Stress shielding and micromotions are the most sign ificant problems occurring at the bone- implants interface due to mismatch of their mechani cal properties. Mechanical metamaterials with their exceptional behavior and c haracteristics, can provide an opportunity to solve the mismatch of mechanical pro perties between the bone and implant. In this study, four types of human femoral hip implants (one solid implant and three lattice-structured meta-implants) have been c onsidered for stress and micromotion analysis. For this aim, at the first stage, a well- known auxetic 3D re-entrant structure was studied analytically, and precise closed-form a nalytical relationships for elastic modulus and Poisson’s ratio of the unit cell were o btained. The analytical solution was validated with the numerical solution based on beam elements. At the second stage, three lattice meta-implants made of 3D re-entrant unit ce lls with positive, negative, and graded distribution of Poisson’s ratio were analyzed and t he stress and strain distributions in implant and the micromotion at the bone-implant int erface were studied. The results of analytical solution for mechanical properties of th e 3D re-entrant structure presented great improvements in comparison with previous anal ytical studies on this structure. Moreover, the implementation of the re-entrant stru cture in the hip implant provided very smooth results for stress and strain distribut ions in the lattice meta-implants and could solve the stress shielding problem which occu rred in the solid implant. The lattice meta-implant based on the graded unit cell distribu tion presented smoother stress-strain distribution in comparison with the other lattice m eta-implants. Moreover, the graded lattice meta-implant gave minimum areas of local st ress and local strain concentration at the contact region of the implants with the interna l bone surfaces. Among all the cases, the graded meta-implant also gave micromotion level s which are the closest to values reported to be desirable for bone growth (40 µm). Keywords: 3D auxetic, Hip implant, Mechanical metamaterial, Porous biomaterials. 3 1. Introduction There are two established methods for femoral impla nt fixation namely cementation [1] and press-fit [2]. The cementless implants have the advantage of providing the possibility of bone ingrowth inside porous implants or porous s urfaces of solid implants. Completely porous implants usually have lower stiffness levels as compared to solid implants thus allowing for better transfer of load from implant t o bone and hence decrease in stress shielding phenomenon which can lead to bone resorpt ion. Initial and long-term stability of femoral implant which is defined as minimizing t he relative motion of the implant with respect to the bone in its vicinity (known as micro motion, micro-movement, or relative motion [3]) is of great importance for long term su ccess of the implant [4]. Due to their more vulnerability to micromotions, the frequency o f aseptic loosening is still higher in cementless femoral stems as compared to cemented ty pes [5]. Some recent researches [6] have shown that an initi al press-fit condition in the range of 0.01 mm and 0.1 mm gives the optimal results for st ability, durability, and bone growth. Depending on their extents, micromotions can have t wo opposite effects on bone ingrowth inside the implant. Micromotion less than a critica l value (30 µm [7], 40 µm [8], 50 µm [9], 90 µm [10], 100 µm [11]) enhances osseointegra tion while micromotions larger than 150 µm [8] results in formation of fibrous tissue i n the external surface of implant. Fibrous tissues are the result of bone tissue repair mechan ism in response to repetitive loading of the implant [12]. Moreover, excessive micromotions can activate osteoclast cells which leads to bone resorption and thus implant failure [ 13, 14]. The first research work on the effect of micromotion on the bone ingrowth in porou s implants were carried out by Cameron et al [15, 16] in which they showed that bo ne growth is disturbed by micromotions. The amount of micromotion is signific antly affected by the direction of the 4 applied load. Cristofolini et al. [17] showed that micromotions usually cause a gap opening in the medial region and shear slippage in the post erior region. One of the promising methods to reduce micromotion can be to manipulate Poisson’s ratio of an implant in different locations and directions . Depending on extent of load an implant receives and the extent of required in-situ relativ e lateral movement, an implant can be designed with desirable distribution of lateral mov ement. This can be accomplished not by normal materials, but by materials which show ex otic properties, commonly known as metamaterials. Metamaterials exhibit properties tha t cannot be found in nature [18]. Recent advances in additive manufacturing has made it possible to have control over the micro- and macro-structural design features of poro us structures [19-25], and manufacture several types of mechanical metamaterials such as P entamodes [23, 26-28], auxetics [29, 30], and materials with negative compressibility [3 1]. One of the most commonly known mechanical metamaterials are auxetics that show neg ative Poisson’s ratio behaviors [32, 33]. It means that unlike normal material which exp and laterally under compressive loading, they shrink laterally. Designing a structure with a wide variety of positi ve and negative Poisson’s ratio at different locations can lead to implants which have lateral displacements in compliance with the surrounding bone. The most famous and appl icable bio-metamaterial structures for this purpose are 3D re-entrant structures which could provide positive or negative values for their Poisson’s ratio depending on their internal struts angles [34]. Such structures due to their wide range of geometrical p arameters (such as strut thickness, two types of internal angles, etc.) can also be