Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

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(c) Deformed shapes of various times Fig. 4.26 Continued. t = 900 s t = 2400 s t = 4280 s 4.9.4 Elastic springbackand viscoelastic ¯uids Non-newtonian ¯ows ± metal and polymer forming 131 In Sec. 4.9.1 we have argued that omission of elastic e€ects in problems of metal or plastic forming can be justi®ed because of the small amount of elastic straining. This is undoubtedly true when we wish to consider the forces necessary to initiate large deformations and to follow these through. There are however a number of problems in which the inclusion of elasticity is important. One such problem is for instance that of `spring-back' occurring particularly in metal forming of complex
132 Incompressible laminar ¯ow shapes. Here it is important to determine the amount of elastic recovery which may occur after removing the forming loads. Some possible suggestions for the treatment of such e€ects have been presented in reference 116 as early as 1984. However since that time much attention has been focused on the ¯ow of viscoelastic ¯uids which is relevant to the above problem as well to the problem of transportation of ¯uids such as synthetic rubbers, etc. The procedures used in the study of such problems are quite complex and belong to the subject of numerical rheology. In this context the work of M. Crochet, K. Walters, P. Townsend and M.F. Webster 154ÿ163 is notable. Obviously the subject is beyond the space limitations of the present book but an essential treatment can be found from the ideas discussed in this volume. 4.10 Direct displacement approach to transient metal forming Explicit dynamic codes using quadrilateral or hexahedral elements have achieved considerable success in modelling short-duration impact phenomena with plastic deformation. The prototypes of ®nite element codes of this type are DYNA2d and DYNA3d developed at Lawrence Livermore National Laboratory. 164;165 For problems of relatively slow metal forming, such codes present some di culties as in general the time step is governed by the elastic compressibility of the metal and a vast number of time steps would be necessary to cover a realistic metal forming problem. Nevertheless much use has been made of such codes in slow metal forming processes by the simple expedient of increasing the density of the material by many orders of magnitude. This is one of the drawbacks of using such codes whose description rightly belongs to the matter discussed in Volume 2 of this book. However, a further drawback is the lack of triangular or tetrahedral elements of a linear kind which could compete with linear quadrilaterals or hexahedra currently used and permit an easier introduction of adaptive re®nement. It is well known that linear triangles or tetrahedra in a pure displacement (velocity) formulation will lock for incompressible or nearly incompressible materials. However we have already found that the CBS algorithm will avoid such locking when the same (linear) interpolation is used for both velocities and pressure. 166 It is therefore possible to proceed in each step by solving a simple Stokes problem to evaluate the lagrangian velocity increment. We have described the use of such velocity formulation in the previous chapter. The update of the displacement allows new stresses to be evaluated by an appropriate plasticity law and the method can be used without di culty as shown by Zienkiewicz et al. 166 In Fig. 4.27, we show a comparison between various methods of solving the impact of a circular bar made of an elastoplastic metal using an axisymmetric formulation. In this ®gure we show the results of a linear triangle displacement (Fig. 4.27b) form with a single integrating point for each element and a similar study again using displacement linear quadrilaterals (Fig. 4.27c) also with a single integration point. This ®gure also shows the same triangles and quadrilaterals solved using the CBS algorithm and very accurate ®nal results (Fig. 4.27d and e).
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The Finite Element Method Fifth edi
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The Finite Element Method Fifth edi
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Dedication This book is dedicated t
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3.7 The performance of two- and sin
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Volume 2: Solid and structural mech
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xiv Preface to Volume 3 this algori
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2 Introduction and the equations of
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10 Introduction and the equations o
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12 Introduction and the equations o
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14 Convection dominated problems We
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16 Convection dominated problems wh
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18 Convection dominated problems ch
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20 Convection dominated problems i.
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22 Convection dominated problems 2
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24 Convection dominated problems 2.
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26 Convection dominated problems ar
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28 Convection dominated problems Th
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30 Convection dominated problems co
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32 Convection dominated problems 2.
