RU2517989C1 - Method for determination of composite material properties - Google Patents

Abstract

FIELD: instrumentation.

SUBSTANCE: method involves generation of vibrations of a composite material sample in the form of a rectangular plate with free edges and determination of frequencies and patterns of natural mode of the plate. The experimentally obtained patterns of vibration modes are divided into three groups: the first group includes plate vibration modes with nodal lines parallel to the smaller side of the rectangular plate, the second group includes plate vibration modes with nodal lines parallel to the bigger side of the rectangular plate, and the third group includes those with nodal lines parallel to both plate sides; composite material properties are defined by going over the values of modulus of elasticity, modulus of shearing and Poisson ratio while substituting them into the mathematic model of the plate and comparing the calculated vibration frequency for each vibration mode to the experimentally obtained frequencies and vibration modes every time. The frequencies and vibration modes referred to the first group help to define the modulus of elasticity with the index of axis parallel to the bigger side - E_x, the frequencies and vibration modes referred to the second group help to define the modulus of elasticity with the index of axis parallel to the smaller side - E_y , the frequencies and vibration modes referred to the third group help to define the moduli of shearing G_xy, G_xz, G_yz . Nine elastic constants (E_x, E_y, E_z, v_xy, v_xz, G_xy, G_xz, G_yz ) are determined in the following sequence: first of all values of modulus of elasticity are gone over, then those of modulus of shearing and at the final stage - those of Poisson ratio. The values of modulus of elasticity, modulus of shearing and Poisson ratio are stopped to be gone over at the calculated identification of all experimentally modes and frequencies of plate vibration.

EFFECT: development of a method to define mechanical properties of orthotropic composite material by generation vibration of the latter.

4 cl, 5 dwg

Description

The invention relates to the field of measurement, in particular the determination of the mechanical properties of materials, and can be used to determine the elastic constants of composite materials in a non-destructive way.

Composite materials have high strength properties, characterized in particular by the ratio of module to weight and stress to weight, have excellent fatigue characteristics and good corrosion properties. These undoubted advantages of composite materials contributed to their widespread adoption in the aerospace industry. Therefore, an understanding of the mechanical behavior of composite materials under loading is the basis for their use as structural materials. Knowledge of the physical constants of structural materials, such as elastic modulus E, shear modulus G, and Poisson's ratio v, has always been of interest to mechanical engineers. With the advent of composite materials and their subsequent widespread use in industry, this problem has become even more relevant and therefore the physical constants of composite materials determined experimentally are of great practical value.

Although composite materials are anisotropic in nature, in many cases their properties are assumed to be homogeneous from the point of view of macromechanics and average mechanical properties are taken into account, which leads to inefficient use of composite materials. To describe the linear relationship between stresses and strains of a unidirectional reinforced composite material in a plane stress state, it is necessary to have five elastic constants: Е _x , Е _y , G _xy v _xy , v _yx .

A known method for determining the mechanical properties of composite materials according to GOST 25.602-80 [1], which consists in a short-term test of samples of composite material for compression with a constant strain rate, which determines:

(напряжение, соответствующее наибольшей нагрузке, предшествующей разрушению образца), МПа;• compressive strength $${\sigma }_{\mathrm{at}}^{\mathrm{from}}$$

(stress corresponding to the largest load preceding the destruction of the sample), MPa;

• elastic modulus of compression E _c - the ratio of stress to the corresponding relative deformation, during compression of the sample within the initial linear portion of the deformation diagram, MPa;

• Poisson's ratio v _c is the ratio of the transverse relative elongation to the longitudinal relative shortening of the sample under compression within the initial linear portion of the deformation diagram.

The specified standard applies to polymer composite materials reinforced with continuous high-modulus carbon, boron, organic and other fibers, the structure of which is symmetrical with respect to their median plane. The disadvantage of this method is the inability to determine the mechanical properties of composite materials reinforced in the transverse direction. In addition, this method provides the determination of the characteristics of the material with only one type of loading - compression.

For composite materials isotropic in the transverse direction, when describing the linear relationship between stresses and strains, it is necessary to determine five elastic constants. Usually they are determined from vibration tests, the results of which (the shape and frequency of oscillations) serve as the initial data necessary for the calculated determination of elastic constants using the finite element method (FEM).

