Mechanical Characterization of Materials and Wave Dispersion: Instrumentation and Experiment Interpretation

Over the last 50 years, the methods of investigating dynamic properties have resulted in significant advances. This book explores dynamic testing, the methods used, and the experiments performed, placing a particular emphasis on the context of bounded medium elastodynamics. Dynamic tests have proven to be as efficient as static tests and are often easier to use at lower frequency. The discussion is divided into four parts. Part A focuses on the complements of continuum mechanics. Part B concerns the various types of rod vibrations: extensional, bending, and torsional. Part C is devoted to mechanical and electronic instrumentation, and guidelines for which experimental set-up should be used are given. Part D concentrates on experiments and experimental interpretations of elastic or viscolelastic moduli. In addition, several chapters contain practical examples alongside theoretical discussion to facilitate the readers understanding. The results presented are the culmination of over 30 years of research by the authors and as such will be of great interest to anyone involved in this field.

• Language: English
• Publisher: Wiley
• Release date: Mar 4, 2013
• ISBN: 9781118621240

Book preview

PART I

Mechanical and Electronic Instrumentation

Chapter 1

Guidelines for Choosing the Experimental Set-up a

From an experimental point of view, the elastic and/or viscoelastic characterization of materials is not necessarily achieved simply by using an existing piece of industrial apparatus.

To begin with, the researcher has to choose the experimental set-up, taking the following items into account:

– the type of wave, whether progressive or stationary;

– the measurement technique;

– the numerical method to calculate the elastic (or viscoelastic) modulus or stiffness coefficient.

In this chapter, choice criteria as well as selection guidelines are presented. The following topics will be discussed in turn:

– choice of matrix coefficient(s) (stiffness or compliance matrix) to be evaluated;

– frequency range in which tests are to be conducted;

– shape and dimensions of the sample;

– temperature range to be adopted;

– viscoelastic properties of the material frequency dependence, damping capacity, etc.

– available previsional calculations (for composite materials) which enable the order, or the range, of elastic constants to be obtained.

1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted

1.1.1. For isotropic materials

The number of elastic constants is reduced to two, chosen from five available elastic constants: Young's modulus, E; shear modulus, G; Poisson's number, ν; volumic dilatation, K; and the stiffness coefficient Ciiii related to an extensional wave. For mechanical applications at low and medium frequency range (f ≤ 10,000 Hz), a compliance matrix [S] is preferred.

Table 1.1. The two classes of tests to be selected when the text material is isotropic

CH_-2.gif

In Table 1.1 the two classes of tests¹ permitting the evaluation of a compliance matrix [S] and a stiffness matrix [C] are presented. A bending wave enables the Young's modulus to be obtained, and a torsional wave, the shear modulus. A bending wave is preferred to an extensional wave for many practical reasons²:

– the ease with which measurements are effected;

– a bending wave dispersion is completely portrayed by the fourth order equation of motion (Mindlin–Timoshenko's equation or, with restriction at lower a frequency range, Bernoulli–Euler's equation);

– an extensional wave is the other possibility. However, at medium and higher frequency ranges, a sixth order equation of motion is referred to and consequently it is more difficult to handle the characteristic functions.

The ultrasonic method is easy to carry out. A thick plate sample must be chosen so as to produce, with some care, progressive waves (extensional or shear) in the samples. The wavelength Λ through thickness h satisfies the following inequality:

[1.1]

1.1.2. For anisotropic materials

The number of elastic constants depends on the degree of symmetry of the material. Remember that the number of different constants required for various materials is as follows:

– orthotropic material (wood): 9 constants;

– quasi-transverse (tetragonal) material: 6 constants;

– transverse isotropic material (long fibers regularly distributed in resin matrix): 5 constants;

– quasi-isotropic (cubic) material: 3 constants.

If preliminary information about the material is known (for example the degree of material symmetry [CHE 10]), the samples (number and shape) can be tailored with respect to the symmetry axis of the material. For a rod sample, its axis can be chosen to be coincident or different from the axis of symmetry of the material. For plates in ultrasonic tests, the material axis of symmetry may be collinear (or not) with the plate axis along the thickness, and the propagation direction of waves in any direction is obtained by transducer orientations.

