What is Poisson's ratio?
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. Let us establish a relation among the elastic constants E,G and u. Consider In actual practice no real material has value of Poisson's ratio negative. Thus, the. materials simple relations exist between elastic constants (Young's modulus E, shear modulus G.
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. 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.
Therefore the constrained modulus Cis comparable to the bulk modulus and is much larger than the shear or Young's modulus of rubber. Practical example - cork in a bottle. An example of the practical application of a particular value of Poisson's ratio is the cork of a wine bottle.
The cork must be easily inserted and removed, yet it also must withstand the pressure from within the bottle. Rubber, with a Poisson's ratio of 0. Cork, by contrast, with a Poisson's ratio of nearly zero, is ideal in this application.
Practical example - design of rubber buffers. How does three-dimensional deformation influence the use of viscoelastic rubber in such applications as shoe insoles to reduce impact force in running, or wrestling mats to reduce impact force in falls?
Young's modulus - Wikipedia
Solution Refer to the above analysis, in which deformation under transverse constraint is analyzed. Rubbery materials are much stiffer when compressed in a thin layer geometry than they are in shear or in simple tension; they are too stiff to perform the function of reducing impact. Compliant layers can be formed by corrugating the rubber to provide room for lateral expansion or by using an elastomeric foam, which typically has a Poisson's ratio near 0. Corrugated rubber is used in shoe sneaker insoles and in vibration isolators for machinery.
Foam is used in shoes and in wrestling mats. Practical example - aircraft sandwich panels. The honeycomb shown above is used in composite sandwich panels for aircraft.
The honeycomb is a core between face-sheets of graphite-epoxy composite. Such panels are usually flat. If curved panels are desired, the honeycomb cell shape must be changed from the usual regular hexagon shape, otherwise the cells will be crushed during bending.
Several alternative cell shapes are known, including those which result in a negative Poisson's ratio.
Practical example - Interpreting compression tests on blocks of flexible material. The material is constrained at contact surfaces by the compression device, so the Poisson effect cannot freely occur. 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.
Most metals and ceramics, along with many other materials, are isotropicand their mechanical properties are the same in all orientations.