**1. Introduction**

298 Biomaterials – Physics and Chemistry

Rodriguez-Lorenzo, L. M., Vallet-Regi, M., Ferreira, J. M. F., Ginebra, M. P., Aparicio, C. &

Roveri, N. & Palazzo, B. (2006). Hydroxyapatite Nanocrystals as Bone Tissue Substitute. In

Sakhno, Y., Bertinetti, L., Iafisco, M., Tampieri, A., Roveri, N. & Martra, G. (2010) Surface

Sato, K., Kogure, T., Iwai, H. & Tanaka, J. (2002) Atomic-Scale {101-0} Interfacial Structure in

Suda, H., Yashima, M., Kakihana, M. & Yoshimura, M. (1995) Monoclinic <--> Hexagonal

Tadic, D., Beckmann, F., Schwarz, K. & Epple, M. (2004) A novel method to produce

Tilocca, A. (2010) Models of structure, dynamics and reactivity of bioglasses: a review. *Journal of Materials Chemistry,* Vol.20, No.33, pp.6848-6858, ISSN 0959-9428. Tilocca, A. & Cormack, A. N. (2007) Structural effects of Phosphorus Inclusion in Bioactive

Tilocca, A. & Cormack, A. N. (2009) Surface Signatures of Bioactivity: MD Simulations of 45S and 65S Silicate Glasses. *Langmuir,* Vol.26, No.1, pp.545-551, ISSN 0743-7463. Toulhat, N., Potocek, V., Neskovic, M., Fedoroff, M., Jeanjean, J. & Vincent, V. (1996)

Ugliengo, P., Viterbo, D. & Chiari, G. (1993) MOLDRAW: Molecular Graphics on a Personal Computer. *Zeitschrift fur Kristallographie,* Vol.207, No.1, pp.9-23, ISSN 0044-2968. Wahl, D. A., Sachlos, E., Liu, C. & Czernuszka, J. T. (2007) Controlling the processing of

Weiner, S. & Wagner, H. D. (1998) The material bone: structure-mechanical function

Wierzbicki, A. & Cheung, H. S. (2000) Molecular modeling of inhibition of hydroxyapatite

Young, R. A. & Brown, W. E. (1982). Structures of Biological Minerals. In *Biological* 

*Science: Materials in Medicine,* Vol.18, 201-209, ISSN 1573-4838.

Verlag, ISBN 978-0387115214, Berlin, Heidelberg, New York.

Vol.60, 159-166, ISSN 1097-4636.

ISSN 1551-2916.

ISSN 1520-6106.

0033-8230.

6600.

pp.73-82, ISSN 0166-1280.

ISBN 978-3-527-31389-1, Weinheim.

Vol.114, No.39, pp.16640-16648, ISSN 1932-7447.

Vol.99, No.17, pp.6752-6754, ISSN 0022-3654.

*Biomaterials,* Vol.25, No.16, pp.3335-3340, ISSN 0142-9612.

Planell, J. A. (2002) Hydroxyapatite ceramic bodies with tailored mechanical properties for different applications. *Journal of Biomedical Materials Research A,*

*Tissue, Cell and Organ Engineering,* C.S.S.R. Kumar (Ed.), pp. 283-307, Wiley-VCH,

Hydration and Cationic Sites of Nanohydroxyapatites with Amorphous or Crystalline Surfaces: A Comparative Study. *The Journal of Physical Chemistry C,*

Hydroxyapatite Determined by High-Resolution Transmission Electron Microscopy. *Journal of the American Ceramic Society,* Vol.85, No.12, pp.3054-3058,

Phase Transition in Hydroxyapatite Studied by X-ray Powder Diffraction and Differential Scanning Calorimeter Techniques. *The Journal of Physical Chemistry,*

hydroxyapatite objects with interconnecting porosity that avoids sintering.

SIlicate Glasses. *The Journal of Physical Chemistry B,* Vol.111, No.51, pp.14256-14264,

Perspectives for the study of the diffusion of radionuclides into minerals using the nuclear microprobe techniques. *Radiochimica Acta,* Vol.74, No.1, pp.257-262, ISSN

collagen-hydroxyapatite scaffolds for bone tissue engineering. *Journal of Materials* 

relations. *Annual Review of Materials Science,* Vol.28, No.1, pp.271-298, ISSN 0084-

by phosphocitrate. *Journal of Molecular Structure: THEOCHEM,* Vol.529, No.1-3,

*Mineralization and Demineralization,* G.H. Nancollas (Ed.), pp. 101-141, Springer-

Mechanical properties are a primary quality factor in many materials ranging from liquids to solids including foods, cosmetics, certain pharmaceuticals, paints, inks, polymer solutions, to name a few. The mechanical properties of these products are important because they could be related to either a quality attribute or a functional requirement. Thus, there is always a need for the development of testing methods capable to meet various material characterization requirements from both the industry and basic research.

