NAFEMS Selected Benchmarks for Material Non-linearity

Published on: **Mar 3, 2016**

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- 1. AVAVAVAVAV 1 ‘ N FEM AVAVA VAVAV AVAVAVA VAV Committed to Professional Development AVAVAVAVAV for Engineering Analysis & Simulation _ NAFEMS Selected Benchmarks for Material Non-Linearity D Linkens Published by NAF EMS Ref: R0026 REV 1 wwW. nafems. Org
- 2. ACKNOWLEDGEMENTS The NAFEMS Non-linear Working Group has the broad remit to improve the reliability of non-linear finite element analysis in industry. To that end it has embarked on a number of initiatives, including development of benchmarks, workshops, educational texts and guidance documents. This report falls into the former category. It describes a total of 10 benchmark problems covering small strain non-linear material behaviour (i. e. creep and plasticity). The work was performed by D. Linkens of WS Atkins under the auspices of the NAFEMS Non-linear Working Group which comprised, N. C. Knowles WS Atkins (Chairman) R. Andrews The Welding Institute M. A. Crisfield Imperial College T. K. Hellen Nuclear Electric E. Hinton University of Swansea P. A. Lyons FEA Ltd. B. S. Marsden AEA Technology P. Newton MacNeal-Schwendler D. Phillips Glasgow University P. S. White GEC Alsthom Dr. P. S. White, in particular, contributed generously with his time, interest and constructive discussions and Dr. B. S. Marsden provided a number of corroborative solutions. STRUCOM & WS Atkins provided copies of ANSYS and ASAS-NL respectively. / /Claims N. C. Knowles Chairman NAFEMS Non-linear Working Group January 1993 (5)
- 3. SUMMARY A set of benchmarks is presented for ﬁnite element assemblies modelling non-linear material behaviour with small deformations. The test cases cover both plasticity and creep and are intended to have clear industrial relevance. Each test is described in a standard single page summary which includes authoritative target solutions. A companion volume [Ref. 1] provides fuller details. (ii)
- 4. TABLE OF CONTENTS ACKNOWLEDGEMENTS (1) SUMMARY 1 (ii) TABLE OF CONTENTS (iii) 1. INTRODUCTION 1 2. SCOPE OF TESTS 2.1 Criteria for Benchmarks 2.2 Overview of Tests 2.3 Reporting Format 3. REFERENCES 4. DETAILS OF EACH TEST lU‘l-l>C0l)l. 'i (iii)
- 5. INTRODUCTION This document provides details of 10 benchmark tests for finite element assemblies modelling non-linear material behaviour with small deformations. Its aim is to provide a compendium of simple, authoritatively documented tests covering the more common types of material non-linear phenomena encountered in industrial practice. Emphasis is on quasi-static small-displacement small- strain continuum behaviour of ductile metals. Collectively the tests may be used for a number of purposes including: (1) to check the ability of a finite element program to produce acceptable solutions to problems involving material non—linearity. (2) to exercise a program's ability to model physical behaviour. (3) to educate inexperienced users about the potential pitfalls and difficulties that may arise when modelling non-linear phenomena. The overall philosophy has been to select problems with a clear industrial relevance, which can be modelled satisfactorily with the capabilities of robust general purpose commercial codes and which, moreover, are both physically simple and computationally undemanding. The problems have been drawn from an earlier survey [Ref. 2], but in most instances have been "tuned" to make them more discriminating or to improve their clarity and ease of interpretation. The document is, quite deliberately, fairly concise. Essential details of each test are provided in a standard format in Section 4, which is intended to be self explanatory. However, some further information covering the organisation and background of the various tests is given in Section 2. A companion volume to this report [Ref. 1] gives more details about the development of the tests and provides more complete details of the solutions.
- 6. 2. SCOPE OF TESTS 2.1 Criteria for Benchmarks In selecting the tests the objective has been to achieve: OOOO optimal technical cover of relevant topics minimal duplication ease of reporting maximum use of published relevant experience The requirements for a satisfactory benchmark were set out in the earlier work. Paraphrasing [Ref. 2], these are: 1. A test should be precisely and unambiguously stated in terms of geometry, material behaviour, loading and boundary conditions. Material laws should be fully defined and of the type likely to be available in commercial finite element programs. A reliable solution should be available with enough detail for all aspects of an attempted solution to be checked. This ‘reference’ solution may either be of a closed form or a well corroborated numerical solution (preferably by alternative/ independent numerical approaches). The model size should be reasonable. This depends on the context of the problem but it should be less than those of practical engineering problems involving the same phenomena. Each problem should have educational merit. The collection, as a whole, should cover a good overall range of both breadth and technical complexity. Ideally tests should be discriminating and there should be some non-obvious potential for bad results if a poor model or inadequate program is used.
