G These slides are designed based on the book: Finite Elements in Plasticity Theory and Practice, D.R.J. Owen and E. Hinton, 1970, Pineridge Press Ltd., Swansea, UK. 1
Course Content: A INTRODUCTION AND OVERVIEW Numerical method and Comuter-Aided Engineering; Physical roblems; Mathematical models; Finite element method. B REVIEW OF 1-D FORMULATIONS Elements and nodes, natural coordinates, interolation function, bar elements, constitutive equations, stiffness matrix, boundary conditions, alied loads, theory of minimum otential energy; Plane truss elements; Examles. C PLANE ELASTICITY PROBLEM FORMULATIONS Constant-strain triangular (CST) elements; Plane stress, lane strain; Axisymmetric elements; Stress calculations; Programming structure; Numerical examles. 2
D ELASTIC-PLASTIC PROBLEM FORMULATIONS Iterative solution methods,1-d roblems, Mathematical theory of lasticity, matrix formulation, yield criteria, Equations for lane stress and lane strain case; Numerical examles E PHYSICAL INTERPRETATION OF FE RESULTS Case studies in solid mechanics; FE simulations in 3-D; Physical interretation; FE model validation.. 3
OBJECTIVE To learn the use of finite element method for the solution of roblems involving elasticlastic materials. 4
Stress-strain diagram 600 P Tegasan ( MPa) 400 200 A o l o l o +l 0 0.0 0.1 0.2 0.3 0.4 0.5 Terikan Ujikaji A Ujikaji B P Effects of high straining rates Effects of test temerature Foil versus bulk secimens 5
Elastic-Plastic Behavior Characteristics: (4) Post-yield deformation occurs at reduced material stiffness (2) Onset of yield governed by a yield criterion (3) Irreversible on unloading (1) An initial elastic material resonse onto which a lastic deformation is suerimosed after a certain level of stress has been reached 6
Monotonic stress-strain behavior Tyical low carbon steel Engineering strain Engineering stress True or natural strain d True stress dl l e l l l o S o dl l P A o P A l l o ln l l o For the assumed constant volume condition, Then; S 1 e A o ln ln 1 A Al A o l o e 7
Plastic strains Elastic region Hooke s law: E log log E (1) log Tegasan ( MPa) 600 400 200 SS316 steel Plastic region: n K log log K nlog Ujikaji A Ujikaji B 0 0.0 0.1 0.2 0.3 0.4 0.5 Terikan Engineering stress-strain diagram True stress-strain diagram 8
Plastic strains - Examle Non-linear /Power-law SS316 steel K n 700 600 747.3 0.199 STRESS, (MPa) 500 400 300 200 100 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 STRESS, (MPa) 1000 PLASTIC STRAIN, 100 = K n log K = 2.8735 n = 0.1992 r 2 = 0.9772 0.01 0.1 1 PLASTIC STRAIN, 9
Ideal Materials behavior Elstic-erfectly lastic Elstic-linear hardening erfectly lastic 10
Simulation of elastic-lastic roblem Requirements for formulating the theory to model elastic-lastic material deformation: stress-strain relationshi for ost yield behavior a yield criterion indicating the onset of lastic flow. exlicit stress-strain relationshi under elastic condition. 11
Elastic-lastic roblem in 2-D In elastic region: Ckl kl Ckl kl ik jl il jk, are Lamé constants Kronecker s delta: 1 0 if if i i j j 12
Elastic case for lane stress and lane strain (Review) Strain Stress u x x v y y xy u v y x x y xy Hooke s Law D 2 D Elasticity matrix 1 v 0 E v 1 0 1 v 1v 0 0 2 1 v v 0 E D v 1 v 0 1v1 2v 1 0 0 2 v Plane stress Plane strain 13
Yield criteria: Stress level at which lastic deformation begins k f k() is an exerimentally determined function of hardening arameter The material of a comonent subjected to comlex loading will start to yield when the (arametric stress) reaches the (characteristic stress) in an identical material during a tensile test. Other arameters: Strain Energy Secific stress comonent (shear stress, maximum rincial stress) 14
Yield criteria: Stress level at which lastic deformation begins k f Yield criterion should be indeendent of coordinate axes orientation, thus can be exressed in terms of stress invariants: J J J 1 2 3 ii 1 2 1 jk ki 3 15
Plastic deformation is indeendent of hydrostatic ressure f J J k 3 2, J 2, J 3 are the second and third invariant of deviatoric stress: 1 3 kk deviatoric hydrostatic + 16
