On quantifying dynamic behavior of architected metal/polymer TPMS/lattices-based interpenetrating phase composites | Scientific Reports

Scientific Reports volume 15, Article number: 4253 (2025) Cite this article

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This article presents the numerical analysis of architected metal/polymer-based interpenetrating phase composites (IPCs) to study their effective mechanical properties and dynamic behavior using finite element (FE) simulation. In this, we considered four types of Triply periodic minimal surfaces (TPMS) and lattice architectures, including gyroid, primitive, cubic, and octet, to form architected IPCs. The aluminum alloy is used for the TPMS/lattice reinforcing phase, and epoxy as a reinforced phase. The periodic boundary conditions were applied using FE analysis to compute the effective properties, while these properties were utilized to investigate the dynamic analysis of composite structures considering free vibration, wherein actual and homogenized models are compared. Our results reveal that the effective properties of IPCs increase with respect to the volume fraction of respective architectures in conjunction with enhanced natural frequency and less deformation. Moreover, we conducted a comparative study between these newly architected metal/polymer IPCs and conventional composites.

Composite materials are created by blending two or more distinct materials to achieve a favorable balance of material properties1. These properties, including electrical, magnetic, and mechanical characteristics, can be tailored to specific applications through meticulous design and production. To optimize the performance of mechanical devices, it is crucial to integrate desired material traits and connectivity patterns between their constituent phases2,3. In the field of composite structure design and analysis, effective properties were commonly utilized. Composite materials can be represented by an index of two numbers based on the connectivity patterns between their constituent phases. For example, 1–3 represents unidirectional composites, where the first phase is one-dimensionally (1D) connected, while the second is three-dimensionally (3D) connected. Thus, 0–3 (particulate composite), 1–3 (unidirectionally fiber-reinforced composite), and 2–2 (often referred to as bidirectional composites) types represent the most common and fundamental arrangements of the reinforcement and matrix materials in two-phase composite3. Modulating the properties of an individual phase or characteristics allows one to tailor the behavior of the composite. For instance, one may use the finite element (FE) method to study the effective characteristics of heterogeneous and composite materials by considering their microstructures and material phase reinforcements. Hence, the pursuit of creating composite materials with superior properties has driven advancements in FE micromechanics4,5,6. In addition to composites with different connectivity patterns, including 0–3, 1–3, and 2–2, recently, 3–3 composites have garnered noteworthy interest from the research community because of their promising architecture, demonstrating better structural performance than others.

In this study, we consider the following types of composites: conventional composites (including uniform and randomly distributed unidirectional (UD) composites and particulate composites) and novel 3–3 interpenetrating phase composites (IPCs). The random UD composites consist of fibers that vary randomly in their orientation (i.e., with specific mean alignment angle between fiber direction and orientation axis) embedded within a matrix material. The latter (IPCs) includes triply periodic minimal surfaces (TPMS): gyroid, primitive, cubic, and octet lattice as reinforcing phases (refer to Fig. 2). TPMSs exhibit various characteristics inspired by nature, such as multifunctionality, mathematical accuracy, optimized structure for mechanical uses, and extreme properties in geometry, thermal behavior, and electrical attributes7,8,9,10. Notably, some mechanical and transport properties, such as thermal, electrical, and fluid properties, compete within minimal surfaces as a result of optimizing for multiple functions9,11. Additionally, minimal surfaces minimize surface tension locally, thereby reducing surface energy and residual stresses12. The minimal residual stresses in TPMSs make them ideal as 3D reinforcements in matrix materials, facilitating the creation of IPCs. Overall, these TPMSs exhibit zero mean curvature, minimal surface stress, surface tension, and surface energy9. These surfaces can exhibit endless periodicity and extend into three dimensions, creating IPCs with 3–3 connectivity, which enhances their overall properties. These minimal surfaces were considered potential structures for lightweight materials13. Originally developed as mathematical functions, these patterns were subsequently found to be common in biological systems, including beetle shells, weevils, butterfly wings, scales, and crawfish skeletons7,14,15,16.

The fundamental purpose behind utilizing IPCs is to amalgamate the characteristics of two materials in an interconnecting manner, thereby harnessing the full potential of both constituents. IPCs deviate from conventional composites, such as those reinforced with random/periodic particles17, fibers18, and platelets19,20, wherein the reinforcements were solitary, irregular, and segregated from one another within the matrix material. Moreover, periodic IPCs exhibit less directional dependence compared to fiber-reinforced composites. The interconnectivity and continuity of the TPMS within the matrix are anticipated to enhance transport properties (such as diffusion, thermal, and electrical) and improve strength and damage tolerance by effectively obstructing crack propagation. IPCs exhibit markedly different mechanical properties compared to conventional composites such as fiber and particulate composites. Unlike traditional composites, which depend largely on their continuous matrix material with embedded discrete phases, IPCs demonstrate unique mechanical behavior21,22. IPCs, on the other hand, are materials with multiple phases, with each phase forming a fully interconnected 3D network. As a result, each component of IPCs can efficiently impart its distinctive properties to the overall properties of IPCs through its interconnected structure23,24,25,26. Most extensively researched IPCs consist of metal and ceramic phases27,28,29,30,31. However, the ceramic phase is typically brittle and shows minimal hysteresis in basic loading–unloading tests, which may not fully satisfy the needs of military and civil engineering applications. To overcome this limitation, blending polymers into the pores of metals to create metal-polymer IPCs presents a promising method for producing new high-hysteretic damping materials with favorable mechanical characteristics32,33,34,35,36,37. Interpenetrating metal/polymer composites exhibit characteristics such as impact resistance, vibration damping, electromagnetic shielding, and phase-changing capabilities, making them suitable for applications in transportation, energy, aerospace, aviation, and construction industries33,38,39. However, the aforementioned studies only considered metallic foams, not lattices with specific internal architecture. Therefore, the next paragraph explains work related to lattice/TPMS and their respective IPCs.

