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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion The simulation of time dependent behaviour of cement bound materials with a micro-mechanical model Robert Davies and Dr Anthony Jefferson BRE Institute of

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

The simulation of time dependent behaviour of cement bound materials with a micro-mechanical model

Robert Davies and Dr Anthony Jefferson

BRE Institute of Sustainable Engineering, Cardiff University

SSCS2012 /Aix en Provence,France /May 29-June 1 2012

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Introduction

Micromechanical models: Simple mechanisms considered at micro (meso) scales which are expected to capture the macroscopic behaviour Viable alternative to phenomenological models

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Introduction

Micromechanical models: Simple mechanisms considered at micro (meso) scales which are expected to capture the macroscopic behaviour Viable alternative to phenomenological models Inelastic strain may derive from: shrinkage creep micro-cracking differential thermal expansion ageing Need to simulate inelastic behaviour in the matrix phase alone

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Elastic two-phase composite

Two phase composite idealisation with Spherical inclusions(Eshelby solution with modified Mori-Tanaka averaging) Penny-shaped micro-cracks(Budiansky and O‘Connell) ¯ σ = fΩ · σΩ + fM · σM (1) ¯ ε = fΩ · εΩ + fM · εM + εa (2)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Elastic two-phase composite

Two phase composite idealisation with Spherical inclusions(Eshelby solution with modified Mori-Tanaka averaging) Penny-shaped micro-cracks(Budiansky and O‘Connell) ¯ σ = fΩ · σΩ + fM · σM (1) ¯ ε = fΩ · εΩ + fM · εM + εa (2) Average stress-strain relationship ¯ σ = DMΩ : ¯ εe = DMΩ : (¯ ε − εa) (3) DMΩ = (fΩ · DΩ : TΩ + fM · DM) · (fΩ · TΩ + fM)−1, (4a) TΩ = I2s + SΩ : AΩ, (4b) AΩ = [(DΩ − DM) : SΩ + DM]−1 : (DΩ − DM) (4c)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Non-linear behaviour within the composite

Inelastic strains in the inclusion DΩ : (εo + εc + εIN) = DM : (εo + εc + εIN − ετ) (5a) εc = SΩ : (ετ + εIN) (5b)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Non-linear behaviour within the composite

Inelastic strains in the inclusion DΩ : (εo + εc + εIN) = DM : (εo + εc + εIN − ετ) (5a) εc = SΩ : (ετ + εIN) (5b) Inelastic strains in the matrix Secant moduli method (Weng) DΩ : (εo + εc) = DSecant : (εo + εc − ετ) (6a) εc = Ssecant : ετ (6b) Elastic constraint method (Weng) DΩ : (εo + εc + εIN) = DM : (εo + εc + εIN − ετ) (7a) εc = SΩ : (ετ − εIN) (7b)

(Nemat Nasser and Hori(1993), Mura (1987) and Weng (1988))

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Two material problem with matrix undergoing transformation strain εIN. Ω remains elastic.

Matrix phase σM = DM : εM = DM : (εo+εc−εIN) (8) Inclusion phase σΩ = DΩ : εΩ = DΩ : (εo+εc) (9) Consistency equation DΩ : (εo + εc) = DM : (εo + εc − ετ) (10) Constrained strain εc = SΩ : (ετ − εIN) (11) Transformation eigenstrain ετ = AΩ : (εo − SΩ : εIN) (12)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Individual phase and composite equations

σM = DM : εMel = DM : (εM − εIN) (13) σΩ = DΩ : TΩ : (εM − SΩ : εIN) (14) Substitution and rearranged total strain equation εM = (fΩ · TΩ + fM)−1 : (¯ ε + fΩ · TΩ · SΩ : εIN) (15) Constitutive equation ¯ σ = DMