Willam Modeling of Concrete Materials and Structures

8/3/2019 Willam Modeling of Concrete Materials and Structures

1/11

MODELING OF CONCRETE MATERIALS AND STRUCTURES

Kaspar WillamUniversity 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


2/11

STRUCTURAL EQUILIBRIUM

Out-of-Balance Force Calculation:

Out-of Balance Residual Forces: RRR(uuu) = FFFint FFFext 000

Internal Forces: FFFint =

e

V

BBBtdV

External Forces: FFFext =

e

VNNNtbbbdV +

e

SNNNttttdS

Rate of Equilibrium:

e

V

BBBtdV =

e

V

NNNtbbbdV +

e

S

NNNttttdS

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 (shouldinclude 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; Heunsmethod at midstep, and Runge-Kutta h/o methods at intermediate stages).


3/11

INCREMENTAL SOLVERS

Euler Forward Integration: Classical Tangent stiffness approach

Internal forces: FFFext =

e

V BBBtdV

Tangential Material Law: = EEEtan

Tangential Stiffness Relationship:

KKKtanuuu = FFF where KKKtan =

e

V

BBBtEEEtanBBBdV

Incremental Format:un+1

unKKKtanduuu = FFFn+1 FFFn

Euler Forward Integration: KKK

n

tan =

e

V BBB

t

EEE

n

tanBBBdV

KKKntanuuu = FFF where EEEntan = EEEtan(tn)

Note: Uncontrolled drift of response path if no equilibrium corrections areincluded at each load step.



4/11

2. ITERATIVE SOLVERS

Picard direct substitution iteration vs Newton-Raphson iteration withint = tn+1

tn

Robustness Issues: Range and Rate of Convergence?

Newton-Raphson Residual Force Iteration: RRR(uuu) = 000

Truncated Taylor Series Expansion of the residual RRR around uuui1 yields

RRR(uuu)i = RRR(uuu)i1 + (RRR

uuu)i1[uuui uuui1]

Letting RRR(uuui) = 000 solve

(RRR

uuu)i1[uuui uuui1] = RRR(uuu)i1



5/11

NEWTON-RAPHSON EQUILIBRIUM ITERATION

Assuming conservative external forces: RRRuuu

=

e

V

BBBt dduuu

dV

Chain rule of differentiation leads to

d = EEEtand = EEEtan BBBduuu such thatdduuu

= dd

dduuu

= EEEtan BBB.

Tangent stiffness matrix provides Jacobian of N-R residual iteration,

RRRuuu = KKKtan where KKKtan =

V

BBBtEEEtanBBBdV

Newton-Raphson Equilibrium Iteration:

KKKi1tan [uuui uuui1] = RRR(uuu)i1

For i=1, the starting conditions for the first iteration cycle are,

KKKntan[uuu1 uuun] = FFFn+1

V

BBBtn

First iteration cycle coincides with Euler forward step, whereby each equilibriumiteration requires updating the tangential stiffness matrix.

Note: Difficulties near limit point when det KKKtan 0


6/11

RADIAL RETURN METHOD OF J2-PLASTICITY I

Mises Yield Function:F(sss) =

1

2sss : sss 1

32Y = 0

Associated Plastic Flow Rule:

p = sss where mmm =F

sss= sss

Plastic Consistency Condition:

F =F

sss: sss = sss : sss = 0

Deviatoric Stress Rate:

sss = 2G [eee eeep] = 2G [eeesss]

Plastic Multiplier:

=sss : eee

sss : sss


7/11

RADIAL RETURN METHOD OF J2-PLASTICITY II

Incremental Format:

sss = 2G [eeesss]Elastic Predictor-Plastic Corrector Split:

(a) Elastic Predictor: ssstrial = sssn + 2Geee(b) Plastic Corrector: sssn+1 = ssstrial

2Gssstrial = [1

2G]ssstrial

Full Consistency: Fn+1 =1

2sssn+1 : sssn+1 132Y = 0

Quadratic equation for computing plastic multiplier 1 =? and 2 =?

1

2[ssstrial 2Gssstrial] : [ssstrial 2Gssstrial] = 1

32Y

Plastic Multiplier: min =1

2G[1

2

3

Ytrial:trial

]

Radial Return: represents closest point projection of the trial stress stateonto the yield surface. Final stress state is the scaled-back trial stress,

sssn+1 =

23

Yssstrial : ssstrial

ssstrial


8/11

GENERAL FORMAT OF PLASTIC RETURN METHOD

Incremental Format:

= EEE : [mmm]Elastic Predictor-Plastic Corrector Split:

(a) Elastic Predictor: trial = n + EEE : (b) Plastic Corrector: n+1 = trial EEE : mmm

Full Consistency: Fn+1 = F(n + EEE : EEE : mmm) = 0

(i) Explicit Format: mmm = mmmn (or evaluate mmm at mmm = mmmc or mmm = mmmtrial)

Use N-R for solving =? for a given direction of plastic return e.g. mmm = mmmn.

(i) Implicit Format: mmm = mmmn+ where 0 < 1 (mmm = mmmn+1 for BEM).

Fn+1 = F(n + EEE : EEE : mmmn+) = 0Use N-R for solving =? in addition to unknown mmm = mmmn+



9/11

ALGORITHMIC TANGENT STIFFNESS

Consistent Tangent vs Continuum Tangent:

Uniaxial Example:

= Etan where Etan = E0[1 0

] hence = 0[1 eE00

]

Fully Implicit Euler Backward Integration: Etan = En+1tan in t = tn+1

tn,

n+1 = n + E0[1 n+1

0][n+1 n]

Algorithmic Tangent Stiffness: Relates dn+1 to dn+1 at tn+1

dn+1 = Ealgtan dn+1 where Ealgtan =

E0[1 n+10 ]1 + E0

0

Ratio of Tangent Stiffness Properties:E

algtan

Etan= 1

1+E00 0.7



10/11

ALGORITHMIC TANGENT STIFFNESS OF J2-PLASTICITY

Incremental Form of Elastic-Plastic Split:

sss = 2G [eeesss]Fully Implicit Euler Backward Integration: for t = tn+1 tn,

sssn+1 = sssn + 2G[eeen+1

eeen]

2Gsssn+1

Relating dsssn+1 to deeen+1 at tn+1, differentiation yields,

dsssn+1 = 2G deeen+1 d2Gsssn+1 2Gdsssn+1

Algorithmic Tangent Stiffness Relationship:

dsssn+1 =2G

1 + 2G[III sssn+1 sssn+1

sssn+1 : sssn+1] : deeen+1

Ratio of Tangent Stiffness Properties:GGG

algtan

GGGconttan= 1

1+2Gwhen

||sssn+1

| | | |sssn

||.



11/11

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

KKKtan 0 (ill-conditioning).CPPM for Computational Plasticity:

Analytical Radial Return solution available for J2-plasticity andDrucker-Prager. Generalization leads to explicit and implicit plasticreturn strategies which are nowadays combined with the incrementalhardening and incremental stress residuals in a monolithic Newtonstrategy.

Algorithmic vs Continuum Tangent:For quadratic convergence tangent operator must be consistent with

integration of constitutive equations. The algorithmic tangentcompares to secant stiffness in increments which are truly finite.

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Willam Modeling of Concrete Materials and Structures