Modeling and Analysis of a Thermal Expansion and Poisson’s Ratio Integrated Tunable Metamaterial Structure

Abstract

The tunable coefficient of thermal expansion (CTE) and Poisson’s ratio (PR) properties of metamaterials help address issues caused by drastic temperature variations and external loads. In this work, we propose a novel bimaterial thermal expansion and PR integrated tunable 2D metamaterial structure. Under certain parameter constraints, the structure based on an Al alloy/low carbon steel (LCS) combination demonstrates a wide tunability, with the CTE ranging from −47 to 28 ppm/°C and the PR varying from −14.8 to 7.3. A general thermoelastic equation is adopted to establish the relationship between temperature, external force, and displacement, which is then assembled into a theoretical model. Through theoretical analysis and numerical simulations, the underlying mechanisms of the proposed 2D metamaterial structure’s CTE, PR, and their relationship with geometric parameters and elastic modulus ratios are revealed. CTE and PR experiments are conducted to validate the theoretical modeling. Finally, the coupling relationship between CTE and PR is revealed.

1. Introduction

Mechanical metamaterials are materials that acquire unconventional properties through the deliberate design of microstructures [1,2]. Since their ability to achieve negative CTE [3,4,5,6,7,8], negative Poisson’s ratio(NPR) [9,10,11,12,13], and complex spatial deformations [14,15] that are difficult to achieve with conventional materials, mechanical metamaterials have attracted increasing interest in recent years.

Different microstructure designs endow mechanical metamaterials with diverse mechanical properties. Among the various performance indices, CTE and PR are of particular importance. Mechanical metamaterials with tunable CTE can effectively regulate structural deformation induced by temperature changes, and thus exhibit significant application potential in aerospace engineering, precision instruments, and electronic packaging. Mechanical metamaterials with NPR tend to contract toward the loading or impact region under external loads, thereby exhibiting energy absorption and cushioning capabilities.

Thermal expansion metamaterials can be categorized into tensile-dominated and bending-dominated types based on their deformation characteristics. Tensile-dominated thermal expansion metamaterials are typically developed from the bimaterial triangular model proposed by Miller [16,17,18,19]. Bending-dominated thermal expansion metamaterials are commonly derived from the bimaterial beam model proposed by Timoshenko [20,21,22,23,24]. NPR metamaterials are classified into re-entrant, perforated, and chiral types based on their deformation mechanisms. Typical re-entrant structures include re-entrant hexagonal [25] and re-entrant triangular configurations [26,27], in which NPR is associated with geometric rotation and unfolding of the unit cells when an external load is applied. Perforated structures exhibit unconventional deformation through the introduction of holes with specific shapes and distributions [28,29,30]. Chiral structures exhibit NPR due to geometric coupling among rotating units [31,32].

In practical engineering applications, structures are commonly subjected to the combined effects of thermal and mechanical loads. To meet the requirements of multi-field coupled operating conditions and to improve the overall material performance, integrating multiple mechanical properties within a single material has been considered an effective approach. A commonly adopted design strategy is to combine tensile-dominated thermal expansion with re-entrant structures [33,34,35,36,37]. For example, Ai et al. [38,39] designed several mechanical metamaterials with simultaneously tunable CTE and PR based on bimaterial star-shaped re-entrant structures. Wei et al. [11,40] combined bimaterial triangular units with re-entrant triangular configurations to develop a series of 2D and 3D metamaterials. In addition, bimaterial curved beams have been introduced into chiral and anti-chiral structures to achieve integrated tunability of CTE and PR [41,42,43,44,45].

Considering the pros and cons of tensile-dominated and bending-dominated metamaterials, it is preferable to make modifications to tensile-dominated structures to achieve an extended tunable range of CTE and PR. In this work, tensile-dominated thermal expansion metamaterials and re-entrant NPR metamaterials are combined within a modified bimaterial rhombic structure incorporating a symmetric arrow-shaped configuration, resulting in a novel two-dimensional metamaterial with tunable CTE and PR. The remainder of this paper is organized as follows: Section 2 introduces the architectural design of the proposed metamaterial and establishes the corresponding theoretical and finite element models. Section 3 systematically investigates the effects of geometric parameters and material combinations on CTE and PR. Section 4 summarizes the main findings of this study.