suitabl e candidates for designing the porous implants with appropriate mechanical properties. Th e behaviors and mechanical properties for the noted structure have been analyz ed analytically and numerically in the 5 literature in several works [35-37]. Availability o f exact analytical relationships for this structure can help to design and optimize the well- matched structure of porous implants. In recent years many researchers have focused on th e effects of geometry, porosity, material, micro-architecture, loading, coating, etc . on the micromotion extents in implants especially total hip replacements (THRs). Chen et a l. [38] evaluated two types of femoral stem (anatomic and straight) and investigated their micromotions relative to the femur using finite element (FE) analysis. In another simi lar study, Bieger et al. [39] studied the primary stability and stress-shielding effect of so me stem prostheses namely Fitmore, CLS, and Mayo. Camine et al. [40] developed a µ-CT-based technique to measure gap and micromotion during compressive loading at all exter ior locations of femoral stem. Wang et al. [41] examined a fully three-dimensional opti mally graded lattice hip implant to minimize the bone resorption and bone–implant micro motion. In a recent study, Kolken et al. [42] presented conceptual hybrid meta-biomat erials with a combination of negative and positive Poisson’s ratio structures. Their resu lts demonstrated improvement in implant–bone contact micromotion under axial loadin g condition. Kolken et al [42] implemented well-known re-entrant and honeycomb str uctures for generating implant hybrid structure in order to minimize the chance of wear and to improve implant fixation at both sides of the hybrid implant faces. In this study, the concept presented by Kolken et a l [42] has been improved by applying a gradient distribution of 3D re-entrant structure in the hip implant. For this aim, first, analytical relationships for mechanical properties of 3D re-entrant unit cell are derived and then four types of implants (solid implant; met a-implant with positive Poisson’s ratio; meta-implant with negative Poisson’s ratio; and gra ded meta-implant) are designed and studied using FE analysis. As the porous implants a re constructed by lattices of metamaterials (auxetics), they are referred to as m eta-implant in the rest of the paper. 6 The implants are then designed based on the anatomy of the femur bone to have better initial adjustment and fixation for hip implants. A geometrical model of the femur bone is created precisely with considering both the canc ellous and cortical bones to obtain accurate results. Finally, the stress and strain di stributions of the implants as well as the micromotions at bone-implant interfaces (considerin g both gap and sliding distances) are analysed and compared between the four designs. 2. Materials and methods In the first part of this section, analytical relat ionships for main mechanical properties of 3D re-entrant unit cell namely elastic modulus and Poisson’s ratio are derived. The newly developed analytical relationships are more precise as compared to existing relationships available in the literature [34, 37]. In the second part of this section, the detailed procedure of numerical modelling and analysis of solid and po rous femoral hip meta-implants is described. The details of implementing the derived analytical relationships into creating meta-implants with positive, negative, and graded P oisson’s ratios are also presented. 2.1 Analytical analysis 2.1.1 Stiffness matrix derivation In this subsection, analytical relationships for el astic modulus and Poisson’s ratio of the re-entrant unit cell are derived as functions of el astic properties of the constituent material ( and ) and the geometrical parameters of the unit cell. A 3D re-entrant unit cell which is considered for obtaining analytical soluti ons is shown in Figure 1. The parts of the neighbor unit cells which share the same struts with the considered unit cells are distinguishable with light red color in this figure . The approach of deriving the analytical relationships for the 3D re-entrant structure is th e linear superposition method which has been introduced and implemented for solving other s tructures such as rhombic 7 dodecahedron [43], truncated cube [44], octahedral [45], rhombicuboctahedron [46], etc. To implement the superposition method for obtaining analytical solutions for the 3D re- entrant structure, it is first needed to define the system of equations for the structure in the matrix form of =  . In this equation,  is the force vector containing the external forces acting on the structure, is the stiffness matrix of system, and the   is the displacement vector of the different joints of the unit cell. Since the analytical relationships (say mechanical properties) for the s tructure are different in the global x and y directions, we will consider two different ty pes of loading condition to obtain the mechanical properties in these two distinct directi ons, see Figure 2. Figure 1: 3D re-entrant unit cell with highlighted shared struts 8 Figure 2: Loading conditions and geometrical parame ters of the re-entrant unit cell According to the four vertical (xy, yz, and two bis ectors of xy and yz) and one horizontal (xz) symmetry planes of the re-entrant structure an d also two considered loading conditions in the x and y directions, the behavior of re-entrant unit cell can be modelled completely by 12 distinct degrees of freedom (DOF), see Figure 3. The stiffness matrix will then be a 12×12 symmetrical matrix. To calcul ate the mechanical properties in two directions, it is necessary to solve the system of equations for two distinct loading conditions such that =  or =  . It is worth noting that since we tried to consider a very general condition for the 3D re-ent rant structure, each unit cell consists    9 of four strut types with lengths ( ,  , ℎ, ℎ) and three types of cross-sectional area (, , ) after excluding the shared parts of neighbors str uts, see Figure 3. The unit cell is fully defined with two unique angles  and  and lengths  and . Angle  is defined as the angle between strut  and the horizontal plane, and angle  is defined as the angle between strut  and the horizontal plane (Figure 3). By considerin g the total occupied volume of the re-entrant unit cell ( = ) and strut thickness ! for square cross-section as well as the above-mentioned geomet rical assumptions, the relative density of 3D re-entrant unit cell could be obtained readil y as: "=  #$  =4ℎ+4ℎ+ℎ+16 +8  =3!