Calculating poissons ratio proof?

Question

I am new to this.
$$\sigma = d(d)/d/(dl/L) f$$ $$\sigma= \frac{d(diameterf)/diameter}{d(length)/length}$$ Now volume being = V $$V= constant$$ for a rod $$(\pi d^2/4)*(l)=V$$ $$=> d^2 \ l = constant$$ $$(d^2)(l)=(d+ d(d))^2(l-dl)$$ $$d^2l=(d^2+2d*d(d)+(d(d))^2)(l-dl)$$ From here it does not seem above ratio is constant seemingly?Then where is main mistake?

Answer 1

As has been already noted in the comments, there is no reason to assume that the volume during a deformation remains constant. Poisson ratio is an experimentally measured constant, and it is a constant only within the framework of linear elasticity. (In fact, it can be shown that in linear approximation and for isotropic material, one can describe all the deformations using only two constants: e.g., Young modulus and Poisson ratio, or the two Lamé coefficients.)

Update
Regarding my last statement, one could take a look in Wikipedia article on linear elasticity. For more detailed explanations one needs to read a book on the theory of elasticity, with a solid theoretical underpinnings. Sadd's book is pretty readable, while Landau&Livshits volume is the standard reference (Contrary to their other volumes this one is not very theoretical - the theoretical elasticity books dive right away into metric tensors and complex analysis, which are relevant mainly beyond the linear regime.)