# Shear modulus and elastic relationship tips

### Rheology and deformation mechanisms

Young's modulus is defined by the relation()σl=EelThus E is the ratio of .. For fracture propagation, the rock must physically split at the fracture tip. KEYWORDS: AFM; elastic properties; Young's modulus; shear modulus; normal contact stiffness; lateral contact stiffness; periodic lateral movement between tip and sample and relationship G D E/2⊳1 C v⊲, where v is the Poisson ratio. Young's modulus or Young modulus is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain .. "Kevlar Technical Guide": 9. ^ M. Staines, W. H. Robinson.

Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however all solid materials exhibit nearly Hookean behavior for small enough strains or stresses. If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear.

Otherwise if the typical stress one would apply is outside the linear range the material is said to be non-linear. Steelcarbon fiber and glass among others are usually considered linear materials, while other materials such as rubber and soils are non-linear. However, this is not an absolute classification: For example, as the linear theory implies reversibilityit would be absurd to use the linear theory to describe the failure of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure.

In solid mechanicsthe slope of the stress—strain curve at any point is called the tangent modulus.

It can be experimentally determined from the slope of a stress—strain curve created during tensile tests conducted on a sample of the material. Directional materials[ edit ] Young's modulus is not always the same in all orientations of a material.

Most metals and ceramics, along with many other materials, are isotropicand their mechanical properties are the same in all orientations.

However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become anisotropicand Young's modulus will change depending on the direction of the force vector. Also, the sensitivity of the nanoindentation measurement to accurately measure anisotropic elastic behavior is shown [ 15 — 17 ] to be a function of the indenter tip shape. For example, a triangular indenter can give values that are much higher than axisymmetric tips such as spheroconical, but with less sensitivity than that obtained using axisymmetric shapes.

Nanoindentation measurements with Berkovich diamond tips may not quantify the full effect of anisotropic behavior; that is, the anisotropic response can be underestimated in both modeling and experimental efforts. Similarly, different effects are found [ 1819 ] for the effect of tip shape on plastic flow wherein spheroconical shapes evidence work hardening of material during indentation as opposed to well-defined pyramidal indentations that produce a near-perfect plastic response.

The torsion-resonance mode of atomic force acoustic microscope is used [ 20 ] to measure the elastic constants of anisotropic materials. A piezoelectric device is excited using an alternating current voltage to induce vibrations in the atomic force microscopy AFM cantilever while the tip is in contact with the sample surface.

### Young's modulus - Wikipedia

Indentation elastic modulus is extracted from the tip-surface interaction assuming Hertzian contact mechanics. In a similar technique [ 21 ], the deflection of the AFM cantilever is used to determine the localized modulus. Vibrating reed measurements [ 22 ] have similarities to the AFM technique where the major difference is that the sample along with the substrate in the vibrating reed method is exposed to piezoelectric vibrations whereas the probe cantilever is vibrated in the AFM technique.

The oscillating bubble method [ 23 ] is another technique for measuring surface elasticity of liquids. The tapping mode elastic modulus measurement technique [ 24 — 29 ] provides another relatively new method for measuring the reduced elastic modulus.

The measurement is based on the oscillation frequency of a probe that is in elastic contact with the surface. The tapping mode technique is nondestructive, which provides an advantage of measuring the elastic modulus prior to plastic deformation whereas nanoindentation measures the elastic-plastically deformed material.

Vapor deposited coatings of cubic metals can have a single-growth texture that is of interest for assessing the sensitivity of the measurement technique to anisotropic elastic behavior in nanocrystalline metals such as nickel.

## Journal of Nanotechnology

For this purpose, polycrystalline samples provide a basis for measurement calibration along with single-crystal wafer specimens. The new solution can be used to simulate the magnitude of the shift in the resonant frequency as a function of the reduced elastic modulus using the cantilever bending stiffness and the indenter tip radius of curvature. Materials and Methods 2. Materials The materials for the tapping mode measurement should have smooth surfaces. If there is a high amplitude to short-wavelength surface roughness, then multiple point contact can occur which could appear to artificially stiffen the mechanical response.

## Dynamics: Rheology and deformation mechanisms

The metals are coated onto polished flats such as semiconductor-grade silicon and sapphire wafers. In brief, the 0.

The measured surface temperature of the substrate remains below K during the deposition process. The deposition rates and thickness accumulation during deposition are recorded using calibrated Au-coated quartz-crystal microbalances.

Details of the structural characterization for the Ta, V, and Ni sputter deposited coatings are reported elsewhere [ 3035 ]. The sputter deposited metal coatings have a single growth orientation of close-packed planes. The body-center-cubic bcc metals of Ta and V areand the face-center-cubic fcc Ni is TEM imaging in bright field, in the electron-diffraction mode, and the use of high resolution lattice imaging conditions [ 3637 ] enable a closer look at the grain size, grain boundary structure, and defect structure.

Relation between Elastic Constants - Strength of Materials

Selected area diffraction patterns SADPs confirm the growth for the Ta and V coatings as well as the growth for the Ni coating. Also, the SADPs indicate that the coatings are polycrystalline in plane, that is, randomly oriented parallel to the surface. The grain sizes determined from the average width to the columnar growth as seen in the bright field TEM images range from 14 to 20 nm for the Ta and V coatings to less than nm for the Ni coating.

AFM images [ 35 ] of the Ta and V surfaces indicate a smooth surface with less than a nm variation in amplitude over 1 m length scales, that is, a surface smooth for establishing a Hertz-contact condition. Semiconductor-grade, single-side polished wafers of silicon SiSiand sapphire Al2O3 Also, polished polycrystalline samples of polycarbonate pCquartz SiO2hydroxyapatite HAand sapphire Al2O3 are used as reference standards with known elastic constants for the establishment of the calibration curve from the tapping mode measurements.

Tapping Mode Experiment The tapping mode use [ 242628 ] of a nanoprobe, in the configuration of a scanning force microscope, provides a method for measuring elastic deformation.

Both the 14 kHz resonant frequency of the free-standing cantilever with a spring constant of The frequency and amplitude of the probe oscillation are quantitatively measured as a function of the vertical displacement of the probe tip.