ℎ+!ℎ+8!+8!  (1) where ,  and  are: =!4⁄ , =!2⁄ , =! (2) 10 Figure 3: Considered degrees of freedom for analyti cal solution of the structure To obtain the elements of -th column of the stiffness matrix , we displace DOF - while keeping the other DOFs fixed. By calculating the of forces that should be applied to each DOF to create the above-mentioned displacement, the elements of the -th column of the stiffness matrix could be obtained. Two well-known beam theories namely Euler-Bernoulli and Timoshenko theories will be implemented for ana lytical analysis of the 3D re-entrant unit cell. The parameters ./ and 0/ have been used for summarizing the relationships. The term ./ denotes the axial tension/compression term which i s identical for both Euler- Bernoulli and Timoshenko theories and it is defined as ./=1234 2. The term 0/ denotes the lateral deformation term which is different in two theories. It is defined as 0/=3 452 26 and   11 0/= 726 89: 4;2<72 =>2?4 for Euler-Bernoulli and Timoshenko beam theories, respectively (see [47] for more details). By considering these terms and i mplementing the superposition method, the final stiffness matrix for the 3D re-entrant st ructure could be obtained as =    where  ,  and  matrices are given by:  = ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡8. 0 −8. 0 0 40 0 −40 −8. 0 4(2.+ 2 sin  .+ sin  .I+ 2 cos  0L+ cos  0I) 4C cos  sin  (−.I+ 0I) 0 −40 4 cos  sin  (−.I+ 0I) 4(cos  .I+ 0+ 20+ sin  0I) 0 0 −8(sin  .+ cos  0L) 0 0 0 0 −80 0 0 8 cos  sin  (−.+ 0L) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −4(sin .I+cos 0I) 2sin 2(.I−0I) ⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤ (3)  = ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡0 0 0 0 0 0 0 0 −8(sin  .+ cos  0L) 0 8 cos  sin  (−.+ 0L) 0 0 −80 0 0 16(.+ sin  .+ cos  0L) 4 sin 2 (.− 0L) 4 sin 2 (.− 0L) 0 4 sin 2 (.− 0L) 8(cos  .+ 0+ sin  0L) 0 0 4 sin 2 (.− 0L) 0 8(cos  .+ 0+ sin  0L) 0 0 0 0 8. 0 0 0 0 −8(sin  .+ cos  0L) 8 cos  sin  (−.+ 0L) 0 −8. 0 0 −80 0 0 0 0 0⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤ (4)  = ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡0 0 0 0 0 0 0 0 0 0 0 −4(sin  .I+ cos  0I) 0 0 0 2 sin 2 (.I− 0I) 0 −8(sin  .+ cos  0L) 0 0 0 8 cos  sin  (−.+ 0L) 0 0 0 0 −80 0 0 −8. 0 0 40 0 −40 0 0 4(2.+ 2 sin  .+ sin  .I+ 2 cos  0L+ cos  0I) 4 cos  sin  (−.I+ 0I) −4(sin  .I+ cos  0I) −40 4 cos  sin  (−.I+ 0I) 4(cos  .I+ 0+ 20+ sin  0I) 2 sin 2 (.I− 0I) 0 −4(sin  .I+ cos  0I) 2 sin 2 (.I− 0I) 4(.L+ 2 sin  .I+ 2 cos  0I)⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤ (5) 2.1.2 Elastic properties relationships All the unknown DOFs / (- = 1 !R 12 ) can be obtained as functions of the external forc e ST (which could be considered as  or  based on which direction we want to calculate the mechanical properties). This can be done by inv erting the final stiffness matrix = 12    and multiplying it by the force vector. The elasti c modulus of unit cell can be calculated according to the basic definition of elastic modulus as follows: ST =ST ST ST UST (6) where ST , ST , ST , UST , and ST are the applied load, length, cross-sectional area , deformation in the load direction, and elastic modu lus of the re-entrant unit cell, respectively. By considering ST =  , UST = 2 , ST =  and ST = , the elastic modulus of re-entrant unit cell in the y direction can be obtained as: =. 2  (7) In addition, Poisson’s ratio of the unit cell for t his loading condition can be found as:  =  W (8) By calculating  and  from the solution of system of equations and inser ting them in equations (7) and (8), the normalized elastic modul us and Poisson’s ratio relationships of the re-entrant unit cell for Euler-Bernoulli theory are obtained as: X,3Y =32 !Lcos  Z16 !cos +(+2!−2!cos 2+16 !sin )[ (9) \ ,3Y =(−+2!+2!cos 2)tan  Z16 !cos +(+2!−2!cos 2+16 !sin )[ (10) where EB denotes the Euler-Bernoulli beam theory. B y considering ST =  , UST = 2  , ST =  and ST =  , the elastic modulus of the re-entrant unit cell i n the x direction can be calculated as: = 2  (11) In addition, Poisson’s ratio of the unit cell for t his loading condition can be found as:  = 1 11 W (12) 13 Similar to the y direction, by substituting the res ults of calculated  and  into equations (11) and (12), the normalized elastic mod ulus and Poisson’s ratio relationships of the re-entrant unit cell for Euler-Bernoulli the ory could be found as: X ,3Y =−64 !Lcos (+2!−2!cos 2)  (−L−14 !−2!L+(L−!L)cos 2+2!(−+!)cos 4+!Lcos 6) (13) \  ,3Y =(L−2!L−4!cos 2+2!Lcos 4)sin 2 (−L−14 !−2!L+(L−!L)cos 2+2!(−+!)cos 4+!Lcos 6) (14) By considering 0/= 726 89: 4;2<72 =>2?4 in the stiffness matrix (Equations (3), (4) and (5 )), all the relationships could be transformed into relationshi ps based on Timoshenko beam theory. After this transformation, the elastic modulus and Poisson’s ratio relationships in the y direction for Timoshenko beam theory can be obtaine d as follows: X,_ =32`!Lcos  (16`!cos  + (`+ 4!+ 2`!+ 4! − 2!(−2 + ` − 2) cos 2 + 16`!sin )) (15) \ ,Y =(`(− 2!) + 4!(1 + ) − 2!(` − 2(1 + )) cos 2) tan  (16`!cos  + (`+ 4!+ 2`!+ 4! − 2!(−2 + a − 2) cos 2 + 16`!sin )) (16) The shear coefficient factor of Timoshenko beam the ory for square cross-section used in the relationships is ` =b (<c ) <c. Similarly, the relevant equations for Timoshenko be am theory in the x direction can be obtained as: X ,_ =(64 `!Lcos (`(+2!)+4!(1+)−2!