the lattice micro-structures. 2.1. Tensile monotonic tests Three monotonic tensile tests were carried out on standard specimens in order to obtain the monotonic mechanical properties. The test were conducted following the ISO-6892 standard on round specimen with a diameter of 6 mm, a length of reduced section of 30 mm and a gage length of 25 mm. The test piece was obtained by machining a bar printed using the same set of parameters used for producing the lattice specimens. The mean value of mechanical properties is reported in Table 1. In order to study the material behaviour in a condition as close as possible to the printed material in the lattice structures, additional tensile monotonic tests were carried out on dog-bone specimens with a rectangular cross section with a width of 3 mm, a thickness t of 0.6 mm, 0.8 mm and 1.0 mm and a parallel gauge length of 12 mm. The specimens were tested in an as-manufactured condition in order to study the effect of the surface roughness on the monotonic tensile properties. The tests were conducted in displacement control on a Deben Microtest machine with a crosshead speed of 0.3 mm/min. The longitudinal strain was measured by means of an axial estensometer with a gauge length of 3 mm. The experimental results are reported in Table 1. About dog bone specimen, the main variability is on the elongation at failure with a coefficient of variation CV = 0.24 . The elastic modulus, the yield strength and the ultimate tensile strength appear to be less affected by a 2

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periodic structure, eight cubic cells were needed to be repeated in the structure as a unit cell. The third lattice-structure was based on a simple SC unit cell made of 12 struts (one for each edge of the cube) which was rotated by 45° compared to the horizontal plane. For all three structures tested, the cube constituting the base unit cell had a side length of l = 2.4 mm and a strut diameter of ϕ = 0.6 mm. The slenderness λ of the cell, defined as the ratio l/ ϕ , was equal to 4. The unit cells have different as-designed relative density ρ∗ that are: ∗ ∗ ∗ ρSC − BCC = 36%, ρSC 8 − FCC = 28% and ρSC = 11.5% . It is somewhat important to remark that according to Maxwell’s criterion the three cells are very different. A generalisation of the Maxwell rule in three dimensional space, [42], can be used to discriminate stretching and bending dominated structures and it is based on the coefficient M, that is defined as M = b − 3j + 6 , where b is the number of the struts and j is the number of joints in the structure. The SC-BCC has M > 0 (hyperstatic structure), the SC8-FCC has M = 0 (isostatic truss), while the SC has M < 0 (mechanism, bending dominated) [28].

Table 1 Tensile monotonic properties of bulk material. Spec. ID

t [mm]

E [GPa]

Rp02

Rm [MPa]

A% [%]

[MPa] M1.3.1 M2.3.1 N2.3.08 M2.5.08 M2.3.06

1.0 1.0 0.8 0.8 0.6

63.1 64.7 54.1 59.3 58.1

136 163 151 141 144

345 343 357 354 352

5.4 5.0 6.6 3.3 4.9

Mean – μ Standard deviation – σ Coefficient of variation – CV = μ/ σ

59.9 4.2 0.07

147 10.5 0.07

350 6.0 0.02

5.0 1.2 0.24

Standard specimens (mean value)

66.9

241

378

6.6

specimen thickness in a range from 0.6 to 1 mm. Conversely, the mechanical properties of thin specimens are different from those of standard specimens, as regards the yield strength. Due to this high difference and considering the size of the struts of the lattice structures, the mechanical properties of small specimens will be used for finite element analyses and for predicting fatigue strength.

3.2. Specimens shape and dimension For the lattice structures based on SC-BCC and SC8-FCC unit cell, two different specimen geometries were considered: smaller cubic specimens (4 × 4 × 4 specimens) with 4 cubic unit cells on each side for a total of 64 cells (Fig. 2a and 2b) and higher specimens (4 × 4 × 12 specimens) with a total of 192 cells, 4 on each side of the cross section and 12 at the height (Fig. 2c and 2d). The SC specimens had 4 cubic cells on one side of the cross section, 5 cubic cells on the other side and 10 unit cells at the height of the specimen (Fig. 2e). At the top and bottom of each specimen, a thin skin of solid material (with a thickness of 0.6 mm) was designed to have a better alignment between the compression plates and the specimens and to uniformly distribute the load. A Renishaw AM250 SLM system was used for the production of the samples by suppling a gas atomised AlSi7Mg spherical powder with a particle size ranging from 20 μm to 63 μm . The SLM printer is characterised by a single mode fiber laser. The maximum laser power was set at 200 W with a point distance equal to 75 μm . The exposure time varied from 140 μs (for borders, hatches and up skin) to 50 μs (for fill contours, down skin borders, down skin fill contours and down skin hatch). The applied layer thickness was equal to 25 μm and a meander scanning strategy with an increment value of rotation angle of 67 degrees was used to print the specimens by means of partially overlapped melting spots: the hatch distance was 100 μm and the hatch offset in the fill pattern was 30 μm . Meander always moved towards the gas flow and the pattern repeated every 180 layers. The specimens were rotated for the printing process in order to minimize for all the struts the printing