A typical technique is to excite vibrations of a composite material sample, as a rule, by the first tone, measure the natural frequency and substitute its value in the frequency equation, which relates the natural resonant frequency to the dimensions, mass and elastic constant of the material. Solving the frequency equation, determine the elastic constant. Typically, the samples have a simple geometric shape in the form of beams and rods, which are used to determine elastic constants, such as Young's modulus E (elastic modulus), shear modulus G (elastic modulus of the second kind), and Poisson's ratio v. Thus, to determine the elastic constants it is always necessary to fulfill two conditions. First, from the experiment to obtain the resonant frequency of the sample material. Second, substitute the frequency in the theoretical formula and calculate the elastic constant. This procedure is prescribed in ASTM standard E1875-00e1 [2], when the oscillation is excited by a sinusoidal signal, and in ASTM standard E1876-01 [3], when the vibration is excited by the shock method.

Standards [2, 3] relate to direct methods when a single test is carried out on samples of simple geometric shapes, such as a beam or a rod, and using the well-known formula for each case, the elastic modulus or Poisson's ratio is calculated. They are difficult to apply to more complex samples in the form of rectangular plates, since the natural frequencies of the oscillations of the plates depend on several elastic constants, for the determination of which it is necessary to know not only the natural frequencies, but also the modes of vibration. For isotropic round plates with free edges, it is possible to apply the techniques proposed in the indicated standards [2, 3], but they are unacceptable for plates made of orthotropic materials.

There are also known methods for determining the elastic modulus by exciting sample vibrations for cellular ceramic materials [4], heterogeneous refractory and carbon materials [5], layered composite materials [6], and also multilayer composite coatings [7]. These methods also do not provide an accurate determination of the elastic characteristics of orthotropic materials.

Since the solution of the equations of motion for rectangular plates cannot be obtained in an analytical form, one has to resort to numerical methods. In the literature there are many examples of the application of various mathematical models, techniques that combine theoretical and experimental data for determining elastic constants [8].

Closest to the proposed technical solution is a method for determining the four elastic constants of a composite material by resonance testing of a rectangular plate with free edges [9]. Using the Ritz method, the authors obtained a frequency equation with four elastic constants. If you know the exact value of the frequency from the oscillations and the four elastic constants Е _x , Е _y , G _xy and v _xy , then, substituting them into the equation, we obtain the identity, i.e. the right side of the equation will be equal to the left. Therefore, to determine the four elastic constants, it is necessary to determine the eigenfrequencies of the four plate vibration modes and compose a system of four equations, each of which must be substituted with the numerical value of the experimentally determined frequency ω and the indices m and n corresponding to the wave form that was excited at this frequency.

The specified method is as follows. From a composite material, the mechanical characteristics of which must be determined, a sample is made in the form of a thin plate. Suspend the plate in such a way as to allow excitation of free vibrations. The shock method excites vibrations of the test plate. The resonant frequencies of the plate oscillations are determined. The values of the indicated resonant frequencies are substituted into the system of equations (mathematical model) and four elastic constants E _x , E _y , G _xy and v _{xy are} determined.

The disadvantage of this method lies in the impossibility of determining nine elastic constants that fully characterize the mechanical properties of an orthotropic composite material, namely Е _x , Е _y , Е _z , v _xy , v _xz , v _yz , G _xy , G _xz , G _yz . Therefore, this method cannot provide an accurate determination of the mechanical properties of an orthotropic composite structural material.

The objective of the present invention is to improve the accuracy of measuring the mechanical properties of an orthotropic composite structural material by determining the nine elastic constants of the sample of the latter in a dynamic manner. An additional object of the invention is to reduce the amount of time required to measure the characteristics of a composite sample.

The technical result is the creation of a method for determining the mechanical properties of an orthotropic composite material by exciting the oscillations of the latter.

The problem is solved in that in a method for determining the characteristics of a composite material in which vibrations of a sample of composite material made in the form of a rectangular plate are excited, the natural frequencies of the plate are recorded, based on which the characteristics of the composite material are determined using the mathematical model of the plate, according to the invention, registering the natural frequencies of the plate experimentally obtain the corresponding patterns of vibration modes plate. The obtained patterns of vibrational modes and the corresponding frequencies of natural vibrations of the plate are divided into three groups, the first of which includes the vibrational modes of the plate with nodal lines parallel to the smaller side of the rectangular plate, the second group - the vibrational modes with nodal lines parallel to the larger side of the plate, and to the third - with nodal lines parallel to both sides of the plate. In this case, the characteristics of the composite material are determined by sorting the values of the elastic modulus, shear modulus, and Poisson's ratio, substituting them into the mathematical model of the plate and comparing each time the calculated vibration frequency for each waveform with the frequencies and waveforms obtained experimentally, moreover, from the frequencies and waveforms referred to the first group, determining a modulus of elasticity index axis parallel to the greater side - E _x, the frequencies and waveforms of the second group mode is determined s elasticity index axis parallel to the short side - E _y, the frequencies and forms of the third group - shear moduli _{G xy, G xz, G yz} .

