# Isotropic material poisson ratio

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Some remarks on the range of Poisson's ratio in isotropic linear elasticity.Transversely isotropic materials have a plane of symmetry in which the elastic properties are isotropic. If we assume that this plane of symmetry is, then Hooke's law takes the form.

This leaves us with six independent constants. However, transverse isotropy gives rise to a further constraint between and which is. Therefore, there are five independent elastic material properties two of which are Poisson's ratios.

For the assumed plane of symmetry, the larger of and is the major Poisson's ratio. The other major and minor Poisson's ratios are equal. But we miseducate young children when we assume that their learning abilities are comparable to those of older children and that they can be taught with materials and with the same instructional procedures appropriate to school-age children. Home Contact Privacy. Transversely Isotropic Materials Transversely isotropic materials have a plane of symmetry in which the elastic properties are isotropic.

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If we assume that this plane of symmetry is, then Hooke's law takes the form where we have used the plane of symmetry to reduce the number of constants, i. The symmetry of the stress and strain tensors implies that This leaves us with six independent constants. However, transverse isotropy gives rise to a further constraint between and which is Therefore, there are five independent elastic material properties two of which are Poisson's ratios.

Source s : Wikipedia Materials Creative Commons.The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. Most materials have Poisson's ratio values ranging between 0. Soft materials,  such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.

## Poisson's ratio in linear isotropic classical elasticity

For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0. Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching.

It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare cases,  a material will actually shrink in the transverse direction when compressed or expand when stretched which will yield a negative value of the Poisson ratio. A perfectly incompressible isotropic material deformed elastically at small strains would have a Poisson's ratio of exactly 0.

Most steels and rigid polymers when used within their design limits before yield exhibit values of about 0.

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Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed and glass is between 0. Some materials, e. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubeszigzag-based folded sheet materials,   and honeycomb auxetic metamaterials  to name a few, can exhibit one or more Poisson's ratios above 0.

Assuming that the material is stretched or compressed in only one direction the x axis in the diagram below :. If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives. The above formula is true only in the case of small deformations; if deformations are large then the following more precise formula can be used:. For a linear isotropic material subjected only to compressive i.

Thus it is possible to generalize Hooke's Law for compressive forces into three dimensions:.

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In the most general case, also shear stresses will hold as well as normal stresses, and the full generalization of Hooke's law is given by:. The Einstein notation is usually adopted:. For anisotropic materials, the Poisson's ratio depends on the direction of extension and transverse deformation. Poisson's ratio has a different number of special directions depending on the type of anisotropy.

### Poisson’s Ratio: Definition, Formula, Unit, Symbol, FAQs

Orthotropic materials have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff and strong along the grain, and less so in the other directions. Then Hooke's law can be expressed in matrix form as  . The Poisson's ratio of an orthotropic material is different in each direction x, y and z. However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent.

There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations.

We can find similar relations between the other Poisson's ratios. Transversely isotropic materials have a plane of isotropy in which the elastic properties are isotropic.For a linear isotropic material subjected only to compressive i. Thus it is possible to generalize Hooke's Law for compressive forces into three dimensions:. These equations will hold in the general case which includes shear forces as well as compressive forces, and the full generalization of Hooke's law is given by:.

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Isotropic Materials For a linear isotropic material subjected only to compressive i. Thus it is possible to generalize Hooke's Law for compressive forces into three dimensions: or whereand are strain in the direction of, and axisand are stress in the direction of, and axis is Young's modulus the same in all directions:, and for isotropic materials is Poisson's ratio the same in all directions:, and for isotropic materials These equations will hold in the general case which includes shear forces as well as compressive forces, and the full generalization of Hooke's law is given by: where is the Kronecker delta and.

For exactly isotropic materialsand for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient This ratio arises because volume is composed of Thus, in an isotropic materialfor small differential changes, one-third of the volumetric expansion is in a single axis Main Site Subjects.

