strain types principal strains stress orthogonal system body equal elastic
CHAPTER X bnp6rfeet Concurrences of two Stress or Strain Def. The concurrence of any stresses or strains of two stated types is the proportion which the work done when a body of unit volume experiences a stress of either type, while acquirirg a strain >f the other, bears to the product of the numbers measuring the stress and strain respectively.
Cur. 1. In orthogonal resolution of a stress or strain, its component of any stated type is equal to its own amount multiplied by its concurrence with that type; or the stress or strain of a stated type which, along with another or others orthogonal to it, have a given stress or strain for their resultant, is equal to the amount of the given stress or strain reduced in the ratio of its concurrence with that stated type.
Con 2. The concurrence of two coincident stresses or strains is unity ; or a perfect concurrence is numerically equal to unity.
Con 3. The concurrence of two orthogonal stresses and strains is zero.
Cur. 4. The concurrence of two directly opposite stresses or strains is -1.
Cur. 5. If a, y, z, 7), C, are orthogonal components of any strain or stress r, its concurrences with the types of reference are respective] y six orthogonal types of reference, and 1', ne, a', g', V thorn of the other.
Coe. 7. The most convenient oprci lieation of a type for strains or stresses, being in general a statement of the components, according to the types of reference, of a unit strain or stress of the type to be specified, liecomes a statement of its concurrences with the types of reference when these are orthogonal.
Esdnudes. - (1) Tlic mutual concurrence of two simple longitudinal strains or stresses, inclined to one another at an angle II, is cost 0.
Hence the components of a simple distortion (3 along two rectangular axes in its plane, and two ethers bisecting the angle between these taken as axes of component simple distortions, are respectively, if Ii be the angle between the axis of elongation in the given distortion and in the first component type.
The mutual concurrence of a simple longitudinal strain and a simple distortion is if a and 13 be the angles at which the direction of the longitudinal strain is inclined to the line bisecting the angles between the axes of the distortion; It is also equal to if (ts and denote the angles at which the direction of the longitudinal strain is inclined to the axis of the distortion.
The mutual concurrence of a simple longitudinal strain and of a uniform The specifying elements exhibited in Example (7) of the preceding Chapter are the concurrences of the new system of orthogonal types described in Example (3) of Chap. IX. with the ordinary system, Examples (I) and (2), Chap. IX.
To transform the specification (x, y, 6 7), (') of a stress or strain with reference to one system of types into (xi,a4,a6) with reference to another system of types. Let (a„ b„ e„ e,, fit gli be the components, according to the original system, of a unit strain of the first type of the new system ; let (a2, 2, L'2, c2, ca, f2, Us) be the corresponding specification of the second type of the new system; and so on. Then we have, for the required formuhe of transformationx=aizi-da,x,-i-a3x3+a,x4-Ea,x,+a,ix6, 611% d-btzrzi- bsra +br:rt +bars +bsxs = gtxf+g2x2+g3x3+gaxs+g5x5+gaxe Erample. - The transforming equations to pass from a specification (x, In a paper on the Thermo-elastic Properties of Matter, published in the first number of the Quarterly Mathematical Journal, April 1855, and republished in the Philosophical Magazine, 1877, second half year, it was proved, from general principles in tho theory of the Transformation of Energy, that the amount of work (w) required to reduce an elastic solid, kept at a constant temperature, from one stated condition of internal strain to another depends solely on these two conditions, and not at all On the cycle of varied states through which the body may have been niacin to pass in effecting the change, provided always there has been no failure in the work required to be done upon it is _ • The stress which must be applied to its surface to keep the body in equilibrium in the state (e, y, z, 1, n, C) must therefore be such that it would do this amount of work if the body, under its action, were to acquire the arbitrary strain dx, dy, dx, dl, dn, ; that is, it must be the resultant of six stresses: - one orthogonal to the five strains dy, dz, d4, do, dC, and of such a magnitude as to do the n, C) of the strains, the amounts of the six stresses which fulfil those conditions will (Chapter XI.) be given by the equations and the types of these component stresses are determined by being orthogonal to the fives of the six strain-types, wanting the first, the second, &c., respectively.
