PLASTIC ANALYSIS IN FEM
INTRODUCTION TO COMPUTATIONAL MECHANICS OF MATERIALS Civil Engineering, 1st cycle studies, 7th semester
elective subject academic year 2014/2015
Institute L-5, Faculty of Civil Engineering, Cracow University of Technology
Jerzy Pamin Adam Wosatko
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Lecture contents
Elasticity vs plasticity Plastic flow theory Computational plasticity
Simulations of plastic deformations References:
A. Ganczarski i J. Skrzypek.
Plastyczność materiałów inżynierskich. Podstawy, modele, metody i zastosowania komputerowe.
Skrypt PK, Politechnika Krakowska, Kraków, 2009.
R. de Borst and L.J. Sluys.
Computational methods in nonlinear solid mechanics.
Lecture Notes, Delft University of Technology, Delft, 1999.
DIANA Finite Element Analysis - User’s manual, release 7.2.
TNO Building and Construction Research, Delft, 1999.
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Linear elasticity
Hooke’s law Tensor notation σ = D e : σ ij = D ijkl e kl
Voigt’s notation
σ = D e , σ =
σ x
σ y
σ z
τ xy
τ yz
τ zx
, =
x
y
z
γ xy
γ yz
γ zx
Isotropy
D e = D e (E , ν)
E 1 σ
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Weak form of governing equation
Principle of virtual work (small deformations assumed):
Z
V
δ T σdV = Z
V
δu T bdV + Z
S n
δu T tdS
Boundary condition: u = d on S e
Voigt’s notation:
– generalized strain vector σ – generalized stress vector u – generalized displacement vector b – mass force vector
t – traction vector
d – imposed displacement vector
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Elasticity versus plasticity
Micromechanical background
crystal lattice shear dislocation slip
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Tensile bar test for steel
Stress-strain diagram
1 proportionality limit 2 elasticity limit
3 yield strength σ y (upper, lower limit)
4-5-6 plastic hardening
Figure from the book of Ganczarski and Skrzypek [1]
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Tensile bar test for steel
Stress-strain diagram
Diagram σ– depends on:
I material
I loading rate (speed of the process)
I environment temperature spring steel, stainless steel, aluminium alloy, low-carbon steel, brass, copper
Figure from the book of Ganczarski and Skrzypek [1]
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Simplified diagrams for plastic materials
stiff-plastic
elastic-plastic
Figure from the book of Ganczarski and Skrzypek [1]
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Plastic state in a beam
A B C
displacement P
A
+
-
σ y
σ y
σ y
σ y
σ y
σ y
+
- -
+ C B
force
elastic material
plastified
elastic material
plastified
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Plastic flow theory
Load-carrying capacity of a material is not infinite, during deformation irreversible strains occur
Notions of plasticity theory
I Yield function f (σ) = 0
- determines the limit of elastic response
I Plastic flow rule ˙ 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 = ∂σ ∂f )
I Plastic hardening f (σ − α, κ) = 0 kinematic (κ = 0) or isotropic (α = 0)
I Loading/unloading conditions:
f ≤ 0, ˙λ ≥ 0, ˙λf = 0 (unloading is elastic)
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Plastic flow theory
Response is history-dependent, hence constitutive relations written in terms of rates (infinitesimal increments)
Plastic flow when f = 0 and ˙f = 0
(plastic consistency condition) Additive decomposition
˙ = ˙ e + ˙ p Bijective mapping
˙
σ = D e ˙ e
Introduce flow rule
˙
σ = D e ( ˙ − ˙λm) Consistency
˙f = ∂σ ∂f σ + ˙ ∂κ ∂f ˙κ
Hardening modulus h = − 1 ˙
λ
∂f
∂κ ˙κ Substitute ˙ σ into n T σ − h ˙λ = 0 ˙
Determine plastic multiplier
˙λ = h+n n T T D D e e ˙ m
Constitutive equation
˙ σ = h
D e − h+n D e mn T T D D e m e
i
˙
Tangent operator D ep = D e − h+n D e mn T D T D e m e
Time integration necessary at the point level
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
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 2 σ
I Yield function
e.g. with isotropic hardening f (σ, κ) = p3J 2 σ − ¯ σ(κ) = 0 κ - plastic strain measure ( ˙κ = 1 ¯ σ σ T ˙ p = ˙λ)
I Associated flow rule
˙ p = ˙λ ∂σ ∂f
I Hardening rule e.g. linear
¯
σ(κ) = σ y + hκ h - hardening modulus
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Plastic flow theory
Yield functions for metals:
Coulomb-Tresca-Guest i Huber-Mises-Hencky (HMH)
Insensitive to hydrostatic pressure p = 1 3 I 1 σ
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Plastic flow theory
Yield functions for soil:
Mohra-Coulomb i Burzyński-Drucker-Prager (BDP)
Sensitive to hydrostatic pressure
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Computational plasticity
Return mapping algorithm
→ backward Euler algorithm (unconditionally stable) 1) Compute elastic predictor
σ tr = σ t + D e ∆
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 σ tr
σ t
f = 0
Iterative corrections still necessary unless radial return is performed.
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Tension of perforated plate
Force-displacement diagram
0 200 400 600 800 1000 1200 1400 1600
0 0.5 1 1.5 2
Displacement [mm]
plasticity with hardening
F o rce [N]
ideal plasticity
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Tension of perforated plate
Deformation and invariant J 2
Ideal plasticity Plasticity with hardening
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Brazilian split test
Elasticity, plane strain
Deformation, vertical stress σ yy and stress invariant J 2 σ
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Brazilian split test
HMH plasticity
Final deformation and stress σ yy
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Brazilian split test
HMH plasticity
Final strain yy and invariant J 2
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Brazilian split test
Load-displacement diagrams
Perfect plasticity Hardening plasticity
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
Burzyński-Drucker-Prager plasticity
I 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 = 3−sin ϕ 6 sin ϕ , β = 3−sin ϕ 6 cos ϕ ϕ - friction angle
c p (κ) - cohesion
I Plastic potential f p = q + α p α = 3−sin ψ 6 sin ψ
ψ - dilatancy angle Nonassociated flow rule
˙ p = ˙λm, m = ∂f ∂σ p
I Plastic strain measure
˙κ = η ˙λ, η = (1 + 2 9 α 2 ) 1 2
I Cohesion hardening modulus h(κ) = ηβ ∂c ∂κ p
c
pq
p ϕ HMH
BDP
Huber-Mises-Hencky yield function is retrieved for sin ϕ = sin ψ = 0
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education
Slope stability simulation
Gradient-enhanced BDP plasticity
Evolution of plastic strain measure
Project ”The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union