2. Model and Method

2.1. Structure Introduction

The tensile-dominated bimaterial triangular model proposed by Miller et al. [16] is shown in Figure 1a. In the isosceles triangle structure, the base beam is made of a material with a high CTE, while the side beams are made of a material with a low CTE. With the material combination determined, the CTE of this structure depends on two geometric parameters: the base beam length D and the angle θ₁ between the base beam and the side beams. By mirroring two bimaterial triangles along the base beam, a bimaterial rhombus structure is obtained. The regulatory range of CTE is limited, and it cannot regulate PR in such a structure. Accordingly, a new metamaterial unit cell structure is proposed in this work, as shown in Figure 1b. Starting from a bimaterial rhombic structure, a longitudinal tensile deformation is applied, thereby transforming the rhombus into a scissor-like configuration. Based on this scissor mechanism, a symmetric arrow-shaped structure is subsequently integrated, resulting in a unit cell that preserves the original structural symmetry [46]. Compared to the bimaterial rhombus structure, the proposed metamaterial structure has more tunable geometric parameters and enables the regulation of both CTE and PR from negative to positive.

The proposed metamaterial fully exploits structural symmetry in its design, modeling, and deformation behavior. In the design stage, a symmetric arrow-shaped configuration is integrated into the modified bimaterial rhombic structure, ensuring geometric symmetry of the unit cell. This symmetry not only preserves uniform mechanical response but also enables coordinated regulation of both CTE and PR. In the modeling stage, the structural symmetry allows the unit cell to be simplified into a quarter model, significantly reducing computational cost while maintaining accuracy in both theoretical analysis and finite element simulations. In terms of deformation behavior, the symmetric arrangement ensures that thermal expansion and mechanical loading induce coordinated and uniform deformation patterns, avoiding asymmetric distortions and enabling predictable tuning of macroscopic properties.

As shown in Figure 1b, the structure has four geometric parameters, including three angular parameters θ₁, θ₂, θ₃ and the length parameter D. The CTE, elastic modulus, and density of the blue beams are represented by α_L, E_L, and ρ_L, respectively, while the CTE, elastic modulus, and density of the red beams are represented by α_H, E_H, and ρ_H, respectively. All beams have a rectangular cross-section of 2 mm × 3 mm. Based on the proposed structure, a planar metamaterial, as shown in Figure 1c, can be constructed.

2.2. Modeling Theory

In this paper, the stiffness matrix method is used to analyze the equivalent CTE and PR of the metamaterial unit cell structure. Based on the symmetrical characteristics of the overall unit cell structure, the equivalent representation of a quarter of the structure is used to simplify the modeling process. As shown in Figure 2d, constraints are applied so that the deformation of the equivalent structure matches that of the overall unit cell structure. The specific constraint conditions are as follows: the displacement of point 1 in the X direction is restricted to zero, the displacement of point 3 in both the X and Y directions is restricted to zero, the displacement of points 4 and 5 in the Y direction is restricted to zero, and the rotations at points 1, 3, 4, and 5 are also restricted to zero.

The theoretical modeling process follows the methodology presented in previous work [15], and the general thermoelastic equation for the unit cell structure is:

$$\left\{\begin{array}{c}{P}_{\overline{{\mathsf{\nu }}_1}}\\ P_{\overline{{\mathsf{\mu }}_2}}\\ P_{\overline{{\mathsf{\nu }}_2}}\\ M_{\overline{{\mathsf{\phi }}_2}}\\ P_{\overline{{\mathsf{\mu }}_4}}\\ P_{\overline{{\mathsf{\mu }}_5}}\end{array}\right\}=\left[\begin{array}{cccccc}{K}_{11}& K_{12}& K_{13}& K_{14}& K_{15}& K_{16}\\ K_{21}& K_{22}& K_{23}& K_{24}& K_{25}& K_{26}\\ K_{31}& K_{32}& K_{33}& K_{34}& K_{35}& K_{36}\\ K_{41}& K_{42}& K_{43}& K_{44}& K_{45}& K_{46}\\ K_{51}& K_{52}& K_{53}& K_{54}& K_{55}& K_{56}\\ K_{61}& K_{62}& K_{63}& K_{64}& K_{65}& K_{66}\end{array}\right]\left\{\begin{array}{c}\overline{{\nu }_1}\\ \overline{{\mu }_2}\\ \overline{{\nu }_2}\\ \overline{{\phi }_2}\\ \overline{{\mu }_4}\\ \overline{{\mu }_5}\end{array}\right\}-\left\{\begin{array}{c}{T}_1\\ T_2\\ T_3\\ T_4\\ T_5\\ T_6\end{array}\right\}$$

(1)

$$\alpha ={\displaystyle \frac{\overline{{\nu }_1}}{L_{\mathrm{y}}\mathrm{\Delta }T}}$$

(2)

$$\nu =-{\displaystyle \frac{{\epsilon }_{\mathrm{y}}}{{\epsilon }_{\mathrm{x}}}}=-{\displaystyle \frac{\overline{{\nu }_1}/L_{\mathrm{y}}}{\overline{{\mu }_5}/L_{\mathrm{x}}}}$$

(3)

Equation (1) simultaneously considers both external loads and temperature variations, and the specific expressions of K₁₁–K₆₆ are shown in Appendix A. The equivalent CTE of the structure is calculated using Equations (1) and (2), while the equivalent PR is calculated using Equations (1) and (3). Here, Ly represents the equivalent length of the structure in the Y direction, and Lx represents the equivalent length of the structure in the X direction.

