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Failure mechanisms of additively manu-
factured porous biomaterials: Effects of
porosity and type of unit cell
J. Kadkhodapour, H. Montazerian, A.Ch.
Darabi, A.P. Anaraki, S.M. Ahmadi, A.A.
Zadpoor, S. Schmauder
PII: S1751-6161(15)00205-2
DOI: http://dx.doi.org/10.1016/j.jmbbm.2015.06.012
Reference: JMBBM1506
To appear in: Journal of the Mechanical Behavior of Biomedical Materials
Received date:5 March 2015
Revised date: 7 June 2015
Accepted date:
13 June 2015
Cite this article as: J. Kadkhodapour, H. Montazerian, A.Ch. Darabi, A.P.
Anaraki, S.M. Ahmadi, A.A. Zadpoor, S. Schmauder, Failure mechanisms
of additively manufactured porous biomaterials: Effects of porosity and
type of unit cell, Journal of the Mechanical Behavior of Biomedical Materials, http:
//dx.doi.org/10.1016/j.jmbbm.2015.06.012
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1
Failure mechanisms of additively manufactured porous
biomaterials: Effects of porosity and type of unit cell
J. Kadkhodapour
a,b*
, H. Montazerian
a
, A. Ch. Darabi
c
, A. P. Anaraki
a
,S. M.
Ahmadi
d
, A. A. Zadpoor
d
, S. Schmauder
b
a
Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran
b
Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart,
Stuttgart, Germany
c
Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
d
Department of Biomechanical Engineering, Delft University of Technology, Mekelweg 2, Delft 2628CD, The
Netherlands
*
Corresponding Author, Email: javad.kad@srttu.edu, Tel: +98-21-22970052
2
Abstract
Since the advent of additive manufacturing techniques, regular porous biomaterials have emerged as
promising candidates for tissue engineering scaffolds owing to their controllable pore architecture and
feasibility in producing scaffolds from a variety of biomaterials. The architecture of scaffolds could be
designed to achieve similar mechanical properties as in the host bone tissue, thereby avoiding issues such
as stress shielding in bone replacement procedure. In this paper, the deformation and failure mechanisms
of porous titanium (Ti6Al4V) biomaterials manufactured by selective laser melting from two different
types of repeating unit cells, namely cubic and diamond lattice structures, with four different porosities
are studied. The mechanical behavior of the above-mentioned porous biomaterials was studied using
finite element models. The computational results were compared with the experimental findings from a
previous study of ours. The Johnson-Cook plasticity and damage model was implemented in the finite
element models to simulate the failure of the additively manufactured scaffolds under compression. The
computationally predicted stress-strain curves were compared with the experimental ones. The
computational models incorporating the Johnson-Cook damage model could predict the plateau stress and
maximum stress at the first peak with less than 18% error. Moreover, the computationally predicted
deformation modes were in good agreement with the results of scaling law analysis. A layer-by-layer
failure mechanism was found for the stretch-dominated structures, i.e. structures made from the cubic unit
cell, while the failure of the bending-dominated structures, i.e. structures made from the diamond unit
cells, was accompanied by the shearing bands of 45°.
Keywords: Cellular biomaterials, Ti6Al4V, lattice structures, the Johnson-Cook damage model, selective
laser melting, bone substitutes.
3
1. Introduction
The explorations of biomaterials have advanced tissue engineering scaffolds that can be replaced with
damaged tissues while retaining biological activities in the absence of original living tissues. Currently,
autograft and allograft are the most commonly used bone grafting techniques in orthopedic surgeries
owing to their superior osteoconductive properties and ability to be integrated with the host bony tissue
(Kuremsky, Schaller, Hall, Roehr, & Masonis, 2010; Poehling et al., 2005). Autologous bone grafts are
osteogenic, osteoinductive, osteoconductive, and biocompatible (Keating & McQueen, 2001); however,
they are not without their drawbacks as they are prone to postoperative pain and morbidity due to the
extra procedures in donor sites. Furthermore, many treatments are needed to prevent disease transmission,
graft failures, and laxity, especially in the case of allograft procedures (Mirelis et al.; Young Iii & Toth,
2006). Hence, many attempts have been made in tissue engineering to design substitutions as scaffolds
made of biocompatible materials such as coralline hydroxyapatite (Mygind et al., 2007), collagen based
materials (Offeddu, Ashworth, Cameron, & Oyen, 2015), calcium sulfate, and bioactive glasses to
mechanically match the host tissue. Moreover, tissue engineering scaffolds are expected to provide
suitable conditions for cell ingrowth, cell migration, and differentiation (Billström, Blom, Larsson, &
Beswick, 2013). It is frequently claimed in the literature that the cell culture state of scaffolds can be well
related to the fluid permeability of cellular architecture (Dias, Fernandes, Guedes, & Hollister, 2012;
Syahrom, Abdul Kadir, Abdullah, & Öchsner, 2013; Truscello et al., 2012).
Morphological parameters such as pore architecture, pore size, relative density, as well as mechanical
properties of the base material have been found as the most effective factors for governing the mentioned
requirements. Stochastic architectures have shown localized deformations owing to their internal
imperfections (Cansizoglu, Harrysson, Cormier, West, & Mahale, 2008), while higher specific
mechanical properties have been observed for non-stochastic geometries (Queheillalt & Wadley, 2005).
Most of the studies on open cell non-stochastic cellular structures are focused on 3D lattice-based
geometries such as diamond (S. M. Ahmadi et al., 2014), honeycomb (Ajdari, Jahromi, Papadopoulos,
Nayeb-Hashemi, & Vaziri, 2012), octahedral (Sun, Yang, & Wang, 2012), rhombic dodecahedron (Horn
4
et al., 2014), tetrakaidecahedral (Zargarian, Esfahanian, Kadkhodapour, & Ziaei-Rad, 2014), and
crystalline lattices (Karamooz Ravari, Kadkhodaei, Badrossamay, & Rezaei, 2014). Recently, many
studies have utilized triply periodic minimal surfaces (TPMS) as a promising approach in scaffold
designing owing to their unique mechanical and biological features in addition to their capacity in
producing functionally gradient porous structures (Almeida & Bártolo, 2014; Yang, Quan, Zhang, &
Tian, 2014; Yigil et al., 2013; Yoo, 2011). Melchels et al. (Melchels et al., 2010) compared cell seeding
capability of stochastic pore architecture resulted from salt leaching with gyroid structure and reported
more than 10-fold permeability improvement using gyroid structure. They observed better cell penetration
in the center of the scaffold with gyroid structure, while the scaffolds in random architectures were
covered with a cell sheet on the outside. Mechanical aspects of cellular materials with random and regular
pore architectures were also assessed by Cansizoglu et al. (O. Cansizoglu, D. Cormier, O. Harrysson, H.
