STRAINED ELASTIC SOLIDS 435 



one part of the increase of energy in the element of volume. 

 The other pairs of faces when treated similarly yield further 

 parts of the energy increase, viz. hklYybfo,, and hklZ^bfz. Now let 

 us turn to the shears and fix our attention for the moment on 

 the faces of the element which are parallel to the plane OXY and 

 are separated by the distance I in the direction of OZ. A little 

 thought will show that one of these faces has moved in a shearing 

 manner relatively to the other by an amount which is the vector 

 sum of a component U{ezi + en) parallel to OX and a com- 

 ponent Z5(e23 + 632) parallel to OY. (A glance at equation (10) 

 will remind the reader that the "shear" of hues parallel origi- 

 nally to OX and OZ is 5[e5/(e3ei)'] which is substantially 

 6(e3i + eia) ; the "shear" practically measures the small change in 

 the (right) angle between OZ and OX.) We can again simplify 

 our argument by conceiving one of the hk faces fixed and the 

 other slipping over it by amount Uf^ in the direction of OX. 

 The shearing pull across this face by the surrounding matter in 

 the element is hkXz in this direction. (The face is perpendicular 

 to OZ and the pull is in the direction OX.) Thus the work done 

 on this account is hklXz^fh- Similar reasoning yields hklYz^fi 

 for the other component. Each of the other pairs of faces 

 treated in a similar manner would yield similar terms ; the faces 

 parallel to OYZ would yield hklYxdfe and hklZxSf^, and the 

 faces parallel to OZX would yield hklZydfi and hklXr^fe.. It 

 would seem that in order to obtain the increase of energy asso- 

 ciated with the shearing movements, we ought to add these six 

 terms. This is, however, one of the pitfalls of this simple 

 method which we are using so as to evade advanced analytical 

 operations. If we adopted this procedure we should obtain 

 twice the correct increase associated with the shears, and it is 

 not difficult to realize that this is so. For a shear of one Z-face 

 past the other Z-face (meaning the faces perpendicular to the 

 direction OZ) in the direction parallel to OX involves of necessity 

 a shear of an X-face past the other X-face in the direction 

 parallel to OZ. Either shear is one of two alternative ways of 

 describing the resulting distortion. Now our method of cal- 

 culating the work done in this case really requires us to conceive 

 the element of volume as isolated and sheared either by a shear- 



