1 Introduction
We are interested in revealing the manifestation of the anisotropic properties of elastic materials by a collinear propagation of weakly nonlinear quasi-shear plane waves. We analyze the effect of an anisotropy of the elastic material on the form of evolution equations for transverse wave amplitudes. It is known that changes in the amplitudes of weakly nonlinear transverse or quasi-transverse waves in isotropic elastic media are governed by cubically nonlinear evolution equations [1], and quadratic nonlinearity cannot occur in these equations [1]. In the language used in acoustics, this fact is often formulated as follows: second harmonic generation is impossible for shear waves in isotropic materials. However, the presence of fibers in a nonlinear material changes the situation drastically. It turns out that choosing a special direction of transverse wave propagation in an anisotropic material may actually cause the generation of a second harmonic. This manifests itself with the presence of quadratically nonlinear terms in the evolution equations for shear wave amplitudes. Among particular fiber directions we are especially interested in a threefold symmetry acoustic axis. When shear waves propagate along such chosen fiber directions, a quadratically nonlinear coupling occurs in a special way: evolution equations that describe the coupling of a pair of transverse waves are particularly simple in this case because they contain only one parameter responsible for a nonlinear coupling. Furthermore, coupled equations can be transformed into a single inviscid complex Burgers equation.
Using the expansion of weakly nonlinear geometric optics we have shown [2] what the quadratically nonlinear coupled equations for quasi-shear waves in anisotropic materials look like. Our aim in this work is to derive equations similar to those in [2] but by a different method. Here, unlike the presentation in [2] (see also [3‐8]), where the method of weakly nonlinear geometric optics was used, we will apply a double-scale expansion. Instead of introducing a new fast variable and applying geometric optics-type asymptotics, which leads to evolution equations with three independent variables, we introduce here just two new independent variables: a slow time variable \(\tau \) and a characteristic variable \(\theta \). We concentrate on a special fiber direction of wave propagation, namely, a threefold symmetry acoustic axis.
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The paper is organized as follows. After the introduction we present the main equations. Next, in Sect. 3, the double-scale asymptotics is applied and the main evolution equations are derived. The modeling equations for a pair of shear waves propagating along a special fiber direction, which is the threefold symmetry acoustic axis, are then reduced to a single inviscid complex Burgers equation. Some properties of this equation are discussed. In the following section we illustrate our general considerations on examples of a cubic crystal and an icosahedral quasicrystal. The paper ends with concluding remarks.
2 Main equations
2.1 Equations of motion of a continuum
The equation of motion written in Lagrangian coordinates has the formwhere the first Piola–Kirchhoff stress tensor iswith the deformation gradient \({{\mathbf {F}}}\) and the Green strain \({{\mathbf {E}}}\). Here we neglect body forces. The particle velocity is related to the displacement byand the deformation gradient is given by the formulawhile the Green strain tensor isWe supplement Eq. (1) by the relation between the time derivative of the deformation gradient and the gradient of velocity that follows from (3) and (4):Therefore, Eqs. (1) and (6) form our system of conservation laws describing the motion of a continuum. We take \(\rho _0 = 1\) for simplicity.
$$\begin{aligned} \rho _0 \frac{\partial {{\mathbf {v}}}}{\partial t}= \nabla \cdot {{\mathbf {T}}}, \end{aligned}$$
(1)
$$\begin{aligned} {{\mathbf {T}}} = {{\mathbf {F}}}\frac{\partial W}{\partial {{\mathbf {E}}}}, \end{aligned}$$
(2)
$$\begin{aligned} {{\mathbf {v}}} = \frac{\partial {{\mathbf {u}}}}{\partial t}, \end{aligned}$$
(3)
$$\begin{aligned} {{\mathbf {F}}} = {{\mathbf {I}}} + \nabla {{\mathbf {u}}}, \end{aligned}$$
(4)
$$\begin{aligned} {{\mathbf {E}}} = \frac{1}{2} ({{\mathbf {F}}}^\mathrm{{T}}{{\mathbf {F}}} - {{\mathbf {I}}}). \end{aligned}$$
(5)
$$\begin{aligned} \frac{\partial {{\mathbf {F}}}}{\partial t} = \nabla {{\mathbf {v}}}. \end{aligned}$$
(6)
2.2 Constitutive relations
The elastic material is specified by its strain energy \(W\), which is assumed to have the following expansion for small strains:where the summation convention on repeated subscripts \(a,b,\cdots \) is assumed. The symmetry of the strain implies the relationswhich imply no more than 21 and 56 second- and third-order moduli, respectively.
$$\begin{aligned} W = \frac{1}{2!}\, c_{abcd}\, E_{ab}\, E_{cd} + \frac{1}{3!}\,c_{abcdef}\, E_{ab}\, E_{cd}\, E_{ef} + \cdots , \end{aligned}$$
(7)
$$\begin{aligned} c_{abcd} = c_{cdab} = c_{bacd}, \qquad c_{abcdef} = c_{cdabef} = c_{cdefab} =c_{bacdef}, \end{aligned}$$
(8)
Equations (2), (4), and (7) together imply that the first Piola–Kirchhoff stress iswhereandNote that \(M_{abcdef}\ne M_{bacdef} \), which implies that the asymmetry of \(\mathbf T\) is a second-order effect.
$$\begin{aligned} T_{ab} = c_{abcd}\, \frac{\partial u_c}{\partial x_d} + \frac{1}{2} M_{abcdef}\, \frac{\partial u_c}{\partial x_d} \frac{\partial u_e}{\partial x_f} + \frac{1}{6} N_{abcdefgh}\, \frac{\partial u_c}{\partial x_d} \frac{\partial u_e}{\partial x_f}\frac{\partial u_g}{\partial x_h} + \cdots ~, \end{aligned}$$
(9)
$$\begin{aligned} M_{abcdef} = c_{abcdef} + c_{abdf} \, \delta _{ce} + c_{bcdf} \, \delta _{ae} + c_{bdef} \, \delta _{ac} \end{aligned}$$
(10)
$$\begin{aligned} N_{abcdefgh}&= \frac{1}{2} c_{abcdefgh} + c_{chbcdef} \, \delta _{ag}+ c_{abdfgh} \, \delta _{ce} + \frac{1}{2} c_{abcdfh} \,\delta _{eg} \nonumber \\&\quad + \frac{1}{2} c_{bdefgh} \, \delta _{ac} + c_{bdhf} \, \delta _{ce} \, \delta _{ag} + \frac{1}{2} c_{bdfh} \, \delta _{eg} \, \delta _{ac}. \end{aligned}$$
(11)
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2.3 Plane waves
For plane waves, \({{\mathbf {u}}}\) depends only on one component of \({{\mathbf {x}}}\), say \(x = {{\mathbf {x}}}\cdot {{\mathbf {n}}}\), so thatThus,whereThe \(3\times 3\) matrix \({\mathbf {B}} = {{\mathbf {B}}}({\mathbf {n}})\) iswhere \({\varvec{\Lambda }}({{\mathbf {n}}})\), \({\varvec{\Psi }}({{\mathbf {n}}})\), and \({\varvec{\Pi }}({{\mathbf {n}}})\) are, in components,
or, in short,
$$\begin{aligned} \nabla {{\mathbf {u}}} = {{\mathbf {m}}}\otimes {{\mathbf {n}}},\quad \text{ where }\quad {{\mathbf {m}}} = \frac{\partial {{\mathbf {u}}}}{\partial x}. \end{aligned}$$
(12)
$$\begin{aligned} \frac{\partial {{\mathbf {w}}}}{\partial t} + {{\mathbf {A}}}({{\mathbf {w}}}) \frac{\partial {{\mathbf {w}}}}{\partial x}= {{\mathbf {0}}}, \end{aligned}$$
(13)
$$\begin{aligned} {{\mathbf {w}}} = \left( \begin{array}{c} {{\mathbf {v}}} \\ {{\mathbf {m}}} \end{array} \right) , \qquad {{\mathbf {A}}} ({{\mathbf {w}}}) = - \left( \begin{array}{lr} {{\mathbf {0}}}&{}\quad {{\mathbf {B}}}\\ {{\mathbf {I}}} &{}\quad {{\mathbf {0}}} \end{array} \right) . \end{aligned}$$
(14)
$$\begin{aligned} {{\mathbf {B}}}({\mathbf {n}}) = {\varvec{\Lambda }}({{\mathbf {n}}}) + {\varvec{\Psi }}({{\mathbf {n}}}){{\mathbf {m}}} + \frac{1}{2} {\varvec{\Pi }}({{\mathbf {n}}}){{\mathbf {m}}}{{\mathbf {m}}} + \cdots , \end{aligned}$$
(15)
$$\begin{aligned}&\Lambda _{ac}({{\mathbf {n}}}) =c_{abcd}n_bn_d,\end{aligned}$$
(16a)
$$\begin{aligned}&\Psi _{ace}({{\mathbf {n}}}) = M_{abcdef} n_bn_dn_f,\end{aligned}$$
(16b)
$$\begin{aligned}&\Pi _{aceg}({{\mathbf {n}}}) = N_{abcdefgh}n_bn_dn_fn_h, \end{aligned}$$
(16c)
$$\begin{aligned} B_{ac} =c_{abcd}n_bn_d + M_{abcdef} n_bn_dn_f \,m_e+\frac{1}{2} N_{abcdefgh}n_bn_dn_fn_h\, m_em_g + \cdots . \end{aligned}$$
(17)
2.4 Christoffel tensor
Matrix \({{\mathbf {A}}}({{\mathbf {0}}})\) has the following form:The positive-definite property of the second-order moduli implies that \({\varvec{\Lambda }}\), also known as the Christoffel or acoustical tensor, has in general the following spectral form:where \(\alpha _j >0\), and \(\{ {{\mathbf {k}}}_1,{{\mathbf {k}}}_2,{{\mathbf {k}}}_3\}\) is an orthonormal triad of vectors. The six eigenvalues of \({{\mathbf {A}}}({{\mathbf {0}}})\) therefore split into three pairs (which may coincide) with opposite signs:
The corresponding right and left eigenvectors of \({{\mathbf {A}}}({{\mathbf {0}}})\) are, respectively,Note that \({{\mathbf {l}}}_{i} \cdot {{\mathbf {r}}}_{j} = \delta _{ij}\).
