Measuring elastic constants of arbitrarily shaped samples using resonant ultrasound spectroscopy

One of the most important undertakings for materials is the measurement of the elastic behavior. As derivatives of the free energy with respect to atomic displacements, the elastic properties are closely connected to the thermodynamic properties of the material. Elastic behavior is a sensitive probe of the lattice environment in which all solid state phenomena occur, particularly in the vicinity of a phase transition. A useful method for measuring elastic properties is resonant ultrasound spectroscopy (RUS). Some novel materials to which RUS might be applied are often fragile or chemically reactive so that they cannot be polished into the shapes required by conventional RUS; for such cases a finite element method may be used. In this paper a discussion and test of a finite element method for RUS with arbitrarily shaped samples is provided.     

Determination of the dynamic elastic moduli and internal friction using thin resonant bars

Analysis of the data from the resonant-bar technique for determining wave speed and internal friction is presented. Internal friction has been included in the longitudinal and torsional wave equations and the analytical solution has been obtained. In determining the acoustical constants of lossy materials, a broad frequency spectrum is used that includes many resonances and not just data at or near the individual resonances. Corrections due to the mass, length, stiffness, and damping of the transducers are also presented. The theoretical solutions are compared with the measured magnitude and phase response data for torsional, longitudinal, and flexural measurements and are shown to be in good agreement.     

Wood elastic characterization from a single sample by resonant ultrasound spectroscopy

The goal of this paper is to propose an experimental method allowing the identification of the complete elastic tensor of anisotropic biological materials such as wood using only one sample. To do so, two complementary methods are used. First, the wood eigen-directions are defined from a sample of spherical shape that is then cut into a cube in a way to perform resonant ultrasound spectroscopy (RUS). The method is successfully applied on a reference beech sample with known orthotropic directions. A comparison of the identified elastic constants with those from the literature and some inferred from ultrasonic transmission measurements is given.     

Sensitivity of the resonant ultrasound spectroscopy to weak gradients of elastic properties

The applicability of resonant ultrasound spectroscopy on materials with weak spatial gradients in elastic coefficients and density is analyzed. It is shown that such gradients do not affect measurably the resonant spectrum but have a significant impact on the modal shapes. A numerical inverse procedure is proposed to explore the possibility of reconstructing the gradients from experimentally obtained modal shapes. This procedure is tested on synthetic data and applied to determine the gradient of the shear modulus in a continuously graded silicon nitride ceramic material. The results are in a good agreement with the gradient calculated for the examined material theoretically as well as with the results of other experimental methods.     

Sub-gigahertz laser resonant ultrasound spectroscopy for the evaluation of elastic properties of micrometric fibers

The cross-section eigenmodes of micrometric cylinders were measured in the range of several tens of MHz to about 0.5 GHz. The vibrations were excited using subnanosecond laser pulses. The cross-section eigenmodes were simulated using finite element modeling in a 2D geometry. Using the method of resonant ultrasound spectroscopy, the vibration spectrum of an aluminum wire of diameter 33μm served to determine Young’s modulus and Poisson’s ratio with a precision of 0.7% and 0.3%, respectively. The calculated and measured frequencies of cross-section eigenmodes were fitted with a precision better than 0.5% in the 50–500 MHz range.     

Modal resonant ultrasound spectroscopy for ferroelastics

Recent experimental and theoretical improvements of resonant ultrasound spectroscopy (RUS) are summarized to investigate elastic constants of phases in shape memory alloys. The proposed inversion procedure, described in this work, is particularly suitable to reliable evaluation of the temperature dependence of elastic constants of low-symmetry ferroelastic materials which may be strongly elastically anisotropic and tend to exist in twinned forms. The method is applicable even for the evaluation of single-crystal elastic constants from RUS measurements on microtwinned crystals, since it involves a homogenization algorithm based on the macroscopic deformation response of the layered structure. This potentially allows performing meaningful acoustic studies on samples with a general submicron-size layered structure.     

Determination of elastic moduli of GaN epitaxial layers by microindentation technique

It is demonstrated that Young’s modulus of epitaxial gallium nitride layers can be determined by the microindentation of their growth surface. The technique is based on the solution of the Hertz problem for the elastic indentation of a steel sphere into the studied surface. It is established that the isotropic approximation applied in this case is justified and leads to the satisfactory results. The microhardness measurements of epitaxial layers are carried out.     

