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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.

https://doi.org/10.59972/3p9d3dh6


R0018, January 1992

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


Preface

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).

Contents

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

References


Introduction to Nonlinear Stress Analysis

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Hodge, P.E. and White, S.H., A quantitive comparison of flow and deformation theories of plasticity, J. Appl. Mech., 17, 180-184, 1950.

Holsgrove, S. and Lyons, P., Benchmark tests for two-dimensional thin beams and axisymmetric shells, NAFEMS, Department of Trade and Industry, NEL, Glasgow, 1988.

Biezeno, C. B. and Grammel, G., Engineering dynamics, vol II: Elastic problems of single machine elements, Blackie and Son Ltd., London, 1956.

Crook, A.J.L., Hinton, E. and Stagg, K.G., Modelling fracture behaviour and fracture in deep level mining, Proc. of CARE 88 (Conference on Applied Rock Engineering), The Institution of Mining and Metallurgy, 45-53, 1988.

Chakrabarty, J., Theory of plasticity, McGraw-Hill, New York, 1987.

Spencer, A.J.M., Continuum mechanics, Longman, Harlow, 1980.

Malvern, L.E., Introduction to the mechanics of a continuous medium, Prentice-Hall, Englewood Cliffs, 1969.


Geometrically Nonlinear Finite Element Analysis

Spencer, A.J.M., Continuum mechanics, Longman, London, 1980.

Malvern, L.E., Introduction to the mechanics of a continuous medium, Prentice-Hall, Englewood Cliffs, 1969.

Fung, Y.C., Foundations of solid mechanics, Prentice-Hall, Englewood Cliffs, 1965.

Hibbitt, H.D., Marcal, P.V. and Rice, J.R., A finite element formulation for problems of large strain and large displacement, Int. J. Solids and Structures, 6, 1069-1086, 1970.

McMeeking, R.<. and Rice, J.R., Finite element formulations for problems of large elasto-plastic deformation, Int. J. Solids and Structures, 11, 601-616, 1975.

Wood, R.D., Lecture notes on 'Nonlinear continuum mechanics and associated finite element procedures', Dept. of Civil Engineering, University College of Swansea, U.K.

Mattiasson, K., On the co-rotational finite element formulation for large deformation problems, Dept. of Structural Mechanics, Chalmers University of Technology, 1983.

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

Crisfield, M.A., Nonlinear finite element analysis of solids and structures - volume 1: essentials, John Wiley, Chichester, 1991.


Time Independent Material Nonlinearities

Hill, R., The Mathematical theory of plasticity, Clarendon Press, Oxford, 1950.

Nayak, G.C. and Zienkiewicz, O.C., Elasto-plastic stress analysis. A generalization for various constitutive relations including strain softening, Int. J. Num. Meth. Engng., 5, 113-135, 1972.

Argyris, J.H. and Scharff, D.W., Methods of elasto-plastic analysis, Proceedings ISD-ISSC Symposium, Stuttgart also ZAMP, 23, 517-551, 1972.

Krieg, R.D. and Krieg, D.B., Accuracies of numerical solution methods for the elastic-perfectly plastic model, J. Pressure Vessel Tech., ASME, 99, 510-515, 1977.

Owen, D.R.J. and Hinton, E., Finite elements in plasticity - theory and practice, Pineridge Press, Swansea, 1980.

Marques, J.M.M., Stress computation in elastoplasticity, Eng. Comput., 1, 42-51, 1984.

Simo, J.C. and Taylor, R.L., Consistent tangent operators for rate independent elastoplasticity, Comp. Meth. Appl. Math. Engng., 48, 101-118, 1985.

Schreyer, H.L., Kulak, R.F. and Kramer, J.M., Accurate numerical solutions for elastic plastic models, J. Pressure Vessel Tech., ASME, 101, 226-234, 1979.

Ortiz, M. and Popov, E.P., Accuracy and stability of integration algorithms for elasto-plastic relations, Int. J. Num. Meth. Engng., 21, 1561-1576, 1985.

Crisfield, M.A., Finite elements and solution procedures for structural analysis: Vol.1 Linear analysis, Pineridge Press, Swansea, 1986.

