Additively Manufactured Functionally Graded Lattices: Design, Mechanical Response, Deformation Behavior, Applications, and Insights

Abstract

Flora and fauna have evolved to distribute their structural mass efficiently in response to their environment. Inspired by this structural efficiency, functionally graded lattices (FGL) are an emerging subset of non-uniform lattices that employ density gradients for a function-driven mechanical response. These gradients are controlled by stepwise or continuous changes in the geometry or topology of the lattice unit cells. FGLs have the capacity for multifunctionality, facilitating high compliance and energy absorption, or moderate strength and stiffness depending upon the specific gradient. These novel lattice structures have been utilized for a range of applications, including biomimetic implants, heat dissipation, and impact absorption. The fabrication of FGLs with complex internal topologies is facilitated through additive manufacturing (AM) using materials such as metals, polymers, and composites. The mechanical properties of these lattices have been examined through compressive testing. The elastic modulus and the yield stress are reported to range from 0.009 GPa to 6.0 GPa, and from 0.38 MPa to 424 MPa for relative densities between 10% and 80%, respectively. Energy absorption is reported to supersede conventional uniform lattices by up to 30%. By accumulating and assessing the mechanical, geometric, and topological data from the FGL literature, this review will systematically classify and explore the viability of these novel structures for real-world applications.

Introduction

Cellular materials are a benchmark of structural design efficiency. Achieving mainstream attention in Robert Hooke's Micrographia in 1665, the microstructure of some flora and fauna was discovered to be comprised of porous hierarchical architectures, with Hooke coining the term ‘cell’.¹ These findings contrasted with the fully dense materials commonly implemented in structural applications. Nevertheless, cellular materials were considered as curiosities until the 1920s, when microstructural design commenced with stochastic metallic foams.² These foams were manufactured with both open- and closed-cell topologies fabricated through methods like space holding, sponge replication, vapor deposition, hot isostatic pressing, or powder metallurgy.² These cellular structures were typically employed as space fillers due to their high-energy absorptive capabilities at extremely low densities. This interest in stochastic cellular materials encouraged greater research into microstructural engineering and enabled the conceptualization of periodic cellular materials known as honeycombs and lattices. A honeycomb is a prismatic tessellation of 2D geometry, named after the resemblance to a honeycomb produced by bees. Similarly, a lattice is a type of cellular material defined by an ordered arrangement of self-tessellating nodes and struts. Unique from stochastic cellular materials, which were generally classified as energy-absorptive materials, lattices could achieve high energy-absorptive features or strong mechanical properties depending upon the specific unit cell topology selected.³ The mechanical potential of cellular materials was quantified by Gibson, Ashby, and colleagues in the 1980s.^4,5,6,7 The multifunctionality of the mechanical capabilities of lattices was examined by Deshpande in 2001.⁸ These research contributions highlighted the theoretical potential of lattices as structural materials, with the prospect of exceeding dense structural material capabilities at very low densities. However, the complex internal geometry of cellular materials made fabrication difficult by conventional manufacturing means.

Consequently the field of lattice design grew exponentially with the onset of additive manufacturing (AM). The first metal lattice was fabricated through laser powder bed fusion (LPBF) in 2005, achieving high strength and rigidity.⁹ Now, these lattices include various feedstock materials including ceramics, composites, metals, and polymers, and include a range of high-strength and energy-absorptive unit cell topologies as quantified by formal meta-analyses.^3,10 With the field of AM lattice design maturing, the optimization of lattice structures is becoming increasingly important in engineering research, with applications in a range of industries, including aerospace,^11,12 civil,¹³ automotive,¹⁴ biomedical,^15,16 and thermal¹⁷ engineering. Widespread adoption by industry has generated significant research into the optimization of novel lattice types for energy absorption and mechanical strength.

Conventional periodic lattices have been designed and mechanically assessed for the past 20 years,⁹ and over this period specific structures have stood out for certain characteristics, including FCCZ lattices for high specific strength,¹⁸ and BCC and TPMS gyroid lattices for high energy absorption.¹⁹ However, these conventional structures have largely been limited to a constant geometry and topology due to requirements of self-tessellation. Self-tessellation prohibits the redistribution of material from areas of low strain to areas of high strain, as all the unit cells are the same, disallowing for structurally efficient lattices. As a result, conventional uniform lattices are also limited to a singular deformation process, generally defined by either stretch- or bending-dominated deformation modes,⁸ which is typically divided into three distinct stages; linear elastic deformation, plastic deformation, and, densification.²⁰ This limits their energy-absorptive capabilities and possible responses to applied loads.

Conversely, by treating lattices as an accumulation of individual unit cells that form a larger structure, material may be redistributed to create a density gradient for optimum applicability to a number of specific functions. Functionally graded lattices embody this material redistribution. As displayed in Fig. 1, FGLs may be designed by varying unit cell sizes, unit cell numbers, strut diameters, or unit cell topologies to create a density gradient.^21,22,23,24 This enables changes in the relative density of specific layers of unit cells, which can be arranged in either a continuous or stepwise gradient.

Fig. 1

Functionally graded lattice unit cell variants reported within the literature. The collated images are reprinted with permission from Refs. 22, 23,25, 27, 28, 34,35,36,37,38,39, and from Refs. 21, 24 under the terms of the Creative Commons CC BY license.

The redistribution of density to specific areas of the lattice enables optimized structures which can efficiently respond to loads, possess high mechanical strength and stiffness, and undergo global deformation, with catastrophic failure by shearing. FGLs can also be designed for high energy absorption facilitated by the localized deformation of lattice layers from the lowest density segments to the highest. Furthermore, the unit cell arrangement in FGLs can be designed to closely resemble the microstructure of many natural cellular structures. FGLs ability to include both dense and highly porous regions has attracted interest in biomedical applications as a method to replicate the structure of cortical and cancellous bones.^{25,26,27,28,29} Consequently, a large portion of studies have been designed as cylindrical lattices, differing from the conventional cubic design, to better resemble the cross-section shape and mechanical response of human bone. Thermal applications of FGLs are as equally extensive, with the unique geometry facilitating an optimum transfer and dissipation of heat,^30,31,32 and enhancing the ability of components to endure high thermal stress effects.³³

FGLs may provide an opportunity for function-specific lattice optimization; however, their mechanical capabilities, geometrical properties, and topological designs have not been fully quantified. Consequently, no systematic conclusions have been derived from their manufacture, mechanical properties, or failure modes. Recent experimental studies have enabled the accumulation of a significant volume of data that should allow a methodical assessment of the field. This review will explore the relevant geometries, topologies, deformation responses, and mechanical properties of FGLs to provide insight into and application to the novel subset of lattice design.

Designing Density Gradients

While regular lattice structures maintain uniform topologies of self-tessellating unit cells, FGLs can be considered as a combination of changing unit cells that create a density gradient. This gradient is generally achieved through geometric variations to the unit cell size, pore size, strut diameter, isovalue offset for triply periodic minimal surfaces (TPMS) structures, or changes in topology throughout the layers of the lattice. There are three major types of gradings observed within the FGL literature that classify as functionally graded lattices (Fig. 2). These are continuous, stepwise, and hybrid, the first two categories are geometry-based FGLs, while the hybrid lattice is topological. Continuous grading is an arrangement of constantly changing unit cell layers along one or more directions.^26,28,40,41 Visually, these lattices may be characterized by smooth grading transitions, as displayed in Fig. 2a.³⁵ An FGL may reasonably be considered continuous if the number of unit cell layers is equal to or less than the number of relative density changes along the grading direction. If the number of layers exceeds the number of relative density changes, then the lattice is considered a stepwise grading. Stepwise grading occurs in discrete increments, often involving groups of identical unit cell layers before variation.^{25,27,29,42,43} Visually stepwise lattices can be distinguished by staggered transitions, as observed in Fig. 2b.²² The third grading classification is hybrid, and these FGLs are uncommon and involve multi-morphological lattice arrangements that combine multiple types of unit cell topologies, often using hybrid transitional topologies when surface-based topologies are used.^23,36,44 These uncommon FGLs, exemplified in Fig. 2c,²³ typically involve two types of unit cells, each comprising half the layers of the lattice. These novel lattice structures enable unique mechanical properties and deformation characteristics that remain largely unexplored.

