poission ration for biaxial and traxial deformationpt.ppt

n mechanics, Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain. It is named after Siméon Poisson and is denoted by the Greek letter ‘nu’, It is the ratio of the amount of transversal expansion to the amount of axial compression for small values of these changes.Poisson’s ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here,

Compressive deformation is considered negative
Tensile deformation is considered positive.It will compress in the middle. If the original length and breadth of the rubber are taken as L and B respectively, then when pulled longitudinally, it tends to get compressed laterally. In simple words, length has increased by an amount dL and the breadth has increased by an amount dB.

In this case,





The formula for Poisson’s ratio is,





where,

εt is the Lateral or Transverse Strain

εl is the Longitudinal or Axial Strain

ν is Poisson’s Ratio

The strain on its own is defined as the change in dimension (length, breadth, area…) divided by the original dimension.

Read More: Longitudinal Strain

Poisson’s Effect
When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.

Poisson’s Ratio Values for Different Material
It is the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force. There can be a stress and strain relation that is generated with the application of force on a body.

For tensile deformation, Poisson’s ratio is positive.
For compressive deformation, it is negative.
Here, the negative Poisson ratio suggests that the material will exhibit a positive strain in the transverse direction, even though the longitudinal strain is positive as well.

For most materials, the value of Poisson’s ratio lies in the range, 0 to 0.5.

A few examples of poisson’s ratio are given below for different materials.Concrete 0.1 – 0.2
Cast iron 0.21 – 0.26
Steel 0.27 – 0.30
Rubber 0.4999
Gold 0.42 – 0.44
Glass 0.18 – 0.3
Cork 0.0
Copper 0.33
Clay 0.30 – 0.45
Stainless steel 0.30 – 0.31
Foam 0.10 – 0.50In materials science and solid mechanics, Poisson's ratio ν (nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading.In materials science and solid mechanics, Poisson's ratio ν is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. In materials science and solid mechanics, Poisson's ratio ν (nu) is a measure of the Poisson effect, the deformation (expansion or contract

1. POISSON’S RATIO
BIAXIAL and TRIAXIAL
DEFORMATION

2. POISSON’S RATIO
•When a bar is subjected to a tensile loading
there is an increase in length of the bar in
the direction of the applied load, but there is
also a decrease in a lateral dimension
perpendicular to the load. The ratio of the
sidewise deformation (or strain) to the
longitudinal deformation (or strain) is called
the Poisson's ratio and is denoted by . For
most steel, it lies in the range of 0.25 to 0.3,
and 0.20 for concrete.

3. POISSON’S RATIO
•where is strain in the x-direction and and
are the strains in the perpendicular direction. The
negative sign indicates a decrease in the
transverse dimension when is positive.

4. BIAXIALDEFORMATION
•If an element is subjected simultaneously by
tensile stresses, and , in the and
directions, the strain in the direction is
and the strain in the direction is .
Simultaneously, the stress in the direction
will produce a lateral contraction on the
direction of the amount or . The
resulting strain in the direction will be

5. TRIAXIALDEFORMATION
•If an element is subjected simultaneously by
three mutually perpendicular normal stresses ,
, and , which are accompanied by strains
, , and , respectively,

6. •223.Arectangularaluminumblockis100mm
longintheXdirection,75mmwideintheY
directionand50mmthickintheZdirection.Itis
subjectedtoatriaxialloadingconsistingofa
uniformlydistributedtensileforceof200KNinthe
Xdirectionanduniformlydistributedcompressive
forcesof160KNintheYdirectionand220KNin
theZdirection.Ifv=1/3andE=70GPa,determine
asingledistributedloadingintheXdirectionthat
wouldproducethesameZdeformationasthe
originalloading.
PROBLEMS

7. •224.Aweldedsteelcylindricaldrummadeof
10mmplatehasaninternaldiameterof1.20m.
Byhowmuchwillthediameterbechangedbyan
internalpressureof1.5MPa?AssumePoisson’s
ratiois0.30andE=200GPa.
PROBLEMS

8. •226.A150mmlongbronzetube,closedatits
ends,is80mmindiameterandhasawall
thicknessof3mm.Itfitswithoutclearanceinan
80mmholeinarigidblock.Thetubeisthen
subjectedtoaninternalpressureof4MPa.
Assumingv=1/3andE=83x10
3
MPa,determine
thetangentialstressinthetube.
PROBLEMS