In recent advances, Yang et al.40 investigated the fatigue properties of Ti-6Al-4 V gyroid graded lattice structures (GLSs) produced by laser powder bed fusion (LPBF), focusing on the effect of gradient design under lateral loading. Their study found that GLSs improved fatigue life by 1.21 to 1.67 times compared to uniform structures, exhibited various fracture modes, and reduced stress concentration around crack tips, which slowed crack propagation. Jin et al.41 explored the fabrication of Ni–Ti multicell interlacing gyroid lattice structures using LPBF, revealing their ultra-high hyperelastic response, ideal for applications requiring high elasticity and resilience. Their study emphasized the potential of these structures in industries like aerospace, automotive, and healthcare due to their unique mechanical properties and the precision of LPBF. In this, they also addressed the challenges of optimizing process parameters to enhance part quality and reduce defects during fabrication. Recently, Viet et al.42 implemented different TPMS to investigate the influence of various factors on the mechanical properties of functionally graded (FG) TPMS-based titanium implants interconnected with ingrown bone under uniaxial compression. Xie et al.43 also implemented the same strategy for enhancing orthopedic implants using TPMS-based IPCs. Song et al.44 investigated the mechanical behavior of IPCs through experiments and numerical simulations, revealing that the synergistic effect of the constituent phases significantly impacts their strength, toughness, and energy absorption. Guo et al.45 studied the primitive/epoxy IPCs, highlighting their high strength, extended plateau stress for energy absorption, and enhanced strength. Recently, Singh et al.46 explained the hybrid manufacturing process in conjunction with additive manufacturing for the fabrication of metal/metal and metal/ceramic IPCs. In this, they implemented PBF and FDM processes for creating steel and ceramic reinforcement, respectively. Furthermore, researchers47,48 have investigated the use of TPMS-based IPCs in thermal management applications, such as heat exchangers and phase change energy storage systems. These IPC-based thermal management components exhibited superior heat dissipation and durability, addressing the critical thermal challenges in the next-generation automotive technologies. Most recently, Alkunte et al.49 systematically presented the comprehensive review on FG metamaterials. They explained different types of lattices and TPMS structures and their possible applications while considering FG properties. These presented studies mostly focused on polymer and ceramic based IPCs and therefore, the current study aims to investigate the mechanical characteristics of metal/polymer-based IPCs with novel architecture.

The primary objective of this article is to explore the properties of different architectured IPC composite materials that can be utilized in industrial applications, including the automobile, defense, and aerospace sectors. Figure 1 systematically illustrates the different applications and properties of IPCs. Among different types of materials, aluminum-based composites have garnered considerable attention from researchers due to their exceptional strength-to-mass ratio, low density, and corrosion resistance50. While various research works have conveyed the improved performance of different TPMS or lattice-based composites, to the best of the authors’ knowledge, there is no existing literature focusing on epoxy matrix and aluminum alloy as reinforcing metal phases within these composites (including TPMS, conventional particles, fibers, and lattices). And this knowledge gap has motivated our exploration of IPCs. Various TPMS and lattice structures, such as diamond (D), neovius (N), gyroid (G), I-graph and wrapped package-graph (IWP), faces of a Rhombic-Dodecahedron (F-RD), Fischer-Koch S (FK-S), primitive (P) TPMSs, cubic (C), Kelvin (K) cell and octet (O) lattices have been documented in the literature. These structures were characterized by specific mathematical equations that determined their shape51,52. Here, we use FE modeling of thickened TPMSs and lattices as the second (stiff) phase (refer to Fig. 2) to improve the effective properties of softer materials (the first phase is the epoxy matrix).

Schematic of different properties and application of architected IPCs.

Schematics of (a and b) TPMS-based IPCs, (c and d) lattice-based IPCs, (e–h) TPMS and lattice reinforcing phase, and (i and j) uniform and random particle composites and (k and l) uniform and random UD composites.

It is anticipated that this research will pave the way for further studies exploring additional mechanical properties of IPCs and their adaptation in different engineering sectors. Our study employs numerical simulations to compare the effective elastic parameters of conventional composites with those of TPMS/lattice-based IPCs incorporating epoxy and aluminum alloy as constituent materials. Among the different TPMS structures, we focus on gyroid (G)- and primitive (P)-TPMS structures due to their ability to withstand large strains before failure. In Fig. 2 schematics of TPMS- and lattice-based IPCs (now referred to as G-IPCs, P-IPCs, O-IPCs, and C-IPCs), TPMS and lattice reinforcing phase and conventional composites, including uniform and random particle/UD conventional composites, are shown. We compared these composites to understand better how to implement them in future applications. Therefore, after predicting all effective properties of conventional and novel architectured composites, we also investigated the modal analysis of composite structures where different support conditions. In this, we also considered two different models: actual and homogenized. Such modal analyses may be helpful in designing and optimizing structures in terms of vibration and resonance.

The homogenization method is a micromechanical technique used to determine the effective elastic properties of a composite material. This method replaces a heterogeneous material with a homogeneous medium that has equivalent elastic characteristics. The resulting effective properties represent the elastic behavior of this homogeneous medium. A micromechanical model of the representative volume element (RVE) comprising the individual constituents (hard reinforcing phase and soft matrix phase) is utilized to determine these effective properties. Figure 3(a-c) illustrates a single RVE of P-IPC extracted from P-IPC bulk composites with the representation of boundary and loading conditions applied on its faces, to predict the effective stiffness, to be explained in detail in the subsequent Section. Figure 3(d) is a schematic of O-IPC presenting the octet reinforcing and epoxy matrix phases.

