A New Phenomenological Model for the Crushing Failure Mechanism Lattice Structures

A new phenomenological model for lattice structures that exhibit crushing‐like failure mechanisms is presented. The model estimates the compressive stress–strain curves of such lattices based on their relative density. The model is derived from the underdamped oscillator's general equation and the rheological model's properties. The model applicability is tested on five circular cell lattice structures having various relative densities, as well as a body‐centered‐cubic and a square cell lattice, for further validation. The model accurately captures the curves’ profile qualitatively and quantitatively compared with the experimental data. A relationship between each parameter and the relative density is established to extend the model's functionality as fourth‐order polynomial equations. Additionally, the energy absorption values between the measured data and the model up to the crushing of the first layer are in relatively good agreement ranging between 2.68% and 25.3%, proving the model's effectiveness. Overall, the new phenomenological model can estimate essential features of the lattice structures based solely on the relative density while reducing the time and the cost needed during the design phase.


Introduction
Manmade cellular materials, otherwise known as lattice structures, are gaining much traction in leading engineering fields due to their superior characteristics and properties, which are unachievable by their constituent materials. Lattice structures are unit cells packed periodically to fill a given design space. It is perhaps easy to achieve the proper set of properties when dealing with lattice structures, as they are highly optimizable and can be made of almost any material. For instance, the acquired mechanical response will be completely different by altering the unit cell topology or the constituent material. On the other hand, tuning the structure's parameters (i.e., the relative density and the number of cells) will alter its mechanical properties. [1] Accordingly, lattices are currently used in a wide range of applications, including lightweight load-bearing structures, [2,[2][3][4] protective equipment, [5,6] orthopedic implants, [7][8][9] conformal cooling, [10][11][12] vibration absorbers, [13][14][15][16][17] etc.
Under uniaxial compression, the lattice's cells exhibit either bending-or stretchingdominated behavior. For bendingdominated cells, the typical stress-strain curve follows a specific trait consisting of three different regions: 1) linear elastic, 2) plateau, and 3) densification, as shown in Figure 1a. The stress-strain curve of the stretching-dominated structures, Figure 1b, on the other hand, is characterized by an additional postyield softening region (after the linear-elastic region). Depending on the base material and the cell topology, the failure mechanism of both types of structures can be identified as either yielding, buckling, or crushing. Accordingly, following the different collapse mechanisms, Gibson and Ashby [18] derived a positive power relationship between the lattice structures' relative density and the mechanical properties (relative modulus and relative strength) following the different collapse mechanisms. The failure mechanism is a key characteristic in evaluating the overall performance of any type of lattice structure. Also, the same type of lattice structure can exhibit different failure responses. For instance, Teimouri and Asgari [19] investigated the difference between the failure response of uniform and functionally graded faced-centered cubic lattice structures. Their results showed smoother and longer plastic plateaus in the uniform models, with the absence of the early peak force in the compressive stress-strain curve of the functionally graded lattice. Similarly, Mahbod et al. [20] demonstrated that the effective elastic mechanical properties, the crushing behavior of the structures, and the energy absorption properties are highly dependent on the unit cell geometry and the relative density of the individual layers. Jin et al. [21] studied the failure and energy absorption of four different lattice structures and concluded that structures with the stretchingdominated deformation mode often yield better mechanical properties under dynamic loading. Maskery et al. [22] proved experimentally the importance of the cell size in determining the failure mechanism of double-gyroid specimens made of aluminum alloy. Their findings showed that large cell size failed due to crack growth or localized fracture, while structures of small cells either fail in a layer-wise manner (crushing behavior) or diagonally along the shear band. The number of layers is another factor that alters the lattice's failure response. According to Lei et al., [23] increasing the number of layers adversely affects the ultimate DOI: 10.1002/adem.202201695 A new phenomenological model for lattice structures that exhibit crushing-like failure mechanisms is presented. The model estimates the compressive stressstrain curves of such lattices based on their relative density. The model is derived from the underdamped oscillator's general equation and the rheological model's properties. The model applicability is tested on five circular cell lattice structures having various relative densities, as well as a body-centered-cubic and a square cell lattice, for further validation. The model accurately captures the curves' profile qualitatively and quantitatively compared with the experimental data. A relationship between each parameter and the relative density is established to extend the model's functionality as fourth-order polynomial equations. Additionally, the energy absorption values between the measured data and the model up to the crushing of the first layer are in relatively good agreement ranging between 2.68% and 25.3%, proving the model's effectiveness. Overall, the new phenomenological model can estimate essential features of the lattice structures based solely on the relative density while reducing the time and the cost needed during the design phase. strength and stiffness due to the high number of unconstrained cells at the boundaries.
At the macroscopic level, lattice structures are treated as a continuum or a homogenized material whose behavior can be described by constitutive equations. Few phenomenological models have been developed to describe the typical cellular solids' compressive stress-strain curve. Goga and Hučko [24] modeled the three regions mentioned earlier for solid foams using basic mechanical components. The authors used the Maxwell model, which consists of a spring (k) and a dashpot (c) in series to model the linear elastic region and two more springs (k P and k D ) in parallel to model the remaining two regions. The entire system is depicted in Figure 2, and the established stress-strain relationship is shown in Equation (1), where k D is implicitly expressed by the nonlinear function γð1 À e ε Þ n . Furthermore, a curve fitting between the modeled and the measured stress-strain data is used by the authors to obtain linear approximations of the model parameters with respect to the relative density. Olejnik and Awrejcewicz [25] integrated that foam equation into their design to model the response of the thorax when energized by air-last over-pressure. Alzoubi et al. [26] used a constitutive viscoelastic polymeric foam model based on both Kelvin and Maxwell models to capture the compression curves of phase-change graphite composites. The polymeric foam model equation can be employed to design new energy storage composites with tailorable mechanical and thermal characteristics. A similar approach was also used in the work of Gheitaghy et al. [27] to model the behavior of carbon nanotube pillars that exhibit foam-like behavior following indentation tests. Liu and Subhash [28] developed a constitutive model capable of capturing the profile of the nonlinear stress-strain characteristics of polymer foams under large deformation. Similarly, Jeong [29] derived a constitutive model for polymeric foams having a four-parameter modulus function that accounts for the strain rate effect.
A perfectly linear plateau region is rarely obtained for metallic lattice structures, especially for lattices in their as-built conditions. Usually, the plateau region is characterized by fluctuations. Hence such a model does not perfectly fit. Additionally, lattices that fail in a layer-wise crushing manner generally do not exhibit any plateau region. Consequently, this work aims to develop a phenomenological model capable of accurately capturing the complex shape of the compressive stress-strain curves of lattice structures characterized by their crushing failure mechanism.

