© T. Li et al., Published by EDP Sciences, 2025

1 Introduction

Porous metallic materials [1] are a complex research area with different starting points for different applications. Porous metallic materials consist of a solid metal and a large number of pore gaps, which separate the solid part of the metal into many small parts, which can also be called porous foam metal [2,3]. There are countless research directions and ideas on the structural form of the pore, the mechanical properties of the material itself as affected by the various parameters of the pore, and the effect on biocompatibility in bone implantation [4–7], and as a result, there is a wealth of research results on porous materials both at home and abroad.

Stamp et al. [8] prepared aggregate materials with an average pore size of 440 µm and a porosity of 71% using SLM beams with CpTi (grade 1) and used them to fabricate final shape orthopedic components. The results show that SLM beam overlap is a promising technique for fabricating final shape functional bone growth materials. Song Li et al. [9] investigated the design and preparation of graded porous materials, introduced graded porous materials, and reviewed their application scenarios. Porous metal scaffolds [10] can be successfully implanted into the human body as artificial bone replacement materials due to their excellent mechanical properties and biocompatibility [11]. Wang et al. [12] used at 1000 W laser power and scanning speed to successfully prepare Ti walls with high porosity (∼70%), interconnected Ti walls, and an open porous structure with macroscopic pores (∼200 to ∼500 µm) using the porous titanium scaffolds. Currently, biomedical porous titanium alloys [13] have become the main medical porous metal for products and manufacturing of surgical implants and orthopedic devices worldwide. Ai-hua YU et al. [14] designed and fabricated hierarchically structured titanium scaffolds with different porosities by selective design and studied their manufacturability, microstructure, mechanical properties and permeability. Medical titanium alloys have been used in human joint implants since the 1960s, and have experienced the development from Ti-6Al-4V to the new β-titanium alloys, which have the advantages of light weight, high strength, and corrosion resistance, and have developed rapidly in human implant materials [15–17]. Medical metal materials due to the high modulus of elasticity (the modulus of elasticity of titanium alloy is 100–110 Gpa [18], the modulus of elasticity of human bone is 1–30 GPa) and other characteristics of the elastic modulus of human bone and metal materials, resulting in implantation of “stress shielding”, especially in the joints, spine, The difference in elastic modulus between human bone and metal materials is large, which leads to “stress shielding” after implantation, especially in joints, spine, and traumatized areas, causing bone stress resorption, and ultimately loosening and fracture of the implant [19]. Zhang S et al. [20] prepared porous Ti6Al4V alloy implants using laser additive manufacturing technology, and changed the pore size (250–450 µm) by adjusting the scanning spacing to meet the needs of human bone. Experiments showed that the increase in pore size led to a decrease in compressive strength and modulus of elasticity. Additively manufactured (AM) porous structures [21] are a new class of biomaterials that show many advantages over traditional biomaterials [22–25]. In addition, additive manufacturing technology is capable of creating customized implants based on CT images of patients, and the main techniques for rapid prototyping of metal parts [26] include selective laser sintering (SLS), laser electroless net molding (LENS), selective laser melting (SLM) [27], and selective electron beam melting (SEBM) [28]. Among them, selective laser melting technology has been widely emphasized in foreign countries due to good densification, metallurgical bonding and high precision of molded parts.

At present, foreign researchers on laser additive manufacturing (3D printing) preparation of porous metal materials more research [29], the preparation of porous metal specimens are mainly focused on biocompatible medical porous metal implants, due to there is no certainty of the optimal porous structure, and affects the porous structure of the biological and mechanical properties of the factors are too much, so this paper in the premise of the physical experiments are not carried out, the main search for Therefore, this paper, without conducting physical experiments, mainly explores the effects of porosity and pore size on the mechanical properties of porous materials, and finds out the general pattern of change and connection between these factors.

2 Materials and methods

2.1 Materials

In this paper, two sets of control in vertical and horizontal directions are set up and 3D modeling techniques as well as analog simulation techniques are applied to explore and study them theoretically. Parametric porous materials as shown in Table 1 are created according to the research idea.