(a and b) Single RVE of P-IPC extracted from P-IPC bulk composites, (c) boundary and loading conditions to predict the effective stiffness, and (d) O-IPC.

To model IPCs, we utilized the TPMS architectures differently, i.e., the reinforcing phases consist of TPMS sheets created by thickening the TPMS, while the reinforced phases are formed by deducting these TPMS from their surrounding cubic structure53. The integration of the reinforcing and reinforced phases produces IPCs with continuous and interconnected periodic structures. The methodology for developing the TPMS-IPCs is outlined in Fig. 4. Different numerical and analytical models were available for calculating the effective properties of composite materials53,54,55,56,57,58,59,60. These analytical methods were typically used for symmetric geometries and did not account for generalized loading conditions. Other numerical techniques, such as semi-analytical and mean-field methods, have been reported in the literature but local field fluctuations were often overlooked. For complex shapes and loading conditions, FE-based numerical methods are well-suited, as they allow for sufficient domain discretization. The present study employs the FE method on a unit cell model, representing an RVE of unit dimensions, to determine the homogenized effective stiffness properties. This is achieved by utilizing appropriate periodic boundary conditions to the RVE. The RVE, extracted from the composite, possesses effective properties identical to the bulk composites. For the 0–3 and 1–3 composites, the RVEs were generated by uniform and random distribution of spherical particles and UD fibers of varying sizes within a unit cell of the epoxy matrix. While modeling, the reinforcing and reinforced phases are presumed to be perfectly bonded.

Methodology implemented to create CAD and FE models of architected IPCs.

We have created FE models to determine the effective properties of architected IPCs and conventional composites using ANSYS 2023R1. The FE modeling of 3D RVE was carried out with ten-noded tetrahedral element “SOLID 187” which has three displacement degrees of freedom (DOF) in x, y, and z directions, and “CONTA174” element was used for contact interactions. CONTA174 is generally utilized to model sliding and contact interactions between 3D deformable and target surfaces. It is employed in 3D coupled-field contact analyses and is suitable for pair-based and general contact scenarios.

The RVE of the composites was homogenized and examined under different boundary conditions. Though our P-IPCs have cubic symmetry, we considered the FE model for composites considering transversely isotropic behavior and evaluated their effective independent elastic constants: \({\text{C}}_{11}^{\text{eff}}\), \({\text{C}}_{12}^{\text{eff}}\), \({\text{C}}_{13}^{\text{eff}}\), \({\text{C}}_{33}^{\text{eff}}\), \({\text{C}}_{44}^{\text{eff}}\) and \({\text{C}}_{66}^{\text{eff}}\). Subsequently, if we were to consider different natures and symmetry of composite, such as cubic, isotropic, transversely isotropic, orthotropic, or even anisotropic, their effective properties can be easily calculated. Moreover, in case of orthotropic composites, one can easily convert the stiffness coefficient to elastic constants using the relation between the compliance matrix, elastic constant, and Poisson’s ratio. The constitutive relations of composites are given by Eq. (1) in which \({\text{C}}_{\text{ij}}^{\text{eff}}\) is the effective elastic constant of composites.

To calculate the volume averaged stress \(\left\{{\overline{\upsigma } }_{\text{ij}}\right\}\) and strain \(\left\{{\overline{\upvarepsilon } }_{\text{ij}}\right\}\) in the RVE, volume averaging was employed over the entire volume under the imposed mechanical loading. This process involves integrating the relevant quantities over the volume of the RVE, and the equations used for calculating these averaged quantities are as follows:

where V denotes the volume of the RVE.

In this study, IPCs have cubic symmetry and thus, they can be fully described using three independent constants, \({\text{C}}_{11}^{\text{eff}}={\text{C}}_{33}^{\text{eff}}, {\text{C}}_{12}^{\text{eff}}={\text{C}}_{13}^{\text{eff}} \text{and} {\text{C}}_{{4}{\text{4}}}^{\text{eff}}={\text{C}}_{66}^{\text{eff}}\). Moreover, the periodic fiber-reinforced UD composites exhibit transversely isotropic behavior wherein \({\text{C}}_{66}^{\text{eff}}=\left({\text{C}}_{11}^{\text{eff}}-{\text{C}}_{12}^{\text{eff}}\right)/2\) while periodic particle-reinforced composites exhibit isotropic behavior. As IPCs have cubic symmetry with different degrees of anisotropy, we characterize them with the Zener anisotropy index \((\text{A})\)61,62:

From Eq. (1), it is evident that if there is a single normal strain present at any point in the composite, with the other strain components being zero, then three normal stresses arise. The ratio of each of these normal stresses to the normal strain yields an effective elastic coefficient. Therefore, a single simulation can compute three such elastic coefficients at that point. To determine a specific effective elastic coefficient from the FE model, appropriate boundary conditions must be prescribed on the faces of the RVE, discussed as follows.

In this, we first discuss the results determined from the FE analysis to see the variation of the effective elastic properties of distinct metallic TPMSs and lattices and their respective metal/polymer IPCs as well as conventional composites. Then, these effective properties are utilized to study the modal analysis of composite structures considering different support conditions. We also carried out a comparative analysis of all these composites. The properties of materials utilized in this analysis are enlisted in Table 1.

