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On the fatigue behavior of additive manufactured lattice structures
Accepted Manuscript On the Fatigue Behavior of Additive Manufactured Lattice Structures Ali Zargarian, Mohsen Esfahanian, J. Kadkhodapour, Saeid Ziaei-Rad, Delaram Zamani PII: DOI: Reference:
S0167-8442(18)30223-4 https://doi.org/10.1016/j.tafmec.2019.01.012 TAFMEC 2165
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
9 May 2018 2 November 2018 11 January 2019
Please cite this article as: A. Zargarian, M. Esfahanian, J. Kadkhodapour, S. Ziaei-Rad, D. Zamani, On the Fatigue Behavior of Additive Manufactured Lattice Structures, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.tafmec.2019.01.012
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On the Fatigue Behavior of Additive Manufactured Lattice Structures Ali Zargariana , Mohsen Esfahaniana , J. Kadkhodapourc,d , Saeid Ziaei-Rada , Delaram Zamanib,∗ a Department
of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran c Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran d Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany
b Department
Abstract This paper studies the effect of different factors on high cycle fatigue failure of additive manufactured lattice structure by means of numerical simulation and compare the results with experimental data in the literature. Finite element method in conjunction with failure event-based algorithm is used to simulate high cycle fatigue. The proposed algorithm is efficient and accurate for our purpose. It is shown that relative density, the fatigue strength of the bulk material and cell geometry determined the coefficient of power law function while the solid distribution and bulk material fatigue properties affect the exponent. A power law expression is proposed to predict the fatigue strength of the cellular material. The effect of different parameters on constants of the equation is discussed in detail. The results of this study render the need for performing a vast amount of experiment to determine fatigue of properties of cellular materials. Keywords: Finite Element, Fatigue, Additive Manufacturing, Lattice Structure
∗ Corresponding
author Email address: [email protected] (Delaram Zamani)
Preprint submitted to Journal of Additive Manufacturing
December 2, 2017
1. Introduction Cellular solids are a branch of material that is made of a network of edges and faces which construct distinct cells. They usually categorized as open cell and closed cell. In open cell cellular materials, the edges (struts, ligaments) are 5
the main constructive elements which meet at vertices. In close cell ones, the faces (walls) are the main constructive elements which meet at edges. The fluid could pass through the open cell structure while it is not impossible for closed cell ones. This characteristic along with other features such as low weight, high stiffness to weight ratio and flexibility in tailoring mechanical properties made
10
them an absolute choice of orthopedic surgery for bone replacement material. Such bio-materials should have enough permeability to let cell growth and let implant and living tissue fuse together. On the other hand, they should have elastic properties that match elastic properties of bone to avoid stress shielding effect. Gibson was one of the pioneers who made a comprehensive study on
15
structure and properties of cellular materials. The cellular materials are abundant in nature and could inspire researchers to invent new artificial materials base on natural ones. Meyers et al. [1] conducted an exhaustive review of the structure and properties of biological cellular materials. The method of selective laser melting which is
20
an additive manufacturing process (AM) is known as a modern technique particularly used in orthopedics applications to fabricate asymmetric and complex three-dimensional samples. This method achieved by fusing fine metal powders together. The fast developments in additive manufacturing techniques open new horizons for the production of open cellular solids with desirable mechanical
25
properties [2, 3, 4, 5, 6, 7]. During daily activities, bone replacement implants go under fluctuating forces. Consequently, fatigue failure of the implant should be considered in the design stage. The aluminum foams are one of those cellular materials that their fatigue behavior studied extensively [8, 9, 10, 11, 12, 13, 14, 15]. The aluminum foams have no biocompatibility, and their fatigue behavior should
30
be investigated to use in structural applications mainly as the core of sandwich
2
panels. Although the structure of the open cell and closed-cell foam is different, three different step could be observed in their fatigue failure. In the first step, within few cycles, the strain increases rapidly. In the second step, the accumulative 35
strain does not change significantly. This step takes 104 to 106 cycles. In the third step, cumulative strain raises exponentially, which eventually causes failure of the specimen in a few cycles. In most cases, the fatigue experiments were performed under fully compressive loading and the ratio of maximum to minimum load is about to 0.1 to ensure that the specimens remain in their place
40
during the test. Open cellular cell materials could be categorized as stretch dominated and bending dominated. In the former ones, the struts undergo axial force when a macro stress applied to media. In latter ones, the macro stress results in bending moment in the structure. The stretch dominated structures are stiffer than
45
bending dominated ones. The aluminum foams could be considered as bending dominated structures. Thus, macroscopic fluctuating uni-axial stress results in the bending moment which results in tension stress in the outer fiber of struts. The tension stress causes initiation and propagation of cracks in the struts up to final fracture of struts. By successive failure of struts, the structure loses
50
its load bearing capacity and the stiffness decreases which result in the final collapse of the whole structure. Biocompatibility, high strength to weight ratio and other attractive mechanical properties made Titanium alloys an excellent choice for orthopedic applications. New developments in additive manufacturing techniques of metals have
55
made making lattice structures that mimic mechanical properties of the bone possible. The fatigue behavior of titanium scaffolds was the subject of many topics of research [5, 16, 17, 18, 19]. The result of these studies shows that the fatigue performance of Titanium lattice structure is comparable to aluminum foams. The endurance limit of Ti-6Al-4V scaffolds is about 0.1-0.25 of their
60
yield strength. This value is less than the fatigue strength of bulk titanium alloy which is about 0.4 yield strength [16]. This conspiracy could be contributed 3
to the unpolished surface of struts, notch sensitivity of titanium alloy, significant porosity in struts, residual stress, and microstructure. The mechanical treatment to improve fatigue strength (such as shot peening) is not possible in 65
a lattice structure, and heat treatment could be used instead. Leuders et al. [20] showed that hot isostatic pressing treatment could considerably increase the fatigue life of Ti-6Al-4V scaffolds. Furthermore, S. Zhao et al. [21] investigated three kinds of meshes (cubic, G7, and rhombic dodecahedron) of Ti-6Al-4V alloys to illustrate the relationship between cell morphology and it is compressive
70
fatigue behavior. Fatigue mechanism for these three groups is the interaction of cyclic ratcheting and growth of fatigue crack on struts due to the bucketing and bending deformation of struts. The biological responses of 3D printed porous titanium alloy mesh structures constructed by electron beam melting was investigated by K. C. Nune et al. [22] which showed conductivity to osteoblast
75
functions. Some principal factors that have an impact on fatigue behavior such as stress relieving (SR), hot isostatic pressing (HIP) and chemical etching (CE) as well as some beneficial tools to improve fatigue performance were studied by Brecht Van Hooreweder et al. [23]. In our previous work, we proposed an algorithm to predict fatigue strength of
80
titanium porous lattice structures [24]. The similar methodology was used by Hedayati et al. [25] to predict fatigue behavior of porous biomaterials. The fatigue properties of cellular solids (like their other mechanical properties) are mainly affected by four factors: the bulk material’s mechanical properties, relative density, the geometry of cells, distribution of material within a structure
85
which define the shape of walls or struts. Studying the effect of these factors on fatigue properties of lattice structures is very costly and time-consuming. This is due to the fact that fatigue test needs many samples and the cost of the additive manufactured samples are relatively high (approximately 100$ per specimen). On the other hand, fatigue testing is very time-consuming and test-
90
ing equipment is more expensive than monotonic testing. In the current study, numerical simulation was employed to study the effect of various parameters on the endurance limit of additively manufactured lattice 4
structure. the result of this study could be expanded to other porous materials with open cell structure. In particular those with bending dominating struc95
ture. In the following section, the geometry of different lattice structure and assumption in generating finite element model is discussed. furthermore, the details of proposed fatigue algorithm is presented. In continue, the result of finite element simulation is presented and the effect of different parameters on fatigue performance is examined. Finally, a conclusion of the result is made and
100
a simple yet effective expression is proposed to predict fatigue life of cellular materials.
