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78. A question of great importance in the physical theory of the elasticity of solids, "What changes are produced in the moduluses of elasticity by permanent changes in its molecular condition," has occupied the attention, no doubt, of every 66 naturalist " "who has studied the subject, and valuable contributions to its answer by experiment had been given by Wertheim and other investigators, but solely with reference to Young's modulus. In 1865 au investigation of the effect on the torsional rigidity of wires of different metals, produced by stretching them longitudinally beyond their limits of elasticity, was commenced in the physical laboratory of the university of Glasgow in its old buildings in 1865. The following description of experiments and table of results is extracted from the paper by W. Thomson "On the Elasticity and Viscosity of Metals," already quoted (section 30), with reference to viscosity and fatigue of elasticity.

"To determine rigidities by torsional vibrations, taking advan tage of an obvious but most valuable suggestion made to me by Dr Joule, I used as vibrator in each case a thin cylinder of sheet brass, turned true outside and inside (of which the radius of gravitation must be, to a very close degree of approximation, the arithmetic mean of the radii of the outer and inner cylindrical surfaces), supported by a thin flat rectangular bar, of which the square of the radius of gravitation is one-third of the square of the distance from the centre to the corner. The wire to be tested passed perpendicularly through a hole in the middle of the bar, and was there firmly soldered. The cylinder was tied to the middle of the bar by light silk thread so as to hang with its axis vertical. Each wire, after having been suspended and stretched with just force enough to make it as nearly straight as was necessary for accuracy, was vibrated. Then it was stretched by hand (applied to the cross bar soldered to its lower end) and vibrated again, and stretched again, and so on till it broke.' The experiments were performed with great care and accuracy by Mr Donald M'Farlane. results, as shown in the accompanying table, were most surprising." The highest and lowest rigidities found for copper in the table are as follows :—

"

than that of Cagniard-Latour, a most important and interest-
ing investigation might be made.
accurate determination of the Young's modulus for the particular
The results, along with an
No. 2.
case, give (sec. 47) the modulus of compression, and the rigidity
cylinders, to be subjected to longitudinal pull, and (after the
Regnault suggested the use of hollow instead of solid
heim, adopting this excellent suggestion, obtained seemingly very
manner of the bulb and tube of a thermometer) a capillary tube
to aid in measuring changes of volume of the hollow; and Wert-
accurate results for brass and glass which are given in the tables of
section 77.

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317-4 1.962

8.835

5.040

425.9 x 106

Copper

315.6

81771 8-155

442.3 x 106

235.5

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Copper 7

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Copper

256.5 1-6145 8.90

4.2226 463.5 x 106

267.9

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292.2

4.5625 453.3 x 10 4.915 446.2 x 10° 5.240 445.5 x 106

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Soft Iron 316.8

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X 106 × 106

Highest rigidity 473 x 106, being that of a wire which had been oftened by heating it to redness and plunging it into water, and which was found to be of density 8.91.

Lowest rigidity 393'4 x 106, being that of a wire which had been rendered so brittle by heating it to redness surrounded by powdered charcoal in a crucible and letting it cool very slowly, that it could scarcely be touched without breaking it, and which had been found to be reduced in density by this process to as low as 8.674. The wires used were all commercial specimens-those of copper_being all, or nearly all, cut from hanks supplied by the Gutta Percha Company, having been selected as of high electric conductivity, and of good mechanical quality, for submarine cables.

It ought to be remarked that the change of molecular condition produced by permanently stretching a wire or solid cylinder of metal is certainly a change from a condition which, if originally isotropic, becomes æolotropic as to some qualities, and that the changed condi tions may therefore be presumed to be reolotropic as to elasticity. If so, the rigidities corresponding to the direct and diagonal distortions (indicated by No. 1 and No. 2 in fig. 14) must in all proba bility become different from one another when a wire is permanently stretched, instead of being equal as they must be when its substance is isotropic. It becomes, therefore, a question of extreme interest to find whether rigidity No. 2 is not increased by this process, which, as is proved by the experiments above described, diminishes, to a very remark. able degree, the rigidity No. 1. Tho most obvious experiment, and indeed the only practicable experiment, adapted to answer this question, for a wire or round bar is that of CagniardLatour, in which an accurate determination of the difference produced in the volume of the substance is made by applying and removing longitudinal traction within its limits of elasticity. With the requisite apparatus, which must be much more accurate

No.2

Fig. 14.

