11(5 SYSTEMS OF FORCES. [CHAP. VII. 



121. Deformable bodies, strain. A system of particles 

 which is not rigid can move so that the distance between two 

 particles changes. Whenever this can happen a particular con- 

 figuration is chosen as a standard, and, in a configuration in 

 which the distance between any two particles is greater or less 

 than that in the standard, the system is said to be strained. 



For a continuous system of particles forming a deformable body 

 it can be shown that the motion of the body can be precisely 

 specified by the following method: Choose one particle of the 

 body, one line of particles going out from the chosen particle, 

 one plane of particles passing through the chosen line. In the 

 deformation of the body the line and the plane respectively become 

 a curve and a curved surface of changing forms, but the curve has 

 a tangent line, and the surface a tangent plane, passing through 

 the chosen particle. The particle, the tangent line, and the 

 tangent plane determine a frame relative to which all the particles 

 of the body have at any time definite positions. The determi- 

 nation of the position of this frame relative to the frame of 

 reference is, as we have seen in Article 114, the same problem 

 as the determination of the position at any time of a rigid body. 

 The determination of the positions of other particles of the body 

 relative to this frame can be shown to depend upon the determi- 

 nation of six quantities continuously variable from point to point, 

 and known as the components of strain. The theory which effects 

 this determination is the Theory of Elasticity. 



In the case of a solid body, for which it may be assumed that 

 the deformation is always small, the above method can be applied 

 with success. But it is manifestly less applicable in the case of a 

 fluid, or of a plastic solid undergoing finite deformation. 



In all cases, however, the equations of motion of a system of 

 particles, obtained in Article 107, have to be applied to every 

 portion into which the system can be divided. To succeed in this 

 process we must consider more closely the resultants of internal 

 forces between parts of a system. 



122. Stress. Conceive that in a continuous (or discontinuous) 

 system of particles, exerting forces on each other, a plane is drawn, 

 and a closed curve marked on the plane. Some of the lines 

 between particles cross the plane within the curve. Let the 



