
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39251673
71111
10.1038/s41598-024-71111-0
Article
A new method for identifying elastic parameters of isotropic materials based on square specimens
Zhang Longxin 1
Zhang Wenbin 190322507@qq.com

2
Xu Han 1
Ma Yaxing 1
1 https://ror.org/00xyeez13 grid.218292.2 0000 0000 8571 108X Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, 650500 China
2 grid.411157.7 0000 0000 8840 8596 College of Mechanical and Electrical Engineering, Kunming University, Kunming, 650214 China
9 9 2024
9 9 2024
2024
14 2105121 6 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
This paper proposes a new impulse excitation technique using a square plate. First, the functional relationship between the modal frequency of the specimen and the geometrical dimensions and mechanical parameters was established by using the finite element method. Then, the continuous functional relationship derived by a homotopy method allowed the frequency ratios to be related to the thickness-to-length ratio and Poisson’s ratio. By measuring the frequency ratios and thickness-to-length ratio, Poisson’s ratio could be calculated using this functional relationship. When the density and Poisson’s ratio were known, Young’s modulus could be identified inversely in conjunction with the finite element analysis. Finally, a comparison test between this method and the traditional impulse excitation technique was designed and implemented, and the results showed that this method has advantages in both testing efficiency and accuracy. The study provides a new idea for system identification, which has important application value and promotion significance.

Keywords

Young’s modulus
Poisson’s ratio
Homotopy method
Impulse excitation technique
Subject terms

Computational methods
Materials science
Mathematics and computing
Support Program for Xingdian TalentsYNWR-QNBJ-2018-349 Zhang Wenbin issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

With the advancement of science and technology, new types of materials are emerging and have been widely used in the aerospace field. Materials can be categorized into isotropic and anisotropic. The elastic parameters of isotropic materials include Young’s modulus and Poisson’s ratio, which usually vary from material to material. By testing the elastic parameters of a material, engineers can assess the reliability of the material, i.e., whether the material will experience performance degradation or failure during long-term use1–3. However, the measurement results of the same specimen can be affected by the material preparation process and the temperature of the testing environment, independent of the magnitude of the external force and the geometry of the object. Therefore, it is of great practical significance and scientific value to improve the testing accuracy of elastic parameters.

In the last hundred years, at room temperature, elastic parameters have been tested by static, ultrasonic and dynamic methods4. The impulse excitation technique (IET) is a common non-destructive testing technique, which is included in ASTM (American Society for Testing and Materials) for its simplicity of device, versatility, efficiency and flexibility5. The IET does not cause damage to the specimen, which makes it possible to use it for brittle materials that are difficult to measure by static methods, thus extending the range of test objects. The IET has been widely used in scientific research for composites6–11, glass12,13, rock14, ceramics15,16, and concrete17,18. In the ambient testing environment, scholars have utilized IET to test novel materials for product design, material selection, quality control, and failure analysis. For example, some scholars doped the material components to study the changes in the mechanical properties of the materials to meet the needs of product mechanical properties8,11,18. However, with the gradual increase of the service temperature of materials, the study of the mechanical properties of materials at room temperature could no longer meet the growing demand19. The High Temperature Impulse Excitation Technique (HTIET) had been developed to temporarily fulfill this urgent need. The properties of materials may decay in a high-temperature environment6,7,12–14,16,17. For example, concrete, a common building material, in the event of a fire, the strength of the building structure may decrease as the temperature increases17, which may lead to a catastrophic event. The mechanical properties of rock are affected by the heating–cooling cycle, which in turn increases the risk of landslides, rockfalls, and stone building collapses14.

The IET usually uses a rectangular sample as a specimen and picks up the flexural and torsion modal frequencies using a vibration picker to invert the elastic parameters of the material5. The flexural and torsion modal frequencies of a rectangular specimen have different nodal positions, resulting in different support methods for the sample during the bending and torsion frequency tests, which usually requires the operator to adjust the support position. At room temperature, the accuracy of IET measurements is generally affected by the flatness of the plate and the precision of the operator’s placement of the support position. In high-temperature environments, since the sample is in a closed furnace, it is not possible to adjust the support method during the test, resulting in the same specimen not being able to obtain all of its elastic parameters in a single measurement. Therefore, there is an urgent need for a test method that can obtain all the elastic parameters of a specimen in one test without adjusting the support position.

The traditional IET needs to examine the support of vibrating specimens to complete the testing of all elastic parameters, and the accuracy is also limited. To solve these problems, this paper adopts a square specimen instead of the rectangular specimen and deduces a new test method based on modern control theory and homotopy method, combined with the finite element method (FEM). The elastic parameters of materials can be measured quickly and accurately. In addition, since the zero amplitude positions of torsion frequency and anti-clastic frequency are located in the center of the square plate surface, a small circular cross-section close to the center point is used as a fixture, so that the operator can obtain all the mechanical parameters of the same specimen at one time without moving the specimen, regardless of whether the specimen is at room temperature or high-temperature. This not only avoids the problem of errors due to unstandardized operations but also solves the incompleteness of testing in high-temperature environments. Finally, by designing comparative experiments, it is verified that the method proposed in this paper has advantages in testing efficiency and accuracy compared with the traditional IET.