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34 Convection dominated problems ha
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36 Convection dominated problems an
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38 Convection dominated problems ac
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40 Convection dominated problems An
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42 Convection dominated problems U
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44 Convection dominated problems (a
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46 Convection dominated problems sp
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48 Convection dominated problems ag
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50 Convection dominated problems To
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52 Convection dominated problems C
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56 Convection dominated problems an
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60 Convection dominated problems Ma
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62 Convection dominated problems 47
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3 A general algorithm for compressi
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66 A general algorithm for compress
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Page 93 and 94: 80 A general algorithm for compress
Page 105 and 106: 92 Incompressible laminar ¯ow Cons
Page 107 and 108: 94 Incompressible laminar ¯ow and,
Page 109 and 110: 96 Incompressible laminar ¯ow V/V
Page 111 and 112: 98 Incompressible laminar ¯ow The
Page 113 and 114: 100 Incompressible laminar ¯ow 1.5
Page 115 and 116: 102 Incompressible laminar ¯ow p 1
Page 117 and 118: 104 Incompressible laminar ¯ow �
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Page 129 and 130: 116 Incompressible laminar ¯ow (b)
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Page 139 and 140: 126 Incompressible laminar ¯ow U U
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Page 149 and 150: 136 Incompressible laminar ¯ow 28.
Page 151 and 152: 138 Incompressible laminar ¯ow 71.
Page 153 and 154: 140 Incompressible laminar ¯ow 114
Page 155 and 156: 142 Incompressible laminar ¯ow 153
Page 157 and 158: 144 Free surfaces, buoyancy and tur
Page 183 and 184: 170 Compressible high-speed gas ¯o
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182 Compressible high-speed gas ¯o
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7 Shallow-water problems 7.1 Introd
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220 Shallow-water problems reduced
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222 Shallow-water problems Approxim
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224 Shallow-water problems These si
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226 Shallow-water problems h = 2 η
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228 Shallow-water problems Surface
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230 Shallow-water problems FL: 578
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232 Shallow-water problems B Fig. 7
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234 Shallow-water problems Time = 0
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236 Shallow-water problems Fig. 7.1
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238 Shallow-water problems where no
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240 Shallow-water problems 14. T.D.
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8 Waves Peter Bettess 8.1 Introduct
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244 Waves the familiar form where
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246 Waves of 10 nodes or thereabout
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248 Waves and the Astley wave envel
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250 Waves agreement with the analyt
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252 Waves Fig. 8.3 General wave dom
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254 Waves Relative errors (%) 10 8
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256 Waves (The indirect boundary in
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258 Waves where m is the number of
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260 Waves Fig. 17.6 of Volume 1. La
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262 Waves to using such coordinate
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264 Waves r/a 5 4 3 2 1 0 0 2 4 6 8
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266 Waves 8.16 Three-dimensional ef
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268 Waves wave elevations in shallo
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270 Waves integrals. Preliminary re
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272 Waves 39. R.J. Astley. FE mode
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9 Computer implementation of the CB
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276 Computer implementation of the
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292 Appendix A Again substituting t
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294 Appendix B As usual the domain
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296 Appendix B 5. While the piecewi
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Appendix C Edge-based ®nite elemen
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Appendix D Multigrid methods It is
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Appendix E Boundary layer±inviscid
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304 Appendix E In Fig. E.2, Cp is t
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Author index Page numbers in bold r
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Gerdes, K. 259, 264, 265, 272 Ghia,
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Nakos, D.E. 146, 165 Narayana, P.A.
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Wu, J. 67, 90; 102, 108, 112, 137;
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316 Subject index Blurring of the s
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318 Subject index Convected coordin
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320 Subject index Elements ± cont.
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322 Subject index Flows: buoyancy d
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324 Subject index Incompressible St
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326 Subject index Metals: hot, 120
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328 Subject index Prescribed tracti
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330 Subject index Shock interaction
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332 Subject index Theorem ± cont.
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334 Subject index Wave ± cont. ste
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