Revealing the apparent and local mechanical properties of heterogeneous lattice: a multi-scale study of functionally graded scaffold

ABSTRACT Functionally graded scaffold (FGS) can flexibly regulate the mechanical properties of bone scaffold and holds great promise for offering multifunctional responses of orthopaedic implants. Heterogeneous FGSs constructed by skeletal and sheet triply periodic minimal surfaces (TPMSs) have been proposed in this study. The diversified deformation mechanisms of TPMS-FGSs showed superior mechanical stability, and energy absorption efficiency was enhanced by 3.0–79.0% and 2.6–16.8% compared to uniform skeletal and sheet TPMS, respectively. The graded structure of TPMS-FGSs altered the large-scale 45° shear failure to layer-wise or zigzag failure mode. Moreover, the comprehensive reformation of strain distribution and crack propagation in transition region under small compressive strain was experimentally and numerically studied. The results shed light on the global and local mechanical regulation mechanism of TPMS-FGS. GRAPHICAL ABSTRACT


Introduction
Segmental bone defect, either from trauma, tumour or infection requires surgical management and interventions to reconstruct the integrity of the skeletal system.Bone scaffold is, therefore, needed to bear the physiological load and temporally serve as the substrate material for cell attachment, differentiation and proliferation.The graded and bioactive physiological environment of segmental bone defect poses rigorous demand on the design and characterisation of bone tissue scaffold (Du et al. 2019).Functionally graded scaffold (FGS), manufactured by using additive manufacturing (AM) technology, has been gaining momentum from both the fundamental and technological perspectives due to its diversified response behaviours and multi-functionality (Herath et al. 2021;Kolken et al. 2022;Yang, Wei, and Mao 2022).Extensive studies made attempts to biomimetically reproduce the hierarchical structure of human bone, especially the loadbearing bone tissue such as hip and femur, by using diverse lattice structures (Zhang et al. 2019;McGregor et al. 2021;Tan et al. 2021).By properly encoding the spatial distribution field of structural parameters such as pore size, strut thickness and porosity, mechanical properties and cellular microenvironment can be precisely regulated to reduce stress shielding effect and promote bone tissue regeneration (Wang et al. 2016).
Recently much effort has been devoted to develop favourable design strategies for FGS.Changing the strut thickness of lattice structure has been the most widely used method because of its easy accessibility and good structural continuity; this design strategy provides a porosity gradient and, therefore, the mechanical properties can be altered according to a rule of power law exponent (Gibson and Ashby 1989).Various types of unit cells (lamellar structure (Yang et al. 2020), polyhedral frame (Li et al. 2020), stochastic truss (Wang et al. 2018), etc. and multiple design rules including step-wise (Zhang et al. 2019), linear (Soro et al. 2021), exponential (Zhang et al. 2018a) and sigmoid (Xiong et al. 2021;Yang et al. 2021) structure thickness gradient are proposed, the effects of magnitude and direction of structure gradient on the apparent properties (elastic modulus, yielding strength, permeability) and failure modes at large compressive strain have been validated.Properly assigning the spatial arrangement of porosity showed a potential of weakening large-scale buckling and preventing largescale shearing fracture plane, as verified by our previous work (Zhang et al. 2019).For the purpose of further improving the performance diversity and biological compatibilities, other FGS design strategies, including but not limited to changing periodicity and anisotropy were also investigated (Zhang et al. 2020;Wang et al. 2021).Although the regulatory effect of graded structures on the large-scale motion, deformation and failure mode were confirmed experimentally and numerically, the lattice structure requiring high strength is usually attained at low porosity, leading to squeezed pore structures or over-thicken struts and further negatively impacts the cell growth, permeability and media transfer activities and cause severe strain concentration under biological loading.Therefore, FGS that made up by multiple types of unit cells has become a new research direction.The spatially-changing topology structure significantly enhances the flexibility and diversity of the mechanical behaviours of FGS (Pham et al. 2019;Liu, Lertthanasarn, and Pham 2021) and comprehensively meets the mechanical and biological needs in bone tissue engineering.
More recently, mathematically defined triply periodic minimal surface (TPMS) has been increasingly used in FGS design due to its excellent overall mechanical performance, high permeability and large surface area.TPMS possesses an average curvature close to zero; similar structure were found extensively exist in nature such as trabecular bone (Callens et al. 2020) and knobby starfish (Yang et al. 2022).TPMS can be expressed using implicit equations, endowing it with great availability and design flexibility (Feng et al. 2020).Specifically, multiple structural gradient (period (Zhang et al. 2020), amplitude (Liu et al. 2018), torsional displacement (Chen et al. 2018)) are proposed by modifying the form of a trigonometric function.The FGS construction method of altering the constant term in TPMS equations showed impressive convenience in defining skeletal and sheet TPMS structures (Kapfer et al. 2011;Fan et al. 2021), as well as controlling the spatial distribution of structure thickness (Yoo 2011).This intrinsic mathematical property of TPMS provides the possibility of integrating the high specific surface area and strength of sheet TPMS together with sufficient space for tissue regeneration of highlyporous skeletal TPMS, making it a favourable candidate for FGS application.Moreover, sheet TPMS generally exhibits a semi-stretching dominated deformation mechanism (Abueidda et al. 2016;Liu et al. 2022), whereas skeletal TPMS can be altered from bending dominated to stretching dominated depending on its topology (Abou-Ali et al. 2020), therefore, a continuous and smooth transition of deformation mode can be achieved by coupling skeletal and sheet TPMSs together.TPMS-FGS not only simultaneously possess excellent mechanical and biological performance, but also show great design flexibility in deformation pattern adjustment.Therefore, it is of great interest to develop a heterogeneous TPMS-FGS based on skeletal TPMS and sheet TPMS and provide a deeper insight into its biomechanical performance.Specifically, the precise characterisation of localised deformation and regulatory role of graded structures on the mechanical behaviour of FGS urgently require further comprehensive investigation.
Most studies on the deformation features of TPMS-FGS adopted compression test as the main experimental characterisation method, the compressing process generally goes until a large engineering strain is reached.Researchers focused on the apparent mechanical properties such as elastic modulus and yield strength in elastic stage (small compressive strain), whereas deformation and failure pattern are usually discussed in plateau or densifying stage (large compressive strain) (Zhang et al. 2019;Bai et al. 2021;Wang et al. 2021).This study mode is often applicable for bulk materials, but for FGSs, the complex structure of which experiences significant local strain concentration and premature failure in the initial stage of compressing (Feng et al. 2021), these early ununiformity and destruction are considered to be fatal for the subsequent overall yielding and failure (Cao et al. 2018;Wang et al. 2021).Hence, the mechanical response of FGS at small compressive strain is worth to be revealed.Digital image correlation (DIC) method is increasingly accepted as a powerful tool for shape, motion and deformation measurement of lattice structures.By tracking the motion of the natural rough surface of as-built AM samples (Tan et al. 2021) or the painted speckle patterns (Xu et al. 2021), intuitionistic and quantitative characterisation of strain distribution can be obtained.The accuracy and reliability of DIC analysis highly depend on the quality of speckle pattern (Dong and Pan 2017).However, the irregular morphology of as-built part is intrinsically determined by the selective laser manufacturing (SLM) technology, while the painted speckles is bedeviled by low controllability and shape irregularity.These technical limitations hinder the highly accurate characterisation of unit-scale or subunit-scale deformation.Few studies experimentally reveal the high-precision strain distribution feature of TPMS-FGS in the elastic stage with improved DIC speckle pattern.
On the other hand, the local mechanical behaviour of lattice material is sensitive to the structural parameters and loading conditions (du Plessis et al. 2022).Given the fact that lattice structures are mostly composed of periodically arranged unit cells, the graded structure of FGS inevitably alters its periodic feature of deformation pattern.Therefore, the spatial-temporal asynchronous of mechanical behaviour, which is triggered by the structural variation of FGS, serves as a decisive factor in controlling the apparent mechanical response.When the loading goes in the gradient direction, the stress-strain curve shows a continuous or step-wise uptrend (Zhou, Jin, and Du 2020).The buckling and collapse firstly emerged in highly porous part and gradually extend to low porosity region.While the structural gradient perpendicular to the loading direction showed the potential to regulate the plateau stage and failure mode (Zhao et al. 2018;Kas and Yilmaz 2021).Although the graded structure significantly influenced the elastic deformation and yielding and thus determined the subsequent deformation and failure, a more accurate characterisation of strain distribution across the graded structure, especially the unit cell scale or sub-unit cell scale structure, is still urgently needed.
Herein, we hybridised skeletal-TPMS based and sheet-TPMS based structures with continuously changing porosity by using sigmoid functions, four types of TPMSs, including Diamond, Gyroid, I-WP, and Primitive were used to construct heterogeneous TPMS-FGS.The quasistatic mechanical properties (elastic modulus, yielding strength, deformation behaviour, failure mechanism, etc.) were systematically investigated from perspectives of material microstructure and graded structural designs.The evolution mechanism of Skeletal & Sheet-TPMS-based graded structure fracture and failure were firstly revealed by a high precision digital image correlation (DIC) method and numerical study.The present study sheds light on the versatile bone tissue engineering methodology with favourable design flexibility and feasible mechanical properties.

