## Abstract

We investigated free-vibration acoustic resonance (FVAR) of two-dimensional St Venant–Kirchhoff-type hyperelastic materials and revealed the existence and structure of colour symmetry embedded therein. The hyperelastic material is isotropic and frame indifferent and includes geometrical nonlinearity in its constitutive equation. The FVAR state is formulated using the principle of stationary action with a subsidiary condition. Numerical analysis based on the Ritz method revealed the existence of four types of nonlinear FVAR modes associated with the irreducible representations of a linearized system. Projection operation revealed that the FVAR modes can be classified on the basis of a *single colour* (black or white) and three types of *bicolour* (black and white) magnetic point groups: , , and . These results demonstrate that colour symmetry naturally arises in the finite amplitude nonlinear FVAR modes, and its vibrational symmetries are explained in terms of magnetic point groups rather than the irreducible representations that have been used for linearized systems. We also predicted a *grey colour* nonlinear FVAR mode which cannot be derived from a linearized system.

## 1. Introduction

Free-vibration acoustic resonance (FVAR) is a characteri- stic feature of solids because it expresses their *eigen* vibration. Ever since the pioneering work of Rayleigh [1,2] and Ritz [3,4], this phenomenon has attracted much attention from both scientists and engineers. In 1977, Demarest [5] investigated FVAR of rectangular parallelepiped-shaped Si and pointed out that the second-order elastic constants *C*_{ijkl} can be obtained from the FVAR frequencies *ω*_{i}. Soon after, Ohno [6] conducted an ultrasound spectroscopy experiment and determined the elastic constants from the inverse analysis of *ω*_{i}. Some improvements followed, and, in consequence, this technique is now established as resonant ultrasound spectroscopy (RUS) [7–13]. RUS has several advantages over previous *C*_{ijkl} measurement methods: (i) the complete set of *C*_{ijkl} tensor is obtained from an FVAR spectrum of a single crystal specimen, (ii) it can be applied to small (millimetre-order) specimens, and (iii) the measurement accuracy is sufficiently high; in general, the inaccuracy is less than 0.1%. To the author’s knowledge, this is the state-of-the-art technique for determining the second-order elastic constants of a solid. As shown in a following section, the theory of RUS is written within a framework of linear elasticity and the calculus of variations. Because a linear approximation is used, the FVAR state can be obtained immediately from an eigenvalue equation. However, in exchange for this theoretical simplicity, this approximation eliminates every phenomenon that originates from the nonlinearity of a solid. For instance, RUS cannot be used to determine the third-order elastic constants *C*_{ijklmn}, which are responsible for unharmonic interaction between atoms in a crystal. It also fails to explain the experimentally observed nonlinear interaction between FVAR modes [14]. In fact, FVAR of a nonlinear hyperelastic material is still relatively little understood and, to the best of the author’s knowledge, further investigation is needed. We recently investigated FVAR of one-dimensional nonlinear hyperelastic material and revealed that it shows several characteristic features, such as an amplitude dependence of FVAR frequency, nonlinear excitation of higher harmonics and symmetry breaking [15]. All of these phenomena originate from the geometrical and constitutive nonlinearities, and therefore never appear in a linearized system. In this study, we extend the theory and investigate FVAR of a two-dimensional St Venant–Kirchhoff hyperelastic material.

This paper is organized as follows. In the next section, we present a variational formulation for FVAR of the hyperelastic material. It includes definition of the action integral, first variation, Euler–Lagrange equation and some natural boundary conditions. We also introduce a basis function that satisfies time reversal symmetry and time periodicity conditions. Section 3 is devoted to linearization, deriving an eigenvalue equation that is equivalent to the conventional RUS theory. The results of numerical analysis are presented in §4. In this section, we show the vibration patterns of nonlinear FVAR modes and reveal the existence of *colour symmetry* embedded therein. In addition, we demonstrate that magnetic point groups precisely explain the structure of the colour symmetry by projection operations to the basis function. Furthermore, we predict the existence and vibrational symmetry of a *grey colour* nonlinear FVAR mode. Some concluding remarks are presented in §5.

