Estimation of elastic properties of an additively manufactured lattice using resonant ultrasound spectroscopy^a)

Selective laser melting (SLM) is an additive manufacturing process in which a three-dimensional (3D) object is created by selectively melting layer upon layer of a base metal powder. A comprehensive review of this technology is presented by Yap et al.¹ The design and build process is controlled by computer enabling the creation of a wide range of near net, non-stochastic periodic structures possible. The SLM process allows engineers and designers to control thermal, chemical, dynamic, and geometrical characteristics of a structure from a base material. Using SLM, structures can be tailored to achieve unique design and engineering requirements.

In addition to design and fabrication advances, there exists a strong emphasis for the development of improved approaches for the characterization and inspection of these additively manufactured structures. The traditional approach for measuring linear elastic material properties falls into two general categories, quasi-steady state methods, using tensile testing apparatus to measure the stress-strain behavior,² and dynamic methods, using elastic wave propagation measurements.^3–6 In the former case, engineered sample specimens are created to ensure controlled uniform strain states as the sample is exposed to increasing stress loading. The recorded stress-strain curve is then used to determine the elastic properties of the sample.² The challenges with quasi-steady measurements (tensile and compression testing) are that the properties of the lattices that the designer is interested in are at two different length scales. The individual struts in the lattice can be several orders of magnitude smaller than the aggregate structure, thus introducing uncertainties as to how the larger structure will behave under operational loading conditions.⁷ 

The problem of determining the mechanical properties of 3D lattice structures is complicated by the interactions between the topology of the lattice, unit cell size, relative lattice density, and uniformity of individual strut dimensions.⁸ The topology of the unit cell controls whether the lattice is elongation dominated (octet and similar types of symmetry) or bending dominated as in the case of a rhombic dodecahedron (RD).^9,10 Control of these aspects are important to the designer as the application may require trade-offs between the two feature spaces,¹¹ leading to even more reliance on developing a comprehensive understanding of the static and dynamic behaviors of these lattice systems.

An analysis of the lattice type of structures has been investigated by numerous researchers, incorporating a wide range of analytical and numerical tools. Babaee et al.,¹² developed an analytical approach, using beam theory to analyze the unit cell of several lattice geometries. However, extending this analysis to larger structures requires consideration of the periodicity of the lattice topology. Analytical-numerical approaches using beam theory and Floquet-Bloch principles to analyze dispersion characteristics of lattice structures have been investigated.¹³ Fully numerical solutions using finite elements (FEs) to model the individual truss elements of the unit cell are routinely used to model and analyze complicated lattice geometries.^14,15

Model verification and measurements pose significant challenges for lattice structure materials. When applying traditional quasi-static testing approaches to these materials, experimental data are usually recorded along single dimensions using standard testing protocols. The results are limited to measuring strains along the dimension to which the extensometers are applied, which tends to be in the direction of the applied force.¹⁶ The complex mechanical testing apparatus can pose significant measurement challenges with the potential to introduce large uncertainties into the results. Experimental determination of the elastic properties associated with anisotropic materials complicates the measurement process even further. Special sample geometries and mounting fixtures are required to analyze these reduced symmetry materials. Typically, multiple measurements are required on geometrically consistent samples in the various symmetry planes to isolate the elastic moduli.

Dynamic approaches using propagating elastic waves, such as ultrasonic time-of-flight measurements, have similar requirements for anisotropic materials in that multiple sample paths and geometries are required to measure the sound propagation and determine the elastic coefficients along the various symmetry axes.¹⁷ Lattice structures pose the additional challenge that propagating wavelengths are band limited and large relative to the sample structures. The potential differences in length scales between the unit cell of the lattice and the larger structure constrain the propagation of the associated wavefield. Higher frequencies simply will not propagate effectively and at lower frequencies, the length scales become impractical.

