Third-order Elastic Stiffness Research Articles

Periodic solutions of a graphene based model in micro-electro-mechanical pull-in device

Phase plane analysis of the nonlinear spring-mass equation arising in modeling vibrations of a lumped mass attached to a graphene sheet with a fixed end is presented. The nonlinear lumped-mass model takes into account the nonlinear behavior of the graphene by including the third-order elastic stiffness constant and the nonlinear electrostatic force. Standard pull-in voltages are computed. Graphic phase diagrams are used to demonstrate the conclusions. The nonlinear wave forms and the associated resonance frequencies are computed and presented graphically to demonstrate the effects of the nonlinear stiffness constant comparing with the corresponding linear model. The existence of periodic solutions of the model is proved analytically for physically admissible periodic solutions, and conditions for bifurcation points on a parameter associated with the third-order elastic stiffness constant are determined..

A Lumped-Parameter Model for Nonlinear Waves in Graphene

A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applications.

Elastic and nonlinear stiffness of graphene: A simple approach

The recent experiment C. G. Lee et al. [ Science 321, 385 (2008)] on the Young's modulus and third-order elastic stiffness of graphene are well explained in a very simple approach, where the graphene is described by a simplified system and the force constant for the nonlinear interaction is estimated from the Tersoff-Brenner potential.

Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene

We measured the elastic properties and intrinsic breaking strength of free-standing monolayer graphene membranes by nanoindentation in an atomic force microscope. The force-displacement behavior is interpreted within a framework of nonlinear elastic stress-strain response, and yields second- and third-order elastic stiffnesses of 340 newtons per meter (N m(-1)) and -690 Nm(-1), respectively. The breaking strength is 42 N m(-1) and represents the intrinsic strength of a defect-free sheet. These quantities correspond to a Young's modulus of E = 1.0 terapascals, third-order elastic stiffness of D = -2.0 terapascals, and intrinsic strength of sigma(int) = 130 gigapascals for bulk graphite. These experiments establish graphene as the strongest material ever measured, and show that atomically perfect nanoscale materials can be mechanically tested to deformations well beyond the linear regime.

Harmonic and anharmonic components of third-order elastic constants

The values of the third-order elastic stiffness constants ${c}_{\mathrm{IJK}}$ contain contributions not only from the anharmonic components of the interatomic potentials but also from the use of Lagrangian strain and from the harmonic (Hooke's law) components under finite deformation. The contribution from Lagrangian strain is general, while the finite-deformation contribution depends on the structure of the crystal. These harmonic components are calculated for some group IV and III-V tetrahedral semiconductors within a valence force-field model.

PRACTICAL NON-LINEAR ELASTICITY THEORY FOR LARGE STRAINS

We present a theory of non-linear elasticity, of finite deformation, in which real stress-strain relationships are set up and defined thermodynamically. Relationships between these quantities and the standard third-order elastic stiffness coefficients c IJK are given. The utility of these quantities is demonstrated by easy derivations of the pressure coefficients of the second-order elastic constants c IJK , and of the acoustic velocities in materials under elastic strain.

Effective thermodynamic elastic constants under finite deformation

Nonlinear elasticity is required to account for mechanical and acoustic properties of bodies under finite elastic deformation. R. N. Thurston and K. Brugger gave equations [Phys. Rev. 133, A1604 (1964)] relating experimental acoustic velocities in a strained elastic body to the thermodynamic third-order elastic constants cIJK given in data books. However, no correct and complete set of expressions has been given for the general case, to obtain effective elastic constants from the cIJK to solve mechanical or physical problems. By considering two deformations applied to an elastic body, effective thermodynamic second-order constants cIJ under finite strain are defined thermodynamically and correctly related to the Lagrangian and Eulerian third-order elastic stiffness constants, cIJK.