Structural design of TPMS samples
The design strategy in the present study seeks to blend the Skeletal TPMS-based and Sheet TPMS-based scaffold with an increasing density, intending to acquire a biomechanical gradient similar with trabecular and cortical bone.The basic TPMS equations are illustrated in Equations (1-4).
f Diamond (x, y, z) = cos(ax) cos (ay) cos (az) − sin(ax)sin(ay)sin(az) (1) f Gyroid (x, y, z) = cos(ax) sin (ay) + cos(ay)sin(az) f I−WP (x, y, z) = 2(cos(ax) cos (ay) + cos(ay)cos(az) +cos(az)cos(ax)) − (cos (ax) + cos (ay) + cos (az)) (3) where a is the constant coefficient related to the period of the TPMS structure.In the present study, the value of a was taken as 1.33π so that the unit cell size is set to 1.5 mm.FGS is generated and exported by performing TPMS equation transformation and Boolean calculation.The solid part of the TPMS-based structure is identified by adjusting the constant value t and in equation Equation (5): The cellular structures with different porosities were combined using the Sigmoid function to achieve a smooth S-shaped gradient structure.A graded skeletal TPMS-based FGS can be expressed as in Equation (6).
Sigmoid function provides a S-shaped structural gradient and controllable transition boundary (Vijayavenkataraman, Kuan, and Lu 2020; Yang et al. 2021).The relative density gradient of TPMS-FGS was achieved by using a sigmoid function a(x, y, z), as shown in Equation (7).kG(x,y,z)  (7) In the present study, the structural gradient of the FGSs was on X-axis and the middle-point of the Sshaped transition lay in the G(x, y, z) = x − 7.5 plane.The gradient control parameter k was set to 0.5 to provide a smooth and steady structure variation.
In Boolean calculation, firstly, a skeletal based-TPMS structure with density ranging from 25% to 65% is achieved by coupling sigmoid function with TPMS equations, then a graded inner part with a similar transition mode of apparent density (0-20%) is removed to create a heterogeneous structure with relative density increases from 25% to 45%, as respectively indicated by translucent part and coloured part in Figure 1(a), the density values mentioned above are modulated by modifying the constant value t in TPMS equations.Considering the selective laser melting (SLM) processing limit of feature size, which is 200 μm in this case, the inner skeletal part between x = 0 and x = 7.5 (the structures marked red in Figure 1(a)) has been cut off to avoid pore blocking and structure fracture.By doing so, the morphological irregularity caused by manufacturing defects is effectively reduced and the biomechanical response is more stable and reliable.The same design strategy has been applied to Diamond, I-WP and Primitive (Figure 1(b)).The skeletal, sheet and graded TPMS samples were referred as X-Sk, X-Sh and X-FGS (X indicates the TPMS type D, G, I and P, which correspond to Diamond, Gyroid, I-WP and Primitive TPMSs, respectively) in the following sections (Figure 2).
All samples are exported in STL format and the external dimension of the sample is 15 × 15 × 15 mm 3 .The post-processing of the STL file was conducted in Geomagic Studio.A self-developed script is used to evaluate the morphology of TPMS-based FGS, as shown in Figure 1(a), the graded structure generally exhibits negative Gaussian curvature, similar to that of trabecula.Finally, the FGSs are fabricated by using SLM.

