STRAINED ELASTIC SOLIDS 475 



special cases where wider possibilities can exist. If, for instance, 

 in the state of strain the three principal stresses are equal to 

 one another, the "stress quadric" is a sphere; all sets of three 

 axes are principal; there are no shearing stresses for any axes. 

 (See case (3), Gibbs, I, bottom of page 195.) This is in fact the 

 case of ''hydrostatic stress" referred to frequently in these 

 pages by Gibbs. In such a state the form of the solid does 

 not matter. Immersed in a fluid throughout which there 

 exists a constant pressure a sohd will be in a homogeneous state 

 of strain compatible with the condition of hydrostatic stress, 

 that is, the condition in which there are no shears and the stress 

 over any surface is normal to it and is of the pressure type. 

 (The reader should not misconceive the phrase "homogeneous 

 state of strain." This implies that an, an, • • • ass have values 

 which are severally constant throughout the solid. But there is 

 no implication, for instance, that an = a22 = 0,33- It should be 

 clearly recognized that this is not necessarily the case even for a 

 state compatible with hydrostatic stress. It would be so, no 

 doubt, if the solid were isotropic in nature; in that event all linear 

 contractions or extensions would be equal and no shears would 

 exist, but for crystalline solids the more general nature of the 

 stress-strain relations would permit of wider conditions of 

 strain, even if for any set of axes Xx, Yy, Zz were equal to one 

 another, and the remaining stress-constituents zero.) If, how- 

 ever, one is to maintain the rectangular parallelopiped of solid 

 material, imagined by Gibbs at this juncture, in equilibrium in a 

 general homogeneous state of strain, one must arrange for 

 different pressures on the different pairs of faces. So if the 

 solid is in contact with a fluid of suitable pressure at one pair of 

 opposite faces, it cannot be so at the other two pairs. It must 

 be constrained by some other surface forces (pressural or 

 tensional) on these faces to maintain the assigned state of strain. 

 If these constraints are released and the fluid comes into contact 

 with all six faces there will be an immediate change to another 

 state of homogeneous strain compatible with the condition of 

 hydrostatic stress. In such a change there will be a diminution 

 of intrinsic energy of strain, since all release of constraints if 

 followed by movement converts potential energy into kinetic 