2.2. Three point bending fatigue tests Three-point bending fatigue tests were performed to measure the fatigue strength at N = 107 cycles for the bulk material in the as-built condition. The specimens geometry, shown in Fig. 1, had a length of 60 mm, height of 10 mm and width of 6 mm. The specimen had a radius of 20 mm with a minimum resistant section of 6 × 6 [mm]. Specimens were printed in horizontal direction (layer deposition along the height) with the curvature directed upwards: printing parameters were the same as lattice samples (see Section 3.2). After 3D printing, the upper and the lower surfaces of the specimens were machined in order to ensure parallelism of the surfaces, whereas the central radius surface was kept as-built in order to obtain fatigue properties with a surface condition similar to that of micro-lattice structures. The tests were conducted in load control with a stress ratio of R = 0.1 at a frequency of 50 Hz under an Instron E10000 testing machine with a maximum load capacity of 10 kN. The run-out condition was set to 107 cycles. Failures occurred in slightly different positions along the curved region of the specimen: a FEM analysis of the test setup allowed us to determine the effective local maximum stress at any failure location. The effective stress was then calculated averaging the stress over a distance of 75 μm (see discussion in Section 4.2). The experimental test results are shown in Fig. 1: they have been interpolated applying the least squares analysis as suggested by ASTM-E739, [39]. The mean value of fatigue strength at 107 cycles was calculated as s0 = 53 MPa with a scatter (determined from the linear part of the diagram [40]) of 8.7%. 3. Experimental tests – Micro-lattice structures 3.1. Cell types Three different periodic truss structures were considered. The selected lattice-structures were designed by combining elementary cubic truss lattices including the Simple Cubic (SC), Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) lattices. The first lattice structure was a combined SC-BCC lattice, resulting in a cubic cell with 16 struts (the edges and the space diagonals of the cube [41]. The second one was a combined SC8-FCC, that is essentially a 3D-Warren truss formed by combination of octet-truss and cubic truss [1,41]). To obtain a

Fig. 1. Three point bending tests results on bulk material. 3

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Fig. 2. Micro-lattice specimens shape and geometry.

3.3. Monotonic tests in compression

angle compared to the vertical direction for the sake of achieving a better quality. In particular, SC-BCC and SC8-FCC specimens were rotated by 22.5° compared to the vertical direction and to one of the other two directions. The SC specimens were rotated by 45°. Optical microscopy analyses were performed to examine the microstructure of lattice specimen after polishing to 1 μ m grit size and etching. Fig. 3 shows an optical micrograph of a SC-BCC sample. At the strut intersection it is possible to note the melt pools and the heat affected zone. Especially for horizontal struts, the influence of cooling rate on grains dimensions and porosity is evident. Due to build orientation, the horizontal struts are more in contact with the powder bed resulting in a slower cooling rate. In the down skin the level of porosity and surface roughness is higher. The microstructure becomes finer as the distance from the powder bed increases. A high value of surface roughness, i.e. high value of stress concentration at geometrical irregularities can be detrimental to fatigue strength [24,25].

Several static compression tests were run on the Instron E10000 electrodynamic testing machine. The monotonic tests in compression were performed in displacement control, with a rate of 0.5 mm/min. During the test, loading/unloading ramps were performed at different load levels to calculate the stiffness of the specimens. In order to measure the displacement during the tests two laser transducers were applied on two opposite sides of the compression plates. The displacements measured by means of the two transducers were compared with the displacement measured by the Linear Variable Displacement Transducer (LVDT) of the testing machine. In Fig. 4a the results of a 4 × 4 × 4 SC-BCC specimens in terms of force and displacement are shown. The difference between the LVDT and the two laser transducers is mainly due to the stiffness of the testing machine Km . The value of Km was estimated by considering the testing machine

Fig. 3. Optical micrograph of a SC-BCC sample. Details of microstructure at: (a) strut intersection; (b,c) vertical struts; (d,e) horizontal struts; (f) node. 4

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Fig. 4. Monotonic test on SC-BCC1 (4 × 4 × 4) specimen. (a) Force-displacement curve obtained with different transducers: LVDT and laser transducers. (b) Forcedisplacement curve obtained considering the stiffness of the testing machine.

and the specimen as a system of two springs in series with a stiffness equal to Km and Ks respectively. The value of the stiffness of the testing machine was calculated for each specimen and for each loading/unloading ramp showing a constant value of approximately 105 N/mm. The displacement of the specimen can be obtained starting from the applied force and the displacement provided by the LVDT:

δs = δLVDT

F − Km

Table 2 Stiffness ratio between 4 × 4 × 12 and 4 × 4 × 4 specimens at each loading/ unloading ramp performed during the tests (Loading/unloading ramp 2–1–2 kN: from 2 to 1 kN, 3–2–3 kN: from 3 to 2 kN, 4–3–4 kN: from 4 to 3 kN, see Fig. 4(a). Loading/unloading ramp

a=

(1)

The results for specimen SC-BCC1 are shown in Fig. 4b. By removing the contribution of the stiffness of the testing machine, the difference between the monotonic curves, the one based on LVDT and the others on the laser transducers, becomes negligible. This simple procedure was adopted for obtaining the specimen stiffness during the fatigue tests because laser transducers cannot be applied due to high test frequency.