The FE mesh convergence study was carried out to verify the accuracy of the results. As discussed before, we used “tetrahedral” mesh type as “SOLID 187” is used for 3D modelling of IPCs. We performed meshing of IPC structures using conformal and periodic meshing options. In conformal meshing, coincident topologies are shared, meaning the mesh nodes at the interfaces are also shared between adjoining elements. In periodic meshing, the mesh generated on opposite faces of the RVE is identical in all directions where the geometry exhibits periodicity. The relative error between successive results determined for every different mesh size is enlisted in Table 2. We considered different sizes of elements ranging from 1 mm to 0.1 mm (coarsest to finest) for analyzing the effects of meshing on results. The summary of these results can be found in Table 2 and schematically illustrated in Figs. 5 and 6. These results indicate that beyond a specific size of elements, there is no substantial alteration in the prediction of elastic constant. Figure 5 shows the different element sizes (0.1 and 0.5) used to discretize the P-TPMS reinforcing phase utilized in IPC. Therefore, we used a fine-sized element to mesh IPC RVE.

Mesh analysis using different element sizes.

Convergence analysis using different element sizes.

After the mesh convergence study, we validated our FE modeling with earlier reported results for P-IPCs and enlisted in Table 3. These results show no significant difference for different volume fractions, indicating that our FE modeling can be implemented to calculate the effective properties of different types of IPCs and conventional composites.

In this, we considered IPCs to be composed of Primitive-TPMS (P-TPMS) with Al alloy and epoxy matrix, as shown in Fig. 3. Therefore, the procedure for determining the effective properties of P-IPCs is explained below.

To determine \({\text{C}}_{11}^{\text{eff}}\text{ and }{\text{C}}_{12}^{\text{eff}}\), the RVE must undergo the states of strain wherein only normal strain \({\varepsilon }_{11}\) is present; and all other strain components are zero \(\left( {\overline{\varepsilon }_{22} = \overline{\varepsilon }_{33} = \overline{\varepsilon }_{23} = \overline{\varepsilon }_{13} = \overline{\varepsilon }_{12} = 0} \right)\) These strains can be achieved by restricting/confining the surfaces of RVE \((x^{ + } /x^{ - } ,y^{ + } /y^{ - } ,z^{ + } /z^{ - } )\) as follows:

Here, \(\text{x},\text{y},\text{ and z}\) signify the coordinates equivalent to 1, 2, and 3-axes, respectively. The dimensions of RVE are represented by length (l), width (a), and height or depth (b), while small displacement applied according to prescribed boundary and loading conditions are \(\text{u},\text{ v},\text{ and w}\) in \(\text{x}-,\text{y}-,\text{ and z}-\) directions of RVE (Fig. 3c). Under prescribed boundary conditions, a uniform normal displacement (\(\text{u}\)) is applied on the surface \(({\text{x}}^{+}=\text{a})\) of the RVE to subject it to only of \({\overline{\varepsilon }}_{11}\). Using Eq. (2), the average stresses and strain (\({\-{\sigma }}_{11}\text{,}{ \-{\sigma }}_{22}\text{ and }{\overline{\varepsilon }}_{11}\)) can be obtained to calculate the values of \({\text{C}}_{11}^{\text{eff}}\text{ (= }{\-{\sigma }}_{11}\text{/ }{\overline{\varepsilon }}_{11})\) and \({\text{C}}_{12}^{\text{eff}} (={\-{\sigma }}_{22}\text{ / }{\overline{\varepsilon }}_{11})\). The distributions of stresses and strains in each individual phase of P-IPC are depicted in Fig. 7 when the displacement is applied to the RVE along the \(\text{x}\)-direction. These figures also show the deformations of all respective phases, as demonstrated in Fig. 7(d) with dotted lines.

Distributions of (a, b, c) stresses and (d, e, f) strains in each phase of P-IPC along the x-direction.

To determine \({\text{C}}_{13}^{\text{eff}}\text{ and }{\text{C}}_{33}^{\text{eff}}\), the RVE shown in Fig. 3c must undergo the states of strain wherein only normal strain \({\upvarepsilon }_{33}\) is present and all other strain components are zero. These strains can be achieved by restricting/confining the surfaces of RVE at five boundary surfaces \(({\text{x}}^{+}/{\text{x}}^{-}=0\); \({\text{y}}^{+}/{\text{y}}^{-}=0\); \({\text{z}}^{-}=0)\). A uniform normal displacement (\(\text{w}\)) along the z-direction needs to be applied on the surface (\({\text{z}}^{+}=\text{l}\)) of the RVE such that it is subjected to \({\overline{\varepsilon }}_{33}\) only (while \({\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{22}={\overline{\varepsilon }}_{23}={\overline{\varepsilon }}_{13}={\overline{\varepsilon }}_{12}=0)\). The average stresses and strains (\({\-{\sigma }}_{11}\text{,} \, {\-{\sigma }}_{33},\text{ and }{\overline{\varepsilon }}_{33}\)) can be determined by using Eq. (2). Using Eq. (1), the values of \({\text{C}}_{33}^{\text{eff}}({=\-{\sigma }}_{33} /{\overline{\varepsilon }}_{33})\) and \({\text{C}}_{13}^{\text{eff}}{(=\-{\sigma }}_{11} /{\overline{\varepsilon }}_{33})\) can be determined. The distributions of stresses and strains produced in P-IPC are depicted in Fig. 8 when the displacement is applied to the RVE along the \(\text{y}-\text{ and z}-\) directions.

Distributions of stresses and strains in P-IPC when displacement is applied along (a) \(\text{y}-\) and (b) \(\text{z}-\) direction.

