Page:EB1911 - Volume 09.djvu/159

The fact is referred to in the distinction between “perfect” and “imperfect” elasticity; and the limitation which must be imposed upon the forces in order that the elasticity may be perfect leads to the investigation of “limits of elasticity” (see §§ 31, 32 below). Steel pianoforte wire is perfectly elastic within rather wide limits, glass within rather narrow limits; building stone, cement and cast iron appear not to be perfectly elastic within any limits, however narrow. When the limits of elasticity are not exceeded no injury is done to a material or structure by the action of the forces. The strength or weakness of a material, and the safety or insecurity of a structure, are thus closely related to the elasticity of the material and to the change of size or shape of the structure when subjected to forces. The “science of elasticity” is occupied with the more abstract side of this relation, viz. with the effects that are produced in a body of definite size, shape and constitution by definite forces; the “science of the strength of materials” is occupied with the more concrete side, viz. with the application of the results obtained in the science of elasticity to practical questions of strength and safety (see Strength of Materials).

2. Stress.—Every body that we know anything about is always under the action of forces. Every body upon which we can experiment is subject to the force of gravity, and must, for the purpose of experiment, be supported by other forces. Such forces are usually applied by way of pressure upon a portion of the surface of the body; and such pressure is exerted by another body in contact with the first. The supported body exerts an equal and opposite pressure upon the supporting body across the portion of surface which is common to the two. The same thing is true of two portions of the same body. If, for example, we consider the two portions into which a body is divided by a (geometrical) horizontal plane, we conclude that the lower portion supports the upper portion by pressure across the plane, and the upper portion presses downwards upon the lower portion with an equal pressure. The pressure is still exerted when the plane is not horizontal, and its direction may be obliquely inclined to, or tangential to, the plane. A more precise meaning is given to “pressure” below. It is important to distinguish between the two classes of forces: forces such as the force of gravity, which act all through a body, and forces such as pressure applied over a surface. The former are named “body forces” or “volume forces,” and the latter “surface tractions.” The action between two portions of a body separated by a geometrical surface is of the nature of surface traction. Body forces are ultimately, when the volumes upon which they act are small enough, proportional to the volumes; surface tractions, on the other hand, are ultimately, when the surfaces across which they act are small enough, proportional to these surfaces. Surface tractions are always exerted by one body upon another, or by one part of a body upon another part, across a surface of contact; and a surface traction is always to be regarded as one aspect of a “stress,” that is to say of a pair of equal and opposite forces; for an equal traction is always exerted by the second body, or part, upon the first across the surface.

3. The proper method of estimating and specifying stress is a matter of importance, and its character is necessarily mathematical. The magnitudes of the surface tractions which compose a stress are estimated as so much force (in dynes or tons) per unit of area (per sq. cm. or per sq. in.). The traction across an assigned plane at an assigned point is measured by the mathematical limit of the fraction ${\text{F}}/{\text{S}},$ where ${\text{F}}$ denotes the numerical measure of the force exerted across a small portion of the plane containing the point, and ${\text{S}}$ denotes the numerical measure of the area of this portion, and the limit is taken by diminishing ${\text{S}}$ indefinitely. The traction may act as "tension," as it does in the case of a horizontal section of a bar supported at its upper end and hanging vertically, or as "pressure," as it does in the case of a horizontal section of a block resting on a horizontal plane, or again it may act obliquely or even tangentially to the separating plane. Normal tractions are reckoned as positive when they are tensions, negative when they are pressures. Tangential tractions are often called “shears” (see § 7 below). Oblique tractions can always be resolved, by the vector law, into normal and tangential tractions. In a fluid at rest the traction across any plane at any point is normal to the plane, and acts as pressure. For the complete specification of the “state of stress” at any point of a body, we should require to know the normal and tangential components of the traction across every plane drawn through the point. Fortunately this requirement can be very much simplified (see §§ 6, 7 below).