- 7. 2.2 Overview of Tests Some 10 tests have been worked up as definitive benchmarks and are presented in Section 4. They are summarised in the accompanying Table A (Page 6), in terms of the various attributes and non-linear phenomena they display. Test 1, the "Torsional creep of circular shaft", is intended to be a simple uncomplicated test which exercises a program's fundamental creep analysis capabilities. Both fonrvard deformation (load control) and relaxation (displacement control) are illustrated. Test»6 is an extension of this to a square shaft. It introduces the physical phenomena of warping and the opportunity for stress redistribution on the cross; section. For the majority of f. e. programs a 3-D model will be required. Tests 2 & 3 involve cyclic thermal loading of a two bar assembly. These tests are designed as simple illustrations of the various limit states in the "Bree" diagram, e. g. elastic shakedown, alternating plasticity and ratcheting. Test 4 extends these concepts to a bending situation. Test 5 (a punch indenting a solid block) is modelled as a plane strain problem. It therefore involves multi-axial yield criteria; also stress concentration and redistribution effects are significant. Test 7 is the collapse behaviour of a square plate under uniform pressure loading, and is the one test for which solutions based on a non-continuum formulation (i. e. plate theory) are presented. Both "gross-section" and "layered" yield criteria are investigated. Test 8 involves a fairly complicated thermal and pressure history in a long axisymmetric cylinder. The geometry and loading give rise to creep, relaxation, stress redistribution and plasticity under conditions of non-radial and reversed loading. Test 9 is similarly challenging. It is a good example of two dimensional thermal ratcheting (in a hollow cylinder with an eccentric bore subjected to internal pressure and cyclic thermal gradient).
- 8. 2.3 Test 10 deals with long-term creep redistribution effects in an internally pressurised hollow sphere. Although strictly one dimensional (spherically symmetric) most f. e. codes will treat it as 2-D using axisymmetric elements described in cylindrical coordinates. The required spherical symmetry is a good check on the robustness of the temporal integration scheme. Reporting Format Each test is described in a "standard" format in Section 4. The intention is that this single page should communicate all salient aspects of the test and be sufficiently detailed to allow independent analysts to reproduce the solution. Details of the latter are deliberately concise - but quantitative values are presented in all cases. Fuller explanations of the background to the test, the physical behaviour and details of the solutions are contained in the companion volume to this document [Ref. 1]. Some explanation of the terminology used to qualify the solutions is appropriate. In all cases an attempt has been made to establish a definitive, reliable solution, either using a closed-form solution to an accepted theory or by two independent numerical solutions. Independent corroboration has been obtained by using published results, or running analyses on ASAS-NL, ABAQUS, or ANSYS as appropriate. Where such solutions have been obtained they are described as "Reference" solutions. In addition "Target" solutions are also presented. These have been obtained with ASAS-NL using the f. e. model described on the test summary sheets. ‘(The "Target" f. e. model is intended to be indicative of good practice but such labelling is necessarily subjective). Occasionally, it has not proved possible to obtain independent numerical solutions that corroborate each other to an acceptable degree. In these cases no solution is labelled as "Reference" and a somewhat subjective decision has been made as to which solution to present for comparison. However, in all cases, the source is indicated.
- 9. REFERENCES 1. Linkens, D. , "Selected Benchmarks for Material Non-Linearity - Vol II'', DTI Contract NEUF/618, WS Atkins Science & Technology, G1475/R01, Issue 1. 2. Jakeman, R. R., and White, P. S., "Review of Benchmark Problems for Non- linear Material Behaviour", DTI Contract NEL/ F/397, GEC Engineering Research Centre, ERC(W) 12.0752, Aug. 1987.
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- 11. DETAILS OF EACH TEST Single page summaries of 10 tests are presented on the following pages: Test 1 Torsional Creep of Circular Shaft A. Relaxation at Constant Twist B. Forward Creep at Steady Twist Rate Test 2 Cyclic Thermal Loading of Two—bar Assembly. A. Approach to Shakedown B. Alternating Plasticity C. Approach to Shakedown (More General Case) Test 3 I Thermal Ratcheting of Two—bar assembly Test 4 Thermal Ratcheting of Uniform Beam Test 5 Plane Strain Rigid Punch A. Strain Hardening Material _ B. Perfectly Plastic Material Test 6 Torsional Creep of Square Shaft A. Relaxation at Constant Twist B. Forward Creep at Steady Twist Rate Test 7 Square Plate under Uniformly Distributed Load A. Simply Supported Edges B. Clamped Edges Test 8 Non-proportional Loading of Thick Cylinder Test 9 Thermal Ratcheting of Eccentric Tube Test 10 Thermally Induced Creep of lntemally Pressurised Sphere.