There is a wide range of mechanical tests in the market with a wide price range. However, there is an increasing interest in finding new methods for mechanical characterization of materials specifically capable to be adapted to in-line instruments. Acoustic/vibration methods have gained considerable attention and several instruments designed and built in government labs (e.g. Pacific Northwest National Laboratory and Argon National Laboratory) have been made commercially available.

To measure mechanical properties of material a number of conventional techniques are available, which in some cases may alter or change the sample during testing (destructive testing). In other tests the strains/deformations applied are so small that the test can be considered non-destructive. Both types of test are based on the application of a controlled strain and the measurement of the resulting stress, or viceversa. Different types of deformations, e.g. compression, shear, torsion are used to test these materials.

Depending on the type of material, different conventional techniques utilized to measure its mechanical properties can be grouped as viscosity measurement tests (liquid properties), viscoelasticity measurement tests (semiliquid/semisolid properties), and elastic measurement tests (solid properties).

Acoustics based techniques can be used for all types of material and the following sections discuss in detail how these techniques have been adapted and used to measure materials whose properties range from liquids to solids. Some of the applications discussed in this chapter are based on the basic impedance tube technique. Applications of this technique for

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 301

wave moving through the liquid. In order to have manageable equations to estimate the viscosity of the liquid from acoustic measurements using these systems it is important to generate planar standing waves from which acoustic parameters can be readily obtained. Mert et al. (2004) described a model and the experimental conditions under which the

If a tube with rigid walls is considered the propagation of unidirectional plane sound waves though the liquid contained in the rigid tube can be described by the following equation

> 2 2 2 2 2 1 *p p <sup>C</sup> t x*

where *p* is the acoustic pressure, *t* is time, *x* distance and *C*1 the speed of the sound in the testing liquid. The solution of Equation (1) can be expressed in terms of two harmonic waves travelling in opposite directions and whose composition gives place to the formation of

ˆ ˆ ( ) ( ) ( ,) *i t kL x i t kL x*

*p+* is the amplitude of the wave traveling in the direction *+x* whereas *p-* is the amplitude of the wave traveling in the direction *–x* and *i* is the imaginary number equal to 1 . The

from integration of the pressure derivative respect to the distance *x*, an equation that is

By applying the boundary conditions such that at *x* = 0 the fluid velocity is equal to the

condition is derived from the practical situation of using an air space on the other end of the tube, which mathematical provides a pressure release condition at *x = L*, written as *p(L, t) = 0*.

> <sup>010</sup> v v <sup>010</sup> and ˆ ˆ 2 cos 2 cos *C C P P*

*k L k L*

<sup>0</sup> is the density of the liquid, in repose, and *v0* the amplitude of the imposed wave. An analysis made by Temkin (1981) showed that two absorption mechanisms produce the attenuation of the sound energy. One is due to the attenuation effects produced by the dilatational motion of the liquid during the acoustic wave passage. The other term arises from the grazing/friction motion of the liquid on the wall of the tube. It can be shown that the attenuation due to wall effects is significantly larger than the attenuation due to the

includes the attenuation

is the frequency of the wave. The corresponding wave velocity can be obtained

 

> 

 (4)

*pxt p e pe* 

(1)

(2)

(3)

in Equation 3 can be calculated as:

due to the viscosity of the

), which

0 *<sup>p</sup> vxt dt x* 

. The other boundary

assumption of standing planar waves stands.

standing waves (Kinsler et al., 2000):

complex wave number 1 <sup>ˆ</sup>

liquid,

yields:

where 

*k Ci* /

ˆ ˆ ( ) ( ) ( ,) *i t kL x i t kL x vxt P e Pe*

With these boundary conditions the coefficients *P+* and *P-*

velocity of the piston that creates the wave, which is 0 (0, ) *i t v t ve*

known as the Euler equation (Temkin, 1981) and calculated as <sup>1</sup> ( , )

 

(Kinsler et al., 2000):

material characterization in air have been around for almost a century. The technique is described in basic acoustics text books such as Kinsler et al., (2000) and Temkin (1981). Impedance tube methods based on standing waves and the transfer function method have been accepted as standard methods by the American Society for Testing and Materials (ASTM 1990 and 1995), thus they will not be discussed in this chapter. Instead the chapter will focus on liquids, semiliquid and semisolid materials, many of them exhibiting viscoelastic properties, i.e. those properties that are more representative of biomaterials behavior.