1.1.2.1. Orthotropic material

Figure 1.1 shows three rods (of rectangular or square section) which are fabricated with a rod axis collinear with one material axis. From these three samples, six compliance matrix coefficients can be obtained:

[1.2]

[1.3]

Three remaining non-diagonal coefficients are to be evaluated. For this purpose, three other rod samples are fabricated. These samples are off-axis rods. The angles formed by the rod axis and the reference axis related to the material must be optimized, as in Figure 1.1(b).

The three off-axis samples permit the three non-diagonal compliance coefficients to be evaluated.

Figure 1.1. Orthotropic material samples to be fabricated: a) three rods whose axes are respectively collinear with one of the symmetry axis of the orthotropic material. b) Off-axis rods whose axes are in the planes (1, 2), (2, 3), (3, 1) delimited by the symmetry axes of the material. Rod axes make angles φi ≠ 0, i= (1, 2, 3)

CH_-6.gif

Attention should be focused on the accuracy of the angles φi with which the rod samples in Figure 1.1(b) must be made. As the rod axes do not coincide with the symmetry axes of the material, we have to deal with the change of reference axes for tensor components of the fourth order in formulae giving non-diagonal compliance coefficients, since a weak variation of angle φI gives rise to an important variation of power four (see [CHE 10] Chapter 1, pp. 29-31) of the trigonometric functions. For plates tailored for ultrasonic measurement, we have to deal with a Christoffel's tensor of power 2: the accuracy of the angle φ formed between the normal coordinate system tied up to the plate sample and one of the symmetry axes of the material intervenes in a Christoffel's tensor of power 2, (see [CHE 10] Chapter 10, pp. 517-523) consequently this influence is less critical than the aforementioned compliance matrix coefficients.

1.1.2.2. Transverse isotropic material

For artificial composite materials, a transverse isotropic symmetry is usually adopted [CHE 10]. This involves uniaxial long fibers periodically distributed in a resin matrix. Figure 1.2 presents two rod samples whose z axes are, respectively, collinear with symmetry axis 3 and an off-axis rod, whose z axis makes an angle θ with the material plane (1, 2). The third sample is an off-axis whose z axis makes an angle θ ≠ 0 with material axis 3. The two first samples, 1 and 2, enable calculations of the following compliance coefficients:

[1.4a]

[1.4b]

Figure 1.2. Transverse isotropic material with. the material isotropic plane (1, 2) and material symmetry axis 3. Samples 1 and 2 allow four independent compliance coefficients to be obtained. Sample 3 enables calculation of S23

CH_-9.gif

The third sample allows a material non-diagonal coefficient to be obtained and consequently Poisson's numbers ν32 and ν23

[1.5]

The samples used in Figure 1.2 are fragile during fabrication, as well as during measurement manipulation:

- for sample 3, the accuracy of the non-diagonal compliance coefficient strongly depends on the accuracy of the angle θ;

- if the material is strongly viscoelastic (i.e. if the damping capacity is high (tanδ)), the experimenter must be careful when using both the vibration and ultrasonic progressive wave techniques. Results obtained from the two techniques cannot be used concurrently to evaluate the remaining matrix coefficients. Since the working frequencies of waves are not in the same frequency range, the calculations might give rise to large errors.

The second reason to avoid this mixing method is that stationary waves require the use of a compliance matrix even though ultrasonic progressive waves concern a stiffness matrix. Matrix inversion is possible if a complete set of experimental stiffness coefficients has already been obtained.

Table 1.2 shows the two classes of testing methods. Details concerning dimensions of samples will be discussed later.

Table 1.2. For transverse isotropic material, five stiffness (or compliance) coefficients should be determined. The two classes of testing methods are used concurrently

CH_-11.gif

1.1.2.3. Orthotropic materials

For ultrasonic testing of massive materials, such as wood⁴ or artificial three-dimensional composites, three thick plates can be cut (one with an axis collinear with the material's symmetry axis (or trunk axis): a radial plate; one plate at the peripheral; and a tangential plate, perpendicular to material symmetry axis) although difficulty exists in fabricating off-axis plates. For vibration tests on rods, readers should consult Figure 1.1.

1.2. Influence of frequency range

This question is related to the viscoelastic behavior of material. The following remarks might be helpful for experimenters.

1.2.1. The Williams-Landel-Ferry method

It is useful to use the William-Landel-Ferry method to obtain artificial enlargement of the frequency range by using temperature as a variable parameter. The applicability of this method (presented in detail elsewhere: see [CHE 10], Chapter 10) must be valid. It concerns the correspondence (temperature-frequency) principle.

