Wave propagation in a nonlinear elastic medium

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 95-96

Section 1.3.7.3. Wave propagation in a nonlinear elastic medium

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.7.3. Wave propagation in a nonlinear elastic medium

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As an example, let us consider the case of a plane finite amplitude wave propagating along the [x_1] axis. The displacement components in this case become $[u_1 = u_1(X_1,t);\quad u_2 = u_2(X_1,t)\semi\quad u_3 = u_3(X_1,t). ]$ Thus, the Jacobian matrix $[\alpha_{ij}]$ reduces to $[J = \pmatrix {\alpha_{11} &0 &0\cr \alpha_{21} &0 &0\cr \alpha_{31} &0 &0\cr}. ]$

The Lagrangian strain matrix is [equation (1.3.6.8)] $[S = \textstyle{1\over 2}\displaystyle \left(J^T J - \delta \right). ]$ The only nonvanishing strain components are, therefore, $[\eqalign{S_{11} &= \textstyle{1\over 2}\displaystyle \left(\alpha_{11}^2 + \alpha_{21}^2 + \alpha_{31}^2\right) -1\cr &= {\partial u_1\over \partial X_1} + \textstyle{1\over 2}\displaystyle \left[\left({\partial u_1\over \partial X_1}\right)^2 + \left({\partial u_2\over \partial X_1}\right)^2 + \left({\partial u_3\over \partial X_1}\right)^2\right]\cr S_{12} &= S_{21} = \textstyle{1\over 2}\displaystyle{\partial u_2\over \partial X_1}\cr S_{13} &= S_{31} = \textstyle{1\over 2}\displaystyle{\partial u_3\over \partial X_1}\cr} ]$ and the strain invariants reduce to $[I_1 = S_{11} ; \ \ I_2 = - (S_{12}S_{21} + S_{13}S_{31}); \ \ I_3 = 0. ]$

1.3.7.3.1. Isotropic media

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In this case, the strain-energy density becomes $[\eqalignno{\Phi &= \textstyle{1\over 2}\displaystyle (\lambda+2\mu)(S_{11})^2 + 2 \mu(S_{12}S_{21} + S_{13}S_{31}) + \textstyle{1\over 3}\displaystyle (l+2m)(S_{11}){}^3&\cr &\quad + 2m S_{11}\left(S_{12}S_{21} + S_{13}S_{31}\right). &(1.3.7.6)\cr} ]$ Differentiating (1.3.7.6) with respect to the strains, we get $[\openup4.5pt\eqalign{{\partial \Phi \over \partial S_{11}} & = (\lambda+2\mu)S_{11} + (l+2m)(S_{11}){}^2 + 2m \left(S_{12}S_{21} + S_{13}S_{31}\right)\cr {\partial \Phi \over \partial S_{12}} & =2 \mu S_{21} + 2m S_{11}S_{21}\cr {\partial \Phi \over \partial S_{13}} & = 2\mu S_{31} + 2m S_{11}S_{31}\cr {\partial \Phi \over \partial S_{21}} & = 2 \mu S_{12} + 2m S_{11}S_{12}\cr {\partial \Phi \over \partial S_{31}} & =2\mu S_{13} + 2m S_{11}S_{13}.\cr} ]$ All the other $[{\partial \Phi / \partial S_{ij}} = 0]$ .

From (1.3.7.5), we derive the stress components: $[\openup6pt\eqalign{T_{11} &= \alpha_{1k}{\partial \Phi \over \partial S_{1k}}\semi \ \ T_{12} = \alpha_{2k}{\partial \Phi \over \partial S_{1k}}\semi \ \ T_{13} = \alpha_{3k}{\partial \Phi \over \partial S_{1k}}\semi \cr T_{21} &= \alpha_{1k}{\partial \Phi \over \partial S_{2k}}\semi \ \ T_{22} = \alpha_{2k}{\partial \Phi \over \partial S_{2k}}\semi \ \ T_{23} = \alpha_{3k}{\partial \Phi \over \partial S_{2k}}\semi\cr T_{31} &= \alpha_{1k}{\partial \Phi \over \partial S_{3k}}\semi \ \ T_{32} = \alpha_{2k}{\partial \Phi \over \partial S_{3k}}\semi \ \ T_{33} = \alpha_{3k}{\partial \Phi \over \partial S_{3k}}.\cr} ]$ Note that this tensor is not symmetric.

