G These slides ae designed based on the book: Finite Elements in Plasticity Theoy and Pactice, D.R.J. Owen and E. Hinton, 970, Pineidge Pess Ltd., Swansea, UK.
Couse Content: A INTRODUCTION AND OVERVIEW Numeical method and Compute-Aided Engineeing; Physical poblems; Mathematical models; Finite element method. B REVIEW OF -D FORMULATIONS Elements and nodes, natual coodinates, intepolation function, ba elements, constitutive equations, stiffness matix, bounday conditions, applied loads, theoy of minimum potential enegy; Plane tuss elements; Examples. C PLANE ELASTICITY PROBLEM FORMULATIONS Constant-stain tiangula (CST) elements; Plane stess, plane stain; Axisymmetic elements; Stess calculations; Pogamming stuctue; Numeical examples. 2
D ELASTIC-PLASTIC PROBLEM FORMULATIONS Iteative solution methods,-d poblems, Mathematical theoy of plasticity, matix fomulation, yield citeia, Equations fo plane stess and plane stain case; Numeical examples E PHYSICAL INTERPRETATION OF FE RESULTS Case studies in solid mechanics; FE simulations in 3-D; Physical intepetation; FE model validation.. 3
OBJECTIVE To lean the use of finite element method fo the solution of poblems involving elasticplastic mateials. 4
Stess-stain diagam 600 P Tegasan ( MPa) 400 200 A o l o l o +l 0 0.0 0. 0.2 0.3 0.4 0.5 Teikan Ujikaji A Ujikaji B P Effects of high staining ates Effects of test tempeatue Foil vesus bulk specimens 5
Elastic-Plastic Behavio Chaacteistics: (4) Post-yield defomation occus at educed mateial stiffness (2) Onset of yield govened by a yield citeion (3) Ievesible on unloading () An initial elastic mateial esponse onto which a plastic defomation is supeimposed afte a cetain level of stess has been eached 6
Monotonic stess-stain behavio Typical low cabon steel Engineeing stain Engineeing stess Tue o natual stain d Tue stess dl l e l l l o S o dl l P A o P A l l o ln l l o Fo the assumed constant volume condition, Then; S e A o ln ln A Al A o l o e 7
Plastic stains Elastic egion Hooke s law: E log log E () log Tegasan ( MPa) 600 400 200 SS36 steel Plastic egion: n K p log log K nlog Ujikaji A Ujikaji B 0 0.0 0. 0.2 0.3 0.4 0.5 Teikan Engineeing stess-stain diagam Tue stess-stain diagam 8
Plastic stains - Example Non-linea /Powe-law SS36 steel K n p 700 600 747.3 p 0.99 STRESS, (MPa) 500 400 300 200 00 0 0.0 0. 0.2 0.3 0.4 0.5 0.6 STRESS, (MPa) 000 PLASTIC STRAIN, p 00 = K n log K = 2.8735 n = 0.992 2 = 0.9772 0.0 0. PLASTIC STRAIN, p 9
Ideal Mateials behavio Elstic-pefectly plastic Elstic-linea hadening pefectly plastic 0
Simulation of elastic-plastic poblem Requiements fo fomulating the theoy to model elastic-plastic mateial defomation: stess-stain elationship fo post yield behavio a yield citeion indicating the onset of plastic flow. explicit stess-stain elationship unde elastic condition.
Elastic-plastic poblem in 2-D In elastic egion: ij Cijkl kl Cijkl ij kl ik jl il jk, ae Lamé constants Konecke s delta: ij 0 if if i i j j 2
Elastic case fo plane stess and plane stain (Review) Stain Stess u x x v y y xy u v y x x y xy Hooke s Law D 2 D Elasticity matix v 0 E v 0 v v 0 0 2 v v 0 E D v v 0 v 2v 0 0 2 v Plane stess Plane stain 3
Yield citeia: Stess level at which plastic defomation begins k f ij k() is an expeimentally detemined function of hadening paamete The mateial of a component subjected to complex loading will stat to yield when the (paametic stess) eaches the (chaacteistic stess) in an identical mateial duing a tensile test. Othe paametes: Stain Enegy Specific stess component (shea stess, maximum pincipal stess) 4
Yield citeia: Stess level at which plastic defomation begins k f ij Yield citeion should be independent of coodinate axes oientation, thus can be expessed in tems of stess invaiants: J J J 2 3 ii ij ij 2 ij jk ki 3 5
Plastic defomation is independent of hydostatic pessue f J J k 3 2, J 2, J 3 ae the second and thid invaiant of deviatoic stess: ij ij ij 3 kk deviatoic hydostatic + 6
Yield Citeia Fo ductile mateials: Maximum-distotion (shea) stain enegy citeion (von-mises) Maximum-shea-stess citeion (Tesca) Fo bittle mateials: Maximum-pincipal-stess citeion (Rankine) Moh factue citeion 7
Maximum-distotion-enegy theoy (von Mises) deviatoic hydostatic + 2 2 2 2 2 2 3 3 2 Y Fo plane poblems: 2 2 2 2 2 Y 8
Maximum-nomal-stess theoy (Rankine) 2 ult ult 9
Tesca and von Mises yield citeia 20