K s,(12)

2–1–2 kN

3–2–3 kN

4–3–4 kN

0.4

0.37

0.35

K s,(4)

theoretical value of 1/3, see Table 2. The theoretical value is achieved only in the last loading/unloading ramp at the maximum load level. This is probably due to the presence of a non-linear region, caused by a take-up of slack and alignment or seating of the specimen, at the lower load levels. This is mainly a local effect due to the contact between the compression plates and the specimen’s skins. Due to the different number of cells in the height, the relative weight of this local effect on the global stiffness is more pronounced in the 4 × 4 × 4 specimen. (see Fig. 6). In order to describe the effect of the contact, a model based on springs in series can be adopted. The main hypothesis is that the central part of the specimen has the same stiffness regardless of the number of cells along the height. Following this hypothesis the total specimen

3.3.1. Specimen stiffness Loading/unloading ramps in the monotonic compression tests were used to evaluate the changes in the specimen stiffness as a function of the applied force. The results for the SC-BCC specimens are shown in Fig. 5a. The total specimen stiffness, Ks , is not constant but it increases with the load level where the loading/unloading ramps have been performed. Moreover, the ratio a between the stiffness of a 4 × 4 × 12, Ks (12) , and a 4 × 4 × 4 specimen, Ks (4) , is different from the

Fig. 5. (a) Total specimen stiffness Ks for 4 × 4 × 4 specimens (Ks (4) ) and 4 × 4 × 12 specimens (Ks (12) ). (b) Stiffness of a single row of cells Ki . 5

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Fig. 6. Evaluation of stiffness by means of DIC.

geometry as obtained from the AM process. The DIC set-up consists of a set of Optem lens mounted on a tripod produced by Manfrotto and three linear Owis manual precision linear stages. The images were acquired using an ALLI Manta camera with a resolution of 1624 × 1264 px and an acquisition rate of 30 fps. The DIC system is connected with a portable computer equipped with a LabView software that integrates the frame acquisition process and the signals record from the load frame (load and displacement). The lens system adopted in the present study enabled us to adopt different visual field depending on the size of the specimen under investigation. The basic cubic cells were monitored with a visual field of approximately 12 mm × 15.4 mm which resulted in a resolution of 9.5 μ m/px; whereas, for the rectangular micro-lattices the lens were changed to allow a visual field of approximately 30 mm × 23 mm according a resolution of 18.5 μ m/px. Following the tests, the acquired videos were unfolded and each frame was correlated with the reference image captured before loading. The correlation procedure was performed with the VIC-2D software selecting only the regions of the micro-lattices. As the speckle pattern was deposited on the as-manufactured geometry, the DIC measurements were performed targeting only the displacement measurements. The strains were calculated by positioning virtual extensometers at specific cell positions. This procedure made it possible to measure bulk micro-lattice elastic modulus and also local cell elastic modulus. By means of the elastic modulus it was possible also to determine the value of the Ki stiffness in each row of cubic cells in the micro-structures considering the displacement of each points of the target surface. Through measurements it is possible to observe that the inner rows of cubic cells in a specimens are uniformly loaded and have the same stiffness Ki . The mean value of Ki obtained from the DIC analysis is around 2.8–2.9 × 105 N/mm, that is close to the value of Ki obtained from the analytical model shown in Eq. (4) which gives values of Ki between 2.75 × 105 N/mm and 2.85 × 105 N/mm.

stiffness, as evaluated by the slope of the force-LVDT displacement curve, can be written as:

1 2 2 = + Ks (4) Kc Ki

(2)

1 2 10 = + Ks (12) Kc Ki

(3)

where Ks (4) and Ks (12) are the total stiffnesses of the 4 × 4 × 4 specimen and the 4 × 4 × 12 specimen respectively. For both specimen geometries, the first and the last row of cubic cells, the ones adjacent to the upper and the lower skins, have been assumed to have the same stiffness K c . Due to the contact, this value is assumed to be different from the value of the stiffness Ki of a generic row of cubic cells in the central part of the lattice structure. By solving the system of Eqs. (2) and (3) it is possible to obtain the stiffness Ki as:

Ki =

8Ks (12) 1−a

=

8aKs (4) 1−a

(4)

