Gibson-Ashby Model Research Articles

Effect of volume fraction and unit cell size on manufacturability and compressive behaviors of Ni-Ti triply periodic minimal surface lattices

Ni-Ti shape memory alloys with triply periodic minimal surface (TPMS) lattice structures have aroused increasing interests due to their great application potential in bone tissue engineering. In this study, the effect of volume fraction and unit cell size on the manufacturability, compressive mechanical properties, and shape memory behaviors after compression of Ni-Ti Gyroid TPMS lattice structures, which were fabricated by laser powder bed fusion ( L -PBF), were systematically investigated. A mathematical model was established to analyze the geometric influence factors on manufacturing fidelity of Ni-Ti Gyroid TPMS lattice structures fabricated by L -PBF. It was revealed that the manufacturing fidelity decreased as the volume fraction and unit cell size decreased due to the aggravation of powder adhesion caused by the increase of specific surface area and overhanging rate of struts. Three power-law equations based on the Gibson-Ashby models were established to describe the dependences of compressive modulus, nominal yield strength, and ultimate strength of Ni-Ti Gyroid TPMS lattice structures fabricated by L -PBF on the volume fraction and unit cell size. All the above mechanical properties showed positive dependences on the volume fraction but negative dependences on the unit cell size. Both the volume fraction and unit cell size had little influence on the martensitic transformation temperatures as well as the total shape recovery ratio of Ni-Ti Gyroid TPMS lattice structures prepared by L -PBF. All the as-built Ni-Ti lattices exhibited high total shape recovery ratios of above 96.5% after 8% compressive deformation at room temperature (RT). But noteworthily, as the volume fraction increased, the pseudoelastic response (i.e., the recovered strain during unloading) rapidly decreased, followingly the recovered strain during heating increased. This was ascribed to the enhanced stability of stress-induced martensite at RT caused by the increased actual strain of local struts at the same overall compressive deformation in the Ni-Ti lattices with higher volume fractions. In contrast, the Ni-Ti lattices with different unit cell sizes exhibited similar strain distribution and consequently similar deformation recovery behaviors. • Manufacturing fidelity decreases as volume fraction and unit cell size decrease. • Mechanical properties show positive dependences on volume fraction. • Mechanical properties exhibit negative dependences on unit cell size. • Both volume fraction and unit cell size hardly affect the total recovered strain. • Superelastic response decreases as volume fraction increases.

Bone Remodeling Around Solid and Porous Interbody Cages in the Lumbar Spine.

Spinal fusion is an effective surgical treatment for intervertebral disk degeneration. However, the consequences of implantation with interbody cages on load transfer and bone remodeling in the vertebral bodies have scarcely been investigated. Using detailed three-dimensional models of an intact and implanted lumbar spine and the strain energy density based bone remodeling algorithm, this study aimed to investigate the evolutionary changes in distribution of bone density (ρ) around porous and solid interbody cages. Follower load technique and submodeling approach were employed to simulate applied loading conditions on the lumbar spine models. The study determined the relationship between mechanical properties and parametrical characteristics of porous body-centered-cubic (BCC) models, which corroborated well with Gibson-Ashby and exponential regression models. Variations in porosity affected the peri-prosthetic stress distributions and bone remodeling around the cages. In comparison to the solid cage, stresses and strains in the cancellous bone decreased with an increase in cage porosity; whereas the range of motion increased. For the solid cage, increase in bone density of 20-28% was predicted in the L4 inferior and L5 superior regions; whereas the model with 78% porosity exhibited a small 3-5% change in bone density. An overall increase of 9-14% bone density was predicted in the L4 and L5 vertebrae after remodeling for solid interbody cages, which may influence disk degeneration in the adjacent segment. In comparison to the solid cage, an interbody cage with 65-78% porosity could be a viable and promising alternative, provided sufficient mechanical strength is offered.

