RU2465551C1 - Measuring method of elastic constants of materials - Google Patents

Abstract

FIELD: machine building.

SUBSTANCE: design load is applied to the specimen installed in grips. Digital hologram of specimen with no load is recorded. The specified loading force is applied and specimen hologram in loaded condition is recorded. Digital holographic interferogram of the specimen is calculated. Normal movements of the specimen surface at bending are measured, and elasticity modulus and shear modulus is measured. The above sequence of actions is repeated using various values of loading force, and average value of elasticity modulus, shear modulus and Poisson number is calculated based on the obtained values.

EFFECT: higher measurement accuracy of elastic constants of materials with simultaneous reduction of time consumption for preparation and performance of measurements and for processing of results.

4 dwg

Description

The invention relates to methods for determining the mechanical properties of materials by applying specified loads, and in particular to methods for determining the static Young's modulus of elasticity (below the modulus of elasticity), shear modulus, and Poisson's ratio. Measurements are carried out while loading the samples in bending.

A known method of determining the modulus of elasticity [USSR Author's Certificate No. 954850, class. G01N 3/08, 1982], based on the fact that a sample of a material having a cross-sectional area F with a strain sensor installed on it is loaded by tension, the signal ΔR from the sensor is measured, which corresponds to the change in voltage in the working section. A sample of a reference material is used having a cross-sectional area F _E and an elastic modulus E _E with a deformation sensor installed on it, which is placed in series with the sample of the material under investigation and loaded simultaneously with it, the signal ΔR _{E is} measured from the sensor installed on the sample of the reference material, and calculated elastic modulus E of the material.

The disadvantages of this method of determining the modulus of elasticity are: the need to use a reference material, the complexity of implementation and insufficient in many practical cases, the accuracy of the determination of the desired values.

The technical problem to which the invention is directed is to increase the accuracy of measuring the elastic constants of materials (elastic modulus, shear modulus and Poisson's ratio) while reducing the time required to prepare and conduct measurements, as well as to process the results.

The problem is solved in that in the method for measuring the elastic constants of materials by loading the calculated load of the sample installed in the grippers, according to the invention, a hologram of the sample is recorded without load, a predetermined loading force is applied and a hologram of the sample is recorded in the loaded state, the digital interferogram of the sample is calculated from the obtained interferogram measure normal displacements of the sample surface during bending and determine the value of the elastic modulus, shear modulus, and Pu coefficient ssona sample material, wherein said sequence of actions carried out repeatedly, using different values of the loading force, and on the basis of the obtained values is calculated the average value of the elastic constants.

Figure 1 presents a diagram of the loading of the sample when measuring the modulus of elasticity.

Figure 2 presents the interferogram (a) and the corresponding field of normal displacements (b) at the shoulder of the load _Lk = 9 mm when measuring the elastic modulus.

Figure 3 presents a view of the final result of measuring the modulus of elasticity of the sample material.

Fig. 4a) shows a holographic interferogram of a sample fixed at three points and loaded with a force P, and Fig. 4b) shows a graph of the normal displacement function depending on the coordinates W = f (x, y) for a given P when measuring the shear modulus.

The method is implemented as follows.

1. Measurement of the modulus of elasticity

The measurements were carried out on a cantilever beam of rectangular cross-section (hereinafter referred to as the sample) with dimensions of 100 × 15 × 2 mm made of an AMts alloy under loading by a pure bend.

Measurement Procedure

1. Using a digital holographic interferometer, a digital hologram of a sample is recorded without load and stored as a separate file in a computer.

2. The loading force P (load 1 gram) is set at a value of L _k equal to 9 mm (Fig. 1).

3. Using a digital holographic interferometer, a digital hologram of a sample is recorded in a loaded state and stored as a separate file in a computer.

4. The holographic interferogram of the sample is calculated and its elastic line is measured during bending.

5. The measured elastic lines of the sample are approximated by a parabola and the coefficient A is determined in the approximating formula:

,

,

where A is the coefficient determined by approximating the elastic line of the beam with a parabola.

6. The value of the elastic modulus is calculated:

,

,

Where

b - sample width

h - sample thickness

P is the magnitude of the loading force

L _k - shoulder application of loading force

A view of the final measurement result is shown in Fig.3.

7. Repeated actions are carried out according to items 1 ... 6. The number of repeated measurements is n = 10.

8. From 10 measurements, the average value of the elastic modulus (formula 1.1), the standard deviation of the measured results (formula 1.2) and the coefficient of variation (formula 1.3) are determined.

Table 1 shows the results of the corresponding measurements.

Table 1 Measurement Number i

(кгс/мм²)Measured value

(kgf / mm ² ) one 6869,164 2 6983,828 3 7017,671 four 6717,334 5 7217,848 6 7304,779 7 6927,579 8 7111,637 9 7188,368 10 6859,686

, рассчитанное по формуле 1.1:Average measured value

calculated by the formula 1.1:

The root-mean-square (standard) deviation S, determined by formula (1.2), is;

S ₉ = 184.68 kgf / mm ² .

