Figure 14.1 Principal Directions for the Stress Tensor EEm are the principal strains that one can express as functions of stresses using the constitutive Equation 10.1 as: e=l'z-nae(②l E This leads to: )=兰所+6+-a+a+a (Note that (dwldv)represents energy per unit volume). The total elastic deformation above is due to the dilatation and distortion of the material.The Von Mises criterion postulates that the material resists to a state of isotropic (or spherical)stress,but will plastify when the distortion energy per unit volume reaches a critical value.One notes that: $)-(供（傺） spherical stress The isotropic portion of the stress state here is written as:It creates a state of isotropic dilatation (Equation 10.1): e=++四） 总c+om+cmd then: d aw 2003 by CRC Press LLCeI eII eIII are the principal strains that one can express as functions of stresses using the constitutive Equation 10.1 as: This leads to: (Note that (dW/dV) represents energy per unit volume). The total elastic deformation above is due to the dilatation and distortion of the material. The Von Mises criterion postulates that the material resists to a state of isotropic (or spherical) stress, but will plastify when the distortion energy per unit volume reaches a critical value. One notes that: The isotropic portion of the stress state here is written as: . It creates a state of isotropic dilatation (Equation 10.1): then: Figure 14.1 Principal Directions for the Stress Tensor e 1 + n E ------------S n E = – --trace( ) S I dW dV-------- Ë ¯ Ê ˆ total 1 2 -- 1 + n E ------------ sI 2 sII 2 sIII 2 ( ) + + n E -- sI + + sII sIII ( )2 – Ó ˛ Ì ˝ Ï ¸ = dW dV-------- Ë ¯ Ê ˆ distorsion dW dV-------- Ë ¯ Ê ˆ total dW dV-------- Ë ¯ Ê ˆ spherical stress = – sI + + sII sIII 3 -------------------------------- e 1 + n E ------------ sI + + sII sIII 3 ------------------------------- Ë ¯ Ê ˆ = n E -- sI + + sII sIII – ( ) dW dV-------- Ë ¯ Ê ˆ spherical stress 1 2 -- 3 sI + + sII sIII 3 ------------------------------- Ë ¯ Ê ˆ ¥ ¥ e Ó ˛ Ì ˝ Ï ¸ = dW dV-------- Ë ¯ Ê ˆ spherical stress 1 2 -- 1 + n E ------------ sI + + sII sIII ( )2 3 -------------------------------------- n E -- sI + + sII sIII ( )2 – Ó ˛ Ì ˝ Ï ¸ = TX846_Frame_C14 Page 274 Monday, November 18, 2002 12:30 PM © 2003 by CRC Press LLC