420 RICE 



ART. K 



sure on the position of the point. But for a sohd medium the 

 conditions are more complex, and we must consider carefully- 

 just how many numerical magnitudes must be given in order to 

 specify the stress at a point, i.e., to indicate what is the stress 

 at the point across any assigned element of surface. We shall 

 see presently that there are six, and, as is readily suggested by 

 the example of a fluid, each of these may vary in value with the 

 position of the point, i.e., be a function of the coordinates of the 

 point. The analysis of the stress at a point proceeds as 

 follows. 



Consider the point P, the displaced position of a point P' in 

 the unstrained state, and let its coordinates referred to the 

 axes OX, OY, OZ (chosen for the strained state) be x, y, z* 

 First let the conceptual dividing surface be parallel to OYZ, 

 i.e., a plane surface at right angles to OX. We can resolve the 

 postulated force across the element of area at P into three 

 components parallel to the axes, and these when divided hy the 

 area of the element we denote by Xx, Yx, Zx, the suffix indicating 

 clearly that the plane surface under consideration is normal to 

 OX. Xx is of the nature of a tension or pressure, while Yx 

 and Zx are "shearing tractions," their directions lying in the 

 dividing surface. Of course each of these in general varies in 

 value with the position of P and so should strictly be written as 



^x{x, y, z), Yx(x, y, z), Zx(x, y, z) 



to indicate their functional dependence on the values of x, y, z; 

 however, for brevity, we drop the bracketed letters, but this 

 point should never be lost sight of. By considering plane 

 surfaces containing P normal to OF and OZ we can introduce 

 components of the forces at P across these surfaces, when 

 divided by the area of the element, as Xy, Yy, Zy and Xz, Yz, 

 Zz. By the aid of these nine quantities we can now express the 

 stress at P across any element of surface containing P whose 

 direction cosines are given, say «, /3, 7. To do so, draw local 

 axes at P (Fig. 5) and let a plane surface whose direction cosines 



* We may from this point onwards drop double accents in symbols for 

 gtrained positions and coordinates as no longer necessary. 