Car. If the types of reference used in expressing the strain of the body constitute an orthogonal system, the types of the component stresses will coincide with them, and each of the concurrences will be unity. Hence the equations of equilibrium of an elastic solid referred to six orthogonal types arc simply Cum-7En XIV. - Reduelion of the Potential Function, and of the Equations of Equilibrium, of an Elastic Solid to their simplest Forms.
If the condition of the body from which the work denoted by to, is reckoned be that of equilibrium tinder no stress from without, and if x, y, z, C be chosen each zero for this condition, we shall have, by Alaclaurin's theorem, te=r12(2', 14 2, E, t)+II3(x, 2, E, K)+ where II, II5, &c., denote homogeneous functions of the second order, third order, &e., respectively. Hence tilto dw&c., will each be a linear function of the strain coordinates, together with functions of higher orders derived from H„ But experience shows (section 37 above) that, within the elastic limits, the stresses are very nearly if not quite proportional to the strains they are capable of producing ; and therefore H3, lee., may be neglected, and we have simply w Y, 2, 6 lb Now in general there will be twenty-one terms, with independent coefficients, in this function; but by a choice of types of reference, that-is, by a linear transformation of the independent variables, we may, in an infinite variety of ways, reduce it to the form to,.--6(Ax2+By2+ Cs7 -I- Fe+ GO+ I1K2).
The equations of equilibrium then become the simplest possible form under which they can be presented. The interpretation can be expressed as follows.
Prop. Au infinite number of systems of six types of strains or stresses exist in any given elastic solid such that, if a strain of any one of those types be impressed on the body, the elastic reaction is balanced by a stress orthogonal to the five others of the same system.
CuarrEn XV. - On the Six Principal Strains of ass Elastic Solid.
tion, we have only fifteen equations to. satisfy ; while we have thirty disposable transforming coefficients, there being five independent elements to specify a -type, and six types to be changed. Any further condition expressible by just fifteen independent equations may be satisfied, and makes the transformation determinate. Now the condition that six strains may be mutually orthogonal is expressible by just as many equations as there are different pairs of six things, that is, fifteen. The well-known algebraic theory of the linear transformation of quadratic functions shows for the case of six variables - (1) that the six coefficients in the reduced form are the roots of a "determinant" of the sixth degree necessarily real ; (2) that this multiplicity of roots leads determinately to one, and only one system of six types fulfilling the prescribed conditions, unless two or more of the roots are equal to one another, when there will be an infinite number of solutions and definite degrees of isotropy among them ; and (3) that there is no equality between any of the six roots of the determinant in general, when there are twenty-one independent coefficients in the given quadratic.
Prop. Hence a single system of six mutually orthogonal types may be determined for any homogeneous elastic solid, so that its potential energy when homogeneously strained in any way is ex pressed by the sum of the products of the squares of the components of the strain, according to those types, respectively multiplied by six determinate coefficients.
Def. The six strain-types thus determined are called the Six Principal Strain-types of the body.
The commences of the stress-components used in interpreting the differential equation of energy with the types of the strain-coordinates in terms of which the potential of elasticity is expressed, being perfect when these constitute an orthogonal system, each of the quantities denoted above by to, b, c, f, g, h, is unity when the six principal strain-types are chosen for ;he coordinates. The equations of equilibrium of an elastic solid may therefore be expressed where x, y, z, C denote strains belonging to the six Principal Types, and P, Q, It, S, T, U the components according to the same types, of the stress required to hold the body in equilibrium when in the condition of having those strains. The amount of work that must be spent upon it per unit of its volume, to bring it to this state from an unconstrained condition, is given by the equation ,RAx2+By2+ co+pEr+ 0,12 ..1.1q2).
Def. The coefficients A, B, C, F, G, Ii arc called the six Principal Elasticities of the body.
The equations of equilibrium express the following propositions :- Prop. If a body be strained according to any one of its six Principal Types, the stress required to hold it so is directly concurrent with the strain.