New test method for Poisson’s ratio

Differential equations of undamped free motion

In the theory of elasticity, the following three equations are involved in solving the frequency response of an isotropic elastic solid under free boundary conditions without considering the damping present in the system20:1 ρ∂u∂t2=∇·σ,

2 ε=12u∇+∇u,

3 σ=D:ε,

where ρ in (kg/m3) is the mass per unit volume, u in (m) is the displacement, t in (s) is the time, σ in (Pa) is the stress tensor, ε in (1) is the strain tensor, ∇u and u∇ in (1) is the displacement gradient, D in (Pa) is the fourth order elasticity tensor. Equations (1), (2), and (3) are obtained by Newton’s laws of motion, strain coordination theory, and generalized Hooke’s law, respectively. Combined Eq. (1), (2), (3), eliminating the reference variables σ and ε , the differential equation for undamped free motion with the basic unknowns u is obtained as follows21:4 ρ∂u∂t2=λ+G∇∇·u+G∇2u,

where λ in (Pa) is the Lame’s constant, and G in (Pa) is the shear modulus. Their conversion relationship with Young’s modulus E in (Pa) and Poisson’s ratio μ in (1) is as follows22:5 λ=μE1+μ1-2μ,

6 G=E21+μ.

Finite element equations for undamped free motion

The FEM utilizes a finite number of simple but interacting elements to approximate an infinitely unknown real system. The FEM is a very effective numerical method for solving Eq. (4).

Under the small deformation condition, Eq. (4) is equated to the algebraic equation23,24:7 M3n×3nU¨(t)3n×1+K3n×3nU(t)3n×1=0,

where n is the total number of nodes, U(t) is in (m) the total node displacement. The system is determined by the overall stiffness matrix K and the overall mass matrix M:8 K=∑e=1HKe=∑e=1H∭VBeTDBedV,

9 M=∑e=1HMe=ρ∑e=1H∭VNeTNedV,

where H is the total number of elements into which the system is divided, e is the element’s number, V is the element’s volume domain, Be in (1/m) is the element’s strain matrix, Ne in (1) is the element’s shape function matrix, and D in (Pa) is a symmetric 6 × 6 matrix related to E and μ:10 D=E(1+μ)(1-2μ)1-μμμ000μ1-μμ000μμ1-μ0000001-2μ20000001-2μ20000001-2μ2.

The solution of Eq. (7) can be assumed as follows:11 U(t)3n×1=ϕ3n×1sinωt=ϕ3n×1sin2πft,

where ϕ is the amplitude vector of the total nodal displacement U(t), f in (1/s) and ω in (rad/s) are the corresponding intrinsic vibration frequency and intrinsic circular frequency, respectively. Substituting Eq. (11) into Eq. (7), the generalized eigenvalue problem is obtained:12 K-ω2Mϕ=0.

If Eq. (12) has a nonzero solution, the determinant of its coefficient matrix is as follows:13 detK-ω2M=0.

Eq. (13) can be written as:14 detKM-1=0.

Solving Eq. (14) yields 3n eigen-solutions ω12,ϕ1,⋯,ω3n2,ϕ3n, where ω1=2πf1,⋯,ω3n=2πf3n represent the 3n intrinsic circular frequencies of the system, and:15 0≤ω1≤ω2≤⋯≤ω3n.

In ANSYS, the intrinsic vibration frequency fi is usually used as the result of the solution, which is called the modal frequency. The generalized eigenvector ϕ1,⋯,ϕ3n represents the 3n intrinsic vibration modes of the system, which are called modal shapes.

Decoupling of undamped finite-degree-of-freedom vibration systems

The FEM treats a continuous system as a system of finite degrees of freedom. In modern control theory, for a given system of finite degrees of freedom, it is always possible to find a nonsingular matrix consisting of eigenvectors23,25:16 ϕ=ϕ1,⋯,ϕ3n.

The state vector ϕ is linearly transformed to get another state vector ϕ¯=ϕϕ. From the symmetry of the stiffness and mass matrices, Eq. (13) decoupled can be expressed as:17 detEρϕTkϕ-ω2ϕTmϕϕ=0,

where k is a function of Poisson’s ratio and geometric parameters and m is a function of geometry. The original eigenvalue problem is transformed into:18 EρϕTkϕ-ω2ϕTmϕ=Eρk1-ω12m100000.00000.00000.00000Eρk3n-ω3n2m3n=0.

For finite-degree-of-freedom systems, the intrinsic circular frequency of the finite element degrees of freedom numbered i=1,⋯,3n can then be expressed as follows:19 ωi=kimiEρ,

where ki and mi are not directly available. In this paper, a specific geometrical object is investigated in the form of a square plate with mutually orthogonal length, width and height, assuming that the following relationship exists for its frequency fi:20 2πfi=hl2Eρ.

With the help of ANSYS study the variation rule of its geometric dimensions and mechanical parameters on its intrinsic frequency.

Influence of material and geometrical parameters of square plates on their modal frequencies

The material parameters entered in ANSYS: density ρ is 1000 kg/m3, Young’s modulus E is 100 GPa, and Poisson’s ratio μ is 0.2. SOLID186 is selected as the element type, and the element size is controlled to be 0.001 m. The subspace iteration method is selected to solve Eq. (12) for the large matrix eigenvalues and eigenvectors (modal frequencies and shapes). The solution type is a free-mode analysis (The first 6 modal frequencies are rigid body modes with zero frequency). In the dynamic analysis, the size of the weight factor of each mode is proportional to the inverse of the frequency of the mode, i.e., the lower the frequency, the higher the weight, and the low-order modal characteristics have a great influence on the dynamic performance of the product. Therefore, this study focuses only on low-order modes and sets the solution order to 9.