Fig. 2

(a) Continuous FGL, with F2BCC unit cell, and vertical grading. (b) Stepwise FGL, with rhombic dodecahedron unit cell and horizontal grading. (c) A hybrid FGL, with cubic and BCCXY unit cells. (d) A radial horizontally graded honeycomb FGL. (e) A multidirectional uniplanar vertically graded BCC FGL. The collated images are reprinted with permission from Refs. 22,23,27,35, and from Ref. 26 under the terms of the Creative Commons CC BY license.

Orientation and Direction of Grading

The orientation of grading has been identified as the primary factor dictating the deformation and mechanical response of an FGL. Within the literature and characterized in Fig. 2, lattices are graded along the horizontal planes (x, y), and vertical planes (z), from the outside and the center. FGLs with gradings considered to be in the horizontal plane are graded perpendicular to the direction of the applied load. These FGLs can be graded radially (Fig. 2d) from or to the centroid in either a cylindrical or cartesianal coordinate system, depending upon the lattice cross-section shape, and are generally designed for mechanical strength. Conversely, lattices considered to be graded along the vertical plane are graded parallel to the direction of the applied load. These FGLs are typically designed for energy-abortive applications. As observed in Table I, unidirectional and multidirectional uniplanar gradings have been implemented for every type of FGL. As implied by the name, unidirectional gradings involve designing the lattice with either a continuous or stepwise grading in a single direction, while multidirectional uniplanar grading involves grading the lattice in more than one direction along a single plane (Fig. 2e). The deformation modes that dictate these mechanical responses are discussed further in Section 5, while Table I lists the FGLs produced in the literature using AM techniques and demonstrating the three mentioned grading principles.

Table I Summarized geometric and topological parameters in the FGL literature, where (N) is a nominal/design value, (E) is an experimental value, and (AVG) is an average value

Functionality Versus Exploration

Experimental research within the field of density gradient lattices (DGL) is commonly misidentified as work on functionally graded lattices (FGLs). FGLs are non-uniform lattices designed with a density gradient dictated by a given function, whether that be mechanical, biological, or thermal.

Function dictates the geometry, topology, and density gradient. Conversely, DGLs are lattice structures designed with a changing geometry or topology that facilitates a density gradient, though not in response to a specific function. The outcomes of DGL assessments may provide advancements in the field of a specific FGL; however, they can only be considered exploratory when used for this purpose, while FGLs are functionally-driven experiments.

Grading Equations

There are several approaches to grading lattices that have been presented in the literature. These methods are summarized in Table II, with nomenclature in Table III, along with specific constraints adopted or inherited, and design variables used to control the grading.

Table II Summary of grading methods, grouped by function order, presented the FGL literature. See Table III for a list of nomenclature

Table III Nomenclature used in Table II

Stepwise Grading

Many early investigations into FGLs chose a simple method of varying the density of subsequent layers within a lattice. Stepwise grading, while simple to implement, has been limited to single-axis grading, and to date has been primarily employed in exploratory studies, with desired changes in relative density being achieved linearly over a discrete number of steps. Typically, this is achieved by varying the size of the struts in the unit cells comprising each layer group, as changes in strut diameters are simple to implement and do not lead to distortion of unit cell elements, though similar results can be achieved by altering the unit cell’s aspect ratio by varying the length of one side, as seen in Table II, entries 4, 5, 7, and 10. Common among stepwise methodologies were specific design targets, such as biomimetic lattices with mechanical properties similar to bone for use as implants, and can be considered soft constraints on the methods presented.

Linear Grading

Having established the efficacy of FGLs, the focus moved to generalized methods to realize a true continuous density gradient in lattice structures. TPMS lattices were extensively employed, being easily controlled by manipulating the isovalue offset, denoted as $t^{*}$ in Eq. 1, and $c_{{\left( {p,g} \right)}}$ in Eq. 2. Similar to stepwise grading, efforts were focused on a change in element strut or surface thickness, as opposed to changes in aspect ratio. Where presented in the literature, grading functions have been presented along the build axis, which is aligned with the model’s z-axis, with a common linear form of ${\text{thickness}}\left( z \right) = mz + c$, which directly scale the strut size or surface offset in the case of TPMS-based lattices. This form likely extends to entries 11–15, and 18 and 19 in Table II, as the continuous linear nature of grading is discussed despite a grading function not being stated.

$$ t^{*} \left( z \right) = mz + c $$

(1)

$$ c_{{\left( {p,g} \right)}} \left( z \right) = m_{{\left( {p,g} \right)}} z + C_{{\left( {p,g} \right)}} $$

(2)

where $m$ and $m_{{\left( {p,g} \right)}}$ represent the desired gradient, and $c$ and $C_{{\left( {p,g} \right)}}$ are the offset to achieve the unit cell boundary size at normalized length $z = 0$ and $z = 1$, respectively

Hybrid and Polynomial Grading

With stepwise and linear functions limited to single-axis grading primarily by scaling struts or surface thicknesses, investigations have turned towards methods for grading multi-axially, non-linearly, and by topology transition. Al-Ketan et al.³⁶ presented several methods for grading network-based TPMS-based lattices linearly and continuously. Relative density grading was achieved with a linear function, relating the isovalue offset, $c\left( {x,y,z} \right)$, to the position on the z-axis as seen in Eq. 3:

$$ c\left( {x,y,z} \right) = \pm \left( {mz + c} \right) $$

(3)

where $c$ is the offset for the unit cell boundary size at normalized length $z = 0$ and $z = 1$.

Inspired by the works of Liu et al.,⁴⁴ the unit cell size grading is subject to the constraints $\dot{u}_{x} = \dot{v}_{y} = \dot{w}_{z}$ to prevent distortion of the structure, and, under this constraint, a linear method is presented which directly alters the surface functions (Eqs. 4 and 5), by making parameters $\alpha x$, $\beta y$, and $\gamma z$ functions that vary along the $x$, $y$, and $z$ axes resulting in Eqs. 6 and 7:

$$ \varphi_{G} \equiv \sin X\cos Y + \sin Y\cos Z + \sin Z\cos X = c $$

(4)

$$ \varphi_{D} \equiv \cos X\cos Y\cos Z - \sin X\sin Y\sin Z = c $$

(5)

$$ {\text{where}} \; X = 2\pi u_{\left( x \right)} ; \; Y = 2\pi v_{\left( y \right)} ;\, Z = 2\pi w_{\left( z \right)} ;\, u_{\left( x \right)} = \alpha x; v_{\left( y \right)} = \beta y; w_{\left( z \right)} = \gamma z $$

$$ \alpha_{\left( z \right)} = \beta_{\left( z \right)} = k_{1} z + C_{1} ,$$

(6)

$$ {\text{where}}\; k_{1} = \frac{m - 1}{{Z_{{{\text{max}}}} - Z_{{{\text{min}}}} }},\, C_{1} = - Z_{{{\text{min}}}} k_{1} + 1 $$

$$ \gamma_{\left( z \right)} = \frac{{k_{1} }}{2}z + C_{1} + \frac{{C_{0} }}{z},$$

(7)

$$ {\text{where }}\;C_{0} = \frac{1}{2}k_{1} Z_{{{\text{min}}}}^{2}$$

where $\alpha , \beta ,$ and $\gamma$ control the unit cell size, and $Z_{{{\text{min}}}}$ and $Z_{{{\text{max}}}}$ are the lower and upper heights of the lattice region for grading respectively.

