Effect of Process Parameters and Geometry on the Quasi-Static and Dynamic Behavior of Additively Manufactured Lower-Extremity Metallic Lattice Bone Implants: A Design-of-Experiment Study

Abstract

Additively manufactured metallic lattice biomaterials have revolutionized the mechanical properties of lower-extremity bone and joint implants. Most designers have followed a quasi-static approach, where material properties are solely represented by their elastic modulus. In reality, however, the human body experiences repeated dynamic and impact loads in vivo, and bone acts as a passive shock absorber and wave modulator. Most importantly, bone cells sense no load under quasi-static loading and must rather be subjected to impact loads at frequencies near their natural frequencies and high enough accelerations to conduct optimum mechanotransduction. This indicates the necessity of developing a dynamic design strategy that further considers damping and natural frequency. This research is an attempt to study the dynamic performance of selective laser melted lattice implants with the help of design of experiments and finite-element method (FEM). In many cases, lattice implants exhibit up to 80% similar values of elastic modulus and natural frequency to the bone. Regarding the damping, however, the similarity rarely reaches 10%. For the most part, the dynamic material properties of the implants are more significantly affected by their geometry and printability rather than process parameters. Damping decreases with power and increases with porosity and scanning speed. Natural frequency increases with power and remains almost constant with scanning speed. Generally, any change to the process parameters and geometry that improves the elastic modulus affects the natural frequency and damping directly and inversely, respectively. Capturing the main trends of dynamic behavior successfully, FEM accuracy is controlled by the geometry and hence can be distorted by manufacturing defects. All in all, there is a balance between the elastic modulus and damping. Dynamic performance improves with porosity, albeit up to an optimum point.

Graphical Abstract

Keywords

lattice materials damping material dynamics bone implant additive manufacturing selective laser melting process parameters design of experiment

Introduction

Thanks to their highly tunable biomechanical and architectural properties, additively manufactured metallic lattice meta-materials have been largely regarded as load-bearing bone substitutes. Most literature on the design of selective laser melted [Selective Laser Melting (SLM)-ed] metallic bone implants has followed a quasi-static approach, where the elastic performance of a material is solely represented by its elastic modulus. It has been widely accepted that implants shall exhibit material properties that are as close as possible to the bone properties. However, the term “properties” has been limited to elastic modulus only. In reality, in-vivo loading of the human body is mostly dynamic, implying that the bone has crucial dynamic roles. This highlights the existing research gap regarding the dynamic design of implants considering the material damping and natural frequency.^1–3

Yu et al.,⁴ Jiang et al.,⁵ Munford et al.,⁶ and Xu et al.⁷ followed quasi-static approaches. They emphasized that elastic modulus matching improves stress and strain distribution across the bone and prevents stress shielding to avoid bone loss, implant loosening, etc. They further suggested that porosity (or relative density), unit cell topology and its array, microstructural geometry (pore and strut size), and process parameters are the most promising controlling factors.

The human body is a dynamic structure where bones, especially in the lower extremities, are exposed to impulsive impacts and vibrational loads caused by the ground reaction force, muscle system, and inertia during physiological activities such as walking, running, and jumping.^8–10 Biomechanics of the lower limb during normal activities reveals that in the tibia and femur (along with knee and hip joints), load and acceleration can normally reach up to ∼2–8 times the body weight (even 25 for severe activities) and up to 8 g, respectively.^8–12 Also, bone experiences strain rates around 0.001–0.1 s⁻¹ in normal activities and up to 25 s⁻¹ in harsh impact conditions.^12–15 Further, physiological frequencies are mostly within ∼0.1–30 Hz.^3,16,17 As a passive shock absorber and wave modulator, bone is basically there to regulate these effects.^3,18 On top of all, bone is a mechanosensitive tissue that exhibits load sensitivity based on a dynamic scenario at the cell level. Osteocyte, as a main mechanical sensor in a mechanotransduction process, measures the bone force with its cytoplasm and dendrite deformation with two mechanisms, namely impact-induced fluid shear and impact elastic wave propagation resulting from walking, etc., and then regulates other cell behaviors.^19–23 Accordingly, when the body is at rest, no load is sensed. Load sensation has an essential role in bone life time like modeling or remodeling.

