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.
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:
\sigma_{ij}=C_{ijkl}\;\varepsilon_{kl} \qquad i,j,k,l\; \in \;\{1,2,3\} |
The stresses \sigma_{ij} are visualized for an infinitesimal element with the edge lengths dx_i in Figure 1. The strains \varepsilon_{kl} are labeled the same way. C_{ijkl} represents the fourth order elasticity tensor.
Figure 1: Infinitesimal element of edge size dx_iwith occurring stresses \sigma_{ij}. Source: Own work, CC-BY |
Both, the second order tensors for stress and strain as well as the elasticity tensor are symmetrical, i.e. \sigma_{ij} = \sigma_{ji} etc.
Hooke's Law can also be written in matrix form using Voigt's notation:[1]
\begin{gather} \begin{bmatrix} \sigma_{11} \\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23} \\ \sigma_{13}\\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112}\\ & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212}\\ & & C_{3333} & C_{3323} & C_{3313} & C_{3312}\\ & & & C_{2323} & C_{2313} & C_{2312}\\ & sym. & & & C_{1313} & C_{1312}\\ & & & & & C_{1212} \end{bmatrix} \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\,\varepsilon_{23}\\ 2\,\varepsilon_{13}\\ 2\,\varepsilon_{12}\\ \end{bmatrix} \end{gather} |
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 C_{ijkl} will be represented in Voigt's matrix notation. In the following table crystal structures with an increasing number of symmetries are shown:[1]
Triclinic: \alpha \neq \beta \neq \gamma \neq 90° a \neq b \neq c 21 independent coefficients Symmetry: - no symmetry | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: Fiber-reinforced-plastics, minerals (e.g. talc or albite), superconducting iron-arsenides[2][3] | ||
Monoclinic: \alpha < 90° \beta = \gamma = 90° a \neq b \neq c 13 independent coefficients Symmetry: - reflection about x-yplane | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: Minerals (e.g. gypsum)[4] | ||
Orthotropic: \alpha = \beta = \gamma = 90° a \neq b \neq c 9 independent coefficients Symmetry: - reflection about all three orthogonal planes | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: Fiber-reinforced-plastics, round timber[5] | ||
Transversely isotropic: 5 independent coefficients Symmetry: - reflection about all three orthogonal planes - rotation about z-axis | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: Fiber-reinforced-plastics, round timber[6] | ||
Cubic: \alpha = \beta = \gamma = 90° a = b = c 3 independent coefficients Symmetry: - reflection about all three orthogonal planes - 90° rotation about all three axes | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: | ||
Isotropic:
2 independent coefficients Symmetry: - symmetric to all reflections and rotations | Source: Own work, CC-BY | Elasticity matrix:
Examples of materials: Metals, glasses, polymers, concrete[9] |
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 \lambda and \mu:
\begin{gather} C= \begin{bmatrix} \lambda+2\mu& \lambda & \lambda & 0 & 0 & 0 \\ & \lambda+2\mu & \lambda & 0 & 0 & 0 \\ & & \lambda+2\mu & 0 & 0 & 0 \\ & & & \mu & 0 & 0 \\ & sym. & & & \mu & 0 \\ & & & & & \mu \end{bmatrix} \end{gather} |
The Lamé parameters can be expressed and converted by the Young’s modulus E, the shear modulus G and the Poisson’s ratio \nu as follows:
\begin{align} \lambda = \frac{E\nu}{(1+\nu)\,(1-2\nu)}\;, \qquad \mu=G=\frac{E}{2(1+\nu)} \end{align} |
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].
For isotropic materials, the propagation velocity of pressure waves (P-waves) can be determined by Young’s modulus E, Poisson’s ratio \nu and the mass density \rho using [12]
v_p= \sqrt{\frac{E\,(1-\nu)}{\rho\,(1-\nu)\,(1+\nu)}} |
The velocity of shear waves (S-waves) is given by
v_s= \sqrt{\frac{E}{2\,\rho\,(1+\nu)}} |
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 E_\parallel and E_\bot, respectively. Additionally, this material shows two Poisson's ratios. \nu_{\bot,\parallel}denotes the Poisson's ratio for the deformation perpendicular to the fiber direction with an applied load in the fiber direction. \nu_{\bot,\bot} 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 G_\parallel is used. With these parameters one can determine the values of the elasticity matrix:
&C_{1111}=\frac{E_\bot\,\left(E_\bot\,{\nu_{\bot,\parallel}}^2-E_\parallel\right)}{\left(\nu_{\bot,\bot}+1\right)\,\left(2\,E_\bot\,{\nu_{\bot,\parallel}}^2-E_\parallel+E_\parallel\,\nu_{\bot,\bot}\right)}\\ &C_{3333}=\frac{E_\parallel\,\left(E_\parallel\,\nu_{\bot,\bot}-E_\parallel\right)}{2\,E_\bot\,{\nu_{\bot,\parallel}}^2-E_\parallel+E_\parallel\,\nu_{\bot,\bot}}\\ &C_{2323}=G_\parallel\\ &C_{1212}=\frac{E_\bot}{2\,\nu_{\bot,\bot}+2}\\ &C_{1133}=\frac{E_\bot\,E_\parallel\,\nu_{\bot,\parallel}}{E_\parallel-2\,E_\bot\,{\nu_{\bot,\parallel}}^2-E_\parallel\,\nu_{\bot,\bot}} |
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 (v_{p,\parallel,\parallel},v_{p,\bot,\bot},v_{p,45°})and the velocities of two shear waves(v_{s,\parallel,\bot},v_{s,\bot,\bot}) are investigated. v_{p,45°} 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.
v_{p,\bot,\bot}=\sqrt{\frac{C_{1111}}{\rho}}\\ v_{p,\parallel,\parallel}=\sqrt{\frac{C_{3333}}{\rho}}\\ v_{s,\parallel,\bot}=\sqrt{\frac{C_{2323}}{\rho}}\\ v_{s,\bot,\bot}=\sqrt{\frac{C_{1212}}{\rho}}\\ v_{p,45°}=\frac{1}{4}(\frac{C_{1133}}{\rho}+v_{p,\bot,\bot}^2+2\,v_{s,\parallel,\bot}^2)^{\frac{1}{2}} |