An alternative approach for determining elastic material properties uses the fact that the vibrational modes (e.g., resonances) of an elastic structure are related to the elastic moduli through a solution of an eigenvalue problem.^18,19 The process, referred to as resonant ultrasound spectroscopy, has been well documented in the scientific literature as a method to determine the elastic coefficients of a material by measuring the natural frequencies of a sample, such as a regular parallelepiped, and comparing these measurements to computed solutions of the eigenvalue problem.²⁰ The model parameters (elastic coefficients) are iterated to minimize the difference between the measured resonance peaks and the computed values in a least squares sense. A wide variety of materials, ranging from high symmetry (isotropic) to low symmetry anisotropic solids over significant temperature ranges, have been effectively analyzed using resonant spectral ultrasound (RUS) methods.^21,22 RUS presents an opportunity to analyze a lattice structure over a wider frequency range, encompassing not only the resonant behavior of the unit cell but also interactions at larger structural scales. This is due to the fact that the system can be driven at discrete frequencies long enough to ensure that the system reaches a dynamic equilibrium whereby even weakly coupled resonance modes are excited above the measurement noise floor.²³ 

In this article, RUS is used to measure the elastic material coefficients of lower symmetry metal additive manufactured (AM) lattices. The investigation will focus on comparing the elasticity coefficients, which are determined using the RUS process on numerical and experimentally measured eigen-modes. The evaluation is performed on a representative AM lattice sample with a homogeneous base material and defined topological characteristics of the unit cell. In Secs. II–IV, an overview of the theoretical concepts associated with RUS and the inversion process is presented. A brief description of the experimental process is given to show how the experimental eigen-frequencies are determined. Similarly, a 3D FE model of the lattice sample will be used to calculate numerical eigen-frequencies. Both numerical and experimental eigen-frequencies will then be inverted, using a stochastic optimization process, to calculate the effective elastic coefficients for the lattice sample. Generalized stress and strain relations are utilized to relate the resonant spectral measurements back to engineering properties that can be measured using traditional tensile testing apparatus. Finally, a discussion of the results will conclude this article.

Resonant ultrasound spectroscopy relies on the ability to accurately estimate the natural frequencies of an elastic solid. Generalized development of the background theory has been thoroughly represented in the literature cited above. Migliory and Sarrao²⁴ have authored a comprehensive text, highlighting the history, theoretical developments, and experimental applications of RUS for a wide variety of scientific fields. A brief outline of the theory is presented here.

Consider an arbitrary shaped elastic solid with a volume, V, surface area, S, density, ρ, and an elastic tensor, c_ijkl. The surface has the additional requirement that it is free of any imposed displacements and tractions. For this idealized system, Hamilton's principle states that the time integral of the Lagrangian, L, defined as the difference between the kinetic and potential energies of the system (L = KE − PE), is constant. Thus,

where the kinetic and potential energies are defined as

and

where u_i represents the displacement field, c_ijkl is the elastic stiffness tensor, ρ is density, e^iwt is the time dependence, and the indical notation follows Einstein's convention of summing over repeated indices. An exact solution for Eq. (1) exists for only a few specific cases with isotropic materials and very simple geometries such as cubes and spheres. Approximate solutions using the Galerkin²⁵ or Ritz²⁶ methods are required for anisotropic materials with complicated geometries. The Ritz approximation is an effective solution for eigenvalue problems as long as the Lagrangian remains stationary for small perturbations of the eigen-functions. A solution to Eq. (1) can be approximated with a suitable set of basis functions, $ϕα$⁠, defined as

where a_iα represents a vector of coefficients and α = (l,m,n) is a set of nonnegative integers. Substitution of Eq. (4) into the Lagrangian results in the following system of equations:

with $E$ and $Γ$ defined as

and

The volume integrals defined in Eqs. (6) and (7) are solvable, provided a suitable set of basis functions, ϕ_iα, can be defined. Equation (5) can now be posed as an eigenvalue problem

which can be solved with standard mathematical solvers.^27,28

Visher et al.²⁹ developed a simple and very computationally efficient set of basis functions from the product of powers of the coordinate axes.