Many scholars highlighted various dynamic aspects of bone. Garner et al.¹⁸ showed that bone acts as a highly viscoelastic material and its damping is crucial for shock absorption and modulation and filtering of elastic waves. Hansen et al.¹² and Uniyal et al.²⁴ showed that bone properties are sensitive to loading and strain rates. Turner et al.,²² Lanyon et al.,²⁵ and Aiello et al.²⁶ emphasized that bone cells’ physiology is sensitive to excitation frequency, strain rate, and acceleration. Radin et al.²⁷ showed that a bone is exposed to mechanical waves at frequencies up to 2 kHz during walking. Shikata et al.²⁰ and Wu et al.²³ observed that bone cells must be excited at frequencies around their natural frequency before they can exhibit the best cellular activity. Wang et al.²⁸ commented that bone cells (e.g., osteoblast and osteocyte) have natural frequencies around <100 Hz. Warden et al.¹⁰ mentioned that athletes of impact-load sports (e.g., gymnastics) enjoy better bone mineral density and effective excitation, as compared to active sports like swimming. Therefore, bone is not a static load-bearing element but provides dynamic roles. Thus, since implants are designed to mimic bone’s mechanical behavior, implant designers must carefully consider the dynamic implications of strain rate and loading frequency rather than solely focusing on the conventional quasi-static strategy.

To the best of our knowledge, no systematic study has reported on the role of process parameters in the damping of SLM-ed lattice materials. This sheds light on two novelties of the present research: (1) following a dynamic approach to implant design and (2) exploring the effect of process parameters on elasto-dynamic performance of SLM-ed lattice biomaterials.

Materials and Methods

Lattice implant selection and design

In-vivo performance of lattice implants (or scaffolds) depends on their mechanical properties, architecture, and surface characteristics.¹³ Importantly, they must be manufacturable, considering the technical limitations of the current Additive Manufacturing (AM) machines. So, these are the main constraints of this study.

As a prime figure, lower body implants shall be able to bear enough load and exhibit mechanical properties as close as possible to bone.¹³ In a broad classification, lattice structures are either stretch-dominant or bending-dominant, being designed for load-bearing and energy-absorption applications, respectively.^29,30 As a well-known and promising stretch-dominant truss topology in quasi-static analyses,³¹ simple cubic (SC) topology was used in this study, being grounded on strut-based high-stiff, high-strength unit cells (Fig. 1A).

As a manufacturing method, SLM has known technological limits regarding pore and strut size. For small holes, clogging and ovality may occur due to sintering (partial fusion) and adhesion of free powder slags.^30,32,33 For small struts, defects (e.g., missing strut, void, microcrack, waviness, varying cross-section, etc.) may form when approaching the laser beam, depending on the melt pool dimensions.^30,32–35 In this work, following a trial-and-error procedure, the limits for strut and hole size were set to ∼360 µm.

3D microstructural features include pore size and porosity (i.e., strut thickness), which are crucial for bone cell fate prediction. A desirable implant must be osteoconductive and osteoinductive and promote mechanotransduction, vascularization, cell ingrowth, cell metabolism, and communication.^13,30 For the most part, previous researchers investigated, via in vitro and in vivo studies, pore sizes in the range of 100–1000 µm and porosities in the range of 30–90% to find optimum values.^31,36,37 In this work, computer-aided design (CAD) porosity was investigated in the range of 50–80%.

Applying the above constraints, an admissible triangular design space was developed (green area in Fig. 2). Many authors have ended up suggesting optimum pore sizes around 600 µm.³¹ Since a full-factorial design-of-experiment (DOE) model needs a rectangular domain, the gray area in Figure 2 (EFGH) was selected with the parameters listed in Table 1.

FIG. 1.

(A) 3D CAD model for single SC unit cell, and (B) as-built SLM-ed lattice specimens for uniaxial compression test (block) and modal impact hammer test for damping and natural frequency analysis (beam). CAD, computer-aided design; SC, simple cubic; SLM-ed, selective laser melted.

FIG. 2.

Admissible design area in green for SC unit cell and selected rectangular domain for full-factorial DOE study in gray (EFGH region). DOE, design-of-experiment.

Table 1.

Rectangular Design Domain Corner and Center Points

Point	Pore size (µm)	Strut size (µm)	CAD porosity (%)
E	530	360	64
F	530	530	50
G	700	530	60
H	700	360	73
I	615	445	61

CAD, computer-aided design.

As a popular load-bearing orthopedic biomaterial, austenitic 316L stainless steel was selected because of its wide availability, good printability, and acceptable biocompatibility. Metallurgically, it exhibits a face-centered cubic single-phase crystal structure (γ-Fe).^5,13,38,39

For the compression tests, specimens were designed with arrays of 10 × 10 × 10 unit cells according to ISO 13314–2011. To properly account for the damping and natural frequency of the lattice materials, we herein considered the limits of the manufacturing machine volume. Accordingly, blocks of 80 × 7.3 × 7.3 mm in dimensions were fabricated. The blocks were 15 mm solid for clamping, 3 mm solid for excitation, and remained lattice in length (Fig. 1B).