## 2. Variational formulations

### (a) Action integral

Let us consider a two-dimensional St Venant–Kirchhoff hyperelastic material that is defined by *Ω*={(*x*_{1},*x*_{2})|−*L*_{i}<*x*_{i}<*L*_{i}, *i*=1,2}. The hyperelastic domain *Ω* is assumed to be a homogeneous and stress-free reference configuration. The domain is subjected to a displacement *u*_{i}=*u*_{i}(*x*_{1},*x*_{2},*t*) owing to FVAR. Note that the function *u*_{i}(*x*_{1},*x*_{2},*t*) is expressed using the Lagrange description. Let *ρ*=const. be the mass–density defined on *Ω* and let *u*_{i,j}=∂*u*_{i}/∂*x*_{j} and *u*_{i,t}=∂*u*_{i}/∂*t* be the spatial and material time derivatives of the displacement, respectively. Then, the kinetic energy density , nonlinear strain energy density and Lagrangian density of the hyperelastic material are written as follows:
2.1Hereafter, we assume the summation convention for repeated indices. In (2.1), *λ* and *μ* are called the Lamé constants and *E*_{ij} is the Green–Lagrange strain tensor, which is a symmetric tensor defined in the reference configuration. It is written in terms of the displacement gradient *u*_{i,j} in the following form:
2.2The third term on the right-hand side (r.h.s.) of equation (2.2) is called the geometrical nonlinearity and is responsible for all the nonlinear phenomena considered in this study. The strain energy density defined in (2.1) is isotropic and frame indifferent and includes only two elastic constants, *λ* and *μ*. Because of its simplicity, it would be appropriate to investigate finite amplitude nonlinear FVAR as a first approximation. For later use, we introduce the linearized strain energy density ,
2.3where *ϵ*_{ij} is called the Cauchy strain tensor. The energy density is frequently used in the theory of isotropic linear elasticity. Integration of the Lagrangian density over the rectangular domain *x*_{i}∈*Ω* and a certain time interval *t*∈(*t*_{1},*t*_{2}) yields the action integral *I* of the hyperelastic material,
2.4Obviously, this is a functional of the displacement function: *I*=*I*[*u*_{i}].

### (b) The first variation

According to the mathematical settings we made in §2*a*, the St Venant–Kirchhoff hyperelastic material is a holonomic system and is free from any dissipation; in fact, the energy conservation law holds because the Lagrangian density has time translational symmetry. Therefore, the principle of stationary action must hold. This principle states that, among the continuously differentiable functions defined on *x*_{i}∈*Ω* and *t*∈(*t*_{1},*t*_{2}), the actual displacement *u*_{i} must satisfy the stationary condition that *δI*=0. Here, the time domain *t*∈(*t*_{1},*t*_{2}) is unfixed; therefore, the stationary condition *δI*=0 poses not an ordinary but a variable-domain-type variational problem. To simplify the analysis, we set *t*_{1}=0 and *t*_{2}=2*π*/*ω*, where *ω*(>0) is a resonance frequency that we determine throughout the analysis. Although the frequency *ω* causes the length of the interval |*t*_{2}−*t*_{1}|=2*π*/*ω* to change, the time *t*_{1} at which the interval starts is still fixed at *t*_{1}=0. For this reason, we simultaneously consider the variation of the independent and dependent variables, such that
2.5where *φ*=const. and *ψ*_{i} is an arbitrary but continuously differentiable function. In (2.5), the first transformation indicates a rigid time translation that varies the time interval such that (*αφ*,2*π*/*ω*+*αφ*). The second transformation is an ordinary variation of the function *u*_{i}. After some calculations, the first variation *δI* of the action integral ends up with the following form [16]:
2.6where . The expression in the first parentheses on the r.h.s. of (2.6) represents the Euler–Lagrange equation in strong form. Because the mass density *ρ* is a constant in the reference configuration, we have . On the other hand, from the constitutive equation, we have and , where *T*_{ij} represents the first Piola–Kirchhoff stress tensor. Hence, a necessary condition for *δI*=0 yields a system of quasi-linear wave equations:
2.7The wave equation (2.7) has time reversal symmetry because its time derivative is of the second order. The remaining integrands in (2.6) are concerning the boundary conditions. From the expression in the second parentheses, we have the following natural boundary conditions:
2.8The first two conditions indicate that the displacement velocities *u*_{i,t} must vanish at both the beginning and end of the time interval. The last condition indicates the time periodicity of Lagrangian density. Similarly, from the expression in the last parenthesis in (2.6), the following natural boundary conditions should be satisfied:
2.9Let *n*_{i} and *T*_{i} be the outward unit normal vector and the first Piola–Kirchhoff stress vector defined on the surface ∂*Ω*, respectively. From the boundary condition (2.9) and the Cauchy relationship, *T*_{i}=*T*_{ji}*n*_{j}, we have *T*_{1}=0 for *x*_{1}=±*L*_{1} and *T*_{2}=0 for *x*_{2}=±*L*_{2}. This result indicates that the hyperelastic material is free from surface normal traction during vibration. This is why the stationary solution is understood as a surface traction-free vibration or the so-called *free vibration*.

