Elasticity tensor in anisotropic materials

Maximilian Ludwig, winter semester 2020/21

The elasticity tensor of a material connects the strains resulting from the deformations of a material with the stresses, which occur due to these deformations. In structural analysis this relation between the strains and the stresses is called the constitutive law. The elasticity tensor can be interpreted as a measure for the stiffness of a material and is therefore crucial for the propagation of elastic waves. In many materials the stiffness depends on the spatial direction in which a deformation is applied. In this case the elasticity tensor is denoted anisotropic.

• 1 Hooke's Law
• 2 Elasticity tensor for symmetric configurations
• 3 Independent coefficients for isotropic materials
• 4 Wave propagation
  • 4.1 Isotropic material
  • 4.2 Transversely isotropic material
• 5 Literature

Hooke's Law

If a material is deformed elastically, which means the deformations are small and the yield strength is not reached, Hooke's Law is commonly used as the constitutive law. It assumes a linear relation between Cauchy’s strains and Cauchy’s stresses. It can be expressed as a tensor as:

The stresses $//<![CDATA[ \begin{array}{l}\sigma_{ij}\end{array} //]]>$ are visualized for an infinitesimal element with the edge lengths $//<![CDATA[ \begin{array}{l}dx_i\end{array} //]]>$ in Figure 1. The strains $//<![CDATA[ \begin{array}{l}\varepsilon_{kl}\end{array} //]]>$ are labeled the same way. $//<![CDATA[ \begin{array}{l}C_{ijkl}\end{array} //]]>$ represents the fourth order elasticity tensor. 

Both, the second order tensors for stress and strain as well as the elasticity tensor are symmetrical, i.e. $//<![CDATA[ \begin{array}{l}\sigma_{ij} = \sigma_{ji}\end{array} //]]>$ etc.

Hooke's Law can also be written in matrix form using Voigt's notation:^[1]

Elasticity tensor for several symmetry configurations

In general, the elasticity tensor contains 21 independent values, however this number can decrease when there are symmetries in the material behavior. As the use of matrix notation is more intuitive the fourth order elasticity tensor $//<![CDATA[ \begin{array}{l}C_{ijkl}\end{array} //]]>$ will be represented in Voigt's matrix notation. In the following table crystal structures with an increasing number of symmetries are shown:^[1]

Independent coefficients for isotropic materials

In the table above, the elasticity matrix is the same for cubic and isotropic materials. However, the isotropic matrix contains only two independent coefficients. These can be chosen to be the Lamé parameters $//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$:

The Lamé parameters can be expressed and converted by the Young’s modulus $//<![CDATA[ \begin{array}{l}E\end{array} //]]>$, the shear modulus $//<![CDATA[ \begin{array}{l}G\end{array} //]]>$ and the Poisson’s ratio $//<![CDATA[ \begin{array}{l}\nu\end{array} //]]>$ as follows:

Wave propagation

In ultrasonic testing, which is a non-destructive testing method, the propagation speed of elastic waves plays a crucial role. This velocity is used to transform the time measurements into spatial information. In the following the propagation speed of elastic waves is investigated for an isotropic and a transversely isotropic material. For further symmetry configurations the reader is referred to the literature, e.g. ^[10][11].

Isotropic material

For isotropic materials, the propagation velocity of pressure waves (P-waves) can be determined by Young’s modulus $//<![CDATA[ \begin{array}{l}E\end{array} //]]>$, Poisson’s ratio $//<![CDATA[ \begin{array}{l}\nu\end{array} //]]>$ and the mass density $//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ using ^[12]