A comparative investigation on the mechanical properties and cytotoxicity of Cubic, Octet, and TPMS gyroid structures fabricated by selective laser melting of stainless steel 316L

Metallic lattice structures can be fabricated by selective laser melting (SLM) with purposefully designed pores and controlled pore sizes that can bio mimic the natural bone, providing adequate mechanical and biological support for the patients. Strut-based structures, like Cubic, Octet; and sheet-based structures, like triply periodic minimal surface (TPMS) gyroid, have been studied extensively in the past. However, it lacks enough comparative study on the mechanical properties and cytotoxicity among these structures. Therefore, Cubic, Octet, and TPMS gyroid of Stainless steel 316 L (SS316L) are designed, manufactured, and characterized at 40/50/60% relative densities in this study. Moreover, the flowability, density characteristics, and cytotoxicity of SS316L powder are validated to ascertain its suitability for 3D printing and implant application. Based on refining the Gibson-Ashby model, it is possible to predict or design the mechanical properties via adjusting the relative densities. The results indicate these structures demonstrated appropriate Young's modulus and outstanding biocompatibility.

Numerical Investigation of Ti6Al4V Gradient Lattice Structures with Tailored Mechanical Response

Open porous lattice structures have found a wide range of applications as lightweight load‐bearing and energy absorbent structures in various fields. A common requirement for any application is the preliminary assessment of the mechanical response of the proposed architecture. Herein, uniform and continuous functionally graded lattice structures (FGLSs) made of titanium alloy, Ti–6Al–4V, are designed using cubic and pillar octahedron unit cells at overall porosities of 60%, 75%, and 85%. The lattice morphology is modulated using axial, dense‐in, and dense‐out gradient strategies. The mechanical performance of the structures is studied using numerical simulations implementing damage initiation and evolution under quasi‐static compression. The proposed models could properly simulate the mechanical properties and failure behavior of the lattice structures. The designed FGLSs displays a crushing behavior starting from the lower relative density layers toward the higher relative density ones. Cubic and pillar octahedron‐based structures exhibit stretch‐ and bending‐dominated deformation behaviors, respectively. The power‐law analysis verified by the Gibson–Ashby model is used to assess the mechanical response of the FGLSs. The effect of the gradient strategies on the mechanical properties as well as their adaptability for orthopedic implants is discussed in detail.

Compressive mechanical properties and energy absorption characteristics of SLM fabricated Ti6Al4V triply periodic minimal surface cellular structures

This study investigates the compressive mechanical properties and energy absorption characteristics of three types of triply periodic minimal surface (TPMS) Ti6Al4V cellular structures fabricated by selective laser melting (SLM). Based on the SEM observation, the morphology of Ti6Al4V alloy sheet-based TPMS structures was observed, and imperfections of SLM-fabricated cellular structures were investigated. The quasi-static uniaxial compression tests were carried out, and the deformation behavior was recorded by a camera. The result indicated that the compressive mechanical properties had an approximately positive relationship with the relative density of TPMS structures. In addition, the revised Gibson-Ashby prediction models of three sheet-based TPMS structures were established by fitting the compression test results. Furthermore, finite element analysis (FEA) of the compression process was also conducted to facilitate analysis and understanding of the deformation mechanism for TPMS structures. The results also revealed that the energy absorption capacity of TPMS structures increased with the increase in the actual relative density. The mechanical properties, energy absorption, and relative density diagram of sheet-based Ti6Al4V alloy TPMS structures were established to systematically obtain the optimal relative densities of TPMS structures for specific load-bearing and energy absorption applications.

Geometrical parameters and mechanical properties of Ti6Al4V hollow-walled lattices

Hollow-walled lattices are a recent optimization upon the conventional dense-walled lattice design. These novel microstructures can achieve extremely low densities while exhibiting robust technical properties including high ductility, strength, and stiffness. However, the current fabrication process for these structures is ineffective, requiring multiple additive and subtractive manufacturing processes to produce a single specimen. The application of a single step additive manufacturing (AM) method has not yet been evaluated for hollow-walled lattice fabrication. This study aims to assess the viability of such a method, enabling greater geometric control, and providing a single process to reduce both time and cost. The manufacturability of laser powder bed fusion (LPBF) will be explored through Ti6Al4V hollow-walled struts and lattices at varying diameters, lengths, and inclination angles while retaining a constant wall thickness. Quasi static compressional testing will also occur upon the hollow-walled body-centered cubic lattices with a reinforced strut in the Z-axis (BCCZ). These experimental BCCZ lattices have been fabricated at relative densities ranging from 5 to 40%, and strength and modulus values have been observed between 4 and 25 MPa, and 750–3640 MPa, respectively. The resultant data will assist in the generation of the Gibson-Ashby Model to systematically compare LPBF Ti6Al4V hollow-walled lattices to reported dense-walled lattices. This study presents an exploration into the single-step fabrication of metal hollow-walled struts and lattices to broaden the scope of manufacturability and facilitate future advancements.

