MODELING OF CONCRETE MATERIALS AND STRUCTURES
Kaspar Willam
University of Colorado at Boulder
Class Meeting #5: Integration of Constitutive Equations
Structural Equilibrium:
Incremental Tangent Stiffness and Residual Force Iteration
Radial Return Method:
Elastic Predictor-Plastic Corrector Strategy
Algorithmic Tangent Operator:
Consistent Tangent with Backward Constitutive Integration
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
STRUCTURAL EQUILIBRIUM
Out-of-Balance Force Calculation:
u) = F int − F ext → 0
Out-of Balance Residual Forces:
R (u
P R
Internal Forces: F int = e V B tσ dV
P R
P R
t
External Forces: F ext = e V N b dV + e S N tt dS
Rate of Equilibrium:
XZ
e
V
σ dV =
B tσ̇
XZ
e
V
N tḃbdV +
XZ
e
N tṫtdS
S
1. Incremental Methods of Numerical Integration: Path-following continuation
strategies to advance solution within ∆t = tn+1 − tn
(a) Explicit Euler Forward Approach: Forward tangent stiffness strategy (should
include out-of-balance equilibrium corrections to control drift).
(b) Implicit Euler Backward Approach: Backward tangent stiffness
(requires iteration for calculating tangent stiffness at end of increment; Heun’s
method at midstep, and Runge-Kutta h/o methods at intermediate stages).
INCREMENTAL SOLVERS
Euler Forward Integration: Classical Tangent stiffness approach
P R
σ dV
F ext = e V B tσ̇
Internal forces: Ḟ
σ = Ė
E tan˙
Tangential Material Law: σ̇
Tangential Stiffness Relationship:
u = Ḟ
F
K tanu̇
where K tan =
XZ
e
Incremental Format:
R un+1
un
Euler Forward Integration:
B tE tanB dV
V
u = F n+1 − F n
K tandu
K ntan
=
P R
e
t n
B
E tanB dV
V
u = ∆F
F where E ntan = E tan(tn)
K ntan∆u
Note: Uncontrolled drift of response path if no equilibrium corrections are
included at each load step.
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
2. ITERATIVE SOLVERS
Picard direct substitution iteration vs Newton-Raphson iteration within
∆t = tn+1 − tn
Robustness Issues: Range and Rate of Convergence?
Newton-Raphson Residual Force Iteration:
u) = 0
R (u
Truncated Taylor Series Expansion of the residual R around u i−1 yields
R (u
u)i = R (u
u)i−1 + (
ui) = 0 solve
Letting R (u
(
R i−1 i
∂R
u − ui−1]
) [u
u
∂u
R i−1 i
∂R
u − u i−1] = −R
R (u
u)i−1
) [u
u
∂u
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
NEWTON-RAPHSON EQUILIBRIUM ITERATION
Assuming conservative external forces:
R
∂R
u
∂u
=
=
σ d
dσ
u
d du
P R
e
t dσ
σ
B
u dV
V
du
Chain rule of differentiation leads to
σ = E tand = E tan B du
u such that
dσ
σ
dσ
u
du
= E tan B .
Tangent stiffness matrix provides Jacobian of N-R residual iteration,
R
t
R
∂R
=
K
where
K
=
B
E tanB dV
tan
tan
u
V
∂u
Newton-Raphson Equilibrium Iteration:
ui − u i−1] = −R
R (u
u)i−1
K i−1
tan [u
For i=1, the starting conditions for the first iteration cycle are,
Z
u1 − u n] = F n+1 − B tσ n
K ntan[u
V
First iteration cycle coincides with Euler forward step, whereby each equilibrium
iteration requires updating the tangential stiffness matrix.
Note: Difficulties near limit point when det K tan → 0
RADIAL RETURN METHOD OF J2-PLASTICITY I
Mises Yield Function:
1
1 2
s
s
s
F (s ) =
: − σY = 0
2
3
Associated Plastic Flow Rule:
∂F
˙ p = λ̇ s where m =
=s
∂ss
Plastic Consistency Condition:
∂F
Ḟ =
: ṡs = s : ṡs = 0
∂ss
Deviatoric Stress Rate:
ṡs = 2G [ėe − ėep] = 2G [ėe − λ̇ss]
Plastic Multiplier:
λ̇ =
s : ėe
s :s
RADIAL RETURN METHOD OF J2-PLASTICITY II
Incremental Format:
∆ss = 2G [∆ee − ∆λss]
Elastic Predictor-Plastic Corrector Split:
(a) Elastic Predictor: s trial = s n + 2G∆ee
(b) Plastic Corrector: s n+1 = s trial − 2G∆λsstrial = [1 − 2G∆λ]sstrial
“Full” Consistency: Fn+1 = 21s n+1 : s n+1 − 13 σY2 = 0
Quadratic equation for computing plastic multiplier ∆λ1 =? and ∆λ2 =?
1
1
[sstrial − 2G∆λsstrial ] : [sstrial − 2G∆λsstrial ] = σY2
2
3
q
1
[1 − 23 √σ σY:σ ]
Plastic Multiplier: ∆λmin = 2G
trial
trial
“Radial Return”: represents closest point projection of the trial stress state
onto the yield surface. Final stress state is the scaled-back trial stress,
r
2
σY
s n+1 =
s trial
√
3 s trial : s trial
GENERAL FORMAT OF PLASTIC RETURN METHOD
Incremental Format:
σ = E : [∆ − ∆λm
m]
∆σ
Elastic Predictor-Plastic Corrector Split:
(a) Elastic Predictor: σ trial = σ n + E : ∆
E :m
(b) Plastic Corrector: σ n+1 = σ trial − ∆λE
σ n + E : ∆ − ∆λE
E : m) = 0
“Full” Consistency: Fn+1 = F (σ
(i) Explicit Format: m = m n (or evaluate m at m = m c or m = m trial )
Use N-R for solving ∆λ =? for a given direction of plastic return e.g. m = m n.
m = m n+1 for BEM).
(i) Implicit Format: m = m n+α where 0 < α ≤ 1 (m
σ n + E : ∆ − ∆λE
E : m n+α) = 0
Fn+1 = F (σ
Use N-R for solving ∆λ =? in addition to unknown m = m n+α
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
ALGORITHMIC TANGENT STIFFNESS
Consistent Tangent vs Continuum Tangent:
Uniaxial Example:
σ̇ = Etan˙ where Etan
E
σ
− σ0 = E0[1 − ] hence σ = σ0[1 − e 0 ]
σ0
n+1
Fully Implicit Euler Backward Integration: Etan = Etan
in ∆t = tn+1 − tn,
σ n+1
n+1
σ
][n+1 − n]
= σ n + E0[1 −
σ0
Algorithmic Tangent Stiffness: Relates dn+1 to dσ n+1 at tn+1
dσ
n+1
=
alg
dn+1
Etan
where
alg
Etan
alg
Ratio of Tangent Stiffness Properties:
Etan
Etan
=
=
1
E
1+ σ 0 ∆
0
E0[1 −
σ n+1
σ0 ]
1 + Eσ00 ∆
∼ 0.7
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
ALGORITHMIC TANGENT STIFFNESS OF J2-PLASTICITY
Incremental Form of Elastic-Plastic Split:
∆ss = 2G [∆ee − ∆λss]
Fully Implicit Euler Backward Integration: for ∆t = tn+1 − tn,
s n+1 = s n + 2G[een+1 − e n] − ∆λ2Gssn+1
Relating dssn+1 to deen+1 at tn+1, differentiation yields,
dssn+1 = 2G deen+1 − d∆λ2Gssn+1 − ∆λ2Gdssn+1
Algorithmic Tangent Stiffness Relationship:
dssn+1 =
2G
s n+1 ⊗ s n+1
[II −
] : deen+1
1 + ∆λ2G
s n+1 : s n+1
Ratio of Tangent Stiffness Properties:
alg
G tan
G cont
tan
=
1
1+∆λ2G
when ||ssn+1|| ∼ ||ssn||.
Class #5 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
CONCLUDING REMARKS
Main Lessons from Class # 5:
Nonlinear Solvers:
Forward Euler method introduces drift from true response path.
Newton-Raphson Iteration exhibits convergence difficulties when
K tan → 0 (ill-conditioning).
CPPM for Computational Plasticity:
Analytical Radial Return solution available for J2-plasticity and
Drucker-Prager. Generalization leads to explicit and implicit plastic
return strategies which are nowadays combined with the incremental
hardening and incremental stress residuals in a monolithic Newton
strategy.
Algorithmic vs Continuum Tangent:
For quadratic convergence tangent operator must be consistent with
integration of constitutive equations. The algorithmic tangent
compares to secant stiffness in increments which are truly finite.