To realize the hybrid grading of topologies, a sigmoid function was employed. The sigmoid function, represented by $\gamma$ in Eq. 8, allows the creation of a new local topology, $\varphi_{{{\text{MML}}}}$, to allow for a smooth transition from one distinct topology, $\varphi_{{{\text{Surface}}\;1}}$, to another, $\varphi_{{{\text{Surface}}\;2}}$. This local topology’s transition range and severity may be adjusted with the $k$ value, with large values resulting in shorter, sharper transitions while smaller values resulting in longer, smoother transitions. The expression $G\left( {x,y,z} \right)$ is the spatial coordinate set that defines the shape of the transition between topologies.

$$ \varphi_{{{\text{MML}}}} = \gamma \varphi_{{{\text{Surface}}\;1}} + \left( {1 - \gamma } \right)\varphi_{{{\text{Surface}}\;2}} ,\; {\text{where}}\; \gamma \left( {x,y,z} \right) = \frac{1}{{1 + e^{{kG\left( {x,y,z} \right)}} }} $$

(8)

Onal et al.²⁶ demonstrated multi-axial grading of strut-based topologies by defining the diameter, $\emptyset$, of each strut at multiple locations, and then using these points to create the strut using a loft. This strategy allows for a continuous change in strut diameter, controlled by the gradient $A_{{n_{{\left( {x,y,z} \right)}} }}$, throughout the lattice. The general form of the grading equation is presented in Eq. 9, while the simplified form for a linear gradient, where $n = 1$, along the x-axis, is presented in Eq. 10. The term $P_{{\left( {x,y,z} \right)_{0} }}$ is the gradient origin in each relevant axis and allows for the origin of the gradient to be arbitrarily set, allowing for middle-out gradients as well as the more traditional top to bottom gradients seen in stepwise FGLs. The term $\left| {P_{{\left( {x,y,z} \right)}} } \right|$ represents the absolute current coordinate position in the relevant axis, and $c$ is a constant.

$$ \emptyset = c + \mathop \sum \limits_{n = 1}^{{n,\left( {x,y,z} \right)}} A_{{n_{{\left( {x,y,z} \right)}} }} \left( {\left| {P_{{\left( {x,y,z} \right)}} } \right| - P_{{\left( {x,y,z} \right)_{0} }} } \right)^{n} $$

(9)

$$ \emptyset = c + A_{x} \left( {P_{x} - P_{{x_{0} }} } \right) $$

(10)

Liu et al.⁴⁴ presented a set of modified equations for diamond and gyroid surfaces to include tunable parameters for porosity control and cell size, presented in Eqs. 11 and 12:

$$ \varphi_{G} \left( {x,y,z} \right) = \cos \left( {\alpha x} \right) \cdot \sin \left( {\beta y} \right) + \cos \left( {\beta y} \right) \cdot \sin \left( {\gamma z} \right) + \cos \left( {\gamma z} \right) \cdot \sin \left( {\alpha x} \right) + A + R_{G} = 0 $$

(11)

$$ {\text{where}} \; A = 0.08\left[ {\cos \left( {2\alpha x} \right) \cdot \cos \left( {2\beta y} \right) + \cos \left( {2\beta y} \right) \cdot \cos \left( {2\gamma z} \right) + \cos \left( {2\gamma z} \right) \cdot \cos \left( {2\alpha x} \right)} \right] $$

$$ \varphi_{D} \left( {x,y,z} \right) = a + b + c - B + R_{D} = 0 $$

(12)

$$ {\text{where}}\; a = \sin \left( {\alpha x} \right) \cdot \sin \left( {\beta y} \right) \cdot \sin \left( {\gamma z} \right),b = \cos \left( {\alpha x} \right) \cdot \sin \left( {\beta y} \right) \cdot \cos \left( {\gamma z} \right), c = \cos \left( {\alpha x} \right) \cdot \cos \left( {\beta y} \right) \cdot \sin \left( {\gamma z} \right), B = 0.07\left[ {\cos \left( {4\alpha x} \right) + \cos \left( {4\beta y} \right) + \cos \left( {4\gamma z} \right)} \right]$$

where $\alpha , \beta ,$ and $\gamma$ control the unit cell size while $R_{G}$ and $R_{D}$ adjust the porosity, and as such the relative density, of the TPMS. These modified equations were used to create several methods for inducing a density gradient. A method for linear grading along the z-axis is presented in Eq. 13, where $Z_{{{\text{min}}}}$ and $Z_{{{\text{max}}}}$ are the lower and upper heights of the lattice region for taper, respectively, while $R_{{\rho_{1}^{*} }}$ and $R_{{\rho_{2}^{*} }}$ are the corresponding relative densities:

$$ R_{{\left( {G,D} \right)}} = kz + R_{0} , \;{\text{with}}\;{\text{ boundary}}\; {\text{conditions}} \left( {Z_{{{\text{min}}}} ,\; R_{{\rho_{1}^{*} }} } \right){\text{and}} \left( {Z_{{{\text{max}}}} , R_{{\rho_{2}^{*} }} } \right) $$

(13)

Unit cell size grading is achieved as presented in Eqs. 6 and 7, Al-Ketan et al.³⁶ having based their work on derivations by Liu et al. Similar to the work of Al-Ketan et al., hybrid grading is achieved using a sigmoid-type function to create a new local topology, $\varphi_{h} \left( X \right)$, with $k_{i}$ controlling the severity of the transition. The lattice can be composed of an arbitrary number of topologies, $\varphi_{i} \left( X \right)$, with control points, $X_{i}$, placed at the boundaries of desired topology regions:

$$ \varphi_{h} \left( X \right) = \mathop \sum \limits_{i = 1}^{n} \frac{{\varphi_{i} \left( X \right)}}{{e^{{k_{i} X - X_{i}^{2} }} }} $$

(14)

Zhang et al.⁴⁰ presented both linear, $p_{L} \left( {x,y,z} \right)$, and exponential, $p_{E} \left( {x,y,z} \right)$, methods for density grading by porosity control, presented in Eqs. 15 and 16, respectively. While they mention that porosity change is linked to changes in strut diameters, they do not mention the function that relates these two dimensions. Here, $A,B, C, D, a,$ and $b$ are constants:

$$ p_{L} \left( {x,y,z} \right) = Ax + By + Cz + D $$

(15)

$$ p_{E} \left( {x,y,z} \right) = ae^{Ax + By + Cz} + b $$

(16)

A sigmoid function is also presented in Eq. 17 as a means of controlling the porosity change by creating a new local topology, $p_{S} \left( {x,y,z} \right)$, with $k$ controlling the gradient of porosity along the desired axes. The expression $B\left( {x,y,z} \right)$ is the spatial coordinate set that defines the shape of the transition between topologies, presented as a planar boundary located at $z = b$. The terms $\rho_{1}$ and $\rho_{2}$ represent the desired porosities to range between.

$$ p_{S} \left( {x,y,z} \right) = \frac{1}{{1 + e^{{ - k \cdot B\left( {x,y,z} \right)}} }}\rho_{1} + \left( {1 - \frac{1}{{1 + e^{{ - k \cdot B\left( {x,y,z} \right)}} }}} \right)\rho_{2} $$

(17)

$$ {\text{where }}B\left( {x,y,z} \right) = z - b $$

Mostafa et al.⁴⁸ present a method for linear scaling of the unit cell size, $S_{{\left( {x,y,z} \right)}}$, the gradient of which is controlled by $m$ and $c$ is the initial strut diameter, represented by Eq. 18. A relative density method was also developed, whereby the relative density, $\rho_{{\left( {x,y,z} \right)}}$, of the lattice is varied along an axis, presented as a quadratic gradient, with $A^{{}} , B,$ and $C$ as constants, along the x-axis in the form of Eq. 19:

$$ S_{{\left( {x,y,z} \right)}} = m_{{\left( {x,y,z} \right)}} \left( {x,y,z} \right) + c $$

(18)

$$ \rho_{{\left( {x,y,z} \right)}} = A\left( {x,y,z} \right)^{2} + B\left( {x,y,z} \right) + C $$

(19)

Using a combination of both methods allows for the tailoring of both the size and the density of the unit cells comprising the overall lattice, allowing for a variation in pore shape and size while still controlling the relative density of the lattice.