The coefficient of variation ξ determined by formula (1.3) is:

ξ ₉ = 2.63.

9. The loading force P (load 1 gram) is set at a value of L _k equal to 24 mm. Consistently performed actions according to paragraphs 1 ... 8.

Table 2 shows the results of the corresponding measurements.

table 2 Measurement Number i

(кгс/мм²)Measured value

(kgf / mm ² ) one 6717,674 2 6828,177 3 6844,832 four 6764,451 5 7071,464 6 7018,641 7 7164,135 8 6830,893 9 7189,871 10 7079,068

:Average measured value

:

The mean square (standard) deviation S is:

S ₂₄ = 172.54 kgf / mm ² .

The coefficient of variation ξ is:

ξ ₂₄ = 2.48.

10. The loading force P (load 1 gram) is set at a value of L _k equal to 49 mm. Consistently performed actions according to paragraphs 1 ... 8.

Table 3 shows the results of the corresponding measurements.

Table 3 Measurement Number i

(кгс/мм²)Measured value

(kgf / mm ² ) one 6847,665 2 6879,573 3 7019,286 four 7041,327 5 7031,607 6 6907,694 7 7094,436 8 6788,970 9 7055,255 10 6993,248

:Average measured value

:

The mean square (standard) deviation S is:

S ₄₉ = 102.03 kgf / mm ²

The coefficient of variation ξ is:

ξ ₄₉ = 1.46

- среднее значение модуля упругости, S - среднее квадратическое (стандартное) отклонение модуля упругости, ξ - коэффициент вариации.11. Statistical processing of measurement results is performed. The values are calculated:

is the average value of the elastic modulus, S is the mean square (standard) deviation of the elastic modulus, ξ is the coefficient of variation.

The total sample size of the measured E values at three load application arms (L ₉ , L ₂₄ , L ₄₉ ) is 30 measurements.

In this case, the value of the elastic modulus of the alloy AMts:

=6979 кгс/мм²

= 6979 kgf / mm ²

The mean square (standard) deviation S _E is:

S _E = 153.1 kgf / mm ²

The coefficient of variation ξ _E is equal to:

ξ _E = 2.19.

и характеристики рассеяния при определении модуля упругости сплава АМц известным способом при помощи испытательной машины Instron 1185 и заявляемым способом при помощи цифрового голографического интерферометра, а также справочное значение

.Table 4 presents the values

and scattering characteristics when determining the elastic modulus of an AMc alloy in a known manner using an Instron 1185 testing machine and the inventive method using a digital holographic interferometer, and also a reference value

.

Table 4 Reference value Instron 1185 The inventive method

Elastic modulus

7087 kgf / mm ² 6689 kgf / mm ² 6979 kgf / mm ² Standard deviation S _E 202.8 kgf / mm ² 153.1 kgf / mm ² Coefficient of variation ξ _E 3.03 2.19

From the above data it follows that the measurement of the elastic modulus of the claimed method provides an increase in the reliability and accuracy of measurements with minimal requirements for sample preparation and the use of a simple device for fastening and loading the sample.

Moreover, the scattering parameters of the measurement results (standard deviation and coefficient of variation) with the holographic measurement method are significantly lower than when using the Instron 1185 testing machine.

2. Measurement of shear modulus

A rectangular plate with dimensions 110 × 108 × 2.8 (mm) was used as a sample. It was made of AMts material and mounted on three supports, a transverse loading force was applied to its free corner.

1. Using a digital holographic interferometer, a digital hologram of a sample is recorded without load and stored in a computer as a separate file.

2. The loading force P is set (load 2 grams).

3. Using a digital holographic interferometer, a digital hologram of a sample is recorded in a loaded state and stored in a computer as a separate file.

4. The holographic interferogram of the sample is calculated and the normal displacement W is measured at a load of P ₂ at 10 different points on the plate surface with x and y coordinates.

5. Determine the shear modulus at 10 points with x and y coordinates:

Where:

P is the applied load;

h is the plate thickness;

W - normal movement at a point with x and y coordinates,

x and y are the coordinates of the point on the surface of the sample at which the measurement is taken.

6. Calculate the average value of the shear modulus from 10 measurements, the standard deviation, the coefficient of variation.

The measurement results of the shear modulus G at a load of P ₂ = 2 grams.

Table 5 Measurement number X mm Y mm W, μm G, kgf / mm ² one 108.22 106.27 1,1386 2760.49 2 91.37 88.94 0.79996 2776.34 3 78.41 77.76 0,60097 2772.77 four 71.60 69.17 0.48721 2778.14 5 53.78 52.81 0.29032 2673.62 6 51.03 47.14 0.25112 2631.88 7 80.68 52.97 0,42515 2747.22 8 91.21 61.24 0.55729 2739.28 9 61.56 97.04 0.9612 2738.77 10 74.36 74.03 0.54534 2758.79

The average value of the measured shear modulus is:

The mean square (standard) deviation S ₂ is:

S ₂ = 47.98 kgf / mm ²

The coefficient of variation ξ ₂ is equal to:

ξ ₂ = 1.75

6. The loading force P _{4 is set} (load 4 grams).

7. Consistently performed actions according to items 1 ... 6.

The measurement results of the shear modulus G at a load of P ₄ = 4 grams.