### (c) Time periodicity and time reversal symmetry

Let the displacement *u*_{i}(*x*_{1},*x*_{2},0) and displacement velocity *u*_{i,t}(*x*_{1},*x*_{2},0) be the initial state of an FVAR, and let *u*_{i}(*x*_{1},*x*_{2},*t*) be its time evolution as obtained from the quasi-linear wave equation (2.7). Because of the time reversal symmetry of the wave equation, the time reversal displacement *u*_{i}(*x*_{1},*x*_{2},−*t*) should be a solution for the initial state *u*_{i}(*x*_{1},*x*_{2},0) with the inverse velocity −*u*_{i,t}(*x*_{1},*x*_{2},0). According to the boundary condition (2.8), however, the displacement velocity *u*_{i,t} must vanish at *t*=0. Hence, the two initial conditions are equivalent, thereby the trajectories in the +*t* and −*t*-directions are identical. This proves the time reversal symmetry of the displacement
2.10From the third condition in (2.8), we can derive the time periodicity of the strain energy density because and because the kinetic energy identically vanishes at *t*=0 and 2*π*/*ω*. Obviously, a sufficient condition, and presumably the most natural condition for is the time periodicity of displacement,
2.11This condition comes from the time translation condition *φ*≠0 in (2.5) and (2.6), and therefore related to the energy conservation of the system. Equations (2.10) and (2.11) are the fundamental requirements that the displacement of any FVAR modes should satisfy.

Here, we also mention the definition of the vibration period. For the linearized FVAR mode, there is a characteristic time at which all displacement vanishes for all *x*_{i}∈*Ω*. As seen in the following sections, however, nonlinear FVAR modes lack such a characteristic time. Hence, the vibration period should be defined according to the time interval of zero displacement velocity *u*_{i,t}=0 required from the natural boundary condition.

### (d) Numerical analysis by the Ritz method

#### (i) Fourier series expansion

One of the standard ways of investigating the stationary condition of the action integral (2.4) is to solve the Euler–Lagrange equation (2.7) under the conditions of time reversal symmetry and time periodicity, (2.10) and (2.11), respectively. This is, however, an *ill-posed* problem because we have no initial conditions, *u*_{i}|_{t=0} and *u*_{i,t}|_{t=0}, for the quasi-linear wave equation. In addition, a general solution for the wave equation is unknown even in a one-dimensional system [17]. Therefore, we numerically solve the variational problem using a direct analysis based on the Ritz method. In this analysis, we expand the displacement function *u*_{i} by a linear combination of basis functions in such a way that
2.12and
2.13In this expansion, is responsible for the time-dependent component of the displacement *u*_{i}. Because is taken to be an integer, (2.12) and (2.13) satisfy the time reversal symmetry condition, time periodicity condition and natural boundary condition *u*_{i,t}|_{t=0}=*u*_{i,t}|_{t=2π/ω}=0 for all *x*_{i}∈*Ω*. Further, *ϕ*_{m}=*ϕ*_{m}(*x*_{1},*x*_{2}), *φ*_{m}=*φ*_{m}(*x*_{1},*x*_{2}), *χ*_{m}=*χ*_{m}(*x*_{1},*x*_{2}) and *ψ*_{m}=*ψ*_{m}(*x*_{1},*x*_{2}) are the spatial components of the basis function and *a*_{n,m} to *h*_{n,m} are their real-valued coefficients. In this study, we employ the following orthonormal Fourier series:
2.14
2.15
2.16
and
2.17where
2.18In the plane wave expansions (2.12) and (2.13), the indices *m*, *m*_{1} and *m*_{2} denote the order of the basis functions, and their relationship is summarized in table 1. Here, and stand for the maximum order of the basis function for the *x*_{1}- and *x*_{2}-directions, respectively.

In (2.12) and (2.13), the *n*=1 components are ordinary plane waves, and *n*≥2 components are called higher harmonics. The *n*=0 components represent the time-independent displacement. All the plane waves must be in the same phase so as to satisfy the natural boundary condition *u*_{i,t}|_{t=0}=*u*_{i,t}|_{t=2π/ω}=0. Note that the coefficients *d*_{0,0} and *h*_{0,0} are responsible for a rigid body translation. Because they have nothing to do with FVAR, we set *d*_{0,0}=*h*_{0,0}=0 to simplify the analysis. Consequently, the degree of freedom of the basis function is . The remaining boundary condition, (2.9), would be satisfied as owing to the completeness of the Fourier series.

#### (ii) A subsidiary condition

Inserting (2.12) and (2.13) into (2.4) and integrating the Lagrangian density over the variable domain *Ω*×(0,2*π*/*ω*), we obtain an analytical form of the action integral which is not a functional of *u*_{i} but a function of the coefficients of the basis function *I*=*I*(*a*_{n,m},…,*h*_{n,m}). Then, the stationary condition *δI*=0 is equivalent to the th system of the nonlinear algebraic equations such that
2.19This critical condition is, however, insufficient for obtaining an FVAR state because the number of unknown quantities is : the th coefficients of the basis function and a resonance frequency *ω*. Therefore, one equation is missing.

