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Tham khảo tài liệu 'advanced engineering dynamics 2010 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 174 Waves in a three-dimensional elastic solid The rigid body displacement of P relative to p is d roI ilxdr 7.7 or in vector-matrix terms see Appendix 1 assuming the same basis on both sides of the equation du rot Q x dr where 7.8 Q x -a 0 Qr -ạ 0 7.9 0 -Q From Fig. 7.2 it is seen that the rotation about the z axis is _ 1 du dux z T dx dy Ị Thus the total rotation Í2 is given by 7.10 duy duz dz dy 1 dux du. 2 dz dx o dux duy dy dx 0 d dz d dy So in short matrix notation Q ị V x d dz 0 d dx 7.11 7.12 J_ 2 Fig. 7.2 Strain 175 or in indicial notation Q ị - . . 7.13 The elastic displacement will be the total displacement less that due to rotation so d eiastic du tota du roe 7.14 elastic V u T T - j V X M Xj dr 7.15 From Appendix 1 or by direct multiplication we have that V x x u V T - V m t Therefore 7.16 d u 1 V M t T V u T J dr 7.17 The first term in the braces is given in equations 7.3 and 7.5 and the second term is its transpose so half the sum of the two is a symmetric matrix which is the strain matrix e . Hence dw strain e dr where e V u t T V M t J 7.18 7.19 Figure 7.2 shows the geometric definition of shear strain. In indicial notation 7.20 7.21 Note that for i j e y yYy that is half the conventional shear strain whilst is the usual tensile strain. 176 Waves in a three-dimensional elastic solid 7.3 Stress Figure 7.3 shows a tetrahedron with forces acting on all faces. The oblique face has an area A and is acted on by a force Fn FJ Xy FJt e T Fn 7.22 The area on which it acts is e T A 7.23 where A A.y 4r Ạ AZỸ The force on the face whose outward normal is in the X direction is Fxx Ợr.r Fx Êy Ly Ax 7.24 Xxz where ơ is the conventional tensile stress and and TX2 are the conventional shear stresses. Similarly for the forces in the other two directions. The resultant force on the element is Fx Fv F. - Fn p w volume Using equation 7.24 we have Ợrx ty-r xy Ợỵr xz tyz Fn p u volume 7.25 The right hand side is proportional to length3 whilst the left hand side is .