Effects of higher-order anisotropic elasticity using textured polycrystals in three-dimensional wave propagation problems

Increasingly, more precise analysis of components in complicated systems is required to better understand material behavior under high strain-rate deformation. Applications such as warhead/target interactions and crash-worthiness testing are typical examples. Historically, in order to keep these analyses tractable, only a minimal description of the material behavior was included by simplifying the material characterization down to just a few parameters. Recent works have focused on including a greater degree of information regarding the micro- and meso-structure of polycrystals in the modeling of material behavior under a variety of loading conditions. One of the first steps in this effort is the inclusion of a material's crystallographic texture into the estimation of elastic properties. An additional step is the extension of the constitutive modeling to include anisotropic non-linear elastic behavior. Both of these steps are undertaken here in terms of third-order elastic stiffness tensors, as determined for various single crystal and polycrystal test materials of interest, and applied in three-dimensional (3-D) FEM simulation of the collapse of hemispherical shells.

Effects of high pressure on the Bose peak in a-GeS 2 studied by light scattering

High-resolution Brillouin and Raman scattering measurements have been carried out for amorphous germanium disulfide (a-GeS 2) under high pressures (up to 2.5 GPa) using diamond anvil cell (DAC). The pressure dependence of the Bose peak is discussed in comparison with that of the longitudinal acoustic (LA) mode. Third-order elastic stiffness constants for these modes are obtained, which show a large anharmonicity with respect to the Bose peak modes.

Third-order stiffness constants of textured hexagonal crystallite aggregates with an orthotropic macroscopic symmetry

Voigt-type averaging is carried out to obtain the effective second- and third-order elastic stiffness constants of textured hexagonal (6/mmm, 6̄m2 or 6mm point group) crystallite aggregates with orthotropic macroscopic symmetry. The second-order constants agree with previously reported results. The third-order constants are presented in a compact formula. The formula relates the effective third-order stiffness constants to the crystallite orientation distribution coefficients (CODCs) and the single-crystal third-order stiffness constants. If any two of these three sets of parameters are known, information of the third set can be inferred. The sensitivity of the effective third-order stiffness constants to changes of the CODCs is also given.

Elastic Behaviour of Terbium Metaphosphate Glasses Under High Pressures

The elastic and nonlinear acoustic vibrational properties of terbium metaphosphate glasses (Tb2O3)x(P2O5)1?x with x = 0�226,0�247,0�263 and 0�271 (x is the mole fraction) have been determined from measurements of the effects of temperature, hydrostatic pressure, and uniaxial stress on' ultrasonic wave velocity. At temperatures below about 140 K, the elastic stiffness of' (Tb2O3)x(P2O5)1?x glasses becomes anomalously dependent upon temperature, a behaviour usually associated with interactions between acoustic phonons and two-level systems. Except for the (Tb2O3)0�271(P205)0�729 glass, the hydrostatic pressure derivatives of the elastic stiffness and also of the bulk modulus BS of terbium metaphosphate glasses are small and negative. The third-order elastic stiffness tensor components CIJK of the (Tb2O3)0�247(P2O5)0�753 glass between 77 K and 400 K have also been determined. At room temperature, C112, C123 and C144 are positive while C111, C155 and C456 are negative. Both longitudinal and shear acoustic mode Gr�neisen parameters are small and negative: the application of pressure softens the long-wavelength acoustic phonon mode frequencies. The mode softening is enhanced as the temperature is reduced.

