Page:Scientific Papers of Josiah Willard Gibbs.djvu/251

Concerning Solids which absorb Fluids.

There are certain bodies which are solid with respect to some of their components, while they have other components which are fluid. In the following discussion, we shall suppose both the solidity and the fluidity to be perfect, so far as any properties are concerned which can affect the conditions of equilibrium,—i.e., we shall suppose that the solid matter of the body is entirely free from plasticity and that there are no passive resistances to the motion of the fluid components except such as vanish with the velocity of the motion,—leaving it to be determined by experiment how far and in what cases these suppositions are realized.

It is evident that equation (356) must hold true with regard to such a body, when the quantities of the fluid components contained in a given element of the solid remain constant. Let $\Gamma _{a}',\Gamma _{b}'$, etc., denote the quantities of the several fluid components contained in an element of the body divided by the volume of the element in the state of reference, or, in other words, let these symbols denote the densities which the several fluid components would have, if the body should be brought to the state of reference while the matter contained in each element remained unchanged. We may then say that equation (356) will hold true, when $\Gamma _{a}',\Gamma _{b}'$, etc., are constant. The complete value of the differential of $\epsilon _{V'}$ will therefore be given by an equation of the form

$d\epsilon _{V'}=td\eta _{V'}+\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)+L_{a}d\Gamma _{a}'+L_{b}d\Gamma _{b}'+{\text{etc.}}$

(462)

Now when the body is in a state of hydrostatic stress, the term in this equation containing the signs of summation will reduce to $-pdv_{V'}$. ($v_{V'}$ denoting, as elsewhere, the volume of the element divided by its volume in the state of reference). For in this case

$X_{X'}=-p\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right),$

(463)

$\textstyle \sum \sum '\displaystyle \left(X_{X'}d{\frac {dx}{dx'}}\right)=-p\textstyle \sum \sum '\displaystyle \left\{\left({\frac {dy}{dy'}}{\frac {dz}{dz'}}-{\frac {dz}{dy'}}{\frac {dy}{dz'}}\right)d{\frac {dx}{dx'}}\right\}\quad$

$=-pd{\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}&{\frac {dx}{dz'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}&{\frac {dy}{dz'}}\\{\frac {dz}{dx'}}&{\frac {dz}{dy'}}&{\frac {dz}{dz'}}\end{vmatrix}}$

$=-pdv_{V'}.$

(464)