Tesca and von Mises yield citeia 2
Stain Hadening behavio 22
Nomality ule of associated plasticity 23
Notations: Linea elastic poblem: Nonlinea poblem: K Q F 0 K Q F 0 H f 0 H f 0 Thus, iteative solution is equied Fo -D ( vaiable) poblem: 0 H f 24
Iteative Solution Methods The poblem: H f 0 H f 0 0 H f Fo single vaiable Diect iteation / successive appoximation Newton-Raphson method Tangential stiffness method Initial stiffness method 25
Iteative Solution Methods (a) Diect iteation o successive appoximation In each solution step, the pevious solution fo the unknowns is used to pedict the cuent values of the coefficient matix H f 0 H - f Fo (+) th appoximation: H H - f Initial guess 0 is based on solution fo an aveage mateial popeties. Convegence is not guaanteed and cannot be pedicted at initial solution stage. Conveged iteation when - and ae close. 26
Task: to illustate the pocess. Diect iteation method fo a single vaiable Convex H- elation 27
Steps fo Diect Iteation Method Stat with { 0 } Detemine [H( 0 )] Calculate 0 H Repeat until { + } { } f 0 H( 0 ) = -H( 0 ) f + 28
Diect iteation method fo a single vaiable Concave H- elation 29
Assignment # 5 A one degee of feedom poblem can be epesented by H + f = 0 whee f = 0 and H() = 0 (+e 3 ) Design and un a compute algoithm to detemine the solution of the poblem using diect iteation method and Newton- Raphson method. (a) Show the flow chat of the algoithm and desciption of the steps. (b) Tabulate the iteative values of H and and compae the conveged solution fom the two methods. Specify a convegence toleance of %. Ty with 0 = 0.2 Should convege at 7 = -9.444 30
Review of Newton-Raphson Method (fo finding oot of an equation f(x)= 0) Taylo Seies expansion: f 2 x f x x x f x x x f x... 2! Conside only the fist 2 tems: f x f x x x f x Set f(x)=0 to find oots: f x x x x f x f x x f x 0 3
Review of Newton-Raphson Method (fo finding oot of an equation f(x)= 0) Iteative pocedues: x i x i f f x x 32
Newton-Raphson Method The poblem Ψ H f H f 0 0 is the esidual foce If tue solution exists at ψ i N j ψ i Δ j j J Initial guess is based on 0 is based on solution fo an aveage mateial popeties. Conveged iteation when n- and n ae close 33
MMJ53 COMPUTATIONAL METHOD IN SOLID MECHANICS Jacobian Matix 34 H H J ψ h J m k k j i ij j i ij H H J
Newton-Raphson method fo a single vaiable 35
Newton-Raphson method fo a single vaiable 36
Tangential Stiffness Method (Genealized Newton-Raphson Method) Lineaize {} in [H()] such that the tem [H ()] can be omitted Steps: H f Assume a tial value { 0 } Calculate [H( 0 )] Calculate {( 0 )} Calculate Iteate until { } {0} 0 0 0 H 0 0 37
Tangential stiffness method fo a single vaiable 38
Initial Stiffness Method Recall that in Diect Stiffness Method: H f In Tangential Stiffness Method: H This equies complete eduction and solution of the set of simultaneous equations fo each iteation. Use the initial stiffness [H( 0 )] fo subsequent appoximation. 0 H 39
Initial stiffness method fo a single vaiable 40
Uniaxial stess-stain cuve fo elastic-linea hadening Assume d d d e p Define stain-hadening paamete: H d d p 4
MMJ53 COMPUTATIONAL METHOD IN SOLID MECHANICS 42 ) ( L EA K L EA F K e e e A d H A d df A F L d d L d d L p e ; ; H E E L EA L d E d A H d d df K p p ep ) ( H E E L EA K e ep Element stiffness fo elastic-plastic mateial behavio 2 L F Elastic behavio: Elastic-plastic behavio: Element stiffness matix fo elasticplastic mateial behavio
Numeical singulaity issue S Y Slope, H = 0 ( ) EA E K e ep L E H Afte initial yielding, H =0, Thus, [K ep ] (e) = [0] Use initial stiffness method to ensue positive definite [K] 43
LOAD INCREMENT LOOP ITERATION LOOP MMJ53 COMPUTATIONAL METHOD IN SOLID MECHANICS Input geomety, loading, b.c. Pogam stuctue fo -D poblem INITIALIZATION Zeo all aays fo data stoing INC. LOAD SET INDICATOR STIFFNESS Choose type of solution algoithm diect iteation, tangential stiffness, etc [K] (e) ASSEMBLY REDUCTION RESIDUAL [K] {} = {F} Solve fo {} Calculate esidual foce {} fo Newton-Raphson, Initial and Tangential stiffness method N Y Check fo convegence Output esults 44
Incemental stess and stain changes at initial yielding Task: to illustate the pocess. e e H H E f e Check if the element has peviously yielded: H Y Y p 45
Uniaxial elastic-plastic stain hadening behavio 46
Engineeing stess-stain cuve fo Al229 47