CHAPTER II. STRAIN-VELOCITIES.

HOMOGENEOUS STRAIN-FLUX.

WE have already investigated all those velocity-systems which are consistent with rigidity, and shewn how to compound them together. It is probable, however, that no body in nature is ever rigid for so much as a second together. The most solid masonry is constantly transmitting vibrations which it receives from the earth's surface and from the air; these vibrations constitute minute changes of shape. Other minute changes of shape are due to the varying position of attracting bodies, such as the moon. The spins and twists, therefore, which we have investigated are to be regarded as ideal motions, to which certain natural motions more or less closely approximate. The motions of fluids, however, such as water or air, are not even approximately consistent with rigidity, and to describe these we must consider some other velocitysystems. As before, we have to describe ideal motions, which can be dealt with by exact methods, but which only approximately represent the motions which actually take place.

Imagine an elastic string, one end of which is fixed, while the other end moves uniformly along a straight line passing through the fixed end, so that the string is always stretched along the same line. If the strain is always homogeneous, the velocities of any two points on the string will at any moment be proportional to their distances from the fixed end.

Now consider an infinite plane surface, with air on one side of it; and let those particles of air which lie along any straight line perpendicular to the plane be moving

like the particles of the elastic string just considered; that is to say, let the velocity at every point be perpendicular to the plane and proportional to the distance from it. Let a similar motion take place on the other side of the plane, but in the opposite direction; that is, so that both motions are towards the plane, or both away from it. If the velocity at distance x from the plane is ex, and if we suppose the velocity of every particle to remain uniform for one second, then at the end of that second there will be produced a uniform elongation e perpendicular to the plane. For the moment, we may call this velocity-system a stretch perpendicular to the given plane.

Take now three planes intersecting at right angles in a point o, and combine together at every point of space the velocities due to stretches e, f, g perpendicular to these three planes respectively. We shall then have a velocitysystem such that if the velocity of every particle remains uniform for one second, there will be produced a pure homogeneous strain of which e, f, g are the principal elongations.

Lastly, combine with this velocity-system a spin about some axis passing through the point o. The resultant velocity-system has then the following properties.

1. The point o is at rest.

2. The velocities of all points lying in a straight line through o are parallel, and proportional to the distance from 0.

3. If the velocity of every point be kept uniform for one second, there will be produced at the end of that second a homogeneous strain.

Leto be the displacement at the end of one second of a particle whose position-vector from o as origin was originally a, then σ = px, where p is the displacement-function of the homogeneous strain. Hence the equation to the uniform motion of this particle during the second is p = a+tox,

Consequently at the time t = 0, the particle whose position

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193

vector is a has the velocity ox. The motion is therefore such that the velocity at every point is a linear function of the position-vector of the point. Such a velocity-system may be called a homogeneous strain-flux. We may formally define it as follows;

If at any instant the velocity-system of a body be such that by keeping the velocity of each point uniform for one second we should produce a homogeneous strain 4, then at that instant the body is said to have the homogeneous strainАих ф.

If we combine with this velocity-system a translation equal and opposite to the velocity of any point p, the resultant will be a new homogeneous strain-flux with the point p for centre. For if we keep all velocities constant for a second, we shall produce a homogeneous strain together with a translation restoring p to its place; that is, a homogeneous strain in which p is not moved.

It is clear that the resultant of two homogeneous strainfluxes is again a homogeneous strain-flux; but in this term we must include as special cases the motions consistent with rigidity. A twist may be regarded as a homogeneous strain-flux whose centre o is infinitely distant; in the still more special case of a spin, the centre is indeterminate, being any point whatever on the axis.

The latter case is distinguished by the function being a skew function. For let the spin w = pi+qj+rk, then the velocity of any point whose position-vector is p will be Vwp. Consequently we have op Vop, and therefore

$k =Vwk=+qi — pj

so that the matrix of o is

( 0, +r, -q)

-r, 0, +p

+q, -P,

0.

We may now separate any given homogeneous strain

flux into the pure part of it and the spin. For it is evident

(g-g'), (ƒ' —ƒ),

Here the first of the matrices on the right hand belongs to a pure function, and the second to a spin, whose components are (f−ƒ'), \ (g — g'), (h-h'). The resolution cannot be effected in any other way; for to change the spin into any other (not about a parallel axis) we must combine a spin with it. The resultant of the pure strainflux and of this spin reversed will be no longer a pure strain-flux.

CIRCULATION.

Consider a plane curve joining two points p and q. Let a line be drawn through every point of the curve, perpendicular to its plane, representing the component of velocity along the tangent to the

curve at that point. All these lines
will trace out a strip or riband
standing on the curve. The area
of this strip is called the circula-
tion along the curve from p to q. P
When the resolved part of the

velocity is in the direction from q to p, it is to be drawn below the plane, and that part of the area is to be reckoned negative. Hence the circulations from p to q and from q top are equal in magnitude but of opposite sign.

The circulation may also be described as follows. Divide the length of the curve into small pieces, of which SA is one.

Leto be the velocity of some point included

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195

in the piece dλ, then - Sodλ will be the resolved part of this velocity along the curve, multipled by the length of Sλ. The sum - Sod of such quantities for all pieces of the curve may be made to approximate as near as we please to a quantity - Sodλ by increasing the number and diminishing the length of the pieces. This quantity -Soda is called the circulation along the curve from p to q. The second definition is equally applicable to a non-plane curve, on which we cannot draw a riband which shall represent the circulation by its area.

If we suppose the point q to move along the curve pq with unit velocity, the rate of change of the circulation from p to q will be the component along the tangent at q of the instantaneous velocity of the body at q. For if this component remained constant over a unit length of the curve, the change of circulation would be the component multiplied by the unit of length. Thus if s denote the length of the arc pq, and C the circulation from p to q, C=v cos 0, where v velocity at q, and = angle it makes with the tangent to pq.

=

In general, if σ be any vector which has a definite value at every point of space, the quantity - Sodλ is called the line-integral of σ along the curve X; so that we may say that the circulation is the line-integral of the velocity.

a

If an area be divided into parts, the circulation round the whole area is equal to the sum of the circulations round the parts. The area abcd, for example, is made up of abc and acd. The circulation round abc is made up of that along abc and that along ca. The circulation round acd is made up of that along ac and that along cda. Now the circulation along

ca is equal and opposite to that along ac, so that when we put the circulations round the parts together, these two portions destroy one another, and the sum is the circulation round abcd. The same reasoning applies to any number of parts. It it clear that the proposition holds equally good, whether the areas are on a plane or on any other surface.