1.2.1.1. Adjustable temperature

The use of a climatic chamber with positive and negative temperature adjustments is appropriate. A gradient of temperature on the rod is, however, to be avoided if time delay is not respected for temperature stabilization during heating or cooling operation.

1.2.1.2. Choice of frequency range

If a narrow frequency range is adopted, the dimensions of the samples should be chosen so as to obtain measurable amplitude of vibration on the sample. Attention should be focused on the first resonance frequencies.

The choice of frequency range is closely related to wave dispersion. Precaution should be taken to evaluate the wave dispersion correctly before evaluating the viscoelastic dispersion.

1.2.1.3. Ultrasonic tests

The working frequency is that of the transducer itself; it should correspond to the frequency (resonance frequency) of the transducer. The temperature is generally the ambient temperature, except in the case where the transducer coupling medium between the transducer and the sample is not a liquid coupling but a special long rod, at the end of which the sample is glued. The sample can be in a special chamber at high temperature, T > 100°C.

1.3. Dimensions and shape of the samples

If a large volume of material is at the experimenter's disposal, plates and rods can be easily fabricated. However, some rod shapes are better suited to testing and calculations. The following practical considerations are useful.

1.3.1. Square section rod for longitudinal wave

If an extensional wave is adopted, a square section is to be preferred to a rectangular section. The reason for this is that the dispersion curve (velocity versus wave number or frequency) is less pronounced for a square section than in the case of a rectangular section, with flatness coefficient b (width)/h (thickness) < 1.

1.3.2. Rod slenderness

Rod slenderness is defined as the ratio h (thickness)/L (length). If the smallest possible slenderness is chosen, a large number of resonance frequencies is obtained. Higher frequencies are thus more easily obtained.

1.3.3. Imposed shape and size

In many cases it is difficult to obtain the shape and size wished for. In tests on bone, for example, the sample may be small in size, with a curved section. There is no possibility of cutting from a larger sample and one cannot manufacture a flat rod sample. One knows that there is a gradient in the elastic properties from the bone center to the free surface. In this case, the ultrasonic technique with special transducers would be the best method to adopt. Curved samples are often imposed on the experimenter (see [CHE 10], Chapter 10).

1.4. Tests at high and low temperature

Elastic and/or viscoelastic properties of materials change with temperature. A temperature controlled room with adjustable temperature between about -70°C to 250°C would be useful. No temperature gradient on the sample should be accepted. For negative low temperatures, a forced preliminary heating ventilation would be useful to avoid ice condensation on the sample.

Special transducers with special connecting cables are necessary for high temperatures.

1.5. Sample holder at high temperature

The sample holder plays an important role. Caution must be taken in bolt and screw systems to maintain the sample firmly without deforming the sample at the contact zones between sample and holder. Clamping systems require special precautions so as to avoid material creep at high temperatures.

1.6. Visual observation inside the ambient room

A glass window is necessary to examine the sample during tests. The window must be able to withstand high temperature.

1.7. Complex moduli of viscoelastic materials and damping capacity measurements

Measurement techniques deserve the special attention of the experimenter. Measurement techniques change drastically depending on the magnitude of the material damping capacity. With a very low damping coefficient of tan δ ≈ 10-3 measurement at ambient atmosphere is subjected to large errors. The first factor to take into account is air damping around the sample, which is of this order of the material's damping tanδ in the interval (10-3-5.10-3). To avoid this disadvantage, a special set-up with a vacuum system is necessary. Damping of the sample holder is also to be taken into account. Precautions concerning measurement techniques will be examined in Chapter 8.

1.8. Previsional calculation of composite materials

Surprisingly, at first sight, this topic is presented as a useful companion to testing. It merits some explanation. In dynamic tests, the experimenter is often confronted with a problem of the magnitude of the first eigenfrequency, the dimensions and size of the sample being known before a test. If the order of elastic moduli is known in advance it will be a great help for the experimenter to choose the frequency range and to possibly discard resonance frequencies due to parasitic oscillations of the sample holder system and the exciter (see [CHE 10], Chapter 1).

The topics presented above constitute only preliminaries which will be expanded in the following chapters.

1.9. Bibliography

[CHE 10] CHEVALIER , Y., and VINH , J.T. (ed.), Mechanics of Viscoelastic Materials and Wave Dispersion, ISTE Ltd, London and John Wiley & Sons, New York, 2010.