Optical and ultrasonic properties of europium phosphate glasses

To provide an overall picture of the vibrational properties of phosphate glasses containing high concentrations of europium, measurements have been made of their ultrasonic wave velocities and attenuation, optical absorption spectra and laser-induced fluorescence. To examine their valence state, the fluorescence of the glasses has been examined. The spectra do not show any obvious sign of divalent europium ions, only trivalent europium ion fluorescence being observed. Room-temperature absorption spectra of these glasses also provide evidence of only the absorption bands of trivalent europium. The elastic stiffnessesC11 andC44 continue to increase down to low temperatures and the ultrasonic attenuation is characterized by a broad peak, properties which are consistent with thermally activated relaxations of two-level systems. The longitudinal and shear ultrasonic wave velocities decrease under pressure; the hydrostatic pressure derivatives (∂C11/∂P)T, P=0 and (∂C44/∂P)T, P=0 of the elastic stiffness tensor componentsCIJ and (∂B/∂P)T, P=0 of the bulk modulus,B0, are negative. When compressed, the europium phosphate glasses, like their samarium analogues, show the interesting property of becoming easier to squeeze. Measurements, using a pulse superposition technique, of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components of (Eu2O3)0.186(P2O5)0.814 and (Eu2O3)0.20(P2O5)0.80 glasses between 77 and 400 K. The uniaxial and hydrostatic pressure dependences of the elastic constants quantify the cubic term in the strain Hamiltonian and the vibrational anharmonicity of the long-wavelength phonons of these glasses. The acoustic mode Gruneisen parameters are negative: application of pressure induces a decrease in the long-wavelength acoustic phonon mode frequencies. As the temperature is reduced the pressure-induced mode softening becomes enhanced. The hydrostatic pressure derivative (∂C11/∂P)T, P=0 is larger than (∂C44/∂P)T, P=0 over the whole temperature range, and the longitudinal acoustic mode Gruneisen parameter ∥γL∥ is larger than that ∥γS∥ of the shear wave: the longitudinal mode softens more with pressure than the shear mode. Murnaghan's equation-of-state is used to determine the compressionV(P)/V0.

Third-order elastic constants, vibrational anharmonicity, and the Invar behavior of the Fe72Pt28 alloy.

In the ferromagnetic state the alloy ${\mathrm{Fe}}_{72}$${\mathrm{Pt}}_{28}$ shows strong Invar effects governed by magnetoelastic interactions, including a negative hydrostatic pressure dependence of the bulk modulus: it becomes easier to squeeze under pressure. This remarkable behavior has prompted study of its nonlinear acoustic properties by determining its third-order elastic stiffness tensor components (TOEC) from measurements of the influence of uniaxial stress on the velocities of ultrasonic waves between 230 and 370 K, covering the range in which ${\mathrm{Fe}}_{72}$${\mathrm{Pt}}_{28}$ has a negative thermal expansion, a property central to the Invar behavior. To obtain complete sets of the TOEC from 230 K up to the Curie point (367 K), the results have been combined with data for the hydrostatic pressure dependences of ultrasonic wave velocities. The TOEC quantify the first-order anharmonic terms in the interatomic potential and hence the long-wavelength acoustic phonon anharmonicities which are central to the Invar properties of this alloy. The tensor components ${\mathit{C}}_{111}$, ${\mathit{C}}_{112}$, and ${\mathit{C}}_{123}$ are anomalously positive, in accord with the negative values previously determined for the hydrostatic pressure derivatives (\ensuremath{\partial}${\mathit{C}}_{11}$/\ensuremath{\partial}P${)}_{\mathit{T},}$${\mathit{P}}_{=0}$ and (\ensuremath{\partial}${\mathit{B}}^{\mathit{S}}$/\ensuremath{\partial}P${)}_{\mathit{T},}$${\mathit{P}}_{=0}$. Thus, the large stress-induced longitudinal-acoustic-mode softening in the ferromagnetic phase is confirmed, reinforcing the suggestion that the vibrational anharmonicities of the longitudinal acoustic modes, which have large negative mode Gr\"uneisen parameters, play an important part in causing the thermal expansion of ${\mathrm{Fe}}_{72}$${\mathrm{Pt}}_{28}$ to be negative.

Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica

Measurements of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components (TOEC) of vitreous silica between 77 and 293 K. In the absence of such data previous practice has been to assume that these TOEC are independent of temperature; however a significant temperature dependence has been observed. With the exception of the smallest C 456 , each of the TOEC is anomalously positive: a feature consistent with the well established aberrant negative values for the hydrostatic pressure derivatives (∂ C ij /α P ) P =0 of the second-order elastic stiffness tensor components (which reveal softening of the long-wave-length acoustic modes under pressure). The TOEC increase as the temperature is reduced showing that the pressure-induced acoustic mode softening becomes enhanced at lower temperatures. The largest TOEC is C 111 which is 46.7 × 10 10 Pa at 293 K and increases to 99.1 × 10 10 Pa at 77 K. The hydrostatic pressure derivatives of the second-order stiffnesses have been calculated from the TOEC; (∂ C 11 /∂ P ) p =0 is much larger than (∂ C 44 /∂ P ) P =0 over the whole temperature range: the longitudinal acoustic mode softens more with pressure than the shear mode. This effect is emphasised by the finding that the acoustic mode Gruneisen parameters are negative with |γ L | > |γ S |. As the temperature is reduced the longitudinal acoustic mode Gruneisen parameter, γ L , increases in magnitude considerably, reaching a value of −5.5 at 77 K. The mean acoustic mode Gruneisen parameter, γ el , increases to a larger negative value (−3.1 at 77 K) than previously suspected on the basis of the assumption of temperature-independent third-order elastic constants: the discrepancy between the low-temperature limits of the thermal, γ th L , and mean acoustic mode, γ el L , Gruneisen parameters is not quite so large as had been thought. The vibrational anharmonicity of the acoustic modes plays an important part in causing the thermal expansion of vitreous silica to be negative at low temperatures.

The second- and third-order elastic constants of amorphous arsenic

Abstract The hydrostatic and uniaxial pressure and the temperature dependences of the ultrasonic wave velocities in bulk amorphous arsenic (a-As) have been measured by the pulse echo overlap and pulse superposition techniques. The results provide the first complete set of components of the second- and third-order elastic stiffness tensors of an amorphous element. The ultrasound velocities, and elastic stiffnesses of the amorphous material are appreciably smaller than those of the crystalline, rhombohedral allotropic form. The bulk, shear and Young's moduli and the pressure derivatives of the bulk and shear moduli of a-As conform well with trends noted previously for these properties in chalcogenide glasses in terms of the average number of covalent bonds per atom. In marked contrast to the anomalous behaviour of the insulating, silica-based glasses, the pressure derivatives of the bulk and shear moduli of a-As are found to be positive, while the temperature derivatives are negative. This more normal elast...

Effects of acceleration on the resonance frequencies of crystal plates

The changes in the thickness-shear resonance frequencies of circular crystal plates subjected to steady in-plane acceleration of arbitrary direction are studied. A closed-form solution for a circular plate under acceleration with three or more points of mounting is obtained. From this solution, initial stress and strain fields are computed at each and every point of the plate as a function of plate orientation, direction of acceleration, and positions of supports. These fields are then taken into account in the coupled equations of the incremental thickness-shear and flexural vibrations through the second- and third-order elastic stiffness coefficients of the crystal. Due to the space dependence or nonuniformity of the initial fields and the smallness of the frequency changes, (Δf/fo in order of 10−9) a perturbation method is used to calculate the changes in the thickness-shear resonances. The frequency changes as function of the acceleration direction for several configurations of the supports for AT cut of quartz are computed. These are then compared with experimental values of A. W. Werner [Interim Reps. 10−11, Bell Telephone Labs. (1959)] and W. L. Smith [Interim Reps. 12−13, Bell Telephone Labs. (1960)].

Nonlinear effects of initial bending on the vibrations of crystal plates

The changes in the resonance frequencies of circular crystal plates subject to initial bending stresses are studied. The plate is flexed as a cantilever clamped near the edge and stressed by a transverse, concentrated force applied at a point diametrically opposite the support. A system of plate equations which takes into account the nonlinear effects due to the large, initial deflection and to the third-order elastic stiffness coefficients in the stress–strain relations is employed. In obtaining the initial fields caused by bending, strain components are assumed to be small, but large gradient of plate deflection and large rotations of the plate element about x1 and x3 axes are permissble by retaining their quadratic terms in the strain-displacement relations. The changes of resonance frequencies of fundamental thickness-shear vibrations are calculated for rotated Y cuts of quartz. The calculated frequency changes are compared with experimental values as a function of the force position, the azimuth angle ψ, and also as a function of the magnitude of thr force P. Subject Classification :[43]40.24.