2. Numerical simulation The details of finite element modeling and numerical simulation algorithm is presented in this section. 105
2.1. Geometrical and numerical modeling Four open cell structure with regular shape is modeled in this study: Diamond (D), Rhombic Dodecahedron (RD), Kelvin (K) and truncated cuboctahedron (TC). Timoshenko beam theory was used for modeling struts as beams. Each strut is divided into three sections. Head and tail lengths were equal to the nominal radius of the strut. These two sections were divided by two elements each. The area of cross-section of the tail and head at vertices was twice the nominal area of struts’ cross-section and linearly decreases to a nominal value when reaching to the middle section. In this way, the role of joint stiffness on the structure response could be considered. The middle section was divided into three elements. There are many non-uniformities in the additively manufactured lattice structure, for instance, the struts thickness varies in different sites, which could be modeled by changing the cross-section of elements in the middle of struts randomly. The Gaussian distribution is used to assign random radius to strut’s cross-section. Parameter rdev defines normalized standard deviation
5
and maximum deviation was set to 2rdev . Struts may have some waviness in their geometry that causes applying bending moment in presence of axial loads which weakens struts. This irregularity is modeled by moving the section of middle elements randomly by standard deviation r × rdev where r is cross-sections radius. All the elements have the same length of 1000 micrometers. The expression usually used to describe the dependence of relative density to strut geometry with a radius of r and the length l is as follows: r 2 ρ* =C ρs l
(1)
Where geometrical constant C depends on the geometry of cell. The above equation does not consider multiple counting materials in the vertex. Gibson [26] corrected above equation and presented a vertex corrected form: r 2 r 3 ρ* = C1 − C2 ρs l l
(2)
Again, C1 and C2 depend on the cell geometry. For deriving constants, we will assume that each vertex has a volume equal to that of the sphere with strut radius. Assuming that the volume of joint is equal to
4 3 3 πr
and extension of
joint is 0.7rm in each side, where rm is the average radius of struts. C1 and C2 is calculated for Kelvin structure as follows: ρ* Vs = * ρs V 4 Vs = Vvertex + Vstrut = Nvertex ( πr3 ) + Nstrut (l − 2 × 0.7r)πr2 3 4 3 = 12( πr ) + 24(l − 2 × 0.7r)πr2 3 = 24πr2 − 17.6πr3 √ 3 V ∗ = (2 2l) = 22.627l3 r 2 r 3 ρ* Vs 24πr2 − 17.6πr3 = * = = 3.33 − 2.44 ρs 6.158l3 l l V
(3)
The finite element model has 8 × 5 × 5 cell dimension and aspect ratio of 1.6. The experimental specimen in the literature have the height to width ratio of 1.5 ∼ 2. 6
A master node was created in the middle of loading plane and all the other nodes 110
were coupled to it rigidly. The master node at bottom plane was constrained in all degrees of freedom and master node at top plane has the same condition except for translation in loading plane on which a constant force in loading direction was applied to top master node. 2.2. Fatigue failure algorithm In solid materials, fatigue failure is started with crack initiation in first cycles which propagates to the main crack in later cycles and finally lead to fracture of material. During the application of cyclic loading, the constituent struts in the open cell cellular materials undergo cyclic stresses. Struts failure is the result of a fatigue cracks initiation and propagation due to alternating stress. Crack propagation decreases the stiffness of struts which results in stress redistribution in the structure in each cycle. The change in stiffness due to strut failure is much significant compared with the change due to crack propagation in struts. Consequently, stiffness alternation should be considered in each cycle to model fatigue process perfectly. However, this is very time-consuming and impractical in numerical simulation. Fortunately, the effect of crack initiation and propagation on the decrease of structure stiffness could be ignored compared to struts failure. Consequently, fatigue failure was modeled as the continuous failure of struts and structure stiffness assumed to be intact between struts failure and only updated after each struts’ failure. Fatigue properties of struts were derived from data obtained from the literature. In additive manufactured lattice structure, there are numerous parameters that could affect fatigue properties of struts. Some of these parameters will be discussed here. The additive manufacturing process is controlled by various parameters such as energy input of ray, scanning speed, scanning strategy, the size of particles in powder, the temperature of building chamber, layer thickness, etc. All these parameters could affect the quality of built structure and fatigue strength. On the other hand, as the additive manufacturing is a layer 7
by layer process, building new layer results in the heat treatment in previous layers that could alter the microstructure of struts and consequently their fatigue properties. One way to consider all these complicated effects is to conduct mechanical testing directly on extracted struts from built structure. However, this would be not practical as we are interested in predicting fatigue behavior of lattice structure before manufacturing it. Edwards et al. [27] and Edwards et al. [28] conducted a series of fatigue tests to determine fatigue properties specimens made by EBM and SLM process. They did not any surface machining or heat treatment on specimens, so the surface effect and residual stress effect were considered in the results. However, the fatigue properties of struts would much differ from those of bigger specimens due to size effects. For example for aluminum foams Zhou et al. [11] found that strength of struts and the bulk alloy is different and is higher for struts. The following equation was used to express fatigue property of the struts: SN f = As Nf bs
(4)
Where SN f is the fatigue strength in Mpa and Nf is the number of cycles to failure. Mean stress effect is also considered by the Soderberg equation: σa σm + =1 SN f Sy 115
(5)
Where σa is mean stress while σm represents the fluctuating. Yield Strength of the struts, Sy , and Young’s modulus are supposed to be 900 Mpa and 100 GPa, respectively. In the fatigue simulation, each step of simulation corresponds to the failure of one strut. The methodology is described below:
120
1. The life of strut j at stimulation step of i, Nf ij was determined by alternating strength and mean stress derived from step 1 in conjunction with Basquin’s law (Eq. (3)) and Soderberg mean correction method (Eq. (4)). 2. Due to the application of cyclic macro-stress, the remaining life of each strut, (nij ), should be calculated. For this purpose, the Miner’s rule was used, considering damage accumulation of previous failure events. In the 8
following, the nsmin is the life of particular strut with minimum remaining life at simulation of step s. nij
=
Nf ij
1−
i−1 s X n
!
min s N fj s=1
(6)
3. Strut(s) which its remaining life is minimum was eliminated from the ni
model and damage of other struts were increased accordingly, ( Nmin i ). f j
125
The simulation cycles through steps 1 to 4 until abrupt change in the stiffness was observed. The failure criterion is a predefined percentage of stiffness loss or abrupt change in the macro-strain of the finite element model. There is little difference between two criteria and result in similar fatigue life. The calculated life is considered as fatigue life of lattice structure under current cyclic macro stress.