For determining the effective elastic coefficient \({\text{C}}_{44}^{\text{eff}}\), an out-of-plane shear in the \(\text{y}-\text{z}\) plane of the RVE is required to subject it to a pure shear deformation in such a way that the shear strain \({\overline{\varepsilon }}_{23}\) is non-zero, while the remaining strain components are zero (\({\overline{\varepsilon }}_{11}={\overline{\varepsilon }}_{22}={\overline{\varepsilon }}_{22}={\overline{\varepsilon }}_{13}={\overline{\varepsilon }}_{12} =0)\). These strains can be achieved by confining the surface given by \({\text{z}}^{-}=0\) of the RVE and imposing the uniform distributed tangential force on the surface given by \({\text{z}}^{+}=\text{l}\). Consequently, the average shear stress and strain (\({\overline{\upsigma } }_{23}\) and \({\overline{\varepsilon }}_{23})\) induced in the RVE can be calculated using Eq. (2). Finally, \({\text{C}}_{44}^{\text{eff}}\) can be estimated using the relation: \({\overline{\upsigma } }_{23}\text{ / }{\overline{\varepsilon }}_{23}\) while \({\text{C}}_{66}^{\text{eff}}\) is a dependent elastic coefficient which can be calculated using \(\left({\text{C}}_{11}^{\text{eff}}-{\text{C}}_{12}^{\text{eff}}\right)/2\). Figure 9 shows the distributions of stresses, strains, and deformed shapes of RVE considering plane: (a) \(\text{x}-\text{y}\) and (b) \(\text{x}-\text{z}\) and (c) \(\text{y}-\text{z}\). The section view of P-IPC is also presented in the figure to better visualization.

Distributions of stresses, strains and deformed shapes of RVE considering plane: (a) \(x-y\) and (b) \(x-z\) and (c) \(y-z\).

We presented the boundary and loading conditions that can be employed for computing effective properties of any composites with different isotropy and symmetry, such as tetragonal behavior, transversely isotropic, orthotropic, or cubic symmetry. In case of IPCs, the differences in directional properties arise from imperfections in the mesh. Consequently, these various properties are averaged to a single constant to mitigate the impact of such imperfections \(\left(\text{E}={\text{E}}_{\text{x}}={\text{E}}_{\text{y}}={\text{E}}_{\text{z}};\text{G}={\text{G}}_{\text{xy}}={\text{G}}_{\text{xz}}={\text{G}}_{\text{yz}}\text{ and }\mu ={\mu }_{\text{xy}}={\mu }_{\text{xz}}={\mu }_{\text{yz}}\right).\) Therefore, we predicted the effective properties of new metal/polymer-based IPCs and conventional composites. Moreover, we also predicted the effective properties of Al alloy-based G- and P-TPMS as well as C- and O- lattices by varying the volume fraction of Al alloy from 10 to 60%. These architectures can be utilized in lightweight applications to occupy a volume that doesn’t need to meet bulk mechanical specifications. In a complex part, an elementary volume might experience complex loading conditions (such as tensile, shear, etc.) with significant stress and strain gradients. To manage the deformation behavior of the macroscopic part, it’s crucial to tailor the anisotropy of elastic properties at the mesoscopic scale. Thus, the anisotropy of each cubic symmetrical TPMS and lattice is characterized by the Zener ratio (Refer Eq. 3) and illustrated in Fig. 10. The Zener index is unity for an isotropic material \((A=1)\) as \(\text{E}=2\text{G}(1+\upmu )\). Conceptually, it measures how much material deviates from being isotropic. When \(A>1\), \({\text{C}}_{44}^{\text{eff}}\) or \(G\) is predominant, followed by \({\text{C}}_{11}^{\text{eff}}\) and \({\text{C}}_{{1}{\text{2}}}^{\text{eff}}\) (or \(E\)), indicating that the material could be considered "shear friendly." When \(A<1\), \({\text{C}}_{11}^{\text{eff}}\) and \({\text{C}}_{{1}{\text{2}}}^{\text{eff}}\) together dominate over \({\text{C}}_{44}^{\text{eff}}\). This doesn’t directly suggest that the material is “tensile friendly” because of the involvement of two stiffness constants. However, it indicates that the stiffness constant associated to tensile/compressive behavior is more significant than the \({\text{C}}_{44}^{\text{eff}}\) constant.

Zener index of each TPMS and lattice architecture.

As illustrated in Fig. 10, the P-TPMS, C- and O- lattices exhibit high anisotropy, with their Zener indices, particularly at low-volume fractions, deviating significantly from unity. Notably, the Zener index of the G-TPMS remains close to unity across the entire range of volume fractions. From this, one can interpret that G-TPMS nearly behaves like isotropic materials as compared to P-TPMS and O- and C-lattices, while P-TPMS exhibits more anisotropy.

Figures 11 (a and b) demonstrate the change in the effective elastic and shear moduli of G-TPMS and P-TPMS with respect to volume fraction within the range of 10 to 60%. The elastic and shear moduli of TPMS architecture exhibits a nonlinear increase with volume fraction. This behavior arises from the inherent nonlinearity in the elastic response of these structures, attributed to their complex internal geometry. In TPMS architectures, the distribution and connectivity of material and pores are critical in defining their mechanical properties. As the volume fraction increases, the connectivity and interaction between different regions of the material change nonlinearly, affecting the overall stiffness and elastic properties. In Fig. 11a, it is seen that the effective elastic moduli of G-TPMS are greater than those of P-TPMS. This is due to the fact that the G-TPMS has a more complex and interconnected structure compared to the P-TPMS. As discussed in the previous result of the Zener index, G-TPMS may exhibit isotropic mechanical properties while P-TPMS exhibits more anisotropic properties. This means that the material responds more uniformly to stress and strain in different directions, contributing to higher elastic moduli. Therefore, G-TPMS provides more deformation resistance than P-TPMS. Moreover, the interconnected network of G-TPMS results in a more effective load-bearing structure. As we know, the shear modulus describes a material’s resistance to shearing forces. In Fig. 11b, it can be noted that the effective shear moduli are greater for P-TPMS as compared to G-TPMS. This could be due to the complex geometry of G-TPMS architectures, which may allow for more deformation under shear stress (P-TPMS is affected by shear component, i.e., \(A>1\)). Both G- and P-TPMS are porous structures, yet they differ in pore arrangement. G-TPMS architecture generally exhibits a higher degree of pore interconnectivity compared to P-TPMS architecture, which significantly impacts their mechanical properties, including elastic and shear moduli.