- 12. < 1. TORSIONAL CREEP OF CIRCULAR SHAFT A. Relaxation at Constant 'IIvist ATTRIBUTES: - Creep relaxation (displacement control) — 1-dimensional stress ﬁeld —- Power law creep — No warping deﬂections MATERIAL PROPERTIES: — Young's modulus: E = 10.0 — Poisson’s ratio: = 0.3 — Creep law, écq =1ooooo2q where em, = equivalent creep strain rate oeq equivalent stress (Mises) SOLUTION: E | '-‘ 0 V TARGET SOLITHON *‘ REIERENE SOLSUTION EQUIVALWT STRAIN AT EDGE OF SHAFI‘ NAFEMS NON—LINEAR MATERIALS BENCHIVIARKS Case NL1A Issue: ‘L Long prismatic circular shaft. Uniform twist held constant in time: 0.01 radians / unit length from time 0. to 100. FINITE ELEMENT MODEL: (target solution) Four 20-noded brick elements (plus one 16-noded wedge) All nodes on lower face fixed in )(, Y. Z fixed at O, A&B X X, Y displacements prescribed at all nodes on upper face to give rigid body rotation about Z-axis (=0.002). Similarly at mid-side plane (rotation=0.001). Z displacements free. LOAD CHE Moll MAX. 0.11333 -2 MIN. OJZZE-15 CONTOUR KEYS AOJMME-3 BOJDDOE-3 C0.l20tE-2 D IAIHXE-2 30.10018-2 E 0.240184 0 OJIOE-2 HOJIMB ~2 l:0.360tE—l l:0AO0E-1 K0.Ml'lIE—1 EQUIVALENT CREEP STRAINS AT TIIIE = 100.0 REFERENCE SOLUTION: (analytical) Mises equivalent stress (x10‘ 2) at edge of shaft Total Equivalent strain (x10'3) at edge of shaft Elastic Creep 20.0 1.8117 5.7735 1.5702 4.2033
- 13. NAFEMS NONJJNEAR MATERIALS 1. TORSIONAL CREEP OF CIRCULAR SHAFT BENCHMARKS _ Case NLIB B. Forward Creep at Steady Twist Rate Issue: 1. Long prismatic circular shaft. Uniform twist steadily increasing with time: 0.02 radians / unit length / unit time from time 0. to 1.5 ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) — Forward creep (load control) Four 20-noded brick elements (plus one 16-noded wedge) — 1-dimensional stress ﬁeld _. All nodes on low Power law crsep face fixed in X, YTr — No warping deﬂections A z ﬁxgd a, o,A&B_ MATERIAL PROPERTIES: — Young’s modulus: E -— 10.0 — Poisson’s ratio: v = 0.3 — Creep law, é, q =10000oZq Lo X, Y displacements prescribed at all nodes on upper face to where éeq = equivalent creep strain rate give rigid body rotation about Z-axis (=0.004/unit time). . . S’ 'l ‘d—'d I ' =0.002/1)‘ ' . oeq = equivalent stress (Miscs) Z"; ’i's; {;y°: :n': :mSf'm? me (Manon mum) SOLUTION: 0.020 LOAD CAE M0}! MAX. 0.1 l7JE .1 MIN. OMJZG -5 D o v TARGl: ‘l‘SOLUT[ON CON'| 'OLTl KEYS A 0.l000E -1 _. REFEll. E!CE SOLUTION 3 0.10002 -2 C 0JO00E—2 D IIAONE-2 B VJOWE-1 R D. ﬂ)wE -1 CI 0.700%! -2 H OJDME-2 I : osoour-: -1 I: 0.1oNE—l K 0.]! NE -1 EQUIVALENT STRAIN AT EDGE OF SHAFT . _., .. __ EQUIVALENT CREEP STRAIN AT THE = 1.5 0.0 Ill 0.2 OJ 0.4 0.5 0.! 0.7 0.3 0.9 1.0 1.! 1.2 1.3 1.5 1.5 TIME REFERENCE SOLUTION: (implicit analytical) Time Mises equivalent stress (x1O'2) at edge of shaft Equivalent strain (xl0“2) at edge of shaft
- 14. NAFEMS N ON-LIN EAR MATERIALS 2. CY CLIC THERMAL LOADING OF TWO—BAR ASSEMBLY BENCHMARKS Case NLZA Issue: 1 A. Approach to Shakedown Pair of identical bars whose ends are constrained to move together axially. Constant axial load and cyclic temperature variation. ATTRIBUTES: — Elasto-plasticity, kinematic hardening — Elastic shakedown — ‘1-dirnensional’ stress field (discrete values) MATERIALI GEOMETRIC PROPERTIES: LOAD HISTORY: stress Temperature 0 Mechanical Strain E—(1T Bar 1 Bar 2 Modulus of elasticity, E 10000. 10000. Yield stress, oy 10. 10. _ _ Hardening coefficient, (3 0.1 0.1 Applied temperature, Bar 1: + 100. Coefficient of expansion, 01 0.00001 0.00001 _ _ Bar 21 0- C1-osgsecﬁonal area, A 1_ 1_ Applied mechanical load: 15. REFERENCE SOLUTION: (analytical) Meéhamcal Axial Force Mechanical strain Strain 40 (xio-4) (x10—4) 30 Bar 1 Bar 2 Bar 1 Bar 2 .500 7.500 7 500 7.500 .455 4 545 14.545 4.545 .174 10.826 s.264 18.264 .131 3.869 21.307 11.307 .620 11.330 13.797 23.797 _ .583 3.417 25.934 15.834 A““31_ .250 11.750 17.500 27.500 Fbfceln .886 3.114 28.864 18.864 .002 11.998 19.980 29.980 .089 2.911 30.892 20.892 .836 12.164 21.639 31.639 Note: Total strain equals mechanical strain in Bar 2. 10