TPMS-FGS fabrication by selective laser melting
All TPMS samples were built by SLM using a commercial EOS M290 system (EOS GmbH, Germany) with a Yb-fiber laser (400 W, 1064 nm).Ti6Al4V ELI (Grade 23) was used to fabricate TPMS-FGSs (Table S1).The morphology of Ti6Al4V ELI powder (EOS GmbH, Germany) and particle size distribution are shown in Fig. S1.Before sample fabrication, processing parameters including scanning speed, laser power, layer thickness and hatch distance were optimised and set to 240 mm/s, 240 W, 30 and 50 μm, respectively.In order to residual stress reducing purpose, the laser rotates 67 degrees on each layer during SLM processing (Figure 3).Argon protective atmosphere was adopted to keep the oxygen content below 0.1% within the build chamber.After fabrication, the as-built samples were revealed from the titanium substrate using electric wire cutting.

Heat treatment and metallographic observation
Subtransus heat treatment was adopted to investigate the effect of microstructure on the biomechanical performance (Zhang et al. 2018).Samples were heattreated by a vacuum furnace (below 10 3 Pa) with a heating rate of 20°C /min.The temperature was maintained at 900°C for 2 h before furnace cooling, samples were removed from the furnace for air cooling after the furnace temperature reached 200°C (Ruirun et al. 2016).To observe the microstructure, samples were manually grinded up to 2000 grid using SiC paper and then polished up to 1 μm using polishing paste.The samples were immersed in an aqueous solution containing 1 mL HF, 5 mL HNO 3 and 10 mL H 2 O.The microstructure is characterised by using optical microscope.

Structure and morphology characterization
The overall density values of the samples were determined by dry weighing.Prior to dry weighing, all the samples were cleaned in an ultrasonic bath and then dried.The theoretical density of the Ti-6Al-4V alloy was assumed to be 4.43 g/cm 3 and the density value of each sample was calculated by dividing its mass by the mass of a solid circular cylinder with the same outline dimensions.
The three-dimensional model of as-built samples was reconstructed by high-resolution computed tomography (CT) (Phoenix nanotom m system, General Electric).The applied voltage and current were 100 kV and 135 µA, respectively.In order to capture structural details, voxel sizes of 3.5 × 3.5 × 3.5 μm and a scanning time of 20 min were used.Two-dimensional cross-section images of the sample were reconstructed by using built-in software and morphological parameters, including pore size, strut thickness and porosity, were calculated using the substrate plugin BoneJ 1.4.2 in ImageJ and self-made script.After compression, fracture surfaces micrographs were obtained using scanning electron microscopy (SEM, FEI Quanta 200 FEG).

Mechanical test
The biomechanical performance of TPMS-based FGSs was characterised by compression test using a universal material testing machine.According to the mechanical testing standards ISO 13314 for porous and cellular metals, the moving speed of the crosshead was set to 0.075 mm/s (International Standard, I 2011).The loading direction was vertical to the structural gradient direction,  (1,0,0), (1,1,0), (1,1,1) and (−1,−1,1) directions.
establishing a similar loading mode with mid-femur.In the data analysis of stress-strain curve, engineering strain ε was calculated by dividing the displacement of the crosshead by the length of the sample.Elastic modulus was determined by the linear fitting of the stress-strain curve in elastic deformation stage, while yield stress was further determined by the intersection of the stressstrain curve and a line parallel to the linear quasi-elastic curve at a strain offset of 0.2%.The average value of the stresses between 20% and 40% compressive strains was considered to be the plateau strength.

Speckle pattern preparation and DIC analysis
For the purpose of investigating the multi-scale characterisation of TPMS sample deformation, two DIC speckle preparation methods were conducted: (1) To capture the overall deformation of FGS, the surface facing camera lens was carefully polished using 2000 grid SiC paper and covered by ultralight white clay, a glass slide was thereafter pressed on the clay to obtain a smooth plane.Ultralight clay's low elastic modulus and high flexibility make it an appropriate material to fill the pores of FGS without influencing its yielding and failure mode.The black speckle pattern was prepared by spraying black paint using a high-pressure spray gun.Thereupon, a qualified artificial speckle pattern available for DIC analysis has been prepared.The thickness of the clay was kept under 1 mm to improve the observing sensitivity.(2) In order to observe the deformation of subunit-scale structure, a high-resolution speckle pattern was prepared by laser ablation.Firstly, a DIC speckle pattern was designed and exported in TIFF format, then the TIFF images were converted to 8-bit images and imported into the laser manufacturing system.A Ti: sapphire laser regenerative amplifier system with a central wavelength of 800 nm and pulse duration of 35 fs was used to fabricate the high precision speckle pattern.The laser was precisely focused on the polished surface of samples and the speckle size was kept around 10 μm to achieve the desired characterisation accuracy.The yielding and failure evolution was characterised by an inhouse developed 2D digital image correlation (DIC) system.The testing process was recorded by using two high-resolution CCD (charge-coupled device) cameras, the sampling frequency was set to 5 Hz (Figure 4).