1 It is exactly the square root of the mean of their squares. For example, see paper "On Electrodynamic Qualities of Mctals," Philosophical Transactions, 1856, by W. Thomson.

:

Remarks.

6.5325 455.0 x10

1 Only forty vibrations from initial arc of convenient amplitude could be counted. Had been stretched considerably before this experiment.

So viscous that only twenty vibrations could be counted Broke in stretching.

3A piece of the preceding stretched.

The preceding made red-hot in a crucible filled with powdered charcoal and allowed to cool slowly, became very brittle: a part of it with difficulty saved for the experiment.

Another piece of the long (2435 centims.) wire; stretched by successive simple tractions.

A finer gauge copper wire; stretched by successive tractions. 7 A finer gauge copper wire, softened by being heated to redness and plunged in water. A length of 260 centimetres cut from this, suspended, and elongated by successive tractions.

Another length of 260 centimetres cut from the same, and similarily treated.

"One piece, successively elongated by simple tractions till it broke.

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The above results are from Wertheim's "Mémoires " on Elasticity, Ann. de Chim. et Phys., tom xii. (1844). The change in the rigidity-modulus produced by change of temperature was investigated by Kohlrausch. He found that it is expressed by the formula n=no (1-at-ẞt2), where no denotes the value of the rigidity-modulus at 0° C., n its value at temperature t, and a, B. coefficients the values of which for iron, copper, and brass are as follows:

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80. Tempering soft iron by long-continued stress.— Preliminary experiments by Mr J. T. Bottomley towards the investigation promised in section 5 above have discovered a very remarkable property of soft iron wire respecting its ultimate tensile strength. Eight different specimens, tested by the gradual application of more and more weight within ten minutes of time in each case until the wire broke, bore from 43 to 46 b (average 45.2) just before breaking, with elongations of from 17 per cent to 22 per cent. Another specimen left with 43 ib hanging on it for 24 hours, and then tested by the gradual addition of weights during 25 minutes till it broke, bore 491 b before breaking, with elongation of 15 per cent. Another left for 3 days 11 hours 40 minutes with 43 b hanging on it, and then tested by the gradual addition of weights during 34 minutes till it broke, bore 51 Ib just before breaking, with elongation of 14.4 per cent. Another specimen of the same wire was set up with 40 b hanging on it on the 5th of July 1877, on the 6th of July 3 Ib were added, on the 9th 1 b more, and on the 10th more, making in all on this date 451 b. Thenceforward day by day, with occasional intervals of two days or three days, the weight

1 The modulus seems to be a minimum near the temperature of

maximum density,

was increased first by half a pound at a time, and latterly by a quarter of a pound at a time, until on the 3d of September the wire broke with 571 b (elongation not recorded). This gradual addition of weight therefore had increased the tensile strength of the metal by 26.7 per cent. I 81. Experiments made for this article.-There are many subjects in the theory of elasticity regarding which information to be obtained by experiment only is greatly wanted. Several of these have been pointed out above (section 21), and while this article was being put in type, experiments were made in the physical laboratory of the university of Glasgow with a view of answering some of the questions proposed. Mr Donald M'Farlane, besides making the experiments referred to in sections 3 and 21, investigated the effects of applying different amounts of pull to a steel pianoforte wire which had been twisted to nearly its limits of elasticity, and which was kept twisted by means law by showing a diminution of the torsional rigidity, of a couple. The results proved a deviation from Hooke's about 1.6 per cent., produced by hanging a weight of 112 Ib on the wire. Of this 1.2 per cent. is accounted for by elongation and by shrinkage of the diameter, leaving 4 per cent. of diminution of the rigidity-modulus.

of

It was also found that when the wire was twisted far beyond its limits of elasticity, and then freed from torsional stress, a weight hung on it caused it to untwist slightly. When the weight was removed and reapplied again and again, the lower end of the wire always turned in the same direction as the permanent twist when the weight was removed, and in the opposite direction when it was applied. This result shows the development of solotropic quality in the substance of the wire, according to which a small cube cut from any part of it far out from the axis, with two sides of the cube parallel to the length, and the other two pairs of sides making angles of 45° with the length, would show different compressibilities in the directions perpendicular to the last-mentioned pairs of sides.

Another very interesting result, discovered in the course of these experiments, was that when a length of five metres of the steel wire, with a weight of 39 Ib hung upon it, was twisted to the extent of 95 turns, it became gradually elongated to the extent of of the length of the wire; when farther twisted it began to shorten till, when 25 turns had been given (in all 120 turns), the weight had risen from its lowest position through nearly of the length of the wire, so that the previous elongation had beer. diminished by about of its amount.