West, & T. Mahale, 2006). They argued that irregular network geometries with randomized connectivity
had a more similar mechanical behavior to the bone due to the decay of stiffness, especially when nodes
were randomly disconnected. On the other hand, it is intuitively clear that increasing porosity leads to
losing the strength of scaffolds, while it enhances situations for cell ingrowth and nutrient transformation
in addition to helping prevent stress shielding which affects the longevity of the implant (Parthasarathy,
Starly, Raman, & Christensen, 2010). Furthermore, increasing the cell size of porous network has been
observed to decrease mechanical strength and stiffness (Yan et al., 2014; Yan, Hao, Hussein, & Raymont,
2012). Hence, a compromise is needed between strength, stiffness, permeability, and mechanical
properties of the base material in the designing procedure of tissue engineering scaffold.
Giving this scenario, additive manufacturing (AM) techniques are increasingly developed due to their
controllability on material, internal architecture, and consequently mechanical and biological responses of
scaffolds. Among the biocompatible materials such as tantalum, chrome, cobalt, and stainless steel
(Parthasarathy et al., 2010), medical grade titanium alloys, namely Ti6Al4V, has been commercially
advanced for fabricating implants by many methods such as melt processing, powder processing, and
vapor deposition (Harrysson, Cansizoglu, Marcellin-Little, Cormier, & West Ii, 2008). Moreover,
5
producing titanium implants by additive manufacturing is progressively developed, since they are
biologically inert and provide adequate osteointegration properties in addition to exhibiting high
mechanical strength and good corrosion resistance (Jamshidinia, Wang, Tong, & Kovacevic, 2014;
Wieding, Souffrant, Mittelmeier, & Bader, 2013). However, elastic modulus of titanium is almost
114GPa, while for cortical bone, it ranges from 0.5 to 20GPa (Wieding, Wolf, & Bader, 2014) which
implies adequate morphological manipulation of pores to match the host tissue.
Selective laser melting (SLM) is an additive manufacturing process in which powder particles with the
sizes below 40mm are melted selectively to produce solid structures with the materials having melting
point of less than the temperature provided by laser, such as titanium, stainless steel, and nickel-based
super alloys (McKown et al., 2008). The quality of the produced materials can be controlled by powder,
melting and processing parameters, and geometry itself (Gorny et al., 2011). Furthermore, it allows
manufacturing fully dense metal parts without any need for post-processes such as infiltration, sintering,
and hot isostatic pressing (Yan et al., 2012). Brenne et al. (Brenne, Niendorf, & Maier, 2013) revealed
that post-SLM heat treatment can improve ductility in monotonic loading response of Ti6Al4V samples.
Moreover, they reported enhanced energy absorption by 4 point bending test and increased fatigue life
after the post-heat treatment of the samples. Smith et al. (Smith, Guan, & Cantwell, 2013) evaluated finite
element models by comparing their results with the experimental data of compression test on the lattice
materials produced by SLM technique. They modeled lattice models by beam and brick elements and
concluded that key mechanical properties of large lattices composed of a huge number of elements can be
well predicted by investigating their constitutive unit cell. Moreover, the differences between numerical
and experimental results were reduced considering the diameter variations of struts by reverse
engineering of unit cell.
With this picture in mind, the cubic and diamond lattice structures produced by SLM technique were
considered in this paper and their mechanical response was evaluated under quasi-static loading
conditions and compared at the relative densities of 11, 22, 28, and 35% to discuss pore architecture
effects on their compressive behavior. In order to get insight into mechanical predictability of Ti6Al4V
6
lattice structures, the Johnson-Cook damage model was studied through finite element analysis.
Furthermore, the relationships between deformation and mechanical properties of cellular structures were
investigated by scaling laws analysis of specific Young's modulus obtained in the experimental procedure
and the numerical deformation mechanism was compared with the results of scaling laws analysis on the
experimental Young's modulus and compressive strength.
2. Materials and Methods
2.1. Experimental Procedure
In order to study the relationships between deformation mode and mechanical properties, the
experimental data of a previous work (Seyed Mohammad Ahmadi et al., 2015) for the samples with
different deformation mechanisms, namely cubic and diamond, were utilized in this paper. Selective laser
machines (Layerwise, Kueven, Belgium) were used to build the porous structures. The laser processing
parameters and manufacturing procedures were similar to the ones used in our previous studies (Amin
Yavari, Ahmadi, et al., 2014; Amin Yavari, van der Stok, et al., 2014; Amin Yavari et al., 2013). As
previously stated, Ti6Al4V ELI powder (grade 23 according to ASTM F136) was used to produce lattice
structures by laser melting technology. CAD models of the constitutive unit cell for each lattice geometry
was produced and the block structures were generated to achieve cellular structures with approximately
11, 22, 28, and 35% volume fractions. Geometry of unit cells by which the cubic and diamond lattices are
produced, is illustrated in Fig. 1(a). Different values of porosity were generated by changing the diameter
of struts in lattice structures. Geometrical characterization of the structures is presented in Table 1. CAD
models were imported to the additive manufacturing machine and cylindrical lattice samples were
constructed on top of a solid titanium substrate with the rough length of 15 mm and diameter of 10 mm.
Then, the samples were removed from the substrate using electrical discharge machining (EDM).
The quality and micro-structure of the parent material could potentially influence the mechanical
properties of the resulting porous structures. In our previous studies (Amin Yavari et al., 2013), we found
the microstructure of the parent material to be typical of the Ti-6Al-4V alloy. Moreover, the
7
microstructure was largely similar for porous structures with different porosities. The parent material was
of high quality as indicated by the very small percentage of pores (see e.g. Figure 5 in (Amin Yavari,
Wauthle, et al., 2014)) in the parent material. High density of the parent material is crucial when studying
the mechanical properties of additively manufactured porous biomaterials and that is why, in our
experimental studies, we accept the parent material to be of acceptable quality only when its density well
exceeds 90% (typical values of density >99%)”
To evaluate the mechanical response of lattice structures, INSTRON 5985 mechanical testing
machine (100 kN load cell) was used to perform compressive tests under displacement control mode with
the deformation rate of 1.8 mm/min. Compressive tests were repeated for 5 times for each of the samples
and the average value of the resulted stress-strain curves was considered for interpreting compressive
characteristics of the samples. All the tests were performed with standard methods for porous metallic
materials (ISO, 2011). In order to idealize the stress-strain curves, modulus of elasticity was considered as
the slope of linear fits to the experimental data up to 0.02 strain. Further, yield stress was computed by
intersecting the curve with 0.002 offset line parallel to elastic region and plateau stress was calculated as
the average value of stress-strain curve from the onset of plateau region. Stress at the first peak of the
stress-strain curve was also attributed to
max
σ
. More details of the experimental procedure and results
were reported elsewhere (Seyed Mohammad Ahmadi et al., 2015).