130
3. Results and discussion In this section, the effect of different parameters like cell topology, relative density, etc. were investigated. For each model, fatigue life of structure was predicted by numerical simulation at four different stress levels, and the S-N curve was plotted. The S-N curve is linear in logarithmic scale. Consequently, a power-law equation was used to describe fatigue strength: S * = A* Nf *
b*
(7)
The effect of different geometrical and mechanical parameters on A∗ and b∗ is discussed in what follows. 3.1. Effect of fatigue property of bulk material It was assumed that strut material has S-N curve which could be expressed as: SN f = As Nf bs
(8)
S-N curve for different values of As and constant value of bs = −0.3 is depicted in Fig. 1. The strength of cellular structure was increased by increasing fatigue 9
Figure 1: S-N curve for different values of As .
Figure 2: Variation of A∗ vs. As .
strength of the bulk material as expected. Values of A∗ and b∗ are presented for different values of As in Table. 1. The b∗ is almost constant while A∗ increases by the increase in As . for better illustration of data, A∗ is plotted against As in Fig. 2. The A∗ increases linearly by As so their relationship could be simply expressed as: A* = C A As
(9)
In the above, it should be noted that CA is function of relative density, cell topology and struts’ shape. 10
Figure 3: S-N curve for different values of bs .
Figure 4: Variation of A∗ vs. As .
S-N curve for different values of bs and constant value of As = 7000 Mpa is depicted in Fig. 3. Values of A∗ and b∗ are presented in Table 2, for different values of bs . The A∗ is almost constant while b∗ increases by the increase in bs . b∗ is plotted against bs in Fig. 4. As aforementioned, b∗ increases linearly by bs and their relationship could be
11
Figure 5: S-N curve for different cell topology for a relative density of 0.2.
Figure 6: Fatigue strength vs. relative density for different cell topologies.
expressed as: b* = Cb bs 135
(10)
It can be seen that bs is the function of relative density and struts’ shape. It could be concluded that change in fatigue properties of struts directly affect the fatigue properties of the cellular material. 3.2. Effect of cell topology Cell topology could affect fatigue behavior in two ways. First is that different
140
cell topology has a different value of strut thickness at a same relative density
12
Figure 7: S-N curve for different values of rd ev.
Figure 8: Variation of b∗ vs. rd ev.
and second is that angle between nodal loads and strut axis is not same for different cell topologies which result in different values of bending and axial stress. The S-N curves are shown in Fig. 5 for four different cell topology namely, Diamond, Rhombic Dodecahedron, Truncated Cuboctahedron and Kelvin at a 145
relative density of 0.2. The Kelvin structure has the highest fatigue strength while rhombic dodecahedron has the lowest. The values of A∗ and b∗ for different cell topologies and relative densities are presented in Table 3 and Table 4. The results reveal that A∗ is highly dependent on cell topology while b∗ does not change significantly that means it is independent of cell topology. 13
Figure 9: Variation of A∗ vs. rd ev.
Figure 10: S-N curves for different values of ro f f .
150
3.3. Effect of relative density As revealed in Table 3 and Table 4, it could be seen that value of b∗ is independent of relative density and cell topology while A∗ depends on both. Mechanical properties of cellular materials are usually related to those of bulk material and relative density by a power law function in the form of: * n P* ρ =C Ps ρs
(11)
In which, P ∗ and Ps are properties of cellular and bulk materials, ρ∗ and ρs are density of cellular and bulk materials, respectively and parameters C and n are 14
Figure 11: Variation of b∗ vs. ro f f .
Figure 12: Variation of A∗ vs. ro f f .
constants depending on the topology of cells and shape of walls. For fatigue strength, it has been shown that fatigue strength of cellular material is linearly related to the bulk material so we could be written: * nA A* ρ = CA As ρs
(12)
In which, CA and nA are constants depending on topology of cells and shape of walls. As it could be seen in Fig. 6, the value of nA is almost same for all structures and is equal to 2. For Young’s modulus, n, is 2 for bending dominated structures and is 1 for stretch dominated structures. 15
Table 1: Values of b∗ and A∗ for different values of As .
b∗
A∗ (Mpa)
As (Mpa)
-0.33
78
1000
-0.32
225
3000
-0.31
381
5000
-0.33
660
7000
-0.33
935
10000
Figure 13: Displacement-cycles curves for different values of mean stress and same stress amplitude.
155
3.4. Effect of irregularities in struts The deviation of cross-section thickness from mean value is controlled by parameter rdev and the waviness by rof f . The S-N curve is plotted for various values of rdev in Fig. 7 for As = 10000 and bs = −0.3 in Kelvin structure. It is clear that by increasing value of rdev the fatigue strength dramatically
160
decreases. It should be mentioned that the upper limit for rdev is 0.5 which means there is the possibility of struts with zero thickness (broken struts). As reported values for this parameter is less than 0.3, the higher values were not studied. The values of b∗ and A∗ are presented in Fig. 8 and Fig. 9. It shows that fatigue behavior of cellular solids is very sensitive to the variation of a
16
Table 2: Values of b∗ and A∗ for different values of bs .
b∗
A∗ (Mpa)
bs (Mpa)
-0.115
336
-0.1
-0.228
375
-0.2
-0.329
349
-0.3
-0.416
305
-0.4
Figure 14: S-N curve for different mean stress and amplitude stress.
Figure 15: Modified S-N curve by Goodman equation.
17
Table 3: Values of A∗ for different topologies and relative densities.
A∗ K
RD
D
Relative Density
239.883
105
166
0.1
1148.154
380
724
0.2
3235.937
933
1862
0.3
Table 4: Values of b∗ for different topologies and relative densities.
b∗ K
RD
D
Relative Density
-0.45
-0.44
-0.42
0.1
-0.44
-0.43
-0.42
0.2
-0.45
-0.44
-0.42
0.3
Figure 16: Modified S-N curve by Gerber equation.
