MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations

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1 MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic Damage Models: Scalar Damage, Volumetric-Deviatoric Damage Fixed vs Rotating Crack Models: Anisotropic vs Isotropic Damage

2 NONLINEAR ELASTICITY Nonlinear Descriptions: 1. Algebraic Format: Pseudo-Elasticity σ = f(ɛ) reduces to Secant Stiffness Model: σ = E sec : ɛ 2. Integral Format: Hyper-Elasticity σ = W ɛ Given W = W (ɛ) Strainenergy Potential σ = W ɛ 3. Differential Format: Hypo-Elasticity σ = g(σ ɛ) reduces to Tangent Stiffness Model: σ = E tan : ɛ σ P E dσ t = dε σ = f( ε) W( ε) = σdε E s = σ ε σ = dw dε ε ε

3 ALGEBRAIC FORMAT: CAUCHY ELASTICITY Elastic Stiffness: σ = f(ɛ) Representation Theorem of Isotropic Tensor Functions: σ = Φ Φ 2 ɛ + Φ 3 ɛ 2 Three Invariant Response Functions: Φ i = Φ i (I 1, I 2, I 3 ) ɛ Elastic Secant Relationship: σ = E sec : ɛ Matrix Format of Secant Stiffness: Letting Φ 3 = 0 Φ 1 = Φ 1 (I 1 ) ɛ K sec = K(trɛ) and Φ 2 = Φ 2 (J 2 ) ɛ G sec = G(tre 2 ) K s G s K s 2 3 G s K s 2 3 G s K s 2 3 G s K s G s K s 2 3 G s 0 [E] sec = K s 2 3 G s K s 2 3 G s K s G s G s 0 G s G s

4 SCALAR FORMS OF ELASTIC DAMAGE Elastic Damage Relationship: σ = E d sec : ɛ Matrix Format of Secant Stiffness: Letting Φ 3 = 0 (a) Volumetric Damage: K d = [1 d vol ]K 0 with d vol = 1 K s K 0 (b) Deviatoric Damage: G d = [1 d dev ]G 0 with d dev = 1 G s G 0 [E] d sec = K d G d K d 2 3 G d K d 2 3 G d K d 2 3 G d K d G d K d 2 3 G d 0 K d 2 3 G d K d 2 3 G d K d G d G d 0 G d G d Note: Damage is evolutionary process w/ d vol > 0 and d dev > 0.

5 SINGLE SCALAR FORM OF ELASTIC DAMAGE Scalar Damage Relationship: σ = [1 d]e 0 : ɛ Single scalar damage function: d = d vol = d dev = 1 E s E 0 whereby ν = ν 0 = const. [E] d sec = [1 d] K G o K G 0 K G 0 K G 0 K G 0 K G 0 0 K G 0 K G 0 K G 0 G 0 0 G 0 G 0

6 INTEGRAL FORM: GREEN ELASTICITY Hyperelastic Stress: W (ɛ) = ɛ σ : dɛ = ɛ σ = W ɛ W ɛ : dɛ = ɛ dw with W (ɛ) = ɛ dw = 0 Representation Theorem of Isotropic Scalar Functions: σ = W ɛ = W I 1 I 1 ɛ + W I 2 I 2 ɛ + W I 3 I 3 ɛ Three Invariant Response Functions: σ = W W 2 ɛ + W 3 ɛ 2 where W i = W i (I 1, I 2, I 3 ) ɛ with W i I j = W j I i since 2 W I i I j = 2 W I j I i. Assuming W 3 = 0: W (ɛ) = W vol (trɛ) + W dev (tre 2 ) Octahedral Format of Hyperelastic Secant Stiffness: σ oct = 1 3 trσ; τ oct = ( 1 3 trs2 ) 1 2 [ ] [ ] [ ] σoct 3Ks (trɛ) 0 ɛoct = 0 2G s (tre 2 ) τ oct γ oct

7 PRINCIPAL COORDINATE FORMAT OF GREEN ELASTICITY Hyperelastic Stress: σ i = W ɛ i where W (ɛ) = W (ɛ 1, ɛ 2, ɛ 3 ) σ 1 = W ɛ 1 ; σ 2 = W ɛ 2 ; σ 3 = W ɛ 3 Tangential Format: σ i = σ i ɛ j ɛ j = [ 2 W ɛ i ɛ j ] ɛ j σ 1 σ 2 σ 3 = ɛ 1 ɛ 1 ɛ ɛ 1 ɛ 3 ɛ 1 ɛ 1 ɛ 2 ɛ ɛ 2 ɛ 3 ɛ 2 ɛ 1 ɛ 3 ɛ ɛ 3 ɛ 3 ɛ 3 ɛ 1 ɛ 2 ɛ 3 Note #1: Symmetry and Apparent Orthotropy due strain-induced anisotropy. Note #2: Shear Stiffness maintains co-axiality of principal axes: θ σ = θ ɛ. τ 12 τ 23 τ 31 = 1 σ 2 σ 1 2 ɛ 2 ɛ σ 3 σ 2 ɛ 3 ɛ σ 1 σ 3 ɛ 1 ɛ 3 γ 12 γ 23 γ 31

8 DIFFERENTIAL FORM: TRUESDELL ELASTICITY Hypoelastic Stress Format: σ = g(σ, ɛ) Incrementally Linear Hypoelastic Formulation: Tangential Stiffness Format: σ = E tan : ɛ where E tan = E(σ) From Representation Theorem of Isotropic Tensor Functions we find: E tan = C C 2 σ 1 +C 3 σ 2 1 +C 4 1 σ +C 5 σ σ +C 6 σ 2 σ +C 7 1 σ 2 +C 8 σ σ 2 +C 9 σ 2 σ 2 +C 10 [ ] +C 11 [σ σ] +C 12 [σ σ 2 Note: Path-Independence requires that hypoelastic constitutive relations satisfy integrability conditions. σ = E tan (σ) : dɛ dt dt ɛ

9 SMEARED CRACK APPROACH 1. Fixed Crack Approach: Orthotropic material formulation (permanent crack memory) 2. Rotating Crack Approach: Isotropic material formulation (fading crack memory) Additional Crack Compliance: due to crack separation ɛ f = u s ɛ = ν E (trσ) G σ [N ɛf + ɛ f N] In-plane strains vanish due in local [N, S, T ] system such that ɛ f SS = ɛf T T = ɛf ST = 0, where N is the normal vector to initial crack direction.

10 1. FIXED CRACK APPROACH Softening Traction-Separation Model in fixed axes of Orthotropy: t N = E N (ɛ N, ɛ T )ɛ f N and t T = E T (ɛ N, ɛ T )ɛ f T (a) Interfacial Relations of normal components: ɛ NN ɛ SS ɛ T T = 1 E 1 + EC N ν ν ν 1 ν ν ν 1 σ NN σ SS σ T T (b) Interfacial relations of shear components: γ NS γ ST γ T N = 1 G 1 + GC T GC T σ NS σ ST σ T N Note: Shear retention factor relates C T = 1 E T (ɛ N,ɛ T ) to C N = 1 E N (ɛ N,ɛ T )

11 2. ROTATING CRACK APPROACH Compliance Format of Cracking when crack orientation rotates with the principal axes of strain: [N, S, T ] [e 1, e 2, e 3 ] 1 (a) Interfacial relations of principal compliances: C N = E N (ɛ N,ɛ T ) ɛ 1 ɛ 2 ɛ 3 = 1 E 1 + EC N ν ν ν 1 ν ν ν 1 (b) Interfacial relations of shear components: Argument of isotropy when the principal axes of stress coincide with the principal axes of strain. This requires that the tangential shear compliance follows: σ 1 σ 1 σ 3 γ 12 γ 23 γ 31 = σ 2 σ ɛ 3 ɛ 2 σ 3 σ ɛ 1 ɛ 3 σ 1 σ 3 2 ɛ 2 ɛ 1 τ 12 τ 23 τ 31

12 CONCLUDING REMARKS Main Lessons from Class # 2: Nonlinear Hyperelasticity: preserves path-independence, reversibility and energy (no dissipation) Canonical Form of Nonlinear Elastic Behavior: Volumetric-Deviatoric Damage Model for K s G s Smeared Cracking of Concrete: Fixed crack approach (orthotropic format) introduces shear locking - Rotating crack approach minimizes shear locking (isotropic format) Rotating Cracking in Form of Tensile Damage due to C N : ɛ = ν E (trσ) G σ + C N σ 1 [e 1 e 1 ] Note analogy to plastic softening according to Rankine.