Experiments were also made by Mr Andrew Gray and Mr Thomas Gray for the purpose of determining the effects of various amounts of permanent twist in altering the rigidity-modulus and the Young's modulus of wires of copper, iron, and steel. A copper wire, of 3.15 metres in length and 154 centimetre diameter, No. 17 B. W.G., which had a rigidity-modulus of 442 million grammes per square centimetre to begin with, was found to have 420 after 10 turns, showing a diminution in the modulus of of its own amount. The diminution went on rapidly until 100 turns of permanent twist had been given, when the modulus was as low as 385. The diminution of the modnins continued with further twist, but very slowly, up to 1225 turns, when the modulus was found to be 371, showing a diminution to the extent of of its original value! There was little farther change until 1400 turns had been given, when the modulus began to increase. At 1525 turns its value was 373, and at 1625 it was 377. Twenty turns more broke the wire before the torsional elasticity had been again determined.

A piece of iron wire of nearly the same length, about three metres, but of smaller diameter (087 centimetre), showed continued diminution of torsional rigidity as far as

1350 turns of permanent twist, when the diminution had amounted to 14 per cent. of the primitive value, 36 turns more broke the wire before another determination of torsional rigidity had been made.

The steel pianoforte wire also showed a diminution of torsional rigidity with permanent twist, and (as did the copper wire) showed first a diminution and then a slight augmentation. The amount of the diminution in the steel wire was enormously greater than the surprisingly great amount which had been discovered in the copper wire, and the ultimate augmentation was considerably greater in the steel than what it had been in the copper before rupture. Thus after 473 turns of permanent twist the torsional modulus had diminished from 751 million grammes per square centimetre to 414! 95 more turns of permanent twist augmented the rigidity from 414 to 430, and when farther twisted the wire broke before another observation had been made. The vibrator used in these experiments was a cylinder of lead weighing 56 b, which was kept hanging on the wire while it was being twisted, and in fact during the whole of about 100 hours from the beginning of the experiment till the wire broke, except on two occasions for a few minutes, while the top fastening which had given way was being resoldered. The period of vibration was augmented from 39.375 seconds to 51.9 seconds by the twist. The wire took the twist very irregularly, some parts not beginning to show much signs of permanent twist till near the end of the experiment

In two specimens of copper wire of the same length and gauge as those described above, the Young's modulus was found to be increased 10 per cent. by 100 turns of permanent twist.

Five metres of the steel pianoforte wire, bearing a weight of 39 Ib, was in one of Mr M'Farlane's experiments twisted 120 turns, and then allowed to untwist, and 38 turns came out, leaving the wire in equilibrium with 81 turns of permanent twist. Its Young's modulus was then found not to differ as much as per cent. from the value it had before the wire was twisted.

MATHEMATICAL THEORY OF ELASTICITY.1

PART I. ON STRESSES AND STRAINS, CHAPTER I.-Initial Definitions and Explanations. Def. A stress is an equilibrating application of force to a body. Cor. The stress on any part of a body in equilibrium will thus signify the force which it experiences from the matter touching that part all round, whether entirely homogeneous with itself, or only so acros/ a portion of its bounding surface.

Def. A strain is any definite alteration of form or dimensions experienced by a solid.

Examples.-Equal and opposite forces acting at the two ends of a wire or rod of any substance constitute a stress upon it. A body pressed equally all round -for instance, any mass touched by air on all sides experiences a stress. A stone in a building experiences stress if it is pressed upon by other stones, or by any parts of the structure, in contact with it. Any part of a continuous solid mass, simply resting on a fixed base, experiences stress from the surrounding parts in consequence of their weight. The differnt parts of a ship in a heavy Bea experience stresses from which they are exempt when the water is smooth If a rod of any substance become either longer or shorter, it is said to experience a strain. If a body be uniformly condensed in all directions it experiences & strain. If a stone, a beam, or a mass of metal in a building, or in a piece of framework, becomes condensed or dilated in any direction, or bent, or twisted, or distorted in any way, it is said to experience a strain, to become strained, or often in common language, simply "to strain." A ship is said to "strain" if in launching, or when working in a heavy sea, the different parts of it experience relative motions.