$$\begin{aligned} {{\mathbf {A}}} ({{\mathbf {0}}}) = - \left( \begin{array}{lr} {{\mathbf {0}}}&{}\quad {\varvec{\Lambda }}\\ {{\mathbf {I}}} &{}\quad {{\mathbf {0}}} \end{array} \right) . \end{aligned}$$
(18)
$$\begin{aligned} {\varvec{\Lambda }} = \alpha _1 {{\mathbf {k}}}_1\otimes {{\mathbf {k}}}_1 + \alpha _2 {{\mathbf {k}}}_2\otimes {{\mathbf {k}}}_2 + \alpha _3 {{\mathbf {k}}}_3\otimes {{\mathbf {k}}}_3, \end{aligned}$$
(19)
$$\begin{aligned} \lambda _1&= - \sqrt{\alpha _1} = - \lambda _2,\end{aligned}$$
(20a)
$$\begin{aligned} \lambda _3&= - \sqrt{\alpha _2} = - \lambda _4,\end{aligned}$$
(20b)
$$\begin{aligned} \lambda _5&= - \sqrt{\alpha _3} = - \lambda _6. \end{aligned}$$
(20c)
$$\begin{aligned} {{\mathbf {r}}}_{2j-1}&= \begin{pmatrix} - \lambda _{2j-1} {{\mathbf {k}}}_j \\ {{\mathbf {k}}}_j \end{pmatrix}, \qquad {{\mathbf {r}}}_{2j} = \begin{pmatrix} - \lambda _{2j} {{\mathbf {k}}}_j \\ {{\mathbf {k}}}_j \end{pmatrix}, \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad j=1,2,3. \nonumber \\ {{\mathbf {l}}}_{2j-1}&= \frac{1}{2} \begin{pmatrix} - \lambda _{2j-1}^{-1} {{\mathbf {k}}}_j \\ {{\mathbf {k}}}_j \end{pmatrix}, \qquad {{\mathbf {l}}}_{2j} = \frac{1}{2} \begin{pmatrix} - \lambda _{2j}^{-1} {{\mathbf {k}}}_j \\ {{\mathbf {k}}}_j \end{pmatrix}. \end{aligned}$$
(21)
The property \(\alpha _j > 0\) implies that all six eigenvalues \(\lambda _k,\, k = 1, 2, \ldots , 6\) of \({{\mathbf {A}}}({{\mathbf {0}}})\) are real. We assume that all the right and left eigenvectors of \({{\mathbf {A}}}({{\mathbf {0}}})\) form a full set of linearly independent eigenvectors. The system is therefore hyperbolic at the origin. We are interested in the case where matrix \({{\mathbf {A}}}({{\mathbf {0}}})\) has a pair of coinciding eigenvalues. Therefore, we assume that there is a fiber direction \({\mathbf {n}}\) along which a pair of shear waves propagates with a common phase velocity \(\lambda \) at the origin. Such a direction is called an acoustic axis; see, for example, [9]. We can then assume that the Christoffel tensor \({\varvec{\Lambda }}\) from (19) takes the following form [9]:that is, if \( \alpha _1= \alpha _2\) in (20).
$$\begin{aligned} {\varvec{\Lambda }} = \alpha _1 ( {{\mathbf {I}}} - {{\mathbf {k}}}_3\otimes {{\mathbf {k}}}_3 ) + \alpha _3 {{\mathbf {k}}}_3\otimes {{\mathbf {k}}}_3, \end{aligned}$$
(22)
3 Evolution equations
3.1 Asymptotic expansion
In this section we apply a double-scale expansion method to our problem. Let us consider an initial-value problem (IVP) for a quasilinear hyperbolic system:where \(\epsilon \) is a small parameter, \( {\varvec{w}}^\epsilon = [{ \varvec{v}}^\epsilon , { \varvec{m}}^\epsilon ]^\mathrm{{T}}\), and matrix \(\varvec{A}\) is as in (18). We assume that the hyperbolic system (23) is not strictly hyperbolic at the constant state \({\mathbf {0}}\), and that \(\lambda _1({\mathbf {0}}) = \lambda _2({\mathbf {0}}) \equiv \lambda \) is a double eigenvalue of matrix \({{\mathbf {A}}}(0)\) with corresponding left \({{\mathbf {l}}}_k \) and right \({{\mathbf {r}}}_k\) eigenvectors, respectively, \(k = 1,2\), and \(\varvec{l_i} \cdot \varvec{r_j} = \delta _{ij}\), where \(\delta _{ij}\) is Kronecker’s delta.
$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial \varvec{w}^\epsilon }{\partial t}+\varvec{A}({\varvec{w}}^\epsilon )\,\frac{\partial \varvec{w}^\epsilon }{\partial x}=\mathbf {0}, \qquad \varvec{w}^{\epsilon }(0,x)= \epsilon \,\varvec{w}_{1}(0,x), \end{array} \end{aligned}$$
(23)
We assume a Taylor expansion around the zero constant state:where
We seek a solution to IVP (23) around a \({\varvec{0}} \) constant state in the formWe introduce two new variables: a slow time \( \tau = \epsilon t\) and a characteristic \( \theta = x - \lambda t\). We treat them as new independent variables and apply the method of double-scale asymptotics. Let us first calculate derivatives with respect to the old variables \(t\) and \(x\) and express them in terms of the derivatives with respect to the new variables \(\tau \) and \( \theta \). We have
Taking into account (24), (28), and (29) we obtainNow, gathering together terms that appear with like powers of a small parameter \(\epsilon \), we equate these consecutive terms to zero and obtain the following equations:Since we focus on shear waves, we take a solution of these equations in the formwith \(a_j(\tau ,\theta )\) the unknown amplitudes of quasi-shear waves and \(\varvec{r}_j\) the corresponding eigenvectors of matrix \(\varvec{A}({\varvec{0}})\);withThe inhomogeneous algebraic equation (33) has a nonzero solution provided the right-hand side of (33) is orthogonal to the left eigenvectors of matrix \( \varvec{A}({\varvec{0}})\). Let us take two such eigenvectors \(\varvec{l}_1\) and \(\varvec{l}_2 \), which correspond to a pair of shear waves propagating in the reference state with a common phase speed \(\lambda \). From the solvability condition we have in particular
Let us introduce the following notation:for the interaction coefficient. Taking into account (25), (32), and (34) we can write conditions (35) and (36) as follows:
We obtain the pair of coupled nonlinear partial differential equations that are the asymptotic evolution equations for the amplitudes of the pair of quasi-shear elastic waves in the vicinity of a double umbilic point.