As shown in Fig. 1, the results of finite element simulation calculations show the free mode frequencies and vibration shapes of a 100 mm × 100 mm × 5 mm square plate. Figure 1a shows the 7th order mode, two corners on one diagonal vibrate back and forth with a phase difference of π from two corners on the other diagonal; Fig. 1b shows the 8th order mode, one set of opposite edges vibrates back and forth with a phase difference of π from the other set of opposite edges; and Fig. 1c shows the 9th order mode, the four corners and the center area of the plate surface vibrate in phase.Fig. 1 Modal vibration shapes of a square plate. (a) 7th; (b) 8th; (c) 9th.

In this paper, h/l is defined in the range 0<h/l≤1, a plate with h/l = 1 is a square brick. The effect of thickness h on the modal frequency and modal vibration pattern of the square plate is shown in Fig. 2a, when h/l = 0.7, the frequency f9 shows a decreasing trend, which is due to the first change in the direction of the modal vibration pattern (eigenvectors); then when h/l = 0.8, the frequency f9 shows an increasing trend, which is due to the second change in the direction of the modal vibration pattern; and when h/l = 0.8, f8 shows a decreasing trend, which is due to the first change in the direction of the mode vibration pattern. The cross-sectional area l2 is approximately inversely related to the frequency fi as shown in Fig. 2b. The E is proportional to the frequency fi as shown in Fig. 2c. The ρ is inversely proportional to the frequency fi, as shown in Fig. 2d.Fig. 2 Influence of geometrical and mechanical parameters on their modes. (a) thickness; (b) length; (c) Young’s modulus; (d) density.

Poisson’s ratio is a dimensionless parameter which is assumed to be related to the frequency fi:21 2πfi=ψi(μ)hl2Eρ.

In particular, when h/l2 is 1 m/1 m × 1 m, 0.01 m/0.1 m × 0.1 m, respectively, this leads to a mapping relation from geometric parameters to frequency fi that is not unimaginative, and therefore in this paper h/l is included in the function ψi(μ) as a dimensionless dependent variable:22 2πfi=ψi(μ,hl)hl2Eρ.

In order to verify the correctness of the formula, the coefficient ψi(μ,hl) is fixed and the h/l ratio is kept constant, and the simulations are carried out at different values of 1/l. The results show that when Eq. (22) is correct, there is a linear relationship between fi and 1/l as shown in Fig. 3a–c. This simulation result confirms the correctness of Eq. (22) .Fig. 3 Verification results. (a); (b); (c).

The relationship from the set μ,h/l to the frequency fi is shown in Fig. 4. When h/l≤40%, set μ,h/l to the frequency f7,f8,f9 is mapped one-to-one as in Fig. 4a–c.Fig. 4 The set μ,h/l to the frequency fi. (a) f7; (b)  f8; (c) f9.

When h/l≤40%, fi is related to the geometrical and mechanical parameters can be expressed as:23 f7=ψ7(μ,hl)h2πl2Eρ,

24 f8=ψ8(μ,hl)h2πl2Eρ,

25 f9=ψ9(μ,hl)h2πl2Eρ,

where the dimensionless coefficient functions ψ7(μ,hl),ψ8(μ,hl),ψ9(μ,hl) are shown in Fig. 5a–c.Fig. 5 Coefficient function. (a); (b); (c).

Calculation of Poisson’s ratio

Dividing Eq. (24) by Eq. (23), the following functional relationship exists:26 f8f7=ψ87μ,hl,

ψ87μ,hl as shown in Fig. 6a. Similarly, ψ98μ,hl,ψ97μ,hl, and are shown in Fig. 6b and c, respectively.Fig. 6 Relationship between set (µ, h/l) and frequency ratios. (a); (b); (c).

As can be seen from Fig. 6, when h/l is less than 10%, the relationship from set μ,h/l to set f8/f7 is a one-to-one mapping. Since the actual test is usually considered to save material, h/l less than 10% is acceptable. When we make samples h/l is usually not a rational number, in this paper, we use the homotopy method and introduce the parameter26–28:27 phl=-59.6865121095568hl2+16.2937683627739hl-0.0288130972097893.

The successive changes from set μ,h/l to set f8/f7 can be expressed as follows:28 f8f7=φμ,p=1-pψ87μ,0.001+pψ87μ,0.1,

where,29 ψ87μ,0.001=0.0303331630728725μ2+0.127830469008510μ+1.41427221610838,

30 ψ87μ,0.1=0.0101932160287526μ2+0.162230099055876μ+1.44015288291539.

As shown in Fig. 7, the variation from set μ,h/l to set f8/f7 is related. If the parameters h/l,f8/f7 are known we can calculate the Poisson’s ratio of the material by Eq. (28).Fig. 7 Comparison of simulation data with the homotopy method.

Consider the effect of damping

In any practical system there is always a variety of damping, and in order to take into account the effect of the energy dissipating effects of damping on the test results, the modal damping ratio is introduced in the vibration system:31 ζi=ckimi≈lnx(t)x(t+NTd)2Nπ,

where c is the damping coefficient, x is the damped vibration signal, N is the number of cycles, and Td is the period of the quasi-periodic motion in the presence of damping.