Variation of the effective properties (elastic modulus, shear modulus, and Poisson’s ratios) of G-TPMS, P-TPMS, O-lattice, and C-lattices with respect to the volume fractions.

The comparison between the O- and C-lattices for effective elastic and shear moduli with respect to volume fraction is presented in Figs. 11(c and d). In this, we modelled the O- and C-lattices by considering connecting spheres and rounds with specific diameters. From these results, it can be seen that both the effective elastic and shear moduli for O-lattice show greater values for all volume fractions as compared to C-lattice. It is due to fact that O-lattices, with their higher coordination numbers, intricate and interconnected structures as well as increased structural stability, offer enhanced rigidity. This can be advantageous in applications requiring superior load-bearing capabilities, such as structural materials or components subject to high mechanical stress. From Fig. 11(a–d), it can be noticed that G- and P-TPMS architectures often exhibit higher elastic moduli compared to O- and C-lattices due to their unique structural characteristics, such as highly complex and interconnected structure with minimal surface area, which maximizes the material’s stiffness and strength. G- and P-TPMS architectures often feature continuous curvature throughout the structure, which enables efficient load transfer and stress distribution. This continuous curvature minimizes stress concentrations and weak points, contributing to the overall higher modulus of the material. In contrast to TPMS architectures, O- and C-lattices have high symmetry and regularity, with simple repeating unit cells. While this simplicity can facilitate ease of analysis and manufacturing, it also means that the structure lacks the intricate geometry and interconnectedness found in TPMS architectures. As a result, these lattices may not efficiently distribute loads or withstand deformation as effectively as TPMS architectures. Moreover, O- and C-lattices have relatively low surface area per unit volume and more stress concentration zones compared to TPMS architectures. This limited surface area reduces the material’s ability to dissipate energy and withstand external forces, resulting in lower elastic and shear properties. The effective Poisson’s ratio of each TPMS, lattices, and their respective IPCs was also determined. In case of all reinforcing TPMS and lattice architectures, the effective Poisson’s ratios decrease as volume fractions increase, as presented in Fig. 11 (e and f).

Now, we discuss the results for P-IPCs, G-IPCs, C-IPCs, and O-IPCs for their effective properties with volume fraction. All IPCs presented here have different effective properties as each TPMS and lattice architecture behaves distinctively. From Table 4, it can be observed that both the effective elastic and shear moduli are increasing with volume fraction of the reinforcement (10 – 40%) phase. Similar to TPMS and lattice architectures, G-IPCs show greater uniaxial moduli when compared to P-IPCs, C-IPCs, and O-IPCs, whereas P-IPCs show greater shear modulus when compared to G-IPCs, C-IPCs, and O-IPCs. In contrast to elastic properties, in the case of O- and C-IPCs, the effective Poisson’s ratios decrease with an increase in volume fraction of the reinforcement phase, but it appears to be insensitive to variation of volume fraction. Opposite trends are observed in the cases of G- and P-IPCs. First, the effective Poisson’s ratios for both IPCs increased with volume fraction (10 to 20%), but then again, it goes down for 30% and 40% volume fraction. This indicates that the lateral strain increases significantly when compared to the longitudinal strain at a 20% volume fraction. Afterward, it shows the same behavior as per G- and P-TPMS architectures for 30% and 40% volume fractions.

The stress contours for each IPC are depicted in Figs. 7, 8, 9, and 12, illustrating the intensity of stress distribution across reinforcing TPMS/lattices and indicating the relative stress sharing or transfer during loading. It is observed that the stress is most pronounced within the reinforcement, signifying its substantial contribution to the stress response and consequent enhancement of composite stiffness. Notably, P-IPC shows elevated stress areas compared to their complementary parts, indicative of their higher effective uniaxial moduli and the significant contribution of the P-TPMS phase to the stress response. The mechanical response is notably improved by P-IPCs, owing to their support and constraint on the TPMS phase, which leads to increased and more evenly distributed stress levels in the matrix phase. The elastic properties and how the material responds to the stress of the present IPCs are evidently affected by the TPMS architecture phase integrated into the composite.

Distribution of equivalent stresses in IPC and their respective reinforcing and reinforced phases when architecture is loaded along the x-direction.

Examining the stress contours further reveals the high-stress areas in the reinforcing phase that are susceptible to debonding and crack initiation. The noticeable high-stress spots in P-TPMS (Figs. 7–9) suggest potential regions for debonding and cracking if experimental testing is carried out. It is anticipated that surface hot spots may progress towards cracks, whereas internal areas may experience debonding, eventually intersecting with boundaries. However, if cracks were to begin within the matrix, TPMS would impede the propagation of cracks, thereby enhancing damage tolerance. This level of reinforcement interconnectivity and intertwining across the matrix, crucial for achieving such attributes, is effectively achieved by TPMS.