$$\begin{aligned} \varvec{A}({\varvec{w}}^\epsilon ) = \varvec{A}(\varvec{0} + \epsilon \tilde{\varvec{w}}) = \varvec{A}(\varvec{0}) + \epsilon \mathcal {B} \; \tilde{\varvec{w}} + \displaystyle \frac{1}{2}\epsilon ^2 \mathcal {C} \tilde{\varvec{w}}\tilde{\varvec{w}} + \mathcal {O} (\epsilon ^3), \end{aligned}$$
(24)
$$\begin{aligned}&\mathcal {B} \;\tilde{\varvec{w}} \equiv \nabla _{\varvec{u}}\left( \varvec{A}(\varvec{w}) \tilde{\varvec{w}}\right) \!\!\big |_{{\varvec{w}=\varvec{0}} \big .},\end{aligned}$$
(25)
$$\begin{aligned}&\mathcal {C} \tilde{\varvec{w}}\tilde{\varvec{w}} \equiv \nabla _{\varvec{u}} ( \nabla _{\varvec{u} }\left( \varvec{A}(\varvec{w}) \tilde{\varvec{w}} ) \tilde{\varvec{w}}\right) \!\!\big |_{{\varvec{w}= \varvec{0}} \big .}. \end{aligned}$$
(26)
$$\begin{aligned} \varvec{w}^{\epsilon }(t,x) = \epsilon \varvec{w}_1(\tau ,\theta ) + \epsilon ^2 \varvec{w}_2(\tau ,\theta ) + \mathcal {O}(\epsilon ^3). \end{aligned}$$
(27)
$$\begin{aligned} \displaystyle \frac{\partial \varvec{w}^\epsilon }{\partial t}&= \epsilon \frac{\partial {\varvec{w}_1}}{{\partial {\tau }}}\frac{\partial {\tau }}{\partial {t}} + \epsilon \frac{\partial {\varvec{w}_1}}{{\partial \theta }} \frac{\partial {\theta }}{\partial {t}} + \epsilon ^2 \frac{\partial {\varvec{w}_2}}{{\partial \tau }} \frac{\partial {\tau }}{\partial {t}} + \epsilon ^2 \frac{\partial {\varvec{w}_2}}{{\partial \theta }} \frac{\partial {\theta }}{\partial {t}} + \mathcal {O}(\epsilon ^3) \nonumber \\&= \epsilon (- \lambda ) \frac{\partial {\varvec{w}_1}}{\partial {\theta }} + \epsilon ^2 \left( \frac{\partial {\varvec{w}_1}}{{\partial {\tau }}} - \lambda \frac{\partial {\varvec{w}_2}}{{\partial \theta }} \right) + \mathcal {O}(\epsilon ^3),\end{aligned}$$
(28)
$$\begin{aligned} \displaystyle \frac{\partial \varvec{w}^\epsilon }{\partial x}&= \epsilon \frac{\partial {\varvec{w}_1}}{{\partial {\theta }}}\frac{\partial {\theta }}{\partial {x}} + \epsilon ^2 \frac{\partial {\varvec{w}_2}}{{\partial \theta }} \frac{\partial {\theta }}{\partial {x}} + \mathcal {O}(\epsilon ^3) = \epsilon \frac{\partial {\varvec{w}_1}}{{\partial \theta }} + \epsilon ^2 \frac{\partial {\varvec{w}_2}}{\partial {\theta }} + \mathcal {O}(\epsilon ^3). \end{aligned}$$
(29)
$$\begin{aligned} \displaystyle \frac{\partial \varvec{w}^\epsilon }{\partial t}+\varvec{A}({ \varvec{w}}^\epsilon )\,\frac{\partial \varvec{w}^\epsilon }{\partial x} = \epsilon \left( \varvec{A}( {\varvec{0}}) - \lambda \varvec{I} \right) \frac{\partial {\varvec{w}_1}}{{\partial \theta }} + \epsilon ^2\left( \frac{\partial {\varvec{w}_1}}{{\partial {\tau }}} + \mathcal {B} \varvec{w}_1 \frac{\partial {\varvec{w}_1}}{{\partial \theta }}+ (\varvec{A}({\varvec{0}}) - \lambda \varvec{I}) \frac{\partial {\varvec{w}_2}}{{\partial \theta }} \right) + \mathcal {O}(\epsilon ^3). \;\;\;\;\;\;\; \end{aligned}$$
(30)
$$\begin{aligned} \!\!\bullet \quad {\mathcal {O}}(\epsilon )\,\, \hbox {terms vanish} \Leftrightarrow \displaystyle (\varvec{A}({\varvec{0}}) - \lambda _j \varvec{I})\,\frac{\partial {\varvec{w}_1}}{{\partial \theta }}= \mathbf{0}. \end{aligned}$$
(31)
$$\begin{aligned} \varvec{w}_1(\tau ,\theta ) = a_1(\tau ,\theta )\;\varvec{r}_1 + a_2(\tau ,\theta )\;\varvec{r}_2, \end{aligned}$$
(32)
$$\begin{aligned} \!\!\bullet \quad {\mathcal {O}}(\epsilon ^{2})\,\, \hbox {terms vanish} \Leftrightarrow (\varvec{A}({\varvec{0}}) - \lambda \varvec{I})\, \displaystyle \frac{\partial {\varvec{w}_2}}{{\partial \theta }} = - \varvec{\mathcal {F}}, \end{aligned}$$
(33)