- 15. 2. CYCLIC THERMAL LOADING OF TWO'—BAR ASSEMBLY BENCHMARKS Case NLZB B. Alternating Plasticity I 1 ssue: Pair of identical bars whose ends are constrained to move together axially. Constant axial load and cyclic temperature variation. ATTRIBUTES: — Elasto-plasticity, kinematic hardening —- Alternating plasticity — ‘1-diinensional’ stress field (discrete values) MATERIAL/ GEOMETRIC PROPERTIES: LOAD HISTORY: Stress Temperature 0 Mechanical Strain e—aT Bar 1 Bar 2 Modulus of elasticity, E 10000. 10000. Yield stress, cry 10. 10. Hardening coefficient, [3 0.1 0.1 Applied temperature, Bar 1: -"I3 300. Coefﬁcient of expansion, 01 0.00001 0.00001 Bar 2; 0. CIOSS-S¢Cti0I1a1 area, A - - Applied mechanical load: 15. REFERENCE SOLUTION: (analytical) Mechanical strain Strain 100 (X10-4) Bar 1 Bar 2. Bar 1 Bar 2 60 .500 . .500 7 .273 . .727 2 968 . ' - E1. .654 . . 36. .898 . . 78. .917 . . 59. .000 . . 90. .000 . . 60. .000 . . 90. .000 . . 60. .000 . . 90. 3 8 10 Note: Total strain equals mechanical I- O. otnlOU'Iahwt. i)--o . . B ‘ Half_cycle strainm ar2
- 16. NAF EMS NON—LINEA. R MATERIALS 2. ‘CYCLIC THERMAL LOADING OF TWO-BAR ASSEMBLY BENCHMARKS C. Approach to Shakedown (More General Case) fase bfic ssue: Pair of identical bars whose ends are constrained to move together axially. Constant axial load and cyclic temperature variation. This is a more general variant of Test NL2A. ATTRIBUTES: — Elasto-plasticity, kinematic hardening - Elastic shakedown (different parameters for each bar) — ‘1-dimensional’ stress field (discrete values) ‘ MATERIAL / GEOMETRIC PROPERTIES: LOAD HISTORY: Stress Temperature 0 Mechanical Strain a—qT Bar 1 Bar 2 Modulus of elasticity, E 100000. 200000. Yield stress, 0,. 500. 600. Hardening coefficient, B 0.1 0.02 Applied temperature, Bar 1: 3F 300. Coefficient of expansion, (1 0.00002 0.00001 Bar 2: F 150. Cross-sectional area, A 1. 0.75 Applied mechanical load; 700, REFERENCE SOLUTION: (analytical) Mechanical strain (x10‘3) Bar 1 Bar 2 Bar 1 Bar 2 280. SU7. 236. 549. 225. 533. 216. 610. 209. 632. 203. 420. 192. 463. 150. 474. 116. 483. B9. 490. 67. 496. .800 .781 .066 .976 .732 .344 .675 .048 ' .037 .218 .934 IlJL>. l'1Ll'| |l>(-Ja: |hJ0"1O L. J@O'J'| |J'lUIIJl(‘JO
- 17. NAFEMS NON-LINEAR MATERIALS 3. 'I‘HERMAL RATCHETING OF TWO-BAR ASSEMBLY “(‘ﬁ/ ”°““§I*: :5 ase Issue: 1 Pair of identical bars whose ends are constrained to move together axially. Constant axial load and cyclic temperature variation. ATTRIBUTES: — Thermal ratcheting (incremental collapse). - Perfect plasticity (no hardening). — ‘1-dirnensional’ stress field (discrete values). MATERIAL / GEOMETRIC PROPERTIES: LOAD HISTORY: Stress Temperature 0 100 ‘’y E 10 Half- cycle Mechanical Strain 2--aT Bar 1 Bar 2 Modulus of elasticity, E 10000. 10000. ' 10. 10. g(l):1§i: ti: :ftS’0?{3xpanSion (1 000001 000001 Applied temperature, Bar 1: i 100. _ - 1_ 1_ Bar 2: 0. Cross secnonal area’ A Applied mechanical load: 15. REFERENCE SOLUTION: (analytical) Strain 60 (x10—4) Mechanical strain (xl0“) Bar 1 Bar 2 Bar 1 Bar 2 7.500 7.500 7.500 7.500 10.000 5.000 15.000 5.000 5.000 10.000 10.000 20.000 10.000 5.000 25.000 15.000 5.000 10.000 20.000 30.000 10.000 5.000 35.000 25.000 5.000 10.000 30.000 40.000 10.000 5.000 45.000 35.000 5.000 10.000 40.000 50.000 10.000 5.000 55.000 45.000 5.000 10.000 50.000 50.000 H Note: Total strain equals mechanical strain in bar 2. 13