For SC unit cell, the CAD porosity equation as a function of geometry is:

P_{cad} = (1 - \frac{s^{3} + 3 s^{2} \times p}{{(s + p)}^{3}}) \times 100 %

(1)

where

P_{cad}

is CAD porosity,

s

is strut size and

p

is pore size. Also, porosity of SLM-ed part can be calculated as:

P_{slm} = (1 - \frac{W_{L}}{W_{s}}) \times 100 %

(2)

where,

P_{slm}

is experimental porosity,

W_{L}

is the lattice weight, and

W_{s}

is the solid specimen weight. The relative density can be obtained as:

\bar{ρ} = \frac{ρ_{L}}{ρ_{S}} = 1 - P_{L}

(3)

where

\bar{ρ}

is relative density,

ρ_{L}

is the lattice material density,

ρ_{S}

is solid material density, and

P_{L}

is the lattice material porosity.

Material dynamics

The general dynamic equation of motion in the time domain for the studied materials is as follows:¹

m \ddot{u} + c \dot{u} + ku = F

(4)

in which

u

is displacement,

m

is mass constant,

c

is damping constant,

k

is stiffness constant, and

F

is external force. A practical approach to measuring material dynamic property is the free vibration test (F = 0), where the governing equation is as follows:¹

\ddot{u} + 2 {ξ ω}_{n} \dot{u} + ω_{n}^{2} u = 0

(5)

where:¹

ω_{n} = \sqrt{\frac{k}{m}}

(6)

ξ = \frac{c}{2 \sqrt{m}}

(7)

k = f (E . Geometry)

(8)

In the above equations, $ω_{n}$ is natural frequency, $ξ$ is the modal damping ratio, and E is elastic modulus. Another useful expression of damping is tan(δ), usually referred to as loss tangent or loss factor. tan(δ) is suitable for material damping because it is directly proportional to the elastic modulus of the material, as follows:^1,3

\tan (δ) = \frac{E^{''}}{E^{'}} \approx 2 ξ

(9)

in which

E^{'}

is the storage modulus (≈ E) and

E^{''}

denotes loss or artifichial modulus.³ Dynamic properties of lower-extremity bone are compiled in Table 2. Here, damping values at physiological frequencies were taken as reference data.

Table 2.

Dynamic Material Properties of Human Lower-Extremity Bone

Parameter	Bone site/condition	Value
Elastic modulus (GPa)	Femur (Wet, Cortical, Longitudinal)	15,⁴⁰ 17,⁴¹ 17.2,⁴² 15.6–18.3,⁴³ 15 wet & 18.9 dry,⁴⁴ 10–21⁴⁵
	Tibia (Wet, Cortical, Longitudinal)	18.1,⁴² 15⁴⁶
	Fibula (Wet, Cortical, Longitudinal)	18.6,⁴² 15,⁴⁶ 19⁴⁷
Damping; tan (δ)	Dry	0.01 For 0.001–100 Hz^3,18
Damping; tan (δ)	Wet	0.02 For 0.1–100 Hz¹⁸
Density (kg/m3)	Dry	1800¹⁸
Density (kg/m3)	Wet	2100¹³

Manufacturing of lattice implants by SLM

Lattice specimens were fabricated from gas-atomized AISI 316L powder (D10 = 15 μm, D50 = 30 μm, D90 = 45 μm) with a 300W SLM machine (Noura M100P, Noura Co., Iran). The main parameters of manufacturing are listed in Table 3. Only the laser power and scanning speed were varied, and other parameters were set according to the used powder and manufacturer’s recommendations. Once the process was completed, parts were wire-cut by electrical discharge machining (EDM), and loose powders and slags were ultrasonically cleaned in an acetone bath. The ranges of laser power and scanning speed were adjusted via trial and error to ensure that the produced parts had acceptable morphological errors (Table 3). The following equation expresses the volumetric energy density of the laser:

E_{V} = \frac{P}{V \times t \times h}

(10)

where P is the laser power, V is the scanning speed, t is the layer thickness, and h is hatch spacing. According to Table 3, the volumetric energy density was between 72.4 J/mm³ and 95.5 J/mm³ depending on the set parameters.

Table 3.

SLM Parameters

Parameter	Value
Scanning strategy	Meander
Hatch space, h (μm)	60
Layer thickness, t (μm)	30
Laser spot size, S (μm)	80
Power, P (W)	150–172 (according to DOE)
Scanning speed, V (mm/s)	1000–1150 (according to DOE)
Hatch rotation angle (8)	0,90
Laser system	Yb-Fiber (Additive manufacturing grade, λ = 1064 nm)
Laser mode	Continuous, Gaussian spatial distribution
Inert gas	Ar
Oxygen level	0.2 %
Volume contour offset (μm)	30
Contour beam compensation (μm)	67

DOE, design of experiment.