The porous structures were scanned using micro computed tomography (µ-CT). The µ-CT images were
then segmented and used to evaluate the architecture of the porous structures using algorithms that
calculated porosity, strut thickness, and pore size. The details of the scanning protocol, segmentation
algorithms, and algorithms used for calculation of the structural parameters could be found in our
previous studies (Seyed Mohammad Ahmadi et al., 2015; Amin Yavari, van der Stok, et al., 2014; Amin
Yavari et al., 2013).
8
2.2. Simulation and Modeling Procedure
In order to investigate the ability of the Johnson-Cook model in terms of predicting the mechanical
behavior of lattice structures, the 3D FE analysis quasi-static simulations were performed using FE
software. First, the CAD models of cubic and diamond lattice structures were constructed for the analysis
and imported into ABAQUS/CAE. Then, lattice structures were meshed with tetrahedral elements (C3Dd)
automatically.
It is worth mentioning that cell size effect (or sample size to cell size ratio) may lead to high deviations of
numerical results from experimental data, especially at higher strains. However, since cellular materials
generally produce a massive amount of output files, extremely high process cost is resulted during the
early analysis. A previous work (Smith et al., 2013) studied cell size effect and stated that many
properties could be well predicted by simulating lower amounts of unit cells of non-stochastic cellular
materials, rather than the whole structure. Hence, in order to prevent high CPU times for solving
numerical model, 125 unit cells (5×5×5 unit cells patterned along three global coordinates) of lattice
structures were modeled and computationally analyzed. On the other hand, as far as failure behavior of
the structure is concerned, mesh sensitivity plays a key role in the accuracy of finite element model due to
the high sensitivity of results to damage parameters. Thus, mesh sensitivity analysis was carried out in
order to find the suitable mesh size in finite element analysis. The samples with almost 22% volume
fraction was considered for mesh sensitivity analysis and CAD models of cubic porous structures with
100µm composed of 5×5×5 unit cells were discretized by tetrahedral elements with the global seed sizes
of 7, 3, and 2 µm, as illustrated in Fig. 2. For all of the models, an elastic-plastic model was considered
according to the experimental results obtained by (Cain, Thijs, Van Humbeeck, Van Hooreweder, &
Knutsen, 2015). The elastic modulus of 110Gpa and Poisson's ratio of 0.3 was set as elastic properties
and plastic stresses and strains were defined as shown in Fig. 3. Finally, corresponding to each model,
compressive displacement was defined as boundary conditions with the strain rate matching experimental
procedure and ABAQUS explicit code was employed to solve the FEM problems.
9
2.3. Computational Approach
It is obvious that material properties play an important role in the numerical investigation of mechanical
properties for lattice structures, especially at higher strains. In this paper, the material properties of
Ti6Al4V were set according to those reported in (Garciandia, 2009).
In order to evaluate damage in lattice structures, the Johnson-Cook model was employed with the same
approach as in (Johnson & Cook, 1985). In this failure model, after an element exceeds the failure limit, it
will be removed from the structure. The fracture strain-based Johnson-Cook damage model is described
as:
( )
[ ]
*
*
1 2 3 4 5
( exp 1 ( ) 1
failure
D D D D ln D T
ε σ ε
= + + +
where
1
D
,
2
D
,
3
D
,
4
D
, and
5
D
are five material constants in the Johnson-Cook damage model,
1
D
,
2
D
, and
3
D
are damage parameters related to the relationships between failure strain and stress
triaxiality,
4
D
and
5
D
depend on strain rate and temperature, respectively, and
*
σ
is stress triaxiality
defined as hydrostatic stress (
h
σ
) to the equivalent stress (
q
σ
) ratio:
*
h
q
σ
σ
σ
=
Calculating material properties and damage constants of lattice structures (stress-strain curve) is
accompanied by difficulties since the need to frequent and different tensile tests on structures with
different diameters. Since tests were performed at room temperature with the same strain rates,
4
D
and
5
D
were ignored. Moreover, the values of
1
D
,
2
D
, and
3
D
were selected in terms of their fit to
experimental data; consequently, the values of -0.68, 0.73, and -0.25 were found to well agree with the
experimental data, respectively.
10
3. Results and Discussion
3.1. Mesh Sensitivity Analysis
In order to ensure convergence in finite element results, a mesh study was performed considering three
seed sizes of 2, 3, and 7 µ m to mesh the cubic cellular structures with the length of 100µm. Stress-strain
curves corresponding to the defined seed sizes are indicated in Fig. 4. As can be seen, convergence was
found between the mesh sizes of 3 and 2µm. Hence, due to the time consuming run of the models with
2µm mesh size, 3µ m was set for the seed size of the described lattice models; thereby, almost 270000,
320000, and 280000 mesh numbers were resulted corresponding to 3µm mesh size for 22% volume
fraction of cubic and diamond structures, respectively.
3.2. Characterizing Deformation Modes by Scaling Law Analysis of Experimental Data
In order to compare effect of lattice morphology on compressive behavior of scaffolds, stress-strain
curves of experimental compression test for the cubic and diamond lattice structures corresponding to
four volume fractions of almost 11, 22, 28, and 35% are separately represented in Fig. 5. Stress-strain
curves for all the specimens followed the same trend as that of typical cellular materials, in which after
elastic regime, energy absorption was accomplished during plateau region up to the onset of densification.
Cellular materials may go along with strain hardening or strain softening at the onset of plateau regime
(Li, 2006). In the case of lattice structures characterized in this paper, according to the stress fall-downs
after the first peak of stress at the end of the elastic regime, post-yield strain softening behavior was
observed for all the specimens with different relative densities and lattice patterns.
The obtained values from the idealization of experimental stress-strain curves are presented in Table 2. It
should be noted that densification was not seen for all the specimens. Occurrence of densification depends
on deformation and failure mechanism during crushing at higher strains. Lack of densification and
consequently terminal hardening are expected in the cases in which deformation is accompanied by
material separation and highly brittle failure of relatively thin struts. Comparison of collapse behavior
from stress-strain curves showed that the set of the samples with almost 11 and 22% relative densities did
11
not totally meet densification up to 0.5 strain, while the onset of densification can be observed for the
cubic and diamond lattice structures with 28 and 35% volume fractions. Moreover, the variations of stress
during plateau region could be attributed to more trend to brittle failure of internal struts. Hence,
comparing the compressive behavior of specimens in plateau regime can provide an understanding of
failure mechanism of the samples. Monotonicity of stress in plateau was seen to be higher at lower
relative densities for both cubic and diamond samples. Generally, increasing relative density as a result of
increasing the diameters of struts leads to more stress drop due to the internal failure of material and
explains the increased stress variations at higher volume fractions.
