An anisotropic continuum damagemodel for concrete

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

1Aalto University, Department of Civil and Structural Engineering2VTT

3Tampere University of Technology, Department of Mechanical Engineering and Industrial Systems

4 June 2015

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

Anisotropic damage – 4.6.2015 2/18

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

Anisotropic damage – 4.6.2015 3/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Introduction

The non-linear behaviour ofquasi-brittle materials under loadingis mainly due to damage andmicro-cracking rather than plasticdeformation.Damage of such materials can bemodelled using scalar, vector orhigher order damage tensors.Failure of rock-like materials intension is mainly due to the growthof the most critical micro-crackFailure of rock-like materials incompression can be seen as acooperative action of a distributedmicrocrack array

http://mps-il.com

Anisotropic damage – 4.6.2015 4/18

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

Anisotropic damage – 4.6.2015 5/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Ottosen’s 4 parameter model

AJ2σc

+ Λ√J2 +BI1 − σc = 0,

Λ =

{k1 cos[ 13 arccos(k2 cos 3θ)] if cos 3θ > 0k1 cos[ 13π − 1

3 arccos(−k2 cos 3θ)] if cos 3θ ≤ 0.

cos 3θ =3√

3

2

J3

J3/22

, : Lode angle

σc: the uniaxial compressive strengthI1 = trσ: the first invariant of the stress tensorJ2 = 1

2s : s, J3 = det s = 13 trs3 : deviatoric invariants

A,B, k1, k2: material constants

Anisotropic damage – 4.6.2015 6/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Meridian plane & plane stress

σ1

σ2 σ3

σ1

σ2 σ3

(a) (b)

Figure 1: Comparison of the failure surfaces on the deviatoric plane. Green line indicatesthe Mohr-Coulomb criteria with tension cut-off, blue line the Ottosen model and red line theBarcelona model. (a) π-plane, (b) σm = fc.

1

2

3

4

5

6

7

0−1−2−3−4−5 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

(a) (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).

6

Green line = Mohr-Coulomb with tension cut-offBlue line = Ottosen’s modelRed 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

(a) (b)

Kuva 5.11: Murtopintojen vertailu deviatorisella tasolla. Vihrea viiva on vedossa katkaistuMohr-Coulomb, sininen Ottosenin malli ja punainen viiva kuvaa Lublinerin mallia. (a)π-taso, (b)σm = f ′

c.

0−1−2−3−4−5 1

1

2

3

4

5

6

7

θ = 0◦

θ = 60◦σe/f

′c

σm/f′c

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/f′c

σ2/f′c

(a) (b)

Kuva 5.12: Murtopintojen vertailu (a) meridiaanitasoillaja (b) tasojannitystilassa. Vih-rea viiva on vedossa katkaistu Mohr-Coulomb, sininen Ottosenin malli ja punainen viivakuvaa Lublinerin mallia,f ′

t = 0,1; f ′bc = 1,16.

47

π − plane σm = −fc

Green line = Mohr-Coulomb with tension cut-offBlue line = Ottosen’s modelRed line = Barcelona model

Anisotropic damage – 4.6.2015 8/18

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

Anisotropic damage – 4.6.2015 9/18

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 = ρ0ψc − σ : ε. γ > 0. Clausius-Duhem inequality

ϕ(W ;S) W = (Y ,K) Dissipation potential

γ ≡ BY : Y +BKK

Define Y = ρ0∂ψc

∂DK = −ρ0

∂ψc

∂κ,(

ρ0∂ψc

∂σ− ε)

: σ +(D −BY

): Y + (−κ−BK)K = 0.