Numerical simulation
Finite element simulation software ABAQUS 6.14 was adopted in numerical simulation.Material properties such as elastic modulus (115 GPa), Poisson's ratio (0.33) and yielding strength (1162 MPa) of SLM-Ti6Al4V were adopted.The plastic deformation information was extracted from the compressive stress-strain curve.To increase computational efficiency and descend computational resources, FGS models were composed of 5 × 5 × 6 unit cells and 6 unit cells were arranged in gradient direction.The boundary conditions were defined as follows: The upper face of TPMS model was moving downwards at a speed of 0.0375 mm/s, until reaching the maximum compressive strain of 0.1, so as to calculate the micro-deformation mode under small compressive strain.The bottom face was set encastre in the whole simulating process.General contact was adopted and a friction coefficient of 0.1 was determined in simulation.

Morphology, structure and microstructures
The reconstructed models confirmed the integrity and continuity of TPMS-FGSs.As shown in Figure 5(a), the 2D gray-scale images exhibit typical skeletal and sheet features, Skeletal TPMS generally possess larger pore size than that of sheet TPMS.While the continuous bilayer structure of sheet TPMS divides the hollow domain of FGS into two disconnected regions, which endows FGS ample surface area and promotes cell attachment and mass transport.Negative Gaussian curvature, which is often found on trabecular bone, has been detected both in skeletal and sheet region of FGSs (Figure 5(b)).Based on the results of the analysis of manufacture error, the average offset values of FGS-D, FGS-G, FGS-I and FGS-P samples are 67, 33, 28 and 39 μm, respectively.Considering the mean diameter of ti6al4v power particle is 35 μm in this study, and the fact that partial melted particles attached on the surface of FGS could not be fully removed by ultrasonic cleaning, as-built FGSs demonstrate high precision, complexity and consistency.However, the inherent structural anisotropy and morphological features of different types of TPMS still produce distinguishable difference in SLM processability, as shown in Figure 5(c): FGS-G sample shows superior dimensional stability in various directions (1,0,0), (1,1,0), (1,1,1), (−1, −1,0), which can be attributed to the larger pore size and feasible thickness of sheet TPMS (Figures 5(a Figures 5(a) and 6(b)) and shape topology of different types of TPMS.As for P-FGS, the manufacturing error mainly presents in the skeletal region, which is considered to be caused by the fragile connection parts between adjacent unit cells.
The average structural thickness of FGSs generally increases from 248.8 to 457.4 μm to 215.6 to 306.0 μm (Figure 6(a)).Given similar relative density, Sample G-Sh structure shows relatively larger sheet thickness (457.4 ± 10.9 μm) than other three samples without reducing pore size, indicating a better SLM processability for Gyroid structure.Figure 6(c) reveals the changing trend of relative densities of FGSs along the structural gradient direction.The proposed design strategy showed potential in achieving graded relative density (25-55%) while restricting the structure thickness within an appropriate range suitable for SLM manufacturing.It is worth noting that D-FGS and G-FGS possess much more gentle rangeability of porosity both in skeletal and sheet TPMS regions compared to those of I-FGS and P-FGS in (1, 0, 0) direction.The moderate structural transition of lattice scaffold is considered to be more advantageous to avoid abrupt failure and extensive fracture.The apparent density gradually increases by around 30% in X direction and then declines at the transition boundary (X = 7.5).For FGS-P, the sheet TPMS significantly reduces the fluctuation range of apparent density while FGS-I shows a reverse effect.In this case, TPMS type plays a decisive role in determining the structural homogeneity and sheet TPMS does not always outperform skeletal based-TPMS.These findings imply the possibility of constructing heterogeneous TPMS structure with a stable distribution of structural parameters by properly combining specific types of TPMS together.Gyroid shows superior structural performance in processability and homogeneity compare with other designs in this study.

Compressive stress-strain curves of as-built samples
The apparent mechanical behaviours of TPMS samples under quasi-static loading are indicated in Figure 7.It can be inferred from the results of stress-strain curves that the elastic modulus, yielding strength and ultimate strength of sheet TPMS are generally higher than those of skeletal samples.This multiple enhancement of mechanical performance is considered to be caused by the increasing relative density (Gibson and Ashby 1989) and excellent load-bearing ability and mechanical stability of sheet TPMS.The stress-strain curve of the porous scaffold can be generally divided into four stages: elastic deformation, yielding stage, plateau stage and densifying stage.With regards to the skeletal TPMS samples, the fluctuation range of stress in the plateau stage is more significant.Especially for Sample D-Sk, which periodically failed until all struts collapsed and closely huddled together, causing a wavy-like pattern.P-Sk is an extreme case in the present study, an early failure occurred in the initial compression stage and thus the whole structure quickly collapsed and lost load-bearing capability, leading to an evidently lower yielding and compressive strength (Table 1).Sample G-Sk and I-Sk all possess a longer and more stable plateau stage compared with the other two TPMS designs in this study, which further indicate the mild and gradual deformation and failure process.The sheet-like TPMS samples generally possess a compressive strength of around 450 MPa and load reduction after the first massive structure failure (marked by the green arrow in Figure 7).The plateau stage is also significantly shorter than that of skeletal TPMS samples and the densifying stage started at a compressive strain of around 0.3.The small unit cell size (1.5 mm) and high apparent density (38.7-42.5%)contribute to the fast densification of compressing tests.The TPMS-FGS respectively inherits the long and steady plateau stage and load reduction from skeletal based-TPMS and sheet TPMS.
Key apparent mechanical properties such as elastic modulus (E s ), yielding strength (s s ), ultimate compressive strength (s max ), plateau strength (s p ) and compressive fracture strain (1 f ) were extracted and shown in Table 1.The volume fraction of the composing structures and their mechanical properties essentially regulate the apparent mechanical property of TPMS-FGSs.Especially for the case that the structural gradient is perpendicular to the loading direction, the analytical model (Zhao et al.  and strength.