Following the same approach as Ashbey's (Ashby, 2006), specific mechanical properties of cellular
materials can be described by scaling analysis relative to density. Variation of normalized Young's
modulus is reported to be exponentially in relation with density as (
n
s s
EC
E
ρ
ρ
=
). Exponential values
of power fits to data were found to be 0.904 and 1.658 for the cubic and diamond structures,
correspondingly. The obtained power trend was in good agreement with the one attributed to the
structures with stretch dominated and bending dominated deformation, since the exponential value for the
structure with stretch dominated deformation, namely cubic structure, approached 1, while the ones in
which bending was dominated on the deformation, it tended to 2, which can be also described by the
geometry of lattices. The orientation of struts in cubic lattice was all parallel and perpendicular to loading
direction, while in the case of diamond, bending of internal struts was expected owing to their inclination
relative to loading direction.
3.3. Validating Simulation Results by Experimental Data
Computational stress-strain curves for the cubic and diamond structures corresponding to four
volume fractions with the Johnson-cook damage model were compared with the experimental data in Fig.
6. In all of the simulations, the same material property was assigned to each lattice structure. Finite
element results in the elastic regime were found to be in good agreement with the experimental data.
12
Mechanical characteristics of the models obtained by finite element analysis of the Johnson-Cook
plasticity model are presented in Table 1 and compared with the experimental results. As stated in
previous studies, a considerable amount of discrepancy can be explained by variations in diameter during
SLM processes. Although yield strength as well as first peak of stress was in good agreement with the
experimental data for all the samples, computational yield strain was seen to be higher than the
experimental results, which can be attributed to cell size effect as a higher number of unit cells was tested
in the experimental procedure, while 125 unit cells were modeled for FE analysis. In the plastic regime,
although the amplitude of stress variations in numerical results was higher than the experimental data,
plateau stresses (average value of stress in plateau regime) were in good agreement with the one obtained
in the experimental results. The results in Table 2 demonstrate that the Johnson-Cook plasticity model
could properly predict maximum stress (
max
σ
) and plateau stress with relative errors of less than 18%.
Elastic modulus was predicted with relative errors of less than 22% for diamond structure, while in the
case of cubic structure, it was in the range of 28 to 117%. In general, it can be observed by comparing the
computational and experimental data that the Johnson-cook damage model could properly predict the
mechanical characteristics of non-stochastic lattice structures at higher strains. In addition, those lattice
structures with a huge number of unit cells can be predicted considering lower unit cells without losing
accuracy.
On the other hand, the nominal (design) values of the structural parameters of the porous structures were
quite close to the experimental values of the structural parameters determined using µ-CT (Table 1). This
shows the high degree of fidelity of the actual porous structure to the CAD files used for manufacturing.
It is therefore expected that finite element models, which are based on the nominal values of the structural
parameters, are reasonably good representations of the actual porous structure. There are, nevertheless,
some deviations from the nominal values that could potentially influence the finite element results. It is
therefore recommended that the actual porous structure based on reconstructions of µ-CT images be used
in the future studies so as to improve the accuracy of finite element simulations.
13
3.4. Interpreting Failure Mechanism
Whereas different volume fractions of each lattice structure were deformed with the same failure
mechanism, deformation procedure is presented by representing plastic strain contours for the cubic and
diamond with almost 22% volume fraction at different strains in Fig. 7 and Fig. 8, respectively. In order
to illustrate deformation mechanisms and show the relationships between deformation and stress-strain
curves, figures are addressed at their corresponding strain in computational stress-strain curves.
Failure mechanism of the cubic lattice structure in elastic region was accompanied by the uniform
deformation of vertical struts (Fig. 7(a)). Then, deformation was observed to be followed by a layer-by-
layer deformation up to the total collapse of layers such that the first upper layer was crushed initially
(Fig. 7(b)) and the others were collapsed in order up to the end of deformation (Fig. 7(d)). Stress
fluctuations are explained by failure of each layer thanks to the buckling of micro-struts such that each
stress drop corresponds to failure at a specific stage of failure. Since struts withstand axial deformation
owing to their parallel direction relative to loading, higher variations were seen owing to the high release
of energy as each stage of collapse. It shows that higher load bearing capacity can be expected when
deformation is accompanied by buckling of micro-struts. Scaling law analysis of normalized Young's
modulus as well as yield strength was confirmed by the mechanism of failure, as can be seen for stretch
dominated deformation structure.
As previously stated in (Kadkhodapour, Montazerian, & Raeisi, 2014), Diamond lattice structure is
formed of 45° oriented plates placed parallel to sides of an internal pyramid which are connected with -
45° struts at their joints. At the onset of plastic deformation (Fig. 8(a)), tying struts initiated to bend and
resulted in the layer-by-layer crushing of plates in which central plates, namely diagonal plates, were
crushed initially and led to the development of a continuous shear band of 45° around the model (Fig.
8(b)). Subsequently, deformation went along with crushing the outer plates as a result of bending -45°
tying struts (Fig. 8(c)). As shown in the stress-strain curve, stress drops corresponding to each collapse of
layers were discernible. Failure in diamond lattice geometry was seen to be accompanied by shearing of
the micro-struts leading the scaffold to tend to bending dominated deformation. It should be noted that the
14
amplitude of stress variation in plateau region was found to be lower than cubic structure. Hence, it is
deduced that the less energy is released at each stage of failure when shearing failure mechanism as a
result of strut inclination compared with that observed in scaffolds with buckling failure mechanism.
State of deformations for the diamond structures was also found to be in good agreement with the
exponential value of scaling laws, since bending of struts plays a key role in the deformation mechanism
of such structures.
Generally, it can be inferred from the results that the micro-strut orientation plays crucial role in
deformation mechanism and consequently in stress strain behavior of scaffolds. When the struts are
design to be placed parallel to loading direction, buckling is excepted resulting the structure to experience
stretch dominated deformation behavior while the more inclination of micro-struts the more shearing
failure is observed in the whole of the structure. In this case, bending deformation mode dominate the
compressive behavior of scaffold. As well, when bending deformation is dominated on scaffolds, the less
specific mechanical properties is expected compared with the stretch dominated structures.