CHAPTER II.-Homogeneous Stresses and Homogeneous Strains. Def. A stress is said to be homogeneous throughout a body when equal and similar portions of the body, with corresponding lines

The substance of Chap. I-XVI. of this part of the present article was read before the Royal Society by Prof. Wm. Thomson, M.A., F.R.S., April 24, 1856, and published in the Transactions. Chap. XVII, containing the mathematical theory of Waves in an solotropic or isotropic elastic solid, is new.

These terms were first definitively introduced into the Theory of Elasticity by Rankine, and have been found very valuable in writing on the subject. It will be seen that there is a slight deviation from Rankine's definition of the word "stress." It is here applied to the direct action experienced by a body from the matter around it, and not, as proposed by him, to the elastic reaction of the body equal and opposite to that action.

parallel, experience equal and parallel pressure or tensions on corresponding elements of their surfaces.

mutual tension or pressure between the parts of it on two sides of Cor. When a body is subjected to any homogeneous stress, tho the parts on the two sides of any parallel plane; and the former any plane amounts to the same per unit of surface as that between tension or pressure is parallel to the latter.

body is said to be homogeneously strained, when equal and similar A strain is said to be homogeneous throughout a body, or the portions, with corresponding lines parallel, experience equal and similar alterations of dimansions

Cor. All the particles of the body in parallel pianes remain in parallel planes, when the body is homogeneously strained in any way.

Examples-A long uniform rod, if pulled out, or a pillar loaded with a weight, will experience a uniform strain, except near its ends. There will be a sensible heterogeneousness of the strain, because of the end attachments, or other clrcumstances preventing the ends from expanding laterally to the same extent as the middle does.

A piece of cloth held in a plane, and distorted so that a warp and woof, instead of being perpendicular to one another, become two sets of parallels cutting one another obliquely, experiences a homogeneous strain. The strain is heterogetorsion, and heterogeneous as to direction in different positions in a circle round neous as to intensity, from the axis to the surface of a cylindrical wire under the axis.

CHAPTER III.-On the Distribution of Force in a Stress. three rectangular planes, each of which is perpendicular to the Theorem.-In every homogeneous stress there is a system of direction of the mutual force between the parts of the body on its two sides.

X, Y, Z, any three rect
For let P(X), P(Y), P(Z) denote the components, parallel to
angular lines of reference,
of the force experienced per
tion of the solid bounded by
unit of surface at any por-
Q(X), Q(Y), Q(Z), the corre
a plane parallel to (Y, Z);
sponding components of the
surface of the solid parallel
force experienced by any
R(Z), those of the force at
to (Z, X); and R(X), R(Y),
a surface parallel to (X, Y).
equilibrium of a cube of the
Now, by considering the
solid with faces parallel to
the planes of reference (fig.
15), we see that the couple

R(Y) Fig. 15.

of forces Q(Z) on its two faces perpendicular to Y is balanced by
the couple of forces R(Y) on the faces perpendicular to Z. Hence
we must have
Similarly it is seen that

and

Q(Z) = RM. R(X) = P(Z) PMQX).

gential forces respectively perpendicular to X, Y, Z) may be For the sake of brevity, these pairs of equal quantities (being tar: denoted by T(X), T(Y), T(Z).

Consider a tetrahedral portion of the body (surrounded it may through a point O parallel to the planes of the pairs of lines of be with continuous solid) contained within three planes A, B, C, respectively; so that as regards the areas of the different sides we reference, and a third plane K cutting these at angles a, B, y shall have

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A-K cos a B-K cos B, C-K cos y.

The forces actually experienced by the sides A, B, C have nothing to balance them except the force actually experienced by K. Hence those three forces must have a single resultant, and the force on K must be equal and opposite to it. If, therefore, the force on K per unit of surface be denoted by F. and its direction cosines bv l, m, n, we have

F.K.I-P(X)A+T(Z)B+TC, F.K.m-T(Z)A+QMB+T(X)C, F.K.R-TA+T(X)B+R(Z)C;

and, by the relations between the cases stated above, we deduce
\FI-P(X) cos a+T) cos ẞ+T) 00s y
Fm-T(Z) cos a+QT) cos ẞ+T) cos y
FA-T) cos a+T) cos ẞ+R(Z) con y

Hence the problem of finding (a, B, y), so that the force F (1, m, n)
may be perpendicular to it, will be solved by substituting cos a
cos B, cos y for l, m, n in these equations. By the elimination of
cos a, cos B, cos y from the three equations thus obtained, we have
necessarily real, lead, when no two of them are equal, to one and only
the well-known cubic determinantal equation, of which the roots,
one system of three rectangular axes having the stated property.

