Here assumptions themselves act as the demerit to the theory & are self explanatory that why this particular theory is unpopular in comparison to FEM. Further a lot depends on the mathematical assumptions &simplifications
The value of deformation itself is asssumed either to be higher or lower than the initial value. More the assumptions, more would be the errors. Further plane strain conditions mean that, components engaged in multi axial state of stress cannot be probed. Friction effects cannot be neglected in true conditions.
But in reality the material being deformed is heterogeneous most of the time (reinforced tyre, concrete etc). Here to assumptions act as demerits.
Finite element analysis in metal forming
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What is analysis? Imagination384,400 kilometers 200 meters "In 1 million to 10 million years they might be able to make a plane that would fly." -The New York Times, 1903 Application Model Abhishek & Jitendra 2
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Methods of analysis Mode of analysis Analytical & numerical Empirical methods Methods Upper & Slip Line Finite Finite Lower VisioElementary field Similarity plastic theory Element Difference Bound theory theory Method method (FDM) method method (FEM) Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt Abhishek & Jitendra 3
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Elementary plasticity theoryApproach:Establishment of kinetics relative to the process.Establishment of differential equations suiting the process & the simplification. Abhishek & Jitendra 4
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Upper & Lower boundary method Method approximates the values of deforming forces to be higher or lower than actual forces.“Any estimate of the collapse load of a structure made by equating the rate of the energy dissipation internally to the rate at which external forces do work, in some assumed pattern of the deformation will be greater than or equal to the correct load” -W.F. Hosford & R.M CaddellAssumptions of the method: Material being deformed is isotropic & homogenous There is no effect of work hardening William Hosford No friction exists between work piece & tool interface. Plane strain conditions assumed. Abhishek & Jitendra 5
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Slip-line theoryHere flow pattern from point to point while deformation is considered & analyzed. due to frictionlessSlip line refers to the planes of maximum upsetting shear stress which are inclined at 45o to the principle planes. Slip line on due toAssumptions of the method: the Symmetry edge Material being deformed is isotropic & homogenous Symmetry There is no effect of work hardening & strain plane rate. No friction exists between work piece & tool interface. Plane strain conditions assumed. Image courtesy: Lecture notes, Fundamental of solving methods, Effect of temperature , strain rate, & time Prof Dr. –Ing. G. Hirt neglected. Abhishek & Jitendra 6
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Comparison FEM • Material flow analysis & local states of stress & strain described. • Various boundary conditions can be applied. • Multi-axial stress in considerationAnalytical methods• Only Global analysis is done.• Material homogeneity is assumed.• 2-deminsional conditions.• Temperature effects neglected. Abhishek & Jitendra 7
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Finite Element AnalysisA brief history Concept was developed by the works of Richard Courant & Alexander Hrennikoff (early 40’s). Idea was originated to solve complex problems of civil engineering & structural analysis. Richard Courant Idea was promoted by Boeing to compute sweep of airplane wings (mid 50’s). M.J Turner & Ray W. Clough articles established the applications of FEA (mid 50’s). Idea was also used to compute roof of Munich Olympic stadium (late 60’s) Abhishek & Jitendra R.W.Clough 8
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Areas of applications Engineering • Fluid mechanics • Thermodynamics • Metal Forming etc Biological Sciences • Botany • Zoology • Archeological Anthropology • Paleontology General application • Geology • Astrophysics Abhishek & Jitendra 9
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Engineering applicationsNumerical assessment of static & seismic behavior of Department of Atomic & Solid state Hochschule Regensburg,the Basilica of Santa Maria all’Impruneta (Italy) physics University of Cornell Biomechanik Department of electronics & C-blade Forging & Lehrstuhl Numerische Mathematik, Telecommunications, University of Manufacturing Ruprecht-Karls-Universität, Heidelberg Naples,Italy 10 Abhishek & Jitendra
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Hierarchy of FEM Physical ProblemEstablish Finite element model of the physical problem Solve the problem Interpret the result Abhishek & Jitendra 11
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Space IncrementationFinite Elements: Every model is sub-divided into finite elements. Their junction points are called as nodes. Model assumes that forces act at nodes & stresses & strain exist at the finite element. Reliability of FEA depends on number of finite elements. Abhishek & Jitendra 12
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Stiffness MatrixExample: Beams protruding from fixed surface u E A 1 1 u1 F1 L 1 1 u2 F2 K u F Stiffness Coeffecient Displacement Force F1 F2 u1 u 2 E A 1 1 0 u1 F1 F1 (u 2 u1 ) E A L 1 2 1 u2 F2 L E A 0 1 1 u3 F3 F2 (u1 u2 ) L Stiffness Matrix 13
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Space IncrementationElement types: Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt Abhishek & Jitendra 14
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Space IncrementationMeshingNetwork of nodes is called a mesh.There are 2 broad mesh-generation methods. Unstructured( Formed automatically) A Structured (Formed by grid based sub-dividing of geometry) B Abhishek & Jitendra 15
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Space IncrementationMeshing: Accuracy of results always depends on the assumptions. Fine mesh is considered where there are stress & strain gradients. A coarse mesh is used in the areas of reasonably constant stress or areas of interest. Abhishek & Jitendra 16
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Protocols Gaps are not permitted during meshing. Nodes are numbered sequentially. Abhishek & Jitendra 17
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Space IncrementationApproaches:LAGRANGE’S approach Mesh is bound to the material Mesh will be distorted with increasing deformation. Courtesy: FHWA. USA Abhishek & Jitendra 18