4. Conclusions
This study examined the Johnson-Cook plasticity and damage model in terms of predicting mechanical
response and deformation procedure of Ti6Al4V lattice structures produced by selective laser melting.
Lattice structures with cubic and diamond geometries were modelled with the diameters such that four
volume fractions of 11, 22, 28, and 35% were achieved corresponding to each lattice structure. Finite
element models were prepared by repeating 5 unit cells along three global coordinates and consequently
the lattice structures composed of 125 unit cells were achieved. Subsequently, cellular structures with
almost 22% volume fraction were selected to study their convergence in finite element analysis.
In order to analyze the deformation mechanism of the samples, scaling analysis on specific Young's
modulus of the experimental results was performed and compared with the deformation procedure
achieved by finite element analysis of the Johnson-Cook damage model. Cubic lattice as a structure with
stretch dominated deformation represented higher specific mechanical properties; hence, deformation
15
mechanism was found to illustrate the mechanical properties of materials. The exponential value of power
fit to experimental Young's modulus for the cubic and diamond structure was found to be 0.904 and
1.658, which implied the domination of stretching and bending deformation, respectively.
These results of power fits to data were in good agreement with the deformation mechanism of lattices,
since for bending dominated structure (diamond), deformation was accompanied by the shearing band of
45°, while layer-by-layer failure was seen for the cubic structure with stretch dominated deformation.
Approaches such as scaling laws analysis or Maxwell criterion have been previously introduced to
examine the deformation state of lattices and characterize mechanical properties. However, they are
developed to show if bending is dominated on the deformation of a specific architecture or stretching.
Hence, development of methods in which more details regarding deformation mode are quantitatively
provided can improve understanding the effect of deformation and failure mechanism on mechanical
characteristics of cellular materials, since structures with stretch dominated deformation are shown to
provide more specific stiffness and strength, making them appropriate for structural applications, while
bending dominated structures are reported to be more applicable for energy absorption application.
In this study, two different lattice structures were studied and the deformation mechanism of each one
was investigated. The topology and shape of the structures as a source of difference were examined in
detail. Although more detailed investigations have to be done on the material property and damage
mechanism in terms of considering geometries, production process, etc.
Comparing computational stress-strain curves with the experimental ones illustrated good ability of the
Johnson-Cook damage model in predicting stress at the first peak as well as plateau stress (with less than
18% relative error). It can be concluded that there is a potential to predict mechanical behavior of
structures with a huge number of unit cells by modeling their constitutive unit cells to prevent current
restrictions for solving large models. Using simulation methods, a clearer picture was derived for the
deformation pattern and failure mechanism of porous lattice structures of different geometries. It was also
shown that the geometry as a source of difference in mechanical properties can play a role in the
dominant deformation pattern of structure.
16
References
Ahmadi, S. M., Campoli, G., Amin Yavari, S., Sajadi, B., Wauthle, R., Schrooten, J., . . . Zadpoor, A. A.
(2014). Mechanical behavior of regular open-cell porous biomaterials made of diamond lattice
unit cells. J Mech Behav Biomed Mater, 34(0), 106-115. doi:
http://dx.doi.org/10.1016/j.jmbbm.2014.02.003
Ahmadi, S. M., Yavari, S. A., Wauthle, R., Pouran, B., Schrooten, J., Weinans, H., & Zadpoor, A. A.
(2015). Additively Manufactured Open-Cell Porous Biomaterials Made from Six Different
Space-Filling Unit Cells: The Mechanical and Morphological Properties. Materials, 8(4), 1871-
1896.
Ajdari, A., Jahromi, B. H., Papadopoulos, J., Nayeb-Hashemi, H., & Vaziri, A. (2012). Hierarchical
honeycombs with tailorable properties. International Journal of Solids and Structures, 49(11–12),
1413-1419. doi: http://dx.doi.org/10.1016/j.ijsolstr.2012.02.029
Almeida, H. A., & Bártolo, P. J. (2014). Design of tissue engineering scaffolds based on hyperbolic
surfaces: Structural numerical evaluation. Med Eng Phys, 36(8), 1033-1040. doi:
http://dx.doi.org/10.1016/j.medengphy.2014.05.006
Amin Yavari, S., Ahmadi, S. M., van der Stok, J., Wauthle, R., Riemslag, A. C., Janssen, M., . . .
Zadpoor, A. A. (2014). Effects of bio-functionalizing surface treatments on the mechanical
behavior of open porous titanium biomaterials. J Mech Behav Biomed Mater, 36(0), 109-119. doi:
http://dx.doi.org/10.1016/j.jmbbm.2014.04.010
Amin Yavari, S., van der Stok, J., Chai, Y. C., Wauthle, R., Tahmasebi Birgani, Z., Habibovic, P., . . .
Zadpoor, A. A. (2014). Bone regeneration performance of surface-treated porous titanium.
Biomaterials, 35(24), 6172-6181. doi: 10.1016/j.biomaterials.2014.04.054
Amin Yavari, S., Wauthle, R., Böttger, A. J., Schrooten, J., Weinans, H., & Zadpoor, A. A. (2014).
Crystal structure and nanotopographical features on the surface of heat-treated and anodized
porous titanium biomaterials produced using selective laser melting. Applied Surface Science,
290(0), 287-294. doi: http://dx.doi.org/10.1016/j.apsusc.2013.11.069
Amin Yavari, S., Wauthle, R., van der Stok, J., Riemslag, A. C., Janssen, M., Mulier, M., . . . Zadpoor, A.