Here, the different types of composites, such as conventional (0–3 and 1–3) and novel architecture-based (3–3) composites, were considered for comparison study. The conventional composites include uniform and random fiber-reinforced (UD) and particle-reinforced composites wherein uniform particle composites are modelled by considering body centered cubic (BCC) structure (Fig. 2). In this, the random UD composites consist of fibers that vary randomly in their orientation (i.e., with specific mean alignment angle of \(2^\circ\) between fiber direction and x-axis) embedded within a matrix material. The results of these different composites with a reinforcement phase of 20% volume fraction were studied and results are listed in Table 5.

It is widely acknowledged that the effective properties of fiber-reinforced composites vary significantly with the direction of loading. This suggests that the elastic modulus of unidirectional fiber-reinforced composite is at its lowest in the transverse direction, aligning with the modulus of the spherical particle-reinforced composite. In contrast, TPMS-IPCs exhibit identical moduli across all directions. Composites constructed with periodic fiber reinforcements prefer to be aligned in a specific orientation to get the desired performance, as the elastic properties in the transverse direction are notably weaker, and thus, lamination is needed. In contrast, IPCs, characterized by their extensive interconnections in three dimensions, offer a relatively stiff response even in the weaker, i.e., transverse direction. Specifically, the transverse direction of IPCs is stiffer than that of conventional composites. Additionally, we anticipate that the interconnections within these IPCs obstruct fracture propagation, thereby enhancing the damage tolerance of the composite. Conversely, fibers within UD composites introduce significant anisotropy and are susceptible to disorientation and debonding, compromising the integrity and alignment essential for intended loading conditions or applications.

Table 5 shows that the uniform and random UD composites show excellent longitudinal effective elastic properties compared to all novel IPCs and remaining conventional composites (uniform and random). It is due to fact that these UD composites are composed of fibers which are reinforced along the longitudinal axis (along 1- or x-direction). It can also be observed that the effective moduli for all IPCs are greater compared to the transverse elastic moduli of conventional composites. While comparing the same with IPCs, G-IPCs show greater uniaxial moduli when compared to P-IPCs, C-IPCs, and O-IPCs. It is seen that the effective shear moduli for all IPCs are greater as compared to conventional composites, thus they can be utilized in structures resisting shear deformation. In the context of IPCs, P-IPCs show greater shear modulus when compared to G-IPCs, C-IPCs and O-IPCs.

Now, the effective properties determined for all types of different composite architectures are utilized to investigate the dynamic/modal analysis of structures considering free vibration and different edge support conditions such as simply supported (SS), clamped-simply supported (CS) and clamped–clamped (CC). The modal analyses, including mode shapes, were carried out as the possibility of resonance can be determined, which is the main reason for failure in certain structures. Modes are the physical characteristics of every mechanical system and are a function of their stiffness (elastic constants), mass or self-weight (in terms of mass density), and edge support conditions. Every single mode can be described using modal frequency, mode shapes/contours, and damping known as “modal parameters.” Therefore, studying the fundamental frequency is vital for stand-alone structures to withstand vibration, such as an earthquake. The spacing between two successive modes is crucial to avoid aeroelastic failure in an aircraft wing. For instance, we considered the squared plate-like structure (as shown in Fig. 13) with length and width (200 mm) and thickness of 10 mm. Figure 13(a) characterizes the wireframe model of the P-IPC plate, Fig. 13(b) shows their top and side view, and Fig. 13(c) depicts the FE results for the first mode. In this, a single block of a square with a circle depicts a single P-IPC from the whole P-IPC plate. The effective properties of all IPCs and conventional composites are used for modal analysis of plate-like structures.

Representation of actual model of P-IPC plate: (a) wireframe model, (b) solid model, and (c) FE result.

First, we considered P-IPC-based plate models with two different modeling techniques, i.e., actual model and homogenized model with the same effective properties with 20% volume fraction. Modal analysis (free vibration) was carried out considering CC edge support conditions and the comparative results are summarized in Table 6 for both models. These results show excellent agreement with less than ~ 2.5% error. From Table 6 and Figs. 14 and 15, it can be observed that the mode shapes and natural frequencies obtained from both models are very similar, demonstrating the reliability of the homogenized model. In Figs. 14 and 15, contour bar plots represent the mode shapes and the corresponding value of displacement amplitudes of the P-IPC-based plate at its natural frequencies, which are normalized with the mass matrix. Contour plots display the mode shapes by illustrating maximum and minimum displacement areas. For example, red color represents areas of maximum displacement, while blue indicates areas with minimal displacement. Researchers/engineers use these contour plots to identify potential issues, such as resonant frequencies, where the structure may experience excessive vibrations. Understanding mode shapes helps in designing structures to avoid resonant frequencies that could lead to failure or damage. The scale on the color bar shows the deformation of P-IPC plates corresponding to different modes. From these deformation scales, It can also be noted that the P-IPC plate exhibits very little deformation as compared to the conventional composite plate. These homogenized models can be very useful for minimizing the computational cost. Therefore, the homogenized model is used to compare IPC-based and conventional composite plates when subjected to different edge support conditions. Here, the volume fraction of all reinforcing phases is 20%.

First six modes of actual P-IPC plate.

First six modes of homogenized P-IPC plate.