The damped free vibration frequency of a vibrating system in the presence of damping is:32 ωi¯=2πfi¯=1-ζi2ωi=1-ζi22πfi.

Considering the presence of damping then Eq. (28) has the following relationship:33 f¯8f¯7=1-ζ821-ζ72f8f7=1-ζ821-ζ72φμ,p.

If the user needs to take the damping factor into account, the frequency obtained through the microphone needs to be corrected using Eq. (32) to obtain its ideal frequency. In general testing, the damping ratio of the material is close to zero and has a negligible effect on the test results.

Experiment and comparison

Impulse excitation technique

The measurement principle of IET is shown in Fig. 8, and its general test procedure is as follows5:Fig. 8 Impulse Excitation Technique (IET).

Measurement of the mass m, thickness h, width b and length l of the rectangular specimen.

The specimen is supported using support 2 and struck at position 2 using an impact bar, the vibration signal is captured by a microphone, and the signal is analyzed by an analysis system to determine the torsional frequency response ft. The shear modulus can be obtained:34 G=4lmft2bhB1+A,

where B and A are correction factors as follows:35 B=b/h+h/b4h/b-2.52h/b2+0.21h/b6,

36 A=0.5062-0.8776b/h+0.3504b/h2-0.0078b/h312.03b/h+9.892b/h2.

Similarly, the flexural frequency response ff is obtained by supporting the specimen with support1. The Young’s modulus can be obtained as follows:37 E=0.9465mff2l3bh3T1,

when l/h<20, where the correction coefficient T1 is:38 T1=1+6.5851+0.0752μ+0.8109μ2h/l2-0.868h/l4-8.3401+0.2023μ+2.173μ2h/l41.000+6.3381+0.1408μ+1.536μ2h/l2,

when l/h>20, T1 can be simplified as:39 T1=1+6.585hl2,

μ is Poisson’s ratio:40 μ=E2G-1.

Assume an initial Poisson’s ratio and then use an iterative process to balance Eq. (37) and (40) to obtain the Poisson’s ratio.

Square plate elasticity test technique

The square plate elasticity test technique (SPETT) proposed in this study is shown in Fig. 9. The technique is suitable for test specimens with side lengths of 100 mm ~ 200 mm and recommends the use of a center clamping beam with a diameter of 5 mm (a center clamping beam of this size was 3D printed for the subsequent experimental part of this paper). The test flow of SPETT is shown in Fig. 10 with the following steps:Measurement of parameter l using vernier calipers, measurement of h using an external micrometer, measurement of parameter m using an electronic scale

Measuring the torsion frequency f7¯ and anti-clastic frequency f8¯ of the square specimen using a microphone.

Calculate its Poisson’s ratio by using Eq. (28).

Substituting the Poisson’s ratio and density values of the material into ANSYS to obtain its Young’s modulus.

Fig. 9 square plate elasticity test technique (SPETT).

Fig. 10 Test Flowchart.

For the torsional vibration f7 and the anti-clastic vibration f8(The actual frequency measured through the microphone is the actual frequency f7¯,f8¯ in the presence of damping effects.), the zero-amplitude positions are located at the center of the plate surface (shown as blue areas in Fig. 1 (a) (b)). When the diameter of the center support beam is much smaller than the specimen side length l, the measured frequencies f7 and f8 can be regarded as their natural mode frequencies, which are consistent with the results of the free mode frequencies calculated by simulation. In this study, frequency tests were performed using MATLAB software. First, the time-domain sound pressure signal was captured through a professional microphone using the audiorecorder () function. According to the sampling law, in order to prevent distortion, the sampling frequency is set to be more than twice of the highest frequency. The sampling time is set to 8 s, and the number of channels is set to 1. Next, the getaudiodata () function is used to read the captured audio data and convert it to floating-point double-precision type. Finally, the Fast Fourier Transform (FFT) algorithm is called to plot the spectrum of the signal and identify the target frequency from the spectrum.

Comparison experiment

The actual measurement scenario is shown in Fig. 11.Fig. 11 Illustration of SPETT and IET tests.

The elastic material properties of the specimens at room temperature were determined based on the IET using three rectangular specimens of the following dimensions:h × b × l = 0.00198 m × 0.0995 m × 0.1995 m (Specimen structural steel #1).

h × b × l = 0.00370 m × 0.0500 m × 0.1000 m (Specimen carbon steel #2).

h × b × l = 0.00412 m × 0.0498 m × 0.1001 m (Specimen aluminum #3).

Table 1 summarizes the averaged elastic material properties resulting based on IET.Table 1 Results from pretests for IET at room temperature.

Specimen	ρ(kg/m3)	E(GPa)	μ(1)	
structural steel #1	7843.89	219.5	0.255	
carbon steel #2	7740.15	210.9	0.2538	
aluminum #3	2681.7	69.73	0.292	

The elastic material properties of the specimens at room temperature were determined based on the SPETT using three square specimens of the following dimensions:h × l × l = 0.00195m × 0.0995m × 0.0995m (Specimen structural steel #4).

h × l × l = 0.00397m × 0.1010m × 0.1010m (Specimen carbon steel #5).

h × l × l = 0.00306m × 0.0994 m × 0.0994m (Specimen aluminum #6).