(`−2(1+))cos 2))  (17) where: =(`L+ 4`!+ 14`!+ 8!L+ 56`!L+ 2`!L+ 4`! + 16!L + 56`!L + 8!L+ (`(−L+ !L) + 60`!L(1 + ) + 4!L(1 +  )) cos 2 + 2!(a − 2(1 + ))(`(− !) + 2!(1 + )) cos 4 − (4!L− 4`!L+ `!L+ 8!L − 4`!L + 4!L) cos 6)  and 14 \  ,_=−(`+4!(1+))cos 2+2!L(−2+`−2)cos 4)sin 2)  (18) where: =(`L+ 8`!+ 24!L− 8`!L− 2`!L+ 8`! + 48!L − 8`!L + 24!L− 4!(` − 2(1 + )  and =(`L+ 4`!+ 14`!+ 8!L+ 56`!L+ 2`!L+ 4`! + 16!L + 56`!L + 8!L+ (`(−L+ !L) + 60`!L(1 + ) + 4!L(1 +  )) cos 2 + 2!(` − 2(1 + ))(`(− !) + 2!(1 + )) cos 4 − (4!L− 4`!L+ `!L+ 8!L − 4`!L + 4!L) cos 6) . It is worth noting that all the final relationships have been presented for the specific condition of  =  otherwise, the relationships became very lengthy. 2.2 Numerical analysis In the second stage of this study, the focus is on the biomedical application of the analyzed 3D re-entrant structure to improve the micromotion problems in implants, particularly in femoral hip joint implants. In this subsection, the detailed steps of designing and FE analysis of femoral implant constructed from 3D re- entrant lattice structures are described. Three types of porous implants with posi tive, negative and graded Poisson’s ratios are constructed for this aim. 2.2.1 Methodology of the implant design In this study, the upper half of human femur bone w ith a total length of 25.83 cm has been considered for numerical modeling and analysis (Figure 4a). For comparison purposes, first, a solid implant is modelled. It is worth noting that to have more precise numerical analysis, both the cortical and trabecula r parts of femur bone have been considered in the FE modeling. To create a compatib le geometry for femoral implant, the anatomy of the femur bone has been considered as th e basic geometry of the implant. For this aim, the implant external edges and surface (F igure 4b) have been designed in 15 compliance with the internal faces of the cortical part of the femoral bone model (Figure 4a). The total length of the modelled implant was 1 6.72 cm. Implementing this concept minimizes the potential gap occurrence at the femur bone and implant interface, when the bone/implant assembly undergoes external loadin g. The final assembled model consisting of cortical bone, trabecular bone, and s olid implant is shown in Figure 4c. The material properties used for solid implant was = 113.8 and = 0.342 which is based on Ti-6Al-4V as well-known implant material alloy. (a) (b) (c) Figure 4: (a) Femur 3D CAD model (b) Designed Impla nt 3D model, (c) Designed implant assembled into cortical and trabecular bone 2.2.2 Meta-implant design For the case of meta-implants, we need to connect t he mechanical properties of the human femur bone and our analytical solution of the re-en trant structure to provide a lattice structure for the hip implant (stiffness of which a ccommodates to that of human bone). 16 The main mechanical parameters required to create a meta-implant that is compliant with its surrounding femur bone of study are elasti c modulus and Poisson’s ratio. The implant elastic modulus controls the stress flow an d its distribution in the implant, and Poisson’s ratio can affect the micromotion at the i mplant/bone interface. A lot of works in the literature have studied the mechanical prope rties of the human femur bone [48-52]. In this study, the values of 16.7 GPa and 0.155 GPa have been considered for the elastic modulus of respectively the cortical and trabecular bones based on a similar study [53]. In addition, the values of respectively 0.3 and 0.2 5 have been considered for the Poisson’s ratio of cortical and trabecular bones [54]. Accord ing to our analytical solution, the re- entrant structure provides much larger elastic modu lus ranges in the x direction as compared to the y direction (see Results section). Therefore, to gain the elastic modulus of 16.7 GPa for the meta-implant, the analytical re lationships for elastic modulus of the unit cell in the x direction (Equation (17)) has be en used. By considering the elastic modulus value of = 113.8 GPa for Ti-6Al-4V as a well-known implant material alloy, which has been used for normalizing the analytical relationships, the suitable normalized elastic modulus of the re-entrant structure is obta ined as 0.1467. Since the re-entrant structure provides almost same values of elastic mo dulus for positive or negative values of  and  (see Results section), two different hip meta-impl ants with positive and negative Poisson’s ratios are constructed to have d eeper understanding of the effect of Poisson’s ratio on the micromotion at the bone-impl ant interface. 2.2.3 Graded meta-implant Based on the previous studies on the femur implant and the results of this study for solid and femoral meta-implant (with either completely po sitive or negative unit cells) models, it is clear that the stress distribution in the hip implant varies from negative values to positive values due to the tensile and compressive load distribution in the implant (see 17 Results section). The main problem in the meta-impl ants with either completely positive or negative unit cells is that the parts of implant with positive Poisson’s ratio which have tensile stress distribution tend to separate from t he bone surface at the contact surfaces. A behavior also occurs when the parts of implant wi th negative Poisson’s ratio are subjected to compressive stress. This problem can b e resolved by implementing negative Poisson’s ratio unit cell of re-entrant structure i n the areas of implant which have tensile stress distribution on one hand and using negative Poisson’s ratio unit cell of re-entrant structure in the areas of implant which have compre ssive stress distribution on the other hand. To apply this approach to our meta-implant, a gradient structure which varies from positive Poisson’s ratio at one side of the implant to negative Poisson’s ratio at the other side of the implant is designed and constructed. Th e variation of Poisson’s ratio in the re- entrant structure was achieved by changing the valu es of angle  only, to provide smoother distribution of mechanical properties of f inal implant structure. 