2018) produces more accurate results of elastic modulus
where E a and s a are the apparent elastic modulus and strength of the FGS, respectively; f i , E i and s i refer to the volume fraction, elastic modulus and strength of the composing lattice structure.f x indicates the volume fraction of the porous structure at a distance x and E x and s x are the corresponding elastic modulus and strength.The present study adopted an S-shape structural gradient, the transition boundary of which locates at mid-plane of TPMS samples, providing an approximately symmetrical structural change from skeletal based-TPMS to sheet TPMS.The arithmetic average s s of skeletal samples and sheet samples compare well with those of FGS samples: the prediction error of Sample D-FGS, G-FGS, I-FGS and P-FGS are 5.8%, 4.6%, 5.1% and 13.5%, respectively.However, the calculation error of E s using the same method increased to 7.8% (P-FGS) ∼20.1% (D-FGS), which is possibly caused by the processing defects of samples, deformation of crosshead and accuracy limit of displacement sensor.Sample FGS-G showed superior overall mechanical performance with relatively higher E s , s s , s max and s p .This holistic advantage is considered to be closely related to the structural characteristics of Gyroid.As illustrated in section 3.1, the Gyroid possesses excellent uniformity, integrity and processing accuracy compared with other designs, which in turn leads to more stable response behaviour and higher strength.
Skeleton system constantly experiences cyclic loading in daily activities.The energy absorption capability of BTES provides valuable guidance to the FGS design.The energy absorption E abs and energy absorption efficiency w abs are shown in Figure 7(b, e, h, k) and Figure 7(c, f, i, l), respectively.The energy absorption capacity per volume E abs of TPMS samples is determined by integrating the area under compressive stress-strain curves, as interpreted in Equation ( 10).
The energy absorption efficiency, which is considered to be determined by the deformation and fracture mechanism of the lattice structure, is given by Equation ( 11) Compared with skeletal based-TPMS and sheet TPMS structure, the w abs of which varies from 41.48 ± 0.2% (Sample D-Sk) to 72.7 ± 3.0% (Sample G-Sk) and from 63.9 ± 2.5% (Sample D-Sh) to 72.7 ± 3.5% (Sample G-Sh), the TPMS-FGSs exhibit considerable improvement in energy absorption efficiency (74.7-82.7%).The superior energy absorption capability of TPMS-FGS is considered to be caused by the heterogeneous TPMS, which simultaneously possesses diverse elastic/plastic deformation mechanisms and arrests the propagation failure.The deformation transition caused by graded structures will be further elucidated by conducting a DIC analysis and numerical study in Section 3.3.

Effect of heat treatment on the apparent mechanical performance
The acicular martensite α ′ of as-built (AS) TPMS samples gradually transfer to lamellar α+β after sub-transus heat treatment (Zhang et al. 2018).The decomposition of supersaturation solid solution causes grain coarsening and element segregation, further affects the overall mechanical performance of heat treatment (HT) samples.As shown in Figure 8(a), HT samples generally possess lower s s and s max and gentler stress plateau stage.HT samples show superior plasticity and loadbearing capacity, the unloading process is greatly suppressed and the stress-strain curves slowly rise up and exhibit a more significant densifying effect.The plateau strength of G-FGS-HT and I-FGS-HT samples is higher than that of as-built materials.Moreover, the stable stress-strain responding behaviour slightly contributes to the energy absorption efficiency of FGSs, as can be confirmed by Figure 8(a) that w abs increases by 3.8-12.6%.The fracture morphology has also been characterised to illustrate the effect of heat treatment on the failure mechanism (Figure 8(b)).Different from the river-like brittle fracture surface of as-built materials, α+β lamellar structure was observed after sub-transus heat treatment.Vanadium, which is the stabilising elements of β phase, diffused and concentrated at the boundary of α lamellae (Figure 8(b)).The lamellar microstructure was considered to possess superior plasticity and ductility but lower strength that as-built materials (Lütjering 1998;Yan et al. 2018).