Examples. - (1) If a solid be cubically isotropic in its elastic properties, as crystals of the cubical class probably are, any portion of It will, when subject to n uniform positive or negative normal pressure all round its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensation is one of its six principal strains. Three plane distortion( with axes bisecting the angles between the edges of the cube of symmetry are clearly also principal strains, and shire the three eorrespoading principal elasticities are equal to one another, any strain whatever compounded of these three is a principal strain. Lastly, a plane distortion whose axes coincide with any two edges of the cube, being clearly a principal distortion, and the principal elasticities corresponding to the three distortions of this kind being equal to one another, any distortion compounded of them is also a principal distortion.
Hence the system of orthogonal types treated of in Examples (3) Chap. IX., and (7) Chap. X., or any system in which, for (II.), (III.), and (IV.), any three orthogonal strains compounded of them are substituted, constitutes a system of six Principal Strains in a solid cubically isotropic. There are only th•ce distinct Principal Elasticities for such a body, and these are - (A) its modulus of compressibility, (B) its rigidity against diagonal distortion In any of its principal plated (three equal elasticities), and (C) Its rigidity against rectangular distortion* of a cube of symmetry (two equal elasticities).
(2) In a perfectly isotopic solid, the rigidity against all distortions is equal. Hence the rigidity (B) against diagonal distortion must be equal to the rigidity (C) against rectangular distortion, In a cube; and It is easily seen that if this condition is fulfilled for one net of three rectangular planes for which a substance Is isotropic, the Isotropy must be complete. The conditions of perfect or spherical isotropy are therefore expressed In terms of the conditions referred to in the preceding example, with the farther condition B=C.
A uniform condensation In all directions, and any system whatever of Ave orthogonal distortions, constitute a system of six Principal Strains in a spherically isotropic solid. Its Principal Elasticities are simply its Modulus of Compressibility and its Rigidity.
Prop. Unless some of the six Principal Elasticities be equal to one another, the stress required to keep the body strained otherwise than according to one or other of six distinct types is oblique to the strain.
Prop. The stress required to maintain a given amount of strain is a maximum or a maximum-minimum, or a minimum, if it is of one of the six Principal Types.
Cor. if A be the greatest and II the least of the six quantities A, B, C, F, G, II, the principal type to which the first corresponds is that of a strain requiring a greater stress to maintain it than any other strain of equal amount ; and the principal type to which the last corresponds is that of a strain which is maintained by a less stress than any other strain of equal amount in the same body. The stresses corresponding to the four other principal strain-types have each the maximmn-minimum property in a determinate way.
Prop. If a body be strained in the direction of which the concurrences with the principal strain-types are 1, m, n, a, th, v, and to an amount equal to r, the stress required to maintain it in this state will be equal to nr, where 12=(A2g+p,2m2+c2„2+FIXE+GE/y2+1120)1, and will be of a type of which the concii•rences with the principal types are respectively Prop. A homogeneous elastic solid, crystalline or non-crystalline, :subject to magnetic force or free front magnetic force, has neither any right-handed or left-handed, nor any dipolar, properties dependent on elastic forces simply proportional to strains.
COr. The elastic forces concerned in the huniniferous vibrations of a solid or fluid medium possessing the right- or left-handed property, whether axial or rotatory, such as quartz crystal, or tartaric acid, or solution of sugar, either depend on the heterogeneousness or on the magnitude of the strains experienced.
hence as they do not depend on the magnitude of the strain, they do depend on its heterogeneousness through the portion of a medium containing a wave.
Con There cannot possibly be any characteristic of elastic forces simply proportional to the strains in a homogeneous body, corresponding to certain peculiarities of crystalline form which have been observed, - for instance corresponding to the plagiedral faces discovered by Sir John Herschel to indicate the optical character, whether right-handed or left-!landed, in different specimens of quartz crystal, or corresponding to the distinguishing characteristics of the crystals of the right-handed and left-handed tartaric acids obtained by M. Pasteur from racemic acid, or corresponding to the dipolar characteristics of form said to have been discovered in electric crystals.