---

a Chapter written by Jean Tuong VINH

1 Indexes are used for anisotropic composite materials and not for isotropic materials.

2 The wave dispersion of an extensional wave requires a sixth order equation of motion to cover the whole frequency range.

3 Figures in subscript are different in tensorial and matrix notations. For shear moduli Gij, subscripts i and j indicate the plane in which shear stress and strain occur.

4 The elastic properties of wood from the center of a trunk to the bark might present a gradient in mechanical properties which constitutes a particular problem to be solved.

Chapter 2

Review of Industrial Analyzers for Material Characterization a

The issue of instrumentation for material characterization is indeed a very wide subject, covering numerous different areas of concern, such as:

– the mechanical behavior of high polymers;

– the elaboration of artificial composite materials;

– metallic materials for special applications;

– the application of materials in special conditions of temperature and environment;

– the characterization of materials in a large range of temperature and/or frequencies, in view of special applications requiring quantitative information on the viscoelastic behavior of materials;

– correlation between mechanical viscoelastic properties of materials and molecular interpretation, as well as analysis of structure geometry.

There are many books in which the principles of measurement of each of these apparatus are presented in detail, including those by J.D. Ferry [FER 69] and L.E. Nielsen [NIE 74], and I.M. Ward [WAR 71], in a series of chapters, presented a comprehensive discussion of the last item in the list above.

With this book being oriented towards mechanical applications, the mechanical point of view is consequently emphasized. The review of some available industrial analyzers enables the reader to appreciate the mechanical conception of each instrument and understand whether it is suitable for their needs. Often, the mechanical part of the instrument is presented in such a manner that it is hidden or drowned in a complex whole, where the electronic equipment and automatically programmed calculations by computers seem to be the most important part of the analyzer.

Often, the first-time user of this kind of instrument can feel that it is comfortable and easy to use, and almost forgets to ask the main question: is the instrument well adapted for my measurement objectives?

2.1. Rheovibron and its successive versions

2.1.1. Testing of filamentous sample and short rods

Rheovibrons such as DMA 100, 150, 400 (Dynamic Mechanical Analyzers) or VHF 104 allow the measurement of the complex Young's modulus of samples presented as filaments. Figure 2.1 presents the principles of the apparatus. The sample filament is placed in a climatic chamber whose temperature can be adjusted in the range -150°C < T < 300°C.

Figure 2.1. Schematic diagram of a Rheovibron

CH_-12.gif

The forced vibration imposed on the sample is produced by an electromagnetic exciter (on the left of the diagram in Figure 2.1). At the entrance to the oven, a displacement strain gauge transducer measures ∆x imposed to the point i of the filament. The tip k at the right of the sample is connected to a force strain gauge transducer which measures the force Fk. These two transducers give, through two Wheatstone bridges, two electrical tensions Ed and Ef such as:

[2.1a]

[2.1b]

Kd, Ed, KF, EF are proportional constants of the apparatus related to the measurement circuits. From [2.1a] and [2.1b] longitudinal strain and stress are evaluated:

[2.2]

Stars designate complex quantities.

Bringing [2.1] into [2.2], one obtains the complex Young's modulus E*:

[2.3]

The first parenthesis corresponds to a deviation of an electronic instrument, the second and the third parentheses are apparatus constants, and the fourth parenthesis is related to the sample. As the measurement is effected under harmonic regime, there is a phase angle between the two vectors σ* and ε*.

An electric phasemeter permits the measurement of the phase angle between stress and strain. If the electric voltage proportional to σ* is adjusted so that Vσ equals the tension proportional to ε*:

[2.4]

Figure 2.2 shows that the vector joining the two vectors is equal to 2V sin (δE/2).

