tailieunhanh - Nonlinear Finite Elements for Continua and Structures Part 7

Tham khảo tài liệu 'nonlinear finite elements for continua and structures part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | T. Belytschko Lagrangian Meshes December 16 1998 X X2 l0 where l 0 is the initial length of the element. In this example the coordinates X Y are used in a somewhat different sense than before it is no longer true that x t 0 X. However the definition used here corresponds to a rotation and translation of x t 0 . Since neither rotation nor translation effects E or any strain measure this choice of an X Y coordinate system is perfectly acceptable. We could have used the element coordinates as material coordinates but this complicates the definition of physical strain components. The spatial coordinates are given in terms of the element coordinates by x x1 1 - x2 y y1 1 - y x l_Tx 1 x21 1 lyj 1 Ly y2 J 1 or x t x I t NI where N 1 - 1-X X L 0 l0 The B0 matrix as defined in is given by K d V n y- -1 1 E4 ƠX ƠX 10 where Eq. has been used to give 1 -7- . The deformation gradient is given by dX l0 d F x I B T x x2 1 J- 1l t f y y2 - 101 1J j- x 2- x1 y2 -y1 jr x21 y21 The deformation gradient F is not a square matrix for the rod since there are two space dimensions but only one independent variable describes the motion . The only nonzero stress is along the axis of the rod. To take advantage of this we use the nodal force formula in terms of the PK2 stress since 511 is the only nonzero component of this stress. For the nominal stress T11 is not the only nonzero component. The X axis as defined here is corotational with the axis of the rod so 511 is always the stress component along the axis of the rod. Substituting and into Eq. then gives the following expression for the internal nodal forces 4-72 T. Belytschko Lagrangian Meshes December 16 1998 th 1BsF dfl jN xSFdfl J -1 1 A x21 y 2 dfl fl0 fl0 fl 0 L 1J 0 Since the deformation is constant in the element we can assume the integrand is constant so multiplying the integrand by the volume A010 we have f1x f1y int _ A0 - 1 - x21 - y21 .

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