A. (2013). Fatigue behavior of porous biomaterials manufactured using selective laser melting.
Materials Science and Engineering: C, 33(8), 4849-4858. doi:
http://dx.doi.org/10.1016/j.msec.2013.08.006
Ashby, M. F. (2006). The properties of foams and lattices. Philos Trans A Math Phys Eng Sci, 364(1838),
15-30. doi: 10.1098/rsta.2005.1678
Billström, G. H., Blom, A. W., Larsson, S., & Beswick, A. D. (2013). Application of scaffolds for bone
regeneration strategies: Current trends and future directions. Injury, 44, Supplement 1(0), S28-
S33. doi: http://dx.doi.org/10.1016/S0020-1383(13)70007-X
Brenne, F., Niendorf, T., & Maier, H. J. (2013). Additively manufactured cellular structures: Impact of
microstructure and local strains on the monotonic and cyclic behavior under uniaxial and bending
load. Journal of Materials Processing Technology, 213(9), 1558-1564. doi:
http://dx.doi.org/10.1016/j.jmatprotec.2013.03.013
Cain, V., Thijs, L., Van Humbeeck, J., Van Hooreweder, B., & Knutsen, R. (2015). Crack propagation
and fracture toughness of Ti6Al4V alloy produced by selective laser melting. Additive
Manufacturing, 5(0), 68-76. doi: http://dx.doi.org/10.1016/j.addma.2014.12.006
Cansizoglu, O., Harrysson, O., Cormier, D., West, H., & Mahale, T. (2008). Properties of Ti–6Al–4V
non-stochastic lattice structures fabricated via electron beam melting. Materials Science and
Engineering: A, 492(1–2), 468-474. doi: http://dx.doi.org/10.1016/j.msea.2008.04.002
Dias, M. R., Fernandes, P. R., Guedes, J. M., & Hollister, S. J. (2012). Permeability analysis of scaffolds
for bone tissue engineering. J Biomech, 45(6), 938-944. doi:
http://dx.doi.org/10.1016/j.jbiomech.2012.01.019
17
Garciandia, L. E. (2009). Characterization of rapid prototyped Ti6Al4V bone scaffolds by the combined
use of micro-CT and in situ loading. Carlos III De Madrid, Leganés.
Gorny, B., Niendorf, T., Lackmann, J., Thoene, M., Troester, T., & Maier, H. J. (2011). In situ
characterization of the deformation and failure behavior of non-stochastic porous structures
processed by selective laser melting. Materials Science and Engineering: A, 528(27), 7962-7967.
doi: 10.1016/j.msea.2011.07.026
Harrysson, O. L. A., Cansizoglu, O., Marcellin-Little, D. J., Cormier, D. R., & West Ii, H. A. (2008).
Direct metal fabrication of titanium implants with tailored materials and mechanical properties
using electron beam melting technology. Materials Science and Engineering: C, 28(3), 366-373.
doi: http://dx.doi.org/10.1016/j.msec.2007.04.022
Horn, T. J., Harrysson, O. L. A., Marcellin-Little, D. J., West, H. A., Lascelles, B. D. X., & Aman, R.
(2014). Flexural properties of Ti6Al4V rhombic dodecahedron open cellular structures fabricated
with electron beam melting. Additive Manufacturing, 1–4(0), 2-11. doi:
http://dx.doi.org/10.1016/j.addma.2014.05.001
ISO. (2011). Mechanical testing of metals -- Ductility testing -- Compression test for porous and cellular
metals (Vol. 13314).
Jamshidinia, M., Wang, L., Tong, W., & Kovacevic, R. (2014). The bio-compatible dental implant
designed by using non-stochastic porosity produced by Electron Beam Melting® (EBM). Journal
of Materials Processing Technology, 214(8), 1728-1739. doi: 10.1016/j.jmatprotec.2014.02.025
Johnson, G. R., & Cook, W. H. (1985). Fracture characteristics of three metals subjected to various
strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1), 31-48.
doi: http://dx.doi.org/10.1016/0013-7944(85)90052-9
Kadkhodapour, J., Montazerian, H., & Raeisi, S. (2014). Investigating internal architecture effect in
plastic deformation and failure for TPMS-based scaffolds using simulation methods and
experimental procedure. Materials Science and Engineering: C, 43(0), 587-597. doi:
http://dx.doi.org/10.1016/j.msec.2014.07.047
Karamooz Ravari, M. R., Kadkhodaei, M., Badrossamay, M., & Rezaei, R. (2014). Numerical
investigation on mechanical properties of cellular lattice structures fabricated by fused deposition
modeling. International Journal of Mechanical Sciences, 88(0), 154-161. doi:
http://dx.doi.org/10.1016/j.ijmecsci.2014.08.009
Keating, J. F., & McQueen, M. M. (2001). Substitutes for autologous bone graft in orthopaedic trauma. J
Bone Joint Surg Br, 83(1), 3-8.
Kuremsky, M. A., Schaller, T. M., Hall, C. C., Roehr, B. A., & Masonis, J. L. (2010). Comparison of
Autograft vs Allograft in Opening-Wedge High Tibial Osteotomy. The Journal of Arthroplasty,
25(6), 951-957. doi: http://dx.doi.org/10.1016/j.arth.2009.07.026
Li, Q. M. (2006). Compressive Strain at the Onset of Densification of Cellular Solids. Journal of Cellular
Plastics, 42(5), 371-392. doi: 10.1177/0021955x06063519
McKown, S., Shen, Y., Brookes, W. K., Sutcliffe, C. J., Cantwell, W. J., Langdon, G. S., . . . Theobald,
M. D. (2008). The quasi-static and blast loading response of lattice structures. International
Journal of Impact Engineering, 35(8), 795-810. doi: 10.1016/j.ijimpeng.2007.10.005
Melchels, F. P. W., Barradas, A. M. C., van Blitterswijk, C. A., de Boer, J., Feijen, J., & Grijpma, D. W.
(2010). Effects of the architecture of tissue engineering scaffolds on cell seeding and culturing.
Acta Biomaterialia, 6(11), 4208-4217. doi: http://dx.doi.org/10.1016/j.actbio.2010.06.012
Mirelis, J. G., García-Pavía, P., Cavero, M. A., González-López, E., Echavarria-Pinto, M., Pastrana, M., .
. . Escaned, J. Magnetic Resonance for Noninvasive Detection of Microcirculatory Disease
Associated With Allograft Vasculopathy: Intracoronary Measurement Validation. Revista
Española de Cardiología (English Edition)(0). doi: http://dx.doi.org/10.1016/j.rec.2014.07.030
Mygind, T., Stiehler, M., Baatrup, A., Li, H., Zou, X., Flyvbjerg, A., . . . Bünger, C. (2007).
Mesenchymal stem cell ingrowth and differentiation on coralline hydroxyapatite scaffolds.
Biomaterials, 28(6), 1036-1047. doi: http://dx.doi.org/10.1016/j.biomaterials.2006.10.003
18
O. Cansizoglu, D. Cormier, O. Harrysson, H. West, & T. Mahale. (2006). An Evaluation of Non-
Stochastic Lattice Structures Fabricated Via Electron Beam Melting. Paper presented at the
Proceedings of the Solid Freeform Fabrication Symposium, Austin, Texas.