Experimental Section
The new phenomenological model describes the compressive stress-strain curves of the recently developed circular cell-based lattice structures (7 Â 7 Â 7) made of Ti 6 Al 4 V. The manufacturability process parameters and the experimental setup are described in our previous work. [30] In order to provide a relationship between the parameters of the model and the relative density, five structures of different wall thicknesses (0.6, 0.7, 0.8, 0.9, and 1 mm) were considered, Figure 3. Three of those structures were already tested in our previous work (0.6, 0.8, and 1 mm), and two additional structures (0.7 and 0.9 mm) were additively manufactured and tested in order to provide reliable relationships between the different parameters. As shown in Figure 3, all these structures failed in a layer-wise crushing manner and followed the same traits. The latter means it is possible, with the right correlation, to have one phenomenological model applicable to these types of structures with any relative density.  The crushing failure mechanism resembles an underdamped oscillator's response, consisting of a spring, mass, and viscous damper, as depicted in Figure 4a. Figure 4b illustrates the resemblance between the oscillations and the peaks of the crushing layers. The amplitude or, in our case, the stresses decrease similarly in an exponential function having the form Therefore, the oscillator response presented in Equation (2) was employed with some tuning to describe the shape of the compressive stress-strain curves of the CirC lattice structures. The equation can be first transformed from time-displacement  www.advancedsciencenews.com www.aem-journal.com space into strain-stress where x and t become σ and ε, respectively, Equation (3).
Then, the whole response must be elevated above zero. This was achieved by adding a nonlinear spring in parallel with the whole oscillator system with a stiffness value of The stress can then be presented as For parallel components, the stresses were added, and the strains were equal Hence the equation becomes To model the densification region, another spring was added in parallel to the system with a nonlinear stiffness of The corresponding stress was Consequently, the whole system is presented in Figure 5, and the corresponding phenomenological model was expressed as follows Based on the above equation, the influence of the parameters A 0 , b, m, φ, Z, and n, on the shape of the stress-strain curve should be investigated. Another parameter that must be considered is k which represents the stiffness of the spring in the oscillator system. This parameter is included in the equation Additionally, a correlation between some of the indicated parameters and the structures' relative density must be obtained to extend the model's functionality for all the structures.