Topology Optimization

Topology optimization (TO) tools comprise a set of algorithms whose purpose is finding optimal material distributions through the minimization of some sensitivity function (often compliance-based).⁵² Although commonly used for structural applications, TO algorithms have been created and applied to a large range of engineering design problems, such as thermodynamics,⁵³ fluid mechanics,⁵⁴ vibrations,⁵⁵ or optical applications.⁵⁶

TO solutions come in a range of forms depending on the method of TO being used. The result of such optimization algorithms is generally a discretized field, or a function, of density values. This definition for density is generally binarized in some way where a density of 0 represents a void-like material and 1, that of solid material, and can be utilized by a designer to make decisions about a solutions form, ensuring the geometry highly utilizes the material, reducing mass overall.

This technique represents a unique opportunity for functionally graded lattices, whereby the TO solution can be utilized to inform the functional grading of the lattice structure. The simplest way in which this functional grading is applied is by direct mapping of voxel densities within the TO solution to the relative density of a unit cell within the lattice. TO is generally conducted with a technique such as SIMP, allowing density values to vary between 0 and 1 for each voxel within the solution domain and scaled for unit cell relative density, to sit between an upper and lower bound generally determined from manufacturability constraints.

Daynes et al. demonstrated this technique for the generation of FGLs with a BCC unit cell topology, changing the local relative density of the unit cells by selecting an appropriate strut diameter.⁵⁷ TO was conducted for a 3-point bending test scenario, specimens were fabricated, and functionally graded lattices were found to outperform uniform lattice structures with an increase of 119% in strength and 30% in stiffness. Panesar et al. utilized the same technique of mapping variable density values from TO solutions to the relative density of lattice cells, comparing their performance to “intersected lattices” where the discrete solid-void TO solution is intersected with a uniform density lattice, and “scaled lattices” where the density values are rescaled so that there are no void-like regions within the lattice.⁵⁸ The different structures were numerically evaluated for strain energy in a cantilever loading case. Graded lattice structures were found to behave the most desirably. Nguyen et al. also used relative density mapping for functional grading but altered the TO model used to include the anisotropy behavior of AM FDM lattices.⁵⁹ TO was conducted for a 3-point bending load scenario, FGLs fabricated at 5 different build orientations were tested under 3-point bending, and load response was used to validate simulated models. Goel et al. mapped densities from SIMP TO solutions to local cellular relative densities of BCC lattices.⁶⁰ Their work proposes a B-spline surface-based design for the geometry of these functionally graded lattices, improving their geometric connectivity at nodes, as strut-based lattices commonly have geometric discontinuities between cells of differing relative density.

Although mapping TO voxel density to unit cell relative density is a simple and straightforward method for functionally grading a lattice, there is an inherent disconnect between the material properties of the voxel within the TO solution and that of the unit cell. TO solutions have an assumption of isotropic material with a uniform density throughout each voxel, whereas this is not the case for lattice unit cell topologies. Efforts have been made to connect the material properties of the unit cell to its relative density, improving the mapping process. Li et al. developed a modified density-based optimization strategy for determining optimal material distribution for cellular gyroid structures.⁶¹ This allowed the inclusion of material properties and density in the optimization process by mapping one to the other, and in this way the TO solution itself utilized material properties more closely related to that of the lattice structure. Specimens were generated for a Messerschmitt–Bölkow–Blohm (MBB) beam loading condition, numerically simulated, and physically tested. The technique was also utilized for a case study design of a quadcopter rotor arm. Similarly, Cheng et al. proposed a stress-based asymptotic homogenization method to obtain the effective properties of a lattice structure in terms of its relative density.⁶² Cheng et al.’s technique was applied in a simulated manner to the L-bracket and 3-point bending load cases, with 3-point loading cases being validated experimentally. Jansen et al. formulated a hybrid density/level set TO algorithm whereby a level set XFEM TO approach defines the shape of the macro-scale structures, while functionally graded lattice inside this structure is modeled by a homogenized material law that is parameterized by volume densities.⁶³ This method was applied numerically to a double-clamped beam and to MBB beam problems in two dimensions, as well as a three-dimensional case study of an industrial bracket.

TO has also been utilized for the optimization of the unit cell itself. Kim et al. used TO to first optimize the unit cell geometry, adjusting volume fraction TO constraints to generate optimized cells of varying relative density.⁶⁴ These unit cells were then mapped globally to a TO solution for the MBB load case, simulated, and tested physically in a 3-point bending test. Teimouri et al. generated unit cell geometry using TO in the same manner as Kim et al., while also hollowing out these unit cells into thin-walled structures to create novel lattice structures.⁶⁵ Any functional grading of the structure was not conducted using TO, instead carried out linearly layerwise.

In addition to utilizing the density field generated from TO, work has also been carried out by Daynes et al. to spatially grade a lattice structure utilizing principal strain trajectories.^57,66 By aligning the cell boundaries along these trajectories, shear stress can be reduced within the structure, allowing for lightweight and stiff structures to be generated. These solutions do not bear a resemblance to periodic cellular structures, however, and have more commonality with the work of Yi et al. and Liu et al. who have altered TO methods to produce results that resemble trusses.^67,68,69 Although they refer to these solutions as functionally graded lattices, perhaps an alternative naming convention should be adopted to differentiate these solutions from functionally graded lattices with periodic unit cells.

Mechanical Properties

The elastic modulus and yield strength of FGLs are plotted in Figs. 3 and 4 to evaluate their mechanical potential. As seen in both cases, the results are less consistent than those of their energy absorptive properties. Nevertheless, certain FGLs displayed a propensity for high specific strength: for example, the rhombic dodecahedron²³ and F2BCC³⁵ FGLs depicted in Fig. 3. Conversely, their stiffness capabilities, determined by the elastic modulus, all appeared to be weak, as seen in Fig. 3. To remove the influence of the material, the mechanical data were presented as relative values, by dividing the recorded properties by the bulk material properties. The Gibson–Ashby model, discussed in Section 8.1, was then applied to the accumulated FGL data to quantify their structural efficiency against the model’s upper and lower limits to which periodic strut-based lattices generally conform. Data positioned from the lower middle to below the lower boundaries of the model were considered mechanically weak. Conversely, data positioned from the upper-middle to above the upper boundaries of the model were considered mechanically strong. This classic model was implemented as it remains the gold standard for lattice analysis, showing a high predictive correlation with experimental results in significant research.³

Fig. 3

The reported FGL relative elastic moduli plotted against the relative density, as a Gibson–Ashby model; circle elements denote the vertically graded lattices, square elements denote the horizontally graded lattices.

Fig. 4

The reported FGL relative yield strength plotted against the relative density, as a Gibson–Ashby model; circle elements denote the vertically graded lattices, square elements denote the horizontally graded lattices.