Offeddu, G. S., Ashworth, J. C., Cameron, R. E., & Oyen, M. L. (2015). Multi-scale mechanical response
of freeze-dried collagen scaffolds for tissue engineering applications. J Mech Behav Biomed
Mater, 42(0), 19-25. doi: http://dx.doi.org/10.1016/j.jmbbm.2014.10.015
Parthasarathy, J., Starly, B., Raman, S., & Christensen, A. (2010). Mechanical evaluation of porous
titanium (Ti6Al4V) structures with electron beam melting (EBM). J Mech Behav Biomed Mater,
3(3), 249-259. doi: http://dx.doi.org/10.1016/j.jmbbm.2009.10.006
Poehling, G. G., Curl, W. W., Lee, C. A., Ginn, T. A., Rushing, J. T., Naughton, M. J., . . . Smith, B. P.
(2005). Analysis of Outcomes of Anterior Cruciate Ligament Repair With 5-Year Follow-up:
Allograft Versus Autograft. Arthroscopy: The Journal of Arthroscopic & Related Surgery, 21(7),
774.e771-774.e715. doi: http://dx.doi.org/10.1016/j.arthro.2005.04.112
Queheillalt, D. T., & Wadley, H. N. (2005). Cellular metal lattices with hollow trusses. Acta Materialia,
53(2), 303-313.
Smith, M., Guan, Z., & Cantwell, W. J. (2013). Finite element modelling of the compressive response of
lattice structures manufactured using the selective laser melting technique. International Journal
of Mechanical Sciences, 67, 28-41. doi: 10.1016/j.ijmecsci.2012.12.004
Sun, J., Yang, Y., & Wang, D. (2012). Mechanical Properties of Ti-6Al-4V Octahedral Porous Material
Unit Formed by Selective Laser Melting. Advances in Mechanical Engineering, 2012, 1-11. doi:
10.1155/2012/427386
Syahrom, A., Abdul Kadir, M. R., Abdullah, J., & Öchsner, A. (2013). Permeability studies of artificial
and natural cancellous bone structures. Med Eng Phys, 35(6), 792-799. doi:
http://dx.doi.org/10.1016/j.medengphy.2012.08.011
Truscello, S., Kerckhofs, G., Van Bael, S., Pyka, G., Schrooten, J., & Van Oosterwyck, H. (2012).
Prediction of permeability of regular scaffolds for skeletal tissue engineering: A combined
computational and experimental study. Acta Biomaterialia, 8(4), 1648-1658. doi:
http://dx.doi.org/10.1016/j.actbio.2011.12.021
Wieding, J., Souffrant, R., Mittelmeier, W., & Bader, R. (2013). Finite element analysis on the
biomechanical stability of open porous titanium scaffolds for large segmental bone defects under
physiological load conditions. Med Eng Phys, 35(4), 422-432. doi:
10.1016/j.medengphy.2012.06.006
Wieding, J., Wolf, A., & Bader, R. (2014). Numerical optimization of open-porous bone scaffold
structures to match the elastic properties of human cortical bone. J Mech Behav Biomed Mater,
37, 56-68. doi: 10.1016/j.jmbbm.2014.05.002
Yan, C., Hao, L., Hussein, A., Bubb, S. L., Young, P., & Raymont, D. (2014). Evaluation of light-weight
AlSi10Mg periodic cellular lattice structures fabricated via direct metal laser sintering. Journal of
Materials Processing Technology, 214(4), 856-864. doi:
http://dx.doi.org/10.1016/j.jmatprotec.2013.12.004
Yan, C., Hao, L., Hussein, A., & Raymont, D. (2012). Evaluations of cellular lattice structures
manufactured using selective laser melting. International Journal of Machine Tools and
Manufacture, 62, 32-38. doi: 10.1016/j.ijmachtools.2012.06.002
Yang, N., Quan, Z., Zhang, D., & Tian, Y. (2014). Multi-morphology transition hybridization CAD
design of minimal surface porous structures for use in tissue engineering. Computer-Aided
Design, 56(0), 11-21. doi: http://dx.doi.org/10.1016/j.cad.2014.06.006
Yigil, C., Tae-Hong, A., Hoon-Hwe, C., Joong-Ho, S., Jun Hyuk, M., Shu, Y., . . . Ju, L. (2013). Study of
architectural responses of 3D periodic cellular materials. Modelling and Simulation in Materials
Science and Engineering, 21(6), 065018.
Yoo, D.-J. (2011). Computer-aided porous scaffold design for tissue engineering using triply periodic
minimal surfaces. International Journal of Precision Engineering and Manufacturing, 12(1), 61-
71. doi: 10.1007/s12541-011-0008-9
19
Young Iii, S. D., & Toth, A. P. (2006). Complications of Allograft Use in Anterior Cruciate Ligament
Reconstruction. Operative Techniques in Sports Medicine, 14(1), 20-26. doi:
http://dx.doi.org/10.1053/j.otsm.2006.02.009
Zargarian, A., Esfahanian, M., Kadkhodapour, J., & Ziaei-Rad, S. (2014). Effect of Solid Distribution on
Elastic Properties of Open-cell Cellular Solids Using Numerical and Experimental Methods. J
Mech Behav Biomed Mater.
Figure Captions
Fig. 1. Representing produced lattice structures by SLM technique: (a) cubic, and (b) diamond structures
(Seyed Mohammad Ahmadi et al., 2015)
Fig. 2. Overview of meshed cellular structures with the mesh seed sizes of (a) 7µm, (b) 3µm, and (c) 2 µm
Fig. 3. Stress strain curve of Ti6Al4V utilized as the input for FE analysis. (Cain et al., 2015)
Fig. 4. Convergence of stress-strain curves corresponding to the three mesh seed sizes for 22% volume
fraction models of (a) cubic, and (b) diamond lattice structures
Fig. 5. Experimental compressive stress-strain curves for the cubic and diamond models at relative
densities of almost (a) 11%, (b) 22%, (c) 28%, and (d) 35%; Samples are named with the combination of
the first letter of their names together with the corresponding relative density (Seyed Mohammad Ahmadi
et al., 2015).
Fig. 6. Comparing experimental stress-strain data with the numerical simulation data for different
volume fractions of (a) cubic, and (b) diamond structures; Yield strains are found to be more than the
experimental results. In plateau region, higher stress variations with more amplitudes lead to deviations
from the experimental data. Nevertheless, plateau stress and consequently energy absorption capability
are well predicted.
Fig. 7. Deformation procedure for the cubic lattice structure; Layer-by-layer deformation mechanism is
confirmed by stretch dominated deformation in scaling laws.
Fig. 8. Failure mechanism of diamond lattice structure at 22% volume fraction; Continuous sharing band
of 45°, owing to crushing diagonal layers, is observed. Shearing of layers is accompanied by the bending
failure of tying struts perpendicular to the diagonal plates.