DOE study

A full-factorial statistical model was used to organize test data to investigate the effects of the adjusted parameters and their interactions on the output with a minimum number of experiments. Accordingly, a regular 2-level 4-factorial (see Table 4) design was considered (with 2⁴ = 16 corner points), and 4 extra center points were further included to enable statistical checking of the curvature on the fitted model. After all, 20 randomized tests were designed. Data analysis was performed with Design-Expert 12 software, and the variance analysis (ANOVA) was further conducted. According to the null hypothesis, the response was significantly associated with neither input variables nor their interactions. To test the null hypothesis, we focus on the p value by categorizing this statistic at three levels, namely <0.05 (moderate), 0.05–0.1 (weak), and 0.1 < (insufficient). So, parameters with p values below 0.05 and above 0.1 were considered significant and insignificant, respectively. Moreover, results with p values in the range of 0.05–0.1 were identified as arguably significant in this study.⁴⁸

Table 4.

Design-of-Experiment Parameters Along with Their Actual and Coded Values

	Levels
Design factors	−1	+1
A: pore size (µm)	530	700
B: strut size (µm)	360	530
C: power (W)	150	172
D: scanning speed (mm/s)	1000	1145

A first-order polynomial regression model with two-way interactions was employed for predicting the response as follows:

R = β_{0} + \sum_{i = 1}^{4} β_{i} X_{i} + \sum_{i = 1}^{4} \sum_{j = i + 1}^{4} β_{ij} X_{i} X_{j}

(11)

where

X_{i}

and

X_{j}

are independent input parameters,

R

is an arbitrary response,

β_{0}

is a constant coefficient,

β_{i}

refers to linear coefficient, and

β_{i j}

is related to the interaction coefficient. It is worth noting that, for the sake of simplicity, the inclusion of factors in the regression equation was done with model reduction by stepwise backward elimination of insignificant factors while keeping R² in an acceptable range.⁴⁹

Fem

Linear elastic FEM was implemented in ABAQUS 2017 (as an artificial laboratory) to analyze the mechanical properties of micro-sized lattice materials (Fig. 3). Defect-free perfect geometries and isotropic materials were considered. Tetrahedral elements C3D10 were used to mesh the model. The implicit solver with the general procedure was used in a displacement-controlled approach for evaluating the quasi-static runs, constraining the kinetic energy to less than 5% of the internal energy. Damping and natural frequency were analyzed using modal dynamics and frequency procedures, respectively. For each part of the computation, grid independence analysis was performed beforehand to opt for the optimum mesh size. After a couple of preliminary trials, it was found that modeling the platens of the compression testing machine and its friction effect is not necessary for small displacements. It was further figured out that FEM is not sensitive to the size of lattice structures so the model could be run just for single unit cells whenever applicable.

FIG. 3.

FEM setup for (A) quasi-static compression test and (B) modal dynamic model for natural frequency and damping analysis. FEM, finite-element method.

An effective way to resolve the damping matrix in FEM analysis is to treat it as equivalent Rayleigh damping, which is proportional to a combination of stiffness and mass matrixes, as follows:¹

[C] = α [M] + β [K]

(12)

Some authors suggested that the matrix mass-equivalent damping is negligible, making it sufficient to consider just stiffness-equivalent damping. This can be estimated by solving the equation for mode shapes as follows:^1,50

ξ_{i} \approx \frac{{β ω}_{n . i}}{2}

(13)

In modal dynamic analysis, the so-called direct modal or viscous damping ratio (ξ) was used, which can be approximated using Equation 9 for small values.¹ The basic material properties are demonstrated in Table 5.

Table 5.

Reference Material Properties of Solid AISIS 316L and Fresh Bone, as Input Data to Finite-Element Method Analysis

Parameter	AISI 316L	Fresh/wet bone
Elastic modulus (GPa)	146 (measured)	17¹⁸
Density (kg/m³)	7360 (measured)	2100¹³
Poison ratio	0.3	0.3³
Damping, tan (δ)	0.00108 (measured)	0.02¹⁸

Characterization of implants

Elastic modulus measurements were done via quasi-static analysis based on uniaxial compression tests at a constant strain rate of 0.001 s⁻¹ on an INSTRON 5500 (20 kN load cell). Normal human activities like walking and running produce impulsive excitations in the form of impact loading to the lower limb. Structural dynamic features (e.g., damping and natural frequency) were measured by an instrumented modal impact hammer (CRYSTAL Instruments with EDM internal modal software Version 5.0.1.) via a resonant method (Figs. 4 and 5). Accelerations were measured using a contactless laser Doppler vibrometer as an optical measurement system (Laser Point LP01). Indeed, the natural frequency was calculated by a frequency response function in the frequency domain, and the damping was extracted from the free vibration decay curve in the time domain as follows:³

\tan (δ) ≅ \frac{1}{π} \frac{T}{t_{1 / e}}

(14)

in which

T

is the vibration period and

t_{1 / e}

is the time it takes to have

1 / e

attenuation in amplitude.