The yield strength data displayed a potential for load-bearing applications. As displayed in Fig. 4, the majority of the FGL lattices were positioned between the center and upper boundaries of the model, a placement similar to bending-dominated uniform lattices. This is important as the FGLs do not exhibit any noticeable change in strength while redistributing the material to enable a graded configuration. Additionally, the data positioning on the Gibson–Ashby model displays high conformity to the upper and lower empirical limits, indicating that the model remains useful when quantifying the strength properties of FGLs. The horizontally graded lattices (perpendicular to the loading direction) attained a consistently higher strength than the vertically graded specimens, with the honeycomb,²⁷ spiderweb,²⁵ and diamond²⁷ lattices exceeding the upper bound of the Gibson–Ashby model. However, the vertically graded (parallel to the loading direction) could observe a high structural efficiency, with diamond and gyroid⁴⁴ lattices positioned at the upper empirical limit of the model. This indicates that these structures can achieve high strength and large structural strains, enabling load-bearing lattice structures with energy-absorptive functionality as seen in Figs. 4 and 5.

Fig. 5

Comparison between uniform lattice data and FGL data, plotting the reported energy absorption against respective relative densities.

Of the FGLs analyzed, their modulus was the weakest property observed. As displayed in Fig. 3, the reported FGL data were generally positioned at the middle to lower boundaries of the model, indicating a low stiffness. The horizontally graded samples displayed a slightly higher stiffness to the vertically graded samples; however, the variation was minor. This may be partially attributed to the fact that the majority of the FGL lattices were being designed for orthopedic applications, and high stiffness is not an ideal property for bone as it may induce stress shielding. The rhombic dodecahedron²³ unit cell graded horizontally (perpendicular to the loading direction) and F2BCC³⁵ unit cell graded vertically (parallel to the loading direction) were the only two lattices to display high rigidity. Characterized in Fig. 3, the rhombic dodecahedron and F2BCC DGLs were positioned above and at the upper empirical limit of the model, respectively, indicating that the unit cell topology may still fundamentally alter the properties of the graded lattices. The localized deformation common in these structures, and bending-dominated unit cells generally, prevented the FGLs from achieving high mechanical properties. Additionally, the placement of the elastic modulus data indicates poor conformity with the Gibson–Ashby model. Consequently, a new analytical method may be required when quantifying the modulus of FGLs in the future (Sect. “The Gibson–Ashby Model”).

Dynamic Properties

Fatigue performance of a component is dependent on material, geometry, surface, and sub-surface defects, and surface finish, of which the latter two factors are highly dependent on machine operational parameters. Given the specific nature of fatigue properties, data on FGLs is limited although several trends are emerging. Zhao et al. quantified the fatigue properties of EB-PBF Ti–6Al–4V rhombic dodecahedron FGLs, finding that FGLs graded perpendicular to the loading direction fail region by region, caused by crack initiation within the larger strut elements before failing and progressing through the lattice. When compared to uniform lattices of equivalent bulk relative density, FGLs were found to have superior specific fatigue strength at 10⁷ cycles at approximately 4.7 MPa cm³ g⁻¹ compared to 4.3 MPa cm³ g⁻¹ for the strongest uniform lattice.²³ Wang et al. investigated EB-PBF Ti-6Al-4V rhombic dodecahedron FGLs graded parallel to the load, finding that at both high and low stress levels the FGL performed similarly to the uniform lattice that matches the weakest constituting region of the FGL.⁷⁰ Mahmoud et al. noted that the fatigue performance of radially graded cylindrical LPBF Ti-6Al-4V gyroid FGLs were similar to the densest uniform gyroid, both of which were outperformed by the least dense gyroid. This was attributed to increased internal defect rate, stair stepping effects, and partially adhered particles on the surface of struts acting as nucleation points of crack initiation, all defect types are more likely to occur in larger volume struts with higher surface area.⁷¹ Bai et al. determined that both the grading direction and surface quality of LPBF Ti-6Al-4V BCC FGLs had a strong influence on fatigue performance. When grading perpendicular to the load direction, it was noted that strain accumulation in as-manufactured FGLs was lower than when graded parallel to the direction of loading, which indicates superior fatigue resistance. Sandblasting was shown to improve the fatigue performance of FGLs graded parallel to the load, though there was no noticeable effect of FGL graded perpendicular to the load.⁷² Chen et al. found that LPBF elemental tantalum gyroid FGLs show a grading orientation dependency on fatigue properties. When graded perpendicular to the load direction, mechanical and fatigue properties were higher than when graded perpendicular or uniform lattices.⁷³ Yang et al. found that the fatigue life of LPBF Ti-6Al-4V FGL gyroids is increased by up to 167% when compared to uniform gyroid lattices.⁷⁴

Energy Absorption

To quantify the absorption capabilities of the FGLs, the energy absorption and relative density for each of the reported lattices were accumulated and plotted against their corresponding uniform lattice, which can be seen in Fig. 5. A clear improvement over uniform lattices of equivalent average relative density was observed for the FGLs, with a reported increase in energy absorption ranging between 2% and 37%. Furthermore, when plotting the trajectory of the FGL data against the uniform lattice data, it was noticed that, as the relative density decreased, the energy-absorption capabilities for the FGL data consistently deviated further from the uniform trendline, outperforming the conventional lattice structures. Conversely, as the relative density increased, the trajectories begin to converge, which is expected, as the cellular materials gradually become solid and resemble a solid with high porosity. The highest-performing FGLs were radially graded with a higher density at the outer radius and lower density at the core; however, these lattices had the additional benefit of dense supporting columns to reinforce the cellular material.²⁷ These reinforcing components greatly improved the mechanical properties of the lattices and enabled superior energy absorption. However, to evaluate the optimum unit cell for energy absorption, these FGLs were not considered, and the structure with the greatest energy absorption was the TPMS diamond unit cell with a continuous grading along the vertical plane.

The energy absorption capabilities are derived from two primary factors, the grading orientation, and the unit cell type. It has been demonstrated that the grading orientation directly dictates the deformation behavior of the FGL. For vertically graded lattices, localized deformation was observed. This differed from hollow-strut lattices which deformed layer by layer starting at cells in direct contact with the applied load;⁷⁵ instead, regions of the lowest density would undergo deformation, regardless of position. This deformation pattern would occur from lowest density to highest density until the complete structure underwent densification. This localized failure behavior enabled the vertically graded lattices to generally observed higher energy absorptive properties to uniform lattices (Fig. 5). Conversely, the horizontally graded FGLs were more likely to undergo catastrophic failure through global shearing through the lattice (further discussed in Section 6), which would limit their energy absorptive capabilities and closely align their properties to the uniform lattices. The secondary impacting factor is the specific unit cell topology comprising these FGLs. The F2BCC³⁵ and honeycomb²⁷ exemplify how a unit cell type may enhance energy absorption. Furthermore, the high strength observed in Fig. 4 indicates that FGLs may facilitate high-strength, high-energy absorptive structures.

Deformation

Depending on the orientation of grading, FGLs can undergo either localized elastic–plastic deformation or catastrophic failure. If the grading occurs along an axis parallel to the direction of the applied load, deformation due to compression will occur locally at regions of the lowest density. Conversely, if the gradings occur along an axis perpendicular to the direction of the applied load, the lattice shows signs of barreling, which is commonly followed by catastrophic failure in the form of fractures bifurcating the lattice. Localized deformation at areas of low relative density enables the fabrication of highly specific energy-absorptive lattices. This mechanism also allows for highly predictable deformation behaviors and failure points in lattices.