2.3. Model Verification

To verify the correctness of the theoretical model presented in Section 2.2 for the 2D metamaterial unit cell, finite element simulations are performed using ABAQUS. In the theoretical model, only a quarter of the unit cell structure is analyzed to simplify the modeling process. In contrast to the theoretical model, the full 2D metamaterial unit cell is modeled in the finite element simulations. The 2D metamaterial model is created in the Part module. In the Property module, the material properties of the beams are defined, along with the cross-sectional dimensions and beam orientations. The cross-sectional width of the middle beam (Beam④ in Figure 2d) is twice that of the other beams, with the middle beam’s cross-section being (2 mm × 6 mm) and the others being (2 mm × 3 mm). In the Mesh module, the model is meshed with a global size of 1 mm, and the element type is set to B22.

Table 1. Physical properties of four materials.

Material	Elastic Modulus (GPa)	CTE (ppm/°C)	Density (g/cm3)
Nylon	2.90	72	1.16
Invar	148	1.3	8.05
Al alloy	70.3	23.8	2.68
LCS	202	12	7.85

lists the physical properties of the four materials used in this study, assuming that these material properties are independent of temperature variations. To facilitate manufacturing and to mitigate the local buckling effect in the beams, the geometric parameters of the 2D metamaterial unit cell structure are subject to the following range limitations: θ₁ ∈ (20°, 60°), θ₂ ∈ (20°, 160°), θ₃ ∈ (10°, 60°), D ∈ (100 mm, 500 mm), and θ₂ − θ₃ > 5°.

Table 2. Property ratios of material combinations.

Material Category	Elastic Modulus Ratio	CTE Ratio	Density Ratio
Al alloy/LCS	0.348	1.98	0.341
Nylon/LCS	0.0144	6	0.148
Al alloy/Invar	0.475	18.31	0.333
A/B	0.348	6	0.148

provides the material combinations and their respective physical property ratios. The A/B combination represents a hypothetical material combination used for subsequent mechanical performance analysis.

2.3.1. Mesh Convergence Analysis

To ensure that the numerical results are independent of mesh size, a mesh convergence analysis is conducted. A representative metamaterial model is selected, and five different global seed sizes, namely 0.5 mm, 1 mm, 2 mm, 4 mm, and 8 mm, are employed to generate meshes with increasing refinement. Figure 3 shows the distribution of grid seeds under different mesh seed sizes for the structure. For each mesh density, the CTE is calculated under the same boundary and loading conditions. The results are summarized in

Table 3. Mesh convergence study of CTE.

Global Seed Size (mm)	Number of Elements	Simulation CTE (ppm/°C)	Change in CTE (%)
0.5	1184	47.312841716442819	-
1	940	47.312841716442819	0.0000
2	472	47.311729446000847	0.0024
4	234	47.311729446000847	0.0000
8	118	47.289184791079059	0.0476

. It can be observed that the CTE gradually converges as the mesh is refined. For the finest meshes, the differences between successive refinements become negligibly small.

These results indicate that further mesh refinement has an insignificant effect on the predicted CTE, demonstrating that the numerical solutions are mesh-independent. Considering both computational efficiency and accuracy, a global seed size of 1 mm is adopted in all subsequent simulations.

2.3.2. CTE Simulation

To simulate the thermal expansion process of the 2D metamaterial unit cell structure, a temperature field uniformly increasing from 20 °C to 60 °C is applied in the Load module. The boundary conditions for the 2D metamaterial unit cell structure are set as follows: the lower endpoint of the structure is fixed, and the upper endpoint can only move along the Y direction. Using an Al alloy/LCS material combination as an example, when θ₁ = 20° and D = 300 mm, the variation curve of the relative CTE (α/α_H) with respect to θ₂ and θ₃ is shown in Figure 4a. It can be observed that: (i) As θ₂ increases, α/α_H first decreases and then increases; (ii) As θ₃ increases, the range of α/α_H in the negative region decreases, while the range in the positive region increases; (iii) As θ₃ increases, the point of minimum α/α_H shifts to the left. As shown in Figure 4a, three representative size combinations are selected for CTE simulations. The corresponding results in Figure 4b–d demonstrate negative, positive, and quasi-zero thermal expansion, respectively. The black dashed line in the simulation figures represents the initial state at 20 °C, and the colored lines represent the deformed state at 60 °C.