Under the limitations of the existing computer modeling capabilities and analytical capabilities, the overall volume of the uniform square specimen 20 mm ×  20 mm × 20 mm = 8000 mm³ is calculated to calculate the optimal number of holes for different holes under various parameter constraints.

$$\frac{V+x^2\times x\times L^3\times 2}{3\times \left(x^2\times 20\right)}=L^2$$(1)

x²–Number of holes per side; L − Hole Diameter (Side Length); V − pore volume.

According to the principle of conservation of volume, the total volume of the test block is equal to the pore volume plus the remaining volume.

The total pore volume V plus the extra volume due to the holes (the portion of the volume where the holes penetrate both faces but are counted only once) is V + x²  × x × L³ × 2. (Here x² is because the holes are arranged in two dimensions, with x holes on each face, for a total of x² pore surfaces, but each hole penetrates both faces, so it is multiplied by 2, and then multiplied by the number of holes, x, and the volume of the holes, L³ ).

The volume after removing the pores is 3 × (x² × 20) ×  L². Here 3 ×(x² × 20) represents the “effective” portion of the test block divided into three dimensions (the portion with the holes removed), and the height (or depth) of each such portion is L² (where L² actually represents the “area” of the remaining portion). (here L² actually represents the projection of the “area” of the remaining portion in the direction perpendicular to the holes, but since the holes are penetrating and uniformly distributed, the representation can be simplified to this).

Simplify to get:

$$2L^3{x}^3-60L^2{x}^2+V=0.$$(2)

The porous material parameters are shown in Table 1.

2.2 Modeling of porous materials

Grasshopper is a very special modeling software on the rhino platform, and its modeling approach has many obvious differences compared to traditional modeling software. The most important feature is that grasshopper can be modularized through a series of modeling commands (operators) to build a model of the complete generation of logic, and through the computer operation to execute these commands to generate the final model.

For the porous material modeling of truss-like structure, the idea of this modeling is mainly to regard the entities around the regular holes as one bracket, one-dimensionalize the regular brackets, then combine these one-dimensional lines to form the spatial mesh structure of the porous structure, and finally assign the thickness of the one-dimensionalized brackets, according to the way of assigning the thickness, the square tube and the round tube can be generated. The main battery diagram of the truss-like structure part is shown in Figure 1.

For the modeling of bionic porous materials, this modeling idea applies the Voronoi3D algorithm to obtain a closed multiple shear surface patchwork solid by creating the underlying compartmentalized structure without directly obtaining a model. Then the physical simulation plugin kangaroo is applied to realize the design concept with growth development by simulating the physical motion of point, line and mesh type objects. The main battery diagram for the modeling part of the bionic porous material is shown in Figure 2.

The solid sample pieces with different porosities and pore sizes obtained by the above modeling approach are shown in Figure 3.

Fig. 1 Diagram of the main cell in the porous material modeling section of the truss-like structure.

Fig. 2 Diagram of the main cell in the modeling section of the bionic porous material.

Fig. 3 Solid printed sample pieces (a) solid corresponding to 40% porosity and 700 µm pore size, (b) solid corresponding to 60% porosity and 700 µm pore size, (c) solid corresponding to 40% porosity and 500 µm pore size.

3 Results and analysis

The analysis is carried out using Workbench 15.0 interface of ANSYS, where the linear static analysis of porous structure is carried out in this paper to perform the exhaustive mechanical analysis of the linear structure.

3.1 Equivalent modulus of elasticity for porous materials

Under the homogenization idea to solve the modulus of elasticity of porous materials [30], since the modulus of elasticity of dense materials does not change due to the creation of pores in them, while the mechanical properties of the material change after the creation of pores, the pores themselves are not subjected to force, but the modulus of elasticity measured under the force of the porous material produces a change in the modulus of elasticity of the porous material in comparison with that of the raw material. Using the idea of homogenization, the porous material is regarded as a dense material after the change of the elastic modulus of the raw material, and the equivalent elastic modulus is used to explore the law. In this paper, in order to find the equivalent elastic modulus of the porous material, the porous material is equated to a continuous homogeneous medium. Using ANSYS finite element analysis, the continuous uniform medium, when subjected to uniform compressive stress can be calculated to obtain the continuous uniform medium in the compression direction of the tensile deformation variable, the compression deformation variable is equivalent to the porous material is subjected to uniform compressive stress compression deformation variable, so as to theoretically predict the equivalent modulus of elasticity of the porous material.