Localized Deformation of Vertical Graded FGLs

Lattices with vertical grading, which is graded along an axis parallel to the load, exhibit a layer-by-layer deformation process. The deformation process begins with the failure of the unit cell layers of the lowest density. After this localized failure occurs, the subsequent lowest density layer deforms and the process is repeated until each layer has failed in turn and the structure either densifies, as demonstrated in Fig. 6a, or experiences catastrophic failure. This targeted deformation is inclusive of all vertically graded lattices regardless of grading direction, that is, if the lattice is graded outwards, inwards, upwards, downwards, or radially. As long as the grading is parallel to the applied load, failure will occur as described above. As the vertical grading is defined by localized failure, the stress–strain relationship appears jagged and unique from conventional uniform lattices. Each layer possesses an elastic zone, as observed in Fig. 6b, followed by an extended plateau region and densification as the following layer is compressed.³⁵ It is also typical of these lattices to have maximum peaks of up to 4 times that of the initial peak in stress–strain graphs.³⁵ These observations indicate an improved load-bearing capacity and strength recovery that could resist further damage.⁴⁶ Furthermore, the unique deformation behavior of these FGLs facilitates a capability for energy absorption that supersedes uniform lattices.^35,37 Conversely, the stress–strain behavior for uniform lattices observed in Fig. 6b displays a singular elastic zone, the subsequent formation of 45° shear bands begins as the struts undergo brittle fracture, and, as the lattice collapses, up to 95% of its strength may be lost and is unrecoverable.⁵⁰ Densification then follows as the intact cells attempt to re-establish lost strength. This deformation behavior indicates a limited capacity for structural recoverability following the elastic deformation, realizing why the energy-absorptive capabilities of these lattices are superseded by vertical FGLs.

Fig. 6

(a) Localized deformation of an AlSi10Mg continuously graded BCC lattice sample. (b) Comparison between the stress–strain data of an Al-12Si F2BCC uniform lattice and an FGL graded along the vertical plane. The collated images are reprinted with permission from Refs. 35, 50.

Stepwise lattices graded vertically deform with the formation of shear bands.²² This can be expected due to large sections of single grading size before an abrupt change to the following graded section, as well as a higher similarity to uniform lattices in comparison to FGLs. A typical feature of FGL stress–strain graphs is a stress value that recovers and surpasses the initial peak stress before densification.²² The stepwise lattice with vertical grading has limited recovery after the initial peak stress before densification, with the stress–strain response showing greater similarity to uniform lattices. There is also very limited data in comparison to the other graded lattices which positions the stepwise grading as an area of future research.

Catastrophic Deformation of Horizontal Graded FGLs

The deformation behavior of horizontally graded FGLs (gradings perpendicular to the direction of the applied load) indicates greater similarity to uniform lattices. There are two types of horizontal plane gradings present in the literature: radial and unidirectional.

FGLs with unidirectional gradings are typically cubic or rectangular lattices, that are graded along one direction (Sects. “Designing Density Gradients” and “Orientation and Direction of Grading”). They show the same deformation behavior as a uniform lattice, as displayed in Fig. 7. The deformation response of unidirectional graded lattices under uniaxial compression exhibits three distinct regions: linear elasticity, plateau region, and densification. With uniform lattices, the first peak marks the start of the diagonal shear band formation, which is a trait also exhibited in FGLs graded along the horizontal plane.^36,37 This results in a loss of strength in the stress–strain response before entering the plateau region. The deformation continues and has been observed to take place layerwise, after the shear band³⁶ and before reaching densification.

Fig. 7

Side views of (a) uniform (U-GCS); (b) vertically graded (G1-GCS); (c) horizontally graded (G2-GCS)SS316L gyroid lattices; and (d) the stress-strain graph for each of these structures. The collated images are reprinted with permission from Ref. 37.

Cylindrical lattices have also been examined with radial gradings perpendicular to the applied load (Sects. “Designing Density Gradients” and “Orientation and Direction of Grading”). These lattices are primarily designed cylindrically with unit cell layers becoming denser or more porous internally or externally to replicate the architecture and shape of bone for heightened biomimicry. Uniform cylindrical structures exhibit the typical 45° shear band commonly seen in cubic lattices.³⁴ Cylindrical FGLs with radial grading commonly exhibit bulging or barreling when in compression.^27,29,34 The areas with lower densities are the areas in which bulging initiates. There are also instances in which radially graded lattices exhibit a shear band when deforming,^27,29,34 which can be attributed to the uniform grading along the height of the test piece. Axially graded FGLs have a lower tendency to deform in a shear band and generally occur in a layerwise deformation process.³⁴

A stepwise lattice graded horizontally produces a similar response as that of vertically graded stepwise lattices. Cylindrical lattices that are graded radially and in a stepwise fashion deform with the characteristic shear band, typical of uniform lattices.²⁹ This response is expected as the grading is uniform with the load being applied perpendicularly, resulting in a deformation process like that of a uniform lattice. There are limited data on cubic lattices that are graded horizontally in a stepwise fashion, which indicates this is an area of future work and investigation.

Deformation of Hybrid Gradings

Hybrid lattices incorporate multiple unit cell topologies to grade the lattice with either horizontal or vertical gradings. Hybrid lattices that are graded horizontally, typically act as uniform lattices, which is expected, being essentially the combination of two uniform lattices.^23,36 Stress–strain data confirm this, with a performance similar to that of uniform lattices; the structure is unable to recover substantially after the initial peak stress, as seen in Fig. 8b (MML1). Hybrid FGLs graded parallel to the loading direction display a less severe layer-by-layer deformation. As also seen in Fig. 8, the stress–strain response is that of the parallel multi-morphology lattice, where the response increases incrementally after the initial peak. The common layerwise deformation approach is visible, beginning at the connection of the different unit cells.^36,44 Following initial compression, a layerwise deformation system can still be observed before the combination of a 45° shear band and layer compression, as seen in Fig. 8c (MML1T strain = 50%).^36,44 This response is expected with the grading system being a combination of two uniform unit cells with a short sharp transition in topology, resulting in a pseudo-stepwise structure.

Fig. 8

Gyroid/diamond SS316L hybrid FGLs: (a) graded vertically (MML1T); (b) graded horizontally (MML1); (c) stress–strain graphs for MML1T and MML1 FGLs; and (d) observations of the FGLs during the compressive deformation process. The collated images are reprinted with permission from Ref. 36.

Applications

FGLs are lattice topologies designed for their application. Industry interest in FGLs is driven primarily by the ability to tailor the lattice geometry and topology for function-specific mechanical properties, including energy absorption, thermal resistance, expansion and dissipation, and biomimetic applications.

Energy Absorptive Applications

High-porosity and low-density lattices exhibit excellent energy absorption capabilities. FGL structures provide an advancement upon these properties, achieving higher energy absorption at lower relative densities, as displayed in Fig. 5 and Table IV. These results can be derived from the topology and grading orientation of the FGL, as, if densities are graded vertically, the lattice will generally achieve excellent energy absorption. This is due to the deformation occurring sequentially, where layers with the lowest relative densities are the first to undergo deformation and catastrophic failure. This will continue from the lowest density to the highest density until the lattice undergoes complete densification; this is further discussed in Section 6.1 and Section 5. A high energy absorption may enable new or expanded functionality in many engineering sectors, including automotive and aerospace. Uniform lattices have been proposed within crash boxes in automotive vehicles⁷⁶ and flight data storage systems.¹² Both are implemented to reduce the impact and damage to the passengers, pedestrians, and flight data, respectively. Uniform lattices, unlike FGLs, generally deform through catastrophic failure. Consequently, their capacity for energy absorption is limited. If FGLs were instead applied to these applications, a substantial improvement may be attained.