FIG. 4.

Experimental instrument for modal impact hammer test.

FIG. 5.

Examples of (A) a free vibration velocity decay response (related to specimen 12) and (B) a frequency response function (FRF) curve (related to specimen 12).

Results and Discussion

Full-factorial DOE

Design parameters, their levels, and responses are listed in Table 6. Responses include dynamic material parameters. The difference between experimentally measured and FEM-calculated elastic moduli provides a conveniently quantifiable measure of manufacturing defects.

Table 6.

Design-of-Experiment Matrix Properties Along with Experimental and Finite-Element Method Responses

Run					Responses
	Uncoded factors				ElasticModulus, E,(GPa)		Damping,tan (δ)		First natural frequency, ¯ω_n(Hz)		Elastic modulus error (%)
	Pore size (µm)	Strut size (µm)	Power (ω)	Scanning speed (mm/s)	Exp.	FEM	Exp.	FEM	Exp.	FEM	—
1	530	530	172	1145	23.04	43.8	0.0012	0.00117	894	1040	47.5
2	700	360	172	1000	3.37	17.5	0.0018	0.00135	734	941	80.6
3	700	360	150	1000	4.47	17.5	0.0019	0.00135	701	941	74.2
4	530	360	150	1145	3.34	28.8	0.0018	0.00125	723	958	88.6
5	530	360	150	1000	4.10	28.8	0.00135	0.00125	729	958	85.8
6	530	530	150	1000	16.18	43.8	0.00125	0.00117	902	1040	63.2
7	700	360	172	1145	3.36	17.5	0.0015	0.00135	695	941	80.7
8	530	530	150	1145	7.78	43.8	0.0012	0.00117	871	1040	82.3
9	530	360	172	1145	4.88	28.8	0.0015	0.00125	810	958	83.1
10	700	530	172	1145	11.41	30.0	0.0011	0.00128	941	1057	61.1
11	530	360	172	1000	3.40	28.8	00012	0.0015	768	958	90.0
12	700	530	150	1000	12.68	30.0	0.0012	0.00128	821	1057	57.0
13	615	445	161	1072.5	7.68	29.5	0.0011	0.0012	807	1041	74.4
14	615	445	161	1072.5	8.23	29.5	0.0012	0.0012	797	1041	72.6
15	530	530	172	1000	25.17	43.8	0.00116	0.00117	900	1040	43.0
16	700	530	172	1000	14.54	30	0.0011	0.00128	847	1057	50.0
17	700	360	150	1145	1.97	17.5	0.0018	0.00135	652	941	88.0
18	700	530	150	1145	5.85	30.0	0.0011	0.00128	814	1057	80.1
19	615	445	161	1072.5	5.12	29.5	0.0017	0.0012	861	1041	82.9
20	615	445	161	1072.5	8.28	29.5	0.0014	0.0012	777	1041	72.4

E, tan(δ) and ¯ω_n of fresh bone are 17 GPa, 0.02 and 760 Hz, respectively.

FEM, finite-element method.

Results of ANOVA and mathematical regression analysis

Table 7 presents the results of ANOVA in terms of p values, and Table 8 lists the regressed models of the responses.

Table 7.

The p values of Design Factors, Their Interactions, Model, Curvature, and Lack-of-Fit, as Obtained by the ANOVA

Source	Elastic Modulus	Damping, tan (δ)	First natural frequency	Elastic modulus error
Model	<0.0001*	0.0551	0.0017	<0.0001
A (Pore Size)	0.0038	0.2598	0.0220	0.4895
B (Strut Size)	<0.0001	0.0015	<0.0001	<0.0001
C (Power)	0.0024	0.1822	0.0264	0.0009
D (Scanning Speed)	0.0180	0.7327	0.9891	0.0032
AB	0.0088	0.0998	0.4821	0.0570
AC	0.0511 ^**	0.9906	0.5774	0.1088
AD	0.6961^***	0.2181	1.0000	0.1172
BC	0.0031	0.4108	0.8479	0.0013
BD	0.0323	0.4776	0.4903	0.0185
CD	0.0764	0.7152	0.2268	0.0154
Curvature	0.1303	0.9960	0.6298	0.1765
Lack of Fit	0.3103	0.9745	0.7327	0.5578

Bold values show significant parameters (p value < 0.05).