This general solution can be applied to rectangular and even complicated curved geometries, provided a sufficiently large order of the approximation is chosen. The powers {a,b,c} are typically limited to the set satisfying a + b + c ≤ N_p, where N_p is the on the order of ten.

The inverse problem of determining the elastic coefficients from measured resonance spectra is the more useful feature of this approach. However, the forward solution defined in Eq. (8) is an approximation and cannot be inverted. The solution to the inversion process relies on an iterative approach by minimizing a cost function F(a_i), defined as

where for a given set of M measured eigen-frequencies, $fiMeas$⁠, we seek to improve our estimate (i.e., $fiFit$⁠) of the system by iterating the set of elastic parameters, a_i, in the forward model such that Eq. (10) is minimized. The behavior of the objective function is quadratic near extremes, therefore, non-linear approaches to search the parameter space for the optimal solution are required (e.g., least squares and, recently, stochastic methods).^22–24,30

RUS offers a powerful approach for estimating the material properties for samples with simple regular geometric shapes. The samples, however, are typically solid and characterized as homogeneous elastic continuums with a defined crystallographic symmetry. In Secs. II B and II C, this technique will be applied to estimate the anisotropic elastic properties and additively manufactured lattice structure with orthotropic symmetry.

Lattice topologies can be designed with specific features and characteristics, depending on the application space. One common topology used for applications where lightweight, stiffness, and vibrational damping characteristics are sought after is the RD.^9–11 Recent investigations of this topology have shown that its mechanical behavior is bending dominated, providing desirable energy dissipation characteristics.³¹ 

The basic construction of a RD lattice begins with a unit cell comprising 12 identical rhombic faces with 24 edges and 14 vertices. Figure 1 shows the construction of the RD lattice where the entire structure can be fabricated from a series of mirrored reflections of a basic “v” shape shown in Fig. 1(a). Here, the planar angle between the two struts is β = 109.47°. The RD unit cell, shown in Fig. 1(b), is then tessellated into a larger structure with (4 × 4 × 4) repetition. The final structure is sandwiched between two solid plates as shown in Fig. 1(c). Dimensions of the structure are given in Table I. A RD has orthotropic symmetry, resulting in three orthogonal symmetry planes. Materials with orthotropic symmetry have nine independent elastic coefficients.

The computer-aided design (CAD) model provides the geometrical template for the fabrication process. A test sample of this lattice was fabricated by Proto-Labs (Maple Plain, MN), using 316 L stainless steel as the base material with a Youngs modulus, E_o = 205 GPa, Poisson's ratio, ν_o = 0.28, and density, ρ_o = 7800 kg/m³. Details of the fabrication process are available in the open literature and not presented here.^1,31,32

The CAD file also provides the geometrical data required to effectively generate a 3D FE mesh. The mesh is then used with established FE approaches to numerically determined the eigen-frequencies and eigen-modes of the lattice sample.

A 3D FE model of the lattice structure shown in Fig. 1(c) was analyzed using COMSOL Multiphysics.³³ Geometrical dimensions for the model are listed in Table I. Here, m is the actual mass of the lattice structure. The model mass is calculated by multiplying the total internal volume of the CAD model by the density, ρ_o, of the base material. The sample mass was determined by direct measurement. Finally, the effective density of the lattice structure is calculated as ρ_eff = m / V, where V = (L_x × L_y × L_z), and defines the total volume occupied by the sample.

The eigenvalues of the lattice were calculated using a direct solver routine (MUMPS),³⁴ provided in COMSOL. In the numerical solution, the boundary condition on the entire outer surface domain is modeled as a free surface with no displacement or traction boundary conditions. The mesh is refined until the square of the difference between two successive estimates of the highest eigen-frequency divided by the square of the refined eigen-frequency is less than 0.1%. Computational resources limited the upper bound of the modeled eigen-modes to frequencies less than 12 kHz.