Influence of the Model Parameters
To effectively fit the phenomenological model with the experimental data, a clear understanding of the influence of each parameter becomes a must. To achieve the latter, only one parameter was varied at a time while fixing the others. Consequently, Figure 6 shows the influence of the four main parameters of the model (A 0 , b, w, and Z ). A 0 represents the stress amplitude; as its value increases, the peak stresses increase and vice versa. The second parameter, b, controls the damping of the peak stresses. Increasing its value leads to a surge of the damping effect and, conversely, a reduction with its decrease. The third parameter, Z, was used to appropriately model the densification region by either lifting it (high values of Z ) or lowering it (low values of Z ). It should be mentioned that n has a similar influence on the model; nevertheless, the n value is barely modified and is often a positive even number (i.e., 6 or 8). Phi 'φ' was only used to shift the whole model to the right or left. Finally, the value of w determined the number of peaks obtained using the phenomenological model. w is the crucial parameter to tune first, as the number of peaks (layers) is well known beforehand. From Figure 6, it is clear that a gradual increase in the value of w yields more peaks and vice versa. Since w is a function of b, k, and m, these variables were tuned first to obtain a proper preliminary fitting of the model. Subsequently, the other parameters were tweaked to reach the best fit compared with the experimental outcomes.

Results and Discussions
Some of the parameters within the phenomenological model can be substituted by the mechanical properties of the corresponding lattice structure. Mainly, A 0, which represents the first peak stress, is typically the maximum compressive strength of the www.advancedsciencenews.com www.aem-journal.com lattice divided by two (σ max =2). The division by two accommodates the elevation of the original underdamped response above 0. Additionally, k, which represents the stiffness of the spring in the oscillator system, is attributed to the value of the modulus of elasticity E of the lattice. Since a relationship between the mechanical properties and the relative densities of lattice structures can be quickly established, the application of the phenomenological model with these two parameters becomes easier. In the case of the CirC lattice, the relative densities and the mechanical properties of the various wall thicknesses investigated in this study are presented in Table 1. The relationship was obtained by calculating the relative modulus ðE lattice =E Solid Þ and the relative strength ðσ lattice =σ Solid Þ of the lattices. The values of the solid's (Ti6Al4V) modulus and strength used herein are 113 GPa and 1013 MPa, respectively. To calculate the relative density, the total volume of the solid was calculated by first dividing the weight of the lattice by the density of the solid (0.00441 g mm À3 ) and then dividing the outcome by the total theoretical volume (length Â width Â height). Subsequently, the least-square method is employed to fit a polynomial curve and extract the corresponding equation of the relationship between the parameters. The relationships were acquired in fourth-order polynomial equations with a least-square R 2 of 1. The fitted curves and the relations are presented in Figure 7 and Equation (13) and (14), respectively.

