Analysis of Strain

ANALYSIS OF STRAIN Displacements.-13. In the analysis of strain, or distortion, the assumption of continuity is again fundamental. Disregarding all questions of molecular structure, we imagine our ideal material to occupy every point within a certain continuous surface (or within a volume contained by continuous surfaces, when the body considered is hollow)—the "boundary" of the solid body considered. When the material is distorted by the application of force, this boundary surface will assume a different form ; but so long as the material is unbroken, it will remain a continuous surface.

In the unstrained body there will be, at any point defined by co-ordinates x, y, z, a certain "particle" of material. When the body is distorted, this particular particle will in general occupy a different position, which we may define by co-ordinates x+u, v, Z+W ; 11, v, w are termed the component displacements of the particle in question. It is clear that, if we know the component displacements of any two particles P and Q, we can calculate the increase, due to strain, in the distance PQ; and further that, if we know the component displacements of a third point R, we can calculate the change, due to strain, in the angle PQR.

Simple Extension.

i4. Let us imagine, for example, that a cube of the material is distorted in the manner which is in dicated by fig. 5, so that four of its edges are lengthened by the same amount, whilst the other eight edges retain their original lengths and all the faces remain rectangular. If we choose axes Ox, Oy, Oz coinciding with three edges of the cube, as shown, it is clear that this state of strain will occur when every particle undergoes a displacement confined to the direction of Ox and of magnitude proportional to its original distance from the face OABC; then we shall have We say that the material undergoes "simple shearing strain" (or, more briefly, "simple shear") in the (z, x) plane; the mag nitude of the strain is y, the reduction in the angle between lines which were originally parallel to Oz and Ox.

Composition of Simple Strains: Principle of Superposi tion.-16. Suppose that the top face ABHG, after moving to the position A'B'H'G', undergoes a further displacement in its own plane, this time in the direction Oy. We may imagine that the edges OA, CB, EH, FG, and any line of particles which was originally parallel to them, again remain straight and parallel; they now rotate through another angle y; which is the reduction (in circular measure) in the (originally right) angle AOC. We say that a second simple shearing strain has been superposed on the first.

Evidently, only the top and bottom faces of the cube will remain rectangular after this second strain is imposed, and these can have their angles changed by the imposition of a third simple shear. In its final state, the body will be bounded by six paral lelograms, and opposite faces will be similar; that is to say, the cube has distorted into a parallelopiped. Since the lengths of its edges are unaltered by the distortion, three quantities (namely, three angles of shear 71, 'Y2, are required to specify its final shape.

Lastly, we may imagine that the material undergoes three successive simple extensions in the three perpendicular directions Ox, Oy, Oz. Denoting these extensions by el, e3, defined as above, we see that the three sets of parallel edges of the paral lelopiped will be increased in the ratios (I + el), (I e2) and (I These expressions are not strictly correct, but they are sufficiently accurate if ei, e2, e3 are small; for on this under standing the final shape would have been the same if the ex tensions had been imposed first and the shearing strains last. This is the principle of superposition for small strains.

Homogeneous Strain.-17.

In these examples of simple strain, the lame description and diagrams would evidently apply to any cubical portion of the material, however small. In such circumstances the strain is said to be "homogeneous," or uniform.