The first five eigen-modes are shown in Fig. 6. The order of the mode shapes follows closely to what has been presented in Ohno²⁰ for an orthotropic parallelepiped. The first mode is shown in Fig. 6(a) and corresponds to an eigen-frequency of 3996 Hz. This mode shows an aggregate torsional behavior about the z axis. The second mode at 4007 Hz [Fig. 6(b)] has a distinct symmetrical flexural shape and is a doublet due to mid-plane symmetry in the x-y plane. The third mode [Fig. 6(c)] at 4143 Hz is the first of the breathing modes and symmetric about each of the three orthogonal planes [1–2], [2–3], and [1–3]. The fourth mode [Fig. 6(d)] at 5047 Hz is a combination of dilation and symmetric flexure. Finally, the fifth mode [Fig. 6(e)] at 6105 Hz is the first to display a pure shear type of behavior. This mode is also a doublet as a result of the symmetry about the [1–2] plane. These results are consistent with the eigenvalue predictions and measurements of regular shaped parallelepipeds of continuous elastic materials measured in the literature.^19,20

Figure 2 depicts the basic RUS measurement setup.^23,24 A brief explanation will be given here. The shape of the sample is dictated by the ability of the Ritz model to accurately predict the natural frequencies. Basic geometries include cubes, regular parallelepipeds, right cylinders, and spheres. Geometrical precision of these shapes needs to be as high as possible because deviations from parallelism, concentricity, and surface finish can adversely affect the resonance structure.

In this setup, two piezoelectric transducers are utilized to transmit and receive elastic signals from the sample geometry. The upper transducer is driven at a constant frequency while the lower records the driven system response of the sample. The sample is mounted between the transducers in such a manner as to minimize clamping effects. A typical approach is to contact the sample on opposing vertices with minimal loading forces. Here, the goal is to approximate the free surface boundary condition on the sample as required by the eigenvalue problem posed by Eq. (8).

A signal generator sweeps over a range of frequencies that cover a sufficiently large set of resonances of the structure. The drive signal (reference) and the output signal (measured) are then compared using a lock-in amplifier that calculates the in-phase and quadrature relationships between the two signals. When the magnitude of the product of these two signals is at a local maximum, a natural resonance of the system has been identified. Sweeping over a range of frequencies results in a series of peaks, representing the spectra of resonances in that frequency band. The peaks typically have very high Q values (f/Δf > 100), which allows for accurate measurements of the eigen-frequencies.

Figure 3(a) shows the mounted sample in between two piezoelectric transducers in the RUS test setup. An Agilent signal generator model 32333 (Santa Clara, CA) drives the upper transducer with a swept sinusoidal voltage signal at 10 V peak-peak. The transducers are a standard acoustic emission design, Physical Acoustics Corp, model R30A (Princeton, NJ). The lower transducer records a voltage proportional to the sample displacement. A Stanford Instruments 865 A lock-in amplifier (Sunnyvale, CA) records the in-phase and quadrature response of the reference signal and the output of the receiving transducer (lower). The system is controlled through a serial to general-purpose interface bus (GPIB) interface by a laptop. The data acquisition process and the analysis are done in matlab. Similar measurement systems and analysis codes are also available.^35,36 By comparison, Fig. 3(b) shows the same sample in a typical quasi-static compression test.² 

For each data set, the sample was mounted between the measuring transducers [see Fig. 3(a)], and the vibrational response was recorded through the range of frequencies of 3 kHz < f < 15 kHz at 5 Hz intervals. The frequency sweep was chosen slightly larger than the FE model predictions to avoid missing any of the predicted eigen-frequencies. A peak finding routine was used to determine the resonance frequencies in the individual response spectra. Additional information provided by the FE model was utilized to identify missing or poorly defined modes. Between 3 kHz < f < 11 kHz, there are approximately 20 easily identifiable resonance peaks. Another ten resonance peaks are identified using the model as a predictive guide and magnifying the scale of the spectrum. A total of 30 eigen-frequencies were measured for the sample.