20
Table Captions
Table 1. Morphological characterization of lattice structures (Seyed Mohammad Ahmadi et al., 2015)
Table 2. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the
experimental data. Relative densities for the maximum stress at the first peak and plateau stress are
predicted with high accuracies. Experimental data are extracted from (Seyed Mohammad Ahmadi et al.,
2015).
Table 1. Morphological characterization of lattice structures (S. Ahmadi et al., 2014)
Unit cell type Sample
Label
Apparent density (%) Strut diameter (µ m) Pore size (µm)
Dry weighing µ-CT Nominal µ-CT Nominal µ-CT
Cubic
C-10 11±0.1 13 348 451±147 1452 1413±366
C-22 21±0.2 24 540 654±190 1260 1139±359
C-27 26±0.2 28 612 693±200 1188 1155±354
C-35 34±0.1 37 720 823±230 1080 1020±311
Diamond
D-11 11±0.1 11 277 240±46 923 958±144
D-21 20±0.2 21 450 416±65 750 780±141
D-28 26±0.4 28 520 482±70 680 719±130
D-35 34±0.3 36 600 564±76 600 641±137
Table 2. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the
experimental data. Relative densities for the maximum stress at the first peak and plateau stress are
predicted with high accuracies. Experimental data are extracted from (S. Ahmadi et al., 2014).
Sample
Name
Young’s Modulus (MPa) Yield Stress (MPa) Maximum stress
ߪ
௫
(MPa) Plateau stress
ߪ
(MPa)
Experiment
Simulation %Error Experiment Simulation
%Error Experiment
Simulation %Error Experimen
t
Simulation %Error
C-10 1578 2016 28% 29 29 - 30 31 3% 11 13 16%
C-22 3157 3831 21% 63 48 25% 77 72 6% 40 39 10%
C-27 3691 6913 87% 66 67 2% 111 104 5% 64 43 14%
C-35 4836 10513 117% 113 118 5% 185 189 2% 142 133 1%
D-11 511 559 9% 7 12 70% 15 16 6% 8 7 10%
D-21 1505 1468 2% 29 33 14% 47 48 2% 29 25 13%
D-28 2178 2049 6% 32 40 27% 57 59 3% 38 31 16%
D-35 3694 4466 21% 71 78 11% 113 114 1% 82 59 18%
21
Table 3. Morphological characterization of lattice structures (S. Ahmadi et al., 2014)
Unit cell type Sample
Label
Apparent density (%) Strut diameter (µ m) Pore size (µm)
Dry weighing µ-CT Nominal µ-CT Nominal µ-CT
Cubic
C-10 11±0.1 13 348 451±147 1452 1413±366
C-22 21±0.2 24 540 654±190 1260 1139±359
C-27 26±0.2 28 612 693±200 1188 1155±354
C-35 34±0.1 37 720 823±230 1080 1020±311
Diamond
D-11 11±0.1 11 277 240±46 923 958±144
D-21 20±0.2 21 450 416±65 750 780±141
D-28 26±0.4 28 520 482±70 680 719±130
D-35 34±0.3 36 600 564±76 600 641±137
Table 4. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the
experimental data. Relative densities for the maximum stress at the first peak and plateau stress are
predicted with high accuracies. Experimental data are extracted from (S. Ahmadi et al., 2014).
Sample
Name
Young’s Modulus (MPa) Yield Stress (MPa) Maximum stress
max
σ
(MPa)
Plateau stress
pl
σ
(MPa)
Experiment
Simulation %Error Experiment Simulation
%Error Experiment
Simulation %Error Experimen
t
Simulation %Error
C-10 1578 2016 28% 29 29 - 30 31 3% 11 13 16%
C-22 3157 3831 21% 63 48 25% 77 72 6% 40 39 10%
C-27 3691 6913 87% 66 67 2% 111 104 5% 64 43 14%
C-35 4836 10513 117% 113 118 5% 185 189 2% 142 133 1%
D-11 511 559 9% 7 12 70% 15 16 6% 8 7 10%
D-21 1505 1468 2% 29 33 14% 47 48 2% 29 25 13%
D-28 2178 2049 6% 32 40 27% 57 59 3% 38 31 16%
D-35 3694 4466 21% 71 78 11% 113 114 1% 82 59 18%
(a)
Fig. 9. Representing
produced lattice structures by SLM technique
Cubic
Diamond
(a)
Fig. 10. Overvie
w of meshed cellular structures with
22
(a)
(b)
produced lattice structures by SLM technique
: (a) cubic, and
(b) diamond structure
(Ahmadi et al., 2015)
(b)
w of meshed cellular structures with
the mesh seed sizes of (a) 7µm,
(b) 3µm
µm
(b) diamond structure
s
(c)
(b) 3µm
, and (c) 2
23
Fig. 11. Stress strain curve of Ti6Al4V utilized as the input for FE analysis. (Cain, Thijs, Van Humbeeck,
Van Hooreweder, & Knutsen, 2015)
(a)
24
(b)
Fig. 12. Convergence of stress-strain curves corresponding to the three mesh seed sizes for 22% volume
fraction models of (a) cubic, and (b) diamond lattice structures
(a)
25
(b)
(c)
(d)
Fig. 13. Experimental compressive stress-strain curves for the cubic and diamond models at relative
densities of almost (a) 11%, (b) 22%, (c) 28%, and (d) 35%; Samples are named with the combination of
the first letter of their names together with the corresponding relative density (Ahmadi et al., 2015).
Compressive Stress (MPa)Compressive Stress (MPa)
26
27
(a) (b)
Fig. 14. Comparing experimental stress-strain data with the numerical simulation data for different
volume fractions of (a) cubic, and (b) diamond structures; Yield strains are found to be more than the
experimental results. In plateau region, higher stress variations with more amplitudes lead to deviations
from the experimental data. Nevertheless, plateau stress and consequently energy absorption capability
are well predicted.
Stress (MPa)
(a)
(c)
Fig. 15
. Deformation procedure for
confirmed by stretch dominated deformation in scaling laws.
28
(b)
(d)
. Deformation procedure for
the cubic lattice structure; Layer-by-
layer deformation mechanism
confirmed by stretch dominated deformation in scaling laws.
layer deformation mechanism
is
(a)
(c)
Fig. 16
. Failure mechanism of diamond
band of 45°,
owing to crushing diagonal layers
bending failure of tying struts perpendicular to
29
(b)
(d)
. Failure mechanism of diamond
lattice structure at 22% volume fraction;
Continuous sharing
owing to crushing diagonal layers
, is observed. Shearing of layers is
accompanied by
bending failure of tying struts perpendicular to
the diagonal plates.
Continuous sharing
accompanied by
the