Table 4. Comparison of theoretical and simulated CTE.

Thermal Expansion Type	CTE Simulation Results (ppm/°C)	CTE Theoretical Results (ppm/°C)	$\mathbf{Error} \left|\frac{\left|{\mathit{S}}_{\mathbf{CTE}}-{\mathit{T}}_{\mathbf{CTE}}\right|}{{\mathit{T}}_{\mathbf{CTE}}}\right|$ (%)
Negative thermal expansion	−47.4	−47.3	0.211
Positive thermal expansion	25.4	25.5	0.392
Quasi-zero thermal expansion	−0.240	−0.247	2.834

provides the theoretical and simulation results for the three structures along with their relative errors, where S_CTE represents the simulation result for CTE and T_CTE represents the theoretical result. It can be observed that the relative errors for positive and negative thermal expansion are negligible, while the relative error for the quasi-zero thermal expansion is higher. Since the denominator in the error index (T_CTE) is very small, even a small difference between S_CTE and T_CTE leads to a high relative error. The simulation results validate the correctness of the theoretical model.

2.3.3. PR Simulation

To simulate the PR characteristics of the 2D metamaterial unit cell structure, a horizontal displacement of 1 mm in the positive X-direction is applied at the right endpoint of the structure in the Load module. The boundary conditions for the 2D metamaterial unit cell structure are set as follows: the left endpoint of the structure is fixed. Using an Al alloy/LCS material combination as an example, when θ₁ = 20° and D = 300 mm, the variation curve of PR with respect to θ₂ and θ₃ is shown in Figure 5a.

It can be observed that: (i) As θ₂ increases, PR first decreases and then increases, (ii) As θ₃ increases, the adjustable range of PR decreases, (iii) As θ₃ increases, the point of minimum PR shifts to the right. As shown in Figure 5a, three size combinations are selected for PR simulation, and the simulation results are shown in Figure 5b–d, which correspond to NPR, positive PR, and quasi-zero PR, respectively. The black dashed line in the simulation figures represents the initial state before the displacement of the right endpoint, and the colored lines represent the deformed state after the displacement.

Table 5. Comparison of theoretical and simulated PR.

PR Type	PR Simulation Results	PR Theoretical Results	$\mathbf{Error} \left|\frac{\left|{\mathit{S}}_{\mathbf{PR}}-{\mathit{T}}_{\mathbf{PR}}\right|}{{\mathit{T}}_{\mathbf{PR}}}\right|$ (%)
NPR	−10.407	−10.384	0.221
Positive PR	5.671	5.671	0
Quasi-zero PR	0.045	−0.036	25

provides the theoretical and simulation results for the three structures along with their relative errors, where S_PR represents the simulation result for PR and T_PR represents the theoretical result. It can be observed that the relative errors for both positive PR and NPR are very small and negligible, while the relative error for the quasi-zero PR is higher. Since the denominator in the error index (T_PR) is very small, even a small difference between S_PR and T_PR leads to a high relative error. The simulation results validate the correctness of the theoretical model.

In this study, geometric nonlinearity is not considered, and the analysis is conducted under the assumption of small deformation. However, under large deformation conditions, geometric nonlinearity may influence Poisson’s ratio, leading to deviations from the linear prediction.

2.3.4. Elastic Modulus Simulation

To further verify the correctness of the theoretical model for the 2D metamaterial unit cell structure when an external load is applied, load-deformation simulations are conducted. Figure 6 shows the simulation results for a planar 2 × 3 periodic array structure. Symmetric boundary conditions are applied in the Load module, as indicated by the black dashed lines. In the Interaction module, surface-to-surface contact is established between the rigid line and the array structure, and the displacement of the rigid line is set to 10 mm. The array structure is then compressed using vertical and horizontal rigid lines (represented by the red lines in Figure 6). From the legend on the left side of the simulation results, the reaction forces Q₁ (Figure 6a) and R₁ (Figure 6b) on the rigid lines can be obtained. The obtained reaction forces Q₁ and R₁ are substituted into Equations (4) and (5) to calculate the elastic modulus along the X and Y axes, respectively. The theoretical and simulation results for the elastic modulus are listed in

Table 6. Comparison of theoretical and simulated elastic modulus.