A typical response spectrum for the lattice sample is shown in Fig. 4. The measured spectrum has two regions with the lower region (3 kHz < f < 6 kHz) displaying fewer resonance peaks with lower amplitudes, and the upper region (6 kHz < f < 11 kHz) features a higher amplitude, more complex spectrum of identifiable resonance peaks. The amplitude variation is due, in part, to the frequency response of the acoustic emission transducers used to drive and record the sample response. The amplitude variation will not affect the accuracy of the inversion process but it can make identifying low amplitude eigen-frequencies difficult.

At the end of each scan, the sample was removed from the test fixture, remounted using a different set of opposing vertices, and scanned again. The measurement process was conducted 14 times for repeatability and statistical calculations. Figure 5 shows the measured spectra of three consecutive resonance peaks for all measurements. The average of the 14 eigen-frequencies (peak values) is indicated with black data points. The average values for the experimental resonance peaks are given in Table IV, column 2. Care was taken to minimize changes in the loading forces from the transducers on the sample, however, even slight variations in the mounting loads from the test assembly resulted in the frequency shifting as seen in the data. Variation in the measured data is used to estimate the uncertainty of the resonance peaks in Hz. The average standard deviation (uncertainty) was less than ±30 Hz for all measured frequencies.

The first peak in the experimental data is not the first mode shape; it is an artifact from the test system response. This was verified by measuring the system response of the two transducers in direct contact with one another without a sample. The peak remained in the data as did others due to test system resonances. The FE model and the RUS approximation were used as guides to identify true sample resonances. The first sample mode shape is found at 4292 Hz. A similar peak is predicted from the model at 3997 Hz. The mode shape is shown in Fig. 6. For this mode, the lattice is demonstrating an aggregate torsional type of deformation, which is shear dominated and indicates the slowest sound speed for the sample.

The numerical and experimental techniques outlined in Secs. II and III provide the tools to solve the inverse problem defined by Eq. (8). The tacit assumption is that the lattice sample can be treated as a uniform elastic continuum over the frequency range of the eigenvalues investigated. To facilitate this assumption, the upper limit for the eigen-frequencies will be chosen so as to keep the entire range of frequencies in the long wavelength regime relative to the structural response of the lattice.

At the length of the scale of the RD unit cell, the lattice structure is essentially a system of clamped beams.¹² Using Euler-Bernoulli theory, the first bending mode of an individual strut can be estimated as a clamped-clamped round beam³⁷ 

where L_s is the strut length, I = (πr_s⁴/4) is the area inertia, r_s is the beam radius, E_o is the Youngs modulus of the base material, ρ_o is the density, and k² = 1.506π is a constant related to the boundary conditions specific to a clamped-clamped beam. For the 316 L lattice sample described in Table I, the first clamped-clamped resonance of an individual lattice strut is found to be f_strut = 108 kHz. Therefore, to ensure a valid long wavelength approximation for the RUS analysis, the upper frequency for the RUS measurements is limited to be an order of magnitude below the first strut resonance or (f_RUS)^max= 11 kHz. At this frequency, the lattice sample is essentially homogeneous with effective material properties and the implied symmetry defined from the topology of the unit cell.

For a RD lattice structure operating in the long-wavelength limit, the structure is orthotropic with three mutually perpendicular symmetry planes and nine independent elastic coefficients. The elastic stiffness matrix can be expressed as

where the Voight notation has been applied.³⁸ The stiffness matrix, [C], is related to Eq. (8) through the vector, $a¯={C11,C22,C33,C23,C12,C13,C44,C55,C66}$⁠, with ρ calculated as an effective density, ρ_eff, from Table I.