Here, we introduce the *L*^{2} norm of the displacement function *u*_{i} at *t*=0 such that
2.20and impose a subsidiary condition to the variational problem *δI*=0. There are several reasons to introduce the subsidiary condition: (i) it compensates for the missing equation, so the th system of nonlinear algebraic equations is closed, (ii) it defines the vibration amplitude which is useful for numerical analysis, (iii) it rejects the trivial solution *a*_{n,m}=⋯=*h*_{n,m}=0, and (iv) as seen in a following section, the low-amplitude limit of the constrained variational problem is equivalent to the conventional RUS theory (linearized FVAR). However, the th system of algebraic equations is genuinely nonlinear, and hence cannot be solved in an analytical form. Therefore, we first linearize the hyperelastic material and calculate a linearized FVAR mode. Nonlinear equations (2.19) with respect to are then numerically solved around the linearized solution by a convergent calculation based on the Newton method.

## 3. Linearization

### (a) Stationary condition for linearized action integral

The St Venant–Kirchhoff hyperelastic material *Ω* is linearized simply by replacing the strain energy density in the action integral *I* with the linearized one defined in (2.3). In that case, the fourth-rank tensor expressed by becomes a constant because the energy density is written as a quadratic form of the displacement gradient *u*_{i,j}. Because the tensor is a coefficient of ∂^{2}*u*_{k}/∂*x*_{l}∂*x*_{j} in (2.7), the Euler–Lagrange equation becomes a system of linear wave equations. As is well known, the solution of a linear wave equation is expressed as a harmonic plane wave expansion. There is, therefore, no reason to include the higher harmonics (*n*≥2) and time-independent components (*n*=0) in the basis functions. Hence, we set *n*=1 in (2.12) and (2.13) so as to include only the harmonic plane waves in the basis function. Let be a row vector consisting of the coefficients of the basis function:
3.1Then, the linearized action integral *I*^{L} is written as the quadratic form of the vector ** A**:
3.2On the r.h.s. of this equation, the first term comes from the kinetic energy and the second is the linear strain energy. The matrix is identity because we employed orthonormal basis functions. is a symmetric matrix expressed in the following form:
3.3where stands for a symmetric matrix (the details of the matrix are summarized in appendix A). Then, the stationary condition

*δI*=0 is given by ∂

*I*/∂

**=0. This th system of linear equations is equivalent to the eigenvalue problem expressed as 3.4It is obvious from this equation that the FVAR frequency**

*A**ω*of a linearized hyperelastic material can be obtained from an eigenvalue

*ρω*

^{2}of the matrix

**. The corresponding eigenvector**

*Γ***yields the coefficients of the basis function. This is nothing but the theory of conventional RUS. Let us see the eigenvalue problem from a different viewpoint. It is obvious from the orthonormality of the basis functions that . Then, the eigenvalue problem is equivalent to the variational problem**

*A**δI*

^{L}=0 with the subsidiary condition . In our variational formulation, derived in §2, the effect of geometrical nonlinearity disappears as the vibration amplitude approaches zero . In this regard, the present formulation is the proper generalization of the linearized FVAR that constitutes the conventional RUS theory.

### (b) Classification of linearized free-vibration acoustic resonance modes

It is well known that FVAR of a linearized hyperelastic material can be classified by the irreducible representations of group theory [7]. Here, we briefly summarize some fundamental issues because they also play important roles in nonlinear systems. Because the hyperelastic material *Ω* has a rectangular shape, four types of symmetry operations can be applied to it: the identity operation (*E*), *π* rotation (*C*_{2}), reflection about the *y*-axis (*σ*_{y}) and reflection about the *x*-axis (*σ*_{x}). These symmetry operations define the point group . According to the character table [18,19], the point group has four types of one-dimensional irreducible representation, *A*_{1}, *A*_{2}, *B*_{1} and *B*_{2}, and thus has four types of FVAR modes. Figure 1 schematically illustrates the symmetry of the vibration modes. *A*_{1} is called the breathing mode and preserves all the symmetry operations in . *A*_{2} is a shearing mode and the remaining two are called bending modes.