Italic values show parameters with 0.05 < p value < 0.1.

***

Parameters with p value > 0.1 are not significant.

Table 8.

Regression Models of Responses and Least Squares Fitting (R²)

Regression model	R²
Elastic Modulus = 70.38 + 0.2586A + 0.0761B-1.3021C-0.1224D-0.0002A × B-0.0011A × C + 0.0020B × C-0.0002B × D + 0.0011C × D	0.96
Damping = −0.011712 + 0.000019A + 4.31417E-06B-1.41389E-06C + 0.000014D-1.30623E-08A × B-6.68449E-10A × C-1.10548E-08A × D + 4.74599E-08B × C-6.18661E-09B × D-2.42947E-08C × D	0.79
Natural Frequency = 250.06–0.288A + 0.8661B + 2.1363C	0.88
Elastic Modulus Error = −583.016–0.148028A-0.004001B + 6.07384C + 0.472521D + 0.000313A × B-0.005267B × C + 0.000487B × D-0.003918C × D	0.93

Considering the obtained values of elastic modulus and p value, it is evident that the effects of all parameters and their interactions are significant. The regressed model is hence significant, and the obtained levels of lack-of-fit and R² = 0.96 suggest the suitable fit of the mathematical model to the experimental data. The negligible value of the curvature term suggests the linear nature of the changes in response upon varying the inputs.

Focusing on the damping, the obtained p values indicate that only geometrical parameters, including strut size and its interaction with pore size, are significant. The pore size effect can be attributed to the porosity effect. The developed model exhibits an acceptable p value, making it significant. Based on the obtained values of lack-of-fit and R² = 0.79, the regressed model fitted to the experimental data with acceptable accuracy. The model reduction was not practiced for damping as it would reduce the R². Linear approximability of the responses by the inputs was confirmed for the damping as well.

Concerning the natural frequency, the obtained p values indicated the significance of the effects of the strut size, pore size, and laser power. The developed model was significant, and the resulting lack-of-fit and R² = 0.88 highlighted the acceptable fitness of the regressed model to the experimental data. Results showed that the response could be linearly approximated by the input parameters.

Regarding the approximation errors, it was figured out that B, C, D, BC, BD, and CD parameters imposed significant effects, and the regressed model fitted to experimental data reasonably. Further, the curvature term was found to be negligible, suggesting the linearity of the relation between the responses and the input parameters.

To sum up, results suggest that the geometrical parameters are more effective than the process parameters as far as dynamic material properties are concerned. Moreover, porosity was found to contribute to damping significantly.

Comparison of FEM results and experimental data: manufacturing defects

A solid sample was SLM-ed, and its elastic modulus, density, and damping were measured at 7360 kg/m³, 146 GPa, and 0.00108, respectively, and fed to the FEM. Elastic modulus and damping are the most important dynamic material properties. Figure 6 compares measured values and FEM-estimated levels of these two properties. Figure 7 compares errors in estimating elastic modulus on a perturbation plot, as obtained by the DOE study, showing the main effects simultaneously.

FIG. 6.

Experimentally measured and numerically estimated values of elastic modulus and damping for the lattice implants.

FIG. 7.

Perturbation plot of average elastic modulus error (difference between FEM outputs and experimental data, as a conveniently quantifiable measure of manufacturing defects).

The FEM tends to overestimate the elastic modulus by [43.0%, 90.0%] as it is known to assume a perfect CAD model rather than taking possible manufacturing defects into account. This is while an SLM-ed lattice part is inherently full of various defects, such as voids, cracks, imperfect fusion, agglomerated powders, composition alterations, missing struts, varying cross-section, and waviness, which are known causes of effective stiffness loss via such mechanisms as reducing effective strut cross-section, stress concentration, promoting buckling, etc. It is further evident that the error decreases significantly with elastic modulus.

Regarding damping, the obtained errors fell in the range of [−50.0%, +20.0%], increasing with the manufacturing defects. Generally, modal dynamic FEM could predict damping and its trend with acceptable accuracy. Apparently large errors root in the fact that manufacturing defects and their damping mechanisms are not considered in the FEM. Another source of error is that the modal damping procedure employs viscous damping mechanisms where damping is recognized as proportional to velocity rather than strain.

Considering the manufacturing defects error, the perturbation plot points to the strut size, laser power, velocity, and pore size as significantly effective parameters (mentioned in the increasing order of effect). In an SLM printer, there are technological limitations for feasible minimum strut and pore size, near which a jump occurs in the error. Results indicate that the selected minimum admissible strut size of 360 µm was very close to that limitation. Nevertheless, the selected minimum permissible pore size of 530 µm was unsafe as it was too close to the printer limitation. That is to say, the combination of laser beam positioning error and the melt pool dynamics keeps one from properly achieving a strut size of 360 µm (melt pool size has been measured at ∼1.5–3 times the beam diameter,⁵¹ while it is anticipated to be ∼120–240 µm in this work). For too small struts, the exposure time is less than enough for strut fabrication due to the resulting short scanning paths.

