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of any line
164. When the position of the principal axes, and the magni- Elongation tudes of the principal elongations of a strain are given, the of direction elongation of any line of the body, and the alteration of angle of the body. between any two lines, may be obviously determined by a simple geometrical construction,
Analytically thus:-let a-1, B-1, y-1 denote the principal elongations, so that a, B, y may be now the ratios of alteration along the three principal axes, as we used them formerly for the ratios for any three oblique or rectangular lines. Let l, m, n be the direction cosines of any line, with reference to the three principal axes. Thus,
lr, mr, nr
being the three initial co-ordinates of a point P, at a distance
alr, ẞmr, ynr.
Hence the altered length of OP is
(a3l2 + ß3m2 + y®n2)*r,
and therefore the "elongation" of the body in that direction is
For brevity, let this be denoted by −1, i.e.
The direction cosines of OP in its altered position are
and therefore the angles XOP, YOP, ZOP are altered to having
The cosine of the angle between any two lines OP and OP', specified in the initial condition of the body by the direction cosines l', m', n', is
in the altered condition.
ll' + mm' + nn',
in the initial condition of the body, and becomes
(a3l2 + ß3m3 + y2n2)* (a2l'a + ß3m” + y'n'2)1⁄2
Change of plane in the body.
165. With the same data the alteration of angle between any two planes of the body may also be easily determined, either geometrically or analytically.
Let l, m, n be the cosines of the angles which a plane makes with the planes YOZ, ZOX, XOY, respectively, in the initial condition of the body. The effects of the change being the same on all parallel planes, we may suppose the plane in question to pass through 0; and therefore its equation will be
lx + my + nz = 0.
In the altered condition of the body we shall have, as before, έ=ax, n = By, Lyz,
for the altered co-ordinates of any point initially x, y, z. Hence
lξ ͵ mn ηζ
But the planes of reference are still rectangular, according to our present supposition. Hence the cosines of the inclinations of the plane in question, to YOZ, ZOX, XOY, in the altered condition of the body, are altered from l, m, n to
respectively, where for brevity
If we have a second plane similarly specified by l', m', n', in the initial condition of the body, the cosine of the angle between the two planes, which is
ll' + mm' + nn'
in the initial condition, becomes altered to
W mm' nn'
Conical sur face of equal
166. Returning to elongations, and considering that these are
elongation. generally different in different directions, we perceive that all
lines through any point, in which the elongations have any one
face of equal
value intermediate between the greatest and least, must lie on Conical sura determinate conical surface. This is easily proved to be in elongation. general a cone of the second degree.
For, in a direction denoted by direction cosines l, m, n, we have
a2l2 + ß3m2 + y2n2 = 5',
denotes the ratio of elongation, intermediate between a the greatest and y the least. This is the equation of a cone of the second degree, l, m, n being the direction cosines of a generating line.
of no distortion,
167. In one particular case this cone becomes two planes, Two planes the planes of the circular sections of the strain ellipsoid. The preceding equation becomes a2l3 + y2n3 - ẞ2 (1-m3) = 0,
1 - m2 = l2 + n2,
(a3 — ß3) l2 — (B3 — y2) n2 = 0.
The first member being the product of two factors, the equation
1 (a3 — ß3)1 + n (Ba — yo)* = 0,
1 (a3 — ß3)1 — n (ẞ2 — y2) 3 = 0,
This is the case in which the given elongation is equal being the to that along the mean principal axis of the strain ellipsoid. sections of The two planes are planes through the mean principal axis of ellipsoid. the ellipsoid, equally inclined on the two sides of either of the other axes. The lines along which the elongation is equal to the mean principal elongation, all lie in, or parallel to, either of these two planes. This is easily proved as follows, without any analytical investigation.
168. Let the ellipse of the annexed diagram represent the section of the strain ellipsoid through the greatest and least principal axes. Let S'OS, TOT be the two diameters of this ellipse, which are equal to the mean principal axis of the ellipsoid. Every plane through O, perpendicular to the plane of the diagram, cuts the ellipsoid in an ellipse of which
of no distortion, being the circular sections of the strain ellipsoid.
Two planes one principal axis is the diameter in which it cuts the ellipse of the diagram, and the other, the mean principal diameter of the ellipsoid. Hence a plane through either SS', or TT”, perpendicular to the plane of the diagram, cuts the ellipsoid in an ellipse of which the two principal axes are equal, that is to say, in a circle. Hence the elongations along all lines in either of these planes are equal to the elongation along the mean principal axis of the strain ellipsoid.
planes with of volume.
169. The consideration of the circular sections of the strain ellipsoid is highly instructive, and leads to important views with reference to the analysis of the most general character of a strain. First, let us suppose there to be no alteration of volume on the whole, and neither elongation nor contraction along the mean principal axis. That is to say, let ß=1, 1
and y = (§ 162).
Let OX and
OZ be the directions of elongation a-1 and
respectively. Let A be any point of the
body in its primitive condition, and A, the same point of the altered body, so that OA,= αOA.
Now, if we take OC=0A,, and if C, be the position of that point of the body which was in the position C initially, we shall have OC=10C, and therefore
angles COA and C,OA, are equal and similar.
Initial and Hence CA experiences no alteration of length, but takes tion of lines the altered position CA, in the altered position of the body.
of no elongation.
Similarly, if we measure on XO produced, OA' and О4,' equal respectively to OA and OA,, we find that the line C A' experiences no alteration in length, but takes the altered position CA,'. Consider now a plane of the body initially through CA perpendicular to the plane of the diagram, which will be altered into a plane through CA,, also perpendicular to the plane of
OC=OA. Hence the two tri
of no elon
the diagram. All lines initially perpendicular to the plane of Initial and posi the diagram remain so, and remain unaltered in length. AC tion of lines has just been proved to remain unaltered in length. Hence gation. (§ 158) all lines in the plane we have just drawn remain unaltered in length and in mutual inclination. Similarly we see that all lines in a plane through CA', perpendicular to the plane of the diagram, altering to a plane through C,A,, perpendicular to the plane of the diagram, remain unaltered in length and in mutual inclination.
170. The precise character of the strain we have now under consideration will be elucidated by the following:-Produce CO, and take OC and OC, respectively equal to OC and OC, Join CA, CA', CA, and C'A', by plain and dotted lines. in the diagram. Then we see that the rhombus CACA' (plain lines) of the body in its initial state becomes the rhombus C,A,CA, (dotted) in the altered condition. Now imagine the body thus strained to be moved as a rigid body (i.e., with its state of strain kept unchanged) till A, coincides with A, and C, with C', keeping all the lines of the diagram still in the same plane. AC, will take a position in CA' produced, as shown in the new diagram, and the original and the altered parallelogram will be on the same base AC, and between the same parallels AC and CA, and their other sides will be equally inclined on the two sides of a perpendicular to these parallels. Hence, irrespectively of any rotation, or other absolute motion of the body not involving change of form or dimensions, the strain under consideration may be produced by holding fast and unaltered the plane of the body through AC' perpendicular to the plane of the diagram, and making every plane parallel to it slide, keeping the same distance, through a space proportional to this distance (i. e., different planes parallel to the fixed plane slide through spaces proportional to their distances).
171. This kind of strain is called a simple shear. The Simple plane of a shear is a plane perpendicular to the undistorted planes, and parallel to the lines of their relative motion. It