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

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Feb 21, 2023 116 likes •299 views

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

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An anisotropic continuum damage model for concrete

Saba Tahaei Yaghoubi1, Juha Hartikainen1, Kari Kolari2, Reijo Kouhia3

1Aalto University, Department of Civil and Structural Engineering 2VTT 3Tampere University of Technology, Department of Mechanical Engineering and Industrial Systems 4 June 2015

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Outline

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

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

f 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 – 4.6.2015 4/18

SLIDE 5

1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Ottosen’s 4 parameter model

AJ2 σc + Λ

J2 + BI1 − σc = 0,

Λ = k1 cos[ 1

3 arccos(k2 cos 3θ)]

if cos 3θ 0 k1 cos[ 1

3π − 1 3 arccos(−k2 cos 3θ)]

if cos 3θ ≤ 0 . cos 3θ = 3 √ 3 2 J3 J3/2

, : Lode angle σc: the uniaxial compressive strength I1 = trσ: the first invariant of the stress tensor J2 = 1

2s : s, J3 = det s = 1 3trs3 : deviatoric invariants

A, B, k1, k2: material constants

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Meridian plane & plane stress

1 2 3 4 5 6 7 −1 −2 −3 −4 −5 1 θ = 0◦ θ = 60◦ σe/fc σm/fc θ = 0◦ −0.2 −0.4 −0.6 −0.8 −1.0 −1.2 −1.4 −0.2 −0.4 −0.6 −0.8 −1.0 −1.2 −1.4 σ1/fc σ2/fc

Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen’s model Red line = Barcelona model

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Deviatoric plane

σ1 σ2 σ3 σ1 σ2 σ3

π − plane σm = −fc Green line = Mohr-Coulomb with tension cut-off Blue line = Ottosen’s model Red line = Barcelona model

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Thermodynamic formulation

Two potential functions ψc = ψc(S) S = (σ, D, κ) Specific Gibbs free energy γ = ρ0 ˙ ψc − ˙ σ : ǫ. γ 0. Clausius-Duhem inequality ϕ(W; S) W = (Y , K) Dissipation potential γ ≡ BY : Y + BKK Define Y = ρ0 ∂ψc ∂D K = −ρ0 ∂ψc ∂κ ,

∂ψc ∂σ − ǫ

: ˙

σ +

D − BY

: Y + (− ˙

κ − BK) K = 0. ǫ = ρ0 ∂ψc ∂σ , ˙ D = BY , ˙ κ = −BK,

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SLIDE 11

1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Specific model

Specific Gibbs free energy ρ0ψc(σ, D, κ) = 1 + ν 2E

trσ2 + tr(σ2D)
− ν

2E (1 + 1

3trD)(trσ)2 + ψc,κ(κ)

Elastic domain Σ = {(Y , K)|f(Y , K; σ) 0} where the damage surface is defined as f(Y , K; σ) = A ˜ J2 σc0 + Λ

J2 + BI1 − (σc0 + K) = 0,

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Invariants in terms of Y

˜ J2 = 1 1 + ν

EtrY − 1

6(1 − 2ν)(trσ)2

˜ J3 = 2 3(1 + ν)

E [tr(σY ) − trσtrY ] + 1

9(1 − 2ν)(trσ)3

ϕ(Y , K; σ) = IΣ(Y , K; σ) where IΣ is the indicator function IΣ(Y , K; σ) =

(Y , K) ∈ Σ +∞ if (Y , K) / ∈ Σ (BY , BK) =    (0, 0), if f(Y , Kα; σ) < 0,

λ ∂f ∂Y , ˙ λ ∂f ∂K

, ˙

λ ≥ 0, if f(Y , Kα; σ) = 0, ˙ D = ˙ λ ∂f ∂Y , ˙ κ = − ˙ λ ∂f ∂K

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Some results

Uniaxial compression - ultimate compressive stength σc = 32.8 MPa σc0 = 18 MPa, σt0 = 1 MPa, (I1, √J2) = (−5 √ 3σc0, 4σc0/ √ 2) A = 2.694, B = 5.597, k1 = 19.083, k2 = 0.998 K = [a1(κ/κmax) + a2(κ/κmax)2]/[1 + b(κ/κmax)2] a1 = 85.3 MPa, a2 = −12.65 MPa, b = 0.7032

exp. model −ε11/εc −σ11/σc 2 1.5 1 0.5 1.25 1 0.75 0.5 0.25

0.1 0.2 0.3 0.4 0.5 1 Damage −ε11/εc D11 D22 = D33

Experimental results from Kupfer et al. 1969.

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

Young’s modulus and apparent Poisson’s ratio

10 20 30 0.5 1 1.5 2 2.5 E (GPa) −ε11/εc 0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 −σ11/σc νapp Const. Exp.

Biaxial compression

0.4 0.8 1.2 0.4 0.8 1.2 1.6 −σ11/σc −ε11/εc

Eq. (43)
Eq. (44)

Exp. 0.004 0.008 0.012 0.2 0.4 0.6 Damage −ε11/εc D11 = D22 D33

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

1 Introduction 2 Ottosen’s 4 parameter model 3 Thermodynamic formulation 4 Specific model 5 Some results 6 Conclusions and future work

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1 Introduction 2 Ottosen’s model 3 Thermodynamic 4 Specific model 5 Results 6 Conclusions

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!

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