Tham khảo tài liệu 'nonlinear finite elements for continua and structures part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | T. Belytschko Chapter 2 December 16 1998 undeformed initial configuration cannot be established and no counterpart of is available. Box. . Governing Equations for Eulerian Formulation continuity equation mass conservation p d p 0 d t dx momentum equation dv dv A 3 Aơ _fdv dv A dơ Hi vB pAb PU F Fa pb strain measure rate-of-deformation Dx v x constitutive equation in rate form Dơ Dt ơ t x t ơ x x t v x 1 S Dx X t ơ X t etc. t t energy conservation equation same as before The governing equations are summarized in Box . In comparison with the updated Lagrangian formulation we have just discussed four points are noteworthy 1. The mass conservation equation is now written as a partial differential equation the form used with Lagrangian meshes is not applicable because it applies only to material points. 2. The material time derivative for the velocity in the momentum equation has been written out in terms of the spatial time derivative and transport term. 3. The constitutive equation is expressed in rate form the total form cannot be used since the stress and rate of deformation are functions of material coordinates in a history-dependent material. 4. The boundary conditions are now imposed on spatial points which do not move with time. The continuity equation has been written as a partial differential equation because it is not possible to obtain an integral form such as Eq. when the density is a function of spatial coordinates. Therefore the continuity equation must be treated as a separate partial differential equation although there are approximations which enable the continuity equation to be omitted when the density changes little as for a liquid or solid these are discussed in Chapter 7. The constitutive equation needs to be expressed in terms of material coordinates for history-dependent materials so it is treated in rate form in this .