Direction of Extrusion	Elastic Modulus Simulation Results (MPa)	Elastic Modulus Theoretical Results (MPa)	Error (%)
X direction (EX)	262.5	264.1	0.606
Y direction (EY)	23.4	23.5	0.426

, showing good agreement.

$$E_{\mathrm{x}}={\displaystyle \frac{Q_1/A_{\mathrm{y}}}{\overline{{\mu }_5}/L_{\mathrm{x}}}}$$

(4)

$$E_{\mathrm{y}}={\displaystyle \frac{R_1/A_{\mathrm{x}}}{\overline{{\nu }_1}/L_{\mathrm{y}}}}$$

(5)

2.3.5. Stress Concentration Simulation

For mechanical metamaterials with porous structures, stress concentration is often unavoidable and frequently occurs at the junctions between different beams. Therefore, a finite element simulation of the Mises stress is performed for the proposed 2D metamaterial unit cell structure to evaluate this phenomenon. In the Load module, boundary conditions consistent with those used in the thermal expansion simulation are applied. To make the Mises stress more evident, a 0.5 mm displacement is applied to the left endpoint towards the left, and a 0.5 mm displacement is applied to the right endpoint towards the right. Additionally, a temperature change from 20 °C to 200 °C is simulated. The Mises stress results for NPR, positive PR, and quasi-zero PR are shown in Figure 7. The maximum stress is observed to occur at the connection regions of the structure, and in some cases, the maximum stress may exceed the yield strength of the constituent materials (Al alloy: 231 MPa; LCS: 235 MPa). Therefore, in practical applications, stress mitigation strategies such as introducing fillets and applying surface strengthening treatments should be adopted to reduce stress concentration and prevent structural failure.

2.4. Experimental Verification

2.4.1. Experimental Verification of CTE

The CTE experimental study aims to explore the thermal expansion characteristics of the unit cell structure. The experimental setup is shown in Figure 8. It mainly consists of a temperature-controlled heating platform with an emissivity of 0.97, which uniformly heats the sample and monitors temperature changes. The sample is placed on the heating platform and covered with a 6 mm thick quartz plate to reduce thermal convection between the heating platform and the camera. The quartz plate is supported by four black spacers, each 4 mm thick, ensuring no direct contact between the plate and the sample. The image acquisition uses a CMOS camera (Hikvision, Hangzhou, China, 2448 × 2048 pixels) equipped with a 16 mm fixed lens, two LED lights, and a computer. The camera and LED lights are mounted on a vertical bracket, and real-time image capture is achieved through a USB 3.0 interface. The temperature at the contact points of the sample is measured using a thermocouple thermometer (HT9815). Additionally, an infrared thermal camera (HKMICRO, H21pro, Beijing, China) is used to measure the surface temperature distribution of the sample. In the thermal image, the highest temperature is shown to be 86.6 °C and the lowest temperature is 41.6 °C. The thermal displacement before and after heating the sample is analyzed using Ncorr program. Ncorr is an open-source Digital Image Correlation (DIC) tool based on MATLAB (2021a) code, which is used for precise analysis of thermal deformation and thermal expansion characteristics.

The manufactured experimental sample is composed of an aluminum alloy and low-carbon steel combination, which theoretically exhibits negative thermal expansion characteristics. The geometric parameters of the sample are as follows: θ₁ = 30°, θ₂ = 60°, θ₃ = 15°, D = 300 mm, and the cross-sectional dimensions are 3 mm × 3 mm. The sample consists of two parts: the outer main structure and the central component. The outer main structure is fabricated from a low CTE material (LCS), while the central component is made of a high CTE material (Al alloy). The experimental specimen is manufactured using a DK7735 wire electrical discharge machining (EDM) system, and the two components are assembled via dovetail grooves. During the experiments, CTEs of Al alloy and LCS calibration specimens (80 mm × 8 mm × 3 mm) are measured to determine the CTE of the constituent materials. After the metamaterial specimen is heated to the target temperature, images are captured and processed using the Ncorr software to evaluate the effective CTE of the structure. To validate the accuracy of the experimental system, a 6061 aluminum plate with dimensions of 80 mm × 80 mm × 3 mm and CTE of 23.6 ppm/°C in the temperature range of 20–100 °C is also tested. The measured CTE is 23.52 ppm/°C, corresponding to a deviation of 0.34% from the nominal value, which demonstrates the reliability of the experimental setup. Similarly, the measured linear CTEs of the Al alloy and LCS specimens are 33.8 ppm/°C and 20.1 ppm/°C, respectively. These experimentally obtained CTE values are subsequently adopted in the theoretical analysis and finite element simulations.

