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Introduction to Non-Linear Finite Element Analysis

Introduction to Non-Linear Finite Element Analysis

E. Hinton
Department of Civil Engineering, Swansea University, SA2 8PP Wales, UK.

R0018, January 1992

ISBN (Hardcopy): 978-1-874376-00-2


This book was commissioned by the Nonlinear Working Group of NAFEMS and is the work of several authors with varying backgrounds and therefore reflects a variety of views. It is intended for readers with a background in linear finite element (FE) analysis who wish to gain some insight into nonlinear FE stress analysis. Before attempting this text, prospective readers should have first studied an introductory text on linear finite element analysis such as the NAFEMS Finite Element Primer. Although it is recognised that many different types of nonlinear problems may be solved using the FE method, our attention is here focused on nonlinear FE stress analysis and in particular static or quasi-static problems (i.e. situations in which inertia forces may be neglected).

This book is not intended to be a deeply theoretical text. Consequently, some of the explanations rely heavily on simple presentations of complex ideas. Rigour and comprehensiveness have been sacrificed for the sake of clarity and ease of explanation. Boxes are used throughout the text and usually include algorithmic details which the more inquisitive reader may require - this part of the text may be skipped by the more casual reader.

Nonlinear FE stress analysis has its own particular language or jargon. Such jargon is explained as it is introduced in this text, but a glossary of terms used is also provided at the end of the book as a convenient, easily accessible, aide-mémoire. An attempt has been made to standardise on notation - this inevitably differs to some degre from the notation used in the NAFEMS Finite Element Primer as our present task is a little more arduous. Various aspects concerning the notation used in this book are discussed in Chapter 1 and a local notation list is placed at the end of each chapter.

The NAFEMS Nonlinear Working Group has over the years commissioned a number of benchmark tests for nonlinear FE stress analysis. This text will make use of some of these tests as they have been shown to be an excellent means of training novices.

The five authors involved in writing the text included three academics from the University College of Swansea: Ernest Hinton (EH), Richard Wood (RDW) and Nenad Bicanic (NB), and from industry Peter White (PSW) of GEC Alsthom and Trevor Hellen (TH) of Nuclear Electric plc. EH acted as editor and Mike Crisfield of Imperial College reviewed the text. It is also acknowledged that all of the members of the Nonlinear Working Group of NAFEMS provided many valuable comments and contributions: Nick Otter (GEC Alsthom - former chairman), Nigel Knowles (Atkins Engineering Sciences - current chairman), Paul Lyons (FEA), Paul Newton (MacNeal-Schwendler), Rick Leggatt (Welding Institute) and Dave Phillips (Glasgow University).


Introduction to Nonlinear Stress Analysis

pp. 1-40

Geometrically Nonlinear Finite Element Analysis

pp. 41-102

Time Independent Material Nonlinearities

pp. 103-163

Time Dependent Material Nonlinearities

pp. 164-236

Incremental-Iterative Solutions

pp. 237-282

Further Complexities

pp. 283-313

Practical Use of Finite Element Software in Nonlinear Analysis

pp. 314-345


Introduction to Nonlinear Stress Analysis

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Time Independent Material Nonlinearities

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Time-Dependent Material Nonlinearities

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Ding J.L. and Findley W.N., '48h multiaxial creep and creep recovery of 2618 aluminium alloy at 200oC', J. App. Mech., 51, 125-132 (1984).

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White, P.S., 'A benchmark problem involving a plate in cyclic creep and plasticity', CEC Report, contract RA1-099-UK, 1990.

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Beere, W. and Crossland, I.G., 'Primary and recoverable creep 20/25 stainless steel', Acta Metall, 30, 1891-1899, 1982.

Rees, D.W.A., 'Representations of creep deformation with a dominant tertiary influence', In 'Creep and Fracture of Engineering Materials and Structures 3', ed. Wilshire, B. and Evans, R.W., 475-490, Institute of Metals, London, 1987.