Overall deformation and failure modes
The sample-scale deformation response effectively reflects the decisive influence of structure type on the mechanical performance of FGSs.The super light clay covered on the observation surface of samples deforms with the compressed sample.The strain contour of clay intuitively reflects the deformation pattern of lattice structures, a positive 1 1 indicates that the clay film is affected by tensile stress and the lattice structures are moving away from each other, while a negative 1 1 mean the opposite.In this stage, along with the compression test, partial structure yields and strain concentration becomes more significant (Figures 9-12).The local strain concentration of Sample P-Sk is significant and the maximum strain (>4%) is also higher than that of other samples.As the crosshead continuously compressing samples, the stress concentration and local failure mainly distributed on the maximum shear stress plane, as can be confirmed by the 45°shear failure mode (Sample D-Sk in Figure 9 and I-Sk in Figure 11) and V-shape failure mode (Sample G-Sk in Figure 10).Similar failure mode was also confirmed by other studies (Mazur et al. 2017;Kumar et al. 2020).These directional aggregation of early fracture, tearing and buckling initiate the large-scale collapse of the whole structure.For sheet TPMS based samples, the most stretched part is more inclined to form a V-shape pattern (Sample D-Sh, G-Sh and P-Sh).Meanwhile, the maximum strain decreases and distributed more uniformly than that of skeletal TPMS.
Compared to skeletal TPMS and sheet TPMS, FGSs exhibit an intermediate deformation mode: . Sample D-FGS (Figure 9): Compared with G-Sk and G-Sh, 45°shearing band of skeletal TPMS (left side) was retained, whereas evenly distributed strain was observed for sheet TPMS (right side).Finally, a 45°s hear failure across the entire sample emerged at 1 1 = 0.10. .Sample G-FGS (Figure 10): The yielding area of Gyroid TPMS samples tends to accumulate in the central part.
Differ from the near V-shape failure of Sample G-Sk and G-Sh, the upper part of G-FGS folded layerwisely and cause a mild structure collapse when 1 1 reached 0.14. .Sample I-FGS (Figure 11): All high strain regions are distributed in 45°inclined directions.The graded structure of Sample FGS hindered the yielding area from merging in the same direction and formed a zigzag failure mode.Tension-dominant and compression-dominant deformation were, respectively, observed on both sides of transition boundary. .Sample P-FGS (Figure 12): The structural combination of P-FGS prevented the skeletal TPMS from early buckling and crushing.The high strain area of sheet TPMS (right side) gradually moved upward and formed a triangle region, and then integrated with the layer-wise crashing of skeletal structure, finally an S-shape failure mode occurred at 1 1 = 0.14.differing from FGS-I, the strain of skeletal-Primitive (left) is generally lower than that of sheet-Primitive (right).

Local deformation and failure behaviours at small compressive strain
For the purpose of systematically investigate the effects of graded TPMS on the local mechanical performance of FGS, finite element simulations and compression tests with high precision DIC speckle pattern were further conducted.Especially, we studied the local deformation and failure pattern at small compressive strain (<3%).In  this stage, the samples apparently showed a quasi-linear elastic stress/strain relationship, but local yielding was already initiated.The stress first concentrated at the fragile part between unit cells, as shown in Figure 13.
Due to the diversified topology features of four types of TPMSs, the yielding region of TPMS-FGSs gradually evolved into a different yielding pattern.The yielding region of D-FGS showed the characterisation of ).Sheet Gyroid-TPMS showed a weaker stress concentration effect compared to skeletal TPMS, as supported by the top row in Figure 14.With regards to P-FGS, stress is concentrated in the fragile connection between skeletal unit cells and evenly distributed at the surface in sheet TPMS (Figure 14), leading to a significantly different stress level at two sides of the transition boundary (Figure 12).These results demonstrated that the sheet TPMS promoted the stability of FGS and the apparent and local strain/stress level can be effectively controlled by changing the topology features of TPMS.This regulation mechanism is potentially favourable in multifunctional FGS, which responds to mechanical stimuli (Shuai et al. 2020).
A great agreement between the experimental and numerical local strain distribution results of the structural transition region were achieved (Figure 15) at a small compressive strain (≈0.01).In DIC analysis,  Limited by the observation method using CCD, only an in-plane strain state was reflected.A periodically in-plane strain alteration were revealed numerically and experimentally, as shown in Figure 15(a,b).The maximum principal strain measured by DIC is relatively smaller than FE results, because DIC method used in this study is a 2D characterisation strategy and misses the ability to capture out-of-plane strain.It can be inferred from the 1 xx and 1 yy distribution of TPMS-FGSs that the thinner midsection part between the nodal area is generally subjected to more severe vertical compression and transverse stretching (Figure 15(a,b)).Moreover, no abrupt strain fluctuation or significant concentration was found at the transition boundary.Combined with the strain curve data extracted from the near-transition boundary region (Path AB and A ′ B ′ , marked by a white dash line in Figure 16), the mechanical performance of heterogeneous TPMS was further quantitatively illustrated: .Sheet Gyroid slightly reduced the average value of 1 xx and 1 yy without changing the variation period (around 0.75 mm), a similar result was also provided by FE simulations.However, compared with the overall 1 yy increase on path AB, a greater strain drop was observed on path A'B', indicating a more significant strain concentration at the fragile part of sheet Gyroid. .For Sample I-FGS, the peak values of 1 xx and 1 yy appeared at around 45°, 135°, 225°and 315°.Due to the manufacturing limitation of SLM, thinner sheet TPMS possesses worse structural instability and more significant strain concentration