In the experiments, seven independent tests are conducted, and the average value is taken as the final experimental result. The experimental results are summarized in

Table 7. CTE measurement experiment results and error analysis.

Test No	Test CTE (ppm/°C)	Theoretical Error (%)	Simulation Error (%)
Test1	−5.56	25.4	14.6
Test2	−5.95	20.1	8.6
Test3	−7.12	4.43	9.37
Test4	−6.62	11.1	1.69
Test5	−7.06	5.23	8.45
Test6	−5.68	23.8	12.7
Test7	−6.32	15.2	2.92
Mean	−6.33	-	-
Std. Dev.	0.56	-	-

. And the relative errors between the theoretical results and the simulation results of CTE are shown in Figure 9a. The experimental CTE of the unit cell structure is −6.33 ± 0.56 ppm/°C (mean ± standard deviation, n = 7). For comparison, the corresponding theoretical and simulation results are −7.45 ppm/°C and −6.51 ppm/°C, respectively. The maximum deviation of the experimental results from the theoretical prediction is 25.4%, while the maximum deviation from the simulation result is 14.6%. The relative errors between the average experimental result and the theoretical and simulation results are 15% and 2.76%, respectively. Considering possible gaps between the central and peripheral components, manufacturing tolerances, and uncertainties in temperature measurement, these discrepancies are within an acceptable range. Overall, the experimental results show good agreement with both the theoretical and numerical predictions, confirming the validity of the theoretical model and demonstrating that the proposed unit cell structure exhibits negative thermal expansion behavior.

2.4.2. Experimental Verification of PR

The PR experimental study aims to explore the PR characteristics of the unit cell structure. Since the effect of material on PR is minimal, the PR experimental samples are manufactured using 3D printing technology to evaluate their in-plane tensile behavior. The PR experimental sample parameters are as follows: θ₁ = 30°, θ₂ = 50°, θ₃ = 30°, D = 100 mm and the cross-sectional dimensions are 1.2 mm × 3 mm. The experimental setup is shown in Figure 10. The tensile test is conducted using a universal materials testing machine (Suzhou Qiantong Instruments, Suzhou, China, QT-6203A), with the longitudinal deformation of the structure controlled directly by the testing machine, and the lateral deformation measured and recorded using a vernier caliper (SYNTEK, Shanghai, China, JS20-GTG).

In the experiments, seven independent tests are conducted, and the average value is taken as the final experimental result. The experimental results are summarized in

Table 8. PR measurement experiment results and error analysis.

Test No	Test PR	Theoretical Error (%)	Simulation Error (%)
Test1	−5.473	17.77	1.99
Test2	−4.868	4.76	12.82
Test3	−5.830	25.46	4.41
Test4	−5.000	7.60	10.46
Test5	−4.757	2.37	14.81
Test6	−4.780	2.86	14.40
Test7	−4.994	7.47	10.57
Mean	−5.100	-	-
Std. Dev.	0.385	-	-

. And the relative errors between the theoretical results and the simulation results of CTE are shown in Figure 9b. The experimental PR of the unit cell structure is −5.100 ± 0.385 (mean ± standard deviation, n = 7). For comparison, the corresponding theoretical and simulation results are −4.647 and −5.584, respectively. The maximum deviation between the experimental and theoretical results is 14.81%, while the maximum deviation between the experimental and simulation results is 25.46%. The relative errors between the average experimental result and the theoretical and simulation results are 8.67% and 9.75%, respectively. Considering manufacturing tolerances and possible uncertainties in deformation measurement, these discrepancies are within an acceptable range. Overall, the experimental and numerical results validate the theoretical analysis and demonstrate that the proposed unit cell structure exhibits a negative PR.

2.4.3. Uncertainty Analysis and Error Mitigation

A brief uncertainty analysis is conducted to evaluate the reliability of the experimental results for both CTE and PR measurements. The primary sources of uncertainty include temperature measurement, displacement/strain measurement, and manufacturing tolerances. In the CTE experiments, the thermocouple (HT9815) introduces temperature measurement uncertainty, while the infrared thermal camera may also contribute to deviations due to emissivity assumptions and surface reflections. The displacement measurement based on DIC using Ncorr is affected by image resolution (2448 × 2048 pixels), speckle quality, and noise, which may influence the accuracy of strain calculation. In addition, small assembly gaps between the aluminum alloy and LCS components, as well as machining tolerances from wire EDM, can lead to geometric deviations and affect the measured effective CTE.

For the PR experiments, uncertainties mainly arise from deformation measurement using a vernier caliper, which has limited resolution compared to full-field optical methods such as DIC, and from fabrication inaccuracies associated with 3D printing. These factors contribute to the observed deviations between experimental, theoretical, and simulation results.