165
cross-section along the axis of struts. Higher deviation means that sections with very low thickness and very high thickness increases in strut which result in a very high-stress level that causes early failure in struts and lower fatigue strength of the structure. The results show that the difference between fatigue strength is low for values of rdev = 0.05 and rdev = 0.1. It could be concluded
18
Table 5: Values of A∗ and b∗ obtained from experimental studies.
Experiment RhD-EBM
Experiment D-SLM
Experiment RhD-SLM
0.16
0.20
0.25
0.11
0.20
0.28
0.16
0.23
0.30
101
120
82
110
363
612
106
244
727
-0.28
-0.25
0.19
-0.32
-0.32
-0.32
-0.27
-0.29
-0.32
Relative Density A
∗
b∗
Table 6: Values of A∗ and b∗ obtained from experimental studies (continued).
Experiment D-EBM
Experiment TC-SLM
Relative Density
0.16
0.20
0.25
0.20
0.22
0.27
0.33
A∗
101
120
82
960
912
1914
2871
-0.26
0.25
-0.24
-0.33
-0.33
-0.35
-0.35
∗
b
Figure 17: Comparison of numerical simulation and experimental S-N curves for diamond titanium lattice structure made by EBM method [16].
170
that the deviation of 0.1 is a good balance between quality of structure and its fatigue strength. So it could be used as a measure to check the quality of lattice
19
structures. The S-N curve is depicted for different values of rof f in Fig. 10. By increasing of waviness in structure, the fatigue strength decreases. The increase of waviness 175
causes that axial load in strut to create a bending moment in strut due to the offset of the cross-section from medial axis. However, this effect is less important compared to deviation from a mean thickness of struts. The values of b∗ and A∗ are presented in Fig. 11 and Fig. 12. 3.5. Effect of mean stress There is no experimental investigation on the effect of mean stress on the fatigue behavior of lattice structures. The compression loading ratio R was 0.1 in most research. To study this issue, several simulations with the same value of stress amplitude and three different mean stresses were conducted. In conventional metals, compressive mean stress results in higher fatigue life while tensile mean stress has detrimental effects. However, this statement could not be applied to bending dominated cellular material. In such materials, any compressive or tensile stress results in bending moment in struts. The bending moment generates tensile stress on the outer surface of struts which causes propagation of fatigue cracks. Thus both tensile and mean stress has a negative effect on the fatigue strength of bending dominated lattice structures. The response of lattice structure with Kelvin unit cell with a relative density of 0.2 is depicted in Fig. 13 for three different value of compressive means stress. The fluctuating stress is 6.3 Mpa for all cases and mean stress is 7.7, 9.2 and 11.6 Mpa. By increase in mean stress, the fatigue life decreases, as 50 percent increase in mean stress results in 50 percent decrease in fatigue life. To consider mean stress effect, two famous equations, known as modified Goodman and Gerber equation are compared. In this way mean stress-corrected fatigue strength for
20
these two equations could be expressed as: SGoodman = Nf
S a 1 − SSmu
Sa SGerber = 2 Nf 1 − SSmu
(13)
(14)
(15) 180
Where Sm is mean stress value and Su is the ultimate strength of the material. For cellular material, ultimate strength is ambiguous. Thus plateau stress used instead. Experimental data show that yield stress and plateau stress is close to yielding stress for bending dominated structures. The S-N curve for alternating stress, Goodman stress, and Gerber stress are depicted in Fig. 14, Fig. 15 and
185
Fig. 16. The results show that both Goodman and Gerber equation could reflect the influence of mean stress. However, the Goodman equation is preferred because it gives value of b∗ closer to value of bs which we have seen should be almost same for acceptable irregularities in struts.
190
4. Experimental data validation There are many data in the literature that could be used to validate the result of numerical simulations obtained in the previous section. A study carried out by Hrabe et al. [16] was the source of the first dataset. They used electron beam melting techniques (EBM) to build titanium lattice
195
structure. The structures have diamond cell shape and relative density was in the range of 0.17 to 0.4. Li et al. [17] employed electron beam melting to construct titanium mesh structure with the rhombic-dodecahedron unit cell geometry and relative densities in the range of 0.15-0.4. It has to be mentioned that they conducted fatigue test
200
under compressive loading. The result of this study was used as the source of the second dataset. The third
21
dataset was obtained from Yavari et al. [29] study which investigated fatigue life of titanium scaffold made with Selective Laser Melting (SLM) method and with three different cell geometry. The relative density of this structure was in 205
the range of 0.11 to 0.37. As an example, the results of numerical simulation and experimental test is compared in Fig. 17. The values of A∗ and b∗ for experimental studies are presented in Table 5 and Table 6. As predicted by numerical simulations, the results of these experimental stud-
210
ies showed that the relationship between fatigue strength and cycles to failure obeyed the power law. The coefficient of power function was dependent on relative density, geometry and fatigue properties of the bulk material while the exponent was only dependent on the fatigue behavior of the bulk material and struts irregularities which is highly influenced by the manufacturing process.
215
5. Conclusion Effect of different factors on fatigue behavior of additive manufactured lattice structures were studied using a numerical simulation of fatigue process. The result of simulations showed that the relationship between cycles to failure and fatigue strength (S-N curve) obeys the power law. The effect of fatigue properties of struts, relative density, cell topology and solid distribution on coefficient and exponent of power-law function was investigated. The results of simulations showed that the exponent is independent of cell topology and relative density and is just a function of fatigue property of struts and irregularities in the struts. On the other hand, the coefficient of power law was influenced by all mentioned factors. It was shown that the fatigue properties of struts have a direct effect on the fatigue strength of lattice structure and the relationship is linear. The fatigue strength of lattice structure was increased by an increase in relative density. The relationship could be expressed by a power law. The exponent was about 2 for bending dominated structures. The topology of cells had a great influence on fatigue strength of lattice structures. Among topologies
22
investigated in the current study, the Kelvin unit cell has the highest fatigue strength at certain relative density. Finally, it was shown that as the shape of struts become more irregular the fatigue strength of lattice structure decreases rapidly. The influence of mean stress effect was also investigated and it was concluded that Goodman equation could perfectly consider the effect of mean stress. In conclusion, the fatigue behavior of open cell cellular materials could be expressed by the following equation: *
SN f = A
*
b N *f
*
= C A As
ρ* ρs
nA
N *f
Cb bs
(16)
Where CA and nA are constants that depend on cell topology and material distribution within a structure. Constant Cb depends on struts irregularities and could be considered could be considered to be 1 for most structures. Experimental data in the literature support this statement. Consequently CA and nA 220
could be determined by experiment only by conducting fatigue test at 2 different stress level providing strut fatigue properties (As and bs ) is known. If As and bs was unknown there would be necessary to do fatigue test at 4 different stress level due to the fact that there are 4 unknown parameters. This would considerably reduce required fatigue test to determine the fatigue properties of
225
cellular materials that reduce the cost of tests.