A more concrete description of a linearized FVAR mode can be obtained from a projection operation. According to group theory, the projection operator *P*^{β} for a one-dimensional irreducible representation *β* is written in the following form [18,19]:
3.5Here *R* and *P*_{R} stand for an element of the point group and the corresponding symmetry operation, respectively. Further, *g* and *χ*^{β}(*R*)* denote the order of the group and the complex conjugate of the character *χ*(*R*), respectively, for the irreducible representation *β*. For the point group , we have , , and . These operators can be used to project out the irreducible representations from the basis function. The result are summarized as follows:
3.6
3.7
3.8
and
3.9This result completely classifies the FVAR modes of the linearized hyperelastic material. For instance, FVAR of the *A*_{1} mode consists only of the basis functions and . *Γ*^{A1} is a corresponding part of the matrix ** Γ**; namely, the eigenvalues and eigenvectors of the matrix

*Γ*^{A1}yield FVAR of the

*A*

_{1}mode. In linear algebra, this classification is understood as a block diagonalization. Let

**→**

*A***′ be a linear transformation such that 3.10Then, the similarity transformation**

*A***→**

*Γ***′ yields block diagonalization such that**

*Γ***′=**

*Γ*

*Γ*^{A1}⊕

*Γ*^{A2}⊕

*Γ*^{B1}⊕

*Γ*^{B2}, where ⊕ stands for the direct sum. Hence, the sets (

*b*

_{1,m},

*g*

_{1,m}), (

*c*

_{1,m},

*f*

_{1,m}), (

*d*

_{1,m},

*e*

_{1,m}) and (

*a*

_{1,m},

*h*

_{1,m}) span the respective invariant subspaces, forming FVAR modes that are independent of each other. The symmetry of displacement for the respective FVAR modes is summarized in table 2.

## 4. Results and discussion

### (a) Mathematical settings for numerical analysis

As mentioned in §2*d*, we first calculate the resonance frequency *ω* and the coefficients of the basis function ** A** for a linearized FVAR mode by solving an eigenvalue problem: (3.6)–(3.9). The th system of nonlinear algebraic equations, (2.19) with ∥

*u*

_{i}∥

_{L2}=const., is then numerically solved around the linearized solution by convergent calculation using the Newton method. In this study, we employed the dimensionless quantities

*ρ*=1,

*L*

_{1}=1.1,

*L*

_{2}=0.9,

*λ*=1 and

*μ*=0.1 to simplify the numerical analysis. The order of the basis functions is

*N*=3 and ; therefore, the degree of freedom is . The maximum vibration amplitude is set to ∥

*u*

_{i}∥

_{L2}=0.05 to keep the strong ellipticity condition for all

*x*

_{i}∈

*Ω*and

*t*∈(0,2

*π*/

*ω*). Note that we neglected the fourth-order terms of the displacement gradient in the action integral

*I*. This is simply for computational reasons: they consume considerable amount of computational resources, whereas their effects on FVAR are negligible in the present amplitude range. In fact, by numerical analysis on a small system, we confirmed that the fourth-order terms do not influence on the qualitative discussion in this section.

### (b) Free-vibration acoustic resonance patterns

#### (i) *A*_{1}′ mode

Figure 2*a* shows a nonlinear FVAR pattern obtained from the initial data *A*_{1}−1, which has the lowest FVAR frequency in the linearized *A*_{1} mode. Hereafter, we refer to it as the *A*_{1}′−1 mode. Although only half a period, 0≤*t*≤*π*/*ω*, is shown here, the remaining half can be easily expected from the time reversal symmetry of displacement. Figure 2*b*–*e* shows *n*=0–3 components in (*a*), so their summation yields (*a*). To see the low-amplitude displacement, we magnified the original displacement as described in the figure caption. Although the amplitude of the anharmonic components in figure 2*b*,*d*,*e* are small compared with the principal harmonic component in figure 2*c*, they do exist in the nonlinear FVAR mode and are forbidden in a linearized system.

It is obvious from figure 2 that nonlinear FVAR mode (*a*) has *A*_{1} symmetry because its components (figure 2*b*–*e*) has *A*_{1} symmetry for all *t*∈(0,*π*/*ω*). From (3.6), the basis function of the *A*_{1} mode is represented by (*b*_{1,m}*φ*_{m},*g*_{1,m}*χ*_{m}). Hence, a general form of nonlinear FVAR of the *A*_{1}′ mode is expressed by
4.1and
4.2Note that all the other basis functions identically vanish in *A*_{1}′ mode.