Relative Modulus
Similarly, the same approach was taken to model the compressive stress-strain curves of the various CirC lattices. The least-square method was used to estimate the model parameters adequately. The calculated values of the required parameters are listed in Table 2. b is simply the number of layers or the number of layers 1 based on which of those numbers provide the best fit in the calibration process. Accordingly, the parameters b were fixed and set to a value of 6. Similarly, n is a constant parameter equivalent to 6 for all structures except for the circ-1 mm, which has n = 8. A 0 and k, as previously mentioned, are obtained from the mechanical properties. The three other parameters, m (implicitly w), phi, and Z, are tuned to establish the best fit with Figure 6. Influence of the model parameters A 0 , b, w, and Z. www.advancedsciencenews.com www.aem-journal.com the measured data. A comparison between the outcomes of the phenomenological model and the stress-strain curves of the five CirC lattices is clearly presented in Figure 8. It can be observed that the model is reliable in depicting the shape of the stressstrain curves. The peak stresses and the corresponding strain values are quantitatively accurate and well representative of the measured data, especially for high relative density (the CirC-1 mm). As the relative density decreases, discrepancies between both curves can be clearly seen. Nevertheless, the estimated curves can still represent the measured data. They can also be used as preliminary design criteria for a specific application before manufacturing and testing. For instance, it is possible to estimate the specific energy absorption of a given structure with a well-known relative density and determine whether it is sufficient. Table 3 shows the energy absorption of the five lattices up to the crushing of the first layer and highlights the difference between the measured values and the ones obtained using the developed model. The values do not perfectly match as there are some errors in any developed models. Yet, the values can serve as a decent first estimation during the design phase. Estimating the energy absorption up to the densification point is still possible, but the error will increase. All the parameters that have been tuned also depend on the lattice structure's relative density. Therefore, a relationship between each model parameter and the relative density has been established based on the values obtained in Table 2. Similarly, a relationship between the relative density and the measured energy absorption is established to better estimate its value and reduce the error. The same least-square method has been used to fit the curves and extract the corresponding equations. All the equations obtained are fourth-order polynomials with a least-square R 2 of precisely 1. The fitted curves and the relations for the parameters and the energy absorption are presented in Figure 9 and 10 and Equation (15)- (20), respectively. Using these equations, estimating and plotting the compressive stress-strain curves of the CirC lattices become extremely easy, and the estimation of the energy absorption W becomes more accurate. All that needs to be known are the relative density and the number of layers to calculate the variables and apply the model.
W ¼ Àð204, 599 ⋅ RD 4 Þ þ ð173, 308 ⋅ RD 3 Þ À ð54, 049 ⋅ RD 2 Þ þ ð7, 379.8 ⋅ RDÞ À 372.25 (20) The same phenomenological model can be applied to any lattice that exhibits crushing-like failure behavior. [21,[31][32][33] For example, the compressive stress-strain curves of two additional types of lattice structures with similar failure behavior (obtained in the literature) were also modeled for further validation. The two types are 1) the typical body-centered-cubic (BCC) investigated by Jin et al. [34] and 2) the square-cell (SC) lattice studied by Wang et al. [35] The BCC structure was fabricated by selective laser melting (SLM) using Ti 6 Al 4 V powder with the unit cell dimensions being 5 Â 5 Â 5 mm, strut diameter of 0.8 mm, and four layers (four unit cells in the three directions). The reported mechanical properties of the BCC lattice structures are 12 and 306 MPa for the ultimate strength and the modulus, respectively. On the other hand, the SC lattice structure was made of steel 316 L using the same SLM technology, having a    Table 4. Figure 11 presents the outcomes of the tuned model in comparison with the reported compressive stress-strain data of the two types of lattice structures. It is evident that the model was capable of capturing the failure response of the two structures both quantitatively and qualitatively, which proves its  www.advancedsciencenews.com www.aem-journal.com functionality in such an application. Since the data of only one relative density is available for each structure, no relationship can be established at the moment. Generally, an appropriate tuning of the model parameters has to be repeated based on newly measured data. It should be reminded that changing the unit cell size can affect the failure response of the lattice structure. [22] Therefore, this model is better suited for lattices of different relative densities which have a fixed unit cell size or are characterized by a well-known failure response. Overall, such a model can drastically reduce the time needed in the design phase and the money to manufacture prototypes. It can help assess the design features (compressive www.advancedsciencenews.com www.aem-journal.com strength, modulus of elasticity, energy absorption) for a specific application prior to manufacturing.

Conclusion
The modeling strategy for metamaterial lattices changes depending on the scale considered. At the macroscale level, the lattice is treated as a continuum or homogenized material with its own set of effective properties. Therefore, it is potentially possible to establish a relevant constitutive equation for such structures.
In this study, a new phenomenological model has been developed to estimate the mechanical properties and the shape of the compressive stress-strain curves of lattice structures that exhibit crushing-like failure behavior. The response of an underdamped oscillator was used as the basis of the model due to the resemblance between the oscillations and the peaks of the crushing layers. Additionally, rheological properties were added to the system to obtain an accurate profile of the failure mechanism. The compressive stress-strain curves of five circular cell lattice structures with different relative densities have been considered for testing the model's effectiveness. The compression tests showed apparent traits between all structures, indicating the possibility of establishing a relationship between the model parameters and the relative density. The model consists of six parameters that clearly influence the shape of the stress-strain curves. Two parameters are obtained from the mechanical properties of the lattice structures, two are constants (for the circular cell design), and the last two are variables. The mechanical properties and the variables can be easily extracted through fourth-order polynomial equations established herein based on the calibrated values. The model's outcomes were in good agreement with the experimental data, especially for high relative density. Moreover, the estimated energy absorption up to the crushing of the first layer was quantitatively reasonable for all the lattices. The model's applicability was further demonstrated through the modeling of the failure response of two additional types of lattice structures (BCC and SC). The results were comparable to the compressive data of both structures quantitatively and qualitatively. Overall, the model has shown promising results, and it can be used for any type of lattice that exhibits a similar failure mechanism. For large-scale lattice structures, this model can be advantageous in terms of time and money as it allows for estimating all the necessary properties and features for a specific application without the need for numerous unnecessary prototypes.  Figure 11. Modeled and measured compressive stress-strain curves of a) BCC [34] and b) SC lattice structures. [35] www.advancedsciencenews.com www.aem-journal.com