- 18. NAF EMS NON-LINEAR MATERIALS BENCI-IMARKS Case NL4 Issue: 1 4. THERMAL RATCHETING OF UNIFORM BEAM Uniform beam with constant end load and cyclically varying linear through-thickness Temp? -fanlfe temperature gradient. distribution _ _ _ Plane sections are constramed to remain straight and parallel, i. e. no curvature. ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) '- Elasto-plasticity. kinematic hardening B D - Uniform mesh of 20 8-node isoparametric — Cyclic accumulation of plastic strain membrané elements through depth — Elastic/ plastic shakedown — Continuum stress field, 1D variation 1_? '°A1l§‘)‘E':1f1r'Ze“5j? )f_1n‘giSt1il; ’F13rSe5ssed on face AB — X-freedoms on face CD constrained to remain equal MATERIAL PROPERTIES: - Y-freedom suppressed at A Loading: Y — Sin le nodal load a lied at mid-thickne s 3 B 8 PP S tress A C — Prescribed nodal temperatures X LOAD HISTORY: Temperature, T -400 Mechanical Strain E-(IT Modulus of elasticity, E Yield stress, 0,. Hardening coefficient, 0 Coefﬁcient of expansion, (1 0 1 10 Half-cycle Applied mechanical load (constant) 900 per unit width of beam N: SOLUTI0 PLASTIC STRAIN DISTRIBUTION THROUGH DEPTH, sxx(x10‘-3) Iu um In I: In up 1.. In an Lo 1.! 1.0 {,4 I! I! u M u an «.0 u :1 all u u u an —u-uauznumu —unn¢xnunun —nIuI. IuxIn. uII3u . _.. ,g. g,. ._. m.. I o vuunumu " - unuuunuu M c vi-suntan . ,, ,¢, _,, ,,. . u M u an nusaiulaauaaorusn us: no n: a1nno| anuouuuotum"u. ._. ¢., ... ., u. Half-cycle 1 I-I: lf-cycle 2 Half-cycle 9 Half-cycle 1 REFERENCE SOLUTION: (analytical) l-Ialt'—o/ cle number 0 1 Total strain (cmuunt through thickness), e, ,,(xl0’3) ‘mp surface: Elastic Strain, e, u(x10'3) Plastic Strain, £n(xl0'3) Bottom surface: Elastic Strain, :, ,(x10'3) Plastic Strain. ., .,(x1o-3) 14
- 19. NAF EMS NON-LINEAR MATERIALS BENCHMARKS TEST NLSA Issue: 1 5. PLANE STRAIN RIGID PUNCH A. Strain Hardening Material HHH Rigid, frictionless punch is pressed into a deep plate of finite width supported on a frictionless plane. 2D plane strain conditions. Displacement controlled loading. ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) — Elas_to-plasticanalysis. Prescribed Y freedom at - Strain hardening. punch-plate interface — 2-dimensional stress ﬁeld — Non-radial material response MATERIAL PROPERTIES: Symmetry face Stress _ _ (X freedoms suppressed) 0 BE E Strain 8 Young’s Modulus, E 1000 Y displacements suppressed ' Poisson’s ratio, v 0.3 _ _ _ Yield stress, 0,. 1.0 - 20x16 mesh of 8-noded quadratic isoparametric Hardening coefficient, [5 0.1 ’“°‘“b”“‘° °l°m°‘“5- Von Mises yield, associated ﬂow rule ‘ 9-*2 Ga“55i3“ im°31'3ﬁ°“ SOLUTION: APPIJED DELECYDN or PUNCH (xio-2) A A TARGET SOLUTION —— REFENEDCE $0LI'l1ON DIRECFSIRESS (I: Iyy)ATPOINTA I in II I APPLIE DEIECIION OF PUNG-I (X104) REFERENCE SOLUTION: (f. e.) Stress at Point A (x10"1) Strain at Point A (x10‘3) Note: Stresses and strains are extrapolated to Point A using gauss point values from element at its upper left 15
- 20. NAFEMS NON-LINEAR MATERIALS 5 PLANE STRAIN RIGID PUNCH BE"°”M’““‘S B. Perfectly Plastic Material TEST NLSB Issuezl HHH Rigid, frictionless punch is pressed into a -non-x-1 deep plate of finite width supported on a frictionless plane. 2D plane strain conditions. Displacement controlled loading. Point A ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) ' E135‘°'P135ti° “1n31Y5i5 Prescribed Y freedoms at - PCIf8Ct plasticity punch«plate interface — 2-dimensional stress ﬁeld — Non-radial material response — Plastic collapse MATERIAL PROPERTIES: Svmmwvface (X freedoms suppressed) Stress ‘’ E Lx Strain 8 Young’s Modulus, E 1000 _ _ _ Poissonas ratio’ V 03 — 20x16 mesh of 8-noded quadratic isoparametnc Yield stress, oy 1.0 membrane elements. Von Mises yield, associated ﬂow rule ‘ 2-"2 Gaussian integf 31i°n Y displacements suppressed ' SOLUTION: APPLIED DE-‘LECT ION OF PUNCH (X104) I5|1 125 100 ‘)5 A A TARGET SOLU'| '|ON ———-— REFERENCE 5DLU'l10N DIRECTSTRESS (Gyy)ATP0lNTA I0 I 5 APPLHED DEFIECUON OP PUNCH REFERENCE SOLUTION: (f. e.) Stress at Point A (x10‘1) Strain at Point A (xl0‘3) Note: Stresses and strains are extrapolated to Point A using gauss point values from element at its upper left 16
