FEM for elastic-plastic problems

Transcription

1 FEM for elastic-plastic problems Jerzy Pamin With thanks to: P. Mika, A. Winnicki, A. Wosatko TNO DIANA FEAP

2 Lecture scope Physical nonlinearity Plastic flow theory Computational plasticity Simulation of plastic deformations Final remarks References [1] R. de Borst and L.J. Sluys. Computational Methods in Nonlinear Solid Mechanics. Lecture notes, Delft University of Technology, [2] G. Rakowski, Z. Kacprzyk. Metoda elementow skończonych w mechanice kostrukcji. Oficyna Wyd. PW, Warszawa, [3] M. Jirásek and Z.P. Bažant. Inelastic Analysis of Structures. J. Wiley & Sons, Chichester, 2002.

3 Incremental-iterative analysis Nonlinear problem: f ext applied in increments t t + t σ t+ t = σ t + σ Equilibrium at time t + t: where: n e A e T e=1 n e A e T e=1 B T σ t+ t dv = fext t+ t V e B T σ dv = fext t+ t V e f t int = n e e=1 Ae T V e B T σ t dv Linearization of the left-hand side at time t Equation set for an increment: σ = σ( ɛ( u)) K d = f t+ t ext f t int f t int

4 Physical nonlinearity Linearization of LHS at time t: σ = ( ) σ t ( ɛ t ɛ u) u D = σ ɛ, L = ɛ u Discretization: u = N d e K d = f t+ t ext f t int σ = σ( ɛ( u)) Linear geometrical relations Matrix of discrete kinematic relations B = LN independent of displacements Tangent stiffness matrix n e K = A e T B T D B dv A e V e e=1

5 Plastic yielding of material force A B C P σ y - A σ y B σ y - - C displacement σ y σ y σ y elastic material plastified elastic material microscopic level crystal shear dislocation lattice slip equivalent plastic strain distribution

6 Plastic flow theory [1,3] Load-carrying capacity of a material is not infinite, during deformation irreversible strains occur Notions of plasticity theory Yield function f (σ) = 0 - determines the limit of elastic response Plastic flow rule ɛ p = λm - determines the rate of plastic strain λ - plastic multiplier m - direction of plastic flow (usually associated with the yield function m T = n T = f σ ) Plastic hardening f (σ α, κ) = 0 kinematic (κ = 0) or isotropic (α = 0) Loading/unloading conditions: f 0, λ 0, λf = 0 (unloading is elastic)

7 Plastic flow theory Response is history-dependent, constitutive relations written in rates Plastic flow when f = 0 and ḟ = 0 (plastic consistency condition) Additive decomposition ɛ = ɛ e + ɛ p Bijective mapping σ = D e ɛ e Introduce flow rule σ = D e ( ɛ λm) Consistency ḟ = f f σ σ + κ κ Time integration necessary at the point level Hardening modulus f h = 1 λ κ κ Substitute σ into n T σ h λ = 0 Determine plastic multiplier λ = nt D e ɛ h+n T D e m Constitutive equation [ ] σ = D e De mn T D e ɛ h+n T D e m Tangent operator D ep = D e De mn T D e h+n T D e m

8 Huber-Mises-Hencky plasticity Most frequently used is the Huber-Mises-Hencky (HMH) plastic flow theory, based on a scalar measure of distortional energy J σ 2 Yield function e.g. with isotropic hardening f (σ, κ) = 3J σ 2 σ(κ) = 0 κ - plastic strain measure ( κ = 1 σ σt ɛ p = λ) Associated flow rule ɛ p = λ f σ Hardening rule e.g. linear σ(κ) = σ y + hκ h - hardening modulus

9 Response: force-displacement diagrams Ideal plasticity Hardening plasticity

10 Plastic flow theory Yield functions for metals: Coulomb-Tresca-Guest i Huber-Mises-Hencky (HMH) Insensitive to hydrostatic pressure p = 1 3 I σ 1

11 Plastic flow theory Yield functions for soil: Mohra-Coulomb i Burzyński-Drucker-Prager (BDP) Sensitive to hydrostatic pressure

12 Yield functions for concrete (plane stress) Kupfer s experiment Rankine yield function: f (σ, κ) = σ 1 σ(κ) = 0 Inelastic strain measure κ = ɛ p 1

13 Computational plasticity Return mapping algorithm backward Euler algorithm (unconditionally stable) 1) Compute elastic predictor σ tr = σ t + D e ɛ 2) Check f (σ tr, κ t ) > 0? If not then elastic compute σ = σ tr If yes then plastic compute plastic corrector σ = σ tr λd e m(σ) f (σ, κ) = 0 (set of 7 nonlinear equations for σ, λ) Determine κ = κ t + κ( λ) σ t f = 0 Iterative corrections still necessary unless radial return is performed. σ σ tr

14 Brazilian split test Elasticity, plane strain Deformation, vertical stress σ yy and stress invariant J σ 2

15 Brazilian split test Elasticity, mesh sensitivity of stresses Stress σ yy for coarse and fine meshes Stress under the force goes to infinity (results depend on mesh density) - solution at odds with physics

16 Brazilian split test Ideal Huber-Mises-Hencky plasticity Final deformation and stress σ yy

17 Brazilian split test Ideal Huber-Mises-Hencky plasticity Final strain ɛ yy and strain invariant J ɛ 2

18 Brazilian split test Ideal Huber-Mises-Hencky plasticity Force 400 Force This is correct! Displacement Displacement For four-noded element load-displacement diagram exhibits artificial hardening due to so-called volumetric locking, since HMH flow theory contains kinematic constraint - isochoric plastic behaviour which cannot be reproduced by FEM model. Eight-noded element does not involve locking.

19 Brazilian split test Elasticity, plane strain, eight-noded elements Deformation, vertical stress σ yy and stress invariant J σ 2

20 Brazilian split test HMH plasticity Final deformation and stress σ yy

21 Brazilian split test HMH plasticity Final strain ɛ yy and invariant J ɛ 2

22 Burzyński-Drucker-Prager plasticity Yield function with isotropic hardening f (σ, κ) = q + α p βc p (κ) = 0 q = 3J 2 - deviatoric stress measure p = 1 3 I 1 - hydrostatic pressure α = 6 sin ϕ 3 sin ϕ, β = 6 cos ϕ 3 sin ϕ ϕ - friction angle c p (κ) - cohesion Plastic potential f p = q + α p α = 6 sin ψ 3 sin ψ ψ - dilatancy angle Nonassociated flow rule ɛ p = λm, m = f p σ Plastic strain measure κ = η λ, η = ( α 2 ) 1 2 Cohesion hardening modulus h(κ) = ηβ cp κ HMH BDP q βc p Huber-Mises-Hencky yield function is retrieved for sin ϕ = sin ψ = 0 ϕ p

23 Slope stability simulation Gradient-enhanced BDP plasticity Evolution of plastic strain measure

24 Final remarks 1. In design one usually accepts calculation of stresses (internal forces) based on linear elasticity combined with limit state analysis considering plasticity or cracking. 2. In nonlinear computations one estimates the load multiplier for which damage/failure/buckling of a structure occurs. The multiplier can be interpreted as a global safety coefficient, hence the computations should be based on medium values of loading and strength.