To reduce these uncertainties in future work, higher-precision temperature sensors and improved thermal calibration methods can be employed. The use of higher-resolution cameras and the adoption of full-field DIC techniques for PR experiments would significantly improve displacement measurement accuracy. Moreover, adopting precision manufacturing methods with tighter tolerances and improving assembly techniques to eliminate interfacial gaps would further improve the consistency between experimental and theoretical results.

3. Results and Discussion

3.1. Relative CTE Analysis

3.1.1. Influence of Geometric Parameters

Based on the theoretical model and simulation results in Section 2, the CTE of the proposed 2D metamaterial unit cell structure can be adjusted by changing the geometric parameters. Therefore, the material combinations listed in Table 2 are selected to study the relationship between α/α_H and the geometric parameters.

When θ₂ = 20°, θ₃ = 15° and D = 300 mm, the relationship between α/α_H and θ₁ for different material combinations is shown in Figure 11a. It can be observed that α/α_H for all three combinations starts with negative values. The α/α_H for all three combinations monotonically increases as θ₁ increases. The α/α_H of the Al alloy/LCS combination changes from negative to positive, while α/α_H for the other two combinations remains negative. It is hypothesized that the special behavior of the Al alloy/LCS combination curve is related to its small CTE ratio (α_H/α_L). To verify this hypothesis, a hypothetical material combination A/B listed in Table 2 is used. The elastic modulus ratio of this combination is defined as the same as that of the Al alloy/LCS combination, while the CTE ratio and density ratio are the same as those of the Nylon/LCS combination. The α/α_H for the A/B combination is shown by the black dashed line, and its trend is similar to that of the pink dashed line and blue dashed line, confirming the hypothesis.

When θ₁ = 30°, θ₃ = 15° and D = 300 mm, the relationship between α/α_H and θ₂ is shown in Figure 11b. It can be observed that the curves for all three material combinations decrease monotonically in the initial part, with the exception that the α/α_H curve for the Nylon/LCS combination shows a significant decrease towards the end, while the α/α_H curves for the other two combinations show a slight increase in the latter part. It is hypothesized that the special behavior of the Nylon/LCS combination curve is due to its small elastic modulus ratio (E_H/E_L). The α/α_H for the A/B combination is shown by the black dashed line, and its trend is similar to that of the red solid line and blue dashed line, confirming the hypothesis.

When θ₁ = 30°, θ₂ = 90° and D = 300 mm, the relationship between α/α_H and θ₃ is shown in Figure 11c. As θ₃ increases, α/α_H for the Nylon/LCS combination increases monotonically, while CTEs for the other two combinations increase initially and then decrease. The reason is again that E_H/E_L for the Nylon/LCS combination is too small.

When θ₁ = 30°, θ₂ = 20° and θ₃ = 15°, the relationship between α/α_H and D is shown in Figure 11d. It can be observed that as increases, α/α_H for all three combinations decreases monotonically. Since the small E_H/E_L for the Nylon/LCS combination, its α/α_H shows the largest decrease.

3.1.2. Influence of Material Physical Properties

Based on the results shown in Figure 11, it can be observed that the physical properties of the constituent materials also have an impact on α/αH. The geometric parameters are fixed at θ₁ = 30°, θ₂ = 20°, θ₃ = 15°, D = 300 mm. This allows the influence of the physical properties of the constituent materials on α/α_H to be investigated without the interference of geometric variations. The relationship between α/α_H, α_H/α_L and E_H/E_L is shown in Figure 12a and Figure 12b, respectively. It can be observed that α/α_H monotonically decreases as the value of α_H/α_L increases. Excluding the line for α_H/α_L = 1, the other four curves in Figure 12b decrease monotonically. When α_H/α_L = 1, it indicates that the two constituent materials have the same CTE. In this case, during the heating process, the structural components expand proportionally without bending deformation, effectively resulting in an isometric scaling that keeps α/α_H unchanged. When the geometric parameters are fixed, a smaller α/α_H can be achieved by selecting the largest possible values for α_H/α_L and E_H/E_L. Therefore, from the material combinations listed in Table 2, the Al alloy/Invar combination can be selected to achieve a smaller α/α_H. Figure 13 shows the variation of CTE of the structure in the θ₂ − θ₃ parameter space under different material combinations.

3.2. PR Analysis

3.2.1. Influence of Geometric Parameters

Similarly, the PR of the proposed 2D metamaterial unit cell structure can be adjusted by changing the geometric parameters. Material combinations listed in Table 2 are selected to study the relationship between PR and the geometric parameters.