$$\begin{aligned} \displaystyle \varvec{\mathcal {F}} \equiv \, \mathcal {B}\, \varvec{w}_{1}\, \frac{\partial {\varvec{w}_1}}{{\partial \theta }}\, + \frac{\partial {\varvec{w}_1}}{{\partial \tau }}. \end{aligned}$$
(34)
$$\begin{aligned} \varvec{l}_1 \cdot \varvec{\mathcal {F}}&= 0,\end{aligned}$$
(35)
$$\begin{aligned} \varvec{l}_2 \cdot \varvec{\mathcal {F}}&= 0. \end{aligned}$$
(36)
$$\begin{aligned} {} \Gamma ^j_{pq} = \varvec{l}_j \cdot \big ( \nabla _{\varvec{w}}\,\varvec{A}({\varvec{w}} ){\big |_{\varvec{w}=\varvec{0}} \big . }\big )\, \varvec{r}_p \, \varvec{r}_q \end{aligned}$$
(37)
$$\begin{aligned}&\!\!\!\displaystyle { \frac{\partial a_{1}}{\partial \tau } + \Gamma ^1_{11} a_{1}\frac{\partial a_{1}}{\partial \theta } + \Gamma ^1_{21} a_{2}\frac{\partial a_{1}}{\partial \theta } + \Gamma ^1_{12}a_{1}\frac{\partial a_{2}}{\partial \theta } + \Gamma ^1_{22}a_{2}\frac{\partial a_{2}}{\partial \theta } = 0,}\end{aligned}$$
(38)
$$\begin{aligned}&\!\!\!\displaystyle { \frac{\partial a_{2}}{\partial \tau } + \Gamma ^2_{11}a_{1}\frac{\partial a_{1}}{\partial \theta } + \Gamma ^2_{21}a_{2}\frac{\partial a_{1}}{\partial \theta } + \Gamma ^2_{12}a_{1}\frac{\partial a_{2}}{\partial \theta } + \Gamma ^2_{22}a_{2}\frac{\partial a_{2}}{\partial \theta } = 0.} \end{aligned}$$
(39)
3.2 Interaction coefficients
The interaction coefficients can be represented aswhere \({\lfloor ~ \rfloor }\) denotes the floor function, that is, the largest integer less than or equal to a given number. Using Cartesian components with the notation \({{\mathbf {k}}}_j \equiv k_1^j {{\mathbf {e}}}_1 + k_2^j{{\mathbf {e}}}_2 + k_3^j{{\mathbf {e}}}_3\), we can express further the interaction coefficients as follows:The assumption of hyperelasticity (2) implies that interaction coefficients (37) have the following symmetry property:It follows from the preceding formulas that the coefficients have in our case, in addition to the general property (42), the following indicial symmetries:This makes them totally symmetric under the interchange of indices. In particular, note that
$$\begin{aligned} \Gamma ^j_{p,q} = \frac{1}{2} \lambda _j^{-1}{{\mathbf {k}}}_{\lfloor \! \frac{j+1}{2}\!\rfloor } \cdot {\varvec{\Psi }}\, {{\mathbf {k}}}_{\lfloor \! \frac{p+1}{2}\!\rfloor }{{\mathbf {k}}}_{\lfloor \! \frac{q+1}{2}\!\rfloor }, \end{aligned}$$
(40)
$$\begin{aligned} \Gamma ^j_{p,q}&= \frac{1}{2} \lambda _j^{-1}\left[ c_{abcdef}n_bn_dn_f k_a^{\lfloor \! (j+1)/2\!\rfloor } k_c^{\lfloor \! (p+1)/2\!\rfloor } k_e^{\lfloor \! (q+1)/2\!\rfloor }\right. \nonumber \\&\left. \qquad + c_{abcd}n_bn_cn_d \big ( k_a^{\lfloor \! (j+1)/2\!\rfloor } \delta _{pq} + k_a^{\lfloor \! (p+1)/2\!\rfloor }\delta _{jq} + k_a^{\lfloor \! (q+1)/2\!\rfloor }\delta _{jp}\big ) \right] \nonumber \\&= \frac{1}{2} \lambda {_j}^{-1} c_{abcdef}n_bn_dn_f k_a^{\lfloor \! (j+1)/2\!\rfloor } k_c^{\lfloor \! (p+1)/2\!\rfloor } k_e^{\lfloor \! (q+1)/2\!\rfloor } \nonumber \\&\qquad + \frac{1}{2}\lambda {_j} {{\mathbf {n}}}\cdot \big ( {{\mathbf {k}}}_{\lfloor \! (j+1)/2\!\rfloor } \delta _{pq} + {{\mathbf {k}}}_{\lfloor \! (p+1)/2\!\rfloor } \delta _{jq} + {{\mathbf {k}}}_{\lfloor \! (q+1)/2\!\rfloor } \delta _{jp}\big ). \end{aligned}$$
(41)
$$\begin{aligned} \Gamma ^j_{p,q} = \Gamma ^j_{q,p}. \end{aligned}$$
(42)
$$\begin{aligned} \Gamma ^j_{p,q} = \Gamma ^p_{j,q}. \end{aligned}$$
(43)
$$\begin{aligned} \Gamma ^{1}_{2,2} = \Gamma ^{2}_{1,2}&\equiv \Gamma ^{1}_{2},\end{aligned}$$
(44a)
$$\begin{aligned} \Gamma ^{2}_{1,1} = \Gamma ^{1}_{1,2}&\equiv \Gamma ^{2}_{1}. \end{aligned}$$
(44b)
Using the notation \(\Gamma ^{1} \equiv \Gamma ^{1}_{1,1} \) and \(\Gamma ^{2} = \Gamma ^{2}_{2,2}\), and summarizing we have the following lemma.