The velocity of shear waves (S-waves) is given by

Transversely isotropic material

In a transversely isotropic material one can determine different Young’s moduli in the fiber direction and perpendicular to it.^[13] In the following they are denoted by $//<![CDATA[ \begin{array}{l}E_\parallel\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}E_\bot\end{array} //]]>$, respectively. Additionally, this material shows two Poisson's ratios. $//<![CDATA[ \begin{array}{l}\nu_{\bot,\parallel}\end{array} //]]>$denotes the Poisson's ratio for the deformation perpendicular to the fiber direction with an applied load in the fiber direction. $//<![CDATA[ \begin{array}{l}\nu_{\bot,\bot}\end{array} //]]>$ represents the Poisson's ratio for the deformation perpendicular to the fiber direction due to a load which is applied perpendicular to the deformation and to the fiber direction. As the fifth independent coefficient the shear modulus in fiber direction $//<![CDATA[ \begin{array}{l}G_\parallel\end{array} //]]>$ is used. With these parameters one can determine the values of the elasticity matrix:

Knowing the values in the elasticity matrix, it is possible to calculate the velocities of the different wave types in the different directions.^[13] In the following the velocities of three compression waves ($//<![CDATA[ \begin{array}{l}v_{p,\parallel,\parallel}\end{array} //]]>$,$//<![CDATA[ \begin{array}{l}v_{p,\bot,\bot}\end{array} //]]>$,$//<![CDATA[ \begin{array}{l}v_{p,45°}\end{array} //]]>$)and the velocities of two shear waves($//<![CDATA[ \begin{array}{l}v_{s,\parallel,\bot}\end{array} //]]>$,$//<![CDATA[ \begin{array}{l}v_{s,\bot,\bot}\end{array} //]]>$) are investigated. $//<![CDATA[ \begin{array}{l}v_{p,45°}\end{array} //]]>$ relates to the compressional velocity, which propagates at an angle of 45° with respect to the fiber direction. For the other velocities, the second index represents the propagation direction, the third index refers to the polarization direction.

Literature

1. Constitutive Equations. MIT AERO ASTRO, Lecture Notes. Accessed 22.01.2021
2. Schürmann, H.: Konstruieren mit Faser-Kunststoff-Verbunden. Springer, 2007.
3. Sedlák, P. et al.: Determination of All 21 Independent Elastic Coefficients of Generally Anisotropic Solids by Resonant Ultrasound Spectroscopy: Benchmark Examples. Society for Experimental Mechanics, 2014.
4. Britannica, The Editors of Encyclopaedia.: Monoclinic system. Encyclopedia Britannica, https://www.britannica.com/science/monoclinic-system. Accessed 22.01.2021.
5. Mascia, N. and Rocco Lahr, F.: Remarks on orthotropic elastic models applied to wood. Materials Research 2006.
6. Subhani, M., Li, J. & Samali, B.: A comparative study of guided wave propagation in timber poles with isotropic and transversely isotropic material models. Civil Struct Health Monit 2013. https://doi.org/10.1007/s13349-012-0032-1
7. Nakashima, Philip N.H., The Crystallography of Aluminum and Its Alloys. Encyclopedia of Aluminum and Its Alloys ed. George E. Totten, Murat Tiryakioğlu and Olaf Kessler (Boca Raton: CRC Press,16 Nov 2018),488 –586.
8. Pepperhoff W., Acet M.: The structure of iron. In: Constitution and Magnetism of Iron and its Alloys. Engineering Materials. Springer, 2001. https://doi.org/10.1007/978-3-662-04345-5_1
9. Isotropy vs Anisotropy. Matmatch GmbH. München, 2019. Accessed 22.01.2021
10. Tsvankin, I.: Anisotropic parameters and P-wave velocity for orthorhombic media. Geophysics 1997.
11. Zhang, Y. and Chen, D.: Propagation of Acoustic Wave in Crystals. Multilayer Integrated Film Bulk Acoustic Resonators. Springer, 2013. https://doi.org/10.1007/978-3-642-31776-7_2
12. Ashby, M. F. et al.: Material Classes, Structure, and Properties. Nanomaterials, Nanotechnologies and Design, 2009. https://doi.org/10.1016/B978-0-7506-8149-0.00006-4
13. Wong, R. C. K. et al.: Inherent transversely isotropic elastic parameters of over-consolidated shale measured by ultrasonic waves and their comparison with static and acoustic in situ log measurements. Journal of Geophysics and Engineering, 2008. https://doi.org/10.1088/1742-2132/5/1/011