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Def. The three lines thus proved to exist for every possible homogeneous stress are called its axes. The planes of their pairs are called its normal planes; the mutual forces between parts of the body separated by these planes, or the forces on portions of the bounding surface parallel to them, are called the principal tensions. Cor. 1. The Principal Tensions of the stress are the roots of the determinant cubic referred to in the demonstration.

Cor. 2. If a stress be specified by the notation P(X)+ &c., as explained above, its normal planes are the principal olanes of the surface of the second degree whose equation is

P(X)X3+Q(Y)Y2+R(Z)Z2+?T(X)YZ+2T(Y)ZX

and its principal tensions are equal to the reciprocals of the squares of the lengths of the semi-princípal-axes of the same surface (quantities which are negative of course for the principal axis or axes which do not cut the surface when the surface is a hyperboloid of one or of two sheets).

Cor. 3. The ellipsoid whose equation, referred to the rectangular axes of a stress, is

(1 − 2eF)X2+(1 − 2eG)Y2+(1 − 2eH)Z3 = 1,

where F, G, H denote the principal tensions, and any infinitely small quantity, represents the stress, in the following manner :From any point P in the surface of the ellipsoid draw a line in the tangent plane half-way to the point where this plane is cut by a perpendicular to it through the centre; and from the end of the first-mentioned line draw a radial line to meet the surface of a sphere of unit radius concentric with the ellipsoid. The tension at this point of the surface of a sphere of the solid is in the line from it to the point P; and its amount per unit of surface is equal to the length of that infinitely small line, divided by e.

Cor. 4. Any stress is fully specified by six quantities, viz., its three principal tensions (F, G, H), and three angles (0, 4, 4) or three numerical quantities equivalent to the nine direction cosines specifying its axes.

CHAPTER IV.-On the Distribution of Displacement in a Strain. Prop. In every homogeneous strain any part of the solid bounded by an ellipsoid remains bounded by an ellipsoid.

For all particles of the solid in a plane remain in a plane, and two parallel planes remain parallel. Consequently every system of conjugate diametral planes of an ellipsoid of the solid retain the property of conjugate diametral planes with reference to the altered curve surface containing the same particies. This altered surface is therefore an ellipsoid.

Prop. There is a single system (and only a single system, except in the cases of symmetry) of three rectangular planes for every homogeneous strain, which remain at right angles to one another in the altered solid.

Def. 1. These three planes are called the normal planes of the strain, or simply the strain-normals. Their lines of intersection are called the axes of the strain. The elongations of the solid per unit of length along these axes or perpendicular to these planes are called the Principal Elongations of the strain.

Remark. The preceding propositions and definitions are not limited to infinitely small strains, but are applicable to whatever extent the body may be strained.

Prop. If a body, while experiencing an infinitely small strain, be held with one point fixed and the normal planes of the strain parallel to three fixed rectangular planes through the point 0, a sphere of the solid of unit radius having this point for its centre becomes, when strained, an ellipsoid, whose equation, referred to the strain-normals through O, is

(1-2x)X2+(1-2y)Y2 + (1—2x)Z2 = 1,

if x, y, z denote the elongations of the solid per unit of length, in the directions respectively perpendicular to these three planes; and the position, on the surface of this ellipsoid, attained by any particular point of the solid, is such that if a line be drawn in the tangent plane, half way to the point of intersection of this plane with a perpendicular from the centre, a radial line drawn through its extremity cuts the primitive spherical surface in the primitive position of that point.

Cor. 1. For every stress, there is a certain infinitely small strain, and conversely, for every infinitely small strain, there is a certain stress, so related that if, while the strain is being acquired, the centre and the strain-normals through it are unmoved, the absolute displacements of particles belonging to a spherical surface of the solid represent, in intensity (according to a definite convention as to units for the representation!) of force by lines) and in direction, the force (reckoned as to intensity, in amount per unit of area) experienced by the enclosed sphere of the solid, at the different parts of its surface, when subjected to the stress.

Cor. 2. Any strain is fully specified by six quantities, viz., its three principal elongations, and three angles (0, 4, 4), or nine direction cosines, equivalent to three independent quantities specifying its axes.