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S0167-8442(18)30223-4 https://doi.org/10.1016/j.tafmec.2019.01.012 TAFMEC 2165
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
9 May 2018 2 November 2018 11 January 2019
Please cite this article as: A. Zargarian, M. Esfahanian, J. Kadkhodapour, S. Ziaei-Rad, D. Zamani, On the Fatigue Behavior of Additive Manufactured Lattice Structures, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.tafmec.2019.01.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
On the Fatigue Behavior of Additive Manufactured Lattice Structures Ali Zargariana , Mohsen Esfahaniana , J. Kadkhodapourc,d , Saeid Ziaei-Rada , Delaram Zamanib,∗ a Department
of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran c Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran d Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany
b Department
Abstract This paper studies the effect of different factors on high cycle fatigue failure of additive manufactured lattice structure by means of numerical simulation and compare the results with experimental data in the literature. Finite element method in conjunction with failure event-based algorithm is used to simulate high cycle fatigue. The proposed algorithm is efficient and accurate for our purpose. It is shown that relative density, the fatigue strength of the bulk material and cell geometry determined the coefficient of power law function while the solid distribution and bulk material fatigue properties affect the exponent. A power law expression is proposed to predict the fatigue strength of the cellular material. The effect of different parameters on constants of the equation is discussed in detail. The results of this study render the need for performing a vast amount of experiment to determine fatigue of properties of cellular materials. Keywords: Finite Element, Fatigue, Additive Manufacturing, Lattice Structure
∗ Corresponding
author Email address: [email protected] (Delaram Zamani)
Preprint submitted to Journal of Additive Manufacturing
December 2, 2017
1. Introduction Cellular solids are a branch of material that is made of a network of edges and faces which construct distinct cells. They usually categorized as open cell and closed cell. In open cell cellular materials, the edges (struts, ligaments) are 5
the main constructive elements which meet at vertices. In close cell ones, the faces (walls) are the main constructive elements which meet at edges. The fluid could pass through the open cell structure while it is not impossible for closed cell ones. This characteristic along with other features such as low weight, high stiffness to weight ratio and flexibility in tailoring mechanical properties made
10
them an absolute choice of orthopedic surgery for bone replacement material. Such bio-materials should have enough permeability to let cell growth and let implant and living tissue fuse together. On the other hand, they should have elastic properties that match elastic properties of bone to avoid stress shielding effect. Gibson was one of the pioneers who made a comprehensive study on
15
structure and properties of cellular materials. The cellular materials are abundant in nature and could inspire researchers to invent new artificial materials base on natural ones. Meyers et al. [1] conducted an exhaustive review of the structure and properties of biological cellular materials. The method of selective laser melting which is
20
an additive manufacturing process (AM) is known as a modern technique particularly used in orthopedics applications to fabricate asymmetric and complex three-dimensional samples. This method achieved by fusing fine metal powders together. The fast developments in additive manufacturing techniques open new horizons for the production of open cellular solids with desirable mechanical
25
properties [2, 3, 4, 5, 6, 7]. During daily activities, bone replacement implants go under fluctuating forces. Consequently, fatigue failure of the implant should be considered in the design stage. The aluminum foams are one of those cellular materials that their fatigue behavior studied extensively [8, 9, 10, 11, 12, 13, 14, 15]. The aluminum foams have no biocompatibility, and their fatigue behavior should
30
be investigated to use in structural applications mainly as the core of sandwich
2
panels. Although the structure of the open cell and closed-cell foam is different, three different step could be observed in their fatigue failure. In the first step, within few cycles, the strain increases rapidly. In the second step, the accumulative 35
strain does not change significantly. This step takes 104 to 106 cycles. In the third step, cumulative strain raises exponentially, which eventually causes failure of the specimen in a few cycles. In most cases, the fatigue experiments were performed under fully compressive loading and the ratio of maximum to minimum load is about to 0.1 to ensure that the specimens remain in their place
40
during the test. Open cellular cell materials could be categorized as stretch dominated and bending dominated. In the former ones, the struts undergo axial force when a macro stress applied to media. In latter ones, the macro stress results in bending moment in the structure. The stretch dominated structures are stiffer than
45
bending dominated ones. The aluminum foams could be considered as bending dominated structures. Thus, macroscopic fluctuating uni-axial stress results in the bending moment which results in tension stress in the outer fiber of struts. The tension stress causes initiation and propagation of cracks in the struts up to final fracture of struts. By successive failure of struts, the structure loses
50
its load bearing capacity and the stiffness decreases which result in the final collapse of the whole structure. Biocompatibility, high strength to weight ratio and other attractive mechanical properties made Titanium alloys an excellent choice for orthopedic applications. New developments in additive manufacturing techniques of metals have
55
made making lattice structures that mimic mechanical properties of the bone possible. The fatigue behavior of titanium scaffolds was the subject of many topics of research [5, 16, 17, 18, 19]. The result of these studies shows that the fatigue performance of Titanium lattice structure is comparable to aluminum foams. The endurance limit of Ti-6Al-4V scaffolds is about 0.1-0.25 of their
60
yield strength. This value is less than the fatigue strength of bulk titanium alloy which is about 0.4 yield strength [16]. This conspiracy could be contributed 3
to the unpolished surface of struts, notch sensitivity of titanium alloy, significant porosity in struts, residual stress, and microstructure. The mechanical treatment to improve fatigue strength (such as shot peening) is not possible in 65
a lattice structure, and heat treatment could be used instead. Leuders et al. [20] showed that hot isostatic pressing treatment could considerably increase the fatigue life of Ti-6Al-4V scaffolds. Furthermore, S. Zhao et al. [21] investigated three kinds of meshes (cubic, G7, and rhombic dodecahedron) of Ti-6Al-4V alloys to illustrate the relationship between cell morphology and it is compressive
70
fatigue behavior. Fatigue mechanism for these three groups is the interaction of cyclic ratcheting and growth of fatigue crack on struts due to the bucketing and bending deformation of struts. The biological responses of 3D printed porous titanium alloy mesh structures constructed by electron beam melting was investigated by K. C. Nune et al. [22] which showed conductivity to osteoblast
75
functions. Some principal factors that have an impact on fatigue behavior such as stress relieving (SR), hot isostatic pressing (HIP) and chemical etching (CE) as well as some beneficial tools to improve fatigue performance were studied by Brecht Van Hooreweder et al. [23]. In our previous work, we proposed an algorithm to predict fatigue strength of
80
titanium porous lattice structures [24]. The similar methodology was used by Hedayati et al. [25] to predict fatigue behavior of porous biomaterials. The fatigue properties of cellular solids (like their other mechanical properties) are mainly affected by four factors: the bulk material’s mechanical properties, relative density, the geometry of cells, distribution of material within a structure
85
which define the shape of walls or struts. Studying the effect of these factors on fatigue properties of lattice structures is very costly and time-consuming. This is due to the fact that fatigue test needs many samples and the cost of the additive manufactured samples are relatively high (approximately 100$ per specimen). On the other hand, fatigue testing is very time-consuming and test-
90
ing equipment is more expensive than monotonic testing. In the current study, numerical simulation was employed to study the effect of various parameters on the endurance limit of additively manufactured lattice 4
structure. the result of this study could be expanded to other porous materials with open cell structure. In particular those with bending dominating struc95
ture. In the following section, the geometry of different lattice structure and assumption in generating finite element model is discussed. furthermore, the details of proposed fatigue algorithm is presented. In continue, the result of finite element simulation is presented and the effect of different parameters on fatigue performance is examined. Finally, a conclusion of the result is made and
100
a simple yet effective expression is proposed to predict fatigue life of cellular materials.
