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ν ij is the Poisson ratio that corresponds to a contraction in direction j when an extension is applied in direction i. The Poisson 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.
In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.
The way the equation is defined won't give you a poisson's ratio of 0.5 for a perfectly incompressible material. It gives a ratio of 2 as defined in the article. Draw a quick before and after square diagram to see what I mean.
E 1 and E 2 are the Young's moduli in the 1- and 2-direction and G 12 is the in-plane shear modulus. v 12 is the major Poisson's ratio and v 21 is the minor Poisson's ratio. The flexibility matrix [S] is symmetric. The minor Poisson's ratio can hence be found if E 1, E 2 and v 12 are known.
2 Poisson's ratio. 3 Bulk modulus. 4 Shear modulus. 5 References. 6 See also. Toggle the table of contents. Elastic properties of the elements (data page) 1 language.
The resistivity of these materials changes with strain, accounting for the / term of the defining equation above. In constantan strain gauges (the most commercially popular), the effect accounts for 20% of the gauge factor, but in silicon gauges, the contribution of the piezoresistive term is much larger than the geometric terms.
where is the volume fraction of the fibers in the composite (and is the volume fraction of the matrix).. If it is assumed that the composite material behaves as a linear-elastic material, i.e., abiding Hooke's law = for some elastic modulus of the composite and some strain of the composite , then equations 1 and 2 can be combined to give
The earliest published example of a material with negative Poisson's constant is due to A. G. Kolpakov in 1985, "Determination of the average characteristics of elastic frameworks"; the next synthetic auxetic material was described in Science in 1987, entitled "Foam structures with a Negative Poisson's Ratio" [1] by R.S. Lakes from the ...