MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #3: Elastoplastic Concrete Models Uniaxial Model: Strain-Driven Format of Elastoplasticity Triaxial Model: Generalized Format of Elastoplasticity Isotropic Hardening/Softening: Volumetric-Deviatoric Interaction Rotating Plastic Crack Model: Softening Rankine Formulation Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
Fundamental Steps: ELASTOPLASTIC MATERIAL MODEL 1. Additive Decomposition: Elastic-Plastic Partition ɛ = ɛ e + ɛ p Incremental format of elastic stress σ = E : [ ɛ ɛ p ] yields elastoplastic tangent stiffness: σ = E ep : ɛ 2. Yield Condition: Plastic Initiation and Persistence: F (σ) = 0 Plastic consistency condition distinguishes plastic loading from elastic unloading F = F σ : σ = 0 n : σ = 0 3. Flow Rule: Plastic Evolution Equation ɛ p = λm Orientation of plastic flow is defined by m = Q σ and magnitude by plastic multiplier λ > 0 4. Hardening/Softening Rule: Plastic Stiffness H p = F λ normally expressed in terms of an invariant stress-plastic strain (plastic work) relationship E p = dσ eq dɛ p eq Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
UNIAXIAL ELASTOPLASTIC MODEL 1. Deformation Theory of Hencky [1924]: Total secant relationship 2. Flow Theory of Prandtl-Reuss [1928]: Incremental tangent relationship Additive Decomposition: Consequently, ɛ = ɛ e + ɛ p where ɛ e = σ E and ɛ p = σ E p ɛ = σ E + σ E p = Elastoplastic Tangent Stiffness Relationship: σ E ep σ = E ep ɛ where E ep = EE p E + E P Note E ep = when Ep crit = E.
Note: ɛ p = σ E p = 0 0 when E p = 0 UNIAXIAL ELASTOPLASTIC MODEL Use strain rather than stress control: ɛ p = σ E p = E E + E p ɛ Formal Yield Condition: F (σ) = σ σ y = 0 Plastic action (i) when stress path reaches the yield capacity of the material σ = σ y (ii) persistent plastic loading when df dσ E ɛ > 0 for strain control.
Mises Yield Function: IDEAL J 2 -ELASTOPLASTICITY 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
Deviatoric Stress-Strain Relation: IDEAL J 2 -ELASTOPLASTICITY II ṡs = 2G [I s s s : s ] : ėe ṡs = G ep : ėe with G ep = 2G [I s s s : s ] Tangent Stiffness Operator: σ = 1 3 (tr σ)1 + ṡs = K(tr ɛ)1 + G ep : ėe Elastoplastic Tangent Operator σ = K(tr ɛ)1 + G ep : [ ɛ 1 3 (tr ɛ)1] σ = E ep : ɛ with E ep = Λ1 1 + 2G [I s s s : s ] Note: Elastoplastic constitutive structure similar to K G(e) model. Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
SIMPLE SHEAR EXAMPLE von Mises vs parabolic Drucker-Prager: Response when γ 12 > 0 Parabolic Yield Function: Associated Flow Rule: F (I 1, J 2 ) = J 2 + α F I 1 τ 2 y = 0 ɛ p = λ[s + α F 1] Simple Shear: α F = 1 3 [f c f t] = 0 for von Mises, while τ 2 Y = 1 3 f cf t = 1 3 σ2 Y
GENERAL FORMULATION OF ELASTOPLASTIC BEHAVIOR I Kinematic Setting: Decomposition of Total Deformation ɛ = 1 2 [ u + t u], ɛ = ɛ e + ɛ p Elastic Behavior: Hyperelastic concept of free energy potential: Ψ = Ψ(ɛ, ɛ p, κ) σ = Ψ ɛ e and σ = E : [ ɛ ɛ p ] Plastic Yield Condition: F (σ, κ) = f(σ) r y (κ) 0 with n = F σ f(σ) defines the internal stress demand and r y = the material resistance r y = Ψ and ṙ y = H p κ κ Hardening modulus H p characterizes the rate of yield resistance. Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
Plastic Flow Rule: GENERAL ELASTOPLASTIC FORMULATION II ɛ p = λ m with m = Q σ Associated flow when m n (normality of plastic flow). Plastic Consistency Condition: F = 0 Consistency condition enforces the stress path to remain on the yield surface. Kuhn Tucker Condition of Plastic Loading: F 0 λ 0 F λ = 0 Plastic Multiplier: λ = 1 n : E : ɛ h p with h p = H p + n : E : m Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
GENERAL ELASTOPLASTIC FORMULATION III Elastoplastic Stiffness Relation: σ = E : [ ɛ λm] = E : [ ɛ m n : E : ɛ H p + n : E : m ] σ = E ep : ɛ Note #1: Plastic stiffness forms rank one (two) update of the elastic material operator where h p = H p + n : E : m. E ep = E 1 h p E : m n : E Note #2: h p = 0 when softening modulus reaches H crit p = n : E : m. Note #3: Loss of symmetry, E ep E t ep when n m for non-associated flow. Class #3 Concrete Modeling, UNICAMP, Campinas, Brazil, August 20-28, 2007
SIMPLE SHEAR RESPONSE Three Invariant Elastoplastic Concrete Model: Kang and Willam [1999] Effect of Confinement under Strain Control
CONCLUDING REMARKS Main Lessons from Class # 3: Flow Theory of Plasticity: introduces path-dependence, irreversibility and energy dissipation Canonical Form of J 2 Elastoplasticity: Decouples volumetric-deviatoric behavior, see K G(e) model Volumetric-Deviatoric Coupling: Two and three invariant elastoplastic models - Isotropic hardening/softening compares to rotating crack approach (no crack/slip memory) Smeared Cracking in Form of Plastic Softening of Major Strain Component ɛ = ν E (trσ)1 + 1 2G σ + C N σ 1 [e 1 e 1 ] Softening Rankine plasticity is equivalent to rotating crack formulation using elastic damage.