# Shear modulus and young relationship

### Relations Between Moduli

Elastic moduli: E = Young's modulus (direct stress / direct strain). = Poisson's ratio (–lateral strain / direct strain). G = shear modulus (shear stress / shear strain). your suspicion is justified: The stiffness tensor (or matrix if you will) has to be positive definite. In the transversal isotropic case you have 4. Similar studies on the relationship between Young's modulus and the shear modulus or Poisson's ratio for cracked brittle materials have been.

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.

## Shear modulus

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.

### Young's modulus - Wikipedia

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.

### Shear modulus - Wikipedia

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.

There is stress in only one direction but there can be strain in three directions. So Young's modulus E is the stiffness for simple tension, with the Poisson effect free to occur. The physical meaning of Cis the stiffness for tension or compression in the x or 1 direction, when strain in the y and z directions is constrained to be zero.

**Relation between bulk modulus and young modulus (relation between E and K)**

The reason is that for such a constraint the sum in the tensorial equation for Hooke's law collapses into a single term containing only C The constraint could be applied by a rigid mold, or if the material is compressed in a thin layer between rigid platens.

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.

## Young's 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. Anisotropy can be seen in many composites as well.

For example, carbon fiber has a much higher Young's modulus is much stiffer when force is loaded parallel to the fibers along the grain. Other such materials include wood and reinforced concrete.

Engineers can use this directional phenomenon to their advantage in creating structures.