Table 2 summarizes the averaged elastic material properties resulting based on SPETT.Table 2 Results from pretests for SPETT at room temperature.

Specimen	ρ(kg/m3)	E(GPa)	μ(1)	
structural steel #4	7840.88	206.5	0.329	
carbon steel #5	7739.15	202.5	0.294	
aluminum #6	2682.4	68.82	0.328	

From Table 1 and Table 2, it can be seen that the results of Young’s modulus of the new test method proposed in this study are lower than those measured by IET, and the results of Poisson’s ratio are higher than those by IET. At present, the reasons for this phenomenon are unclear. Subsequent chapters 3.4 will analyze comparative tests to reveal the flaws in the IET test theory and explain the root cause of this phenomenon.

In order to effectively compare the accuracy of the two methods, the frequency response of the specimen in Table 2 is predicted using the results of the IET in Table 1 based on ANSYS and compared with the actual frequency response of the specimen in Table 2 (whose damped vibration signals are collected through a microphone, and whose actual frequency response is obtained by signal analysis), as shown in Fig. 12a. Similarly, the frequency response of the specimen in Table 1 was predicted using the results of the SPETT in Table 2 and compared with the actual frequency response of the specimen in Table 1, as shown in Fig. 12b.Fig. 12 Simulation results. (a); (b).

From Fig. 12b, it can be seen that the test results of the SPETT in Table 2 accurately predict the frequency response of the rectangular specimen in Table 1, which is less different from the actual frequency. In the actual selection of specimens, the test found that when the parallelism accuracy of the two specimens is too low, the test results of both methods are unsatisfactory.

The actual frequency response of the specimen is obtained by the following methods. Firstly, the simulation was used to get a preliminary understanding of the mode shapes and mode frequencies of the specimen, and then to find out the zero-amplitude position and the highest amplitude position under different mode shapes. The zero-amplitude position of the specimen at a certain order of modal frequency corresponding to the modal vibration pattern was clamped with a center clamping sorghum and the highest amplitude position was struck with a mallet. In this case, the vibration signal obtained from the microphone must contain the corresponding modal frequency (natural frequency) of the mode shape, and the signal with other frequency components (zero amplitude position not coinciding with the clamping position) will have a lower frequency response compared to the actual frequency response. By comparing the simulation results with the spectrogram (Fast Fourier Transform of signals), the corresponding modal frequencies are identified in the spectrogram.

Error analysis

Due to the difficulty of obtaining ideal test specimens and the difficulty of determining the true values of the actual material parameters, a virtual material model was used in order to more objectively compare the accuracy of the two test methods. This model, whose material parameters are known, ensures the comparability of the two methods and helps us to determine which method gives more accurate results. This paper gives hypothetical ideal specimens as follows:h × b × l = 0.005 m × 0.06 m × 0.12 m , E=60GPa, ρ=2600kg/m3, μ=0.25 (Granite #7).

h × l × l = 0.006 m × 0.1 m × 0.1 m, E=60GPa, ρ=2600kg/m3, μ=0.25 (Granite #8).

h × b × l = 0.005 m × 0.05 m × 0.12 m , E=170GPa, ρ=2329kg/m3, μ=0.28 (Silicon #9).

h × l × l = 0.005 m × 0.1 m × 0.1 m, E=170GPa, ρ=2329kg/m3, μ=0.28 (Silicon #10).

h × b × l = 0.005 m × 0.025 m × 0.12 m , E=25GPa, ρ=2300kg/m3, μ=0.2 (Concrete #11).

h × l × l = 0.004 m × 0.1 m × 0.1 m, E=25GPa, ρ=2300kg/m3, μ=0.2 (Concrete #12).

Since both test methods require the acquisition of free-mode frequency information of the material, we used simulation instead of using a microphone to acquire the frequencies. For this purpose, we designed the test program using MATLAB and tested the virtual specimen. The test results are shown in Table 3.Table 3 Test results and errors for hypothetical ideal specimens.

Specimen	Material	Method	Elastic parameter	Relative error	
E(GPa)	μ(1)	
Eerrors(1)	μerrors(1)	
#7	Granite	IET	60.408	0.212	0.68%	− 15.2%	
#8	SPETT	60.05	0.2505	0.083%	0.72%	
#9	Silicon	IET	171.35	0.255	0.79%	− 8.92%	
#10	SPETT	170.2	0.2804	0.12%	0.14%	
#11	Concrete	IET	25.045	0.259	0.18%	29.5%	
#12	SPETT	25.01	0.2008	0.04%	0.4%	

Table 3 shows that the measurements based on the IET Young’s modulus are relatively reliable, but the Poisson’s ratio has a higher error. Poisson’s ratio is obtained indirectly by Eq. (40). This phenomenon coincides with the results of the comparative tests in chapter 3.3. Eq. (38) shows that the test results of Young’s modulus are affected by Poisson’s ratio, but the effect is negligible. This error is because the relationship between Young’s modulus and the frequency response of bending vibration is derived relatively rigorously, whereas the relationship between shear modulus and the frequency response of torsional vibration is not rigorously derived.