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Space IncrementationEULER’S Approach Mesh is fixed & not bound to the material. Material flows through fixed mesh. Abhishek & Jitendra 19
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Space IncrementationRemeshingWhy is it necessary? Formation of unacceptable shapes due to large local deformations. High relative motion between die surface & deforming material. Large displacement causes computational problems. Difficulties encountered in incorporating die boundary shapes with increase in relative displacement.To overcome above difficulties, periodic redefining of mesh is necessary Abhishek & Jitendra 21
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Space IncrementationRemeshing comprises of following steps:1. Assignment of new mesh system to work piece2. Transfer of information (strain, strain rate, & temperature) from the old to the new mesh through interpolation. Image courtesy: emerald.com Abhishek & Jitendra 22
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SolversFor simulation of metal forming, following 2 solutions are used: Implicit method ( Stable, iterative, high computational effort) Explicit method (conditionally stable, no iteration, less computational effort) Abhishek & Jitendra 23
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Implicit solvers Studies reveal that this solver is useful in smaller & 2D problems. Each time step or increment has to be treated as unconditionally stable process. Large time steps lead to larger iterations & process do not converge. (Newton Raphson method )Non- linear analysis of reinforced Abhishek & Jitendra concrete beam 24
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Implicit Solvers In the implicit approach a solution to the set of finite element equations involves iteration until a convergence criterion is satisfied for each increment. Here computation is divided into several calculation time steps. At the end of each time step(increment) the equilibrium between internal & external load must be reached. Else iteration continues. Abhishek & Jitendra 25
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Explicit solvers The finite element equations in the explicit approach are reformulated as being dynamic. In this form they can be solved directly to determine the solution at the end of the increment, without iteration. Two methods are followed for time step calculations. Abhishek & Jitendra 26
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Explicit solversHere largest allowable time step for a stable solution depends on: Highest Eigen frequency occurring (ωmax )in the system Corresponding damping (ξ) ∆tm ≤ (2/ωmax)* ((1+ξ2)0.5-ξ)Sonic frequency & smallest element Le are estimated as follows: ∆ t ≤Le /C with C=(E/ρ)0.5 To compensate the disadvantage of extremely small time step, will be reduced through increasing the density or shortening the process time. Abhishek & Jitendra 27
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Computational time required for explicit/implicit methods(Calculation time)Complexity Efficiency Implicit Implicit Explicit Explicit Model-size Statics Structural Highly dynamics dynamic implicit: Complexity ~ number of freedom degrees x wave front explicit: Complexity ~ number of freedom degrees Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt
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Non-linearities in FEM Following Non-linearities are encountered during the simulations. Geometrical Non-linearity Material Non-linearity Contact variance (Change of boundary conditions) Friction Abhishek & Jitendra 29
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Geometrical Non-linearity In practical cases it is not uncommon to encounter strain of magnitude 2 or more due to : Large elongation Large rotation Courtesy: MRF tyres, India Portions of rigid body movements Abhishek & Jitendra 30
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Geometrical Non linearityHydroforming Operation Tools Upper part Tube Lower part In consideration of Geometrical nonlinearity geometrical nonlinearity neglected Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt
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Material Non-linearityOccurs when: Transition of elastic to plastic phase Depends on ρ,θ, λ, CPNote: This non-linearity is important when considering thermal effects z.B hot forming or for calculation of temperature Courtesy: COMSOL, USA increase during forming process. Abhishek & Jitendra 32
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Material Non-linearity during tensile test considered Material-nonlinearity (flow curve) Not considered kf = kf ( v, v) Initial mesh kf = 100 N/mm2 = const.(strain hardened!) Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt
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Contact Non-linearityChanging contact changes:a.) Mechanical Boundariesb.) Thermal Boundaries.Types of contacts in metal forming1.) Contacts with rigid tools2.) Contacts with deforming tools Courtesy: ICS, Switzerland3.) Self contact Abhishek & Jitendra 34
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FrictionFriction is non-linear. Friction leads to asymmetrical equation system. This increases the calculation complexity.Categorization:1. τ<µσN -- Sticking friction2. τ=µσN – Slide friction Abhishek & Jitendra 35
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Why is FEM advantageous over other solving methods with frictionwithout friction Abhishek & Jitendra 36
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Comparision with & without friction during upsetting Upsetting without friction with friction Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. –Ing. G. Hirt
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Vote of thanks & ReferencesSincerely indebted to:Prof Dr. –Ing. G. Hirt, Head of the department, IBF, RWTH AachenDipl.-Ing. Simon Seuren, IBF, RWTH AachenInstitute of Metal Forming, RWTH AahcenReferences: Fundamentals of solving methods in metal forming by Prof.Dr.- Ing.G.Hirt Metal forming & finite element method –Atlan,Oh, Kobayashi Manufacturing process III –A.C.Niranjan Comparison of the implicit and explicit finite element methods using crystal plasticity- F.J. Harewood , P.E. McHughWeb resources: National program on technology enhanced learning, Dr. R. Krishnakumar, IIT madras. Abhishek & Jitendra 38
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Thank you for your patience & kindattention! Abhishek & Jitendra 39
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