^^L'allungamento assottiglia l'elastico. Effetto Poisson.

allungare assottiglia

usualmente

quando un materiale viene tirato in una direzione, si contrae nelle direzioni trasversali

viceversa

quando un materiale viene compresso in una direzione, si espande nelle direzioni trasversali

compressive stress
results in deformation,
which shortens the object
but also expands it outwards.

credit: wp/Deformation_(engineering)

wp/Poisson's_ratio

credit: wp/File:Poisson_Coefficient.svg

ν = - ε_trans /ε_longitudinal

yt/Material Properties 101

Variazione di volume. Caso cubo. Caso ∆V=0

V=xyz

dV=d(xyz) = yzdx + xzdy + xydz

supponiamo che

l'allungamento sia nella direzione x: dx>0

il comportamento sia isotropo, quindi dy=dz xzdy = xydz

Se dV=0

yzdx + xzdy + xydz = 0

yzdx = - (xzdy + xydz)

dx = - (dy + dz)

dx = - 2dy

dy= -0.5dx

d(xy)= ydx + xdy differenziale del prodotto

Links inet Poisson ratio

1. http://silver.neep.wisc.edu/~lakes/PoissonIntro2.html

2. Poisson's ratio over two centuries: challenging hypotheses

Most physical parameters in classical physics relate to laws that emerged between the 17th and 19th centuries. Where other physicists are honoured in the names of physical units used to measure these parameters—Pascal, Newton, Coulomb, Ampère, Volta, Dalton, Faraday, Gauss, Hertz, Kelvin—Siméon Denis Poisson (1787–1840) is particularly remembered for a ratio, a dimensionless quantity, which today has surprisingly acquired ubiquitous physical significance.

The phenomenon that Poisson's ratio is based on was well expressed by Thomas Young (1773–1829) in his remarkable 1807 Lectures on Natural Philosophy and the Mechanical Arts:

‘We may easily observe that if we compress a piece of elastic gum in any direction, it extends itself in other directions; and if we extend it in length, its breadth and thickness are diminished’.
ref: T. Young, Course of Lectures on Natural Philosophy and the Mechanical Arts (London, 1807; Taylor & Walton, London, 1845): Lecture 13, ‘On Passive Strength and Friction’, pp. 109–113; squeeze–stretch ratio, p. 105. >>>

This was often called the ‘squeeze–stretch ratio’ but was not specifically defined by Young (figure 1). In the same lecture, though, he described the concepts of stress σ_l, the external pressure, and strain e_l, the resulting fractional distortion (ΔL/L); the stress–strain curve, starting with the linear elastic region from which Young's modulus E = σ_l/e_l became defined; the critical yield point σ_c, beyond which elasticity ceases and plastic flow starts; and the eventual mechanical failure in which a body is quite simply broken

‘by tearing it asunder’.

Young, op. cit. (note 2), Lecture 12, ‘On Pneumatic Equilibrium’, pp. 204–209; compressibility of liquids, p. 209

This overarching description of the mechanics of solids, liquids and gases came largely from observation, but it took 20 years for a mathematical theory to emerge to handle even the elastic regime, first in France in the hands of

Navier (1785–1836)
C. L. M. H. Navier, ‘Mémoire sur les lois de l’équilibre et du mouvement des corps solides élastiques', Mém. Acad. Sci. 7, 375–394 (1827); an extract appeared earlier in Bull. Soc. Philomath. Paris, pp. 177–181 (1823).
and then Cauchy (1789–1857)
A. L. Cauchy, ‘Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non-élastiques', Bull. Soc. Philomath. Paris, pp. 9–13 (1823); published later in full in Exercices de Mathématiques in Oeuvres complètes, Ser. 2 (Gauthier-Villars, Paris, 1828), vol. 8, under ‘Sur les équations qui expriment les conditions d’équilibre, ou les lois du mouvement intérieur d'un corps solide élastique ou non élastique', pp. 227–277.
but rather later in Britain, notably by Stokes (1819–1903).
G. G. Stokes, ‘On the theories of the internal friction of fluids in motion, and the equilibrium and motion of elastic solids’, Trans. Camb. Phil. Soc. 8, 287–319 (1845)