﻿on 
  Material 
  Stresses. 
  491 
  

  

  (1) 
  The 
  nine 
  elements 
  of 
  mechanical 
  stress 
  obey 
  well- 
  

   known 
  laws 
  of 
  resolution. 
  By 
  introducing 
  the 
  notion 
  of 
  infi- 
  

   nitesimal 
  rotations 
  of 
  the 
  coordinate 
  axes 
  about 
  their 
  own 
  

   positions, 
  we 
  can, 
  for 
  our 
  present 
  purposes, 
  replace 
  these 
  

   relations 
  by 
  the 
  three 
  differential-operators*, 
  acting 
  upon 
  

   the 
  elements 
  of 
  stress, 
  the 
  determination 
  of 
  which 
  follows 
  

   from 
  the 
  laws 
  of 
  their 
  resolution 
  ; 
  that 
  is, 
  these 
  operators 
  

   are 
  based 
  on 
  mechanical 
  considerations. 
  

  

  (2) 
  If 
  we 
  now 
  consider 
  any 
  arbitrary 
  vector 
  (which 
  we 
  

   shall 
  speak 
  of 
  conveniently 
  as 
  a 
  virtual 
  or 
  potential 
  dis- 
  

   placement), 
  we 
  may 
  look 
  upon 
  the 
  nine 
  first 
  differential 
  

   coefficients 
  of 
  its 
  components 
  as 
  being 
  replaced 
  by 
  the 
  nine 
  

   elements 
  of 
  virtual 
  or 
  potential 
  strain 
  (including 
  among 
  them 
  

   the 
  three 
  rotations). 
  

  

  These 
  nine 
  virtual 
  strains 
  may 
  now 
  be 
  affected 
  by 
  the 
  

   same 
  infinitesimal 
  rotation 
  of 
  the 
  axes 
  as 
  is 
  employed 
  

   in 
  (1), 
  and 
  the 
  three 
  consequent 
  differential-operators* 
  

   acting 
  upon 
  the 
  elements 
  of 
  strain 
  may 
  be 
  deduced 
  ; 
  in 
  this 
  

   case 
  on 
  geometrical 
  grounds. 
  It 
  will 
  be 
  noticed 
  that 
  the 
  

   forms 
  of 
  the 
  two 
  sets 
  of 
  operators 
  are 
  similar, 
  and 
  may 
  easily 
  

   be 
  made 
  identical. 
  

  

  (3) 
  I 
  shall 
  now 
  introduce 
  the 
  fundamental 
  assumption 
  on 
  

   which 
  the 
  theory 
  is 
  based 
  ; 
  that 
  the 
  elements 
  of 
  a 
  material 
  

   stress 
  are 
  functions 
  of 
  the 
  first 
  differential 
  coefficients 
  of 
  the 
  

   components 
  of 
  some 
  vector 
  quantity 
  ; 
  in 
  other 
  words, 
  functions 
  

   of 
  some 
  set 
  of 
  nine 
  virtual 
  strains. 
  

  

  (I) 
  If 
  we 
  now 
  turn 
  to 
  the 
  two 
  sets 
  of 
  differential-operators 
  

   concerning 
  elements 
  of 
  stress 
  and 
  of 
  virtual 
  strain, 
  we 
  are 
  

   able, 
  in 
  consequence 
  of 
  assumption 
  (3), 
  to 
  express 
  each 
  

   element 
  of 
  stress 
  in 
  terms 
  of 
  the 
  nine 
  elements 
  of 
  virtual 
  

   strain; 
  or, 
  conversely, 
  each 
  element 
  of 
  strain 
  in 
  terms 
  of 
  the 
  

   nine 
  elements 
  of 
  stress, 
  by 
  means 
  of 
  sets 
  of 
  simple 
  partial 
  

   differential 
  equations. 
  

  

  In 
  obtaining 
  these 
  solutions 
  constants 
  will 
  be 
  regarded 
  as 
  

   uniform 
  and 
  isotropic, 
  and, 
  consequently, 
  we 
  shall 
  exclude, 
  

   among 
  other 
  things, 
  the 
  possibility 
  of 
  a 
  crystalline 
  structure. 
  

   AVe 
  shall 
  also 
  suppose 
  the 
  body 
  not 
  to 
  be 
  in 
  a 
  slate 
  of 
  con- 
  

   straint, 
  or 
  otherwise 
  that 
  that 
  state 
  has 
  been 
  eased. 
  I 
  shall, 
  

   further, 
  limit 
  the 
  solutions 
  to 
  relations 
  of 
  a 
  linear 
  form. 
  

   From 
  this 
  it 
  will 
  follow 
  that 
  the 
  relations 
  between 
  the 
  

   elements 
  of 
  stress 
  and 
  of 
  virtual 
  strain 
  agree, 
  in 
  form, 
  with 
  

   the 
  corresponding 
  relations 
  familiar 
  to 
  us 
  in 
  the 
  Theory 
  of 
  

   Elasticity. 
  

  

  (5) 
  Supposing 
  the 
  elements 
  of 
  virtual 
  strain 
  to 
  be 
  expressed 
  

  

  * 
  Manchester 
  Memoirs, 
  Xo. 
  3, 
  vol. 
  ix. 
  (1895), 
  No. 
  1, 
  vol. 
  lxii. 
  (1917). 
  

  

  