Development and numerical implementation of an anisotropic continuum damage model for concrete

Transcription

1 Development and numerical implementation of an anisotropic continuum damage model for concrete Juha Hartikainen 1, Kari Kolari 2, Reijo Kouhia 3 1 Tampere University of Technology, Department of Civil Engineering 2 VTT 3 Tampere University of Technology, Department of Mechanical Engineering and Industrial Systems 15th International Conference on Fracture and Damage Mechanics, September 2016, Alicante, Spain

2 Outline 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

3 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

4 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 Anisotropic damage R. Kouhia /18

5 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

6 Ottosen s 4 parameter model A J 2 + Λ J 2 + BI 1 σ c = 0, σ c { k1 cos[ 1 Λ = 3 arccos(k 2 cos 3θ)] if cos 3θ 0 k 1 cos[ 1 3 π 1 3 arccos( k 2 cos 3θ)] if cos 3θ 0. cos 3θ = 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 R. Kouhia /18

7 (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 = 0 σe/fc θ = (a) σm/fc Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen s model Red line = Barcelona model6 (b) 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 R. Kouhia /18

8 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 = 60 7 Anisotropic damage R. Kouhia σ 2 /f c /18 σ 1 /f c

9 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

10 Thermodynamic formulation Two potential functions ψ c = ψ c (S), S = (σ, D, κ) Specific Gibbs free energy γ = ρ 0 ψ c σ : ɛ, γ 0 Clausius-Duhem inequality ϕ(w ; S), W = (Y, K) Dissipation potential γ B Y : Y + B K K ψ c Define Y = ρ 0 D K = ρ ψ c 0 κ ψ (ρ c ) ) 0 σ ɛ : σ + (Ḋ BY : Y + ( κ B K ) K = 0 ɛ = ρ 0 ψ c σ, Ḋ = B Y, κ = B K Anisotropic damage R. Kouhia /18

11 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

12 Specific model Specific Gibbs free energy ρ 0 ψ c (σ, D, κ) = 1 + ν 2E Elastic domain [ trσ 2 + tr(σ 2 D) ] ν 2E ( trd)(trσ)2 + ψ c,κ (κ) Σ = {(Y, K) f(y, K; σ) 0} where the damage surface is defined as f(y, K; σ) = A J 2 σ c0 + Λ J 2 + BI 1 (σ c0 + K) = 0 Anisotropic damage R. Kouhia /18

13 Invariants in terms of Y J 2 = 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 { 0 if (Y, K) Σ I Σ(Y, K; σ) = + if (Y, K) / Σ (0, ( 0), if f(y, K; σ) < 0 (B Y, B K) = λ f ) f, λ, λ 0, if f(y, K; σ) = 0 Y K Ḋ = λ f Y, f κ = λ K Anisotropic damage R. Kouhia /18

14 2 Ottosen s 4 parameter model formulation 5 Some results and future work Anisotropic damage R. Kouhia /18

15 Some results Uniaxial compression - ultimate compressive stength σ c = 32.8 MPa σ c0 = 18 MPa, σ t0 = 1 MPa, (I 1, J 2 ) = ( 5 3σ c0, 4σ c0 / 2) A = 2.694, B = 5.597, k 1 = , k 2 = 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 = MPa, b = (a) σ11/σc model exp. Damage (b) D 11 D 22 = D 33 (a) Damage D D 22 = D ε11/εc ε 11/ε c Experimental results from Kupfer et al 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 Anisotropic that damage damage is larger R. Kouhia in the planes parallel to the 15/18 loading direction indicating splitting failure mode.

16 E (GPa) Young s modulus (a) and apparent Poisson s ratio (b) σ11/σc Const. Exp ε 11/ε c ν app Biaxial compression Figure 5: (a) The predicted reduce of the Young s modulus E,(b) Apparent Poisson s ratio νapp = ε22/ε11 of the concrete (a) (a) (b) specimen under uniaxial compression plotted using the (b) constitutive model compared to experimental results (Chen, 1982) D D 11 = 11 D = D D 22 = D 33. This is a characteristic property of elastic-brittle materials (Basista, D D ; Murakami and Kamiya, 1997; Nemat-Nasser and Hori, ). In0.8 Fig.7 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 0.4 et al., 1969). Eq. model Although (43) some deviation from0.004 experimental data is 0.4 observed, the model with Eq. hardening exper. (44) variable (43) qualitatively predicts the volumetric change of the material. Exp. Applying the hardening variable (44), however, causes the 0 model to fail to predict the volume increase which will eventually result in positive 0 0 values for V/V, after the material reaches its minimum volume The predicted behaviour ε 11/ε of cc the concrete specimen subjected to equibiaxial ε 11/ε ε 11/ε c c compression is compared to the experimental data (Kupfer et al., 1969), as can Figure be9: seen 8: (a) (a) incomparison Fig.8a. between Although between the predicted the predicted model stress-strain with stress-strain both curves hardening curves for concrete forvariables concrete specimen specimen gives under under equibiaxial an acceptable equibiaxial compression compression approximation with withof experimental hardening the ultimate variables data equibiaxial (Kupfer of equations etcompressive al., 1969), (43) and (b) stress, Predicted (44) and theex- perimental curves of the constitutive model for the concrete specimen under equibiaxial damage-strain magnitude data of (Kupfer the predicted et al., 1969), strain (b) ε Predicted 11 corresponding damage-strain Anisotropic damage to the curves ultimate of R. thekouhia stress constitutive /18 is compression. model for the concrete specimen under equibiaxial compression. approximately 60% smaller than expected. Moreover, as shown in Fig.8b, pre- σ11/σc σ11/σc Damage Damage

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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! Juana Francés ( ) 1960 Anisotropic damage R. Kouhia /18