﻿448 
  Specifying 
  Method 
  for 
  Stress 
  and 
  Strain 
  in 
  Elastic 
  Solid. 
  

  

  Thomson 
  and 
  Tait 
  referred 
  to 
  in 
  the 
  introductory 
  paragraph 
  

   above. 
  By 
  the 
  infinitesimal 
  geometry 
  of 
  the 
  affair, 
  we 
  

   easily 
  find 
  the 
  corresponding 
  alterations 
  of 
  the 
  face-diagonals, 
  

   which 
  according 
  to 
  our 
  present 
  notation 
  are 
  (p 
  — 
  1)X, 
  (p' 
  — 
  l)\, 
  

   (q 
  — 
  1)\, 
  etc., 
  and 
  thus 
  we 
  have 
  as 
  follows 
  : 
  — 
  

  

  j>-i=K/+"0+°) 
  I 
  

  

  

  r 
  -l 
  = 
  i(« 
  + 
  /+«) 
  

   r'-l 
  = 
  i(« 
  + 
  /-c) 
  

  

  for 
  the 
  relation 
  between 
  the 
  two 
  specifications 
  of 
  any 
  

   infinitesimal 
  strain. 
  Adding 
  these, 
  and 
  denoting 
  e+f+g 
  

   by 
  s, 
  we 
  find 
  

  

  p+p' 
  + 
  q 
  + 
  q'-{-r 
  + 
  r' 
  — 
  6 
  = 
  2s 
  . 
  . 
  . 
  (6). 
  

  

  And 
  solving 
  for 
  a, 
  />, 
  c, 
  e, 
  /, 
  </, 
  in 
  terms 
  of 
  p, 
  q, 
  r, 
  p\ 
  q\ 
  r 
  r 
  , 
  

   we 
  have 
  

  

  a=p-p 
  f 
  ; 
  b 
  = 
  q-q' 
  ; 
  c=?'-r' 
  : 
  ■* 
  

  

  e 
  = 
  5— 
  jp— 
  je?' 
  + 
  2 
  ; 
  f=s 
  — 
  q 
  — 
  q 
  r 
  + 
  2 
  ; 
  g 
  = 
  s 
  — 
  r—r' 
  + 
  2) 
  '' 
  

  

  § 
  8. 
  The 
  work 
  required 
  to 
  produce 
  an 
  infinitesimal 
  strain 
  

   <?, 
  /, 
  #, 
  a, 
  b, 
  c, 
  in 
  a 
  homogeneous 
  solid 
  of 
  cubic 
  crystalline 
  

   symmetry 
  is 
  expressed 
  by 
  the 
  following 
  formula 
  : 
  — 
  

  

  •2w 
  = 
  &{e*+f+g*) 
  + 
  2®{f<! 
  + 
  ge 
  + 
  ef) 
  +<a 
  2 
  + 
  /> 
  2 
  + 
  c 
  2 
  ). 
  (8). 
  

  

  This 
  may 
  be 
  conveniently 
  modified 
  by 
  putting 
  

  

  *=i(« 
  + 
  M) 
  ; 
  m=i(«-») 
  .... 
  (9), 
  

  

  where 
  k 
  denotes 
  the 
  bulk 
  modulus 
  and 
  n 
  ly 
  n 
  the 
  two 
  rigidity- 
  

   mocluluses. 
  With 
  this 
  notation 
  (8) 
  becomes 
  

  

  2™.= 
  *(*+/+#)?+frc 
  1 
  [(/^ 
  (10), 
  

  

  The 
  rigidity 
  relative 
  to 
  shearings 
  parallel 
  to 
  the 
  pairs 
  of 
  

   planes 
  of 
  the 
  cube, 
  or, 
  which 
  is 
  the 
  same 
  thing, 
  changes 
  of 
  

   the 
  angles 
  of 
  the 
  corners 
  of 
  the 
  square 
  faces 
  from 
  right 
  angles 
  

   to 
  acute 
  or 
  obtuse 
  angles, 
  is 
  wj. 
  The 
  rigidity 
  relative 
  to 
  

   changes 
  of 
  the 
  angles 
  between 
  the 
  diagonals 
  of 
  the 
  faces 
  from 
  

   right 
  angles 
  to 
  acute 
  or 
  obtuse 
  angles 
  is 
  n. 
  The 
  compressibility 
  

   modulus 
  is 
  h. 
  Using 
  now 
  (7) 
  in 
  (10) 
  we 
  have 
  

  

  2w 
  = 
  ks 
  2 
  + 
  %n 
  x 
  [(q 
  -f 
  q 
  1 
  — 
  r 
  — 
  r') 
  2 
  + 
  (r 
  + 
  r 
  —p 
  —p') 
  2 
  + 
  (p 
  -+ 
  p' 
  — 
  q 
  — 
  </) 
  2 
  J 
  

   + 
  n[(p-p'y-t-(q-q<)"+(r-r 
  f 
  y] 
  ..... 
  (11). 
  

  

  