Introduction to Continuum Mechanics 3 Episode 4

Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 106 Kinematics of a Continuum Thus for small deformation e - Ex Eq Et Eỵỵ Eos E-ỵT This unit volume change is known as dilatation. Note also that e Eịị áivu oAf In cylindrical coordinates du idue u 3 e dr rM r dz In spherical coordinates dur due 2ur 1 du f . 0COtớ n 107 X e - fr 7 r The Infinitesimal Rotation Tensor Equation . dx JX Vu dX can be written dx dX E Q dX where 1 the antisymmetric part of Vu is known as the infinitesimal rotation tensor. We see that the change of direction for JX in general comes from two sources the infinitesimal deformation tensor E and the infinitesimal rotation tensor Q. However for any dX which is in the direction of an eigenvector of E there is no change of direction due to E only that due to Q. Therefore the tensor Í2 represents the infinitesimal rotation of the triad of the eigenvectors of E. It can be described by a vector t4 in the sense that t4 xdX QdX where see Section t4 Q32el 13e2 21e3 Thus Q32 Q13 Q21 are the infinitesimal angles of rotation about e15 e2 and e3-axes of the triad of material elements which are in the principal direction of E. Time Rate of Change of a Material Element Let us consider a material element dx emanating from a material point X located at X at time t. We wish to compute D Dt dx the rate of change of length and direction of the material element dx. From X - x X t we have Time Rate of Change of a Material Element 107 c x x X JXự -x Xự i Taking the material derivative of Eq. i we obtain ii Now D Dt x X í vỢíự V0ự where v X and v x are the material and the spatial description of the velocity of the particle X therefore Eq. ii becomes dx v7x dXự -í Xự ỳx diự -ỹ Kt iii Thus from the definition see Section of the gradient of a vector function we have l j Vxv JX 7 and l ldx vxv dx In Eq. the subscript X in Vxv emphasizes that Vxv is the gradient of the material description of the velocity field V and in .