A Finite Element Investigation on the Design of Mechanically Compatible Functionally Graded Orthopaedic Plate for Diaphyseal Tibia Transverse Fracture

The bidirectional stiffness grading of orthopaedic plate has been investigated for internal fixation of transverse fracture in tibia bone. Highly porous cellular materials made of two types of cubic unit cells were designed. Their effective mechanical properties (elastic moduli and yield strengths) were computationally evaluated using finite element analysis. The Gibson-Ashby model was derived for elastic moduli and yield strengths of the cellular geometries. The modified cellular structure provided the desired minimum stiffness within the selected strut sizes. The developed structures were mapped to seven zones of the orthopaedic plate assigning the bidirectional stiffness grading (longitudinal direction). The effect of functional grading on mechanical stimulation through three stages of healing was studied. The results indicated less stress shielding and improvement in uniformity of distribution of equivalent stress around the newly formed bone. Thus, a functionally designed plate will offer a way to dynamically load the healing tissues by encouraging the patient for a regular walk and exercise. Hence, the generated mechanical signal will significantly help mechanically stimulated bone formation and remodeling in the fractured zone.

Development of a Computational Model of Lattice Structure

This paper deals with the description of the mechanical properties of the plastic lattice structure. These structures are nowadays used in the additive manufacturing. The final goal was to found a mathematical formula, which allowed the FEM simulation of the described lattice structures as the structure were a solid material. The formula stablishes relationship between the original material Young Modulus and weight ratio of the lattice cell. Validation of the formula was performed by the mechanical test. The basis of the study was real testing and measurement, analysis of simulation using the finite element method and the Gibson-Ashby model with some enhancements to get to results within 95% of reliability.

Mechanical and morphological characterization of BCC - derived unit cells for biomedical devices

3D periodic cellular solids attract great attentions for applications in the biomedical field due to the combination of their mechanical and morphological properties.The mechanical behaviour of two classes of periodic cellular structures has been investigated: the first one is a strut-based structure; the second is a triply periodic minimal surfaces (TPMS) structure. Such structures derived from the body centered cubic (BCC) unit cell and were the G7 strut-based and the IWP sheet-based TPMS structure.The study was aimed at the mechanical and morphological characterization of the above cited cells for their application in biomedical devices. By means of the additive manufacturing Electron Beam Melting (EBM) technology, the G7 microlattice structures have been produced in titanium alloy. Compression tests have been carried out on three different levels of relative density, selected to evaluate the effect of this parameter on the mechanical properties.The correlation between relative density and mechanical properties of the G7 structure was defined by the Gibson-Ashby model. SEM analyses have been performed before the experimental tests in order to detect the morphological features and the matching between designed and real parameters, and after the tests to evaluate the failure mechanism of the structures. The experimental tests results were used for the validation of a non-linear finite element model of both G7 and IWP structures with the application of the Johnson-Cook plasticity model. Different cell configurations were obtained by varying the relative density and the cell diameter, in order to evaluate the cell size effect on the mechanical properties. G7 unit cell meets the mechanical and morphological requirements for the application in the biomechanical field. However, the analysis of the collapse mode showed that an improvement of the mechanical behaviour is required. The numerical analyses performed on the IWP unit cell allowed to assess that the structure is suitable for the production of biomedical devices.

Finite element analysis on titanium-tantalum lattice structures fabricated using selective laser melting

In this study, Finite Element Analysis (FEA) is used to study the mechanical properties of titanium-tantalum (TiTa) alloy lattice structures fabricated using selective laser melting (SLM) in previous literature. Based on the actual measured dimension of the experiment samples, the geometry is reproduced and analyzed in ANSYS using two different approaches (beam model, full-3D model). The mechanical properties, such as elastic modulus and yield strength, are then predicted using the two ANSYS models and compared to the experiment results. It is found that both beam model and full-3D model can predict the elastic modulus of the fabricated TiTa parts with high accuracy. However, for yield strength, the predictions from beam model and full-3D model are around 50 % and 65 % in average lower than the experimental data respectively. Finally, in order to validate the design parameter, e.g. porosity levels, the mechanical properties of the lattice structures with porosity from 50 % to 90 % are studied using full-3D FEA model and the results are found to follow the Gibson-Ashby model.