Higher laser energy is expected at higher power and lower scanning speeds. This can heal the defects via such mechanisms as full melting, remelting of previous tracks, crack closure, reduced void, crack formation, etc. It is worth noting that there is a limit for input energy above which one should expect spattering, vaporization, composition alteration, slag adhesion, clogging, keyhole void, etc.^5,52 Results indicate that the limit was not exceeded in this study, so the laser energy followed a monotonically increasing trend.

Results showed that the error decreases with pore size. Indeed, a smaller pore size results in a larger volume of heat-affected powder coupled with shorter cooling time due to the heat accumulation caused by increased effects of start-stop and turning points and longer exposure time. In other words, the free powder will not have enough time to cool down to sub-sintering temperatures. These slags and accumulated particles cause reduced flowability of the powder, resulting in non-uniform and defective distribution of the next powder layer and, hence, more defects.

Effects of different parameters and their interactions on dynamic material properties

Figure 8 illustrates the perturbation plot of responses against individual parameters and their interactions.

FIG. 8.

Perturbation plot of average responses: (A) elastic modulus, (B) damping, and (C) natural frequency.

Elastic modulus analysis

According to Figure 8A, the largest impact on the elastic modulus was that of strut size followed by pore size, laser power, and scanning speed. Based on the experimental data, the results for the lattices (Table 6) differed from those for fresh bone by [−88%, +48%]. Findings suggest that the porosity and printability impose crucial effects on the elastic modulus.

Figure 9A–D shows how interactions between the individual parameters affect the response. Overall, increasing the strut size and/or decreasing the pore size increases the relative density, thereby improving the elastic modulus. Getting away from the technological limit of minimum feasible strut size reduces defects and substantially improves stiffness.

FIG. 9.

Interaction plot of individual parameters as they affect elastic modulus: (A) strut size-pore size, (B) scanning speed-power, (C) power-strut size, and (D) scanning speed-strut size.

In general, higher powers and lower scanning speeds improve elastic modulus by boosting the applied energy and, thus, increasing the effective strut size and printability by complete melting. Indeed, a higher scanning speed can lower the applied energy and, hence, interrupt the stability of the melting process, which causes such defects as humping, balling, and defective fusion. Inadequate laser power can further lead to such defects as defective fusion, trapped powders, voids, and cracks, not to mention the resulting drop in laser energy absorption.

Figure 9A–D suggests that, at a strut size of 360 µm, variation of process parameters cannot improve stiffness because of marginal closeness to the technological limit of the printer.

Damping analysis

The research on the damping of SLM-ed lattice materials is very limited.^50,53,54 Damping refers to the capacity to dissipate vibratory energy by internal friction or hysteretic loss.⁵⁵ In metals, damping is mainly controlled by the strain, temperature, composition, metallurgical structures, and phases, with small contributions from the frequency (or strain rate).^3,53,55–57 Above all, the strain (or amplitude) has a crucial role, so that the damping is strongly correlated to the strain.^55,56 In particular, after a certain threshold, the damping increases rapidly with the strain.⁵⁶ Plastic, rather than elastic, strain results in much higher damping. Metallurgical phase transition during heating and cooling strongly affects the damping.³ Generally, association of damping with frequency is weak in metals.³ Austenitic metals exhibit lower damping capacity compared to ferritic and martensitic ones. Lamellar phases exhibit larger damping capacities.⁵⁸ Multi-phase structures are also known to exhibit higher damping capacities than single-phase structures.⁵³ For particular solutes, atomic motions increase the damping.³ Heat treatment alters the damping capacity of metals, usually decreasing it.^53,58 Strain and stress gradients improve the damping through the microscopic flow of thermal energy in a phenomenon called thermoelastic damping.³ Manufacturing defects increase the damping capacity.⁵⁹ Coulomb damping (or dry friction damping) due to the rubbing effect at cracks, pores, and defective fusions increases the damping.¹ Grain boundary slip results in a higher damping.⁵⁶ Displacement of 1D and 2D defects (e.g., high-density, high-mobility dislocations) boosts the damping.³ Microplastic strains in cracks and pores represent another source of significant increases in damping.⁵³

According to Figure 8B, the most effective parameter on the damping was found to be the strut size, followed by pore size, laser power, and scanning speed. Here the experimental data on the lattice (Table 6) differed from the fresh bone by −94% to −90%. Figure 10 shows how interactions among different individual parameters affect the damping as the response. In general, the results emphasize the critical role of the strain magnitude in the damping. In fact, any change that increases the strain level (e.g., a reduction of elastic modulus, an increase of defects, or a higher porosity) tends to increase the damping. Internal friction mechanisms (e.g., thermoelastic damping, microplastic deformation at defects and corners, coulomb damping in defects, movement of dislocations, etc.) are intensified with strain. Additionally, the lattice-specific Brag scattering and fluid damping increase with the porosity.

