# Derive relationship between bulk modulus and youngs aluminum

### Bulk modulus - Wikipedia

Poisson's ratio: relation to elastic moduli in isotropic solids Aluminum [4] ratio values are derived in terms of the elastic compliances of materials with cubic. The bulk modulus of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting where ρ is density and dP/dρ denotes the derivative of pressure with respect to density (i.e. pressure rate of change with volume). The inverse of the. Simple relations between the first (or second) pressure derivative of the bulk modulus and . have reached 3 GPa [Jackson and Neisler, ; Rigden et al., . Modulus. The elastic constants have been calculated as strain.

Rinde, Poisson's ratio for rigid plastic foams, J.

### Elasticity – The Physics Hypertextbook

Applied Polymer Science, 14, Gray, American Institute of Physics Handbook, 3rd ed. LakesFoam structures with a Negative Poisson's ratioScience, LakesTemperature insensitive negative Poisson's ratios in isotropic alloys near a morphotropic phase boundary, Appl.

Lett, Dec. Bend a bar or plate. Poisson's ratio governs the curvature in a direction perpendicular to the direction of bending. This "anticlastic curvature" is easily seen in the bending of a rubber eraser. Shown here is bending, by a moment applied to opposite edges, of a honeycomb with hexagonal cells. Since the honeycomb is anisotropic, the Poisson's ratio need not lie within the above range.

Poisson's ratio and anisotropy. In anisotropic solids including single crystals, honeycombs, and fibrous composites, physical properties, including Poisson's ratio and elastic moduli, depend on direction. Poisson's ratio can have positive or negative values of arbitrarily large magnitude in anisotropic materials. For orthotropic materials, Poisson's ratio is bounded by the ratio of Young's moduli E as follows. Poisson's ratio in viscoelastic materials The Poisson's ratio in a viscoelastic material is time dependent in the context of transient tests such as creep and stress relaxation.

If the deformation is sinusoidal in time, Poisson's ratio may depend on frequency, and may have an associated phase angle. Specifically, the transverse strain may be out of phase with the longitudinal strain in a viscoelastic solid. Get pdf of a research article on this.

Poisson's ratio and phase transformations Poisson's ratio can vary substantially in the vicinity of a phase transformation.

Typically the bulk modulus softens near a phase transformation but the shear modulus does not change much. The Poisson's ratio then decreases in the vicinity of a phase transformation and can attain negative values.

Phase transformations are discussed further on the linked page.

Poisson's ratio, waves and deformation The Poisson's ratio of a material influences the speed of propagation and reflection of stress waves. In geological applications, the ratio of compressional to shear wave speed is important in inferring the nature of the rock deep in the Earth.

This wave speed ratio depends on Poisson's ratio. Poisson's ratio also affects the decay of stress with distance according to Saint Venant's principle, and the distribution of stress around holes and cracks. Analysis of effect of Poisson's ratio on compression of a layer.

What about the effect of Poisson's ratio on constrained compression in the 1 or x direction?

Constrained compression means that the Poisson effect is restrained from occurring. This could be done by side walls in an experiment.

Also, compression of a thin layer by stiff surfaces is effectively constrained. Moreover, in ultrasonic testing, the wavelength of the ultrasound is usually much less than the specimen dimensions. The Poisson effect is restrained from occurring in this case as well.

In Hooke's law with the elastic modulus tensor as Cijkl we sum over k and l, but, due to the constraint, the only strain component which is non-zero is e Let us find the physical significance of that tensor element in terms of engineering constants.

One may also work with the elementary isotropic form for Hooke's law. Usage[ edit ] Young's modulus enables the calculation of the change in the dimension of a bar made of an isotropic elastic material under tensile or compressive loads. For instance, it predicts how much a material sample extends under tension or shortens under compression.

The Young's modulus directly applies to cases of uniaxial stress, that is tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also used in order to predict the deflection that will occur in a statically determinate beam when a load is applied at a point in between the beam's supports. Other elastic calculations usually require the use of one additional elastic property, such as the shear modulusbulk modulus or Poisson's ratio.

Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. Linear versus non-linear[ edit ] Young's modulus represents the factor of proportionality in Hooke's lawwhich relates the stress and the strain.

## Elasticity

However, Hooke's law is only valid under the assumption of an elastic and linear response. 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.