Discussion
The multi-scale study on the mechanical behaviours of FGS has attracted great attention.Numerous experimental tests have been performed to macroscopically characterise the overall deformation and failure pattern of the samples, some studies pertinently analyzed the local fracture and buckling of the struts under large compressive strain (Maskery et al. 2017;Alberdi et al. 2020;Zhang et al. 2021).However, limited by the observation accuracy and complicated cellular structure, the high precision sub-TPMS unit scale strain distribution of FGS in the elastic deformation stage (small compressive strain) still remains to be tackled.This study presents a multi-scale research method based on uniaxial compression test and FE simulation, combining with global and local DIC analysis.Numerical and experimental results were mutually verified to illustrate the deformation and strain evolution at various scales.
Although manufacturing defect is considered to significantly deteriorate the mechanical response of AM parts (Gu et al. 2021), we found that these local premature failures such as crack nucleation and propagation did not significantly alter the linear elastic property under small compressive strain (<3%).Although local yielding and fracture were observed at compression strain smaller than 2% in all tested FGSs, the compressive stress-strain curve still showed a linear elastic feature.The structural integrity in this stage was still largely maintained and the reduced load-bearing capacity caused by local yielding was temporarily compensated by other intact parts.Moreover, sheet TPMS changed the spatial distribution feature of stress concentration and reduced the risk of premature buckling.These local modulations were finally reflected in the improvement of energy absorption ability and more stable plateau stress in apparent stress-strain curves.Combining with heat treatment, energy absorption efficiency can be further enhanced.
Based on the multi-scale investigation of TPMS-FGSs in the present work, two influencing mechanisms of graded structure strategy on mechanical response can be drawn: . Temporal asynchrony of mechanical property evolution caused by porosity variation.The variation of porosity did not significantly change the strain/ stress variation period or the spatial location of high stress area, the strength improvement of FGS is mainly credited to the superior load-bearing capacity of low porosity part with thicker struts, which helps maintain the structural integrity when weaker parts subject to premature failure.Porosity changing strategy has little effect on the elastic deformation feature but change the post-yielding behaviour mostly through the temporal asynchrony of local yielding.Therefore, the excessive extension of the anisotropic shear band can be regulated by changing the spatial distribution of porosity, leading to a layer-wise or zigzag failure pattern (Figure 17, Rows 1 and 2). .Spatial asynchrony of mechanical parameter distribution caused by topology variation.By constructing heterogeneous lattice structure with different topology designs, especially for those with different deformation modes (bending dominated and stretching dominated), FGS show spatially varying mechanical behaviours when subjects to loading and thus creating a much more flexible deformation and customised failure pattern.This methodology provides radical alternation for young's modulus (Feng et al. 2021), deformation mode (Pham et al. 2019), stressstrain distribution (Zhang et al. 2020) and energy absorption (Wang et al. 2021).In this study, FGS-P possesses columnar (skeletal TPMS) and reticulate (sheet TPMS) strain concentration areas, the involvement of sheet structure improved the plasticity and ductility of TPMS-FGS and prohibited the generation of large-scale shear failure band, as proved by the stress-strain curve and DIC characterisation results (Figure 17, Row 3).Similar structural regulatory effect was also studied by other researchers, Pham proposed the meta-grain lattice structure by coupling different unit cells together (Pham et al. 2019), the embedded 'second phase' prevented the high stress mergence and caused a hardening effect (Figure 17, Row 4).
It should be noted that the angle between loading direction and structural gradient significantly affects the mechanical response of FGS (du Plessis et al. 2022).When the loading direction is parallel with the porosity gradient, the yielding strength is mainly determined by the volume ratio of the high porosity part (Figure 18(a)).Whereas the strength of transversely graded and radially graded FGS (loading direction is vertical to the structural gradient) shows a positive correlation with the apparent density of FGS.(Figure 18(b)).Considering that various unit cells and design strategies were adopted, this finding illustrates that the performance of FGS also strongly depends on the mechanical environment, which brings great challenges for the FGS design in complex or unknown loading conditions.However, this mechanical instability can be partially alleviated by spatially adjusting the topology structure of unit cells, such as the involvement of sheet TPMS, to improve the yield strength and bearing capacity stability of skeletal TPMS.
FGS with diverse design strategies showed enormous potential in energy absorption and tissue engineering.The flexible combination of structures with different porosity or unit cells offers infinite possibilities for mechanical property adjustment.The results of the present study revealed that by rationally tuning the structural characteristics (skeletal and sheet TPMS), energy absorption efficiency, strain concentration, mechanical stability and multi-scale failure pattern could be manipulated.Nevertheless, the intricate spatiotemporal deformation and failure behaviours of FGS need to be further systematically explored to facilitate the intended properties.

Conclusions
A comprehensive experimental and numerical study has been conducted on TPMS-FGSs and multi-scale mechanics of SLM heterogeneous lattice structures were revealed.A high-resolution DIC analysis was performed and the strain distribution of sub-TPMS unit structures was extracted and discussed.The periodic distributed strain of FGS was successfully captured in the elastic deformation stage at a small compression strain (1 < 0.03).The influences of graded TPMS on the distribution and evolution of deformation and failure mechanics were discussed.The following conclusions can be drawn: . All TPMS-FGSs in the present study showed good structural continuity and integrity.Gyroid TPMS outperformed other TPMSs in machinability and dimensional accuracy when a smaller unit cell size was adopted. .Heterogeneous TPMS composed of skeletal TPMS and sheet TPMS exhibits superior energy absorption efficiency and mechanical stability.The graded TPMS prohibits the yielding area propagation across the structural transition boundary.FGS tends to yield in a layer-wise or zigzag manner instead of 45°s hear failure.TPMS-FGS optimised by subtransus heat treatment reduced the yield strength and plateau stress fluctuation, whereas the deformation mode remained similar to that of as-built samples. .The regulation effect of heterogeneous TPMS on local mechanical behaviour was experimentally verified by high-precision DIC analysis based on laser-processing speckle pattern, the experimental characterisation accuracy has been significantly improved.A periodic in-plane strain distribution and comprehensive reformation of strain concentration were experimentally confirmed.Moreover, premature failure, such as crack and buckling, were also identified in early elastic deformation stage of TPMS-FGS. .Compared with adjusting the porosity of AM lattice structures, coupling skeletal and sheet TPMS together showed great potential in altering the mechanical behaviour of FGS both apparently and locally.TPMS-FGS exhibits superiority in mechanical stability, energy absorption, strain concentration reduction, deformation and failure behaviour regulatory.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This  Gang Fang is a professor of Department of Mechanical Engineering, Tsinghua University, Beijing, China.Prof. Fang's research interests include mechanics of materials and metallic materials processing technology.He has authored over 100 peer-reviewed papers.