- 21. NAF EMS NON-LINEAR MATERIALS 6. TORSIONAL CREEP OF SQUARE SHAFT BENCHMARK5 A. Relaxation at Constant Twist Case NLGA Issue: 1 Long prismatic square shaft, ends free to warp. Uniform twist held constant in time: 0.01 radians I unit length from time 0.0001 to 100 ATTRIBUTES: — Creep relaxation (displacement control) — 2-dimensional stress ﬁeld _ Quarter symmetry model — Power law creep with time hardening. of short section of Shaft — Unrestrained warping of cross-section. _ zomoded quadratic brick elements — Warping constrained to be , identical for both — Young s modulus E = 10.0 symmeny faces — Poisson’s ratio v = 0.3 I . — n-plane suppressions on 0; ' — bottom face / t - Prescribed in-plane displacements at all other where écq '* equivalent creep strain rate nodes (=0,(]()2 at point A) 06., equivalent stress (Mises) t time MATERIAL PROPERTIES: —Creeplaw, E“, = 10000 SOLUTION: 0.010 0.010 0.0! 0 0.0!‘ 0.0! I SHEAR §I'RAlN(Vyz)ATP0|N| 'A >mnom~uc: :r. »-4 TARGET SOLUTION: shear stress Shear strain (yyz) at Point A, x10‘2 at Point A <= w>= x1°‘2 17
- 22. NAFEMS NON-LINEAR MATERIALS 6. TORSIONAL CREEP OF SQUARE SHAFT BEN‘-‘HMARK5 B. Forward Creep at Steady Twist Rate Case NL6B V Issue: 1 Long prismatic square shaft, ends free to warp. Uniform twist steadily increasing with time: 0.02 radiansl unit length / unit time from time 0.0001 to 100 ATTRIBUTES : — Forward creep (“load control”) — 2-dimensional stress ﬁeld — Power law creep with time hardening. _ Quam, Symmetry mode] — Unrestrained warping of cross-section. of short section of shaft — 20-noded quadratic brick glernents _ Young’s modulus E = 10.0 -— Warping constrained to be _ identical for both — Poisson’s ratio v = 0.3 Symmetry faces . 0’ -1 - 1 ' __ Creep law’ seq = 10000 7?; n p ane suppressions on bottom face _ _ _ — Prescribed in-plane where 3,. ‘ equivalent creep strain rate djspiacemems at an other Ogq equivalent stress (Mises) . nodes (=0.004/unit time t _ time at Point A) SOLUTION: 0.050 0.045 SHEAR STRAIN (Vyz ) AT POINT A ﬂOl: <|’|1CJI: >-If-i L-2n£_. ;.: iuL. 's. bub»é» amwmwmuoqh u. ... .a. u.u. ... ... i.~ ‘7“'? "I“T". '”? ”.“‘. "‘. "‘2' l>| Jr-I-4r-: -a->->-‘>- Equivalent stresses at time t=1.0 TARGET SOLUTION: Shear Stress Shear strain (yy, )at Point A, x10‘2 at Point A (tyz), x10'2 18
- 23. NAFEMS N ON—LINEAR 7. SQUARE PLATE UNDER UNIFORMLY ggmxs DISTRIBUTED LOAD Case NL7A A. Simply Supported Edges Issue: 1 Thin square plate with simply supported edges (frictionless knife edge supports) Uniformly distributed load over whole area. ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) — Plate bending - Elasto-plastic collapse — 3-dimensional stress field C — Quarter symmetry model. — 2-dimensional shell representation _ 4x4.mesh of 8_noded uadmﬁc isoparametric thick she 1 elements. MATERIAL PROPERTIES: _ . . — Ttapezoidal integration ‘’1' ) B TX Symmetry line Stress < (through thickness), 13 points. Strain B d C 't' Young’s Modulus: E 30000. mm my ondl ions _ Poisson’s Ratio: v 0.3 5Ynnn°n'Y3 Simple Supports: Yield Stress: 03' 30' AB: Y, RX, RZ suppressed BC: Z, RX suppressed ‘IXIsi: ::i3;it‘:1&1f? $vct: :l: '. ADI X. RY, RZ suppressed CD: Z, RY suppressed SOLUTION: UDL (Pressure) —-n-— roluiiaéi (target) ——oo- section solution 3-tuna: -1-in: ::»iL: DEFLECTION AT CENTRE OF PLATE EQUIVALENT PIASTTC STRAIN AT LOWER SURFACE, UDL (PRESSURE)=0.0l88 TARGET SOLUTION: (f. e. layered formulation) Load per unit area (x10"2) 0.0000 1.196 1.482 1.596 1.728 1.805 1.837 1.861 1.877 Central Deflection of Plate 0.0000 0.7184 0.9977 1.213 1.736 2.590 3.296 4.294 5.701 19