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Krieg, R.D., Swearingen, J.C. and Rohde, R.W., 'A physically based internal variable model for rate-dependent plasticity', In 'Inelastic Behaviour of Pressure Vessel and Piping Components', PVP-PB-028, ASME, 1978.

Robinson D.N., 'A Unified Creep-Plasticity Model for Structural Metals at High Temperatures', ORNL/TM 5969, Oak Ridge National Laboratory, 1978.

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Mukherjee, S., 'Boundary Element Methods in Creep and Fracture', Applied Science, London, 1982.

Krieg, R.D., 'Numerical integration of some new unified plasticity-creep formulations', Proceedings 4th SMiRT paper M/6, San Francisco, 1977.

'Advances in Constitutive Laws for Engineering Materials, Vol. 1', ed. Fan, J.H. and Murakami, S. International Academic Publishers, Congquing, China, 1989 (dist. Pergamon, London).


Incremental-Iterative Solutions

Matthies, H. and Strang, G., The solution of nonlinear finite element equations, Int. J. Num. Meth. Engng., 14, 1613-1626, 1979.

Broyden, C.G., Quasi-Newton or modification methods, in Numerical solution of systems of nonlinear equations, (G. Byrne and C. Hall eds.), Academic Press, New York, 1973.

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Simons, J.W., Solution strategies for statically loaded nonlinear structures, Ph. D. thesis, Civil Engineering Department, University of California, Berkeley, 1982.

Cope, M.D., Experimental investigations and non-linear numerical analyses of skewed one-way prestressed concrete bridge decks, Ph.D. thesis, Civil Engineering Department, University of Liverpool, 1987.

Crisfield, M.A., Solution procedures for non-linear structural problems, in Recent advances in non-linear computational mechanics, (E. Hinton, D.R.J. Owen and C. Taylor eds.), Pineridge Press, Swansea, 1982.

Sharifi, P. and Popov, E.P., 'Nonlinear buckling analysis of sandwich arches', ASCE, J. Engng. Mech. Div., 97, 1392-1397, 1971.

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Stricklin, J.A., Haisler, W.E. and Von-Reissman, W.A., 'Evaluation of solution procedures for material and/or geometrically nonlinear structural analysis', AIAA J., 11, 292-299, 1973.

Stricklin, J.A., Haisler, W.E. and Key, J.E., 'Displacement incrementation in nonlinear structural analysis by the self correcting method', Int. J. Num. Meth. Engng., 11, 3-10, 1977.

Batoz, J.L. and Dhatt, G., 'Incremental displacement algorithms for non-linear problems', Int. J. Num. Meth. Engng., 14, 1292-1267, 1979.

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Crisfield, M.A., 'A fast incremental/iterative solutions procedure that handles 'snap through' Comp. and Struct., 13, 55-62, 1981.

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Further Complexities

Newmark, N.M., 'A method of computation for structural dynamics', J. Eng. Mech. Div., ASCE, 85, 67-94, 1959.

Owen, D.R.J. and Hinton, E., 'Finite elements in plasticity', Pineridge Press, Swansea, 1979.

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Geradin, M., Hogge, M. and Idelsohn, S., 'Implicit finite element methods', In Computational methods for transient analysis, (ed. T. Belytschko and T.J.R. Hughes), 417-471, Elsevier Science Publisher, Amsterdam, 1983.

Hitchings, D. (ed.), 'Dynamic finite element primer', NAFEMS, Glasgow, 1991.

Zienkiewicz, O.C., 'Coupled problems - a review', Numerical Methods for Coupled Systems, (ed. R. W. Lewis et al), John Wiley, Chichester, 1984.

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Hitchings, D., 'Contact analysis using finite elements', Institute of Physics Conf. - Contact Stress Analysis - I, Dec 88, I.O.P. Publishing Ltd.

