Equations Of Anisotropic Elasticity Research Articles

Analytic determination of stress fields in cross-ply symmetric composite laminates

A Hamiltonian state space approach for analytic determination of deformation and stress fields in multilayered cross-ply symmetric laminates under extension and bending is presented, in which the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered laminates are satisfied, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates.

Analysis of a Monolithic, Nonperiodic Array of Quartz Crystal Microbalances

We study thickness-shear and thickness-twist vibrations of a crystal plate with a nonperiodic array of surface mass layers for mass sensor applications. A theoretical analysis is performed. The equations of anisotropic elasticity are used. An analytical solution is obtained for the free vibration eigenvalue problem using trigonometric series from which the resonant frequencies and mode shapes are calculated. Results show that, depending on the geometric and physical parameters of the mass layers, the vibration may be mainly trapped under some of the mass layers but not under others.

Overtone frequency spectra for ξ3-dependent modes in AT-cut quartz resonators [Correspondence

We study straight-crested waves and vibration modes with spatial variations along the x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> direction only in an AT-cut quartz plate resonator. The equations of anisotropic elasticity are used. Dispersion relations for face-shear and thickness-twist waves in unbounded plates are plotted. Frequency spectra are obtained for face-shear and thickness-twist vibrations of finite plates in which these modes are coupled by boundary conditions. Most importantly, our analysis produces the frequency spectra for overtone modes which do not seem to have been obtained before for x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> -dependent modes. Numerical results for third- and fifth-overtone AT-cut quartz resonators are presented, showing that higher-order overtone modes are associated with more mode couplings.

Effects of Mass Layer Stiffness and Imperfect Bonding on a Quartz Crystal Microbalance

We study free vibrations of a crystal plate of AT-cut quartz carrying a thin mass layer operating as a quartz crystal microbalance for mass sensing. The mass layer is imperfectly bonded to the crystal plate with its interface described by the so-called shear-slip model that allows a discontinuity of the interface displacement. The effect of mass layer in-plane shear stiffness is also considered. The equations of anisotropic elasticity are used for the crystal plate with the omission of the small elastic constant <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">56</sub> . The mass layer is governed by the plane-stress equations of elasticity. An analytical solution is obtained using Fourier series, from which the resonant frequencies and vibration mode shapes are calculated. The effects of the mass layer in-plane shear stiffness, imperfect interface bonding on resonant frequencies and energy trapping are examined.

Effects of asymmetric and nonuniform mass layers on quartz crystal microbalances

The free vibrations of an AT-cut quartz plate carrying a thin mass layer operating as a quartz crystal microbalance for mass sensing is presented. More general than previous studies, the mass layer may be nonuniform and asymmetric. The equations of anisotropic elasticity are used to model the quartz crystal plate with the omission of the small elastic constant c 56. An analytical solution is obtained using Fourier series from which the resonant frequencies and the vibration mode shapes are calculated. The effects of mass layer nonuniformity and asymmetry on the resonant frequencies and mode shapes as well as the vibration confinement by the mass layer are examined.

Energy trapping of thickness-shear and thickness-twist modes in a partially electroded AT-cut quartz resonator

The thickness-shear and thickness-twist vibrations of a finite and partially electroded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.

Effects of mismatched electrodes on an ATCut quartz resonator

We study thickness-shear and thickness-twist free vibrations of a finite AT-cut quartz resonator with mismatched electrodes. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and vibration confinement resulting from the electrode inertia are calculated and examined.

Thickness-shear and thickness-twist vibrations of an AT-Cut quartz mesa resonator

We study thickness-shear and thickness-twist vibrations of an AT-cut quartz plate mesa resonator with stepped thickness. The equations of anisotropic elasticity are used with the omission of the small elastic constant c(56). An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and energy trapping are calculated and examined. The solution shows that a mesa resonator exhibits strong energy trapping of thickness-shear and thickness-twist modes, and that the trapping is sensitive to some of the structural parameters of the resonator.

Effects of mass layer dimension on a finite quartz crystal microbalance

In this paper, we investigate free thickness-shear and thickness-twist vibrations of a finite crystal resonator of AT-cut quartz carrying a thin mass layer at the central portion of its top surface operating as a quartz crystal microbalance. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes and vibration confinement by the mass layer are calculated and examined.