- 24. NAF EMS NON-LINEAR 7. SQUARE PLATE UNDER UNIFORMLY MATERW5 BENCHMARKS DISTRIBUTED LOAD Case NL7B B. Clamped Edges Issue. 1 Thin square plate with clamped edges (all freedoms restrained) Unifonnly distributed load over whole area. ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) — Plate bending — Elasto-plastic collapse — 3-dimensional stress field C ’ Q“a"°‘ 5Y“““°"Y m°de1- — 2-dimensional shell representation _. 4x4 mesh of gmoded quadmtic isoparametric thick shell elements. MATERIAL PROPERTIES: — Trapezoidal integration ) Symmetry line Sm, “ < (through thickness), 13 points. Strain Young’s Modulus: Poisson’s Ratio: 0.3 Symmetry: Clamped Supports: E v Yield stress: oy 30. Mises yield function AB: Y, RX, RZ suppressed BC: X, Y,Z, RX, RY, RZ suppressed Associated ﬂow m1e_ AD! X. RY, RZ Suppressed CD: X, Y,Z, RY, RY, RZ suppressed SOLUTION: 30000_ Boundary Conditions UDL (Pressure) ' . ... ... ... ... . _. _; . ... ... ... ... ... .. ; . ... ... ... mi. .. ... ... .. . . I E -<I-- lgayued saluting (target) _ . -—| I- Gross section yield solutinn _____ .1.. -__. .a. ... .-. !.. ... ... ... ... ... ... ..é_. ... _._. ... ... ... ?.. _.. ... ... ... ... ._ FWOUMWQIHQ DEFLECFION AT CENTRE OF PLATE EQUIVALENT PLASTIC STRAIN AT LOWER SURFACE, UDL (PRESSURE) =0.038S TARGET SOLUTION: (f. e. layered formulation) Load per unit area (x10'2) 0.0000 2.804 3.107 3.342 3.507 3.621 3.780 Central Deflection of Plate 0.0000 0.6680 0.8358 1.197 1.640 2.268 . 4.416 20
- 25. NAFEMS NON -LIN EAR MATERIALS BENCHMARKS 8. N ON—PROPORTIONAL LOADING OF THICK CYLINDER Long cylinder under plane strain conditions, subjected to temperature transient followed by fluctuating intemal pressure. Plane strain conditions. Stage 1 (elasto-plastic) : Rapid temperature rise of 50 followed by pressurisation Stage 2 (creep) : Temperature and pressure maintained Stage 3 (elasto-plastic) : Rapid reversal of pressure Stage 4 (creep) : Temperature and pressure maintained ATTRIBUTES: FINITE ELEMENT MODEL: Elasto-plastic and creep behaviour Non-proportional loading Non-radial material behaviour 1D stress ﬁeld _ . — Plane strain slice of cylinder represented MATERIAL PROPERTIES: by 10 quadratic isoparametric elements. Young, s Modulus 2.2137 — All Z freedoms suppressed (plane strain) Poisson’s ratio 0.3 Thermal expansion coefficient 1.85E—5 Yield stress 9900. Elasto-plastic constitutive law: Internal Stress 12500. 15200. 17500. 20000. pressure Total strain 0.00958 0.001641 0.003745 0.007059 Plastic strain 0.00039 0.00095 0.00295 0.00615 -3600 Mises yield criterion, isotropic hardening. Temperature transient: Associated ﬂow rule. t 5 0. T = 0 Creep constitutive law: 037 e, = 1.0 x 10~2°o5-1‘ t‘°" ) 0s: <o.1: r = 500 . :('E. o9 r-016 where éc, = equivalent creep strain rate 0.1St<l-0: T = 50 - t( 0-09 ) 0 = equivalent stress t> 1.0: T = 50 RESULTS: TARGET SOLUTION (at inner surface): Stress Equivalent Strain ‘Fine Axial Hoop Equivalent Plastic Creep .000OD+00 .0000D+00 .0000D-01 .4981Do04 .0000Dv0l .264ED+O4 .0O00D+00 .6605Do03 .4000D+00 .6342D+03 .0000D+00 .3094D+O3 .000OD+01 .2227D+03 .0000Do01 .1978D+O2 .0000D+02 . IS27Do02 .0000D+02 .6135D¢02 .9852D+03 .0000Dv02 .7¢90D+02 .4004Do03 .9900D+D2 .2028D+O2 .2194D+03 .9940D+02 .7955D+02 .6496D+03 .000OD+03 .176ED+03 .1953Do04 .0100Do03 . !B87D+03 .6631D+03 .0250D+03 .6651D+03 .2657D+03 .1000Do03 .2S44D¢03 .3D10D¢03 .2000D+03 .0636Do0J .6466D+03 .5000D+03 . E459D+02 .2613D+03 .0000D+03 .135BD+02 .1924D+03 .0000D+03 .9076D+02 .175SD+03 .0000D+00 .201BD9O4 .9634D+03 .3850D4O3 .6133D+03 .2046Dv04 .5293D+03 .4633D+03 .6613D+O3 .0000D+00 .3709D+04 .1207D+04 .3916D+04 .42EBD+04 .5d43D+04 .2072D+04 .6947D+03 .9145D+03 .301SD+03 .7916D+03 .6351De03 .1102D+03 .360SD+04 .1535D+04 .0292D+04 .5714D+03 .004BD+03 .6716D+03 .6118D+03 .5974D+03 .0O00Do00 .2767D-04 .2290D-04 .5277D-04 .5055D-04 .90d0D-04 .9040D-04 .9040D-04 .9040D-04 .9040D-04 .9040D—04 .9040D~04 .9040D-04 .9040D-04 .9040D-04 .9040D—04 .90d0D—04 .9040D-04 .9040D-04 .9040D-04 .9040D-04 .0000Do00 .0000DoO0 .000OD+00 .0000D+0O .0000D¢00 .0000D+00 .0521D-O4 .9d14D-04 .0925D-04 .1771D-03 .6910D-03 .3416D—03 .3416D-03 .3416D—03 .4904D—03 .6009D-03 .8¢09D—03 .0l74D-03 .4067D-O3 .9!0dD—03 .0653D—03 um»----v-r-o--osoulur-nui--N»-r-.9»-o U'| J10»~l~oI-'Ia~lhi| dO ~l~. A~lmrn>->-‘>4:-~l~1uom~ov->-H»->->-o -o~ao~awaIoc~oa~owo»ow~0~o~lammo tntul-Ih; hJlJhJI)lBhlI-lb-‘00I£-IOOOOOO Equlvnlem creep unln Ilqulvulerit plastic strain 21
- 26. 9. THERMAL RATCHETING OF ECCENTRIC TUBE Intemal Pressure ATTRIBUTES: — Elasto-plastic analysis - Ratcheting and alternating plasticity — 2-dimensional continuum stress ﬁeld — Perfect plasticity MATERIAL‘ PROPERTIES: Stress Mechanical Strain a-aT Modulus of elasticity, E 160,000 Yield stress, 0,. 160 Poisson’s ratio, v 0.3 Coefficient of expansion, 0: 0.00002 Von Mises yield criterion. Associated ﬂow rule of plasticity. TARGET SOLUTION: Deflected shape, half-cycle 20 (displacement inagnification fac0or=40) Maximum deflection=0.651 Hoop stress distribution at thinnest section T Half-qcle 19 — - - - Hall-qde20 ' Target solution ° ° Tletarence sruiuion Depth (dhianoeauuwavdrﬂmunud—ihkhmne) tétééeeectr -an -in -in -u -4: 0 II I in lirnnp Stress NAFEMS NON-LIN EAR MATERIALS BENCHMARKS Case NL9 Issue: 1 Short tube of non-uniform thickness subjected to constant intemal pressure and cyclically varying linear through-thickriess temperature gradient. Inside bore is eccentric from outer surface by distance of 2. A two-dimensional section of the tube is taken, under plane stress conditions. FINITE ELEMENT MODEL: 8x24 mesh of 8~noded isoparametric membrane elements Plane stress conditions Y freedoms suppressed on symmetry line Prescribed nodal temperatures LOAD HISTORY: 250 Temperature 01 Steady intemal pressure, p=6.7413 22 Plastic zones, half-cycle 20 Results at thinnest section, half-cycle 20 Eﬂasﬁc Plesuc -159.67 0.003806 -0.00099 D.0DlBOS -121.57 0.003350 -0.00076 0.004610 -03.335 0.003003 —0.0005i 0.000405 -30.123 0.003939 -0.00019 0.004111 0.5455 0.003901 0.000050 0.003910 52.414 0.004000 0.000305 0.003519 101.55 0.004030 0.000533 0.003405 155.90 0.004003 0.000973 0.003109 159.7 0.004114 0.000999 0.003115 159.02 0.004157 0.000997 0.003159 158.91 0.001155 0.000596 0.003109 150.50 0.004222 0.001001 0.003220 157.95 0.004244 0.000995 0.003240 157.57 0.004273 0.001011 0.003251 156.56 0.004297 0.000992 0.003304 155.51 0.004330 0.001031 0.003299
- 27. NAFEMS NON—LIN EAR MATERIALS 10. THERMALLY INDUCED CREEP or INTERNALLY BENCHMARKS PRES SURISED SPHERE Case Nu” Issue: 1 Thick, hollow sphere subjected to constant internal pressure, p=30 Non-uniform temperature (°K) varying with radius R, T=333(1+100/R) Creep from time 0.0 to 101° ATTRIBUTES: FINITE ELEMENT MODEL: (target solution) _ Creep (forward deformation) Ten 8-noded axisymmetric quadrilateral elements — Thermal dependence of material properties — Stress redistribution — Long term numerical stability essential MATERIAL PROPERTIES: -— Young’s modulus: E = 10000. —- Poisson’s ratio: v = 0.25 — Coeff of expansion or = 0.0 — Creep law, écq = 3x10‘5 oggf exp(’ 150°/ T) where éeq equivalent creep strain rate oeq equivalent stress (Mises) T temperature (K) SOLUTION: IIIIIIIIIII IIIIIIIIIHI IIIII a~mmnul! :l E————s{I_I1=H —! !!2EDﬂ2H Eaasgsa--I TIME (LOG 1o) rnvna (LOG 10) Equivalent Suns VARIATION OF RADIAL DISPIAEMENT WITH TIME VARIATION OF EQUIVALENT STRESS WITH TIME 106 107 105 109 4x109 7x109101° E uivalem . 24.49 17.63 13.48 . 11.54 11.47 11.47 11.47 5:55 11.50 15.72 2.0.76 . 17.77 17.61 17.61 17.61 4.07 5.59 9.30 . 19.03 21.10 21.10 21.10 23