For the calculation of Young’s modulus, the relationship between Young’s modulus and the bending resonance frequency response of rods of various cross-sectional shapes was rigorously analyzed by S.P. Timoshenko in the literature29 based on calculus. Subsequently, Von E. Goens made a theoretical approximation to Timoshenko’s formula by introducing a rectangular cross-sectional shape correction factor, as in Eq. (38), and experimentally verified it30. It is this method of calculating Young’s modulus from the bending vibration frequency that is used in practice. Sam Spinner designed comparative tests to analyze the error margins of the Von E. Goens approximation theory.

For the calculation of shear modulus, Roak can only give an approximate Eq. (35) for rectangular cross-sections although he rigorously gave exact expressions for the shear modulus versus torsional vibration frequency response for circular and square cross-Sections31. Pickett and Cady independently investigated the relationship between shear modulus and torsional frequency response of rectangular rods respectively32,33 and obtained different approximate expressions. Among them, Cady’s approximate relationship is34:41 fn=Adnlh2+b2Gρ.

The term n simply assumes that the overtones are integer multiples of the fundamental. Sam Spinner compares Cady’s and Pickett’s approximate theories in34 and finds that there is very little difference in their numerical results. Thankfully, Pickett’s approximation in34,35 was eventually accepted by the ASTM, and Sam Spinner pointed out that the overtones of a rectangular plate are not actually integer multiples of the fundamental. Apparently, the theoretical approximations of both Cady and Pickett were based on the false premise that the overtones are integer multiples of the fundamental, resulting in large errors in Poisson’s ratio test results.

Although the approximate formulas proposed in this paper lack rigorous calculus derivation, the numerical data obtained from finite element simulations are reliable. The theory of the finite element method has been intensively studied by many scholars over the past century. The error in the SPETT Poisson’s ratio is mainly originated from the construction of Eq. (28), while the error in the Young’s modulus is mainly affected by the accuracy of the density measurement.

Conclusion

1. In this paper, a Poisson’s ratio test method is proposed using a square plate as a specimen, which is suitable for specimens with a thickness-to-length ratio of less than 10%. The method can be used in combination with finite element analysis to further calculate the Young’s modulus of the material.

2. The method can overcome the incompleteness of the pulse excitation test under a high-temperature environment, and all the elastic parameters of the same specimen can be obtained in one test without adjusting the support position.

3. Comparison tests are designed to compare the test results of the new method proposed in this paper with the traditional pulse excitation method, and the results show that the method proposed in this paper has higher accuracy and is more convenient to operate.

4. The testing accuracy of the traditional pulse excitation method for calculating Young’s modulus is relatively reliable, but the testing error of Poisson’s ratio is within 30%. The reason is that the relationship between shear modulus and torsional vibration frequency response lacks rigorous argumentation, which leads to a large deviation in the results of Poisson’s ratio.

In the future, higher precision accuracy may be obtained by considering the effects of damping and temperature.

Acknowledgements

This work was sponsored by Support Program for Xingdian Talents. Wenbin Zhang initiated this research.

Author contributions

Longxin Zhang ：wrote the main manuscript text. Wenbin Zhang：Funding provided；reviewed the manuscript. Han Xu：Matlab code provided. Yaxing Ma：Investigations provided.