Mitchell, G.P. and Owen, D.R.J., Numerical solutions to elastic-plastic problems, Eng. Comput., 5, 285-291, 1988.

Runesson, K., Sture, S. and William, K., Integration in computational plasticity, Computers and Structures, 30, 119-130, 1988.

Hinton, E. and Ezzat, M.H., Fundamental tests for two and three-dimensional, small strain, elastoplastic finite element analysis, NAFEMS Report SSEPT, April 1987.

Crisfield, M.A., Nonlinear finite element analysis of solids and structures - volume 1: essentials, John Wiley, Chichester, 1991.


Time-Dependent Material Nonlinearities

Alden T.A., 'Plastic and viscous deformation of metals', Met. Trans A, 16, 375-392, 1985.

Krempl E., 'Cyclic creep - an interpretive literature survey', Welding Research Council Bulletin, 195, 63-123, 1972.

Lemaitre J. and Chaboche J-L., 'Mecanique des Materiaux Solides', Dunod, Paris, 1985.

Hart E., 'Constitutive relations for the nonelastic deformation of metals', J. Eng. Mat. Tech., 98, 193-202, 1976.

Bruhns O.T., Boecke B. and Link F., 'The constitutive relations of elastic-inelastic materials at small strains', Nuc. Eng. Des., 83, 325-331, 1984.

Gittus J., 'Creep, Viscoelasticity and Creep Fracture in Solids', Applied Science, London, 1975.

White P.S., 'Common creep deformation properties among casts of type 316 stainless steel at practical stresses', Transactions 9th SMiRT, L, 247-252, Balkema, Rotterdam, 1987.

Oytana C., Delobelle P. and Mermet A., 'Constitutive equations study in biaxial stress experiments', J. Eng. Mat. Tech., 104, 1-11, 1982.

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).

Gilbert E.R. and Blackburn L.D., 'Creep deformation of 20 percent cold worked type 316 stainless steel', J. Eng. Mat. Tech., 99, 168-180, 1977.

'Concrete and specific aspects of non-metallic materials', Volume Q of 'Transactions of the 10th. SMiRT', AASMiRT, Los Angeles, 1989.

Truesdell C. and Noll W., 'The nonlinear field theories of mechanics', in 'Handbook of Physics' III/3, Springer, Berlin, 1965.

Cho U.W. and Findley W.N., 'Creep and plastic strains of 304 stainless steel at 593oC under step stress changes, considering aging', J. App. Mech., 49, 297-304, 1982.

NAFEMS, 'A Finite Element Primer', DTI, National Engineering Laboratory, Glasgow, 1986.

Bathe K.J., 'Finite Element Procedures in Engineering Analysis', Prentice-Hall, New Jersey, 1982.

Gear C.W., 'Numerical Initial Value Problems in Ordinary Differential Equations', Prentice-Hall, New Jersey, 1971.

Corum, J.M., Greenstreet, W.L., Liu, K.C., Pugh, C.E. and Swindeman, R.W., 'Interim guidelines for detailed inelastic analysis of high temperature reactor system components', ORNL 5014, Oak Ridge National Laboratory, 1974.

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

Henderson, J. and Snedden, J.D., 'Creep recovery of commercially pure copper', NEL Report 296, 1967.

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.

Ion, J.J., Barbose, A., Ashby, M.F., Dyson, B.F. and McLean, M., 'The modelling of creep for engineering design-I', NPL Report DMA A115, 1986.

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.

Delph, T.J., 'Creep relaxation and cyclic behaviour of a beam using a state-variable constitutive model', Nucl. Eng. Design., 65, 411-421, 1981.

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.

Dennis, J.E. and More, J.J., Quasi-Newton methods, motivation and theory, SIAM Review, 19, 46-89, 1977.

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.

Zienkiewics, O.C., 'Incremental displacement in nonlinear analysis', Int. J. Num. Meth. Engng., 3, 587-588, 1971.

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.

Bergan, P.G., 'Solution algorithms for nonlinear structural problems', Comp. and Struct., 12, 497-510, 1980.

Riks, E., 'An incremental approach to the solution of snapping and buckling problems', Int. J. Solids and Structures, 15, 524-551, 1979.