The application of Eq. (4) to SC-BCC specimens is shown in Fig. 5b. Solid lines are referring to a uniform distribution of load and displacement in the whole specimen, i.e. Ki = NKs where N is the number of unit cells at the height of the specimen as it can be obtained by Eq. (4) with the theoretical value of a = 1/3. By using the experimental stiffness ratio, as reported in Table 2, the real stiffness in the central part of the specimen can be obtained independently from the applied load level, dashed line in Fig. 5b. This result is particularly useful for elaborating fatigue test results, where the total stiffness of the specimen, as measured by the LVDT, is still influenced by the non linearity of the contact. The same model has been also applied to the SC8-FCC and SC lattice structures obtaining similar results. 3.3.2. Comparison with DIC To validate the model, some specimens have been analysed by using the full-field DIC technique. DIC technique is based upon an image acquisition process enabling us to collect a sequence of frames pointing to a specific target surface. Following the acquisition process, the images are numerically correlated to track the position of groups (subsets) of pixels and reconstruct the displacement field of the monitored region. To enhance the correlation quality, the specimen surfaces were properly prepared and a speckle pattern was produced. The target specimen surface was then sprayed with a black paint using an Iwata airbrush (nozzle size 0.18 mm). No additional specimen preparation was implemented in order to guarantee the original micro-lattice

3.4. Compression fatigue tests 3.4.1. Experimental set-up and testing procedure Fatigue tests were performed on specimens with the same geometries as those described in Section 3.2 on a Instron E10000 electrodynamic testing machine. All tests were conducted in load control with a stress ratio of R = 0.1 and a frequency of 50 Hz for the 4 × 4 × 4 specimens and a frequency of 40 Hz for the 4 × 4 × 12 and the 4 × 5 × 10 specimens. Test termination was defined when the stiffness decreases by 10% compared to the initial stiffness Ks measured at 10000 cycles or after 107 cycles. The total strain was calculated as the ratio between the displacement measured by the LVDT of the testing 6

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high, 0.37 and 0.36 respectively. Such a high value of variability can be attributed to several factors: testing condition, microstructural effect and different micro-notches severity and position in different specimens. As already pointed out for monotonic test, a non-perfect contact between the machine plates and the specimen skins can affect the global stiffness of the specimen measured as an average value at the height. In order to quantify the effect of the specimen stiffness on the fatigue life, all experimental data has been elaborated in terms of an effective strain range:

machine and the initial height of the specimen. Several stress levels were considered. Two or three specimens were tested at each stress level. Some specimens were scanned by means of the CT scan before and after being tested at different stress levels in order to analyse the failure mode of the SC-BCC, SC8-FCC and SC microstructures under high and low loads and to study the effect of their local irregularities and geometrical defects. The data acquired during the tests were analysed through the same approach proposed for metal foams and cellular structures, [30–34]. For each specimen the total fatigue life has been divided into three stages: ratcheting, damage initiation and damage propagation in one or few struts leading to final failure. Ratcheting is characterised by a constant value of the strain amplitude ∊a and an increase value of the mean strain ∊m . The number of cycle at damage initiation Ni is defined as a 1% increment, dashed line in Fig. 7a, over the initial constant value of the strain amplitude, solid line in Fig. 7a. The average ratcheting ̇ = d ∊m /dN , i.e. the rate of accumulation of inelastic strain strain rate ∊m in the direction of the applied load, is defined as the slope of the line fitting the mean strain from the beginning of the tests up to the number of cycle at damage initiation, as shown in Fig. 7b. The last stage, the coalescence of several cracks leading to failure, is characterised by a sudden increase in the mean strain. The number of cycle to failure is defined as the intersection point between the ratcheting line and the line fitting the last points of ∊m before the end of the test, see Fig. 7b as suggested in [30–34].

Δ ∊eff =

(5)

The scatter has been measured as: N

∑ σlogN =

fi )2 (logNfi − logN

i=1

N−2

(7)

where, in order to compare specimens with a different number of cells at the height, the effective strain range is referring to a single row of cells in the middle section of the specimen, Ki is the stiffness of a single inner row and hi is the height of a single row of cubic cells equal to hi = 2.4 mm and hi = 3.4 mm for SC-BCC/SC8-FCC and SC respectively. For each specimen the stiffness of a single inner row, Ki , has been obtained by using Eq. 4. The value of the stiffness of the specimen was obtained starting from the testing machine signals, LVDT and force transducer, at cycle N = 10, 000 . The value of the ratio a between the total stiffness of the 4 × 4 × 12 and the 4 × 4 × 4 specimens was assumed as the mean value in all fatigue tests, aSC − BCC = 0.43 and aSC 8 − FCC = 0.44 for SC-BCC and SC8-FCC cells respectively. Fig. 9a shows the fatigue test results in terms of effective strain range Δ ∊eff and Nf . The scatter is strongly reduced when the geometry of the micro-structure and the real distribution of the load in the unit cells is taken into account through use of the stiffness. For SC-BCC cell the value of σlogN is reduced from 0.37 to 0.2. The reduction is more pronounced for SC8-FCC and SC cells where the new value of life scatter decreases to 0.23 and 0.07 respectively. The effective strain range at fatigue life of N = 107 cycles is nearly the same regardless of the unit cell type. The ratcheting rate, calculated for each tested specimen as the rate of accumulation of the inelastic strain up to damage initiation, is reported in Fig. 9b. The ratcheting rate appears to be linearly correlated with the number of cycle to failure in a log-log scale with a high value of the coefficient of determination R2 as reported in Fig. 9b. Fatigue life appears to be driven by the strain accumulation in the early stage of the test. At a fatigue life of 107 cycles the ratcheting rate is in the order of 10−8 %/cycle regardless of the unit cell. These results, if further confirmed by new experimental tests, could be used to develop a fatigue detection method based on the strain accumulation. The evolution of the damage in the lattice structures can be studied