Funding

Support Program for Xingdian Talents,YNWR-QNBJ-2018-349

Data availability

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

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Surya Kiran M Parametric study on factors influencing the stiffness of honeycomb sandwich panels using impulse excitation technique J. Sandw. Struct. Mater. 2017 21 1 20 10.1177/1099636216686649
Surya Kiran, M. et al. Parametric study on factors influencing the stiffness of honeycomb sandwich panels using impulse excitation technique. J. Sandw. Struct. Mater. 21, 1–20. 10.1177/1099636216686649 (2017).10.1177/1099636216686649
2. Scislo L Szczepanik-Scislo N Quantification of construction materials quality via frequency response measurements: A mobile testing station Sensors 2023 23 1 14 10.3390/s23218884
Scislo, L. & Szczepanik-Scislo, N. Quantification of construction materials quality via frequency response measurements: A mobile testing station. Sensors 23, 1–14. 10.3390/s23218884 (2023).10.3390/s23218884
3. Paolino DS Damaged composite laminates: Assessment of residual Young’s modulus through the impulse excitation technique Compos. Pt. B-Eng. 2017 128 76 82 10.1016/j.compositesb.2017.07.008
Paolino, D. S. et al. Damaged composite laminates: Assessment of residual Young’s modulus through the impulse excitation technique. Compos. Pt. B-Eng. 128, 76–82. 10.1016/j.compositesb.2017.07.008 (2017).10.1016/j.compositesb.2017.07.008
4. Xie MY Li FX Review of the measurement methods for elastic moduli and internal friction of solids Adv. Mech. 2022 52 33 52 10.6052/1000-0992-21-013
Xie, M. Y. & Li, F. X. Review of the measurement methods for elastic moduli and internal friction of solids. Adv. Mech. 52, 33–52. 10.6052/1000-0992-21-013 (2022).10.6052/1000-0992-21-013
5. DIN. ASTM-E-1876–22 Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation of Vibration. 2022.
6. Pihlatie M Mechanical properties of NiO/Ni–YSZ composites depending on temperature, porosity and redox cycling J. Eur. Ceram. Soc. 2009 29 1657 1664 10.1016/j.jeurceramsoc.2008.10.017
Pihlatie, M. et al. Mechanical properties of NiO/Ni–YSZ composites depending on temperature, porosity and redox cycling. J. Eur. Ceram. Soc. 29, 1657–1664. 10.1016/j.jeurceramsoc.2008.10.017 (2009).10.1016/j.jeurceramsoc.2008.10.017
7. Guicciardi S Temperature dependence of the dynamic Young’s modulus of ZrB2–MoSi2 ultra-refractory ceramic composites Scr. Mater. 2010 62 831 834 10.1016/j.scriptamat.2010.02.011
Guicciardi, S. et al. Temperature dependence of the dynamic Young’s modulus of ZrB2–MoSi2 ultra-refractory ceramic composites. Scr. Mater. 62, 831–834. 10.1016/j.scriptamat.2010.02.011 (2010).10.1016/j.scriptamat.2010.02.011
8. Pabst W Gregorová E Malangré D Hostaša J Elastic properties and damping behavior of alumina–zirconia composites at room temperature Ceram. Int. 2012 38 5931 5939 10.1016/j.ceramint.2012.04.045
Pabst, W., Gregorová, E., Malangré, D. & Hostaša, J. Elastic properties and damping behavior of alumina–zirconia composites at room temperature. Ceram. Int. 38, 5931–5939. 10.1016/j.ceramint.2012.04.045 (2012).10.1016/j.ceramint.2012.04.045
9. Song W Zhong Y Xiang J Mechanical parameters identification for laminated composites based on the impulse excitation technique Compos. Struct. 2017 162 255 260 10.1016/j.compstruct.2016.12.005
Song, W., Zhong, Y. & Xiang, J. Mechanical parameters identification for laminated composites based on the impulse excitation technique. Compos. Struct. 162, 255–260. 10.1016/j.compstruct.2016.12.005 (2017).10.1016/j.compstruct.2016.12.005
10. Giaccu GF Meloni D Valdès M Fragiacomo M Dynamic determination of the modulus of elasticity of maritime pine cross-laminated panels using vibration methods WIT Trans. Ecol. Environ. 2017 226 571 579 10.2495/SDP170501
Giaccu, G. F., Meloni, D., Valdès, M. & Fragiacomo, M. Dynamic determination of the modulus of elasticity of maritime pine cross-laminated panels using vibration methods. WIT Trans. Ecol. Environ. 226, 571–579. 10.2495/SDP170501 (2017).10.2495/SDP170501
11. Tognana S Measurement of the Young’s modulus in particulate epoxy composites using the impulse excitation technique Mater. Sci. Eng. A-Struct. 2010 527 4619 4623 10.1016/j.msea.2010.04.083
Tognana, S. et al. Measurement of the Young’s modulus in particulate epoxy composites using the impulse excitation technique. Mater. Sci. Eng. A-Struct. 527, 4619–4623. 10.1016/j.msea.2010.04.083 (2010).10.1016/j.msea.2010.04.083
12. Sibil A Study of damage of high zirconia fused-cast refractories by measurement of Young’s modulus Mater. Sci. Eng. A-Struct. 2009 521–522 221 223 10.1016/j.msea.2008.09.135
Sibil, A. et al. Study of damage of high zirconia fused-cast refractories by measurement of Young’s modulus. Mater. Sci. Eng. A-Struct. 521–522, 221–223. 10.1016/j.msea.2008.09.135 (2009).10.1016/j.msea.2008.09.135
13. Roebben G Assessment of the high temperature elastic and damping properties of silicon nitrides and carbides with the impulse excitation technique J. Eur. Ceram. Soc. 2002 22 2501 2509 10.1016/S0955-2219(02)00111-5
Roebben, G. et al. Assessment of the high temperature elastic and damping properties of silicon nitrides and carbides with the impulse excitation technique. J. Eur. Ceram. Soc. 22, 2501–2509 (2002).10.1016/S0955-2219(02)00111-5
14. Liu W Elastic modulus evolution of rocks under heating–cooling cycles Sci. Rep. 2020 10 13835 10.1038/s41598-020-70920-3 32796913
Liu, W. et al. Elastic modulus evolution of rocks under heating–cooling cycles. Sci. Rep. 10, 13835. 10.1038/s41598-020-70920-3 (2020).32796913 10.1038/s41598-020-70920-3