2. Numerical simulation The details of finite element modeling and numerical simulation algorithm is presented in this section. 105
2.1. Geometrical and numerical modeling Four open cell structure with regular shape is modeled in this study: Diamond (D), Rhombic Dodecahedron (RD), Kelvin (K) and truncated cuboctahedron (TC). Timoshenko beam theory was used for modeling struts as beams. Each strut is divided into three sections. Head and tail lengths were equal to the nominal radius of the strut. These two sections were divided by two elements each. The area of cross-section of the tail and head at vertices was twice the nominal area of struts’ cross-section and linearly decreases to a nominal value when reaching to the middle section. In this way, the role of joint stiffness on the structure response could be considered. The middle section was divided into three elements. There are many non-uniformities in the additively manufactured lattice structure, for instance, the struts thickness varies in different sites, which could be modeled by changing the cross-section of elements in the middle of struts randomly. The Gaussian distribution is used to assign random radius to strut’s cross-section. Parameter rdev defines normalized standard deviation
5
and maximum deviation was set to 2rdev . Struts may have some waviness in their geometry that causes applying bending moment in presence of axial loads which weakens struts. This irregularity is modeled by moving the section of middle elements randomly by standard deviation r × rdev where r is cross-sections radius. All the elements have the same length of 1000 micrometers. The expression usually used to describe the dependence of relative density to strut geometry with a radius of r and the length l is as follows: r 2 ρ* =C ρs l
(1)
Where geometrical constant C depends on the geometry of cell. The above equation does not consider multiple counting materials in the vertex. Gibson [26] corrected above equation and presented a vertex corrected form: r 2 r 3 ρ* = C1 − C2 ρs l l
(2)
Again, C1 and C2 depend on the cell geometry. For deriving constants, we will assume that each vertex has a volume equal to that of the sphere with strut radius. Assuming that the volume of joint is equal to
4 3 3 πr
and extension of
joint is 0.7rm in each side, where rm is the average radius of struts. C1 and C2 is calculated for Kelvin structure as follows: ρ* Vs = * ρs V 4 Vs = Vvertex + Vstrut = Nvertex ( πr3 ) + Nstrut (l − 2 × 0.7r)πr2 3 4 3 = 12( πr ) + 24(l − 2 × 0.7r)πr2 3 = 24πr2 − 17.6πr3 √ 3 V ∗ = (2 2l) = 22.627l3 r 2 r 3 ρ* Vs 24πr2 − 17.6πr3 = * = = 3.33 − 2.44 ρs 6.158l3 l l V
(3)
The finite element model has 8 × 5 × 5 cell dimension and aspect ratio of 1.6. The experimental specimen in the literature have the height to width ratio of 1.5 ∼ 2. 6
A master node was created in the middle of loading plane and all the other nodes 110
were coupled to it rigidly. The master node at bottom plane was constrained in all degrees of freedom and master node at top plane has the same condition except for translation in loading plane on which a constant force in loading direction was applied to top master node. 2.2. Fatigue failure algorithm In solid materials, fatigue failure is started with crack initiation in first cycles which propagates to the main crack in later cycles and finally lead to fracture of material. During the application of cyclic loading, the constituent struts in the open cell cellular materials undergo cyclic stresses. Struts failure is the result of a fatigue cracks initiation and propagation due to alternating stress. Crack propagation decreases the stiffness of struts which results in stress redistribution in the structure in each cycle. The change in stiffness due to strut failure is much significant compared with the change due to crack propagation in struts. Consequently, stiffness alternation should be considered in each cycle to model fatigue process perfectly. However, this is very time-consuming and impractical in numerical simulation. Fortunately, the effect of crack initiation and propagation on the decrease of structure stiffness could be ignored compared to struts failure. Consequently, fatigue failure was modeled as the continuous failure of struts and structure stiffness assumed to be intact between struts failure and only updated after each struts’ failure. Fatigue properties of struts were derived from data obtained from the literature. In additive manufactured lattice structure, there are numerous parameters that could affect fatigue properties of struts. Some of these parameters will be discussed here. The additive manufacturing process is controlled by various parameters such as energy input of ray, scanning speed, scanning strategy, the size of particles in powder, the temperature of building chamber, layer thickness, etc. All these parameters could affect the quality of built structure and fatigue strength. On the other hand, as the additive manufacturing is a layer 7
by layer process, building new layer results in the heat treatment in previous layers that could alter the microstructure of struts and consequently their fatigue properties. One way to consider all these complicated effects is to conduct mechanical testing directly on extracted struts from built structure. However, this would be not practical as we are interested in predicting fatigue behavior of lattice structure before manufacturing it. Edwards et al. [27] and Edwards et al. [28] conducted a series of fatigue tests to determine fatigue properties specimens made by EBM and SLM process. They did not any surface machining or heat treatment on specimens, so the surface effect and residual stress effect were considered in the results. However, the fatigue properties of struts would much differ from those of bigger specimens due to size effects. For example for aluminum foams Zhou et al. [11] found that strength of struts and the bulk alloy is different and is higher for struts. The following equation was used to express fatigue property of the struts: SN f = As Nf bs
(4)
Where SN f is the fatigue strength in Mpa and Nf is the number of cycles to failure. Mean stress effect is also considered by the Soderberg equation: σa σm + =1 SN f Sy 115
(5)
Where σa is mean stress while σm represents the fluctuating. Yield Strength of the struts, Sy , and Young’s modulus are supposed to be 900 Mpa and 100 GPa, respectively. In the fatigue simulation, each step of simulation corresponds to the failure of one strut. The methodology is described below:
120
1. The life of strut j at stimulation step of i, Nf ij was determined by alternating strength and mean stress derived from step 1 in conjunction with Basquin’s law (Eq. (3)) and Soderberg mean correction method (Eq. (4)). 2. Due to the application of cyclic macro-stress, the remaining life of each strut, (nij ), should be calculated. For this purpose, the Miner’s rule was used, considering damage accumulation of previous failure events. In the 8
following, the nsmin is the life of particular strut with minimum remaining life at simulation of step s. nij
=
Nf ij
1−
i−1 s X n
!
min s N fj s=1
(6)
3. Strut(s) which its remaining life is minimum was eliminated from the ni
model and damage of other struts were increased accordingly, ( Nmin i ). f j
125
The simulation cycles through steps 1 to 4 until abrupt change in the stiffness was observed. The failure criterion is a predefined percentage of stiffness loss or abrupt change in the macro-strain of the finite element model. There is little difference between two criteria and result in similar fatigue life. The calculated life is considered as fatigue life of lattice structure under current cyclic macro stress.