Determination of ultrasonic wave velocities and phase velocity dispersion curves of an Inconel 600 plate using resonant ultrasound spectroscopy and leaky Lamb waves

A plate of Inconel 600 was interrogated using the resonant ultrasound spectroscopy (RUS) and the reflected leaky Lamb waves (LLW). It was found that the plate used in the present work has anisotropy in its material properties by the RUS. The longitudinal and the transverse wave velocities of the Inconel 600 plate were determined by the RUS, ultrasonic pulse-echo method and cut-off frequencies of the LLWs. The wave velocities in the direction of thickness determined by the RUS under the assumption of the orthotropic symmetry were quite similar to those obtained by other methods, the pulse-echo method and from cut-off frequencies. The reflected LLW from the plate was measured with varying the incident angle. The dispersion curves obtained from the reflected LLWs show good agreement with the theoretical calculation in general. The mismatches may be caused by anisotropy of the plate.     

Determination of the elastic properties of layered materials using laser excitation of ultrasound

It is proposed to use ultrasonic signals excited by a laser pulse to investigate the elastic properties (impedances, speeds of sound, and densities) of layered media. The results of studying both a model medium with known parameters and a layered composite are reported. The experimental data are in good agreement with the known properties of the samples investigated.     

On the use of hollow tube geometries for resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) can nondestructively obtain the elastic constants of compact specimens, however many materials have hollow cross-sections and frequency analysis of such geometries is required before inclusion in the RUS methodology. Resonant mode shapes of tubes with length equal to diameter and varying ratios of tube inner to outer diameter (Λ) as well as Poisson's ratio (ν) were identified by eigenvalue analysis using a commercial finite element code. Longitudinal and shear RUS experiments were conducted on tubes with Λ varying between 0 and 0.95 and compared to the numerical results. Simulations predict that the fundamental mode transitions from pure torsion to symmetric or antisymmetric ring bending at Λ = 0.3. The frequency of the first torsion mode is invariant to Λ and unequivocal identification of this mode is obscured by overlap of bending harmonics as Λ approaches 0.95. In the context of rapid calculation of isotropic elastic constants, shear moduli were calculated from the first torsional mode and Poisson's ratio was inferred from the Demarest maps of the mode structure's dependence upon Poisson's ratio. An average shear modulus of 27.5 + 1.5 ∕ -0.6 GPa, about 5% larger than literature values for 6061 aluminum, and ν of 0.33 were inferred. Errors are attributed to tube aspect ratios slightly greater than 1 and weak material anisotropy. Existing analytical solutions for ring bending modes derived from shell approximations and for infinitely long tubes under plane strain assumptions do not adequately describe the fundamental modes for short tubes. The shear modulus can be calculated for all Λ using the existing analytical solution.     

Bone micro-damage assessment using non-linear resonant ultrasound spectroscopy (NRUS) techniques: a feasibility study

Non-linear resonant ultrasound spectroscopy (NRUS) is a technique exploiting the significant non-linear behavior of damaged materials, related to the presence of damage. This study shows for the first time the feasibility of this technique for damage assessment in bone. Two samples of bovine cortical bone were subjected to a progressive damage experiment. Damage accumulation was progressively induced in the samples by mechanical testing. For independent assessment of damage, X-ray CT imaging was performed at each damage step, but only helped in the detection of the prominent cracks. Synchrotron micro-CT imaging and histology using epifluorescence microscopy were performed in one of the two samples at the last damage step and allowed detection of micro-cracks for this step. As the quantity of damage accumulation increased, NRUS revealed a corresponding increase in the non-linear response. The measured change in non-linear response is much more sensitive than the change in elastic modulus. The results suggest that NRUS could be a potential tool for micro-damage assessment in bone. Further work has to be carried out for a better understanding of the physical nature of damaged bone, and for the ultimate goal of in vivo implementation of the technique where bone access will be a challenging problem.     

Statistical mechanics of elastic moduli

We consider continuous systems of particles in the framework of classical statistical mechanics and derive a general expression for the static elastic moduli tensor in terms of correlation functions. We find sufficient conditions for the vanishing of the shear modulus. Relationships between these conditions and others insuring translational or rotational invariance are discussed.     

Point load determination of static elastic moduli using laser speckle interferometry

Electronic speckle interferometry (ESPI) is used to determine the Young's modulus E and Poisson ratio ν of an isotropic material. Micron scale deformations of the surface of the block of polymethyl-methacrylate (PMMA) are induced by normal application of a known near-point force. These deformations are recorded in speckle interferometric fringe patterns. An iterative minimum error inversion technique is developed to obtain the elastic properties from the positions of fringe peaks and troughs observed in the fringe patterns. Sensitivity tests of the method on calculated fringe patterns using measured experimental uncertainties suggest the technique will provide measures of the elastic moduli to better than 5%. In an experimental test on a bloc of PMMA (acrylic) the technique gave values of E and ν that differed from corresponding measures obtained using more conventional strain-gauge methods by less than 4%.     