The typical fitting process starts with an initial estimate of the elastic coefficients, and this can be challenging as the solution to Eq. (10) will have numerous local minimums. A typical starting value might be identifying one or two engineering properties from standard tensile testing measurements or from tabulated references of similar polycrystalline materials. These can then be assigned as an initial approximation for the principle axis C_ii terms as well as the off diagonal C_ij terms. An optimization algorithm iterates on the elastic coefficients to minimize the difference between the estimated eigen-frequencies, f ^Fit, and the measured eigen-frequencies, f ^Meas., in Eq. (10). This first fit is typically not accurate, especially for low symmetry materials. For the second iteration, the first estimate for the elastic coefficients can be used as an improved starting guess and the fitting algorithm is run again. Depending on the nearness of the initial estimate to a solution (or local minimum), the optimization algorithm can converge quickly. Other fitting strategies start with a small but continuous subset of the measured eigenvalues, and these are fit using a lower order approximation of Eq. (9). The results of this lower order fit are then used as an initial guess for higher order fits, using more of the measured eigen-frequencies from the measurement. The fit order (Np) and number of measured eigenvalues are increased until a specified convergence of the criteria has been met. The challenge with least-squares approaches to the minimization problem defined by Eq. (10) is that even with isotropic materials (i.e., $a¯={C11,C44}$⁠), the object function has numerous local minimums. Without a “good” initial guess of the parameters, this can lead to erroneous estimates for the elasticity matrix. The problem is compounded on materials with low order symmetries due to the increased number of parameters requiring an initial guess.

An alternative approach has been to apply stochastic techniques inspired by biological systems. One example is genetic based algorithms, which rely on evolutionary improvements in a search process to find the best solution for minimizing a cost function.³⁰ Similar methods using the swarm behavior of insects have also been developed.^39,40 Particle swarm optimization (PSO) makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. The requirements for a close initial guess can be significantly relaxed.

PSO uses a population of “particles” to search the object function [Eq. (10)] for minimum values, based on random combinations of the parameters in the object function. The particle(s) with the best combinations (solutions) become the leader(s), and the remaining particles adjust their position and velocity toward the leaders. Here, a particle is defined as a specific instance of the vector $a¯i$⁠.

At each time step, k, of the search, new sets of random solutions are calculated, based, in part, on a weighting from the leader's position and velocity. Again, the particle(s) with the best solution(s) becomes the leader and the remaining particles adjust their positions and velocities toward them.

The position and velocity of the ith particle in the swarm is calculated using

and

The variable, p_i, represents the best position found by the ith particle, whereas G represents the global best position of all the particles. γ_1,i and γ_2,i are random numbers in the interval [0,1]. The remaining parameters can be “tuned” to improve the searching performance. The process is repeated until all the particles converge to a common minima as defined by a cost function [Eq. (10)].

A representative fit of the experimentally measured eigen-frequencies is summarized in Table II. The effective density is ρ_eff = 0.82 g/cm² from Table I. Optimized tuning parameters for Eqs. (13) and (14), based on the size of the data set (30 eigen-frequencies) and the number of unknowns (n = 9, orthotropic symmetry), are [S = 63, w = 0.6571, ϕ_p = 1.6319, ϕ_g = 0.6239].⁴¹ The average experimental resonance frequency, $fiMeas$ of the sample lattice is listed in the first column of Table II. The corresponding fitted frequencies $fiFit$⁠, determined from the PSO, are shown in the second column, and the difference between the measured and fitted values is given in the third column. The stochastic nature of the optimization process cannot guarantee an exact solution; however, the quality of the fitted results can be assessed using a measure of the total error, given as a percentage

which is recognized as the cost function given by Eq. (10) with an inverse squared weighting of the measured frequency. The PSO fitting process was conducted in two phases; in the first phase, wide boundary limits on the elastic coefficients (⁠$0.01 GPa<a¯i<1.0 GPa$⁠) conditions were estimated based on prior measurements taken from the literature.^8–14,41,42 The second phase utilized a refined search range for a_i, centered on the best solution from the first phase as a starting point and with a ± variance on each elasticity coefficient of 10%. The second optimization phase was repeated ten times on the measured experimental data. The total error of the fitted results to the experiment for the first 30 consecutive eigenvalues is 0.78%. The resulting vector of experimentally measured elasticity coefficients, $āi$⁠, which produced this fit, is shown in the first row of Table III. Table III summarizes the results for the five separate fits of the experimentally measured eigen-frequencies. The total error for each fit, ε_i, is tabulated in the last column. The mean and uncertainty of each of the orthotropic coefficients and the total average error are summarized in the bottom two rows. The fit results show the mean of the total error to be ε^Exp = 1.24% with a standard deviation of 0.4%.

