TAILIEUCHUNG - Mechanics.Of.Materials.Saouma Episode 13

Tham khảo tài liệu ' episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5 Plastic Yield Conditions Classical Models yiait 12 In terms of the principal stresses those invariants can be simplified into 11 Ơ1 Ơ2 Ơ3 12 ơ1ơ2 Ơ2Ơ3 Ơ3Ơ1 13 Ơ1Ơ2Ơ3 13 Similarly J1 S1 S2 S3 J2 S1S2 S2S3 S3S1 J3 S1S2S3 Physical Interpretations of Stress Invariants 14 If we consider a plane which makes equal angles with respect to each of the principal-stress directions n plane or octahedral plane the normal to this plane is given by toct Ơ1 Ơ2 Ơ3 The vector of traction on this plane is and the normal component of the stress on the octahedral plane is given by or _Ơ1 Ơ2 Ơ3 Ơoct toct n 3 I1 Ơhyd Ơoct 311 3 15 Finally the octahedral shear stress is obtained from 222 2 2 2 Ơ1 Ơ2 Ơ3 Toct toct Ơoct 3 3 3 ơi Ơ2 Ơ3 2 9 Upon algebraic manipulation it can be shown that 9ró2ct ơ1 Ơ2 2 ơ2 Ơ3 2 ơ1 Ơ3 2 6J2 or and finally the direction of the octahedral shear stress is given by cos 39 V2 3 Victor Saouma Mechanics of Materials II 3D PLASTICITY _ laiL 16 The elastic strain energy total per unit volume can be decomposed into two parts U U1 U2 where U1 12 Dilational energy E U2 T- J2 Distortional energy E Geometric Representation of Stress States Adapted from Chen and Zhang 1990 17 Using the three principal stresses Ơ1 Ơ2 and Ơ3 as the coordinates a three-dimensional stress space can be constructed. This stress representation is known as the Haigh-Westergaard stress space Fig. . 18 The decomposition of a stress state into a hydrostatic pỗịj and deviatoric Sj stress components can be geometrically represented in this space. Considering an arbitrary stress state OP starting from 0 0 0 0 and ending at P ơ1 Ơ2 Ơ3 the vector OP can be decomposed into two components ON and NP. The former is along the direction of the unit vector 1 3 1 1 3 1 1 3 and NP1ON. 19 Vector ON represents the hydrostatic component of the stress state and axis OỆ is called the

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