15. Bruls RJ Hintzen HT The temperature dependence of the Young’s modulus of MgSiN2, AlN and Si3N4 J. Eur. Ceram. Soc. 2001 21 263 268 10.1016/S0955-2219(00)00210-7
Bruls, R. J. & Hintzen, H. T. The temperature dependence of the Young’s modulus of MgSiN2, AlN and Si3N4. J. Eur. Ceram. Soc. 21, 263–268 (2001).10.1016/S0955-2219(00)00210-7
16. Roebben G The innovative impulse excitation technique for high-temperature mechanical spectroscopy J. Alloys Compd. 2000 310 284 287 10.1016/s0925-8388(00)00966-x
Roebben, G. et al. The innovative impulse excitation technique for high-temperature mechanical spectroscopy. J. Alloys Compd. 310, 284–287. 10.1016/s0925-8388(00)00966-x (2000).10.1016/s0925-8388(00)00966-x
17. Bahr O Young’s modulus and Poisson’s ratio of concrete at high temperatures: Experimental investigations Mater. Des. 2013 45 421 429 10.1016/j.matdes.2012.07.070
Bahr, O. et al. Young’s modulus and Poisson’s ratio of concrete at high temperatures: Experimental investigations. Mater. Des. 45, 421–429. 10.1016/j.matdes.2012.07.070 (2013).10.1016/j.matdes.2012.07.070
18. Thomaz WA Comparative study of dynamic and static Young’s modulus of concrete containing basaltic aggregates Case Stud. Constr. Mater. 2021 10.1016/j.cscm.2021.e00645
Thomaz, W. A. et al. Comparative study of dynamic and static Young’s modulus of concrete containing basaltic aggregates. Case Stud. Constr. Mater.10.1016/j.cscm.2021.e00645 (2021).10.1016/j.cscm.2021.e00645
19. Heritage K Impulse excitation technique for dynamic flexural measurements at moderate temperature Rev. Sci. Instrum. 1988 59 973 974 10.1063/1.1139761
Heritage, K. et al. Impulse excitation technique for dynamic flexural measurements at moderate temperature. Rev. Sci. Instrum. 59, 973–974. 10.1063/1.1139761 (1988).10.1063/1.1139761
20. Lu, M. W. & Luo, X. F. Fundamentals of elasticity theory, 2nd ed. (Tsinghua University Press, 2001).
21. Zhang, L. & Zhang, M. Fundamentals of vibration and sound, 1st ed. (Harbin Engineering University Press, 2016).
22. Wang G., Ding, G. & Yang, Jie. Elastic mechanics, 3rd ed. (Tsinghua University Press, 2015).
23. Wang, X. & Shao, M. Basic principles and numerical methods of the finite element method, 2nd ed. (Tsinghua University Press, 1996).
24. Li, S. & Xiao, Z. Elastic mechanics and finite elements, 1st ed. (China Machine Press, 2018).
25. Liu, B. & Tang, W. Modern control theory, 3rd ed. (China Machine Press, 2006).
26. He J-H Homotopy perturbation technique Comput. Methods Appl. Mech. Eng. 1999 178 257 262 10.1016/s0045-7825(99)00018-3
He, J.-H. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262. 10.1016/s0045-7825(99)00018-3 (1999).10.1016/s0045-7825(99)00018-3
27. Lee MK An analytical model for computing the sound power of an unbraced irregular-shaped plate of variable thickness Sci. Rep. 2018 8 15355 10.1038/s41598-018-33645-y 30337652
Lee, M. K. et al. An analytical model for computing the sound power of an unbraced irregular-shaped plate of variable thickness. Sci. Rep. 8, 15355. 10.1038/s41598-018-33645-y (2018).30337652 10.1038/s41598-018-33645-y
28. Lee MK Natural frequencies of thin rectangular plates using homotopy-perturbation method Appl. Math. Modell. 2017 50 524 543 10.1016/j.apm.2017.05.050
Lee, M. K. et al. Natural frequencies of thin rectangular plates using homotopy-perturbation method. Appl. Math. Modell. 50, 524–543. 10.1016/j.apm.2017.05.050 (2017).10.1016/j.apm.2017.05.050
29. Timoshenko SP On the transverse vibrations of bars of uniform cross section Phil. Mag. Ser. 1922 6 43 125 131 10.1080/14786442208633855
Timoshenko, S. P. On the transverse vibrations of bars of uniform cross section. Phil. Mag. Ser. 6(43), 125–131 (1922).10.1080/14786442208633855
30. Goens VE Uber die Bestimmung des Elastizitatsmoduls von Staben mit Hilde von Biegung Schwingungen Ann. Phys. 1931 10.1002/andp.19314030602
Goens, V. E. Uber die Bestimmung des Elastizitatsmoduls von Staben mit Hilde von Biegung Schwingungen. Ann. Phys.10.1002/andp.19314030602 (1931).10.1002/andp.19314030602
31. Raymond J Roark, formulas for stress ancl strain 1943 McGraw-Hill Publishing Co
Raymond, J. Roark, formulas for stress ancl strain (McGraw-Hill Publishing Co, 1943).
32. Pickett G Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders Proc. ASTM 1945 45 846 865
Pickett, G. Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders. Proc. ASTM 45, 846–865 (1945).
33. Walter Guyton Cady, Piezoelectricity, p. 114, 1st cd. (McGraw-Hill Publishing Co., Inc., New York, N. Y., 1946).
34. Spinner S Comparison of theoretical and empirical relations between the shear modulus and torsional resonance frequencies for bars of rectangular cross section J. Res. Natl. Bur. Stand. 1958 60 459 464 10.6028/JRES.060.047
Spinner, S. et al. Comparison of theoretical and empirical relations between the shear modulus and torsional resonance frequencies for bars of rectangular cross section. J. Res. Natl. Bur. Stand. 60, 459–464. 10.6028/JRES.060.047 (1958).10.6028/JRES.060.047
35. Spinner S A comparison of experimental and theoretical relations between Young’s modulus and the flexural and longitudinal resonance frequencies of uniform bars J. Res. Natl. Bur. Stand. 1959 64A 147 155 10.6028/jres.064A.014
Spinner, S. et al. A comparison of experimental and theoretical relations between Young’s modulus and the flexural and longitudinal resonance frequencies of uniform bars. J. Res. Natl. Bur. Stand. 64A, 147–155. 10.6028/jres.064A.014 (1959).10.6028/jres.064A.014