Figure 1 .
Figure 1.Biomimetic strategy and graded structures of TPMS-FGS: (a) biomimetic strategy, design CAD model, as-built sample and Gaussian curvature distribution based on Gyroid TPMS (from top to bottom).(b) Graded structures composed of skeletal and sheet TPMS based on diamond, Gyroid, I-WP and primitive TPMSs.

Figure 3 .
Figure 3. (a) Ti6Al4V powder used in the present study; (b) laser scanning strategy in SLM fabrication; (c) as-built samples on the substrate.
) and 6 (b)).While the distribution of larger manufacturing deviations (>2 D) are found in sheet region of FGS-D and FGS-I.This mismatch on TPMS thickness arises from the inherent structural limitations such as smaller pore size (

Figure 4 .
Figure 4. DIC speckle pattern preparation by using (a) spray painting and (b) laser processing.

Figure 6 .
Figure 6.Comparative study of design values (CAD) and reconstructed models of as-built samples (micro-CT): structural thickness (a) and pore size (b) of skeletal and sheet samples; (c) relative density variation along structural gradient and (d) total surface area of TPMS-FGSs.

Figure 7 .
Figure 7. Stress-strain curves(a, d, g, j), energy absorption E abs (b, e, h, k) and energy absorption efficiency w abs (c, f, i, l) at compression fracture strain 1 f of as-built Diamond, Gyroid, I-WP and Primitive TPMSs (from top to bottom).The green arrows in (a, d, g, j) indicate the loading dropping after stress reaches the ultimate compressive strength s max .

Figure 8 .
Figure 8.(a) Stress-strain curves and energy absorption efficiency (w abs ) at 20% compressive strain of as-built (AB) and heat treated (HT) TPMS-FGSs; (b) fracture surface of AB sample (up), HT sample (middle) and EDS results of HT fracture surface (bottom).

Figure 13 .
Figure 13.The Mises-stress distribution of Sample D-FGS, G-FGS, I-FGS and P-FGS (from left to right) at compressive engineering strain of 0.2%, 0.5% and 1.0%.

Figure 15 .
Figure 15.DIC and FE calculated strain distribution ((a) 1 xx , (b) 1 yy and (c) 1 1 ) of TPMS-FGSs at compression strain 1 = 0.01.The central axis of struts of I-WP does not locate on the observation plane so that only the strain of nodal region was observed.

Figure 16 .
Figure 16.Strain variations of TPMS-FGSs along path AB and A ′ B ′ : (a) data extraction paths adopted in DIC analysis.A circular path was adopted for I-FGS, scale bar is 1 mm.(b) 1 xx , and 1 yy variation with increasing compressive strain along path AB and A ′ B ′ .Y indicates the distance from point A and u represents the clockwise rotational angle during data extraction.Strain concentration and crack initiation are marked by pink arrows.

Figure 17 .
Figure 17.Multiple FGSs with different deformation and failure modes, the lattice materials with different structure parameters are marked in cyan and purple.

Figure 18 .
Figure 18.Relationship between apparent elastic modulus/yield strength and relative density.ρ indicates the relative density of FGS, ρ* is the lowest density of FGS.(a) structural gradient is parallel with loading direction (Choy et al. 2017; Han et al. 2018; Liu et al. 2018; Zhang et al. 2018a; Zhou, Jin, and Du 2020); (b) structural gradient is perpendicular to the loading direction (Zhang et al. 2019; Zhang et al. 2020; Kas and Yilmaz 2021; Xiong et al. 2021).

Xiangyu
Zhang received Ph.D. in Mechanical Engineering from Tsinghua University in 2019, and now is the postdoctoral researcher at the Beijing Institute of Technology.He has been engaged in the study of structure design and mechanical performance of additive manufactured lattice materials.Lan Jiang received Ph.D. in Mechanical Engineering from Beijing Institute of Technology in 2000, and now is the Changjiang Distinguished Professor at the Beijing Institute of Technology.He received National Outstanding Young-Scientist Award, National Natural Science Foundation of China.He was elected the National Leading Researcher for S&T Innovations, the Leader of Innovation Group, Ministry of Education, China.Prof. Jiang served as the Panel Chair, National Key R&D Program of Additive Manufacturing and Laser-Based Manufacturing, China.He is a Fellow of ASME, OSA and ISNM.His research interests are the mechanisms, methodologies, measurements and applications of ultrafast laser micro/nano manufacturing.Xingchen Yan is an associate professor in Guangdong Academy of Sciences and a host of China Association for Science and Technology (CAST) Young Talent Support Program.He has been engaged in the study of additive manufactured high-performance metal materials with more than 80 SCI research papers including Additive Manufacturing, Acta Materialia, etc. Zhipeng Wang received his PHD from Beijing Institute of Technology.His areas of interest include ultrafast laser processing, spatial light shaping and micro/nano optical devices fabrication.Xiaowei Li is a professor in the Beijing Institute of Technology.He has been engaged in the research of mechanisms and methodologies of laser micro/nano manufacturing, laser manufacturing technology and its applications.He has published over 90 SCI research papers in journals including Advanced Materials, Light:Science&Applications, ACS Photonics, Optics Letters etc.

Table 1 .
Apparent mechanical properties of TPMS samples.
work was supported by National Natural Science Foundation of China [grant number 52105293], the China Postdoc-