# Shear modulus and elastic relationship problems

### Chapter 6. Mechanical Properties of Metals

Stress is force per unit area - strain is the deformation of a solid due to stress. 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. If E is elastic modulus, G is shear modulus (modulus of rigidity), and ν is Poisson's ratio, the relationship is Related Questions (More Answers Below) What is the relationship of modulus of rigidity to Poisson's ratio and modulus of elasticity?.

### Shear modulus - Wikipedia

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.

## Elasticity

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.

### What is Poisson's ratio?

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.

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. Calso governs the propagation of longitudinal waves in an extended medium, since the waves undergo a similar constraint on transverse displacement. Anelasticity Here the behavior is elastic but not the stress-strain curve is not immediately reversible.

It takes a while for the strain to return to zero. The effect is normally small for metals but can be significant for polymers. Elastic Properties of Materials Materials subject to tension shrink laterally. Those subject to compression, bulge. The ratio of lateral and axial strains is called the Poisson's ratio n. If the stress is too large, the strain deviates from being proportional to the stress. The point at which this happens is the yield point because there the material yields, deforming permanently plastically.

Hooke's law is not valid beyond the yield point. The stress at the yield point is called yield stress, and is an important measure of the mechanical properties of materials.

**Elastic Constants and their Relationship - Lecture - 5**

In practice, the yield stress is chosen as that causing a permanent strain of 0. The yield stress measures the resistance to plastic deformation. The reason for plastic deformation, in normal materials, is not that the atomic bond is stretched beyond repair, but the motion of dislocations, which involves breaking and reforming bonds. Plastic deformation is caused by the motion of dislocations. When stress continues in the plastic regime, the stress-strain passes through a maximum, called the tensile strength sTSand then falls as the material starts to develop a neck and it finally breaks at the fracture point Fig.

Note that it is called strength, not stress, but the units are the same, MPa. For structural applications, the yield stress is usually a more important property than the tensile strength, since once the it is passed, the structure has deformed beyond acceptable limits.

The ability to deform before braking. It is the opposite of brittleness. Ductility can be given either as percent maximum elongation emax or maximum area reduction. Capacity to absorb energy elastically.

The energy per unit volume is the area under the strain-stress curve in the elastic region. Ability to absorb energy up to fracture. The energy per unit volume is the total area under the strain-stress curve.

It is measured by an impact test Ch. True Stress and Strain When one applies a constant tensile force the material will break after reaching the tensile strength.

## Shear modulus

The material starts necking the transverse area decreases but the stress cannot increase beyond sTS. The ratio of the force to the initial area, what we normally do, is called the engineering stress. If the ratio is to the actual area that changes with stress one obtains the true stress. Elastic Recovery During Plastic Deformation If a material is taken beyond the yield point it is deformed plastically and the stress is then released, the material ends up with a permanent strain.

If the stress is reapplied, the material again responds elastically at the beginning up to a new yield point that is higher than the original yield point strain hardening, Ch. The amount of elastic strain that it will take before reaching the yield point is called elastic strain recovery Fig.

Compressive, Shear, and Torsional Deformation Compressive and shear stresses give similar behavior to tensile stresses, but in the case of compressive stresses there is no maximum in the s-e curve, since no necking occurs.

Hardness Hardness is the resistance to plastic deformation e. Thus, it is a measure of plastic deformation, as is the tensile strength, so they are well correlated.