Application of three-dimensional resonant acoustic spectroscopy method to rock and building materials.

This paper presents the experimental and theoretical results of applying resonant acoustic spectroscopy (RAS) to determine elastic parameters and losses in such consolidated granular materials as rock and building bricks. First, the theoretical aspects of the RAS method are outlined. A computer code for the rectangular and cylindrical samples was developed and tested. The results of experiments on specimens of rock and ceramic brick are then described. Finally, a modification of the previously published RUS algorithm is presented which permits a significant reduction in computing time for elongated samples.     

Zero-frequency elastic moduli of uniform fluids

A statistical mechanical treatment of equilibrium elasticity of a uniform fluid phase based on density functional theory is presented. Bulk expressions for the stress tensor and the zero-frequency elastic moduli tensor involving the direct correlation function are found.     

Third-order elastic moduli of single-layer graphene

In the framework of the Keating model with allowance made for the anharmonic constant of the central interaction between the nearest neighbors μ, analytical expressions have been obtained for three third-order independent elastic constants c _ijk (μ, ζ) of single-layer graphene, where ζ = (2α − β)/(4α + β) is the Kleinman internal displacement parameter and α and β are the harmonic constants of the central interaction between the nearest neighbors and the noncentral interaction between the next-nearest neighbors, respectively. The dependences of the second-order elastic constants on the pressure p have been determined. It has been shown that the moduli c ₁₁ and c ₂₂ differently respond to the pressure. Therefore, graphene is isotropic in the harmonic approximation, whereas the inclusion of anharmonicity leads to the appearance of the anisotropy.     

Applying nonlinear resonant ultrasound spectroscopy to improving thermal damage assessment in concrete

Nonlinear resonant ultrasound spectroscopy (NRUS) consists of evaluating one or more resonant frequency peak shifts while increasing excitation amplitude. NRUS exhibits high sensitivity to global damage in a large group of materials. Most studies conducted to date are aimed at interrogating the mechanical damage influence on the nonlinear response, applying bending, or longitudinal modes. The sensitivity of NRUS using longitudinal modes and the comparison of the results with a classical linear method to monitor progressive thermal damage (isotropic) of concrete are studied in this paper. In addition, feasibility and sensitivity of applying shear modes for the NRUS method are explored.     

Determination of absolute chirality using resonant X-ray diffraction

We have demonstrated that resonant diffraction experiments using the circularly polarized X-ray beam absolutely determines the crystal chirality. Both berlinite and quartz crystals belonging to space group P3₂21 show higher azimuth-constant intensity for the negative (?) helicity beam than that for the the positive (+) helicity beam for space-group forbidden reflection 001. The relation is opposite for quartz crystal belonging to space group P3₁21. Theoretical calculation shows that this relation completely agrees with the experimental findings for the enantiomorphic space-group pair P3₁21 and P3₂21. This method is applicable to chiral motifs that occur in biomolecules, liquid crystals, ferroelectrics, and multiferroics, etc.     

Computation of elastic moduli of graphene monolayer in nonsymmetric formulation using energy-based approach

In the paper, elastic moduli of finite-sized graphene monolayers are computed in a nonsymmetric formulation using the lattice statics approach. The motion of atoms due to their interaction is not considered, lattice stability is not studied. The presence of covalent binding is assumed to preserve material structure and all atoms are assigned displacements that correspond to a homogeneous deformation gradient tensor. As a result, the deformation kinematics of graphene is strictly controlled and the material response is defined using a variant of the interatomic interaction potential of the Mie family. The dimensionless parameters of the potential are identified using the coincidence criterion of the experimentally determined Poisson ratio of graphene with an estimated value. The obtained potential parameters are used to determine the elastic properties of a graphene monolayer in a nonsymmetric formulation for low strains and low temperatures. It is shown that the graphene monolayer under homogeneous deformation goes to a nonequilibrium state. In order to provide the potential energy minimum of the specimen in the deformed state, it is necessary to assign displacements to a part of graphene atoms that form one of its “triangular” sublattices relative to atoms of another sublattice, with each sublattice being deformed homogeneously.