tailieunhanh - Advanced mechanics of materials (1993-John Willey) Episode 2

Tham khảo tài liệu 'advanced mechanics of materials (1993-john willey) episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 42 2 THEORIES OF STRESS AND STRAIN since for the principal axes ơxx ơì ƠYY ơ2 ƠZZ ơ3ĩ and ƠXY ƠYZ ơzx 0- See Eqs. . It follows that since are invariants under rotation of axes we may refer Eqs. to arbitrary x y z axes by replacing Il9 I2 13 by their general forms as given by Eqs. . Thus for arbitrary x y z axes roct 3 xx T ơyy T zz 9ĩổct - Vyy 2 Kx - zz 2 - zz 2 6ữỉy 6ơỉz 6ơịz The octahedral normal and shear stresses play a role in certain failure criteria Sec. Mean and Deviator Stress Experiments indicate that yielding and plastic deformation of many metals are essentially independent of the mean normal stress ơm where __ xx Vyy ơzz __ gj ơ2 3 _ 1 r ơm 3 3 371 Comparing Eqs. and we note that the mean normal stresss ơm is equal to ơoct. Most plasticity theories postulate that plastic behavior of materials is related primarily to that part of the stress tensor that is independent of ơm. Therefore the stress array Eq. is rewritten in the following form T Tm Td where T symbolically represents the stress array Eq. and and The array Tm is called the mean stress tensor. The array Td is called the deviator stress tensor since it is a measure of the deviation of the state of stress from a hydrostatic stress state that is from the state of stress that exists in an ideal frictionless fluid . Let x y z be the transformed axes that are in the principal stress directions. Then xx 1 Gyy 2 ZZ @3 xy XZ Gyz 0 TRANSFORMATION OF STRESS PRINCIPAL STRESSES 43 and Eq. is simplified accordingly. Application of Eqs. to Eq. yields the following stress invariants for Td Ji 0 J Ỉ2 ịỉỉ i 0i - O2 2 ơ2 - 73 2 ff3 - ffj2 J3 3 1 2 27 A 2ơi - Ơ2 - ơ3 2ơ2 - ơ3 - 7i 2 73 - Ơ - 72 The principal directions for Td are the same as those for T. It can be shown that since Ji 0 Td represents a state of pure shear. The principal values of the deviator tensor Td are _ _ 2oj - ff2 - ơ3 ơi - ơ3 ơj - ơ2 Ơ1 - ơ- .