In a second analysis, the RD lattice structure depicted in Fig. 6 was modeled with COMSOL Multiphysics.³³ A 3D eigenvalue solution was used to obtain the resonant frequencies over a similar frequency range as the experimental test. In the model, as in the experiment, the base material of the lattice was 316 L stainless steel with a Youngs modulus, E = 205 GPa, Poisson's ratio, ν = 0.28, and a density, ρ_o = 7.8 g/cm³. The effective density of the modeled sample is calculated using Eq. (13) and results in ρ_eff = 0.69 g/cm³.

Table IV shows the model and the fitted resonance peaks. Again, 30 resonance peaks (M = 30) are used to perform the fit. The calculated value of each resonance peak from the FE model is shown in column 1 of Table IV. The fitted results from RUS are shown in the second column with the difference between the fitted values and the measured value shown in the third column. The fourth column is used to assess the fit[Eq. (15)]. The resulting vector of elasticity coefficients, $āi$⁠, which produced this fit, is shown in the first row of Table V. As with the experimental measurements, five separate fits of the numerical eigenvalues are tabulated in Table V. The total relative error of the fitted results to the model for the first 30 consecutive eigenvalues is ε^Model = 1.13% with a standard deviation of 0.01%.

The elastic coefficients tabulated in Tables III and V can be combined into engineering coefficients related to the basis vectors defined in Fig. 1. Engineering coefficients are useful as they are typically measured directly using standard quasi-static tensile and compression measurments.² In addition, longitudinal and shear sound speeds are typically calculated from engineering moduli and Poisson ratios. The engineering coefficients, in terms of the Voight coefficients, are related through the following set of equations:³⁸ 

The resulting engineering coefficients are listed in Table VI. Based on the mean and standard deviation of the fitted elastic coefficients from Tables III and V and Eq. (16), the average uncertainty for all of the engineering coefficients is determined to be ±0.03 GPa.

In this investigation, the elastic properties of a metal lattice sample are determined by inverting a Ritz approximation of the eigenvalue problem, defined by a sample geometry using either experimentally or numerically determined eigen-frequencies. The validity of this approach is based on the assumption that in the long wavelength limit, the lattice with its regular discrete structure can be approximated as a homogeneous elastic continuum with symmetry implied from the lattice. For the case of a lattice based on a rhomboidal dodecahedron unit cell, the implied symmetry is orthotropic with three symmetry planes and nine independent elastic coefficients. Solutions to the eigenvalue problem have typically been challenged by the number of local minimum and the requirement of a very good initial guess. By leveraging stochastic optimization approaches (i.e., particle swarm), the necessity for an accurate initial guess is significantly relaxed. The result is a potentially robust and stable method for estimating the elastic coefficients for low symmetry lattice structures.

A comparison of the elastic coefficients determined by inverting Eq. (10), using numerically and experimentally generated eigen-frequencies, are within reasonable agreement. Differences exist between the model and the sample lattice. These differences are due, in part, to variations between the as-built and the as-modeled properties of the lattice structure. The model has no variation; it is generated by mirroring exact geometrical features through symmetry planes and utilizing periodicity. The sample lattice is built up with the same CAD file uses in the FE model; however, the resulting AM sample is slightly different in both of the individual dimensions, sample dimensions, and strut cross sections. The “as-built” and “as-modeled” irregularities are a result of the idiosyncrasies with the SLM fabrication process, which favors growth normal to the build plane.⁴³ There may also be differences associated with the presumed base material properties, such as porosity, grain structure, and lack of fusion in the structure, as these were not measured directly. Finally, as illustrated in Fig. 5, frequency shifting of the peaks adds to the overall measurement uncertainty. It is the author's belief that this shifting is primarily due to changes in loading on the sample by the RUS transducers. Boundary coupling effects are an ongoing area of investigation.