Table IV The achievable mechanical properties of lattices sourced from the literature

Thermal Applications

The high surface area to volume intrinsic to lattices is an attractive property in the thermal engineering space. FGLs enhance their effectiveness with targeted geometries and TO to enhance control and the elimination of heat. The geometry of FGLs has been controlled to design thermal materials for enhanced thermal stress endurance,³³ expansion,⁷⁷ and dissipation^30,31,32 performances, in comparison to conventional technologies. In turbine blades, TPMS gyroid, primitive, and diamond FGLs have been implemented as space fillers to reduce the impact of high thermal stress effects. The optimum distribution of the relative density deviated between 20 and 90%, and the placement of high- and low-density unit cells were distributed to best respond to thermomechanical loads. The outcomes of this experiment saw reductions in: the weight of the blade by 33.41–40.32%; the stress rate by 25.52–48.55%; and the deformation rate by 7.35–19.58%.³³ Additionally, heat sink FGLs have also been designed to enhance the thermal conductivity through variations in the lattice’s relative density from the bottom to the top of the structure, with the lattice struts becoming thinner further away from the heat source. Conventional fin-based dissipative structures have been superseded by the exploratory FGLs in FEA simulations.³⁰

Biomedical Applications

A clear focus of FGL research is orthopedic applications. Human bone is comprised of both cortical and cancellous layers, and cortical bone has high strength, stiffness, and density. It forms the structure surrounding the bone marrow and is responsible for the structural integrity of the skeleton. Conversely, cancellous bone is contained within the external cortical layer and is sponge-like with low mechanical properties and high compliance. The hierarchical topology and geometry of bone prevent uniform lattices from replicating the structure with a high degree of biomimicry; however, with the application of functionally graded lattices, this may be achieved. FGLs have been proposed for orthopedic implants due to their energy-absorptive capabilities, mechanical properties, and biocompatibility.^{25,26,27,28,29} The high propensity for biomimicry is observed in cell cultures from calcified matrix staining. These experiments indicate that all the tested FGL structures supported mineral production and displayed better osteointegration than available mainstream dental implants.²⁵ For the FGL samples,²⁷ the elastic modulus properties (3.79–10.99 GPa) were within the range for human bone, and the yield strength properties (110.82–423.82 MPa) were high enough to satisfy the biomechanical requirements for orthopedic applications. Furthermore, in comparison to conventional uniform lattice designs for surgical implants, an 11% increase in energy absorption was observed for the FGL designs.²⁸

Clinical Applications

Several clinical applications have been developed that apply FGLs to reduce potential bone reabsorption attributed to stress shielding. Arabnejad et al. proposed a simplified 2D model and genetic algorithm to minimize bone reabsorption and interface stresses in a hip implant, two common methods of failure. When comparing solid and FGL implants, the FGL implant experiences 76% less bone reabsorption and 50% lower interface stresses.⁷⁸ He et al. identified a method of density gradient optimization that first uses TO to match the structural compliance of a healthy femur. Partially solid regions produced by TO are replaced with lattice structures whose struts vary in cross-section area along their span to create a mass minimizing continuously varying density gradient. This FGL design resulted in 57% lower “stress shielding increase”, a measure of an implant’s impact on the surrounding bone, when compared to a generic implant.⁷⁹ Kladovasilakis et al. designed solid and FGL hip implants according to ASTM F2033-12, with the solid stem replaced by lattices with a nominal relative density of 50%. Lattice optimization was achieved by sectioning the lattice region into volumes of high stress, assigned a relative density of 65%, and low stress volumes which were allowed to vary in relative density to maintain the nominal bulk relative density. The ungraded Schwarz diamond topology, when compared to the solid implant with a factor of safety (FOS) of 2.5, saw a 62.4% decrease in FOS, while the FGL Schwarz diamond lattice saw a more modest 16.8% decrease in FOS, while both lattice implants had 38.5% lower mass.⁸⁰ Xu et al. used TO and a genetic algorithm to produce a Burch–Schneider cage, an implant used in total hip arthroplasty surgery. The entire solid implant volume was replaced with an octet-truss lattice, to which the density is made to vary by controlling strut cross-section areas. The results showed a homogenization of stresses and a reduction in peak Von Mises stresses of 75% in the FGL implant when compared to the solid implant.⁸¹

Insights and Outcomes

Following the systematic assessment of the mechanical, geometrical, and topological properties of reported FGL lattices, insights, and outcomes were derived from trends and deficiencies. These findings should provide future research opportunities for FGLs. The insights and outcomes are discussed below.

The Gibson–Ashby Model

The relative yield strength versus relative density graph displayed in Fig. 4, indicates that the Gibson–Ashby model retains a high accuracy to the FGL data. Conversely, the relative modulus versus relative density graph, found in Fig. 3, observes relatively low conformity to the Gibson–Ashby model. The FGL performance is unique when compared to meta-analyses of uniform lattice data,³ where BCC lattices are the only FGL unit cell variant to consistently fall below the model’s lower boundaries for relative modulus. Consequently, a variant of the Gibson–Ashby model may be required for the prediction of the relative elastic modulus for FGLs. Specifically, the Gibson–Ashby model was developed with respect to periodic or stochastic cellular structures, rather than for the intentionally non-uniform cellular structures inherent to FGL. Nevertheless, until a new model is designed, the upper empirical limit of the Gibson–Ashby model remains as the most commonly applied benchmark to assess outstanding cellular materials.

Mechanical Properties

In terms of structural efficiency, the graded lattices observe a moderate relative strength and low relative stiffness as highlighted within the Gibson–Ashby model in Figs. 3 and 4. However, as displayed by the high efficiency of the rhombic dodecahedron,²³ the choice of the unit cell may be an impacting factor. Consequently, the fabrication of graded lattices with well-known high-strength unit cell topologies like the octet truss⁷⁰ or FCCZ¹⁸ should occur to assess whether the lattice grading reduces their structural efficiency. Furthermore, with the recent advancements in LPBF hollow-strut lattices that have achieved high strengths to consistently surpass the upper empirical limit of the Gibson–Ashby model,^82,83,84 it may be valuable to apply the hollow cross-section or vary the cross-sectional shape⁸⁵ of the FGL topology to surpass the performance limits of the uniform lattices and the upper empirical limit of the Gibson–Ashby model for both strength and stiffness.

FGLs can achieve energy-absorptive properties that supersede uniform lattices while maintaining competitive strength that conforms to the middle to upper boundaries of the Gibson–Ashby model. However, these structures display a low stiffness, which falls below the lower boundaries of the analytical model. These properties make the structures ideal for orthopedic applications, strengthening these structures and optimizing their geometry to facilitate a high level of biomimicry in terms of strength and architecture. Furthermore, the low stiffness of these structures could potentially eliminate the concern for stress shielding. The high energy absorption may also find consistent use within damage resistance and shock absorption applications.

Deformation

The deformation capabilities of FGLs are directly dependent upon the direction of grading. Reported uniaxial compression testing data indicate that FGLs graded vertically deform locally. Conversely, FGLs graded horizontally fail catastrophically. The latter deformation does not enable new deformation outcomes, with deformation behavior closely resembling uniform lattices. The local deformation is of particular interest; however, this behavior should enable controllable deformation, with structures deforming at specific regions of the lattice where the relative density is lowest. This level of control is unprecedented and should enable highly customizable deformations for specific applications. For example, FGLs could be implemented as space fillers to absorb impact within vehicles in the case of collision. They could be arranged to progressively deform and absorb the impact to reduce potential harm to the passengers.

Inconsistent Terms

A noticeable inconsistency is observed within the literature between stepwise and continuous grading definitions. It is common to observe incorrect definitions of continuously graded lattices when they are stepwise. Continuously graded FGL should be graded proportionately to the number of unit cell layers in the direction of grading. This incorrect terminology confuses the mechanical capability of each FGL variant and may lead to false outcomes. Consequently, this review has redefined many reported FGLs and the definitions are defined within Section 1.1 and Section 1.2 and the changes may be observed in Table I.

Geometric Grading

Currently, when grading an FGL, either the unit cell size or all of the strut thicknesses within the unit cell are altered, regardless of whether the grading is stepwise or continuous. As research in the field of FGLs advances, there is a significant opportunity for a new class of localized DGLs, and by extension FGLs, in the form of strut-graded lattices. In grading struts, a density gradient is formed within the unit cells that form the bulk lattice. This density gradient redistributes the material away from the strut midspans and towards the nodes.

