An anisotropic continuum damage model for concrete Saba Tahaei - PowerPoint PPT Presentation

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 2015

1 Introduction 2 Ottosen’s model Outline 3 Thermodynamic 4 Specific model 5 Results 1 Introduction 6 Conclusions 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 2/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 3/18

1 Introduction 2 Ottosen’s model Introduction 3 Thermodynamic 4 Specific model 5 Results The non-linear behaviour of 6 Conclusions 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 http://mps-il.com microcrack array Anisotropic damage – 4.6.2015 4/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 5/18

1 Introduction 2 Ottosen’s model Ottosen’s 4 parameter model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions AJ 2 � + Λ J 2 + BI 1 − σ c = 0 , σ c � k 1 cos[ 1 3 arccos( k 2 cos 3 θ )] if cos 3 θ � 0 Λ = cos 3 θ ≤ 0 . k 1 cos[ 1 3 π − 1 3 arccos( − k 2 cos 3 θ )] if √ cos 3 θ = 3 3 J 3 , : Lode angle 2 J 3 / 2 2 σ c : the uniaxial compressive strength I 1 = tr σ : the first invariant of the stress tensor 3 tr s 3 : deviatoric invariants J 2 = 1 2 s : s , J 3 = det s = 1 A, B, k 1 , k 2 : material constants Anisotropic damage – 4.6.2015 6/18

1 Introduction 2 Ottosen’s model Meridian plane & plane stress 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions σ 2 /f c σ e /f c θ = 60 ◦ σ 1 /f c 7 − 1 . 4 − 1 . 2 − 1 . 0 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 6 − 0 . 2 5 − 0 . 4 θ = 0 ◦ θ = 0 ◦ 4 − 0 . 6 3 − 0 . 8 2 − 1 . 0 1 σ m /f c − 1 . 2 − 5 − 4 − 3 − 2 − 1 0 1 − 1 . 4 Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen’s model Red line = Barcelona model Anisotropic damage – 4.6.2015 7/18

1 Introduction 2 Ottosen’s model Deviatoric plane 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions σ 1 σ 1 σ 2 σ 3 σ 2 σ 3 π − plane σ m = − f c Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen’s model Red line = Barcelona model Anisotropic damage – 4.6.2015 8/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 9/18

1 Introduction 2 Ottosen’s model Thermodynamic formulation 3 Thermodynamic 4 Specific model 5 Results Two potential functions 6 Conclusions ψ c = ψ c ( S ) S = ( σ , D , κ ) Specific Gibbs free energy ψ c − ˙ γ = ρ 0 ˙ σ : ǫ . γ � 0 . Clausius-Duhem inequality ϕ ( W ; S ) W = ( Y , K ) Dissipation potential γ ≡ B Y : Y + B K K ∂ψ c ∂ψ c Y = ρ 0 K = − ρ 0 ∂κ , Define ∂ D ∂ψ c � � � � ˙ ρ 0 ∂ σ − ǫ : ˙ σ + D − B Y : Y + ( − ˙ κ − B K ) K = 0 . ∂ψ c ˙ ǫ = ρ 0 ∂ σ , D = B Y , κ = − B K , ˙ Anisotropic damage – 4.6.2015 10/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 11/18

1 Introduction 2 Ottosen’s model Specific model 3 Thermodynamic 4 Specific model 5 Results Specific Gibbs free energy 6 Conclusions ρ 0 ψ c ( σ , D , κ ) = 1 + ν − ν tr σ 2 + tr( σ 2 D ) 3 tr D )(tr σ ) 2 + ψ c ,κ ( κ ) � � 2 E (1 + 1 2 E Elastic domain Σ = { ( Y , K ) | f ( Y , K ; σ ) � 0 } where the damage surface is defined as f ( Y , K ; σ ) = A ˜ J 2 � ˜ + Λ J 2 + BI 1 − ( σ c0 + K ) = 0 , σ c0 Anisotropic damage – 4.6.2015 12/18

1 Introduction 2 Ottosen’s model Invariants in terms of Y 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions 1 ˜ 6 (1 − 2 ν )(tr σ ) 2 � � E tr Y − 1 J 2 = 1 + ν 2 ˜ E [tr( σY ) − tr σ tr Y ] + 1 9 (1 − 2 ν )(tr σ ) 3 � J 3 = � 3(1 + ν ) 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 α ; σ ) < 0 ,  ( B Y , B K ) = � λ ∂f λ ∂f � ˙ ∂ Y , ˙ , ˙ λ ≥ 0 , if f ( Y , K α ; σ ) = 0 ,  ∂K λ ∂f λ ∂f D = ˙ ˙ κ = − ˙ ∂ Y , ˙ ∂K Anisotropic damage – 4.6.2015 13/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 14/18

1 Introduction 2 Ottosen’s model Some results 3 Thermodynamic 4 Specific model Uniaxial compression - ultimate compressive stength σ c = 32 . 8 MPa √ √ ( I 1 , √ J 2 ) = ( − 5 5 Results σ c0 = 18 MPa , σ t0 = 1 MPa , 3 σ c0 , 4 σ c0 / 2) 6 Conclusions A = 2 . 694 , B = 5 . 597 , k 1 = 19 . 083 , k 2 = 0 . 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 = 0 . 7032 1.25 D 11 0.4 D 22 = D 33 1 0.3 Damage − σ 11 /σ c 0.75 0.2 0.5 model exp. 0.1 0.25 0 0 0 0.5 1 0 0.5 1 1.5 2 − ε 11 /ε c − ε 11 /ε c Experimental results from Kupfer et al. 1969. Anisotropic damage – 4.6.2015 15/18

Young’s modulus and apparent Poisson’s ratio 1.2 1 Introduction 30 2 Ottosen’s model 1 3 Thermodynamic 0.8 E (GPa) − σ 11 /σ c 20 4 Specific model 0.6 Const. Exp. 5 Results 0.4 10 6 Conclusions 0.2 0 0 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 − ε 11 /ε c ν app Biaxial compression 0.012 D 11 = D 22 1.2 D 33 0.008 − σ 11 /σ c Damage 0.8 0.004 Eq. (43) 0.4 Eq. (44) Exp. 0 0 0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 − ε 11 /ε c − ε 11 /ε c Anisotropic damage – 4.6.2015 16/18

1 Introduction 1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 2 Ottosen’s 4 parameter model 5 Results 6 Conclusions 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work Anisotropic damage – 4.6.2015 17/18

1 Introduction 2 Ottosen’s model Conclusions and future work 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions 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 – 4.6.2015 18/18