Lemma 1
The evolution equations for the amplitudes of shear waves propagating along an acoustic axis are
The nonlinear terms in the equations involve four coefficients: \(\Gamma _1,\, \Gamma _{2},\, \Gamma ^{2}_{1}\), and \(\Gamma _{2}^{1}\), the last two of which determine the coupling between the equations.
$$\begin{aligned} \frac{\partial a_1}{\partial \tau } + \frac{1}{2} \bigg ( \Gamma _1 \frac{\partial a_1^2}{\partial \theta } + 2 \Gamma ^{2}_{1} \frac{\partial (a_1 a_{2})}{\partial \theta } +\Gamma ^{1}_{2} \frac{\partial a_{2}^2}{\partial \theta } \bigg )&= 0,\end{aligned}$$
(45a)
$$\begin{aligned} \frac{\partial a_{2}}{\partial \tau } + \frac{1}{2} \bigg ( \Gamma ^{2}_{1} \frac{\partial a_1^2}{\partial \theta } +2 \Gamma ^{1}_{2} \frac{\partial (a_1 a_{2})}{\partial \theta } +\Gamma _{2} \frac{\partial a_{2}^2}{\partial \theta } \bigg )&= 0. \end{aligned}$$
(45b)
3.3 Threefold axis
Evolution equations simplify even more if we assume that the propagation direction is a threefold symmetry axis. In this case we obtain that \(\Gamma _1 + \Gamma ^1_{2} = 0 \) and that \( \Gamma _{2} = \Gamma _1^{2}=0 \) (see [6] or [2] for details). Therefore, we can formulate the following lemma.
Lemma 2
The evolution equations for the amplitudes of shear waves propagating along an acoustic axis that is a threefold symmetry axis are
Only one coefficient, \(\Gamma _{2}\), determines the coupling between the evolution equations when the propagation direction is a threefold symmetry axis.
$$\begin{aligned} \frac{\partial a_1}{\partial \tau } + \frac{1}{2} \Gamma _{2} \bigg ( \frac{\partial a_1^2}{\partial \theta } -\frac{\partial a_{2}^2}{\partial \theta } \bigg )&= 0,\end{aligned}$$
(46a)
$$\begin{aligned} \frac{\partial a_{2}}{\partial \tau } - \Gamma _{2} \frac{\partial (a_1 a_{2})}{\partial \theta }&= 0. \end{aligned}$$
(46b)
In the next lemma we reveal the formula for the coefficient \(\Gamma _{2}\).
Lemma 3
The coefficient \(\Gamma _{2}\) from (46) can be expressed in the formwhere \(G_2\) is expressed entirely in terms of third-order elastic constants:
$$\begin{aligned} \Gamma _{2} = \lambda ^{-1}_{2}G_2, \end{aligned}$$
(47)
$$\begin{aligned} G_2 = \frac{1}{2} c_{abcdef}n_bn_dn_f k_a^{1} k_c^{1} k_e^{2}. \end{aligned}$$
(48)
The proof follows from the explicit formula for the coefficient \(\Gamma _{2}\) by a simple inspection. Since \( {{\mathbf {n}}} \cdot {{\mathbf {k}}}_1 = 0\), hence the last term in (41) disappears and we obtain (47) and (48); see [2] for more details.
By introducing a complex variable, system (46) can be transformed into the single complex Burgers equation. We will show how to do this in the next section.
3.4 The complex Burgers equation
To simplify the consideration, let us introduce the following notation \(a_1(\tau ,\theta ) \equiv u(t,x), a_2(\tau , \theta ) \equiv v(t,x)\), and let the subscript denote a partial derivative; so for example,and similarlyMoreover, assume that the coefficient \(\Gamma _{2}\) is normalized to 1.
$$\begin{aligned} \frac{\partial a_1 }{\partial \tau } \equiv u_t, \;\;\;\;\; \frac{\partial a_1}{\partial \theta } \equiv u_x, \end{aligned}$$
$$\begin{aligned} \frac{\partial a_2 }{\partial \tau } \equiv v_t, \;\;\;\;\; \frac{\partial a_2}{\partial \theta } \equiv v_x. \end{aligned}$$
Under these assumptions, system (46) reduces toSuppose we add the first equation of (49) to the second equation multiplied by the imaginary \(\mathrm{i}\). We then obtainNow, assuming that \(U \equiv u + \mathrm{{i}} v\), we have that the complex conjugate squared \( \overline{U}^2 = u^2 - 2 \mathrm{i} u\,v - v^2\), hence system (49) can be written as a single complex Burgers equation:Similarly, by subtracting the second equation of (49) multiplied by imaginary \(\mathrm{i}\) from the first one, we obtain the complex conjugate of (51):We can therefore formulate the following lemma.