Comparison of foam models from regular and irregular arrays of Gibson-Ashby open-cells

The paper studies effective elastic properties of foam or cellular materials modeled by a set of Gibson-Ashby open cells with regular or irregular structures. Currently, there are many papers that present results of studying cellular materials using theoretical, numerical and experimental methods. However, these papers consider either regular lattices, or a single cell, or representative volume models based not on the Gibson-Ashby models. In this paper, in addition to the regular lattice, irregular structures were numerically studied. A mathematical formulation of the homogenization problem based on the energy equivalence of a foam-like material and on a homogeneous comparison medium is described. Formulations of six boundary value problems are presented. The solutions of these problems allow us to determine a complete set of effective stiffness modules for foams with different types of physical and geometric anisotropies. All stages of the numerical study were implemented in the ANSYS finite element package. Two algorithms for forming solid-state and finite-element models of irregular Gibson-Ashby lattices with small and large porosity are described in detail. As an example, numerical calculations are carried out for polycarbonate foams. The values of the effective elastic modules for regular and irregular lattices and for the Gibson-Ashby analytical model are compared. The results of numerical experiments showed that the Gibson-Ashby model describes the behavior of highly porous materials quite well (for porosity more than 75%), but this model gives a less satisfactory prediction in case of lower porosity. It is noted that for a large number of cells, regular and irregular lattices statistically give similar results for effective modules. However, for individual structures of irregular lattices, especially with strongly differing cells in individual directions, the effective moduli can have significantly different values, and the effective homogeneous medium can have pronounced anisotropic properties. These effects are due to geometric anisotropy and stress concentration in long connecting beams and at the joints of beams of various sizes in highly irregular Gibson-Ashby lattices. Examples of such lattices are given. We analyze the scatter of value for relative modules, which characterizes the anisotropy of such structures.

Investigation on strength and failure behavior of ceramic foams prepared from silicoaluminous industrial waste under uniaxial compression

This paper experimentally investigated the compressive strength-porosity and compressive strength-bulk density dependence of ceramic foams fabricated using coal bottom ash (CBA), bauxite residue, and low-grade albite. The compressive stress-strain curves of ceramic foams with different bulk densities under pressure were compared to reveal the difference of their fracture mechanisms. The compressive strength-porosity trend of CBA foamed ceramics with a porosity range of 68-87% complied with the Rice model and could be expressed by an exponential equation σf=53852.28×exp-0.105P. However, for the porosity levels exceeding 90%, a linear equation σf=54.33-0.57P was provided the most accurate data fitting. The compressive strength-bulk density dependence of ceramic foams also satisfied the Gibson-Ashby model expressed by σf=73.54ρ∗3/2-26.64ρ∗. Finally, the variation of fracture mechanisms of compressed ceramic foams with their bulk density was revealed. Under one-dimensional compression stress, low-density ceramic foams (ρ ≤ 0.40 g/cm3) underwent plastic-like deformation, while high-density ones (ρ ≥ 0.60 g/cm3) exhibited brittle fracture behavior.

Mechanical behavior of Al–Si10–Mg gyroid surface with variable topological parameters fabricated via laser powder bed fusion

Metallic cellular materials with triply periodic minimal surface (TPMS) topologies have great potential for lightweight and multi-functional applications. However, the effect of level-set values, which can control the topology of TPMS, on the performance of TPMS have not been fully understood because previous studies have mainly focused on the shell thickness, cell size and periodicity of TPMS. In this paper, Al–Si10–Mg gyroid surfaces with three different level-set values are fabricated by using selective laser melting (SLM). Micro computed tomography (Micro-CT) is used to reconstruct the 3D models of as-built samples, then the geometric deviations and surface quality are investigated. Mechanical behavior and energy absorption characteristics of these gyroid surfaces are investigated both experimentally and numerically, which provide a solid basis for predicting and controlling the deformation behavior in gyroid surfaces. Afterwards, the Gibson–Ashby model is used for understanding the effect of the level-set value on Young's modulus and plateau stress of gyroid surfaces. Finally, the relationship between level-set values and densification strain and energy absorbed per unit volume is also presented to gain insight into the role of level-set values on the energy absorption capability of the gyroid surfaces. Such understanding provides a new method to develop desired properties of implants or energy absorption applications by selecting an optimal combination of level-set value, thickness, periodicity etc.