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Bathe, K-J. and Chaudhary A., 'A solution method for planar and axisymmetric contact problems', Int. J. Num. Meth. Engng., 21, 65-88, 1985.

Pascoe, S.K. and Mottershead, J.E., 'Linear elastic contact problems using curved elements and including dynamic friction', Int. J. Num. Meth. Engng., 26, 1631-1643, 1988.

Knott, J.F., 'Fundamentals of Fracture Mechanics', Butterworth Press, London, 1973.

Chell, G.C. (ed), 'Developments in Fracture Mechanics - I', Applied Science Publications, Barking, England, 1979.

Griffith, A.A., 'The phenomena of rupture and flow in solids', Transactions, Royal Soc. London, 221, 163-198, 1920.

Rice, J.R., 'A path-independent integral and the approximate analysis of strain concentration by notches and cracks', J. Appl. Mech., 35, 379-386, 1968.

Hellen, T.K. and Blackburn, W.S., 'Non-linear fracture mechanics and finite elements', Eng. Comput., 4, 2-14, 1987.

Li, F.Z., Shih, C.F. and Needleman, A., 'A comparison of methods for calculating energy release rates', Eng. Fract. Mech., 21, 405-421, 1985.

Sussman, T. and Bathe, K.J., 'The gradient of the finite element variational indicator with respect to polar coordinates', Int. J. Num. Meth. Engng., 21, 763-774, 1985.

Haber, R.B. and Koh, H.M., 'Explicit expressions for energy release rates using virtual crack extensions', Int. J. Num. Meth. Engng., 21, 305-315, 1985.

Hellen, T.K., 'Virtual crack extension methods for non-linear materials', Int. J. Num. Meth. Engng., 28, 929-942, 1989.

Latzko, D.G.H., 'Post-Yield Fracture Mechanics', 2nd Edition, Elsevier Applied Science Publishers, London and New York, 1984.

Riedel, H. and Rice, J.R., 'Tensile cracks in creeping solids', 12th Conf. Fracture Mechanics, ASTM STP700, 112-130, 1980.

Li, F.Z., Needleman, A. and Shih, C.F., 'Characterisation of near tip stress and deformation fields in creeping solids', Int. J. Fract., 36, 163-186, 1988.

Remzi, E.M. and Blackburn, W.S., 'Automatic crack propagation studies in T-junctions and cross bars', Eng. Comput., to be published.

Practical Use

Hutchinson, J.W., 'Singular behaviour at end of tensile crack in hardening material', J. Mech. Phys. Solids., 16, 13-31, 1968.

Rice, J.R. and Rosengren, G.F., 'Plane strain deformation near crack tip in power-law hardening material', J. Mech. Phys. Solids, 16, 1-12, 1968.

Crisfield, M.A., Hunt, G.W., and Duxbury, P.G., 'Benchmark tests for geometric nonlinearity', NAFEMS, Ref. SPGNL, Oct 1987.

Holsgrove, S. and Lyons, P., 'Benchmark tests for 2D beams and axisymmetric shells with geometric nonlinearity', NAFEMS, Ref. FEBNLGBAS(S), Mar 1989.

Owen, D.J.R., Nayak, G.C., Kfouri, A.P. and Griffiths, J.R., 'Stresses in a partly yielded notched bar', Int. J. Num. Meth. Engng., 6, 63-73, 1973.

Owen, D.J.R. and Goncalves, O.J.A., 'Substructuring techniques in material nonlinear analysis', Comp. and Struct., 15, 205-213, 1982.

Hellen, T.K., 'Use of substructuring in nonlinear analysis', Eng. Comput., 1, 343-350, 1984.

Lee, S.L., Manuel, F.S. and Rossow, E.C., 'Large deflection and stability', J. Eng. Mech. Div., ASCE, 94, 521-547, 1968.

Cite this book

E. Hinton, Introduction to Non-Linear Finite Element Analysis, R0018, NAFEMS, 2000,

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AuthorHinton. E
Date 1st January 1992


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