FIG. 10.

Interaction plot of individual parameters as they affect damping: (A) strut size-pore size and (B) scanning speed-power.

On top of all, the large difference in damping between the lattice implant and fresh bone can significantly reduce the implant’s ability to mimic the shock-absorbing, wave-modulating, and wave-filtering functions of the bone in vivo.

Natural frequency analysis

Natural frequency is proportional to $\sqrt{k / m}$ . The measured value of the natural frequency for the solid SLM-ed specimen was about 1002 Hz, while the FEM predicted the fresh bone natural frequency at 760 Hz. The experimentally measured natural frequency for the lattice implant (Table 6) differed from the estimated natural frequency for the fresh bone by [−14%, +23%]. Generally, the FEM tended to overestimate the experimental data.

According to Figure 8C, the natural frequency was most affected by the strut size, followed by the pore size, laser power, and scanning speed. Figure 11 shows how interactions of the individual parameters affect the natural frequency, as the response. Figure 11A clearly shows that the natural frequency decreases with porosity. In general, findings indicated that all mechanisms increasing the stiffness have the same effect on the mass and improve the natural frequency.

FIG. 11.

Interaction plot of individual parameters as they affect natural frequency: (A) strut size-pore size and (B) scanning speed-power.

Stiffness-loss map

Figure 12 presents the stiffness-loss map for all lattice specimens along with the solid sample (s), dry and fresh bone samples, and some conventionally fabricated solid lower-body biomaterials in log scale.

FIG. 12.

Stiffness-loss map of SLM-ed lattices and solid AISI 316L with respect to dry and fresh bone and some conventionally manufactured lower-body biomaterials.

In general, the porosity is seen to lower the elastic modulus while improving the damping, getting the lattice implants closer to the fresh bone. However, there is seemingly an optimum range beyond which any further improvement of damping sacrifices elastic modulus. Indeed, Ashby limit law² is valid here and cannot be ignored. There is a balance between the elastic modulus and damping, highlighting the technological challenge of developing materials that offer simultaneously better levels of both properties.

To sum up, as far as bone implant requirements are concerned, the results of elastic modulus are promising, though the damping needs more effort. Moreover, all specimens were seen to fall behind a threshold line in the stiffness-loss domain (shown as the dashed red line in Fig. 12); the threshold line is expressed as $E \times {\tan (δ)}^{6.04} = 10^{- 15.51}$ , a constraint that cannot be exceeded with the employed printing system in this research.

Conclusions

The quasi-static approach is currently dominant in bone implant elastic design, which is based only on elastic modulus. An innovative attempt to improve the design philosophy of lower-body bone lattice implants by considering the dynamic nature of elastic modulus, damping, and natural frequency was presented. Knowing that conclusions may depend on the printing system (i.e., printer type, powder quality, process parameters, lattice geometry, etc.), the following conclusions can be drawn based on the studied system: (1)

The similarity of the elastic modulus and natural frequency between lattice implants and fresh bone can be as high as 80% and above. This is while the similarity falls to levels below 10% when it comes to damping.

(2)

Geometrical features, rather than process parameters, were found to impose larger impacts on dynamic material properties. The mechanisms that improve elastic modulus tend to decrease the damping while enhancing the natural frequency.

(3)

Damping is mainly influenced by strut size and its interaction with pore size, which can be attributed to the porosity effect. In general, it improves with porosity. Moreover, although the effect of process parameters on damping is little, it increases with the scanning speed and decreases with the laser power. Printability and manufacturing defects significantly affect the damping.

(4)

Natural frequency increases with the strut size and laser power and decreases with the pore size. Scanning speed has little effect on the natural frequency.

(5)

In general, dynamic properties improve with porosity up to an optimum point. Printability plays a key role within an optimum region. There is a balance between elastic modulus and damping, and this balance can be expressed by the threshold line expressed as $E \times {\tan (δ)}^{6.04} = 10^{- 15.51}$ , which could not be exceeded in this study.

Acknowledgement

All the authors confirm that this research is supported by Amirkabir University of Technology, which is an institution that is primarily involved in education and research.

Authors’ Contributions

All authors contributed equally to the research.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

No funding was received for this article.

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