26 rows of re- entrant unit cell with dimensions of 5×5×5 mm 3 have been used for generating the graded meta-implant. 2.2.4 FE analysis The FE analyses of the implants under static compre ssive loading condition were performed using the ANSYS Workbench software. To be tter simulate the real human femur loading condition, the femur bones were angle d at 12˚ flexion and 12˚ adduction, and the lower part of the femur bone (middle shaft) was fixed in all the directions. The femoral head was loaded by applying 2000 N compress ive load to the head of femur implant. The contact condition between the cortical and trabecular bones was considered as bonded constraint. On the other hand, to track t he micromotion between the implant external surfaces and bone internal surfaces, the c ontact condition between the bone and the implant was considered as frictional contact su rfaces. The friction coefficient of 0.3 18 was applied for the frictional contact conditions [ 55]. 3D solid elements were used for discretization of the model in the ANSYS Workbench software, and fine mesh size with smooth transition was used for providing better mes h quality specially at the edge of the implants. The FE model of the meta-implant and the assembled model with femur bone are presented in Figure 5. The total number of elem ents for solid implant assembly was 463,634. Since the lattice structures in the meta-i mplants consist of a large number of thin struts ( ! = 1.04 mm), the mesh size for meta-implants was reduced t o about 1/4 of that in solid implant. As a result, the total numbe r of elements for meta-implants with positive and negative Poisson’s ratio were 1,393,74 5 and 1,563,915, respectively. For the case of graded meta-implant, due to its geometrical complexity and computational costs, the mesh size needed to be increased, and the final total elements number reached 866,600 elements. (a) (b) (c) 19 Figure 5: (a) CAD model and (b) meshed FE model of meta-implant based on 3D re-entrant structure. (c) Final assembly of the imp lant and femur bone model with its corresponding mesh 3. Results 3.1. Effect of relative density on the unit cell mech anical properties Variations in normalized elastic modulus /  for the special case of  =  = 22.5° is demonstrated in Figure 6. The results show that in all ranges of relative density, elastic modulus is almost quadruple in the x direction as c ompared to that in the y direction (Figure 6a). For instance, at relative density of 0 .6, relative elastic modulus based on Timoshenko beam theory in the x direction is 0.117, while it is 0.03 in the y direction. The elastic modulus curve based on Timoshenko beam theory is always lower than the curve obtained based on Euler-Bernoulli beam theory (Figure 6a). Their difference, however, does not exceed 14.76%. In the y direction , the curve obtained by Yang et al. [37] theory overlaps with the Timoshenko results ob tained here. However, in the x direction, the curve obtained by Yang et al. [37] o verlaps neither with the Timoshenko analytical results nor with the Euler-Bernoulli ana lytical results obtained in this paper (Figure 6a). Nevertheless, in high relative densiti es (i.e. " > 0.3 ), the results of Yang et al. theory in the y direction are close to the curv es obtained in this paper. It is important to note that the FE results always overlap with the analytical results based on Timoshenko beam theory (Figure 6a). As for the Poisson’s ratio, the results show that e ven though \ is always negative, relative density has very little effect on its valu e (Figure 6b). In fact, \ only changes in the range of -0.2143 and -0.4132. However, changing relative density from 0.6 to almost zero changes \  from -0.8393 to -2.414. Timoshenko beam theory usu ally predicts greater negative Poisson’s ratio (in absolute sense) as com pared to Euler-Bernoulli beam theory. Regardless of direction, the curves obtained by Yan g et al. [37] don’t have any overlap 20 with neither the analytical Timoshenko results nor with the analytical Euler-Bernoulli results. More prominent difference between our resu lts and those of Yang et al. [37] is for \  . While Yang et al. [37] theory gives constant valu e of -2.414 for \  in all the relative densities, the analytical results in current work s how that \  changes from -0.8393 to - 2.414 (Figure 6b). Similar to elastic modulus resul ts, the FE results always overlap with the analytical results based on Timoshenko beam the ory. (a) 21 (b) Figure 6: Mechanical properties: (a) normalized ela stic modulus, (b) Poisson’s ratio 3.2. Effect of internal angle on the unit cell mechan ical properties The internal angles  and  are other important factors in geometrical charact eristics of the unit cell. Due to the fact that the re-entrant structure is homogenized by considering =   and ℎ= ℎ , in conjunction with the similarity of the relatio nships for these conditions, the angles  and  are set equal to each other here as well as throug hout the study. The ratio of struts thickness to the height (or length) of the unit cell for which the mechanical properties graphs are plotted (Figure 7) is iW= 0.16 . As for the elastic modulus, the internal angle has an almost negligibl e effect in the y direction. However, in the x direction, the relative elastic modulus in  = 0° is almost 8-fold of that in  = 42.5° (0.1024 as compared to 0.0126). It must be noted th at the elastic modulus curves are semi-symmetrical with respect to  = 0° (Figure 7a). As for the Poisson’s ratio, the internal angle has significant effect on both \  and \ (Figure 7b). \  is completely symmetrical and \ is almost symmetrical with respect to the point (, \ )= (0, 0) . The little asymmetrically observed for \ is caused by the differences in the lengths ℎ and ℎ for positive and negative extents of  and  which has effects on mechanical properties. While from  = −42.5° to  = 42.5° , \ changes steadily (almost linearly) from 0.8227 to -0.5701, \  has more extreme variations in the angle range (Figure 7b). \  changes from 1.162 to -1.162 in a smaller angle ra nge of  = −21.25° to  = 21.25° . 