An anisotropic continuum damage model for concrete Saba Tahaei Yaghoubi 1, Juha Hartikainen 1, Kari Kolari 2, Reijo Kouhia 3 1 Aalto University, Department of Civil and Structural Engineering 2 VTT 3 Tampere University of Technology, Department of Mechanical Engineering and Industrial Systems 4 June 215
Outline 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 2/18
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 3/18
Introduction The non-linear behaviour of quasi-brittle materials under loading is mainly due to damage and micro-cracking rather than plastic deformation. Damage of such materials can be modelled using scalar, vector or higher order damage tensors. Failure of rock-like materials in tension is mainly due to the growth of the most critical micro-crack Failure of rock-like materials in compression can be seen as a cooperative action of a distributed microcrack array http://mps-il.com Anisotropic damage 4.6.215 4/18
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 5/18
Ottosen s 4 parameter model A J 2 + Λ J 2 + BI 1 σ c =, σ c { k1 cos[ 1 Λ = 3 arccos(k 2 cos 3θ)] if cos 3θ k 1 cos[ 1 3 π 1 3 arccos( k 2 cos 3θ)] if cos 3θ. cos 3θ = 3 3 2 J 3 J 3/2 2, : Lode angle σ c : the uniaxial compressive strength I 1 = trσ: the first invariant of the stress tensor J 2 = 1 2 s : s, J 3 = det s = 1 3 trs3 : deviatoric invariants A, B, k 1, k 2 : material constants Anisotropic damage 4.6.215 6/18
(a) (b) Meridian plane & plane stress Figure 1: Comparison of the failure surfaces on the deviatoric plane. Green line indicates the Mohr-Coulomb criteria with tension cut-off, blue line the Ottosen model and red line the Barcelona model. (a) π-plane, (b) σm = fc. θ = σe/fc θ = 6 7 5 4 3 2 1 1 (a) 6 5 4 3 2 1 σm/fc 1.4 1.2 1..8.6.4.2.2 Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen s model Red line = Barcelona model6 (b).4.6.8 1. 1.2 1.4 Figure 2: Comparison of the failure surfaces in the (a) meridian plane and in (b)plane stress. Line colours as in Fig.1. Black dots indicate the experimental results (Kupfer et al., 1969). σ2/fc σ1/fc Anisotropic damage 4.6.215 7/18
Deviatoric plane σ 1 σ 1 σ 2 σ 3 σ 2 σ 3 π plane σ m = f c (a) (b) Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen s model Red line = Barcelona model Kuva 5.11: Murtopintojen vertailu deviatorisella tasolla. Vihreä viiva on vedossa katkaistu Mohr-Coulomb, sininen Ottosenin malli ja punainen viiva kuvaa Lublinerin mallia. (a) π- taso, (b) σ m = f c. σ e /f c θ = 6 7 Anisotropic damage σ 2 /f c 4.6.215 8/18 σ 1 /f c
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 9/18
Thermodynamic formulation Two potential functions ψ c = ψ c (S) S = (σ, D, κ) Specific Gibbs free energy γ = ρ ψ c σ : ɛ. γ. Clausius-Duhem inequality ϕ(w ; S) W = (Y, K) Dissipation potential γ B Y : Y + B K K ψ c Define Y = ρ D K = ρ ψ c κ, ψ (ρ c ) ) σ ɛ : σ + (Ḋ BY : Y + ( κ B K ) K =. ɛ = ρ ψ c σ, Ḋ = B Y, κ = B K, Anisotropic damage 4.6.215 1/18
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 11/18
Specific model Specific Gibbs free energy ρ ψ c (σ, D, κ) = 1 + ν 2E Elastic domain [ trσ 2 + tr(σ 2 D) ] ν 2E (1 + 1 3 trd)(trσ)2 + ψ c,κ (κ) Σ = {(Y, K) f(y, K; σ) } where the damage surface is defined as f(y, K; σ) = A J 2 σ c + Λ J 2 + BI 1 (σ c + K) =, Anisotropic damage 4.6.215 12/18
Invariants in terms of Y J 2 = 1 1 + ν J 3 = 2 3(1 + ν) [ EtrY 1 (1 2ν)(trσ)2] 6 { E [tr(σy ) trσtry ] + 1 (1 2ν)(trσ)3} 9 ϕ(y, K; σ) = I Σ(Y, K; σ) where I Σ is the indicator function { if (Y, K) Σ I Σ(Y, K; σ) = + if (Y, K) / Σ ( (, ), if f(y, K α; σ) <, (B Y, B K) = λ f ) f, λ, λ, if f(y, Kα; σ) =, Y K Ḋ = λ f Y, f κ = λ K Anisotropic damage 4.6.215 13/18
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 14/18
Some results Uniaxial compression - ultimate compressive stength σ c = 32.8 MPa σ c = 18 MPa, σ t = 1 MPa, (I 1, J 2 ) = ( 5 3σ c, 4σ c / 2) A = 2.694, B = 5.597, k 1 = 19.83, k 2 =.998 K = [a 1 (κ/κ max) + a 2 (κ/κ max) 2 ]/[1 + b(κ/κ max) 2 ] a 1 = 85.3 MPa, a 2 = 12.65 MPa, b =.732 (a) σ11/σc 1.25 1.75.5.25 model exp. Damage.4.3.2.1 (b) D 11 D 22 = D 33 (a) Damage.4.2 D D 22 = D.5 1 1.5 2.5 1.2 ε11/εc ε 11/ε c Experimental results from Kupfer et al. 1969. Figure 6: (a) Predicted damage-strain curves of the constitutive model for the concrete men under uniaxial compression, (b) Predicted damage-strain curves of the constitutive Figure 2. (a) Stress-strain diagram in uniaxial compression. for the concreteexperimental specimen under data uniaxial fromtension. Ref. [9], (b) Damage evolution in uniaxial compression. Notice that Anisotropic damage is damage larger in the planes parallel 4.6.215 to the 15/18 loading direction indicating splitting failure mode.
E (GPa) 3 2 1 Young s modulus (a) and apparent Poisson s ratio (b).5 1 1.5 2 2.5 σ11/σc 1.2 1.8.6.4.2 Const. Exp..1.2.3.4.5 ε 11/ε c ν app Biaxial compression Figure 5: (a) The predicted reduce (a) of the Young s modulus E,(b) Apparent Poisson s ratio(b) νapp = ε22/ε11 of the concrete specimen under uniaxial compression plotted using the constitutive model compared to experimental results (Chen, 1982)..12 1.2 D 11 = D 22 D 22 = D 33. This is a characteristic property of elastic-brittle materials (Basista, D 33 23; Murakami and Kamiya, 1997; Nemat-Nasser and Hori,.8 1993). In Fig.7.8 the volumetric strain-stress curves obtained by the solution of the constitutive equations and implementation of the hardening variables of equations (43) and (44) are illustrated and compared to the experimental observations (Kupfer.4 Eq. (43).4 et al., 1969). Eq. Although (44) some deviation from experimental data is observed, the model with hardening Exp. variable (43) qualitatively predicts the volumetric change of the material. Applying the hardening variable (44), however, causes the model to fail to predict the volume increase which will eventually result in positive values for V/V, after the material reaches its minimum volume..4.8 1.2 1.6.2.4.6 ε 11/ε c ε 11/ε c The predicted behaviour of the concrete specimen subjected to equibiaxial Figure compression 8: (a) Comparison compared between to the thexperimental predicted stress-strain data (Kupfer curves et foral., concrete 1969), specimen as can under be seen equibiaxial in Fig.8a. compression Although with hardening the model variables with both of equations hardening (43) and variables (44) and gives experimental data (Kupfer et al., 1969), (b) Predicted damage-strain curves of the constitutive an acceptable approximation of the ultimate equibiaxial compressive stress, the model for the concrete specimen under equibiaxial compression. Anisotropic damage magnitude of the predicted strain ε 11 corresponding to the ultimate stress is 4.6.215 16/18 σ11/σc Damage
2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage 4.6.215 17/18
Conclusions and future work Continuum damage formulation of the Ottosen s 4 parameter model Can model axial splitting Implementation into FE software (own codes, ABAQUS) Development of directional hardening model Regularization by higher order gradients Thank you for your attention! Anisotropic damage 4.6.215 18/18