CHAPTER V.

GENERAL THEOREMS.

STRESS-STRAIN EQUATIONS DEDUCED FROM CAUCHY'S
MOLECULAR HYPOTHESIS.

59. Statement of the Hypothesis.

We proceed to investigate, after the manner of Cauchy1, the stress across a small plane area arising from the forces (supposed insensible at sensible distances), that act between the individual pairs of a system of particles homogeneously arranged, when the force between two whose masses are m and m' placed at a distance r is mm'x(r). The nature of the homogeneity of the arrangement can be described by stating that all the particles have the same mass, and, if P, P', Q be the positions of any three particles, and a line QQ be drawn from Q equal and parallel to PP', there will be a particle at Q'.

60. Evaluation of stress-components.

Now, as in ch. I., let x, y, z be the coordinates before strain of a point P, at which there is a particle of mass M, and x+ §, y+n, z + those of another point Q at which there is a particle m of equal mass, r the distance between them, and λ, μ, v the direction-cosines of PQ, so that

Through P draw a plane parallel to the plane yz, and let m1, m,,... denote particles on the side of the plane where x is

De la pression ou tension dans un système de points matériels'. Exercices de Mathématiques, 1828.

x=

greater than the x of P, and mí, má,... particles on the other side of the plane. Describe round P any small curve in the plane const. through P whose radii vectores are all sensibly greater than the greatest distance at which the force between two particles is sensible, and let s be the area of this curve; then, in the notation of ch. II., if

be the sums of components, parallel to x, y, z, of all the forces that cross the plane within the curve s, P, U, T will be the component stresses at P across a plane parallel to x. Now these sums of components are the sums of such quantities as

mimix (r) riji mi mix(g) Mij, mimjx(rj) vi........(3),

Xij›

where m; and m; are the masses of two particles on opposite sides of the plane, r the distance between them, and Xij, Mij, vij, the direction-cosines of this line, and the summation must be extended to all pairs so situated that the line joining them crosses s, and the distance between them does not exceed the greatest distance at which the force is sensible (called by Cauchy the "radius of the sphere of molecular activity").

Now there will be a particle m whose distance r from M is Tij, and such that the line joining M, m is parallel to the line joining m, m, and therefore the force across s arising from the force between me and my will have components

...........

Mmx(r)λ, Mmx(r)μ, Mmx(r) v ..................................(4). The summation may be taken by first summing for all the pairs of particles (mi, mj) that have the same r, λ, μ, v, and are so situated that the line joining them crosses s, and then summing for all the directions λ, μ, v on which pairs of particles are met with, and lastly summing for all the particles on each such line (λ, μ, 1) whose distance apart is not greater than the radius of the sphere of molecular activity. The first summation will be made by multiplying the expressions (4) by the number of particles contained in a cylinder standing on s whose height is rλ; this number is psrλ/M, where p is the density, or mass per unit volume, of the system of particles, and thus we get for the component stresses per unit area across the plane parallel to (yz) through M, the sums of such quantities as

ρmλεχ (r), ρηrλμχ (r), ρηrλνχ (η)............ (5).

Now it is clear that, if the summation be extended to all directions round M in which particles are met with, the force between any pair m¿, m' will have been counted twice, and we thus get P=pΣ[mrλ2x(r)], U={pΣ[mrλμx(r)], T={pΣ[mrλvx(r)]...(6), where the summations refer to all particles m, whose distance from M is not greater than the radius of the sphere of molecular activity.

61.

Stress in terms of strain.

Now let the system be displaced so that M comes to

and m comes to

(x+u, y + v, z+w),

(x+u++du, y+v+n+dv, z+w+5+dw),

then, since m is very near to M, we may express du, dv, dw in the

and use the notation e, f, g, a, b, c, ▲ of strain-components.

Let r become r (1+e), then, by (33) of art. 11,

€ = eλ2 + ƒμ2 + gv2 + aμv + bvλ + cλμ

..(8).

Also rλ is the difference of the x's of m and M, and this becomes

and in like manner we may write down the values of rμ, rv after

Thus P, U, T become

P=1pΣ m

1

r (1+e){X(r)+erx' (r)} {rλ+8(ra)}2

1

U=p'Σ| m ¡{x(r)+erx'(r)} {rλ+d(rλ)}{ru+d(rμ)} ||(12).

r (1+ε)

T=1pm

1

r(1+e)

{x(r)+erx'(r)}{rλ+d(rλ)} {rv+d(rv)}

We shall put down P and U, we get

P = {p'Σ [mrλ2x (r)]

ди

ди

(r)]ɔ̃y Σ[mrλνχ

ди

+ p { Σ [mrλ° x (r)] ou + Σ [mr\px (r)]am + Σ [mrdox (r)]au} +}p'Σ[mr{rx'(r)−x(r)}λ2(eλ2+ƒμ2 + gv2 + aμv+bvλ +cλμ)].......(13).

[mrλ2x (r)] +Σ [mrλμx (r)] õy + Σ [mrλvx (r)] az)

ди

მყ

+ }p' {Σ [mrμλx (r)] 0% + + Σ [mrμ3x (r)]0% +Σ[mrμvx(r)] oz

дох

+}p'Σ[mr{rx' (r)− x (r)} λμ (eλ2+ƒμ2+ gv2+ aμv+bvλ + cλμ)] (14).

In like manner the other four stresses can be put down.

Now suppose the initial state of the system is one of zero stress, or that the system is disturbed from the natural state, then we see that all the 6 quantities such as

Σ[mrλοχ (r)], Σ[mrλμχ (r)]....

.(15)

must vanish identically, and, therefore, the expressions of the six stresses in terms of the strains are such quantities as the last lines of the right hand sides of (13) and (14). In these, neglecting squares of the strains, we may put p for p', and thus writing for shortness

r {rx' (r) − x (r)} = $ (r)..................................................(16),

...(17).

we find such expressions as P=1pΣ [mp (r) λ2 (eλ2 +ƒμ2+gv2 + aμv+bvλ+cλμ)]{ U = +pΣ [mp (r) λμ (eλ2 +ƒμ2+gv2 + aμv + bvλ+ cλμ)]S Hooke's Law follows at once, and the elastic constants are such expressions as

If all the stress-equations similar to the above be written down, and the coefficients compared with the elastic constants c of art. 31, it will be found that

as in equations (20) of that article.

The particular result for isotropic solids is that λ=μ, and consequently σ = 1, as stated in art. 28.

62. The Thermo-Elastic Equations.

Consider a solid strained by unequal heating. Suppose that, when the temperature of any part is increased by t, the force between two particles m, m' is increased by a quantity of the form mm'Kt, where K is independent of the configuration. Then, referring to the investigation of art. 60, we see that we have to add to the expressions for the stresses the sum of all such quantities as

m¿m¡'Ktλ, m¿m; Ktμ, m¿m¡Ktv,

where m¿, m¡ are the equal masses of particles in a line crossing the area s; and, as before, the stresses thence arising are given by such equations as

P = {pΣ [mrλ2Kt], _U=}pΣ[mrλμKt].

We should find in this way the stresses given by such equations as (17), each increased by a quantity, which is the product of t and a constant depending on the material.

In case the particles of the system are distributed symmetrically in all directions, the terms contributed by t to the tangential stresses will disappear, and the terms contributed to the normal stresses will all be equal, so that the stresses will consist of

(1) a hydrostatic pressure proportional to the change of

temperature,

(2) elastic stresses like those of (17) due to the strains. The equations of equilibrium hence deduced will be three of the form

where is a constant, and P... are component stresses due to the strains, and are the same functions of the strains as occur when t is constant.