Because under the influence of porous structure, whether the stress-strain of porous materials is linear when different material properties are assigned is also a question that needs to be investigated experimentally, under the limitation of the simulation conditions, we regard the stress-strain of porous and hollow materials as linear without experimental exploration and make theoretical predictions under this condition.

Then, according to the definition of the modulus of elasticity as $E=\frac{\sigma }{\epsilon }$. Because the homogenization idea is used to solve the equivalent modulus of elasticity so the value of σ needs to be recalculated according to the situation. Let: the area of the hole part of the surface of the test block is Smm², then we have:

$${\sigma }^{\prime }=\frac{3\times \left(20\times 20-S\right)}{20\times 20}=3-0.0075S$$

(where 3 is the applied 3 Mpa and 20 is the side length of the test block).

Then there is:

$$E=\frac{{\sigma }^{\prime }}{\epsilon }=\frac{3-0.0075S}{\epsilon }.$$(3)

3.2 Preliminary finite element analysis of porous materials

The maximum deformation and equivalent stresses corresponding to porous materials with different porosity and pore size for different material properties assigned to the porous materials are shown in Figure 4.

For assigning material properties to nickel based alloy is analyzed as follows:

From the horizontal comparison, it can be seen that as the porosity increases, the deformation of the porous material as well as the maximum stress generated in the structure also increases and increases significantly. This is consistent with the results of the calculations, because the pore spacing decreases significantly with increasing porosity at constant pore size, resulting in the microscopic pore walls (pore trusses) becoming thinner (finer), which changes their structural strength. At the same time with the increase of porosity, the equivalent modulus of elasticity in the transverse direction is also significantly reduced.

From the longitudinal comparison, it can be seen that in the case of the porosity remains constant and the pore diameter decreases, the mechanical properties of the porous material change insignificantly, and its modulus of elasticity decreases and then increases, and there is no regularity. Because of the reduction of pore diameter, the number of holes increases, resulting in a decrease in the pore spacing also, and therefore the maximum stress produced by it has a tendency to increase.

Comprehensive horizontal and vertical comparisons show that for porous metallic materials endowed with material properties of nickel-based alloys, porosity is the main factor affecting the mechanical properties, and there is a clear tendency for the elastic modulus to decrease with the increase in porosity.

For the porous material given the material property of titanium alloy, the maximum deformation and equivalent force corresponding to the porous material at different porosity and pore size are shown in Figure 5.

The results for assigning the material property of titanium alloy are analyzed as follows:

From the comparison in the transverse direction, it can be seen that as the porosity increases, the deformation of the porous material as well as the maximum stresses generated on the structure also increase, and the increase is significant. At the same time as the porosity increases, the equivalent modulus of elasticity decreases significantly in the transverse direction. This is similar to the change in nickel-based alloys.

From the longitudinal comparison, it can be seen that the mechanical properties of the porous material do not change significantly when the porosity is kept constant and the pore size decreases, and its elastic modulus decreases and then increases with no regularity, which confirms that the results for the nickel-based alloys are not due to errors or analytical mistakes.

Comprehensive horizontal and vertical comparisons show that for porous metallic materials with titanium alloy as the endowed material property, the porosity is the main factor affecting the mechanical properties, and there is a clear tendency for the elastic modulus to decrease with the increase of the porosity.

Fig. 4 Maximum deformation and equivalent force corresponding to porous materials with different porosity and pore size (a) 40% porosity, pore size 700 µm state, (b) 50% porosity, pore size 700 µm state, (c) 60% porosity, pore size 700 µm state, (d) 40% porosity, pore size 600 µm state, (e) 40% porosity (e) 40% porosity, 500µm pore size, (f) 40% porosity, 600µm pore size.