22 (a) (b) Figure 7: Effect of angle on mechanical properties for j = k. lmm , n = o = pmm : (a) normalized elastic modulus, (b) Poisson’s ratio 3.3. Stress and strain distributions The stress and strain distributions in femur bone f illed by solid implant, meta-implant constructed by positive Poisson’s ratio unit cells, meta-implant constructed by negative 23 Poisson’s ratio unit cells, and meta-implant with g raded distribution of unit cells are shown in Figure 8. In the case of solid implant, th ere is a high mismatch of qr, s , and t r between the solid implant and the surrounding bone . In case of the meta-implant constructed by positive unit cells, the mismatch of the noted stress ( qr, s , and t r ) between the implant and the bone is mitigated. Unli ke the solid implant, the local stress distribution in meta-implant does not change unifor mly. In fact, different regions of each unit cell have different stresses. Some parts of ea ch unit cell experience stress levels similar to that of the bone, while some other parts of it e xperience different stress levels. In the case of structure constructed by negative un it cells, there are two improvements as compared to the structure constructed by positive u nit cells. First, the stress and strain distributions in each unit cell becomes much more u niform locally. In other words, the stress or strain level throughout each unit cell re mains almost constant. Secondly, and more importantly, the stress and strain levels in t he case of structure constructed by negative unit cells is very close to that of the bo ne surrounding it at each location. As for the case of graded meta-implant, even though the stress and strain distributions (both locally inside each unit cell and as compared to bone) in the graded meta-implant are much better than the structure constructed by p ositive unit cells, it performs worse than the structure constructed by negative unit cel ls. In addition to stress and strain distributions, the level of stress dominant in the implant is also important. Very high stress levels can lead to early implant failure. As for the case of solid implant, the maximum stress in the implant is |qv |= 68 MPa (Figure 8a) which is relatively low, as it was expected. In case of m eta-implant constructed by positive unit cells, the maximum stress level is |qv |= 2,127 MPa. In case of the meta-implant constructed by negative unit cells, the maximum str ess level is |qv |= 13,817 MPa. In 24 case of the graded meta-implant, the maximum stress level is |qv |= 794 MPa. This shows that the graded meta-implant experiences much lower stress levels as compared to two other meta-implants, and that the stress the gr aded meta-implant experiences is in the range bearable by titanium alloys. 25 Solid implant Meta -implant constructed by positive Poisson’s ratio unit cell Meta -implant constructed by negative Poisson’s ratio unit cell Graded meta -implant (a) q (b) qr 26 (c) y r (d) s 27 (e) sr (f) t r Figure 8: Stress and strain distribution: (a) stress in the z direction, (b) strain in the z direction, (c) stress in the { direction, (d) strain in the { direction, (e) stress in the z{ direction, and (f) strain in the z{ direction 28 3.4. Micromotion Micromotion, another important factor in durability of an implant, is compared between the four types of implant in Figure 9. To get a more qu antitative impression of comparison of micromotion between the implant and the bone, the a verage sliding and gap distances between the implant and the bone at each height of implant is plotted in Figure 10. In general, except at the very top part of the meta-im plant ( | > −0.02 ), the maximum sliding in none of the meta-implants exceed 4.5 µm. Therefo re, the main micromotion comparison between the implants should be made based on the gap between the implant and the bone. Except one point, the solid implant demonstrates ne ar-zero gap in both back and front (the parts in compression and tension) of the implant (F igure 10a,b). Among the meta-implants, the meta-implant constructed by negative unit cells has the lowest level of gap in the front side. The average gap between the meta-implant cons tructed by positive unit cells and the bone is 25.02 µm in the front side and 26.27 µm in the back side. The average gap between the meta-implant constructed by negative unit cells and the bone is 11.36 µm in the front side and 35.43 µm in the back side. Finally, the av erage gap between the graded meta- implant and the bone is 19.78 µm in the front side and 47.65 µm in the back side. Therefore, based on the above-mentioned results, the meta-impl ants constructed by positive Poisson’s ratio and the meta-implants constructed by negative Poisson’s ratio provide the minimum and maximum average values of gap at the front side of the implants, respectively. On the other hand, the meta-implant based on the positive Poisson’s ratio and graded structures present the minimum and maximum average values of g ap at the back side of the implants, respectively. Based on the results of gap and sliding distributio ns presented in Figure 9 and Figure 10, the extent of sliding distance ( ≤ 5 "~ ) at both sides of implants are negligible as compa red to the gap distance extent ( ≤ 75 "~ ) in most parts of the implant. Therefore, the 29 micromotion in implant, which is defined as the vec tor resultant of sliding and gap distances, could be well approximated by only the gap distance of the implant at the contact surface. The distribution of gap distance in the solid impla nt is mostly uniform and varies between 0 and 15 "~ at different heights of the implant at both sides (Figure 10a,b). As shown in Figure 10b, a huge gap of ≥ 200"~ occurs between the solid implant and the bone due to discontinuous contact surface at the back side of t he solid implant at the height of  ≅ −0.02 m. In comparison with the meta-implant constructed by positive Poisson’s ratio unit cells, the distribution of gap distance in the graded meta -implant and the meta-implant constructed by negative unit cells is more uniform (and with sinusoidal shape). At the back side of implant, the gap value of the graded and ne gative Poisson’s ratio meta-implant varies between 0 and 75"~ sinusoidally with the height of the implant. In th e front side, this oscillating behavior could be only observed for the meta-implant with negative Poisson’s ratio which varies between 0 and 25 "~ . 