Page:EB1911 - Volume 09.djvu/160

Fig. 17. A state of stress in which the traction across any plane of a set of parallel planes is normal to the plane, and that across any perpendicular plane vanishes, is described as a state of “simple tension” (“simple pressure” if the traction is negative). A state of stress in which the traction across any plane is normal to the plane, and the traction is the same for all planes passing through any point, is described as a state of “uniform tension” (“uniform pressure” if the traction is negative). Sometimes the phrases “isotropic tension” and “hydrostatic pressure” are used instead of “uniform” tension or pressure. The distinction between the two states, simple tension and uniform tension, is illustrated in fig. 1.

A state of stress in which there is purely tangential traction on a plane, and no normal traction on any perpendicular plane, is described as a state of “shearing stress.” The result (2) of § 6 shows that tangential tractions occur in pairs. If, at any point, there is tangential traction, in any direction, on a plane parallel to this direction, and if we draw through the point a plane at right angles to the direction of this traction, and therefore containing the normal to the first plane, then there is equal tangential traction on this second plane in the direction of the normal to the first plane. The result is illustrated in fig. 2, where a rectangular block is subjected on two opposite faces to opposing tangential tractions, and is held in equilibrium by equal tangential tractions applied to two other faces.
Fig. 2

Through any point there always pass three planes, at right angles to each other, across which there is no tangential traction. These planes are called the “principal planes of stress,” and the (normal) tractions across them the “principal stresses.” Lines, usually curved, which have at every point the direction of a principal stress at the point, are called “lines of stress.”

8. It appears that the stress at any point of a body is completely specified by six quantities, which can be taken to be the ${\text{X}}_{x},{\text{Y}}_{y},{\text{Z}}_{z}$ and ${\text{Y}}_{z},{\text{Z}}_{x},{\text{X}}_{y}$ of § 6. The first three are tensions (pressures if they are negative) across three planes parallel to fixed rectangular directions, and the remaining three are tangential tractions across the same three planes. These six quantities are called the “components of stress.” It appears also that the components of stress are connected with each other, and with the body forces and accelerations, by the three partial differential equations of the type (3) of § 6. These equations are available for the purpose of determining the state of stress which exists in a body of definite form subjected to definite forces, but they are not sufficient for the purpose (see § 38 below). In order to effect the determination it is necessary to have information concerning the constitution of the body, and to introduce subsidiary relations founded upon this information.

9. The definite mathematical relations which have been found to connect the components of stress with each other, and with other quantities, result necessarily from the formation of a clear conception of the nature of stress. They do not admit of experimental verification, because the stress within a body does not admit of direct measurement. Results which are deduced by the aid of these relations can be compared with experimental results. If any discrepancy were observed it would not be interpreted as requiring a modification of the concept of stress, but as affecting some one or other of the subsidiary relations which must be introduced for the purpose of obtaining the theoretical result.

10. Strain.—For the specification of the changes of size and shape which are produced in a body by any forces, we begin by defining the “average extension” of any linear element or “filament” of the body. Let $l_{0}$ be the length of the filament before the forces are applied, $l$ its length when the body is subjected to the forces. The average extension of the filament is measured by the fraction $(l-l_{0})/l_{0}.$ If this fraction is negative there is “contraction.” The “extension at a point” of a body in any assigned direction is the mathematical limit of this fraction when one end of the filament is at the point, the filament has the assigned direction, and its length is diminished indefinitely. It is clear that all the changes of size and shape of the body are known when the extension at every point in every direction is known.

The relations between the extensions in different directions around the same point are most simply expressed by introducing the extensions in the directions of the co-ordinate axes and the angles between filaments of the body which are initially parallel to these axes. Let $e_{xx},e_{yy},e_{zz},$ denote the extensions parallel to the axes of $x,y,z,$ and let $e_{yz},e_{zx},e_{zy},$ denote the cosines of the angles between the pairs of filaments which are initially parallel to the axes of $y$ and $z,$ $z$ and $x,$ $x$ and $y.$ Also let $e$ denote the extension in the direction of a line the direction cosines of which are $l,m,n.$ Then, if the changes of size and shape are slight, we have the relation

$e=e_{xx}l^{2}+e_{yy}m^{2}+e_{zz}n^{2}+e_{yz}mn+e_{zx}nl+e_{xy}lm.$

The body which undergoes the change of size or shape is said to be “strained,” and the “strain” is determined when the quantities $e_{xx},e_{yy},e_{zz}$ and $e_{yz},e_{zx},e_{xy}$ defined above are known at every point of it. These quantities are called “components of strain.” The three of the type $e_{xx}$ are extensions, and the three of the type $e_{yz}$ are called “shearing strains” (see § 12 below).

11. All the changes of relative position of particles of the body are known when the strain is known, and conversely the strain can be determined when the changes of relative position are given. These changes can be expressed most simply by the introduction of a vector quantity to represent the displacement of any particle.

When the body is deformed by the action of any forces its particles pass from the positions which they occupied before the action of the forces into new positions. If $x,y,z$ are the co-ordinates of the position of a particle in the first state, its co-ordinates in the second state may be denoted by ${x+u,y+v,z+w}.$ The quantities, $u,v,w$ are the “components of displacement.” When these quantities are small, the strain is connected with them by the equations

⁠ $e_{xx}=\partial u/\partial x,$⁠	$e_{yy}=\partial v/\partial y,$⁠	$e_{zz}=\partial w/\partial z,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ . ⁠(1)
⁠ $e_{yz}={\frac {\partial w}{\partial y}}+{\frac {\partial v}{\partial z}},$	$e_{zx}={\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}},$	$e_{xy}={\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\cdot$

12. These equations enable us to determine more exactly the nature of the “shearing strains” such as $e_{xy}.$ Let $u,$ for example, be of the form $sy,$ where $s$ is constant, and let $v$ and $w$ vanish. Then $e_{xy}=s,$ and the remaining components of strain vanish. The nature of the strain (called “simple shear”) is simply appreciated by imagining the body to consist of a series of thin sheets, like the leaves of a book, which lie one over another and are all parallel to a plane (that of $x,z$); and the displacement is seen to consist in the shifting of each sheet relative to the sheet below in a direction (that of $x$) which is the same for all the sheets. The displacement of any sheet is proportional to its distance $y$ from a particular sheet, which remains undisplaced. The shearing strain has the effect of distorting the shape of any portion of the body without altering its volume. This is shown in fig. 3, where a square ${\text{ABCD}}$ is distorted by simple shear (each point moving parallel to the line marked $xx$) into a rhombus ${\text{A}}'{\text{B}}'{\text{C}}'{\text{D}}'$, as if by an extension of the diagonal ${\text{BD}}$ and a contraction of the diagonal ${\text{AC}}$, which extension and contraction are adjusted so as to leave the area unaltered. In the general case, where $u$ is not of the form $sy$ and $v$ and $w$ do not vanish, the shearing strains such as $e_{xy}$ result from the composition of pairs of simple shears of the type which has just been explained.