Fig. 5 Maximum deformation and equivalent force corresponding to porous materials with different porosity and pore size (a) 40% porosity, pore size 700 µm state, (b) 50% porosity, pore size 700 µm state, (c) 60% porosity, pore size 700 µm state, (d) 40% porosity, pore size 600 µm state, (e) 40% porosity (e) 40% porosity, 500 µm pore size, (f) 40% porosity, 600 µm pore size.

3.3 Summary and analysis of results

The results for assigning the material property of nickel-based alloy are summarized in Table 2.

The results for assigning the material property of titanium alloy are summarized in Table 3.

Comprehensive horizontal and vertical comparisons show that for porous metallic materials with titanium alloy as the endowed material property, the porosity is the main factor affecting the mechanical properties, and there is a clear tendency for the elastic modulus to decrease with the increase of the porosity.

Focusing on the properties related to the elastic modulus of porous materials and their potential application in biomedical field, Aguilar et al. [31] mention that the simulated value is about 20 GPa when the equivalent porosity is 50%; this paper also shows that the increase of the porosity significantly decreases the elastic modulus of the material, and the elastic modulus of titanium alloy porous materials with 50% and 60% porosity have specific values reflecting this trend, respectively. In terms of the effect of porosity on the elastic modulus, this paper agrees with the conclusion that the value of elastic modulus decreases with the increase of equivalent porosity.

A comprehensive comparison of the porous materials of nickel-based alloys and titanium alloys shows that:

From the analysis of (1) (2), it can be seen that the two porous materials with two material properties have the same trend of mechanical property changes, and the stress distribution under the same structural body is also similar. The difference is the specific values of its deformation and elastic modulus. Due to the different properties of nickel-based alloy and titanium alloy itself, the deformation of titanium alloy is larger than that of nickel-based alloy under the same structural body, which leads to the elastic modulus of titanium alloy being much smaller than that of nickel-based alloy under the same structural body. Therefore, in order not to make the structure of the porous material too complicated for manufacturing, the use of titanium alloy as the carrier of the porous material is mechanically superior to nickel-based alloy.

Of course, the first round of analysis also has many shortcomings, for example, although the elastic modulus of the porous material of titanium alloy reaches the range of the elastic modulus of human cortical bone (10–40 Gpa) at 50% and 60% porosity, it is far from the elastic modulus of human cancellous bone (1–4 Gpa). Further structural exploration of the porous material is needed to verify whether the elastic modulus of human cancellous bone can be achieved.

3.4 Discussion

From the above conclusions we know the porosity, pore size and other parameters on the mechanical properties of porous materials, the simulation results also reached the human cortical bone required modulus of elasticity of the range, but from the cancellous bone there is still a certain gap, so here only carried out in order to make the modulus of elasticity to achieve the exploration of 1–4 Gpa.

By the effect of computer modeling capabilities, the volume of the reduced square test block was 10 mm × 10 mm × 10 mm = 1000 mm³. The porous structural body with a pore size of 700 µm and porosity close to 80% and 90% (not exceeding 90%) is similarly established by equation(2) L²x³ − 60L²x² + V =0. It is known that x should be taken as 10 and 11, but the actual porosity should be recalculated here because the error is too large.

Then there is porosity:

$$E_1=\frac{{0.7}^2\times 10\times 100\times 3-\left(100\times 10\times 2\right)\times {0.7}^3}{1000}\times 100\%=78.4\%$$(4)

$$E_2=\frac{{0.7}^2\times 10\times {11}^2\times 3-\left({11}^2\times 11\times 2\right)\times {0.7}^3}{1000}\times 100\%=86.6\%.$$(5)

The porous structures with porosity of 78.4% and 86.6% and pore size of 700 µm were modeled as shown in Figure 6.