ε = ρ0∂ψc

∂σ, D = BY , κ = −BK ,

Anisotropic damage – 4.6.2015 10/18

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

Anisotropic damage – 4.6.2015 11/18

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

3 trD)(trσ)2 +ψc,κ(κ)

Elastic domain

Σ = {(Y ,K)|f(Y ,K;σ) 6 0}

where the damage surface is defined as

f(Y ,K;σ) =AJ2σc0

+ Λ

√J2 +BI1 − (σc0 +K) = 0,

Anisotropic damage – 4.6.2015 12/18

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) =

{0 if (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

Anisotropic damage – 4.6.2015 13/18

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

Anisotropic damage – 4.6.2015 14/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Some resultsUniaxial compression - ultimate compressive stength σc = 32.8MPaσc0 = 18MPa, σt0 = 1MPa, (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 = 85.3MPa, a2 = −12.65MPa, b = 0.7032(a) (b)

exp.model

−ε11/εc

−σ11/σ

c

21.510.50

1.25

1

0.75

0.5

0.25

0

1

0

0.1

0.2

0.3

0.4

0 0.5 1

Dam

age

−ε11/εc

(a)

D11

D22 = D33

0

0.02

0.04

0 0.02 0.04 0.06 0.08 0.1

Dam

age

ε11/εc

(b)

D11

D22 = D33

Figure 6: (a) Predicted damage-strain curves of the constitutive model for the concrete speci-

men under uniaxial compression, (b) Predicted damage-strain curves of the constitutive model

for the concrete specimen under uniaxial tension.

0

0.2

0.4

0.6

0.8

1

1.2

-0.4 0 0.4 0.8

−σ

11/σ

c

−(∆V )/V

Eq. (43)Eq. (44)

Exp.

Figure 7: Comparison between the predicted volumetric strain-stress curves for concrete spec-

imen under uniaxial compression with hardening variables of equations (43) and (44) and

experimental data (Kupfer et al., 1969).

16

Figure 2. (a) Stress-strain diagram in uniaxial compression. Experimental data from Ref. [9], (b)Damage evolution in uniaxial compression. Notice that damage is larger in the planes parallel to theloading direction indicating splitting failure mode.

References

[1] M. Basista. Micromechanics of damage in brittle solids. In J.J. Skrzypek and A. Ganczarski,editors, Anisotropic Behaviour of Damaged Materials, volume 9 of Lecture Notes in Appliedand Computational Mechanics, pages 221–258. Springer-Verlag, 2003.

[2] U. Cicekli, G.Z. Voyiadjis, and R.K. Abu Al-Rub. A plasticity and anisotropic damagemodel for plain concrete. International Journal of Plasticity, 23:1874–1900, 2007.

[3] A. Dragon and Z. Mroz. A continuum model for plastic-brittle behavior of rock and concrete.International Journal of Engineering Science, 17:121–137, 1979.

[4] M. Fremond. Non-Smooth Thermomechanics. Springer, Berlin, 2002.

[5] P. Grassl and M. Jirasek. Damage-plastic model for concrete failure. International Journalof Solids and Structures, 43:7166–7196, 2006.

[6] P. Grassl, K. Lundgren, and K. Gylltoft. Concrete in compression, a plasticity theory witha novel hardening law. International Journal of Solids and Structures, 39:5205–5223, 2002.

[7] L. Jason, A. Huerta, G. Pijaudier-Cabot, and S. Ghavamian. An elastic plastic damage for-mulation for concrete: Application to elementary tests and comparison with an anisotropicdamage model. Computer Methods in Applied Mechanics and Engineering, 195:7077–7092,2006.

[8] L.M. Kachanov. On the creep fracture time. Iz. An SSSR Ofd. Techn. Nauk., (8):26–31,1958. (in Russian).

[9] H. Kupfer, H.K. Hilsdorf, and H. Rusch. Behaviour of concrete under biaxial stresses.Journal of the American Concrete Institute, 66(8):656–666, August 1969.

[10] J. Lee and G.L. Fenves. Plastic-damage model for cyclic loading of concrete structures.Journal of the Engineering Mechanics, ASCE, 124(8):892–900, August 1998.

Experimental results from Kupfer et al. 1969.

Anisotropic damage – 4.6.2015 15/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Young’s modulus and apparent Poisson’s ratio

0

10

20

30

0 0.5 1 1.5 2 2.5

E(G

Pa)

−ε11/εc

(a)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5

−σ11/σ

c

νapp

(b)

Const.Exp.