Figure 2.2. Measurement of the phase angle δ. Vector B is electrically adjusted so that its absolute value equals the absolute value of vector A

CH_-18.gif

Conversion of sin (δ/2) into tan δ is possible:

[2.5]

The apparatus can accommodate a fiber of length 2 < L < 6 mm, with a sample cross-section area s < 0.2 cm². Since in [2.5] the complex Young's modulus is measured in gain and phase, (E*, δ), the real and imaginary parts of the Young's modulus can be evaluated:

[2.6]

The direct reading of tan δ and the absolute value of E* on the one hand, and the wide temperature range on the other hand, are the main characteristics of this instrument. The measured loss coefficient tan δ is in the range:

[2.7]

2.1.2. Improvement of the Rheovibron: the Rheovibron viscoanalyzer DDV II

The initial conception of the mechanical part of the Rheovibron presents many weak points, detailed as follows:

– the elastic compliance of the mechanical system must be taken into account, including the exciter itself and the connection with the sample, as well as the connection between the sample and the stress gauge;

– at high temperature, samples tend to yield between grips. Correction is necessary;

– Massa [MAS 73] suggested mechanical improvements for the inertia of the mechanical system, and proposed correction factors for varying temperature and frequency as well as for the dimensions of the samples themselves, and even for the nature of the material.

This raises the problem of direct connection of the sample via sample holders and the mechanical environment, which is difficult to introduce into the equation given the complex Young's modulus, and complex additional dynamic systems (see Figure 2.3). These last set of mechanical components are made up of spring, dashpot (which are frequency dependent and temperature dependent) and inertia of exciter and transducer.

Figure 2.3. Having an exciter and a mechanical sample holder introduces two sets of components on each side of the sample, constituted by spring ki, dashpot ηi and mass mi

CH_-22.gif

Modification of the sample holders has been proposed to adapt the instrument for bending tests. Erhard [ERH 70] suggested new holders for shearing tests. Compression tests and bending tests were proposed by Murayama [MUR 67] for measurements on anisotropic materials.

A servo-hydraulic actuator was introduced to increase the dynamic load capacity in magnitude as well as in a lower frequency range (f ≅ 0.1 to 5Hz), and a piezoelectric transducer was introduced for force measurements.

2.1.3. Automated and improved version of Rheovibron by Princeton Applied Research Model 129 A

This model, an automated Rheovibron by Princeton Applied Research, introduces two phase locks in the amplifier and data logging system.

2.2. Dynamic mechanical analyzer DMA 01dB – Metravib and VHF 104 Metravib analyzer

This French apparatus is initially adapted for short samples working in dynamic compression tests. Figure 2.4 shows a material presented as a short cylinder between two masses, m1 and m2. The mechanical holder and sample work as a two degrees of freedom system. The frequency response of the system presents a resonance and anti-resonance. The useful frequency part of the system is between these two extremes. The complex stiffness k* of the sample, magnitude |k*(ω)| and the phase (damping) angle δE are evaluated:

CH_-23.gif

Figure 2.4. A 01dB Metravib instrument: (a) schematic diagram of loading system and displacement measurements x1 and x2; (b) gain and phase response of the system, presenting resonance and anti-resonance. The working frequency interval is between the two zones. |Z| is the transfer function x2*/ x1*and ϕ is the phase angle of the whole system (the sample and additional masses)

CH_-24.gif

2.2.1. Comments

To avoid buckling, a sample working in compression must be short, which raises the question of three-dimensional stress states in the sample. The influence of shear stress as well as the influence of friction at both ends of the sample between the two masses m1, m2 and the sample must be taken into account.

Let us mention that the 01dB Metravib manufacturer proposes a large range of analyzers with a complete set of sample holders which allows compression, tension, bending tests as well as torsion and shearing tests with a variety of shapes and dimensions of samples. In the domain of mechanical analyzers with VHF apparatus, the frequency interval is extended from 100 Hz to 10 kHz. Depending on the adopted version, the ambient temperature of the sample can be chosen from between -50 °C and +110 °C.

2.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus)

Bruel and Kjaer, a Danish company, specializes in electronic dynamic measurements (frequency spectrometers, frequency oscillators and level recorders). This company proposes a mechanical system in which the sample is presented as a vibrating reed working in forced flexural vibrations, also referred to as an Oberst apparatus.

The sample is clamped at one end (Figure 2.5). A force transducer also working as a force exciter is located at the lower end. The displacement transducer is a contactless one, located at a point between the clamping ends.

Measurements of the flexural Young's modulus | E* | and damping tan δE are effected at and around resonance frequencies of the sample:

[2.8]

where ρ is the density, S the area of the cross-section, ω the circular frequency, L the sample length, I inertia of the section in bending (I = bh³/ 12), b width, h thickness and β* the root of the eigenequation deduced from boundary condition at both ends of the sample [CHE 10]. In the case of a weak damping material, at resonance frequency, amplitude of [2.8] is written