Shear-horizontal waves in a rotated Y-cut quartz plate in contact with a viscous fluid

We study shear-horizontal (SH) waves in a crystal plate of rotated Y-cut quartz in contact with a semi-infinite viscous fluid. The crystal plate and the fluid are governed by the equations of anisotropic elasticity and the theory of Newtonian fluids. A transcendental equation that determines the dispersion relations of the waves is obtained. Approximate analytical solutions to the equation are presented for the case of low viscosity fluids and the case of long waves whose wavelength is much larger than the plate thickness. The effects of the fluid viscosity and density on the dispersion relations of the waves are examined. The results obtained are fundamental and useful to the understanding and design of acoustic wave fluid sensors for measuring fluid viscosity or density.

Analysis of a monolithic crystal plate acoustic wave filter

We study thickness–shear and thickness–twist vibrations of a finite, monolithic, AT-cut quartz plate crystal filter with two pairs of electrodes. The equations of anisotropic elasticity are used with the omission of the small elastic constant c 56. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and the vibration confinement due to the electrode inertia are calculated and examined.

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.

Edge effect in laminate composite materials

In other studies dealing with the edge effect in composite materials [1-5] each layer was regarded as an anisotropic homogeneous one, all layers rigidly joined together and the entire specimen in a state of generalized plane strain under tension or flexure. The equations of anisotropic elasticity were solved by the finite-difference method [i, 2], by the finite-element method [3, 4], or by expansion into a double series of Legendre polynomials [5]. In the way these methods have been used, however, they hardly yield a true description of the edge effect with respect to stresses, because they are efficacious only when a bounded function will be the solution and not when the stresses can have singularities, as in the case of composite materials. In the application of these methods, the edge stresses remain bounded, but their maximum increases with increasing accuracy of the method (e.g., with an increasing number of terms retained in the series of orthogonal polynomials [5]) and this indirectly confirms the existence of a singularity. Here it will be demonstrated that in composite materials with characteristics close to real, there may appear a stress singularity, then there will be given examples of calculating the degree of such a singularity as a function of the reinforcement factor and of the fiber lay angle in adjacent layers of the composite

Elastic stresses and self-energies of dislocations of arbitrary orientation in anisotropic media: Olivine, orthopyroxene, calcite, and quartz

Elastic stress fields, self-energies, and energy factors of straight edge and screw dislocations in olivine, orthopyroxene, calcite, and α quartz are calculated numerically from the equations of anisotropic elasticity. Energy factor profiles are used to determine which slip systems are preferred on the basis of elastic self-energies, and the effects of elastic anisotropy on the stresses and self-energies of the dislocations are investigated. The moderate anisotropies of olivine and orthopyroxene lead to results that differ by about 10% from the isotropic elastic solutions. Comparison with observations indicates that the slip direction and glide plane preference of dislocations in these orthorhombic minerals cannot be predicted solely on the basis of anisotropic elasticity theory. In quartz and especially in calcite the large elastic anisotropy does have a major influence on slip system preference, which is consistent with the observed systems. Anisotropic calculations for olivine at 1000°C and for α quartz near the α-β transition temperature failed to reveal any significant changes in slip system preference associated with the effects of higher temperatures.

Uniformly Moving Dislocations in Anisotropic Media

Solutions which correspond to surface waves are given of the general dynamic equations of anisotropic elasticity when all displacements are independent of one coordinate. These solutions are used to discuss the uniform glissile motion of a long straight dislocation line of arbitrary Burgers vector by a generalization of a method previously used by Eshelby for the isotropic case. Although the method used here is generally applicable, explicit solutions are for simplicity given only when the moving dislocation line is so related to the symmetry elements of the crystal that the edge and screw components can be considered separately. The corresponding solutions for static dislocations in anisotropic media and uniformly moving dislocations in isotropic media can be derived from these solutions. The dislocation line is also treated in the Peierls-Nabarro approximation and expressions are derived for the variation with velocity of the widths of the edge and screw components. The results are illustrated by considering the variation with velocity of the widths of dislocations moving in the basal planes of magnesium, zinc and cadmium.