#### (ii) *A*_{2}′ modes

Figure 3*a* shows the nonlinear FVAR pattern of the *A*_{2}′−1 mode obtained from the initial data of *A*_{2}−1. As in figure 2, the vibration pattern in figure 3*a* consists of the summation of the patterns in figure 3*b* (*n*=0) to *e* (*n*=3). As seen from the figure, the principal harmonic component (figure 3*c*) has *A*_{2} symmetry during almost the entire vibration. The only exception is *t*=*π*/2*ω*, at which the symmetry becomes *A*_{1} because all the displacement vanishes. The same holds for the *n*=3 component in figure 3*e*. To our surprise, however, the *n*=even components have *A*_{1} symmetry during the vibration. Similar features have been confirmed in the *A*_{2}′−2 to *A*_{2}′−10 modes. We, therefore, concluded that the symmetry of the *A*_{2}′ mode depends on the parity of *n*: it has *A*_{2} symmetry if *n*=odd whereas it becomes *A*_{1} otherwise. Because the basis function of the *A*_{2} mode is written as (*c*_{1,m}*χ*_{m},*f*_{1,m}*φ*_{m}), the general form of the *A*_{2}′ mode becomes
4.3and
4.4

As mentioned earlier, the *A*_{1} mode preserves all the symmetry operations in the point group . However, the symmetry operations *σ*_{x} and *σ*_{y} are no longer hold in figure 3*c*,*e* as it has *A*_{2} symmetry. Because the *A*_{2}′ mode consists of the summation of the patterns in figure 3*b*–*e*, the vibrational symmetry is *A*_{2} for *t*∈(0,*π*/*ω*)∖*π*/2*ω* and *A*_{1} at *t*=*π*/2*ω*.

#### (iii) *B*_{1}′ and *B*_{2}′ modes

As in the previous two cases, we calculated the FVAR patterns of the *B*_{1}′−1 and *B*_{2}′−1 modes using the initial data of *B*_{1}−1 and *B*_{2}−1. The results are summarized in figures 4 and 5, respectively. As shown in figure 4*c*,*d*, the only symmetry operations in that are applicable to the *n*=odd displacement components are *E* and *σ*_{y}. Therefore, the vibrational symmetry is *B*_{1} for all *t*∈(0,*π*/*ω*)∖*π*/2*ω* and *A*_{1} at *t*=*π*/2*ω*. By contrast, *n*=even components have *A*_{1} symmetry during the entire vibration. For the *B*_{2}′−1 mode, the symmetry is *B*_{2} for *n*=odd and *A*_{1} for *n*=even according to figure 5. Hence, the *B*_{1}′ and *B*_{2}′ modes have features similar to those of the *A*_{2}′ mode. According to (3.8) and (3.9), the basis functions of the *B*_{1} mode are (*d*_{1,m}*ψ*_{m},*e*_{1,m}*ϕ*_{m}) and that of the *B*_{2} mode are (*a*_{1,m}*φ*_{m},*h*_{1,m}*ϕ*_{m}). Hence, the general forms of the two nonlinear FVAR modes are written in the following forms:
4.5
4.6
4.7
and
4.8

### (c) Magnetic point groups

We address two questions in this study. The first question is why the basis functions have the specific symmetries described in (4.1)–(4.8). The second is whether there are any other FVAR modes that cannot be classified as *A*_{1}′, *A*_{2}′, *B*_{1}′ or *B*_{2}′. In this section, we consider these two questions in terms of group theory.

#### (i) *A*_{1}′ mode

Let us first consider the *A*_{1}′ mode. As shown in figure 2 and in (4.1) and (4.2), the symmetry of the FVAR mode is *A*_{1} for any order of *n*. In this regard, it has the simplest vibration structure among the four types of nonlinear FVAR modes. Because the *A*_{1} mode preserves all the symmetry operations, the point group is responsible for the vibrational symmetry of the *A*_{1}′ mode. Now let *P*^{A1′}=*P*^{A1} be the projection operator of the *A*_{1}′ mode and let ** u** be a basis function component in vector form:
4.9where

*X*

_{α}=

*X*

_{α}(

*x*

_{1}) is an even function when

*α*is even and vice versa if

*α*is odd. The same holds for

*Y*

_{β}=

*Y*

_{β}(

*x*

_{2}). Further,

**e**

_{1}=(1,0) and

**e**

_{2}=(0,1) are orthonormal basis vectors in the two-dimensional space. Then, the operator

*P*

^{A1′}projects out the basis function in such a way that To satisfy the condition

*P*

^{A1′}

**=**

*u***, we have**

*u**α*

_{1}=odd,

*β*

_{1}=even,

*α*

_{2}=even,

*β*

_{2}=odd, irrespective of the parity of

*n*. This result indicates that

*u*

_{1}is an odd function for the

*x*

_{1}-direction and an even function for the

*x*

_{2}-direction. Similarly,

*u*

_{2}is even for the

*x*

_{1}-direction and odd for the

*x*

_{2}-direction. By comparing the results with table 2, we immediately see that this is nothing but the

*A*

_{1}symmetry. We therefore conclude that the point group and projection operator

*P*

^{A1′}explain the vibrational symmetry of the

*A*

_{1}′ mode.