4. In general let $\nu$ denote the direction of the normal drawn in a specified sense to a plane drawn through a point ${\text{O}}$ of a body; and let ${\text{T}}\nu$ denote the traction exerted across the plane, at the point ${\text{O}},$ by the portion of the body towards which $\nu$ is drawn upon the remaining portion. Then ${\text{T}}\nu$ is a vector quantity, which has a definite magnitude (estimated as above by the limit of a fraction of the form ${\text{F}}/{\text{S}}$) and a definite direction. It can be specified completely by its components ${{\text{X}}_{\nu },{\text{Y}}_{\nu },{\text{Z}}_{\nu },}$ referred to fixed rectangular axes of ${x,y,z.}$ When the direction of $\nu$ is that of the axis of $x,$ in the positive sense, the components are denoted by ${{\text{X}}_{x},{\text{Y}}_{x},{\text{Z}}_{x};}$ and a similar notation is used when the direction of $\nu$ is that of $y$ or $z,$ the suffix $x$ being replaced by $y$ or $z.$

5. Every body about which we know anything is always in a state of stress, that is to say there are always internal forces acting between the parts of the body, and these forces are exerted as surface tractions across geometrical surfaces drawn in the body. The body, and each part of the body, moves under the action of all the forces (body forces and surface tractions) which are exerted upon it; or remains at rest if these forces are in equilibrium. This result is expressed analytically by means of certain equations—the “equations of motion” or “equations of equilibrium” of the body.

Let $\rho$ denote the density of the body at any point, ${\text{X, Y, Z,}}$ the components parallel to the axes of ${x,y,z}$ of the body forces, estimated as so much force per unit of mass; further let ${f_{x},f_{y},f_{z}}$ denote the components, parallel to the same axes, of the acceleration of the particle which is momentarily at the point (${x,y,z}$). The equations of motion express the result that the rates of change of the momentum, and of the moment of momentum, of any portion of the body are those due to the action of all the forces exerted upon the portion by other bodies, or by other portions of the same body. For the changes of momentum, we have three equations of the type

$\iiint \rho {\text{X}}dx\,dy\,dz+\iint {\text{X}}_{\nu }d{\text{S}}=\iiint \rho f_{x}dx\,dy\,dz,$

(1)

in which the volume integrations are taken through the volume of the portion of the body, the surface integration is taken over its surface, and the notation ${\text{X}}_{\nu }$ is that of § 4, the direction of $\nu$ being that of the normal to this surface drawn outwards. For the changes of moment of momentum, we have three equations of the type

$\iiint \rho (y{\text{Z}}-z{\text{Y}})dx\,dy\,dz+\iint (y{\text{Z}}_{\nu }-z{\text{Y}}_{\nu })d{\text{S}}=\iiint \rho (yf_{z}-zf_{y})dx\,dy\,dz.$

(2)

The equations (1) and (2) are the equations of motion of any kind of body. The equations of equilibrium are obtained by replacing the right-hand members of these equations by zero.

6. These equations can be used to obtain relations between the values of ${{\text{X}}_{\nu },{\text{Y}}_{\nu },\dots }$ for different directions $\nu .$ When the equations are applied to a very small volume, it appears that the terms expressed by surface integrals would, unless they tend to zero limits in a higher order than the areas of the surfaces, be very great compared with the terms expressed by volume integrals. We conclude that the surface tractions on the portion of the body which is bounded by any very small closed surface, are ultimately in equilibrium. When this result is interpreted for a small portion in the shape of a tetrahedron, having three of its faces at right angles to the co-ordinate axes, it leads to three equations of the type

${\text{X}}_{\nu }={\text{X}}_{x}\cos(x,\nu )+{\text{X}}_{y}\cos(y,\nu )+{\text{X}}_{z}\cos(z,\nu ),$

(1)

where $\nu$ is the direction of the normal (drawn outwards) to the remaining face of the tetrahedron, and $(x,\nu )\dots$ denote the angles which this normal makes with the axes. Hence ${\text{X}}_{\nu },\dots$ for any direction $\nu$ are expressed in terms of ${\text{X}}_{x},\dots$ When the above result is interpreted for a very small portion in the shape of a cube, having its edges parallel to the co-ordinate axes, it leads to the equations

${\text{Y}}_{z}={\text{Z}}_{y},$⁠${\text{Z}}_{x}={\text{X}}_{z},$⁠${\text{X}}_{y}={\text{Y}}_{x},\dots$

(2)

When we substitute in the general equations the particular results which are thus obtained, we find that the equations of motion take such forms as

$\rho {\text{X}}+{\frac {\partial {\text{X}}_{x}}{\partial x}}+{\frac {\partial {\text{X}}_{y}}{\partial y}}+{\frac {\partial {\text{Z}}_{x}}{\partial z}}=\rho f_{x},\dots$

(3)

and the equations of moments are satisfied identically. The equations of equilibrium are obtained by replacing the right-hand members by zero.