then: which can be compared with Equation 14.1.Thus, aw distorsion The shear stress t also appears as the shear characteristic of the distortion energy. One recognizes in Equation 14.2 the presence of the first and second scalar invariants of the stress tensor independent of the coordinate system.In coordinate axes other than the principal directions,the second invariant can be written as: (o1102-t2)+(o203-t23)+(03011-ti) One then has for any coordinate system: distorsion …-3(o1102-t2)+(o203-t)+(03011-ti)} then: aw d元 =元o-a+(aa- …+(o8-0)+6(t2+3+1)} The elastic domain (where the distortion energy is below a certain critical value) can then be characterized by the condition: a{(c11-022)2+(022-03)+(03-011)2. …+6(2+t3+亏i)}<1(14.3) To determine the constant,a uniaxial test is sufficient;in effect if one denotes by o.the elastic limit obtained from a tension-compression test,one has: a2o6=1 then: a=112o2 2003 by CRC Press LLCthen: which can be compared with Equation 14.1. Thus, The shear stress t also appears as the shear characteristic of the distortion energy.
One recognizes in Equation 14.2 the presence of the first and second scalar invariants of the stress tensor independent of the coordinate system. In coordinate axes other than the principal directions, the second invariant can be written as: One then has for any coordinate system: then: The elastic domain (where the distortion energy is below a certain critical value) can then be characterized by the condition: (14.3) To determine the constant, a uniaxial test is sufficient; in effect if one denotes by se the elastic limit obtained from a tension–compression test, one has: then: t 2 1 3 -- sI 2 sII 2 sIII 2 sI + + sII sIII 3 ------------------------------- Ë ¯ Ê ˆ 2 + + – Ó ˛ Ì ˝ Ï ¸ = dW dV-------- Ë ¯ Ê ˆ distorsion 1 2G ------- 3 2 --t 2 Ë ¯ Ê ˆ = s11s22 t 12 2 ( ) – s22s33 t 23 2 ( ) – s33s11 t 31 2 + + ( ) – dW dV-------- Ë ¯ Ê ˆ distorsion 1 6G ------- s11 + + s22 s33 ( ) = { 2º º 3 s11s22 t 12 2 ( ) – s22s33 t 23 2 ( ) – s33s11 t 31 2 – ( ) + + ( ) – } dW dV-------- Ë ¯ Ê ˆ distorsion 1 12G ---------- s11 – s22 ( )2 s22 – s33 ( )2 = { + º º s33 – s11 ( )2 6 t 12 2 t 23 2 t 31 2 + } + ( ) + + a s11 – s22 ( )2 s22 – s33 ( )2 s33 – s11 ( ) { + + 2º º 6 t 12 2 t 23 2 t 31 2 + } ( ) + + < 1 a 2se 2 ¥ = 1 a 1/2se 2 = TX846_Frame_C14 Page 276 Monday, November 18, 2002 12:30 PM © 2003 by CRC Press LLC