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Introduction to Continuum Mechanics 3 Episode 5

Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 146 Change of Volume due to Deformation Fe3 dA dAo Fe3 Fe3 X Fe2 iii Recall that for any vectors a b and c a bx c determinant whose rows are components of a b and c. Therefore Fe3 Fe1 X Fe2 det F iv Eq. iii becomes Fe3 dAn dAo det F v Using the definition of transpose of a tensor Eqs. ii become 1 FTn e2 Frn 0 vi and Eq. v becomes T dAo _ vii e3 F Tn r det F Thus Frn is in the direction of e3 so that _T dAn _ _ vii Frn - detF e3 1 Therefore dA n dAo det F F 1 7 e3 3.28.4 Equation 3.28.4 states that the deformed area has a normal in the direction of F-1 7 e3 and with a magnitude given by dA dAo det F I F-1 r e31 3.28.5 In deriving Eq. 3.28.4 we have chosen the initial area to be the rectangular area formed by the Cartesian base vectors ex and e2 it can be shown that the formula remains valid for any material area except that e3 be replaced by the normal vector of the undeformed area no. That is in general dA n dAo det F F-1 Tno 3.28.6 3.29 Change of Volume due to Deformation Consider three material elements i x 1 dS JX 2 JS2e2 and JX 3 JS3e3 Kinematics of a Continuum 147 emanating from X. The rectangular volume formed by dX dx and dX at the reference time to is given by dVo dS1dS2dS3 3.29.1 At time t dX deforms into diỉ1 Mố1 dxf deforms into dx 2 FdXpi and dX deforms into dĩỉ FdX and the volume is dV FdX 1 FdX 2 X FdX 3 dSr dS2 ds3 Fej Fe2 X Fe3 dro Fei -Fe2xFe3 3.29.2 That is dr detF dFo 3.29.3 Since c FrF and B FFr therefore detc detB detF 2 3.29.4 Thus Eq. 3.29.3 can also be written as dV VdetC dVo VdetB dVo 3.29.5 We note that for an incompressible material dV dVo so that detF detc detB 1 3.29.6 We note also that due to Eq. 3.29.3 the conservation of mass equation can be written as p 3.29.7 Example 3.29.1 Consider the deformation given by 1 1X1 2 - 3X3 1 3 k2X2 i a Find the deformed volume of the unit cube shown in Fig. 3.14. b Find the deformed area of OABC. c Find the rotation tensor and the axial vector of the antisymmetric part of the rotation tensor. .