Stochastic approach for determining properties of randomly structured materials: Effects of network connectivity

Nanoporous (NP) materials have promising potential uses in energy storage, catalysis, and high-radiation environments. However, wide application of NP materials—or other emerging materials with similarly complex and/or random structures—requires an understanding of and predictive models for structure–property relationships governing materials behavior. Structural complexity and randomness have been a longstanding challenge to understanding and modeling such materials. Here, we describe the Kentucky Random Structures Toolkit (KRaSTk), which addresses this challenge by allowing for the stochastic generation of sets of many model representative volume elements (mRVEs) from a physically-motivated geometric seed description, and the calculation (via FEM) of both expected bulk effective properties and length-scale-dependent distributions of local properties. We demonstrate the applicability of this approach by showing that changes in network connectivity in a random, ligamented NP material have an effect on both elastic and bulk modulus that is independent of solid fraction, and is related to the presence of short-range disorder. The effect of varying network connectivity can be accounted for by adding a power law term to the Gibson-Ashby model for the stiffness of porous structures, such that E*/Es=CEϕ2(NC*−3)nE and K*/Ks=CKϕ2(NC*−3)nK, for NC* the effective network connectivity. These findings highlight the potential of the stochastic mRVE approach implemented in KRaSTk for modeling and exploring the effects of complex and random structure that drive novel behaviors in emerging porous, nanocrystalline, or additively manufactured materials.

Improved mechanical properties and energy absorption of Ti6Al4V laser powder bed fusion lattice structures using curving lattice struts

A porous lattice structure with highly controllable mechanical properties, low weight, and high strength is the most promising option for many fields, such as the aerospace, automobile, and biomedical industries. However, the most common and critical issue is the excessively high stress concentration at the struts’ nodes when the lattice structure is loaded. Thus, a curving lattice design strategy is proposed, maintaining a light structure, through which either a circular or elliptical arc is used in the lattice struts to obtain new forms of curving lattice structures. By establishing theoretical models and preparing samples by laser powder bed fusion (L-PBF), combined with scanning electron microscopy (SEM), quasi-static uniaxial compressive experiments, finite element analysis (FEA), and Gibson-Ashby models, excellent results were obtained. The stress distribution of the original under the loads was altered in the curving lattice structure and the stress concentration at the nodes of the struts was effectively relieved, dramatically improving its mechanical properties. Compared with the original structure, the specific elastic modulus and specific compressive strength of curving lattice structures were maximally increased by 213.7% and 126.2%, respectively. Meanwhile, the curving lattice structure had a superior energy absorption capacity, with a significant increase of 92.9%.

Excellent Specific Mechanical and Electrical Properties of Anisotropic Freeze-Cast Native and Carbonized Bacterial Cellulose-Alginate Foams.

Native and carbonized freeze-cast bacterial cellulose-alginate (BC-ALG) foams possess an ice-templated honeycomb-like architecture with remarkable properties. Their unique pore morphology consists of two levels of porosity: 20-50 μm diameter pores between, and 0.01-10 μm diameter pores within the cell-walls. The mechanical properties of the BC-ALG foams, a Young's modulus of up to 646.2 ± 90.4 kPa and a compressive yield strength of up to 37.1 ± 7.9 kPa, are high for their density and scale as predicted by the Gibson-Ashby model for cellular materials. Carbonizing the BC-ALG foams in an inert atmosphere at 1000-1200 °C in a second processing step, both pore morphology and mechanical properties of the BC-ALG remain well preserved with specific mechanical properties that are higher than those reported in the literature for similar foams. Also the electrical conductivity of the BC-ALG foams is high at 1.68 ± 0.04 S cm-1 at a density of only 0.055 g cm-3, and is found to increase with density as predicted, and as a function of the degree of carbonization determined by both carbonization temperature and atmosphere. The property profile makes freeze-cast BC-ALG foams and their carbonized foams attractive for energy applications and as a sorbent.

Chemically Functionalized Cellulose Nanocrystals as Reactive Filler in Bio-Based Polyurethane Foams.