In Fig. 16, the colored bar plot is depicted to see a change in natural frequency with respect to different modes. From results, it is noted that the natural frequency of IPC-based plates is higher than that of conventional composite plates, illustrated in Fig. 16(a-e). This is obviously because the stiffness of IPC-based plates is higher than conventional composites. Similarly, the natural frequencies of plates with CC support conditions are higher than those of SS and CS supports. In case of a conventional composite plate, we considered CS support conditions in two different ways: (i) clamped in the x-direction and simply supported in the y-direction of the plate, denoted by C-x/S-y and (ii) C-y/S-x. From Fig. 17, it can be observed how support condition with respect to direction plays a significant role in the natural/resonant frequency of the plate. The effect of directional support is observed to be insensitive in the case of uniform and random particles as compared to uniform and random UD composites. It is due to the fact that in the case of UD composites, fibers are oriented in specific directions, i.e., either in x- or y- direction. Moreover, these results are significantly affected by the stiffness of composites in specific directions. Therefore, in the case of uniform and random UD composites, the natural frequency considering C-x/S-y conditions is greater than C-y/S-x. This is because the stiffness in C-x/S-y is more than in C-y/S-x. From Fig. 17, it is clearly observed that there is no significant difference between mode 3 and mode 4 of uniform UD (C-x/S-y) and uniform UD (C-y/S-x). The natural frequency at clamped support condition is higher than simply supported conditions which is obvious reason for higher frequencies in respective structures. Besides this, there are distinct reasons for the changing natural frequencies for different structures at different modes such as:

Clamping along the x-axis provides a higher initial stiffness, and even though the fibers are not aligned along this direction, the boundary condition alone supports higher resistance against bending in the x-direction. This boosts the first and second natural frequencies in C-x/S-y relative to C-y/S-x.

In C-y/S-x, where the clamped direction aligns with the fiber orientation along y, the stiffness in the x-direction (simply supported) does not restrict bending as much in the lower modes, leading to lower first and second frequencies.

For higher modes, such as the third and fourth, the effect of fiber orientation becomes more pronounced as the mode shapes start involving more nodal patterns and bending along both x- and y- directions (i.e., potentially coupled twisting motions).

In C-x/S-y, because the fibers are aligned along the simply supported y-axis, the plate has reduced stiffness in the x-direction for these higher modes, resulting in lower frequencies for modes 3 and 4.

Modal analysis of homogenized IPC and conventional composite plate considering different support conditions: (a and b) CC, (c and d) SS, and (e) CS.

Modal analysis of conventional composite plate for CS conditions.

In summary, it can be noticed that the natural frequencies of plates are affected by edge support conditions, bending or twisting or mixed modes as well as fiber orientations.

The above comparative result shows that these architected composite plates show enhanced natural frequency when compared with conventional composite plates. Thus, it can be noticed that these overall results aim to establish benchmarks for designing and analyzing metal-polymer IPCs with various reinforcement topologies and base material combinations, extending beyond those studied in this research.

This article introduced metal/polymer-based interpenetrating phase composites (IPCs) with different TPMS and lattice architectures using finite element (FE) modeling, offering a new research direction to explore their properties and applications. TPMS and lattice structures were analyzed to develop IPCs and assess their effective properties by employing the periodic boundary conditions. These effective properties were then utilized to study free vibration analysis of different composite structures considering different edge support conditions. The proposed IPCs demonstrate enhanced robustness and superiority compared to conventional composites. The effective elastic and shear moduli of TPMS, lattices, and their IPCs increase with volume fractions, while Poisson’s ratio decreases. G-TPMS exhibits higher elastic moduli than P-TPMS, and both outperform O- and C-lattices due to their complex and interconnected structures. G-TPMS shows nearly isotropic behavior, while P-TPMS is more anisotropic according to the Zener index. These architectured IPCs offer balanced stiffness in all directions compared to conventional composites. Moreover, modal analysis reveals that TPMS/lattice-based IPCs have higher natural frequencies than conventional composites, regardless of edge support, and P-IPC plates exhibit significantly less deformation. The results obtained from homogenized and actual models are found to be in good coherence, while the former model can be useful for different analyses, which saves more computational time.

This research provides insights and explanations for developing advanced architected composite structures in materials science and engineering, paving the way for future innovative design strategies for high-performance structures. This novel class of composites presented in this article will introduce innovative design pathways for developing high-performance fiber network composites. Thus, the continuing research in this area persistently explores new applications and seeks to improve the performance of these materials for practical, real-life situations. Characterizing the mechanical behavior of these architected materials and their innovative structures will be an active area of research aimed at enhancing their use in commercial applications such as mechanical, aerospace, automotive, thermal, biomedical, electronics, chemical, acoustic, energy storage and optical domains. Future endeavors will prioritize experimental validation of the reported findings, focusing on fabricating the presented metal/polymer TPMS-IPCs through additive manufacturing techniques like 3D/4D printing. Additionally, there will be a particular emphasis on exploring TPMS-based foams.

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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KBS, AS, and KL acknowledge funding and support by the Directed Research Projects Program (Lightweight and Performance Enhancing Materials: Multifunctional Materials—8434000508) of the Research and Innovation Center for graphene and 2D materials (RIC2D) of Khalifa University.

Department of Aerospace Engineering, Khalifa University of Science and Technology, 127788, Abu Dhabi, UAE

K. B. Shingare & Kin Liao

Department of Mechanical Engineering, Khalifa University of Science and Technology, 127788, Abu Dhabi, UAE

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K.S. wrote the main manuscript and performed validation, carried out simulations on software, and A.S. and K.L. supervised, provided resources, project administration, and edited the manuscript.

Correspondence to Kin Liao.

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Shingare, K.B., Schiffer, A. & Liao, K. On quantifying dynamic behavior of architected metal/polymer TPMS/lattices-based interpenetrating phase composites. Sci Rep 15, 4253 (2025). https://doi.org/10.1038/s41598-024-84303-5

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DOI: https://doi.org/10.1038/s41598-024-84303-5

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