Mechanical Characterization of Materials and Wave Dispersion

Instrumentation and Experiment Interpretation

Yvon Chevalier, Jean Vinh Tuong, Yvon Chevalier, Jean Vinh Tuong

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848211933
• eBook ISBN: 9781118621240
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

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.

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 C_iiii 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

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.

Fi...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Acknowledgements
• PART I: Mechanical and Electronic Instrumentation
• PART II: Realization of Experimental Set-ups and Interpretation of Measurements
• List of Authors
• Index