2

Non-linear Finite Element Analysis
of Solids and Structures

~~ ~

Volume 2: Advanced Topics

To

Kiki, Lou, Max, ArabeIIa
Gideon, Gavin, Rosie and Lucy

Non-linear Finite Element Analysis
of Solids and Structures

Volume 2: ADVANCED TOPICS

M.A. Crisfield

ImperialCollegeof Science,
Technologyand Medicine, London, UK

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Contents


Preface xiii

10 More continuum mechanics 1


10.1 Relationshipsbetween some strain measuresand the structures

10.2 Large strains and the Jaumann rate

10.3 Hyperelasticity

10.4 The Truesdell rate

10.5 Conjugate stress and strain measureswith emphasis on isotropic

conditions 10

13

10.6 Further work on conjugate stress and strain measures 14


10.6.1 Relationshipbetweeni: and U 15

10.6.2 Relationshipbetweenthe Bio! stress, B and the Kirchhoff stress, T
10.6.3 Relationshipbetween U, the i ’ s and the spin of the Lagrangian 15

triad, W, 16

10.6.4 Relationshipbetween €, the A’s and the spin, W, 17

10.6.5 17

10.6.6 Relationshipbetween6,the 2’s and the spin, W, 17

18

Relationshipbetween €and E 19

10.6.6.1 Specific strain measures 20

22

10.6.7 Conjugatestress measures 24


10.7 Using log,V with isotropy

10.8 Other stress rates and objectivity

10.9 Special notation

10.10 References

11 Non-orthogonalcoordinates and CO-and contravarianttensor 26

components

11.1 Non-orthogonalcoordinates 26


11.2 Transformingthe componentsof a vector (first-ordertensor) to a new set of 28

base vectors 30


11.3 Second-ordertensors in non-orthogonalcoordinates 30

11.4 Transformingthe componentsof a second-ordertensor to a new set of 31

32

base vectors
11.5 The metric tensor
11.6 Work terms and the trace operation