Application of group theory to free oscillations of an anisotropic rectangular parallelepiped.

Mochizuki, Etsuko

doi:10.4294/jpe1952.35.159

Cited by 69 publications

(41 citation statements)

References 7 publications

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“…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.…”

Section: (B) Classification Of Linearized Free-vibration Acoustic Resmentioning

confidence: 99%

Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

Tarumi

2013

Proc. R. Soc. A.

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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: C 2v (C 2v ), C 2v (C 2 ), C 2v (C 1h ) and C 2v (C 1h ). 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.

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Section: (B) Classification Of Linearized Free-vibration Acoustic Resmentioning

confidence: 99%

Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

Tarumi

2013

Proc. R. Soc. A.

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“…Because the task is to excite resonances, it is important to drive the sample at a low-symmetry location to excite as many modes as possible. The lowest symmetry point on a RP sample is the corner, thus this is the most desirable point to drive and detect, an important principle discovered by Demarest [11] and Anderson et al [12], and derived, grouptheoretically, by Mochizuki [21]. Moreover, the corners have a low mechanical impedance so that touching them with a transducer has minimal (less than a 10-5 fractional frequency shift) effect on the free-surface boundary conditions if the contact force is low (10 3 dynes or less).…”

Section: Hardwarementioning

confidence: 99%

Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids

Migliori

Sarrao

Visscher

et al. 1993

Physica B: Condensed Matter

565

361

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The mechanical resonant response of a solid depends on its shape, density, elastic moduli and dissipation. We describe here instrumentation and computational methods for acquiring and analyzing the resonant ultrasound spectrum of very small (0.001 cm 3 ) samples as a function of temperature, and provide examples to demonstrate the power of the technique. The information acquired is in some cases comparable to that obtained from other more conventional ultrasonic measurement techniques, but one unique feature of resonant ultrasound spectroscopy (RUS) is that all moduli are determined simultaneously to very high accuracy. Thus in circumstances where high relative or absolute accuracy is required for very small crystalline or other anisotropic samples RUS can provide unique information. RUS is also sensitive to the fundamental symmetry of the object under test so that certain symmetry breaking effects are uniquely observable, and because transducers require neither couplant nor a flat surface, broken fragments of a material can be quickly screened for phase transitions and other temperature-dependent responses.

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“…One additional advantage of using global approximation functions over the beam cross-section and the length is the potential to group the specific approximations according to the physical character of the displacement components they are being used to represent [26,34]. Various conditions of cross-sectional symmetry, axial wavelength, and grouping of the various approximation functions can drastically reduce the size of the matrix eigenvalue problem.…”

Section: The Ritz Modelmentioning

confidence: 98%

Elasticity-based free vibration of anisotropic thin-walled beams

Heyliger¹

2015

Thin-Walled Structures

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Application of group theory to free oscillations of an anisotropic rectangular parallelepiped.

Abstract: Free oscillations of a homogeneous, anisotropic, elastic, rectangular parallelepiped are studied by means of the group theory. Normal modes are classified and selection rules for excitation are obtained.

Cited by 69 publications

References 7 publications

Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

Emergence of colour symmetry in free-vibration acoustic resonance of a nonlinear hyperelastic material

Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids

Elasticity-based free vibration of anisotropic thin-walled beams

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