# Elastic constants and their relationship

### Elastic constants, relation between elastic constants

enter image description here. Consider a solid cube, subjected to a Shear Stress on the faces PQ and RS and complimentary Shear Stress on faces QR and PS. For this case, there are only two independent elastic constants! There are then this equation is a useful relationship between E and G. We find for many metals. Get Study Notes on Elastic Constants for GATE, ESE and other Mechanical Different types of stresses and their corresponding strains within elastic limit are related which are Proof of relation between elastic constant.

Sokolnikoff, Mathematical theory of elasticity. Krieger, Malabar FL, second edition, Wang, Single crystal elastic constants and calculated aggregate properties: Rinde, Poisson's ratio for rigid plastic foams, J.

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.

- Young's modulus
- Elastic modulus
- Strength of Materials by V. Ramasamy, P. Purushothama Raj

Since the honeycomb is anisotropic, the Poisson's ratio need not lie within the above range. Poisson's ratio and anisotropy.

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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.

### Young's modulus - Wikipedia

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.

**Relation between Elastic Constants - Strength of Materials**

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.

### Elastic Constants Relation Homework Help, Assignment Help, Online Tutor, Tutoring

The Poisson effect is restrained from occurring in this case as well. 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. 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. 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.