One obtains then by replacing: 7 W d元 listorsion -去栏+听+-o生+ .-1+y(g+c+C E 3 then: aw 1 (14.1) d元 distorsion 3 One can rewrite as following the quantity in brackets: 影d++品-oa-ou-no} 3(c+i+m-3c0i+m+omo} d 6G(+Ou+om)2... distorsion (14.2) …-3(C10+00m+0o)} Remarks:If one denotes as n the direction making the same angle with each of the principal directions (following Figure 14.1),one observes on the face with the normal n a stress o such that:=(n) that is: Gl3 {o}= o/3 cm/3 which can be decomposed as: ■A normal stress:on=言.7 then: O= 1+Ou+Ou 3 The above consists of the average or isotropic part of the stress tensor.2 ■A shear stress: t=-on Recall the expressionthat constitutes the first scalar invariant of the stress tensor. 2003 by CRC Press LLCOne obtains then by replacing: (14.1) One can rewrite as following the quantity in brackets: (14.2) Remarks: If one denotes as the direction making the same angle with each of the principal directions (following Figure 14.1), one observes on the face with the normal , a stress such that: = S( ) that is: which can be decomposed as:
A normal stress: sn = ◊ The above consists of the average or isotropic part of the stress tensor.2
A shear stress: 2 Recall the expression sI + sII + sIII that constitutes the first scalar invariant of the stress tensor. dW dV-------- Ë ¯ Ê ˆ distorsion 1 2 -- 1 + n E ------------ sI 2 sII 2 sIII 2 ( ) + + n E -- sI + + sII sIII ( ) – 2º Ó Ì Ï = º 1 + n E ------------ sI + + sII sIII ( )2 3 -------------------------------------- n E -- sI + + sII sIII ( )2 – + ˛ ˝ ¸ then: dW dV-------- Ë ¯ Ê ˆ distorsion 1 4G ------- sI 2 sII 2 sIII 2 ( ) + + sI + + sII sIII ( )2 3 – -------------------------------------- Ó ˛ Ì ˝ Ï ¸ = 2 3 -- sI 2 sII 2 sIII 2 { } + + – sIsII – sIIsIII – sIIIsI 2 3 -- sI + + sII sIII ( )2 3 sIsII + + sIIsIII sIIIsI { } – ( ) dW dV-------- Ë ¯ Ê ˆ distorsion 1 6G ------- sI + + sII sIII ( ) = { 2º º 3 sIsII + sIIsIII + sIIIsI – ( ) } n n s s n { } s sI/ 3 sII/ 3 sIII Ó / 3˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = s n then: sn sI + + sII sIII 3 = ------------------------------- t s2 sn 2 = – TX846_Frame_C14 Page 275 Monday, November 18, 2002 12:30 PM © 2003 by CRC Press LLC