#### (ii) *A*_{2}′ mode

We next consider the *A*_{2}′ mode. As mentioned earlier, vibrational symmetry of this mode depends on the parity of *n*. However, the linearized projection operator *P*^{A2} is independent of the parity, as seen in (3.5). Therefore, unlike the case of the *A*_{1}′ mode, we fail to project out (4.3) and (4.4) from (4.9). According to Neumann’s principle, the symmetry of a macroscopic physical property must be explained from a point group. This implies that we should, at first, construct a suitable point group that is responsible for the finite amplitude FVAR of the *A*_{2}′ mode.

To simplify the analysis, we introduce a time defined as . Then, the time interval of figure 3 is expressed as . The vibrational symmetry of the *A*_{2}′ mode is *A*_{2} for and *A*_{1} at . Therefore, the point group for the *A*_{2}′ mode seems to be for . However, this is incorrect. Let us introduce a time reversal operator *T*, which acts on such that , and define the composition maps by *Tσ*_{x}=*T*°*σ*_{x}=*σ*_{x}°*T* and *Tσ*_{y}=*T*°*σ*_{y}=*σ*_{y}°*T*. Then, the operators *Tσ*_{x} and *Tσ*_{y} also preserve the vibrational symmetry. Note that the operator *T* is equivalent to *E* at ; therefore, the set {*E*,*C*_{2},*Tσ*_{y},*Tσ*_{x}} is responsible for the vibrational symmetry of the *A*_{2}′ mode for the entire time interval including . Therefore, the proper set of symmetry operations is {*E*,*C*_{2},*Tσ*_{y},*Tσ*_{x}}. In group theory, this is called the *bicolour* (black and white) magnetic point group [18]. Because , it is frequently expressed as . The *A*_{2}′ mode is invariant under any symmetry operations in the magnetic point group, we set *χ*(*R*)=1 so as to obtain the totally symmetric projection operator
4.10Here, we briefly summarize the action of the time reversal operator *T* on the basis function (4.9). Because the operator *T* reverses the sign of , direct calculation yields . Thus, the operator *P*^{A2′} project out the basis function (4.9) in such a way that
To satisfy the condition *P*^{A2′}** u**=

**, the following symmetry must hold:**

*u**α*

_{1}=even,

*β*

_{1}=odd,

*α*

_{2}=odd and

*β*

_{2}=even for

*n*=odd and

*α*

_{1}=odd,

*β*

_{1}=even,

*α*

_{2}=even,

*β*

_{2}=odd for

*n*=even.

#### (iii) *B*_{1}′ and *B*_{2}′ modes

Similarly, the vibrational symmetry of the *B*_{1}′ mode can also be written as a bicolour magnetic point group . On the other hand, for the *B*_{2}′ mode, the symmetry is expressed by . The totally symmetric projection operators of the two bicolour magnetic point groups are written as
4.11The operator *P*^{B1′} project out the basis function (4.9) in such a way that
Similarly, the operator *P*^{B2′} yields
From the conditions *P*^{B1′}** u**=

**and**

*u**P*

^{B2′}

**=**

*u***, we obtain the symmetry of the basis functions for the**

*u**B*

_{1}′ and

*B*

_{2}′ modes, similar to the case of the

*A*

_{2}′ modes.

#### (iv) Symmetry of nonlinear free-vibration acoustic resonance modes

Table 3 summarizes the vibrational symmetry of the *A*_{1}′, *A*_{2}′, *B*_{1}′ and *B*_{2}′ modes obtained from the projection operations. A comparison with table 2 immediately reveals that the symmetry depends on the parity of *n*. More precisely, it is identical to that of the respective linearized mode if *n*=odd, whereas it becomes *A*_{1} if *n*=even. Obviously, this result shows excellent agreement with that obtained from the numerical analysis: (4.1)–(4.8). From this result, we conclude that the classification of nonlinear FVAR modes should be based on the magnetic point groups. As previously mentioned, an irreducible representation has been used to classify the linearized FVAR modes. However, the present numerical analysis clearly demonstrated that this classification no longer holds for finite amplitude nonlinear FVAR modes. This is because of the colour symmetry. A linearized FVAR mode generally consists only of harmonic plane waves. However, because of the geometrical nonlinearity, the excitation of higher harmonics and time-independent components naturally occurs in the nonlinear FVAR modes. The anharmonic displacement components possess *colour symmetry* in their vibrational patterns; therefore, magnetic point groups rather than the conventional irreducible representation should be used to classify the nonlinear FVAR modes.