$$\begin{aligned} \begin{array}{l} \displaystyle u_t + \frac{1}{2}(u^2 - v^2)_x = 0, \qquad v_t - (u\,v)_x = 0. \end{array} \end{aligned}$$
(49)
$$\begin{aligned} u_t + \mathrm{i} v_t + \frac{1}{2}(u^2 - 2 \mathrm{i} u\,v - v^2)_x = 0. \end{aligned}$$
(50)
$$\begin{aligned} U_t + \frac{1}{2}\left( \overline{U}^2\right) _x = 0. \end{aligned}$$
(51)
$$\begin{aligned} \overline{U}_t + \frac{1}{2}\left( U^2\right) _x = 0. \end{aligned}$$
(52)
Lemma 4
Definition 1
We say that the periodic function \(U(t,x)\) with period \(L\), \(\left( U(t,x) = U(t,x + L)\right) \), has a threefold symmetry if for all \(t\) and \(x\)
$$\begin{aligned} U\left( t,x + \frac{L}{3}\right) = \mathrm{{e}}^{2\pi \mathrm{{i}}/{3}}U(t,x). \end{aligned}$$
(53)
Lemma 5
If \( U(t,x)\) satisfies the complex Burgers Eq. (51), then so does \(V(t,x) \equiv \mathrm{{e}}^{2\pi \mathrm{{i}}/{3}}U(t,x)\).
Proof
We have \(\overline{V}\,^{2} = \mathrm{{e}}^{-4\pi \mathrm{{i}}/{3}} \overline{U}\,^{2}\), soFrom Lemma 5 and Definition 1 it follows that a periodic function with a threefold symmetry satisfies (49). Some solutions to the complex Burgers Eqs. (49) were studied by Noelle in his thesis [10]. These equations reveal interesting features like, for example, the formation of nonclassical–undercompressive shock waves. The problem of stability of such shocks for complex Burgers equations was investigated in [11].
$$\begin{aligned} V_t + \frac{1}{2}\left( \overline{V}^2\right) _x = \mathrm{{e}}^{2\pi \mathrm{{i}}/{3}}U _t + \frac{1}{2}\mathrm{{e}}^{-4\pi \mathrm{{i}}/{3}}\left( {\overline{U}}^2\right) _x = \mathrm{{e}}^{2\pi \mathrm{{i}}/{3}}\,\left[ U_t + \frac{1}{2}\left( \overline{U}^2\right) _x\right] = 0. \end{aligned}$$
Let us now revisit system (49) and prove the following lemma. \(\square \)
Lemma 6
System (49) is not genuinely nonlinear.
Proof
We write system (49) in a quasilinear form:where \(\varvec{W} = [\,u, v\,]^\mathrm{{T}}\), andThe eigenvalues areWe haveThe corresponding eigenvectors have the formThe genuine nonlinearity condition means thatWe haveOne can easily check that the genuine nonlinearity is lost if \(v^2 = 3u^2\) for \(u < 0\) in the first family and \(u > 0\) in the second family. \(\square \)
$$\begin{aligned} \frac{\partial \varvec{W}}{\partial t} + \displaystyle \varvec{C}(\varvec{W})\, \frac{\partial \varvec{W}}{\partial x} = \varvec{0}, \end{aligned}$$
(54)
$$\begin{aligned} \displaystyle \varvec{C}(\varvec{W}) = \displaystyle \left( \begin{array}{cc} u &{}\quad -v \\ -v &{}\quad -u \end{array} \right) . \end{aligned}$$
$$\begin{aligned} \begin{array}{lrr} \lambda _1(\varvec{W}) = -\sqrt{u^2+v^2} = -|\varvec{W}|, \qquad \lambda _2 (\varvec{W}) = \,\sqrt{u^2+v^2} = \,|\varvec{W}|. \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{lcr} \displaystyle \nabla _{\varvec{W}}\,\lambda _1(\varvec{W}) = -\frac{1}{|\varvec{W}|}(u, v), \qquad \displaystyle \nabla _{\varvec{W}}\,\lambda _2(\varvec{W}) = \, \frac{1}{|\varvec{W}|}(u, v). \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{lrr} {\varvec{r}}_1(\varvec{W}) = \left[ - u + |\varvec{W}|, v\right] , \qquad {\varvec{r}}_2(\varvec{W}) = \left[ - u - |\varvec{W}|, v\right] . \end{array} \end{aligned}$$
$$\begin{aligned} \displaystyle \nabla _{\varvec{W}}\,\lambda _j(\varvec{W}) \cdot \varvec{r}_j \ne 0 \;\;\;\; j = 1,2. \end{aligned}$$
$$\begin{aligned} \begin{array}{lcl} \displaystyle \nabla _{\varvec{W}}\,\lambda _1(\varvec{W}) \cdot \varvec{r}_1 = \displaystyle \frac{u^2 - v^2 - u\,|\varvec{W}|}{|\varvec{W}|}, \qquad \displaystyle \nabla _{\varvec{W}}\,\lambda _2(\varvec{W}) \cdot \varvec{r}_2 = \displaystyle \frac{v^2 - u^2 - u\,|\varvec{W}|}{|\varvec{W}|}. \end{array} \end{aligned}$$
4 Applications
In this section we apply our theory to a cubic crystal and an icosahedral quasicrystal. First, we simplify the notation for elastic constants. We use Brugger’s second-order \(c_{ij}\) and third-order \(c_{ijk}\) elastic constants (see [12]) instead of \(c_{abcd}\) and \(c_{abcdef}\), respectively, where the change of indices is done according to the following rule:
$$\begin{aligned} \begin{array}{lcrclcrclcl} 11 &{} \rightarrow &{} 1 &{} \ \ \ \ \ &{} 22 &{} \rightarrow &{} 2 &{} \ \ \ \ \ &{} 33 &{} \rightarrow &{} 3 \\ 23 &{} \rightarrow &{} 4 &{} \ \ \ \ \ &{} 32 &{} \rightarrow &{} 4 &{} \ \ \ \ \ &{} 13 &{} \rightarrow &{} 5 \\ 31 &{} \rightarrow &{} 5 &{} \ \ \ \ \ &{} 12 &{} \rightarrow &{} 6 &{} \ \ \ \ \ &{} 21 &{} \rightarrow &{} 6. \end{array} \end{aligned}$$
(55)
4.1 Cubic crystal
There are five cubic crystal classes, denoted by 23, \(m3\), 432, \(m3m\), and \(\overline{4}3m\). All of these cubic crystals possess trigonal symmetry with respect to the four cube diagonals. All cubic crystals also have a diagonal symmetry about the three cubic axes. Only two of the cubic crystal classes have an additional tetragonal symmetry about the cubic axes. Each symmetry diminishes the number of elastic stiffness constants. The number of constants for the most symmetric of cubic crystals is reduced to three second-order constants and six third-order constants [5]. We will be interested in the most symmetric of cubic crystals – those belonging to the class \(m3m\). This is motivated by the fact that many crystals appearing in nature belong to this class, for example, all metals of cubic symmetry, Au, Fe, Ni, Ag, and ionic crystals such as, for example, NaCl, KCl, LiF, AgCl, and many more.