(1) Giving material properties as nickel-based alloys

A. Porosity of 78.4

As shown in Figure 7, it can be seen from the analytical results that in the case of 78.4% porosity and 700 µm pore size, the maximum deformation produced is 0.00078308 mm, the maximum stress is 20.397 MPa, and its equivalent elastic modulus is calculated to be E = 19538 Mpa = 19.54 Gpa.

B. Porosity of 86.6%

As shown in Figure 8, from the analytical results, it can be seen that in the case of porosity of 86.6% and pore size of 700 µm, the maximum deformation produced is 0.0011424 mm, the maximum stress is 29.956 MPa, and its equivalent elastic modulus is calculated to be E = 10691 Mpa = 10.69 Gpa.

(2) Assigning material properties as titanium alloy

A. Porosity of 78.4%

As shown in Figure 9, from the analytical results, it can be seen that in the case of porosity of 78.4% and pore size of 700 µm, the maximum deformation produced is 0.0015593 mm, and the maximum stress is 20.46 MPa, and its equivalent elastic modulus is calculated to be E = 8560 MPa = 8.56 Gpa.

B. Porosity of 86.6%

As shown in Figure 10, it can be seen from the analytical results that in the case of porosity of 86.6% and pore size of 700 µm, the maximum deformation produced is 0.0022832 mm, the maximum stress is 30.298 MPa, and its equivalent elastic modulus is calculated to be E = 4123 MPa = 4.12 Gpa.

The elastic modulus of human cortical bone is 17–20 Gpa [32], and the elastic modulus of cancellous bone is 3.2–7.8 Gpa [33]. From the above analysis, it can be seen that the elastic modulus of the stainless steel tetrahedral porous material reaches the level of cortical bone when the porosity is close to 80% under the condition of the pore diameter of 0.7mm, the elastic modulus of the titanium alloy is only different from that of the cancellous bone by 4 Gpa, and the pore ratio is close to 90%. The elasticity of nickel alloy is close to the lowest value of cortical bone, and the elasticity of titanium alloy reaches the level of cancellous bone. Because we believe that the stress-strain of materials are linear, we can boldly predict that the elastic modulus of titanium alloy can reach the level of general cancellous bone (1–4 Gpa) when the porosity reaches 87–90%, and the elastic modulus of nickel alloy can reach the range of the elastic modulus of human cortical bone when the porosity reaches the level of 70–87%. Y. Alex et al. [34] have shown that a porous structure with high porosity is not realistic for tetrahedral nickel-based alloys, and that a small pore spacing value is of no practical significance, since a complex structure would need to be redesigned in order to reach the level of cancellous bone in human bone.

This paper is that by combining 3D modeling, simulation techniques (Grasshopper, ANSYS Workbench) and homogenization ideas, the concepts of truss-like structures and biomimetic porous materials are introduced, and the mechanical properties of porous materials are explored in depth by using advanced techniques, such as Voronoi3D and kangaroo, which enable parametric studies and special Focusing on biomedical applications, the article systematically investigates the effects of porosity and pore size on the elastic modulus and overall mechanical properties of porous materials, and also focuses on biomedical applications, especially the range of elastic modulus of human bone, which provides an optimization basis for titanium alloy porous materials and other porous materials, which is of great practical significance.

Fig. 6 Modeling of porous structure with different porosities. (a) 78.4% porosity and 700 µm pore size state (b) 86.6% porosity and 700 µm pore size state.

Fig. 7 (a) Maximum deformation (b) Equivalent force.

Fig. 8 (a) Maximum deformation (b) Equivalent force.

Fig. 9 (a) Maximum deformation. (b) Equivalent force.

Fig. 10 (a) Maximum deformation (b) Equivalent force.