Cellulose Nanocrystals, CNC, opportunely functionalized are proposed as reactive fillers in bio-based flexible polyurethane foams to improve, mainly, their mechanical properties. To overcome the cellulose hydrophilicity, CNC was functionalized on its surface by linking covalently a suitable bio-based polyol to obtain a grafted-CNC. The polyols grafted with CNC will react with the isocyanate in the preparation of the polyurethane foams. An attractive way to introduce functionalities on cellulose surfaces in aqueous media is silane chemistry by using functional trialkoxy silanes, X-Si (OR)3. Here, we report the synthesis of CNC-grafted-biopolyol to be used as a successful reactive filler in bio-based polyurethane foams, PUFs. The alkyl silanes were used as efficient coupling agents for the grafting of CNC and bio-polyols. Four strategies to obtain CNC-grafted-polyol were fine-tuned to use CNC as an active filler in PUFs. The effective grafting of the bio polyol on CNC was evaluated by FTIR analysis, and the amount of grafted polyol by thermogravimetric analysis. Finally, the morphological, thermal and mechanical properties and hydrophobicity of filled PUFs were thoughtfully assessed as well as the structure of the foams and, in particular, of the edges and walls of the cell foams by means of the Gibson–Ashby model. Improved thermal stability and mechanical properties of PU foams containing CNC-functionalized-polyol are observed. The morphology of the PU foams is also influenced by the functionalization of the CNC.

Compression properties and optimization design of SLM Ti6Al4V square pore tissue engineering scaffolds.

The tissue engineering technology provides a new way to solve bone defect. Porous scaffolds supply support and adhesion space for cells. Design of pore structure of scaffolds is one of the key points in tissue engineering scaffolds, because the structure affects the performance of scaffolds directly. In this paper, mechanical properties of square porous Ti6Al4V scaffolds are studied. By finite element simulation, it can be found that the support structure in vertical direction assumes main force, so the structure can be optimized through relative density mapping (RDM) method. The modified arch structures can improve bearing effect of structure with the same porosity. The designed structures are obtained by selective laser melting. Results of compressive strength indicate that the compressive strength decreases with the increase of porosity. When the porosity is between 40% and 60%, the error of compressive strength calculated by Gibson-Ashby model is below 8%. Moreover, the optimized structure clears a better bearing effect, and the bearing capacity can be increased by 20%-30% under the same porosity.

Precipitation-induced transition in the mechanical behavior of 3D printed Inconel 718 bcc lattices

This work aims to investigate the effect of microstructure on the mechanical behavior of metallic lattice structures. With that purpose, Inconel 718 bcc lattices were manufactured by selective laser melting and their room temperature compressive behavior was compared in the as-built and aged conditions. A distinct transition from bending-dominated to stretch-dominated behavior takes place following the aging treatment as the as-built supersaturated solid solution (containing a minor fraction of nanoparticles) transforms into a microstructure with a comparatively large fraction of precipitates with dimensions spanning several hundred nanometers. Precipitation leads to a stiffer and stronger structure, with a higher capacity to absorb energy. The obtained mechanical property results are compared to the predictions of the Maxwell criterion and the Gibson-Ashby model for bcc lattices. It is shown that microstructure variations explain, at least partially, the low correlation between the bcc experimental data in the literature with the model predictions.

Evaluation of mechanical properties of Ti-25Nb BCC porous cell structure and their association with structure porosity: A combined finite element analysis and analytical approach for orthopedic application.

Ti-based alloys have been commonly employed in manufacturing implants for orthopedic applications. Binary Titanium-Niobium (Ti-25Nb) alloy is a promising material for potential applications in orthopedics because of their lower elastic moduli and superior biocompatibility than the conventional Ti-based alloys. Implants with porous structures encourage bone ingrowth and reduce the effect of stress-shielding further. This study is aimed at establishing the relationship between the mechanical performance and structural parameters of porous body-centered-cubic (BCC) structures made up of Ti-25Nb (25% by wt.). Solid models of BCC porous structures were constructed (unit cell size: 2 mm; overall size: 8 × 8 × 8 mm3). Finite element analysis (FEA) of the BCC structures with porosity ranging from 29% to 79% (seven porosities) was conducted under tension, bending, and torsional loads. The Gibson-Ashby model and Exponential regression model were also employed to determine the stiffness of the above porous structures. The functional relationships between effective Young's modulus, effective yield strength, and porosity generated from both the models were found to match the FEA results well. Results indicated that porosity in the range of 50%-70% can be used to design graded porous stems to mimic the mechanical properties of cortical bone.