These essential features of the method provide a solution to the problem with the achievement of the claimed technical result, since:

- experimental obtaining patterns of plate vibration modes during registration of the frequencies of its own vibrations provides the identification of vibration modes characteristic of a given composite material, which affects the accuracy of determination (measurement) of the mechanical properties of an orthotropic composite structural material;

- the separation of experimentally obtained modes of vibration and the corresponding frequencies of the natural oscillations of the plate into three groups, the first of which includes the vibration modes of the plate with nodal lines parallel to the smaller side of the rectangular plate, the second group - the vibration modes with nodal lines parallel to the larger side of the plate and to the third, with nodal lines parallel to both sides of the plate, it ensures the distribution of experimentally obtained patterns of plate vibration modes in groups characterizing individual components of the mechanical properties of the composite material, in particular, the elastic modulus E, the shear modulus G, and the Poisson's ratio v, sorting of experimentally obtained vibration modes and the corresponding frequencies of natural vibrations of the plate, which is necessary in the future for more accurate determination of nine elastic constants of the sample.

-determining the characteristics of the composite material by sorting the values of the elastic modulus, shear modulus and Poisson's ratio, substituting them into the mathematical model of the plate and comparing each time the calculated vibration frequency for each waveform with the frequencies and waveforms obtained experimentally, and determining the frequencies and waveforms It referred to the first group modulus index axis parallel to the greater side - E _x, the frequencies and waveforms of the second group - elasticity module Indus som axis parallel to the short side - E _y, the frequencies and forms of the third group - shear modulus G _xy, G _xz, G _yz reduces the probability of error in determining specific characteristics and provides thereby improving the accuracy of measuring mechanical properties of orthotropic composite structural material, and also reduces the amount of time needed to measure the characteristics of a given composite material sample.

In the particular case of the implementation of the claimed method to increase the accuracy of measuring the mechanical properties of an orthotropic composite structural material by reducing the likelihood of determining nine elastic constants of a composite material sample, iterating over the values of the elastic modulus, shear modulus and Poisson's ratio and substituting them into the mathematical model of the plate, first enumerating the values of the elastic modulus, then the shear modulus and, at the final stage, the Poisson's ratio.

In another particular case of the implementation of the claimed method, improving the measurement accuracy is achieved in that the enumeration of the elastic modulus, shear modulus and Poisson's ratios is completed at the time of the calculated detection of all experimentally obtained forms and frequencies of oscillations of the plate. The calculated identification of all experimentally obtained forms and vibration frequencies of the plate provides the most accurate measurement of the characteristics of the composite material, since it reflects all possible models of the mechanical behavior of this sample of the composite material under its loading.

For a special case of the implementation of the inventive method, which reduces the amount of time required to measure the characteristics of a composite material sample, the enumeration of the elastic modulus, shear modulus, and Poisson's ratios is completed when sufficient accuracy is achieved between experimentally obtained and calculated shapes and frequencies of plate vibrations.

The claimed method for determining the mechanical properties of an orthotropic composite material by exciting vibrations of the latter is illustrated by the following detailed description with reference to illustrative materials, where

figure 1 shows the experimentally determined waveforms of an orthotropic plate 200 × 101 × 4 mm with free edges with the first (low) frequencies of natural vibrations (CSK);

figure 2 - experimentally defined vibration modes of the same orthotropic plate with subsequent (higher) frequencies of natural vibrations (CSK);

figure 3 - diagram of the nodal lines of the forms and frequencies of oscillations of the plate shown in figures 1 and 2, which are assigned to the first and second groups;

figure 4 - diagram of the nodal lines of the forms and frequencies of oscillations of the plate shown in figures 1 and 2, which are assigned to the third group;

5 is a summary table showing the sequence of characterization of composite material.

Before explaining the essence of the claimed method, we note that in the modal analysis there are generally accepted methods for calculating by the finite element method (FEM) the shapes and frequencies of isotropic and orthotropic rectangular plates with different boundary conditions. Typically, a plate with free edges is chosen in order to minimize errors in calculating natural frequencies. For orthotropic plates, in addition to dimensions and specific gravity, the input parameters are nine elastic constants: Е _x , Е _y , Е _z , v _xy , v _xz , v _yz , G _xy , G _xz , G _yz .