3.4.2. Experimental results Fatigue test results for the three different unit cells are summarised in Fig. 8 where the experimental data have been plotted as a function of the applied load range. The number of cycle to damage initiation, Ni , and the number of cycle to failure, Nf , have been obtained as described in Section 3.4.1. Experimental data have been fitted by using a Basquin type equation:

ΔF = A·N fb

ΔF 1 · Ki hi

(6)

fi is the corresponding value obwhere Nfi is the experimental data, N tained on the fitting line at the same stress range and N is the number of experimental results. The values of σlogN for the different cell types are reported in Fig. 8. The scatter for SC-BCC and SC8-FCC cells is very

Fig. 7. Definition of the different stages of damage evolution during a fatigue test. (a) Number of cycles at damage initiation; (b) number of cycles at failure. 7

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Fig. 8. Fatigue results – Force range vs number of cycles to failure. (a) SC-BCC, (b) SC8-FCC, (c) SC. σlogN : scatter (see Eq. (6)).

initiation and the one in damage propagation (failures in different position of a specimen) can be plotted as a function of the number of cycle to failure. In Fig. 10b, the damage maps for the three unit cells are reported, where Ni / Nf is the fraction of fatigue life spent in damage initiation and 1 − Ni / Nf is the fraction of fatigue life spent in damage

plotting the ratio between the number of cycle to damage initiation and the ones to failure. As for the ratcheting rate, the number of cycle to damage initiation appears to be linearly correlated to the number of cycle to failure in a log-log scale: see Fig. 10-a. This result can be used for defining a damage map where the fraction of life spent in failure

Fig. 9. Fatigue results – (a) Effective strain range vs number of cycles to failure (σlogN : scatter, see Eq. (6). (b) Ratcheting rate (R2 : coefficient of determination). 8

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Fig. 10. Fatigue results – (a) Number of cycles at damage initiation vs number of cycles to failure (R2 : coefficient of determination). (b) Damage map: fraction of life spent in damage initiation, Ni / Nf , vs number of cycles to failure.

system was used to perform μCT in the laboratories of Politecnico di Milano. The CT-scan is characterised by a source that can produce an acceleration tension of 160 kV and reach even sub-micron accuracy. The tube voltage was set at 45 kV and the tube current was kept at 110 μA . The specimen was scanned with a table position compared to the detector and to the source that allows us to avoid problems of undesired unsharpness. The 1440 projections performed during the complete rotation of each specimen were correlated by means of the EfX − CT software to reconstruct the 3d-geometry and to correct inaccuracy of beam hardening, images alignment and ring artefacts. The obtained voxel size was 19.2 μm . The post-test μCT was analysed to find out which struts were broken and the positions of these failures in the struts. The results of a single longitudinal slice of the tomography are shown in Fig. 11a. Even if the specimen was a run-out, some strut failures adjacent to the nodes are present. The SEM analysis clearly

propagation. As a general rule, the fraction of life spent in damage initiation increases with the number of cycle to failure, i.e. at a low load level. The damage evolution appears to be different in SC8-FCC cell, where the life spent in damage initiation always appears to be less than SC-BCC or SC cells. Moreover, at fatigue life of N = 107 cycles, the ratio Ni / Nf < 1 leads to the conclusion that, even near the fatigue limit, failure of one or few struts can occur. 3.4.3. Failure analysis Failure specimens were analysed by means of CT-scan, Scanning Electron Microscope (SEM) and Optical microscopy in order to study the failure location and the influence of stress concentration at geometrical micro-notches and of the microstructure. In the following the results for a run-out SC8-FCC specimen will be reported. The specimen SC8-FCC12 was tested at a force range ΔF = 1315 N. Before and after the fatigue test the sample was scanned under a μCT scan. An X-25 NSI

Fig. 11. Failure analysis of SC8-FCC12 specimen tested at run-out condition. (a) Post-test μCT longitudinal slice. In red two failed struts are highlighted. (b) SEM observation of failed struts. Fatigue crack nucleates from local geometrical imperfection. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 9

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characterised by a high value of roughness and porosity and a coarse microstructure in the down skin of the struts. A finer microstructure can be seen in the up-skin of the struts and in the inner part of the nodes: see Fig. 12a. As already pointed out, run-out condition is characterised by several propagating and non-propagating cracks in a limited number of struts. Cracks start at strut intersection near local irregularities and imperfections. This seems to be independently from the microstructure. Several cracks have been observed. All of them originated from the surface where strut intersection or local geometry irregularities act as

shows that the failure starts from a local irregularity where a stress concentration is expected to occur: see Fig. 11b. In the same picture, it is also possible to see the typical size of geometrical irregularities, with a depth in the order of 150–200 μm that might lead to damage initiation. The same specimen has been prepared for optical microscopy analyses for the purpose of studying the effect of the microstructure in the fatigue damage initiation: the results are shown in Fig. 12. Unfortunately, it was not possible to find out the same broken strut as reported in Fig. 11b. As remarked in Section 3.2the microstructure is