Yield Criteria For ductile materials: Maximum-distortion (shear) strain energy criterion (von-mises) Maximum-shear-stress criterion (Tresca) For brittle materials: Maximum-rincial-stress criterion (Rankine) Mohr fracture criterion 17
Maximum-distortion-energy theory (von Mises) deviatoric hydrostatic + 2 2 2 2 1 2 2 3 3 1 2 Y For lane roblems: 2 2 2 1 1 2 2 Y 18
Maximum-normal-stress theory (Rankine) 1 2 ult ult 19
Tresca and von Mises yield criteria 20
Tresca and von Mises yield criteria 21
Strain Hardening behavior The deendence of stress level for further lastic deformation, after initial yielding on the current degree of lastic straining Strain Hardening models Perfect lasticity - Yield stress level does not deend on current degree of lastic deformation 22
Strain Hardening models Isotroic strain hardening -Current yield surfaces are a uniform exansion of the original yield curve, without translation -For strain soften material, the yield stress level at a oint decreases with increasing lastic strain, thus the original yield curve contracts rogressively without translation -Consequently, the yield surface becomes a failure criterion 23
Strain Hardening models Kinematic strain hardening -Subsequent yield surface reserves their shae and orientation but translates in the stress sace. -This results in Bauschinger effect on cyclic loading 24
Bauschinger effect 25
Progressive develoment of yield surface To relate the yield stress, k to the lastic deformation by means of the hardening arameter, Model 1: The degree of work hardening, is exressed as a function of the total lastic work, W W d Increment of lastic strain Model 2: The strain hardening, is related to a measure of the total lastic deformation termed the effective, generalized or equivalent lastic strain: d d 2 3 d d Since yielding is indeendent of hydrostatic stress, (d ii ) = 0), thus: d d d 2 3 d d 26
Isotroic Hardening For a stress state where f = k reresents the lastic state while elastic behavior is characterized by f < k, an incremental change in the yield function is: df f d df df df 0 0 0 Elastic unloading Neutral loading (lastic behavior for erfectly-lastic material) Plastic loading (for strain hardening materials) 27
Elastic-lastic stress-strain relations d d d In lastic region: e d Elastic strain increment: 1 2 d e 2 E deviatoric d kk hydrostatic Plastic Flow Rule: Assumes that the lastic strain increment is roortional to the stress gradient of the lastic otential, Q: d d Q Q d Plastic multilier 28
Associated theory of lasticity Since the Flow Rule governs the lastic flow after yielding, thus Q must be a function of and J, similar to f, thus: J 3 2 d f Q f d Normality condition 29
Associated theory of lasticity Since the Flow Rule governs the lastic flow after yielding, thus Q must be a function of and J, similar to f, thus: J 3 2 d f Q f d Normality condition Particular case of 2 f J 2 f J d d Prandtl-Reuss equations 30
The incremental stress-strain relationshi for elastic-lastic deformation: d d d e d 1 2 d e 2 E d kk f d 31
Uniaxial elastic-lastic strain hardening behavior Definition of term: E T d d 32
Uniaxial elastic-lastic strain hardening behavior Yield condition f k Strain hardening hyothesis H Proortional to J 2 d d H 33
Uniaxial elastic-lastic strain hardening behavior For uniaxial case 0 1, 2 3 d 1 d d d 2 d 3 2 2 3 3 0.5d d Plastic straining is assumed to be incomressible, = 0.5 H d d d d ET ET 1 E - The hardening function, H is determined exerimentally from tension test data. - H is required 34
Uniaxial stress-strain curve for elastic-linear hardening E T d d Assume d d d e Strain-hardening arameter: H d d 35
Numerical solution rocess for non-linear roblems 36
Notations: Linear elastic roblem: Nonlinear roblem: K Q F 0 K Q F 0 H f 0 H f 0 Thus, iterative solution is required For 1-D (1 variable) roblem: 0 H f 37
Iterative Solution Methods The roblem: H f 0 H f 0 0 H f For single variable Direct iteration / successive aroximation Newton-Rahson method Tangential stiffness method Initial stiffness method 38