130
3. Results and discussion In this section, the effect of different parameters like cell topology, relative density, etc. were investigated. For each model, fatigue life of structure was predicted by numerical simulation at four different stress levels, and the S-N curve was plotted. The S-N curve is linear in logarithmic scale. Consequently, a power-law equation was used to describe fatigue strength: S * = A* Nf *
b*
(7)
The effect of different geometrical and mechanical parameters on A∗ and b∗ is discussed in what follows. 3.1. Effect of fatigue property of bulk material It was assumed that strut material has S-N curve which could be expressed as: SN f = As Nf bs
(8)
S-N curve for different values of As and constant value of bs = −0.3 is depicted in Fig. 1. The strength of cellular structure was increased by increasing fatigue 9
Figure 1: S-N curve for different values of As .
Figure 2: Variation of A∗ vs. As .
strength of the bulk material as expected. Values of A∗ and b∗ are presented for different values of As in Table. 1. The b∗ is almost constant while A∗ increases by the increase in As . for better illustration of data, A∗ is plotted against As in Fig. 2. The A∗ increases linearly by As so their relationship could be simply expressed as: A* = C A As
(9)
In the above, it should be noted that CA is function of relative density, cell topology and struts’ shape. 10
Figure 3: S-N curve for different values of bs .
Figure 4: Variation of A∗ vs. As .
S-N curve for different values of bs and constant value of As = 7000 Mpa is depicted in Fig. 3. Values of A∗ and b∗ are presented in Table 2, for different values of bs . The A∗ is almost constant while b∗ increases by the increase in bs . b∗ is plotted against bs in Fig. 4. As aforementioned, b∗ increases linearly by bs and their relationship could be
11
Figure 5: S-N curve for different cell topology for a relative density of 0.2.
Figure 6: Fatigue strength vs. relative density for different cell topologies.
expressed as: b* = Cb bs 135
(10)
It can be seen that bs is the function of relative density and struts’ shape. It could be concluded that change in fatigue properties of struts directly affect the fatigue properties of the cellular material. 3.2. Effect of cell topology Cell topology could affect fatigue behavior in two ways. First is that different
140
cell topology has a different value of strut thickness at a same relative density
12
Figure 7: S-N curve for different values of rd ev.
Figure 8: Variation of b∗ vs. rd ev.
and second is that angle between nodal loads and strut axis is not same for different cell topologies which result in different values of bending and axial stress. The S-N curves are shown in Fig. 5 for four different cell topology namely, Diamond, Rhombic Dodecahedron, Truncated Cuboctahedron and Kelvin at a 145
relative density of 0.2. The Kelvin structure has the highest fatigue strength while rhombic dodecahedron has the lowest. The values of A∗ and b∗ for different cell topologies and relative densities are presented in Table 3 and Table 4. The results reveal that A∗ is highly dependent on cell topology while b∗ does not change significantly that means it is independent of cell topology. 13
Figure 9: Variation of A∗ vs. rd ev.
Figure 10: S-N curves for different values of ro f f .
150
3.3. Effect of relative density As revealed in Table 3 and Table 4, it could be seen that value of b∗ is independent of relative density and cell topology while A∗ depends on both. Mechanical properties of cellular materials are usually related to those of bulk material and relative density by a power law function in the form of: * n P* ρ =C Ps ρs
(11)
In which, P ∗ and Ps are properties of cellular and bulk materials, ρ∗ and ρs are density of cellular and bulk materials, respectively and parameters C and n are 14
Figure 11: Variation of b∗ vs. ro f f .
Figure 12: Variation of A∗ vs. ro f f .
constants depending on the topology of cells and shape of walls. For fatigue strength, it has been shown that fatigue strength of cellular material is linearly related to the bulk material so we could be written: * nA A* ρ = CA As ρs
(12)
In which, CA and nA are constants depending on topology of cells and shape of walls. As it could be seen in Fig. 6, the value of nA is almost same for all structures and is equal to 2. For Young’s modulus, n, is 2 for bending dominated structures and is 1 for stretch dominated structures. 15
Table 1: Values of b∗ and A∗ for different values of As .
b∗
A∗ (Mpa)
As (Mpa)
-0.33
78
1000
-0.32
225
3000
-0.31
381
5000
-0.33
660
7000
-0.33
935
10000
Figure 13: Displacement-cycles curves for different values of mean stress and same stress amplitude.
155
3.4. Effect of irregularities in struts The deviation of cross-section thickness from mean value is controlled by parameter rdev and the waviness by rof f . The S-N curve is plotted for various values of rdev in Fig. 7 for As = 10000 and bs = −0.3 in Kelvin structure. It is clear that by increasing value of rdev the fatigue strength dramatically
160
decreases. It should be mentioned that the upper limit for rdev is 0.5 which means there is the possibility of struts with zero thickness (broken struts). As reported values for this parameter is less than 0.3, the higher values were not studied. The values of b∗ and A∗ are presented in Fig. 8 and Fig. 9. It shows that fatigue behavior of cellular solids is very sensitive to the variation of a
16
Table 2: Values of b∗ and A∗ for different values of bs .
b∗
A∗ (Mpa)
bs (Mpa)
-0.115
336
-0.1
-0.228
375
-0.2
-0.329
349
-0.3
-0.416
305
-0.4
Figure 14: S-N curve for different mean stress and amplitude stress.
Figure 15: Modified S-N curve by Goodman equation.
17
Table 3: Values of A∗ for different topologies and relative densities.
A∗ K
RD
D
Relative Density
239.883
105
166
0.1
1148.154
380
724
0.2
3235.937
933
1862
0.3
Table 4: Values of b∗ for different topologies and relative densities.
b∗ K
RD
D
Relative Density
-0.45
-0.44
-0.42
0.1
-0.44
-0.43
-0.42
0.2
-0.45
-0.44
-0.42
0.3
Figure 16: Modified S-N curve by Gerber equation.