Fig. 12. Optical micrograph of SC8-FCC12 specimen tested at run-out condition. (a) Micrustructure at strut intersection. Two non propagating cracks are highlighted in the red box. (b) Detail A: non propagating crack originated in the coarse microstructure. (c) Detail B: non propagating crack originated in the fine microstructure. (d) Non-propagating crack at strut intersection in the finer microstructure. (e) Passing through crack in a vertical strut originated and propagated in the finer microstructure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 10

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stress raisers, see Fig. 12c and d. The different microstructure and the internal porosity seem to play a secondary role. Similar observations were made on SC-BCC cells. The analysis of tomography, SEM observations and optical microscopy confirmed that, at run-out condition, a very limited number of horizontal struts fail without leading to a significant decrease of the initial stiffness. The fracture surface of a horizontal strut, see Fig. 13a, shows that fatigue crack starts from the surface due to roughness or geometrical imperfections even if an internal porosity is present. Failures are localised near strut intersections in horizontal struts as well as in the diagonal ones, see Fig. 13b. The high value of internal porosity and the coarse microstructure in the down skin of horizontal struts seem to play a secondary role in fatigue crack nucleation.

Table 3 Data of FE models on fatigue test specimens and experimental tests loading conditions.

4. Discussion

Considering a micro-lattice structure, the values of the local stress is affected by the geometry of the unit cell and by the local geometrical irregularities and defects in the printed sample. Predicting fatigue strength in a lattice structure is a complex task. Even if the applied load is uniaxial, the local stress will result in a multiaxial, non-symmetric and non-uniform state of stress. For this reason, a suitable high cycle fatigue criterion must be able to address multiaxility in the presence of a mean stress and a stress gradient. In the adopted procedure multiaxility has been addressed by using a stress amplitude based on the equivalent von Mises stress proposed by Sines [44]. Eventually, the stress gradient has been addressed in the framework of the theory of the critical distance, [45]. The equivalent stress amplitude has been determined for each node as the average in a spherical volume around it. The size of the volume, a sphere with a diameter of 150 μ m was chosen as a representative dimension for fatigue assessment of AlSi7Mg since it corresponds to a value a0 = 75 μm of El-Haddad’s size parameter as determined in [46]. In detail, the value of the equivalent stress amplitude has been calculated in each node on the surface of a FE model by means of the following numerical procedure:

SPEC ID

Number of elements

Load

SC-BCC 2 SC-BCC 5 FCC 12

6,042,004 6,006,316 5,020,492

2500 N 1550 N 1460 N

tensile monotonic tests described in Section 2.1 and applying the maximum load of the experimental tests, as shown in Table 3. 4.2. Fatigue assessment

The effect of the local geometry can be studied through use of a procedure based on a detailed FE analysis of the real geometry obtained by means of a μCT m CT scan and application of a multiaxial fatigue criterion to the local stress field. 4.1. FEM based on the as-manufactured geometries FE models of the 4 × 4 × 4 SC-BCC and 4 × 4 × 4 SC8-FCC compression specimens were developed and analyzed. In order to study the effect of surface defects and geometrical irregularities, the as-manufactured geometries of the specimens were considered. All the FE models were solved by means of the software ABAQUS. The as-printed geometry of each specimen was obtained by means of the μCT scan and by means of the software Mimics which allows us to process the tomography data and to obtain the surface mesh of the specimens. In order to obtain the mesh of the scanned geometry, it is necessary to set two parameters: the threshold value marked on the histogram, determining which voxels represent the material of the measured object and which ones represent the surrounding air, and the elements dimension [24,43]. A convergence study of the effects of the mesh size and of the threshold value on the FE results were performed in a previous work [24]. The mesh dimension of the surface elements was set at most equal to 100 μm (smallest elements around 40–60 μm are used close to surface irregularities) in order to obtain a geometry of the model very close to the specimen as-manufactured micro-structure. The software ABAQUS was then used to generate the volumetric mesh and the final mesh elements were C3D4 (four-node linear tetrahedral) elements. This procedure can yield different numbers of elements in the specimens, due to the geometry irregularities that requires a larger number of small elements to be modelled. A 4 × 4 × 4 SC8-FCC and a 4 × 4 × 4 SC-BCC specimens were modeled with 6,042,000 and 5,020,500 elements, respectively with an elasto-plastic constitutive law based on the AlSi10Mg stress-strain curve obtained by means of the

1. Identification of a surface node (node (i) ). 2. Identification of all nodes inside a sphere centered in node (i) with a diameter of 150 μ m. 3. Determination of the principal stresses for node (i), (σI (i), σII (i), σIII (i)) , as the average value of principal stresses of all nodes identified in step 2. 4. Determination of the amplitude, (σI (i), a, σII (i), a, σIII (i), a) , and mean value, (σI (i), m, σII (i), m, σIII (i), m) , of the principal stresses for node (i) . 5. Determination of the equivalent amplitude based on Von Mises stress:

Fig. 13. Details of failed struts in SC-BCC specimens. (a) Detail of crack nucleation and propagation: fatigue crack starts from the surface regardless of internal porosity. (b) Optical micrograph of an horizontal strut cracks nucleate and progate from the fienr microstrucutre at strut intersections. 11

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Fig. 14. Equivalent stress amplitude in SC-BCC and SC8-FCC specimens. (+) values: σa (i), eq with positive hydrostatic stress, (−) values: σa (i), eq with negative hydrostatic stress. (a) Normal probability plot for three specimens: SC-BCC5 tested at ΔF = 1350N , N = 107 cycles, SC8-FCC12 tested at ΔF = 1315N , N = 107 cycles, SC8-FCC9 tested at ΔF = 1315N , N = 7.9 × 106 cycles. (b) Equivalent stress amplitude distribution in a single unit cell for SC8-FCC12 specimen. (c) Equivalent stress amplitude distribution in a single unit cell for SC-BCC5 specimen. (d) Equivalent stress amplitude distribution in a single unit cell for SC-BCC5 specimen increasing the load to ΔF = 2500N .

σa∗(i) =

2 σI2(i), a + σII2 (i), a + σIII (i), a − σI (i), a σII (i), a − σII (i), a σIII (i), a − σI (i), a σIII (i), a

6. Determination of the equivalent mean stress: (9)

σa (i), eq = sign(σm∗ (i) )·σa (i), eq 7. Correction of the equivalent stress amplitude by means of the Goodman line in order to account for the mean stress effect:

σa (i), eq =

∗ σm (i)

Rm

(12)

The normal probability1 plot of the equivalent stress amplitude normalised compared to fatigue strength of the bulk material is shown in Fig. 14a. Three samples have been analysed: one SC-BCC (SC-BCC5), tested at a run-out at 107 cycles and two SC8-FCC specimens tested at the same load level, SC8-FCC12 run-out at 107 cycles and SC8-FCC12 failed after 7.9 × 106 cycles. For SC-BCC specimen the fatigue strength is slightly exceeded both

σa∗(i) 1−

(11)

The results for a SC-BCC specimen and a SC8-FCC specimen tested at a load level near the fatigue limit, load range of ΔF = 1350 N and ΔF = 1315 N respectively, are plotted in Fig. 14. In order to show the difference between the struts in tension and those in compression the equivalent stress amplitude has been plotted with the sign of the hydrostatic mean stress:

(8)

σm∗ (i) = σI (i), m + σII (i), m + σIII (i), m

σa σ + m =1 s−1 Rm

(10)

Rm being the monotonic ultimate tensile strength of the material. This procedure was repeated for all nodes on the surface. Fatigue crack nucleation is expected to occur in the lattice structure when the local equivalent stress amplitude exceeds the fatigue strength of the material, σa, eq > s−1. The fully reversed fatigue strength s−1 at 107 cycles has been estimated from the fatigue strength in bending at a stress ratio R = 0.1 by means of the Goodman line in the Haigh diagram:

1

For a gaussian (normal) distribution, the cumulative distribution Φ(y ) appears as a straight line (the so called normal probability paper) when it is plotted on a graph y − z , where z = Φ−1 (y ) (inverse of the Φ ). For a data set yi (i = 1⋯n ), the normal probability plot depicts yi − z i pairs where z i have been calculated for the empirical cumulative probability of the data points [47] 12

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We also acknowledge support by the “Excellence Department” grant awarded by MIUR to the Department of Mechanical Engineering at Politecnico di Milano for the DIC equipment.

in tension and in compression with the maximum value of equivalent stress amplitude in tension. The prediction of the model supports the observation that the run-out condition for lattice structures is not characterised by the absence of failure but mainly by a limited number of failed struts as experimentally observed by μCT -CT scan. Similar results have been obtained on SC8-FCC12 specimen. The maximum value in tension is less than the one for SC-BCC specimen. The failure is expected to be in compression where the value of the equivalent stress amplitude is very close to the fatigue strength of the material. It is interesting to note the difference between the two SC8-FCC samples. The run-out sample has a value of the minimum stress amplitude in compression slightly below the fatigue strength of the bulk material. On the contrary, for the failed specimen the number of struts having an equivalent stress amplitude higher than the fatigue strength of the bulk material strongly increases (Fig. 14a). This results seem to confirm that failure is mainly due to the micro-notches at local geometrical imperfections. For both types of cells the maximum values of the equivalent stress amplitude is located in the strut intersections or near the geometrical imperfections, see Fig. 14b and c. By increasing the applied load, the equivalent stress amplitude increases and the fatigue limit of the material is exceeded both in tension and in compression: see Fig. 14d for the SC-BCC cell. Different struts fail in both the vertical and the horizontal direction as experimentally observed by μCT -CT scan observations carried out on broken samples.