Let us consider a cubic crystal of class \(m3m\) in which the strain energy \(W\) is defined by three second-order and six third-order elastic constants [5]:We study the propagation of a pair of shear waves along an axis of threefold symmetry, \({{\mathbf {n}}} = \frac{1}{\sqrt{3}} [1\,1\,1]\), which is also an acoustic axis. We discover that shear wave amplitudes are described by coupled evolution Eqs. (46) with the coefficient \(\Gamma _2\) given by formula (47), with
$$\begin{aligned} W = W (c_{11},\,c_{12},\,c_{44},\, c_{111},\, c_{112},\,c_{144},\,c_{123},\,c_{166},\,c_{456}). \end{aligned}$$
$$\begin{aligned} \displaystyle \lambda _2 = \sqrt{\frac{ c_{11} -c_{12}+c_{44} }{3} } \qquad \hbox {and} \qquad G_2 = \displaystyle \frac{1}{18\sqrt{2}} \big [ c_{111} +2c_{123}-2 c_{456} -3(c_{112}-c_{144}+c_{166 }) \big ]. \end{aligned}$$
4.2 Icosahedral quasicrystal
The discovery of quasicrystals (see [13] and [14]) spurred strong interest in various properties of these strange solids with unusual symmetry axes that are incompatible with periodicity and without translational symmetry in space.
Icosahedral quasicrystals, besides a phonon phase, also have a phason phase, so in general their energy density function is characterized by 5 second-order elastic constants (\(2\) due to the phonon field, \(2\) due to the phason field, and \(1\) associated with phonon–phason coupling) [15] and \(20\) third-order elastic constants (\(4\) due to phonon fields, \(5\) due to phason fields, \(4\) due to phonon–phonon–phason coupling, and \(7\) due to phonon–phason–phason coupling) [16]. Of these third-order constants only four are independent, and, moreover, the following relations are satisfied (see [17] with corrections from [18]):
Following [19] we consider the icosahedral quasicrystal AlPdMn, which is phason-free and is characterized by two second-order and five third-order elastic constants, so it is linearly isotropic but anisotropic in a nonlinear range. The strain energy density function isWe investigate the propagation of collinear transverse waves along the fiber direction, which is a threefold symmetry acoustic axis. In this case, the evolution equations for a pair of such waves reduce to system (46), with the coefficient \(\Gamma _2\) given by formula (47), with \(\lambda _2 = \sqrt{c_{44}}\) and \( G_2 = \displaystyle (4/9) \displaystyle \left( c_{112} - c_{113}\right) .\)
$$\begin{aligned} c_{111}&= c_{123} + 2 c_{144} + 2( c_{155} + c_{166}),\end{aligned}$$
(56)
$$\begin{aligned} c_{112}&= c_{123} + 2 c_{144} + \left( 2 \cos {\frac{\pi }{5}} - 1\right) (c_{155} - c_{166}), \end{aligned}$$
(57)
$$\begin{aligned} c_{113}&= c_{123} + 2 c_{144} + 2 c_{144} + 2 \cos {\frac{\pi }{5}} (c_{155} - c_{166}),\end{aligned}$$
(58)
$$\begin{aligned} c_{456}&= - \frac{1}{2}c_{144} - \frac{1}{2}\left( 2\cos {\frac{\pi }{5}} - 1\right) c_{155} + \cos {\frac{\pi }{5}} c_{166}. \end{aligned}$$
(59)
$$\begin{aligned} W = W(c_{11}, c_{44}, c_{111}, c_{112}, c_{113},c_{456}). \end{aligned}$$
5 Concluding remarks
The possibility of the appearance of quadratically nonlinear coupling in the evolution equations for shear elastic waves propagating along the threefold symmetry acoustic axis direction in a cubic crystal was shown for the first time in our paper [3]; see also [4, 5]. Experimental confirmation of this fact was announced in [20]. Similar coupling was detected in an icosahedral quasicrystal [18, 19]. The problem of the second harmonic generation of shear waves in crystals has been investigated in [21]; see also the recent paper [22].
It seems that a complex Burgers equation appeared for the first time in Lax’s paper [23]. The properties of the solution to this equation were studied by Lax’s Ph.D. student Sebastian Noelle [10]. This equation appeared for a model of various phenomena, for example, as a one-dimensional model for the dynamics of dislocations in crystals [24] and in studying surfaces in \(\mathcal {R}^3\) that arise as limit shapes in random surface models related to planar dimers [25].
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