#### (v) Grey colour nonlinear free-vibration acoustic resonance mode

Next, we consider the number of nonlinear FVAR modes. According to group theory, there exists three types of magnetic point groups: (i) black or white, (ii) grey and (iii) black and white. The first two are *single colour* groups, whereas the latter is a bicolour magnetic point group. The magnetic point groups can be constructed from a conventional point group [18]. For point group , the single colour (black or white) group is , which is identical to itself. Hence, in terms of the magnetic point group, the vibrational symmetry of the *A*_{1}′ mode is expressed by . The bicolour magnetic point group is defined by a coset decomposition of index 2 [18]. Because the point group is known to have three invariant subgroups, , and , and because these subgroups are indeed the coset decompositions of index 2, three types of *bicolour* magnetic point groups exists:
4.12
4.13
and
4.14These point groups are the basis of the *A*_{2}′, *B*_{1}′ and *B*_{2}′ modes, respectively. Finally, the grey magnetic point group is defined by . The projection operation revealed that there exists a grey colour nonlinear FVAR mode whose symmetry is *A*_{1} for *n*=even, whereas for *n*=odd, the displacement component identically vanishes as shown in table 3. Because there is no harmonic displacement component, the vibrational mode cannot be obtained by convergent calculation around a linearized solution. The details of this mode are still unknown.

## 5. Concluding remarks

We investigated FVAR of two-dimensional St Venant–Kirchhoff hyperelastic materials within the framework of nonlinear elasticity and the calculus of variations. The results obtained in this study are summarized as follows:

(i) The action integral

*I*of the hyperelastic material is defined as an integration of the Lagrangian density over the two-dimensional rectangular domain and a variable time interval. The stationary condition*δI*=0 poses a variable-domain-type variational problem, where the variations of time*t*and displacement*u*_{i}should be simultaneously considered.(ii) The Euler–Lagrange equation yields a set of quasi-linear wave equations whose solution is supposed to satisfy the time reversal symmetry. The natural boundary condition for

*δI*=0 requires that (i) the displacement velocity should periodically vanish at the beginning and end of FVAR, (ii) the displacement should satisfy the time periodicity condition, and (iii) the hyperelastic material should be free from surface-normal traction during the vibration.(iii) FVAR of the hyperelastic material has been formulated as the variational problem

*δI*=0 with a subsidiary condition ∥*u*_{i}∥_{L2}=const. To solve the constrained variational problem, we expanded the displacement functions using plane waves, including higher harmonics, and obtained nonlinear algebraic equations by the Ritz method. The nonlinear equations are solved around a linearized solution by convergent calculation using the Newton method. The numerical analysis demonstrated that there exists four types of nonlinear FVAR modes,*A*_{1}′,*A*_{2}′,*B*_{1}′ and*B*_{2}′, which are associated with the irreducible representations of the point group . We also revealed the excitation of anharmonic plane waves as well as a static displacement owing to the geometrical nonlinearity of the hyperelastic material.(iv) Vibrational symmetry of the nonlinear FVAR modes has been summarized in table 3. A comparison with the linearized modes (table 2) immediately reveals that the symmetry depends on the parity of

*n*: it is identical to a linearized one if*n*=odd, whereas it becomes*A*_{1}if*n*=even. This significant difference originates from the geometrical nonlinearity considered in the St Venant–Kirchhoff-type hyperelastic material.(v) The vibrational symmetry of the

*A*_{1}′ mode is expressed by single colour (black or white) magnetic point group . On the other hand, the other three modes are represented by bicolour (black and white) magnetic point groups: , and . Projection operators, derived from the magnetic point groups, precisely explained the vibration symmetry indicating that the magnetic point group is the basis for the classification of finite amplitude nonlinear FVAR modes. According to the results, we concluded that the magnetic point group is the proper generalization of the irreducible representations that have been used to classify the linearized FVAR modes. Existence and vibrational symmetry of a grey colour nonlinear FVAR mode is also predicted.

## Funding statement

This work was supported by JSPS KAKENHI grant no. 23686024.

## Acknowledgements

The author acknowledges Mr S. Yamada for his help on numerical analysis. He also acknowledges Prof. Y. Shibutani for his continuous encouragement and fruitful discussion.

## Appendix A

is a real-valued square matrix. Let be the tensor notation for the (*m*^{α},*m*^{β})th matrix component. Then, the diagonal components of the matrix ** Γ** are expressed as follows:
The relationship between

*m*

^{α}and is given in table 1, and the same holds for

*m*

^{β}and . For instance,

*m*

^{β}=3 indicates that and . Here,

*δ*

_{mαmβ}is the Kronecker delta, indicating that

*δ*

_{mαmβ}=1 when

*m*

^{α}=

*m*

^{β}and

*δ*

_{mαmβ}=0 otherwise. The off-diagonal components are Note that

*γ*

^{81}=

*γ*

^{18},

*γ*

^{72}=

*γ*

^{27},

*γ*

^{63}=

*γ*

^{36}and

*γ*

^{54}=

*γ*

^{45}because

**is a symmetric matrix. , , and are defined by**

*Γ*- Received April 30, 2013.
- Accepted July 30, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.