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijfatigue.2019.06.041. References [1] Doyoyo M, Hu JW. Multi-axial failure of metallic strut-lattice materials composed of short and slender struts. Int J Solids Struct 2006;43(20):6115–39. https://doi.org/ 10.1016/j.ijsolstr.2005.12.001. [2] Park SI, Rosen DW, kyum Choi S, Duty CE. Effective mechanical properties of lattice material fabricated by material extrusion additive manufacturing. Add Manuf 2014;1–4:12–23. https://doi.org/10.1016/j.addma.2014.07.002. [3] Rashed MG, Ashraf M, Mines RA, Hazell PJ. Metallic microlattice materials: a current state of the art on manufacturing, mechanical properties and applications. Mater Des 2016;95:518–33. https://doi.org/10.1016/j.matdes.2016.01.146. [4] Ashby MF, Bréchet YJ. Designing hybrid materials. Acta Mater 2003;51(19):5801–21. https://doi.org/10.1016/S1359-6454(03)00441-5. [5] Vanderesse N, Richter A, Nuño N, Bocher P. Measurement of deformation heterogeneities in additive manufactured lattice materials by Digital Image Correlation: strain maps analysis and reliability assessment. J Mech Behav Biomed Mater 2018;86:397–408. https://doi.org/10.1016/j.jmbbm.2018.07.010. [6] Dantas AC, Scalabrin DH, De Farias R, Barbosa AA, Ferraz AV, Wirth C. Design of highly porous hydroxyapatite scaffolds by conversion of 3D printed gypsum structures – a comparison study. In: Procedia CIRP, vol. 49; 2016. p. 55–60. doi:https:// doi.org/10.1016/j.procir.2015.07.030. [7] Yoo D. Computer-aided porous scaffold design for tissue engineering using TPMS. Int J Precis Eng Manuf 2012;13(4):527–37. https://doi.org/10.1007/s12541-0120068-5. [8] Helou M, Kara S. Design, analysis and manufacturing of lattice structures: an overview. Int J Comput Integr Manuf 2018;31(3):243–61. https://doi.org/10.1080/ 0951192X.2017.1407456. [9] Ozdemir Z, Hernandez-Nava E, Tyas A, Warren JA, Fay SD, Goodall R, et al. Energy absorption in lattice structures in dynamics: experiments. Int J Impact Eng 2016;89:49–61. https://doi.org/10.1016/j.ijimpeng.2015.10.007. [10] Borsellino C, Di Bella G. Paper-reinforced biomimetic cellular structures for automotive applications. Mater Des 2009;30(10):4054–9. https://doi.org/10.1016/j. matdes.2009.05.013. [11] Bracconi M, Ambrosetti M, Okafor O, Sans V, Zhang X, Ou X, et al. Investigation of pressure drop in 3D replicated open-cell foams: coupling CFD with experimental data on additively manufactured foams; 2018. doi:https://doi.org/10.1016/j.cej. 2018.10.060. [12] Beevers E, Brandão AD, Gumpinger J, Gschweitl M, Seyfert C, Hofbauer P, et al. Fatigue properties and material characteristics of additively manufactured AlSi10Mg – effect of the contour parameter on the microstructure, density, residual stress, roughness and mechanical properties. 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5. Conclusion This research has been devoted to the analysis of fatigue properties in compression of three different lattice structures manufactured in AlSi7Mg by SLM. The cells investigated are two stretch dominated cells (namely a body-centered and a face-centered cubic structural units) and a bending dominated cell (a cubic structure rotated at 45°). The fatigue test in compression on lattice specimen is quite complex due to the difficulty in fulfill the geometrical tolerance of the contact surface with the compression plates of the testing machine. This problem can lead to a non-perfect contact resulting in a high dispersion of experimental results. This large scatter is partially an artefact of the testing procedure and it is not only due to a variability of the microstructure nor of the local geometrical imperfections. The scatter of experimental data was significantly reduced where the S-N diagrams were plotted in terms of the real strain for each tested specimen. Fatigue data have been analysed by using an approach originally proposed for metallic foams. The ratcheting rate in the first stage of each test (and for the three series of lattice samples) looks to be significantly correlated to the life of each sample. The fatigue strength at 107 cycles corresponds to a stable strain accumulation in the range of 10−7 − 10−8 [%/ cycle]. Samples survived at 107 cycles show that some struts are failed without a significant decrease of the sample stiffness. Strain concentrations induced by the irregular geometry of the cells control the fatigue strength of the lattice structures investigated. The effect of the local geometry has been studied through use of a procedure based on a detailed FE analysis of the real geometry obtained by means of a μCT m CT scan and application of a multiaxial fatigue criterion to the local stress field. The endurance limit of the cells corresponds to the local fatigue limit at the hot spots of the cells. The failure position predicted by the model is quite in line with the experimental evidence. The model allows to highlight the critical locations in the lattice geometry and it can help us in designing the cells against fatigue failure. Acknowledgments The experiments and the AlSi7Mg specimens are part of the activities carried out at the METAMAT-Lab of Politecnico di Milano. Support of Politecnico di Milano to METAMAT-Lab is gratefully acknowledged. 13

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