Consider the implementation of the claimed method on the example of an orthotropic composite structural material made in the form of a plate with dimensions 200 × 101 × 4 mm and a known specific gravity.

To determine the forms and frequencies of natural vibrations (CSC) of the plate, it is fixed, leaving the edges of the plate free. Fixing can be carried out by suspension, for example, according to the scheme given in the source [9]. After fixing the plate with a shock or other method, free vibrations of the plate are excited and the forms and frequencies of its own vibrations are determined. The experimentally determined forms and frequencies of natural vibrations of a plate with dimensions 200 × 101 × 4 mm are shown in FIGS. 1 and 2. The resulting patterns of vibration are sorted by the location of the nodal lines relative to the larger and smaller sides of the plate, selecting patterns with nodal lines parallel to the first group the smaller side, in the second group - paintings with nodal lines parallel to the larger side and in the third group - paintings with nodal lines intersecting at right angles (see figures 3 and 4). Each picture of the form of vibration corresponds to its own frequency of natural vibrations. The numbers under the diagrams indicate the presence of nodal lines parallel to the short and long sides of the plate, and the value of HSC. For example, the numbers (2 + 0 678) under the upper left picture in Fig. 3 mean that this diagram describes a waveform with two nodal lines parallel to the smaller (short) side of the plate, nodal lines parallel to the larger (long) side of the plate, absent, and ChSK = 678 Hz.

Elastic constants of a rectangular orthotropic plate are determined by numerically modeling the behavior of the test plate on a computer. The created model allows us to calculate the resonant frequencies and waveforms of the plate, if its dimensions, mass and elastic constants are known. In the mathematical model created for the test rectangular orthotropic plate, we substitute nine elastic constants as the initial data by enumeration: Е _x , Е _y , E _z , v _xy , v _xz , v _yz , G _xy , G _xz , G _yz . The calculations begin with the determination of the value of E _x , which is made on the basis of these forms and frequencies of vibrations assigned to the first group. Next, determine the value of E, which is calculated based on the data of the forms and frequencies of vibrations assigned to the second group. Then, by the enumeration method, the value of E _{z is} calculated. Next, we proceed to determine the values of the shear modulus G _xy , G _xz , G _yz , which are calculated on the basis of the data of the vibration modes and frequencies assigned to the third group. At the final stage, we proceed to determine the values of the Poisson's ratio v _xy , v _xz , v _yz .

The main principle of the method is to compare the calculated frequencies for known waveforms with frequencies obtained experimentally, while the elastic constants are considered unknown. The frequency calculations begin by substituting the test values of the elastic constants in the numerical model, and the obtained frequencies are compared with the experimental ones. Then the elastic constants are adjusted and a new series of frequencies is calculated, etc. Enumeration of values is carried out until the difference between the values of the calculated and experimental frequencies reaches a minimum determined by one or another criterion.

The problem of identifying elastic constants can be considered as an optimization problem when the discrepancy between experimentally obtained frequencies of a real sample and frequencies obtained from a mathematical model is minimized, based on the substitution of the assumed values of elastic constants for the mathematical model of the sample, i.e. function is minimized:

$$F\left(X\right)={\displaystyle \sum _{i=\mathrm{one}}^n{\left\{\frac{{\sout{\int }}_e-{\sout{\int }}_t}{{\sout{\int }}_t}\right\}}^2}$$

where F is the objective function of the elastic constants, the number of which may be 4, 5 or more. The natural frequencies obtained experimentally are denoted by ƒ _e , and those obtained by the finite element method - ƒ _t . The objective function uses n experimental frequencies.

Various methods can be used to optimize the objective function - from the least squares method to genetic algorithms [10]. The approach to solving the problem of optimizing the objective function in these cases is not optimal, since it is statistical rather than physical in nature when the equation of motion of the plate under study is solved. Therefore, it is not the minimum difference between the theoretical and experimentally found frequencies that matters, but the maximum possible number of plate vibration modes in a wide frequency range, revealed by substituting the elastic constants in the differential equation of the plate vibrations. The criterion for the completion of calculations will be the moment when the number of vibration modes and their form determined by calculation coincides with those found experimentally in a wide frequency range, i.e. with each step of the calculations, the number of calculated frequencies and waveforms increases sequentially.