30 Solid implant Meta-implant constructed by positive Poisson’s ratio unit cells Meta-implant constructed by negative Poisson’s ratio unit cells Graded meta-implant (a) Front-side gap (b) Back-side gap 31 (c) Front -side gap (d) Back-side gap (e) Front-side sliding 32 (f) Back -side sliding Figure 9: Micromotions in implant faces: (a,c) fron t-side gap, (b,d) back-side gap, (e) front-side sli ding, and (f) back-side sliding 33 (a) (b) (c) (d) Figure 10: Micromotions in implant faces: (a) front -side gap, (b) back-side gap, (c) front-side slidin g, and (d) back-side sliding 34 4. Discussions 4.1. Analytical model Considering the behavior of the re-entrant unit cel l as sum of the behavior of individual DOFs for analytical solution, is one of the most ac curate approaches which is used for obtaining the analytical relationships for strut-ba sed lattice structures and metamaterials. This technique, by considering all possible deforma tions of the struts at the joints in the lattice structure, provides comprehensive and accur ate analytical solutions for the mechanical properties of lattice structures. The re sults of this study also confirmed that the obtained relationships for normalized elastic modul us and Poisson’s ratio of re-entrant structure based on this technique present satisfact ory outcomes in comparison to other approaches presented in the literature [56]. Moreov er, the general analytical solution based on this approach gives this opportunity to implemen t and compare the results of both Euler- Bernoulli and Timoshenko beam theories simply by ch anging the corresponding parameters (./ and 0/). Based on the aforementioned facts, we observed s ignificant improvement in the final analytical solution in comparison with previo us works [34, 37], especially for normalized elastic modulus in the x direction and   . According to the final relationships,   is dependent on the relative density and angles  and . However, the   relationship presented by Yang et al. [37] is independent from r elative density or strut thickness which is far from accurate especially for high values of relative density. 4.2. Implant macro-geometry Hip implant geometry is one of the most important p arameters which can affect the stress- strain and micromotion distribution at the bone-imp lant interface. There are lots of studies in the literature which have focused on the design and optimization of the hip implant geometry to improve the fixation and to reduce the micromotion at implant surface [39, 57]. Using the internal geometry of human femur bone for designing the shape of the implant 35 could be one of suitable methods to have better con tact between implant and bone surfaces. The superiority of implants with bone-compliant geo metry has been shown in previous studies such as [58, 59]. As mentioned in Results s ection (Section 3.4) and the results demonstrated in Figure 10, it is clear that the gap between the bone and implant at their interface for solid implant is significantly low wh ich can be attributed to good match of geometry of implant’s external surface and femur bo ne’s internal surface. Because of its high stiffness as well as extremely low micromotion exte nt both of which create undesirable conditions for bone growth, solid implant is not a good choice for hip implants. 4.3. Stress and strain distributions By considering Ti-6Al-4V alloy for implants materia l, a large difference occurs between the value of elastic modulus of femur bone and solid im plant. This fact causes a nonhomogeneous stress and strain distributions in the implant, esp ecially at the implant bone contact regions. For implantation with cement-less condition (press- fit), sliding and gap distances occurring at the contact surface of the bone and solid implan t during loading lead to local stress and strains at implanted area (Figure 8) which can caus e damage to bone internal surface. Implementing lattice structures gives the opportuni ty of reducing the difference of elastic modulus between the bone and implant. It is worth n oting that the porous structure at the contact surface of the implants and bone also contr ibutes to improvement in bone ingrowth. That is why in early scientific studies as well as in many of today’s industrial implants, solid implants with porous surface are used [60-62]. As s hown in Figure 8b-c, lattice structures with adjusted elastic modulus to the bone elastic m odulus improve the smoothness of stress- strain distribution in the implant especially at th e interface of the bone and implant. Although the sliding between the bone and implant d ue to press-fit condition could still create some local stress and strain mismatch, a sig nificant enhancement in uniformity of stress and strain was observed in the meta-implant in comparison with the solid implant. 36 The main reason of this fact could be the relaxatio n of meta-implant surfaces which occurs due to the local deformations of lattice structures struts at their contact regions with bone. These deformations of strut beams relieve the local strain and stress which is the one of major problems observed in the solid implant. 