For the particular problem discussed here, the three components of the equation of motion are $[\eqalign{\rho u_1'' &= {\rm d}T_{11}/{\rm d}X_1,\cr \rho u_2'' &= {\rm d}T_{21}/{\rm d}X_1,\cr \rho u_3'' &= {\rm d}T_{31}/{\rm d}X_1.\cr} ]$

If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion: $[\eqalign{\rho u_{1}'' &= (\lambda + 2\mu){\partial^{2} u_{1} \over \partial X_{1}^{2}} + [3(\lambda + 2\mu) + 2(l + 2m)] {\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\cr &\quad + (\lambda + 2\mu + m) \left[{\partial u_{2} \over \partial X_{1}}{\partial^{2} u_{2} \over \partial X_{1}^{2}} + {\partial u_{3} \over \partial X_{1}}{\partial^{2} u_{3} \over \partial X_{1}^{2}}\right]\cr \rho u_{2}'' &= \mu{\partial^{2} u_{2} \over \partial X_{1}^{2}} + (\lambda + 2\mu + m) \left[{\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{2} \over \partial X_{1}^{2}} + {\partial u_{2} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\right]\cr \rho u_{3}'' &= \mu{\partial^{2} u_{3} \over \partial X_{1}^{2}} + (\lambda + 2\mu + m)\left[{\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{3} \over \partial X_{1}^{2}} + {\partial u_{3} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\right].\cr} \eqno(1.3.7.7) ]$

1.3.7.3.2. Cubic media (most symmetrical groups)

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In this case, the strain-energy density becomes $[\eqalignno{\Phi &= \textstyle{1\over 2}\displaystyle c_{11} (S_{11})^{2} + c_{44} \left[(S_{12})^{2} + (S_{21})^{2} + (S_{31})^{2} + (S_{13})^{2}\right]&\cr &\quad+ c_{111}(S_{11})^{3} + \textstyle{1\over 2}\displaystyle c_{166} S_{11} \left[(S_{12})^{2} + (S_{21})^{2} + (S_{31})^{2}+ (S_{13})^{2}\right].\cr &&(1.3.7.8)\cr} ]$ Differentiating (1.3.7.8) with respect to the strain, one obtains $[\eqalign{{\partial \Phi \over S_{11}} &= c_{11} S_{11} + 3c_{111}(S_{11})^{2} + \textstyle{1\over 2}\displaystyle c_{166} [(S_{12})^{2} + (S_{21})^{2}\cr&\quad + (S_{31})^{2} + (S_{13})^{2})]\cr {\partial \Phi \over S_{21}} &= 2c_{44}S_{21} + c_{166}S_{11}S_{21}\cr \noalign{\vskip6pt}{\partial \Phi \over S_{31}} &= 2c_{44}S_{31} + c_{166}S_{11}S_{31}.\cr} ]$ All other $[\partial \Phi / S_{ij} = 0]$ . From (1.3.7.5), we derive the stress components: $[\eqalign{T_{11} &= \alpha_{1k}{\partial \Phi \over \partial S_{1k}}\cr \noalign{\vskip5pt} T_{21} &= \alpha_{1k}{\partial \Phi \over \partial S_{2k}}\cr\noalign{\vskip5pt} T_{31} &= \alpha_{1k}{\partial \Phi \over \partial S_{3k}}.\cr} ]$ In this particular case, the three components of the equation of motion are $[\eqalign{\rho u_{1}'' &= \hbox{d}T_{11}/\hbox{d}X_{1}\cr \rho u_{2}'' &= \hbox{d}T_{21}/\hbox{d}X_{1}\cr \rho u_{3}'' &= \hbox{d}T_{31}/\hbox{d}X_{1}.\cr} ]$

If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion: $[\eqalign{\rho u_{1}'' &= c_{11}{\partial^{2} u_{1} \over \partial X_{1}^{2}} + [3c_{11} + c_{111}] {\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\cr &\quad + (c_{11} + c_{166}) \left[{\partial u_{2} \over \partial X_{1}}{\partial^{2} u_{2} \over \partial X_{1}^{2}} + {\partial u_{3} \over \partial X_{1}}{\partial^{2} u_{3} \over \partial X_{1}^{2}}\right]\cr \rho u_{2}'' &= c_{44}{\partial^{2} u_{2} \over \partial X_{1}^{2}} + (c_{11} + c_{166}) \left[{\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{2} \over \partial X_{1}^2} + {\partial u_{2} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\right]\cr \rho u_{3}'' &= c_{44}{\partial^{2} u_{3} \over \partial X_{1}^{2}} + (c_{11} + c_{166}) \left[{\partial u_{1} \over \partial X_{1}}{\partial^{2} u_{3} \over \partial X_{1}^{2}} + {\partial u_{3} \over \partial X_{1}}{\partial^{2} u_{1} \over \partial X_{1}^{2}}\right],\cr} \eqno(1.3.7.9) ]$ which are identical to (1.3.7.7) if we put $[c_{11} = \lambda + 2\mu\semi \quad c_{44} = \mu\semi \quad c_{111} = 2(l + 2m)\semi \quad c_{166} = m. ]$

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 95-96