Figure 5: (a) The predicted reduce of the Young’s modulus E ,(b) Apparent Poisson’s ratioνapp = −ε22/ε11 of the concrete specimen under uniaxial compression plotted using theconstitutive model compared to experimental results (Chen, 1982).

D22 = D33. This is a characteristic property of elastic-brittle materials (Basista,2003; Murakami and Kamiya, 1997; Nemat-Nasser and Hori, 1993).

In Fig.7 the volumetric strain-stress curves obtained by the solution of theconstitutive equations and implementation of the hardening variables of equa-tions (43) and (44) are illustrated and compared to the experimental observa-tions (Kupfer et al., 1969). Although some deviation from experimental data isobserved, the model with hardening variable (43) qualitatively predicts the vol-umetric change of the material. Applying the hardening variable (44), however,causes the model to fail to predict the volume increase which will eventually re-sult in positive values for ∆V/V , after the material reaches its minimum volume.

The predicted behaviour of the concrete specimen subjected to equibiaxialcompression is compared to the experimental data (Kupfer et al., 1969), as canbe seen in Fig.8a. Although the model with both hardening variables givesan acceptable approximation of the ultimate equibiaxial compressive stress, themagnitude of the predicted strain ε11 corresponding to the ultimate stress isapproximately 60% smaller than expected. Moreover, as shown in Fig.8b, pre-dicted damage parallel to the loading direction D11 = D22, compared to damageperpendicular to the loading D33 is unrealistically large. For plotting Fig.8bthe hardening variable of equation (43) is used. Therefore, the new harden-ing variable K of equation (43) does not improve the results obtained by theimplementation of equation (44), when the material is subjected to equibiaxialcompression. However, it should be remembered that the results shown herereflects the poitwise material behaviour, whereas in finite size experiments thedeformation localizes in narrow regions and the measured values are averages.

15

Biaxial compression

0

0.4

0.8

1.2

0 0.4 0.8 1.2 1.6

−σ11/σ

c

−ε11/εc

(a)

Eq. (43)Eq. (44)

Exp.

0

0.004

0.008

0.012

0 0.2 0.4 0.6

Dam

age

−ε11/εc

(b)

D11 = D22

D33

Figure 8: (a) Comparison between the predicted stress-strain curves for concrete specimenunder equibiaxial compression with hardening variables of equations (43) and (44) and ex-perimental data (Kupfer et al., 1969), (b) Predicted damage-strain curves of the constitutivemodel for the concrete specimen under equibiaxial compression.

5. Concluding remarks

In this paper a termodynamically consistent formulation to model anisotropicdamage for quasi-brittle materials is proposed. The model is based on properexpressions for the spesific Gibbs free energy and the complementary form ofthe dissipation potential. Damaging of the material is described by a symmet-ric second order damage tensor. Especially, the failure surface is formulated insuch a way that it will mimic the behaviour of the well known Ottosen’s fourparameter failure surface. The formulation is basicly non-associated, however,the formulation follows closely the one for the standard dissipative solid.

It is well known that strain-softening rate-independent material model willresult in an ill-posed boundary value problem in the softening range. In the finiteelement analysis use of such model produces highly mesh-dependent solutions.A simple remedy would be to make the softening material model dependent onthe size of the finite element. More rigorous approach to regularize the prob-lem would be to include strain-rate effects or spatial gradients into the model.Further improvements would be the inclusion of an anisotropic hardening in themodel to facilitate analysis for cyclic loading.

References

Abaqus, . Theory Manual. Version 6.10.

Basista, M., 2003. Micromechanics of damage in brittle solids, in: Skrzypek,J., Ganczarski, A. (Eds.), Anisotropic Behaviour of Damaged Materials.

17

Anisotropic damage – 4.6.2015 16/18

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

Anisotropic damage – 4.6.2015 17/18

1 Introduction

2 Ottosen’s model

3 Thermodynamic

4 Specific model

5 Results

6 Conclusions

Conclusions and future work

Continuum damage formulation of theOttosen’s 4 parameter modelCan model axial splittingImplementation into FE software (owncodes, ABAQUS)Development of directional hardeningmodelRegularization by higher order gradients

Thank you for your attention!

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