4.4. Micromotion Although decreasing micromotions at implant-bone in terface can have positive influences on the bone ingrowth, eliminating the micromotion from bone and implant interface altogether can have negative effects on bone ingrowth and fixa tion durability. Burke et al. [63] have demonstrated that the micromotions of 40 µm allow t he formation of woven bone within a porous titanium wire surface. Therefore, complete e limination of gap and sliding distances (the main parameters of micromotion) at the implant surface could have negative effects on bone ingrowth at bone-implant interface. This also means that solid implant, due to its very small values of micromotion, can raise some problem s for bone ingrowth in comparison with meta-implants which provide sufficient values of mi cromotion at contact surfaces. On the other hand, the uniform oscillation of gap value be tween 0 and 75 "~ at the back side of the graded meta-implant and the meta-implant constr ucted by negative unit cells could satisfy the required micromotion extent for bone in growth at the bone-implant contact surface. The variation of gap at the front side of the implant constructed by negative Poisson’s ratio unit cell is in the range of 0 to 30 "~ , while this range for the graded meta- implant is 0 to 85 "~ which could meet the suitable average micromotion requirement for successful bone ingrowth. In addition, the large in ternal pores as well as struts deformations of all the meta-implants could mitigate the micromo tion at critical points of the implant such as at  ≅ −0.02 m in which the solid implant demonstrated huge mic romotion values due to the sudden change in the geometry of the imp lant surface. Since most implants available in market for hip implant replacement are manufactured based on predesigned 37 geometries (not anatomical geometries), this fact c ould be considered as a strong advantageous point for meta-implants which are capa ble of mitigating micromotion at regions in which the implant external surface and t he cortical bone internal surface are not well compatible. 4.5. What micro-structure to choose? Thanks to the anatomical geometry of implants and t he adjusted elastic modulus of the meta-implants, all of the meta-implants based on th e introduced unit cell gave acceptable stress and strain levels in the implantation area. Nevertheless, the meta-implant based on graded unit cell distribution presented smoother st ress-strain distribution in comparison with the other meta-implants. Moreover, the graded meta- implant gave minimum areas of local stress and local strain concentration at the contac t region of the implants with the bone internal surfaces. Among all the cases, the graded meta-implant also gave micromotion levels which were the closest to value reported to be desi rable for bone growth (40 µm) [63]. The maximum level of stress in the graded meta-implant was also the lowest among all the meta- implants and within the range that is acceptable fo r most titanium alloys. 4.6. Further improvements One of the most important parameters which affects the accuracy of analytical results is the effect of considering the effective lengths of stru ts in a unit cell. In higher values of relative densities, the struts get thick, and the overlappin g of the struts at the joints should be taken into account in analytical solution. By defining th e effective length of the struts according to their orientation and thickness, the analytical solution could predict more accurate results for actual condition of the lattice structure [64]. Another noteworthy parameter that can enhance the results is calculating the exact relati onship for the relative density of the unit cell. Exact relationship of relative density by con sidering the multiple counting effect of struts gives this opportunity to calculate the exac t value of the strut thickness for a specific 38 mechanical properties of lattice structure [65]. As the final note, the re-entrant unit cell studied in the work, as it shares struts with its n eighboring unit cells, is not an ideal structure for creating a graded structure such as the graded meta-implant designed in this study. Thus, the change of angle in the graded structure c ould causes problems in connecting the shared struts of adjacent unit cells. Hence using a structure such as idealized re-entrant structure [64] could solve the problems and enhance the results for graded meta-implant. 5. Conclusions A hip-joint meta-implant with an appropriate lattic e structure can improve the stress and strain distributions in implant and surrounding bon e tissues. Moreover, designing an accurate and precise graded lattice structure for i mplant can enhance implant deformation as well as provide suitable micromotion for bone in growth at bone-implant interface. This study focused on designing a graded lattice hip met a-implant by implementing a well-known 3D re-entrant auxetic metamaterial to enhance the m icromotions at the bone-implant contact surfaces. To reach this aim, analytical rel ationships for mechanical properties of the 3D re-entrant structure were obtained using a more accurate analytical approach as compared to methods previously used in the literatu re. The results demonstrated a huge improvement in the analytical solution which overla pped with FE results. One solid implant and three meta-implants were designed based on the anatomy of human femur bone. The new exact relationships for mechanical properties o f 3D re-entrant unit cell were implemented to create three meta-implants with posi tive, negative, and graded Poisson’s ratios. The numerical results of the implants under compressive loading condition showed that the meta-implants could provide enhanced stres s and strain distributions in the implant and its contact regions with bone internal surface. Although the micromotion values for all the meta-implants were in appropriate ranges for bo ne ingrowth, the meta-implants with 39 negative and graded Poisson’s ratios gave more comp atible and uniform distribution of micromotion at major regions of bone-implant interf ace.