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Def. 2. A stress and an infinitely small strain related in the manner defined in Cor. 1, are said to be of the same type. The ellipsoid by means of which the distribution of force over the surface of a sphere of unit radius is represented in one case, and by means of which the displacements of particles from the spherical surface are shown in the other, may be called the geometrical type of either.

Cor. Any stress- or strain-type is fully specified by five quantities, viz., two ratios between its principal strains or elongations and three quantities specifying the angular position of its axes.

CHAPTER V.-Conditions of Perfect Concurrence between Stresses and Strains.

Def. 1. Two stresses are said to be coincident in direction, or to tude. The same relative designations are applied to two strains be perfectly concurrent, when they only differ in absolute magnidiffering from one another only in absolute magnitude.

Cor. If two stresses or two strains differ by one Leing reverse to the other, they may be said to be negatively coincident in direction, or to be directly opposed or directly contrary to one another. component of the mutual force between the parts of the body on Def. 2. When a homogeneous stress is such that the normal the two sides of any plane whatever through it is proportional to the augmentation of distance between the same plane and another parallel to it and initially at unity of distance, due to a certain strain experienced by the same body, the stress and the strain are said to be perfectly concurrent; also to be coincident in direction. The body is said to be yielding directly to a stress applied to it, when it is acquiring a strain thus related to the stress; and in the same circumstances, the stress is said to be working directly on the body, or to be acting in the same direction as the strain. Cor. 1. Perfectly concurrent stresses and strains are of the same type.

Cor. 2. If a strain is of the same type as the stress, its reverse will be said to be negatively of the same type, or to be directly opposed to the strain. A body is said to be working directly against a stress applied to it when it is acquiring a strain directly opposed to the stress; and in the same circumstances, the matter round the body is said to be yielding directly to the reactive stress of the body upon it.

CHAPTER VI.-Orthogonal Stresses and Strains.

Def. 1. A stress is said to act right across a strain, or to act orthogonally to a strain, or to be orthogonal to a strain, if work is neither done upon nor by the body in virtue of the action of the stress upon it while it is acquiring the strain.

Def. 2. Two stresses are said to be orthogonal when either coincides in direction with a strain orthogonal to the other.

Def. 3. Two strains are said to be orthogonal when either coincides in direction with a stress orthogonal to the other.

Examples.-(1) A uniform cubical compression, and any strain involving no alteration of volume, are orthogonal to one another.

(2) A simple extension or contraction in parallel lines unaccompanied by any transverse extension or contraction, that is, a simple longitudinal strain," is orthogonal to any similar strain in lines at right angles to those parallels. (3) A simple longitudinal strain is orthogonal to a "simple tangential strain" in which the sliding is parallel to its direction or at right angles to it. (4) Two infinitely small simple tangential strains in the same plane, with their directions of sliding mutually inclined at an angle of 45°, are orthogonal to one another.

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(5) An infinitely small simple tangential strain is orthogonal to every infinitely small simple tangential strain, in a plane either parallel to its plane of sliding or perpendicular to its line of sliding.

CHAPTER VII.-Composition and Resolution of Stresses and of Strains.

Any number of simultaneously applied homogeneous stresses are equivalent to a single homogeneous stress which is called their resultant.. Any number of superimposed homogeneous strains are equivalent to a single homogeneous resultant strain. Infinitely small strains may be independently superimposed; and in what follows it will be uniformly understood that the strains spoken of are infinitely small, unless the contrary is stated.

Examples.-(1) A strain consisting simply of elongation in one set of parallel lines, and a strain consisting of equal contraction in a direction at right angles to it, applied together, constitute a single strain, of the kind which that described in Example (3) of the preceding chapter is when infinitely small, and is called a plane distortion, or a simple distortion. It is also sometimes called a simple tangential strain, and when so considered, its plane of sliding may be regarded as either of the planes bisecting the angles between planes normal to the lines of the component longitudinal strains.

(2) Any two simple distortions in one plane may be reduced to a single simple distortion in the same plane.

(3) Two simple distortions not in the same plane have for their resultant a strain which is a distortion unaccompanied by change of volume, and which may be called a compound distortion.

(4) Three equal longitudinal elongations or condensations in three directions

1 That is, a homogeneous strain in which all the particles in one plane remala fixed, and other particles are displaced parallel to this plane.

The plane of a simple tangential strain," or the plane of distortion in a simple tangential strain, is a plane perpendicular to that of the particles sup posed to be held fixed, and parallel to the lines of displacement of the others.

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