Iterative Solution Methods (a) Direct iteration or successive aroximation In each solution ste, the revious solution for the unknowns is used to redict the current values of the coefficient matrix H f 0 H -1 f For (r+1) th aroximation: H H r1 r -1 f Initial guess 0 is based on solution for an average material roerties. Convergence is not guaranteed and cannot be redicted at initial solution stage. Converged iteration when r-1 and r are close. 39
Task: to illustrate the rocess. Direct iteration method for a single variable Convex H- relation 40
Stes for Direct Iteration Method Start with { 0 } Determine [H( 0 )] Calculate 0 H 1 Reeat until { r+1 } { r } 1 f 0 H( 0 ) 1 = -H( 0 ) f r+1 r 41
Direct iteration method for a single variable Concave H- relation 42
Assignment # 5 A one degree of freedom roblem can be reresented by H + f = 0 where f = 10 and H() = 10 (1+e 3 ) Design and run a comuter algorithm to determine the solution of the roblem using direct iteration method and Newton- Rahson method. (a) Show the flow chart of the algorithm and descrition of the stes. (b) Tabulate the iterative values of H and and comare the converged solution from the two methods. Secify a convergence tolerance of 1%. Try with 0 = 0.2 Should converge at 7 = -9.444 43
Review of Newton-Rahson Method (for finding root of an equation f(x)= 0) Taylor Series exansion: f 2 x f x x x f x x x f x... 1 2! 1 1 1 1 1 Consider only the first 2 terms: f x f x x x f 1 1 x1 Set f(x)=0 to find roots: f x x x x1 f x1 f x1 x1 f x 1 0 1 44
Review of Newton-Rahson Method (for finding root of an equation f(x)= 0) Iterative rocedures: x i1 x i f f x x 1 1 45
Newton-Rahson Method The roblem Ψ H f H f 0 0 is the residual force If true solution exists at r r ψ r i N j1 r ψ i Δ j j r J r r r Initial guess is based on 0 is based on solution for an average material roerties. Converged iteration when n-1 and n are close 46
MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS Jacobian Matrix 47 H H J ψ h J m 1 k r k r j i r j i r r r r r r H H J 1 1 r r r 1
Newton-Rahson method for a single variable 48
Newton-Rahson method for a single variable 49
Tangential Stiffness Method (Generalized Newton-Rahson Method) Linearize {} in [H()] such that the term [H ()] can be omitted Stes: H f Assume a trial value { 0 } Calculate [H( 0 )] Calculate {( 0 )} Calculate Iterate until { r } {0} 0 0 1 0 H 1 0 0 50
Tangential stiffness method for a single variable 51
Initial Stiffness Method Recall that in Direct Stiffness Method: H r r 1 1 f In Tangential Stiffness Method: r r 1 H This requires comlete reduction and solution of the set of simultaneous equations for each iteration. Use the initial stiffness [H( 0 )] for subsequent aroximation. r 0 H 1 r r 52
Initial stiffness method for a single variable 53
Uniaxial stress-strain curve for elastic-linear hardening Assume d d d e Strain-hardening arameter: H d d 54
MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS 55 1 1 1 1 ) ( 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 e ; ; H E E L EA L d E d A d H d df K e 1 1 1 1 1 1 ) ( H E E L EA K e e Element stiffness for elastic-lastic material behavior 1 2 L F Elastic behavior: Elastic-lastic behavior: Element stiffness matrix for elasticlastic material behavior
Numerical singularity issue S Y Sloe, H = 0 ( ) EA E 1 1 K e 1 e L E H 1 1 After initial yielding, H =0, Thus, [K e ] (e) = [0] Use initial stiffness method to ensure ositive definite [K] 56
LOAD INCREMENT LOOP ITERATION LOOP MMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS Inut geometry, loading, b.c. Program structure for 1-D roblem INITIALIZATION Zero all arrays for data storing INC. LOAD SET INDICATOR STIFFNESS Choose tye of solution algorithm direct iteration, tangential stiffness, etc [K] (e) ASSEMBLY REDUCTION RESIDUAL [K] {} = {F} Solve for {} Calculate residual force {} for Newton-Rahson, Initial and Tangential stiffness method N Y Check for convergence Outut results 57
Incremental stress and strain changes at initial yielding Task: to illustrate the rocess. r1 r r r e r e H r 1 r H r1 r1 r E f r e Check if the element has reviously yielded: H r1 r1 Y Y 58
Engineering stress-strain curve for Al2219 59