When θ₂ = 20°, θ₃ = 15° and D = 300 mm, the relationship between PR and θ₁ for different material combinations is shown in Figure 14a. It can be observed that PR for the Nylon/LCS combination starts with a positive value, while PR for the Al alloy/LCS and Al alloy/Invar combinations start with negative values. Additionally, PR for the Nylon/LCS combination decreases monotonically as θ₁ increases, while PR for the other two combinations increases monotonically as θ₁ increases.

When θ₁ = 30°, θ₃ = 15° and D = 300 mm, the relationship between PR and θ₂ is shown in Figure 14b. It can be observed that the curves for all three material combinations increase monotonically. The difference is that the PR curve for the Nylon/LCS combination has a higher initial value and a slower rate of increase, while the PR curves for the other two combinations start lower and increase more rapidly.

When θ₁ = 30°, θ₂ = 90° and D = 300 mm, the relationship between PR and θ₃ is shown in Figure 14c. As θ₃ increases, PR for the Nylon/LCS combination decreases monotonically, while PR for the other two combinations decreases initially and then increases.

When θ₁ = 30°, θ₂ = 20° and θ₃ = 15°, the relationship between PR and D is shown in Figure 14d. It can be observed that as D increases, PR for all three combinations decreases monotonically. Additionally, the rate of decrease in PR for the Nylon/LCS combination is slower, while the rate of decrease for the other two combinations is faster.

A common feature can be identified across all four cases: the PR curve for the Nylon/LCS combination is higher than that of the Al alloy/LCS combination, and the Al alloy/LCS combination is higher than the Al alloy/Invar combination. This trend is consistent with the elastic modulus ratios for the three combinations: the elastic modulus ratio for the Nylon/LCS combination is smaller than that for the Al alloy/LCS combination, and the elastic modulus ratio for the Al alloy/LCS combination is smaller than that for the Al alloy/Invar combination. A preliminary observation indicates that, with fixed geometric parameters and E_H/E_L < 1, an increase in E_H/E_L leads to a reduction in PR. This has been validated in subsequent studies. Figure 15 shows the variation of PR of the structure in the θ₁ − D parameter space under different material combinations.

3.2.2. Influence of Material Physical Properties

According to the theoretical analysis in Section 2, PR of the structure is independent of CTE. Nevertheless, the physical properties of the constituent materials can affect PR based on the results shown in Figure 14. Therefore, a fixed set of geometric parameters (θ₁ = 30°, θ₂ = 20°, θ₃ = 15°, D = 300 mm) is selected to investigate the relationship between the elastic modulus of the constituent materials and PR. The relationship between PR and E_H/E_L is shown in Figure 16. It can be observed that PR decreases and then increases as E_H/E_L increases, with a minimum PR value occurring at E_H/E_L = 2. When the geometric parameters are fixed, selecting an E_H/E_L ratio close to 2 can result in a smaller PR. Therefore, from the material combinations in Table 2, the Al alloy/Invar combination can be selected to achieve a smaller PR.

3.3. Coupling Analysis

The coupling effect discussed in this subsection refers to the paired tunability of the thermal expansion and PR within the unit cell structure. Specifically, different combinations of thermal expansion and PR, including positive thermal expansion with NPR, positive thermal expansion with positive PR, negative thermal expansion with NPR, and negative thermal expansion with positive PR, can be simultaneously realized in the same unit cell architecture. Taking the aluminum alloy/low carbon steel material combination as an example, the achievable ranges of CTE and PR of the unit cell structure are investigated, as shown in Figure 17. The results indicate that CTE can be continuously regulated within a range of −47 to 28 ppm/°C, while PR varies from −14.8 to 7.3.

4. Conclusions

Inspired by traditional bimaterial triangular structures and symmetric arrow configurations, a novel bimaterial mechanical metamaterial structure is proposed. Based on beam element theory and thermoelastic formulations, a theoretical model is constructed and validated by simulations. According to theoretical analysis, it is found that the CTE of the structure is related to geometric parameters, the CTE ratio of the constituent materials, and the elastic modulus ratio, while PR is related to geometric parameters and the elastic modulus ratio of the constituent materials.

Experimental investigations on both thermal expansion and PR are conducted to validate the theoretical predictions. Taking the Al alloy/LCS material combination as an example, the adjustable ranges of CTE and PR for the unit cell structure are studied. It is found that the unit cell structure can achieve a tunable CTE ranging from −47 ppm/°C to 28 ppm/°C and a tunable PR ranging from −14.8 to 7.3, enabling the realization of targeted values for both CTE and PR. This study provides a new direction for the design of mechanical metamaterials with integrated and tunable CTE and PR.