An example of a sequence of calculated iterations is given in the table in FIG. 5. In the indicated table in the column ″ Elastic constants ″ for each calculation option in the first column are the modules Е _x , Е _y , Е _z in the second column - Poisson's ratios v _xy , v _xz , v _yz ; in the third column, the shear moduli G _xy , G _xz , G _yz .

Preliminary values of the moduli E are considered to be found when they fall into the plug formed by the calculated frequencies ω ₁ , <ω _exp <ω ₂ , where ω _exp is the frequency value determined experimentally for the vibration modes of group 1 and group 2.

In a similar way, three shear moduli G are determined in turn. Lastly, the Poisson coefficient is calculated.

The calculation process is considered complete when the vibrational modes of all frequencies found experimentally are determined by calculation.

The implementation of the inventive method provides improved accuracy in determining the characteristics of the composite material, since it allows you to determine nine elastic constants: E _x , E _y , E _z , v _xy , v _xz , v _yz , G _xy , G _xz , G _yz . In this regard, the understanding of the mechanical behavior of composite materials under loading is more complete, and the design and calculation of the stress state of parts made from composite materials will be more accurate.

Information sources

1. GOST 25.602-80 "Methods of mechanical testing of composite materials with a polymer matrix (composites)."

2. ASTM Standard El875-00e1, Standard test method for dynamic Young's modulus, and Poisson's ratio by sonic resonance, Book of Standards, Volume 03.01.

3. ASTM Standard El876-01, Standard test method for dynamic Young's modulus, shear modulus, and Poisson's ratio by impulse excitation of vibration, Book of Standards, Volume 03.

4. US patent No. 7802478, NKI 73/579, 2010.

5. Application US No. 20070157698, NCI 73 / 12.01, 2007.

6. US patent No. 6535076, NKI 73/597, 2003.

7. RF patent No. 2220412, MKI G01N 3/00, 2003.

8. Recent progress in identification methods for the elastic characterization of materials. Leonardo Pagnota, International Journal of Mechanics, Issue 4, Volume 2, 2008, 129-140.

9. US patent No. 5533399, NKI 73/579, 1996.

10. Elastic constants of composite materials by an inverse determination method based on a hybrid genetic algorithm. S.F. Hwang, J. C. Wu, Evgeny Barkanovs, Rimantas Belevicius, Journal of Mechanics, vol. 26, No. September 3, 2010.

Claims (4)

1. A method for determining the characteristics of a composite material in which oscillations of a sample of composite material made in the form of a rectangular plate are excited, the natural frequencies of the plate are recorded, based on which the characteristics of the composite material are determined using the mathematical model of the plate, characterized in that when registering natural frequencies plates experimentally receive the corresponding patterns of vibration modes of the plate, the resulting patterns of the shape of the ring beats and the corresponding natural frequencies of the plate’s vibrations are divided into three groups, the first of which includes the vibrational modes of the plate with nodal lines parallel to the smaller side of the rectangular plate, the second group - the vibrational modes with nodal lines parallel to the larger side of the plate, and the third group - vibration modes with nodal lines parallel to both sides of the plate, and the characteristics of the composite material are determined by sorting the values of the elastic modulus, shear modulus and Poiss coefficient by substituting them into the mathematical model of the plate and comparing each time the calculated vibrational frequency for each waveform with the frequencies and waveforms obtained experimentally, and the elastic modulus with the axis index parallel to the larger side is determined by the frequencies and waveforms assigned to the first group - Е _x , the elastic modulus with the axis index parallel to the smaller side - Е _y is determined by the frequencies and vibration modes of the second group, the shear moduli G _xy , G _xz , G _yz by the frequencies and vibration modes of the third group. 2. The method for determining the characteristics of the composite material according to claim 1, characterized in that when enumerating the values of the elastic modulus, shear modulus and Poisson's ratio and substituting them into the mathematical model of the plate, the elastic modulus is then enumerated, then the shear modulus and, at the final stage, the coefficient Poisson. 3. A method for determining the characteristics of a composite material according to claim 1 or 2, characterized in that the enumeration of the elastic modulus, shear modulus, and Poisson's ratios is completed at the time of the calculated detection of all experimentally obtained plate shapes and vibration frequencies. 4. A method for determining the characteristics of a composite material according to claim 1 or 2, characterized in that the enumeration of the elastic modulus, shear modulus and Poisson's ratios is completed when sufficient accuracy is achieved between experimentally obtained and calculated shapes and frequencies of oscillations of the plate.

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