165
cross-section along the axis of struts. Higher deviation means that sections with very low thickness and very high thickness increases in strut which result in a very high-stress level that causes early failure in struts and lower fatigue strength of the structure. The results show that the difference between fatigue strength is low for values of rdev = 0.05 and rdev = 0.1. It could be concluded
18
Table 5: Values of A∗ and b∗ obtained from experimental studies.
Experiment RhD-EBM
Experiment D-SLM
Experiment RhD-SLM
0.16
0.20
0.25
0.11
0.20
0.28
0.16
0.23
0.30
101
120
82
110
363
612
106
244
727
-0.28
-0.25
0.19
-0.32
-0.32
-0.32
-0.27
-0.29
-0.32
Relative Density A
∗
b∗
Table 6: Values of A∗ and b∗ obtained from experimental studies (continued).
Experiment D-EBM
Experiment TC-SLM
Relative Density
0.16
0.20
0.25
0.20
0.22
0.27
0.33
A∗
101
120
82
960
912
1914
2871
-0.26
0.25
-0.24
-0.33
-0.33
-0.35
-0.35
∗
b
Figure 17: Comparison of numerical simulation and experimental S-N curves for diamond titanium lattice structure made by EBM method [16].
170
that the deviation of 0.1 is a good balance between quality of structure and its fatigue strength. So it could be used as a measure to check the quality of lattice
19
structures. The S-N curve is depicted for different values of rof f in Fig. 10. By increasing of waviness in structure, the fatigue strength decreases. The increase of waviness 175
causes that axial load in strut to create a bending moment in strut due to the offset of the cross-section from medial axis. However, this effect is less important compared to deviation from a mean thickness of struts. The values of b∗ and A∗ are presented in Fig. 11 and Fig. 12. 3.5. Effect of mean stress There is no experimental investigation on the effect of mean stress on the fatigue behavior of lattice structures. The compression loading ratio R was 0.1 in most research. To study this issue, several simulations with the same value of stress amplitude and three different mean stresses were conducted. In conventional metals, compressive mean stress results in higher fatigue life while tensile mean stress has detrimental effects. However, this statement could not be applied to bending dominated cellular material. In such materials, any compressive or tensile stress results in bending moment in struts. The bending moment generates tensile stress on the outer surface of struts which causes propagation of fatigue cracks. Thus both tensile and mean stress has a negative effect on the fatigue strength of bending dominated lattice structures. The response of lattice structure with Kelvin unit cell with a relative density of 0.2 is depicted in Fig. 13 for three different value of compressive means stress. The fluctuating stress is 6.3 Mpa for all cases and mean stress is 7.7, 9.2 and 11.6 Mpa. By increase in mean stress, the fatigue life decreases, as 50 percent increase in mean stress results in 50 percent decrease in fatigue life. To consider mean stress effect, two famous equations, known as modified Goodman and Gerber equation are compared. In this way mean stress-corrected fatigue strength for
20
these two equations could be expressed as: SGoodman = Nf
S a 1 − SSmu
Sa SGerber = 2 Nf 1 − SSmu
(13)
(14)
(15) 180
Where Sm is mean stress value and Su is the ultimate strength of the material. For cellular material, ultimate strength is ambiguous. Thus plateau stress used instead. Experimental data show that yield stress and plateau stress is close to yielding stress for bending dominated structures. The S-N curve for alternating stress, Goodman stress, and Gerber stress are depicted in Fig. 14, Fig. 15 and
185
Fig. 16. The results show that both Goodman and Gerber equation could reflect the influence of mean stress. However, the Goodman equation is preferred because it gives value of b∗ closer to value of bs which we have seen should be almost same for acceptable irregularities in struts.
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4. Experimental data validation There are many data in the literature that could be used to validate the result of numerical simulations obtained in the previous section. A study carried out by Hrabe et al. [16] was the source of the first dataset. They used electron beam melting techniques (EBM) to build titanium lattice
195
structure. The structures have diamond cell shape and relative density was in the range of 0.17 to 0.4. Li et al. [17] employed electron beam melting to construct titanium mesh structure with the rhombic-dodecahedron unit cell geometry and relative densities in the range of 0.15-0.4. It has to be mentioned that they conducted fatigue test
200
under compressive loading. The result of this study was used as the source of the second dataset. The third
21
dataset was obtained from Yavari et al. [29] study which investigated fatigue life of titanium scaffold made with Selective Laser Melting (SLM) method and with three different cell geometry. The relative density of this structure was in 205
the range of 0.11 to 0.37. As an example, the results of numerical simulation and experimental test is compared in Fig. 17. The values of A∗ and b∗ for experimental studies are presented in Table 5 and Table 6. As predicted by numerical simulations, the results of these experimental stud-
210
ies showed that the relationship between fatigue strength and cycles to failure obeyed the power law. The coefficient of power function was dependent on relative density, geometry and fatigue properties of the bulk material while the exponent was only dependent on the fatigue behavior of the bulk material and struts irregularities which is highly influenced by the manufacturing process.
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5. Conclusion Effect of different factors on fatigue behavior of additive manufactured lattice structures were studied using a numerical simulation of fatigue process. The result of simulations showed that the relationship between cycles to failure and fatigue strength (S-N curve) obeys the power law. The effect of fatigue properties of struts, relative density, cell topology and solid distribution on coefficient and exponent of power-law function was investigated. The results of simulations showed that the exponent is independent of cell topology and relative density and is just a function of fatigue property of struts and irregularities in the struts. On the other hand, the coefficient of power law was influenced by all mentioned factors. It was shown that the fatigue properties of struts have a direct effect on the fatigue strength of lattice structure and the relationship is linear. The fatigue strength of lattice structure was increased by an increase in relative density. The relationship could be expressed by a power law. The exponent was about 2 for bending dominated structures. The topology of cells had a great influence on fatigue strength of lattice structures. Among topologies
22
investigated in the current study, the Kelvin unit cell has the highest fatigue strength at certain relative density. Finally, it was shown that as the shape of struts become more irregular the fatigue strength of lattice structure decreases rapidly. The influence of mean stress effect was also investigated and it was concluded that Goodman equation could perfectly consider the effect of mean stress. In conclusion, the fatigue behavior of open cell cellular materials could be expressed by the following equation: *
SN f = A
*
b N *f
*
= C A As
ρ* ρs
nA
N *f
Cb bs
(16)
Where CA and nA are constants that depend on cell topology and material distribution within a structure. Constant Cb depends on struts irregularities and could be considered could be considered to be 1 for most structures. Experimental data in the literature support this statement. Consequently CA and nA 220
could be determined by experiment only by conducting fatigue test at 2 different stress level providing strut fatigue properties (As and bs ) is known. If As and bs was unknown there would be necessary to do fatigue test at 4 different stress level due to the fact that there are 4 unknown parameters. This would considerably reduce required fatigue test to determine the fatigue properties of
225
cellular materials that reduce the cost of tests.
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