The current results are indicating an average uncertainty of approximately 10%, which is larger than the typical 1%–0.5% accuracy quoted for the RUS results.²⁴ The sandwich construction of the sample does diverge from a pure orthotropic symmetry and the symmetry of the sample is not completely captured in the model. The sample may be more accurately modeled with lower monoclinic symmetry.

Even with the stated issues, agreement with similar investigations in the open literature is encouraging. The longitudinal and shear moduli given in Table VI compare favorably with published results based on analytical, numerical, and experimental techniques.^8–14,42,44 Experimental approaches have correlated quasi-static compression measurements to fitted results based on powers of the density ratio of the effective density (E/E_o) = 0.28(ρ_eff / ρ_o)ⁿ with 2 < n < 2.5, using E_o = 205 GPa, ρ_eff = 0.63 g/cm³, ρ_o= 7.8 g/cm³, and n = 2.38, which predicts a modulus of E = 0.14 GPa.⁹ This result is in the range of the predicted result in Table VI. Analytical results predicted by Babaee¹² give a working range of the elastic moduli for this lattice of 0.3 GPa < E < 0.6 GPa. Again, these are similar in range of the estimated values in Table VI. Finally, results of a simple uniaxial compression test, depicted in Fig. 3(b), produced a value for E₃ of 0.3 GPa. ±0.05 GPa, which is also representative of the current RUS predictions.

Calculating Poisson ratios for low symmetry materials is challenging because of the sensitivity of the strain measurements to experimental error. With low symmetry materials, this is compounded by the number of independent strains needed to be measured. The benefit of RUS is once the elasticity coefficients are determined, the Poisson ratios can be calculated directly. Lattice structures based on a dodecahedron unit cell are considered non-auxetic, and the presence of the negative values for ν₃₁ and ν₁₃ suggest otherwise. Interestingly though, negative Poisson ratios on the order of −0.4 have been reported in composite materials with orthotropic symmetry when the axes are rotated 45 deg from the material axis.⁴⁵ The sample lattice in this study also has a 45-deg rotation in the [1–2] plane, which differs from the typical orientations presented in the literature, which have no rotation relative to the basis coordinates.^9–14,43,44

The results in Table VI show large discrepancies between the estimated elastic coefficients based on a FE prediction and those obtained from the experimental measurements. However, the total error of the individual fits that determined these coefficients is on the order of 1%. This would support the assertion that the fitted results are accurate with respect to the model and the experiment independently but due to differences between the modeled lattice and the experimental lattice, the resulting estimates for the elastic coefficients will be different. The resonance spectra of the model and the experiment clearly show this. Other features of the sample lattice, such as the nonuniformity of the struts and the overall dimensions, are not duplicated in the model. It is possible that refinements of the FEM model, possibly generating an as-built mesh as opposed to the as-modeled CAD mesh, would improve the comparison. In either case, the initial assumption of utilizing the Ritz approximation while operating in the long wavelength regime of the sample can produce accurate and repeatable estimates of the elastic coefficients for low symmetry metal lattice structures. However, the long wavelength regime is an important assumption to be aware of. As the length scales of the sample lattice become large relative to the wavelength of the drive signal, the accuracy of the continuum assumption used in the inversion process of RUS will break down, at which point, the elastic and inertial effects of the individual struts and joints become significant, requiring more traditional Galerkin approaches to capture the elastic behavior.

A method to apply RUS to determine the elastic stiffness coefficients of an orthotropic stainless-steel lattice structure has been presented. Current results are showing good agreement between experiments and modeled data. Complicated lattice structures can be analyzed using a continuum based assumption, provided the dynamic response of the system (eigenvalues) is measured in the long wavelength regime. In addition, stochastic methods using particle swarms have been demonstrated to provide, a potentially robust, stable, and efficient method for solving the minimization problem posed by the RUS technique. This is especially true for low symmetry materials in which there can be up to 21 elastic coefficients. Future investigations will be directed toward improving measurement sensitivity and developing better optimization strategies for low symmetry lattice materials.

This work was performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory.