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
Out-of-Balance Force Calculation: STRUCTURAL EQUILIBRIUM Out-of Balance Residual Forces: R(u) = F int F ext 0 Internal Forces: F int = e External Forces: F ext = e V Bt σdv V N t bdv + e S N t tds Rate of Equilibrium: B t σdv = N t ḃbdv + N t ṫtds e V e V e S 1. Incremental Methods of Numerical Integration: Path-following continuation strategies to advance solution within t = t n+1 t n (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 Internal forces: ḞF ext = e V Bt σdv Tangential Material Law: σ = ĖE tan ɛ Tangential Stiffness Relationship: K tan u = ḞF where K tan = e V B t E tan BdV Incremental Format: u n+1 u n K tan du = F n+1 F n Euler Forward Integration: K n tan = e V Bt E n tanbdv K n tan u = F where E n tan = E tan (t n ) Note: Uncontrolled drift of response path if no equilibrium corrections are included at each load step.
2. ITERATIVE SOLVERS Picard direct substitution iteration vs Newton-Raphson iteration within t = t n+1 t n Robustness Issues: Range and Rate of Convergence? Newton-Raphson Residual Force Iteration: R(u) = 0 Truncated Taylor Series Expansion of the residual R around u i 1 yields R(u) i = R(u) i 1 + ( R u )i 1 [u i u i 1 ] Letting R(u i ) = 0 solve ( R u )i 1 [u i u i 1 ] = R(u) i 1
NEWTON-RAPHSON EQUILIBRIUM ITERATION Assuming conservative external forces: R u = e V Bt dσ du dv Chain rule of differentiation leads to dσ = E tan dɛ = E tan Bdu such that dσ du = dσ dɛ dɛ du = E tan B. Tangent stiffness matrix provides Jacobian of N-R residual iteration, R u = K tan where K tan = V Bt E tan BdV Newton-Raphson Equilibrium Iteration: K i 1 tan [u i u i 1 ] = R(u) i 1 For i=1, the starting conditions for the first iteration cycle are, K n tan[u 1 u n ] = F n+1 B t σ n 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 V
Mises Yield Function: RADIAL RETURN METHOD OF J 2 -PLASTICITY I F (s) = 1 2 s : s 1 3 σ2 Y = 0 Associated Plastic Flow Rule: ɛ p = λ s where m = F s = s Plastic Consistency Condition: F = F : ṡs = s : ṡs = 0 s Deviatoric Stress Rate: Plastic Multiplier: ṡs = 2G [ėe ėe p ] = 2G [ėe λs] λ = s : ėe s : s
Incremental Format: RADIAL RETURN METHOD OF J 2 -PLASTICITY II s = 2G [ e λs] Elastic Predictor-Plastic Corrector Split: (a) Elastic Predictor: s trial = s n + 2G e (b) Plastic Corrector: s n+1 = s trial 2G λs trial = [1 2G λ]s trial Full Consistency: F n+1 = 1 2 s n+1 : s n+1 1 3 σ2 Y = 0 Quadratic equation for computing plastic multiplier λ 1 =? and λ 2 =? 1 2 [s trial 2G λs trial ] : [s trial 2G λs trial ] = 1 3 σ2 Y Plastic Multiplier: λ min = 1 2G [1 2 σ Y 3 σtrial :σ ] 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, s n+1 = 2 σ Y s trial 3 strial : s trial
Incremental Format: GENERAL FORMAT OF PLASTIC RETURN METHOD σ = E : [ ɛ λm] Elastic Predictor-Plastic Corrector Split: (a) Elastic Predictor: σ trial = σ n + E : ɛ (b) Plastic Corrector: σ n+1 = σ trial λe : m Full Consistency: F n+1 = F (σ n + E : ɛ λe : m) = 0 (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. (i) Implicit Format: m = m n+α where 0 < α 1 (m = m n+1 for BEM). F n+1 = F (σ n + E : ɛ λe : m n+α ) = 0 Use N-R for solving λ =? in addition to unknown m = m n+α
ALGORITHMIC TANGENT STIFFNESS Consistent Tangent vs Continuum Tangent: Uniaxial Example: σ = E tan ɛ where E tan = E 0 [1 σ σ 0 ] hence σ = σ 0 [1 e E 0 σ 0 ɛ ] Fully Implicit Euler Backward Integration: E tan = E n+1 tan in t = t n+1 t n, σ n+1 = σ n + E 0 [1 σn+1 σ 0 ][ɛ n+1 ɛ n ] Algorithmic Tangent Stiffness: Relates dɛ n+1 to dσ n+1 at t n+1 σ 0 ] dσ n+1 = E alg tan dɛ n+1 where E alg tan = E 0[1 σn+1 1 + E 0 σ 0 ɛ Ratio of Tangent Stiffness Properties: Ealg tan E tan = 1 1+ E 0 σ 0 ɛ 0.7
ALGORITHMIC TANGENT STIFFNESS OF J 2 -PLASTICITY Incremental Form of Elastic-Plastic Split: s = 2G [ e λs] Fully Implicit Euler Backward Integration: for t = t n+1 t n, s n+1 = s n + 2G[e n+1 e n ] λ2gs n+1 Relating ds n+1 to de n+1 at t n+1, differentiation yields, ds n+1 = 2G de n+1 d λ2gs n+1 λ2gds n+1 Algorithmic Tangent Stiffness Relationship: ds n+1 = 2G 1 + λ2g [I s n+1 s n+1 s n+1 : s n+1 ] : de n+1 Ratio of Tangent Stiffness Properties: G alg tan G cont tan = 1 1+ λ2g when s n+1 s n.
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 J 2 -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.