Wempner, G.A., 'Discrete approximations related to nonlinear theories of solids', Int. J. Solids and Structures, 7, 1581-1599, 1971.

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

Ramm, E., 'Strategies for tracing non-linear responses near limit points', Non-linear Finite Element Analysis in Structural Mechanics, (Eds. W. Wunderlich, E. Stein and K.J. Bathe), Springer-Verlag, New York, 1981.<.p>

Crisfield, M.A., 'Accelerated solution techniques and concrete cracking', Comp. Meth. Appl. Mech. Engng., 33, 585, 1982.

Crisfield, M.A., 'An arc-length method including line searches and accelerations', Int. J. Num. Meth. Engng., 19, 1269, 1983.

Crisfield, M.A., 'Overcoming limit points with material softening and strain localisation', Num. Meth. for nonlinear problems (Eds. C. Taylor et al), 2, 244-277, Pineridge Press, Swansea, 1984.

Crisfield, M.A., 'Difficulties with current numerical models for reinforced concrete and some tentative solutions', Proc. Int. Conf. on the Computer Aided analysis and Design of Concrete Structures (Eds. F. Damjanic et al), 1, 331-358, Split, Yugoslavia, Sept. 1984.

Crisfield, M.A., 'New solution procedures for linear and nonlinear finite element analysis', The mathematics of finite elements and applications V, Academic Press, London 1985.

Maewal, A. and Nachbar, W., 'Stable postbuckling equilibrium of axially compressed elastic cylindrical shells: a finite element analysis and compression with experimental results.' J. Appl. Mech., 44, 475-481, 1977.

Sabir, A.B. and Lock, A.C., 'The application of finite elements to the large deflection geometrically non-linear behaviour of cylindrical shells', Variational Methods in Engng., 7/54-7/65, Southampton Univ. Press, 1972.

Padovan, J. and Tovichakchaikul, S., 'Self-adaptive predictor-corrector algorithms for static nonlinear structral analysis', Comp. and Struct., 15, 363-377, 1982.

Padovan, J. and Moscarello, R., 'Locally bound constrained Newton-Raphson solution algorithms' Comp. and Struct., 15, 365-377, 1982.

Bicanic, N. and Johnson, K.H., 'Who was '-Raphson'?', Int. J. Num. Meth. Engng., 00, 148-152, 1979.

Raphson, J., 'Analysi Aequationum universalis seu ad aequationes algebraicas resolvendas methodus generalis et expedita, ex nove infinitarum serierum doctrina deducta ac demonstrata', London 1690 (original in British Library, London).

Newton, I., 'De analysi per aequationes infinitas' (1690), in The Mathematical papers of Isaac Newton, Vol.11 (1667-1670), (ed. D.T. Whiteside), Cambridge University Press, 207-247, 1968.

Cajori, F., 'Historical note on the Newton-Raphson method of approximation', American Mathematical Monthly, 18, 29-32, 1911.

Cajori, F., 'A History of Mathematics', Macmillan, New York, 1926.


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.

Bathe, K.J., 'Finite element procedures in engineering analysis', Prentice Hall, New Jersey, 1982.

Hughes, T.J.R., 'The finite element method', Prentice Hall, New Jersey, 1987.

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.

Chan, S.K. and Tuba, I.S., 'A finite element method for contact problems of solid bodies - I', Int. J. Mech. Sci., 13, 615-625, 1971.

Hellen, T.K., 'A gap element facility in BERSAFE', C.E.G.B. Report TPRD/B/1043/R88.

Stadter, J.T. and Weiss, R.O., 'Analysis of contact through finite element gaps', Comp. and Struct., 10, 867-873, 1979.

Padovan, J., Moscarello, R., Stafford, J. and Tabaddor, F., 'Pantographing self adaptive gap elements', Comp. and Struct., 20, 745-758, 1985.

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

Hughes, T.J.R. et al, 'A finite element method for a class of contact-impact problems', Comp. Meth. Appl. Mech. Engng., 8, 249-276, 1976.

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, https://doi.org/10.59972/3p9d3dh6

Document Details

ReferenceR0018
AuthorHinton. E
LanguageEnglish
AudienceAnalyst
TypePublication
Date 1st January 1992
RegionGlobal

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