Source s : Wikipedia Materials Creative Commons.Effects of introducing heterogeneity or anisotropy or nonlinearity or viscoelasticity upon Poisson's ratio are presented. Consistency studies to evaluate isotropic elasticity are reviewed. This range is derived from concepts of stability. For an object without constraint to be stable, the elastic moduli measures of material stiffness must be positive.

Poisson's ratio is interrelated with the moduli; the usual interrelations depend on the assumption of isotropy, linearity and elasticity. A positive bulk modulus implies Poisson's ratio greater than The allowable range is populated by various materials; see pages on Negative Poisson's ratio and Poisson's ratio introduction. Microstructure size is not pertinent to the effect, only the shape. Poisson's ratio is a continuum concept. The classical theory of elasticity has no length scale.

Generalized continuum theories such as Cosserat elasticity account for effects of structural heterogeneity; such theories do have a characteristic length scale, but in isotropic Cosserat elasticity the range for Poisson's ratio is the same as that for classical elasticity e. Eringen, Negative Poisson's ratio is not due to Cosserat elasticity Lakes b. Negative Poisson's ratio is attainable in classical elasticity and does not require the characteristic length scale present in Cosserat or micropolar elasticity.

The range of Poisson's ratio allowed by energy considerations of stability is the same in Cosserat elasticity as in classical elasticity: -1 to 0. As for experimental examples of small structure size, negative Poisson's ratio foams have been made using microcellular foams with small cells. Negative Poisson's ratio has been observed in isotropic polycrystalline materials Dong et al. The pertinent heterogeneity size is on the atomic or molecular scale. All physical solids have such heterogeneity. Large magnitudes of Poisson's ratio occur in oriented honeycomb Gibson and Ashby, and are also known in single crystals. Poisson's ratio should then be described as a function Beatty and Stalnaker, The definition of Poisson's ratio as a material constant is valid for small strain; one can choose to apply a small strain to a flexible material such as rubber or polymer foam. For large strain, there is a strain dependence even in materials such as rubber. The viscoelastic Poisson's ratio can increase or decrease with time, can change sign with time, and it need not be monotonic with time.

Viscoelasticity does not expand or constrict the range of Poisson's ratio. Verification : how do we know if it is isotropic? Perhaps the most elegant way to demonstrate a material is linearly isotropic elastic is via resonant ultrasound spectroscopy RUS. In RUS a compact specimen such as a cube, short cylinder, or rectangular prism is subject to vibration at low amplitude linear range by ultrasonic transducers.

The vibration mode structure is determined and the elastic constants inferred. The mode structure allows one to distinguish isotropic from anisotropic materials and to determine the degree of anisotropy if any. Torsion modes are sensitive to the shear modulus, bending modes to Young's modulus, breathing modes primarily sensitive to bulk modulus, and axial modes to Poisson's ratio and Young's modulus.

This sensitivity allows consistency checks. Such experiments have been done by Demarest on fused quartz with a Poisson's ratio of 0.

## Poisson's Ratio - Transversely Isotropic Materials

It is also possible to test for isotropy via wave ultrasound. Measure the propagation velocity of longitudinal waves in each orthogonal principal direction and also in at least one oblique direction.When a force is applied to a bar it deforms elongates or compresses in the axial longitudinal direction.

At the same time, a deformation is observed in the transverse width direction as well. It is a material property and remains constant. In this image, A tensile force F is applied in a bar of diameter d o and length l o. With the action of this force F, the bar elongates and final length in l. Also, the diameter reduces and the final diameter is d.

Understanding Poisson's Ratio

Both longitudinal and lateral strain are dimension less. However, this is not standardized. Rubber is a typical example. When more authoritative data from tests are not available, normally 0. For example, polyurethane foam.

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This is the reason Corks are ideally used as bottle stopper as it does not expand even when compressed. For design calculation, in absence of data, normally 0. I am a Mechanical Engineer turned into a Piping Engineer. I am very much passionate about blogging and always tried to do unique things. This website is my first venture into the world of blogging with the aim of connecting with other piping engineers around the world. Your email address will not be published.

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