Table 6 Measurement number X mm Y mm W, μm G, kgf / mm ² one 93.47 92.82 1,7685 2681.50 2 84.89 82.62 1,4287 2683.30 3 71.77 70.15 1,0328 2664.55 four 56.21 60.10 0.7076 2609.58 5 92.83 90.56 1.7051 2694.91 6 89.26 57.83 1,0401 2712.72 7 76.79 31.75 0.4998 2666.38 8 53.14 89.42 0.9819 2645,20 9 29.65 80.35 0.4790 2718.60 10 103.19 100.93 2,1135 2693.55

The average value of the measured shear modulus is equal to:

The mean square (standard) deviation S ₄ is:

S ₄ = 32.46 kgf / mm ²

The coefficient of variation ξ ₄ is equal to:

ξ ₄ = 1.21.

8. The loading force P _{6 is set} (load 6 grams).

9. Consistently performed actions according to items 1 ... 6.

The measurement results of the shear modulus G at a load of P ₆ = 6 grams.

Table 7 Measurement number X mm Y mm W, μm G, kgf / mm ² one 92.66 92.66 2,6114 2696.03 2 79.87 100.12 2,4328 2695,33 3 99.31 66.74 2,0294 2678.09 four 83.11 79.06 1,9919 2704.93 5 58.81 100.28 1,809 2673.25 6 49.41 95.74 1.4639 2649.78 7 63.50 67.88 1.3055 2707.40 8 64.48 61.40 1,204 2686.38 9 63,99 37.1 0.71975 2704.69 10 43.58 74.52 1,0008 2660.88

The average value of the measured shear modulus is equal to:

The mean square (standard) deviation S ₄ is:

S ₆ = 19.78 kgf / mm ²

The coefficient of variation ξ ₆ is equal to:

ξ ₆ = 0.74

- среднее значение модуля упругости, S_G - среднее квадратическое (стандартное) отклонение модуля сдвига, ξ_G - коэффициент вариации.10. Statistical processing of measurement results is carried out. Determine the values:

is the average value of the elastic modulus, S _G is the mean square (standard) deviation of the shear modulus, ξ _G is the coefficient of variation.

сплава АМц для полной выборки измеренных значений G.Determination of the average shear modulus

AMts alloy for a complete sample of measured G.

The total sample size of the measured values of G _i at three values of the applied load (P ₂ , P ₄ , P ₆ ) is 30 measurements.

In this case, the average shear modulus of the alloy AMts:

The mean square (standard) deviation S _G is:

S _G = 43.66 kgf / mm ² .

The coefficient of variation ξ _G is equal to:

ξ _G = 1.62.

3. Determination of Poisson's ratio

и модуля сдвига

рассчитывается коэффициент Пуассона ν материалаAccording to the measured values of the elastic modulus

and shear modulus

the Poisson's ratio ν of the material is calculated

и

рассчитывается модуль Пуассона, ν=0,30.Substituting

and

the Poisson modulus is calculated, ν = 0.30.

Table 8 shows the reference values of the measured parameters and compares them with experimental data.

Table 8 Parameter Reference value (kgf / mm ² ) The average measured value (kgf / mm ² ) Relative deviation Shear modulus, G 2691 2700 0.3% Modulus of elasticity, E 7085 6979 1.5% Poisson's ratio, ν 0.3 0.30 -

Based on the obtained measurement results, the following main conclusions can be drawn.

The proposed method allows you to:

1) measure the basic elastic constants of isotropic materials (elastic modulus, shear modulus and Poisson's ratio),

2) to ensure high accuracy and reliability of measurements, reduce the complexity and conduct measurements both in industrial and in laboratory conditions,

3) reduce the requirements for the accuracy of the manufacture of samples for testing (use samples of the simplest form),

4) take measurements using simple devices for fastening and loading samples,

5) to measure the elastic modulus and shear modulus also for orthotropic materials.

Thus, the inventive method will allow you to quickly conduct high-precision express analysis of the properties of structural materials, which is of particular importance in the production of highly loaded and critical parts, assemblies and structures.

Claims (1)

A method for measuring the elastic constants of materials by loading a sample installed in grippers with a design load, characterized in that they record a digital hologram of a sample without a load, apply a predetermined loading force and record a hologram of a sample in a loaded state, calculate a digital holographic interferogram of a sample, measure normal displacements of the sample surface in bending and determine the value of the elastic modulus and shear modulus, and the specified sequence of actions is carried out by secondly, using different values of the loading force, and on the basis of the obtained values, the average value of the elastic modulus, shear modulus, and Poisson's ratio is calculated.

Images