TAILIEUCHUNG - Mechanics Of Saouma Part 7

Tham khảo tài liệu 'mechanics of saouma part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | in Relations in Generalized Elasticity 7 29 Yet we have the elementary relations in terms engineering constants E Young s modulus and V Poisson s ratio then it follows that 11 V Ơ E 22 _ 33 11 11 1 X X X 7 31 E x 3X 2x 2 X x VE E X 1 V 1 2v X G 2 1 V 732 30 Similarly in the case of pure shear in the x1x3 and x2 x3 planes we have Ơ21 ơ12 T all other ơij 0 2 12 T G and the X is equal to the shear modulus G. 31 Hooke s law for isotropic material in terms of engineering constants becomes ơij ij E C t V 1 V ij 1-2V dij kk 1 Vơij - V ơkk or e EE EV or TT y t-2V 1 V V E a E 32 When the strain equation is expanded in 3D cartesian coordinates it would yield xx 1 V V 0 0 0 ơxx yy V 1 V 0 0 0 ơyy 1 V V 10 0 0 ơzz Xxy 2 xy 1 E 0 0 0 1 V 0 0 Txy Xyz 2 yz 0 0 0 0 1 V 0 Tyz Yzx 2 zx 0 0 0 0 0 1 V Tzx 33 If we invert this equation we obtain 1 V V .z Vyy zz Txy Tyz Tzx E 1 v 1 2v 1 V 1 V 0 0 1 xx zz Yxy 2 xy 1 0 G 0 1 0 Xyz 2 yz 0 0 1 Yzx 2 zx V 0 Bulk s Modulus Volumetric and Deviatoric Strains 34 We can express the trace of the stress Ẹ in terms of the volumetric strain I From Eq. ơii XSii kk 2x ii 3X 2x ii 3K ii Victor Saouma Mechanics of Materials II CONSTITUTIVE EQUATIONS Part I Engineering Approach 8 Dian or _ K X 2 p 35 We can provide a complement to the volumetric part of the constitutive equations by substracting the trace of the stress from the stress tensor hence we define the deviatoric stress and strains as as a1 a - - tr a I f - tr e I and the corresponding constitutive relation will be a KeI 2p z p I Ẳ a 3K 2p where p 1 tr a is the pressure and a a pI is the stress deviator. Restriction Imposed on the Isotropic Elastic Moduli 36 We can rewrite Eq. as dW Tij dEj but since dW is a scalar invariant energy it can be expressed in terms of volumetric hydrostatic and deviatoric components as dW pde ơ ij dEij substituting p Ke and ơj .

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