Lattices with strut gradients have been developed which have shown greater elastic modulus and yield strengths than equivalent relative density, uniform strut lattices.^86,87 Simulations, confirmed by experimentation have shown a change in failure with sufficient tapering, as can be seen in Fig. 9, in addition to an increase in yield strength accompanied by a reduction in strain before failure.⁸⁸ This conforms with the established pattern in lattices graded parallel to the applied load of failure occurring layerwise, starting with the region of lowest density.

Fig. 9

Experimental results of a uniform strut lattice (a) and a graded strut lattice (b), highlight a change in failure mode from a 45° shear plane to a horizontal layer-wise failure. The collated images were reprinted with permission from Ref. 88.

Improvements in energy absorption have also been achieved though grading struts, and it was established that gains were greater with lattices of higher relative density, though there are diminishing returns as the degree of grading increased.⁸⁹

Further exploration of unit cell grading should include selectively reinforcing nodes of the lattice at areas of high strain with additional material, application of polygonal or shaped strut cross-sections,^83,84,85 and introducing asymmetry among the constituting struts of a unit cell. Additionally, many of these avenues for research are complimentary, and may potentially be applied in concert with each other and with the more conventional methods listed in this review. These methodologies should all be investigated to enable the greatest design potential and to maximize the functionality of FGLs.

The differentiation between stepwise and continuous grading should also be explored in future experiments. The deformation and mechanical performance of the FGLs appear to be largely dependent upon the orientation of grading. However, the grading classification (e.g., stepwise and continuous) may also play an important role, but, unfortunately, the accessible data are currently insufficient to ensure systematic conclusions regarding their impact. This outcome could enable a greater level of application-specific FGLs.

Materials

Although metal, composite, and polymer FGLs have been fabricated, the vast majority of research has employed the titanium alloy, Ti-6Al-4V, due to the focus on orthopedic implants. The consistency of this singular material has limited the ability to understand how deformation response, mechanical properties, and energy absorption potential differs in FGLs comprised of other materials. Consequently, there remains an opportunity for new studies to report on FGLs that are comprised of new feedstocks to broaden the applications for these novel lattices. Additionally, the lack of ceramic FGL studies should provide an opportunity to compare how materials with higher stiffness respond under compression. This could examine whether localized failure can still occur, or whether the materials defy grading orientation and direction, enabling new outcomes.

Nature-Inspired Design

The inspiration for FGLs may be derived from nature; however, the majority of the gradings are unidirectional arrangements. To achieve higher strengths or energy absorption capabilities, natural flora and fauna should be evaluated as inspiration. The graded structural hierarchy of bamboo, for example, is graded at the macro-scale, the lacuna down to its micro- and nano-scale matrices, as seen in Fig. 10. This allows bamboo to yield a higher strength-to-weight ratio than steel and concrete.⁹⁰ FGLs could, for example, improve the lattice stability of bamboos by acknowledging the relationship between the septa and lacuna, which involves a density increase to significantly improve the strength. It could be theorized that this could be achieved by reinforcing the areas of the lattice that undergo the highest strain with an increased density grading, similar to the higher density of the septa, and reducing the density at areas of low strain, similar to the lacuna. The strength of FGL lattices could be greatly improved without drastic changes to the density. If nature-inspired FGLs are designed that recognize the geometries of certain natural topologies, developed through millions of years of evolutionary optimization, new mechanical properties may be attained.

Fig. 10

The hierarchical (a) macrostructure to the (b) microstructure of bamboo. The collated images are reprinted with permission from Ref. 91, and from Ref. 92 under the terms of the Creative Commons CC BY license.

Stretch Dominated Lattices

According to the Maxwell criterion, every lattice studied within this review is bending-dominated. The Maxwell criterion is used to predict the deformation response of a lattice under an applied load, the two primary classifications being stretch- and bending-dominated responses. Stretch-dominated lattices generally observe high stiffness to applied loads, while bending-dominated lattices observe high compliance and energy absorption. This review represents the significant majority of the current FGL AM designs, as observed in Fig. 1. This indicates a clear research focus on the energy-absorptive capabilities that these novel lattices can achieve. The limited variance of unit cells prevents the mechanical capabilities of FGLs from being completely quantified, as stretch-dominated lattices are typically characterized by high rigidity and low compliance. Consequently, although the mechanical capabilities in both Figs. 3 and 4 appear to be unimpressive, further research into stretch-dominated FGLs may achieve improved results. The lack of unit cell variation also limits the understanding of deformation behavior, as unit cells with high rigidity and lower compliance may inhibit local deformation or display a novel buckling behavior. This deficiency within FGL design presents an exciting opportunity for future research.

Fabrication

FGLs have been primarily focused on orthopedic applications; however, the ability to replicate the complex hierarchical arrangement of nano- and microstructures that comprise human bones are currently unattainable due to limitations in current generation PBF. These architectures range from nanostructures and sub-microstructures, like fibrillar collagen and embedded minerals (< 300 nm to 10 µm), to sub-microstructures and microstructures like Haversian systems and osteons (10—500 µm).⁹³ All the PBF techniques (SLS, LPBF, and EB-PBF) are limited to fabricating feature sizes at a microstructural level. Conversely, the research of hollow-strut lattices has enabled various AM techniques to attain nano-scale feature sizes.⁷⁵ Processes including atomic layer deposition, and electroplating may facilitate the direct replication of the complex nano- and sub-microstructures of bone. Furthermore, by fabricating functionally graded hollow-strut lattices, an additional level of architecture could be attained to facilitate further cellular adhesion.

Summary and Concluding Remarks

Functionally graded lattices have observed significant growth within the past decade. The experimental data and research accomplishments within this field have been accumulated and evaluated in this review. Numerous conclusions have been derived, ranging from the quantification of the multifaceted mechanical potential of the lattices to the impact of grading upon deformation behaviors.

The deformation behavior of FGLs is characterized by either localized or global failure. These deformation modes can achieve high compliance or high structural stability, respectively. The different behaviors are directly dependent upon the orientation of grading. Lattices graded horizontally, that is perpendicular to the load, deform globally, with 45° sheer bands a common occurrence. Conversely, lattices graded vertically, that is parallel to the load, deform locally layer-by-layer. The impact of the unit cell and the material have yet to be fully quantified, and further research is necessary to elucidate effective methods of deformation control.

These deformation behaviors facilitate unique mechanical properties, ranging from high energy absorption to high strength. Both of these properties have displayed the potential to exceed the upper boundaries of the Gibson-Ashby model and to supersede uniform lattices. The elastic modulus of these properties, however, is surprisingly low. Excluding the FGLs which incorporated fully dense cores of reinforced fully dense segments, the TPMS diameter lattice graded vertically displayed the highest energy absorption. In terms of strength, the honeycomb and F2BCC lattices graded along the vertical plane demonstrated the highest yield strength. The only lattice with notable modulus properties was the rhombic dodecahedron graded horizontally.

Initial interest in FGLs was facilitated by their greater capacity for design freedom in contrast to uniform lattices, due to the variety of pore sizes. FGLs enabled orthopedic implants with higher biomimicry and energy absorption with greater control. Now, however, interest should be directed to optimizing the unique mechanical properties of these structures. As demonstrated in this review, they have indicated high mechanical potential, especially for exclusively bending-dominated lattices. By exploiting high-strength, stretch-dominated unit cells like FCCZ and FCCXYZ, hollow-strut lattices may be optimized to a point at which they start to consistently exceed the upper regions of the Gibson–Ashby model. Additionally, by taking inspiration from nature-inspired design, the grading could be distributed to respond to loading conditions, thereby improving the mechanical properties of these structures and expanding functionality to load-bearing applications.

Through this review, we have shown that FGLs display the potential to advance on uniform lattices. Their unique properties should enable a variety of real-world applications to be proposed. As AM technologies continue to evolve, we believe that FGL structures will grow to become a new class of engineered materials in the future.