Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/844

820 820 ELASTICITY Def. The three lines thus proved to exist for every possible homogeneous stress are called Its axes. The planes of their pairs aje 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 ex plained above, its normal planes are the principal planes of the surface of the second degree whose equation is P(X)X2+Q(Y)YHR(Z)2 2 +2T(X)YZ+2T(Y)ZX and its principal tensions are equal to the reciprocals of the squares of the lengths of the semi-principal-axcs of the same surface (quan tities 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 }f a stress, is (1 - 2eF)X 2 +(l - 2eG)YH(l - 1cl)Z- = 1 , where F, G, H denote the principal tensions, and e 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 c. Cor. 4. Any stress is fully specified by six quantities, viz., its three principal tensions (F, G, H), and three angles (8, &amp;lt;p, if/) 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 particles. 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. llemark. 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 0, is (l-2.r)X2 + (l-2y)Y2 + (1-22)7.2 _ lt 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 par ticular 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 1 of force by lines) and in direction, the force (reckoned as to intensity, in amount per unit of area) ex perienced 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 (6, &amp;lt;p, j/), or nine direction cosines, equivalent to three independent quantities speci fying its axes. ... .: 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 sur face 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 jive 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 be perfectly concurrent, when they only differ in absolute magni tude. The same relative designations are applied to two strains differing from one another only in absolute magnitude. Cor. If two stresses or two strains differ by one being reverse to the other, they may be said to be negatively coincident in direc tion, or to be directly opposed or directly contrary to one another. Def. 2. When a homogeneous stress is such that the normal component of the mutual force between the parts of the body on 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 arc 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 coin cides in direction with a strain orthogonal to the other. Def. 3. Two strains are said to be orthogonal when either coin cides in direction with a stress orthogonal to the other. Examples. (Y) 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, &quot;, a simple longitudinal strain,&quot; is orthogonal to any similar strain in lines at right angles to those parallels. (3) A simple longitudinal strain is orthogonal to a &quot;simple tangential strain &quot; l 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, 2 with their directions of sliding mutually inclined at an angle of 45, are orthogonal to one another. (5) An infinitely small simple tangential strain is orthogonal to every in 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 regaided as either of the planes bisecting the angles between planes normal to tl f &amp;lt;g C A C ^rsKS^ionst 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 strain which is a distortion unaccompanied by change of volume, and which may bG (4) 1 Three &quot;q&quot;aal U loiigitudinai n elongations or condensations in three directions 1 That is a homogeneous strnin in which all the particles iu one;plane remain fixed, and other particles are displaced parallel to this plane. 2 &quot;The plane of a simple tanci-ntial strain.&quot; or the plane of distortion in a Himple ai gential strain/is a plane perpendicular to that of the part.c e. op posed to be sheld fixed, and V araliel to the lines of displacement of the others.