4 Conclusion

In this paper, the properties of tetrahedral porous structures under ideal conditions are investigated, focusing on the effects of porosity and pore size on the mechanical properties. The intrinsic properties of the material and the porosity were found to have a significant effect on the elastic modulus. Novel tetrahedral porous bodies with different porosities were created and endowed with the material properties of nickel-based alloys and titanium alloys based on the elastic modulus range of human bone, leading to the following conclusions:

• The effects of 500 µm, 600 µm and 700 µm pore diameters suitable for bone formation as well as 40%, 50% and 60% porosity on the elastic modulus of porous structures were investigated. The intrinsic properties of the material and the porosity are the key factors affecting the elastic modulus, and an increase in porosity significantly decreases the elastic modulus of the material, while the change in pore size has a relatively small effect, but a decrease in pore size still slightly decreases the elastic modulus.

• According to the range of elastic modulus of human bone, it is found that the titanium alloy porous material with 50% and 60% porosity can meet the requirements of cortical bone in terms of mechanical properties, and the elastic modulus reaches 19.54 Gpa, 10.69 Gpa, 8.56 Gpa, and 4.12 Gpa, respectively, and at this time, the nickel-based alloy porous material has been able to reach the level of elastic modulus of human cortical bone. The titanium alloy porous material also reaches the level of human cancellous bone. This provides a strong support for the application of titanium alloy porous material in biomedical field.

• Titanium alloy porous materials are usually preferred over nickel-based alloys based on practical manufacturing feasibility. Because they are less difficult to prepare, nickel-based alloys were found to require higher porosity models to achieve mechanical properties similar to those of titanium alloys. Nickel-based alloys require 70% to 87% porosity to reach the level of cortical bone, whereas titanium alloys can reach the level of cortical bone at 60% to 70% porosity. Titanium alloys can reach the level of human cancellous bone at 87% porosity.

Funding

This study was supported by Key R&D Program of Shandong Province, China (2024TSGC0907).

Conflicts of interest

We declare that we do not have any commercial or associative interests that represent a conflict of interest in connection with the work submitted.

Data availability statement

Data will be made available on request.

Author contribution statement

Tianhua Li: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing − Original Draft, Writing − Review and editing, Visualization. Hailong Ma, Shubo Xu: Resources, Supervision, Project administration, Funding acquisition. Renhui Liu: Investigation. Yuefei Pan: Investigation. Guocheng Ren: Writing − Review & Editing. Jianing Li: Writing − Review and editing. Zhongkui Zhao: Data curation.

References

All Tables

All Figures

	Fig. 1 Diagram of the main cell in the porous material modeling section of the truss-like structure.
In the text

	Fig. 2 Diagram of the main cell in the modeling section of the bionic porous material.
In the text

	Fig. 3 Solid printed sample pieces (a) solid corresponding to 40% porosity and 700 µm pore size, (b) solid corresponding to 60% porosity and 700 µm pore size, (c) solid corresponding to 40% porosity and 500 µm pore size.
In the text

	Fig. 4 Maximum deformation and equivalent force corresponding to porous materials with different porosity and pore size (a) 40% porosity, pore size 700 µm state, (b) 50% porosity, pore size 700 µm state, (c) 60% porosity, pore size 700 µm state, (d) 40% porosity, pore size 600 µm state, (e) 40% porosity (e) 40% porosity, 500µm pore size, (f) 40% porosity, 600µm pore size.
In the text

	Fig. 5 Maximum deformation and equivalent force corresponding to porous materials with different porosity and pore size (a) 40% porosity, pore size 700 µm state, (b) 50% porosity, pore size 700 µm state, (c) 60% porosity, pore size 700 µm state, (d) 40% porosity, pore size 600 µm state, (e) 40% porosity (e) 40% porosity, 500 µm pore size, (f) 40% porosity, 600 µm pore size.
In the text

	Fig. 6 Modeling of porous structure with different porosities. (a) 78.4% porosity and 700 µm pore size state (b) 86.6% porosity and 700 µm pore size state.
In the text

	Fig. 7 (a) Maximum deformation (b) Equivalent force.
In the text

	Fig. 8 (a) Maximum deformation (b) Equivalent force.
In the text

	Fig. 9 (a) Maximum deformation. (b) Equivalent force.
In the text

	Fig. 10 (a) Maximum deformation (b) Equivalent force.
In the text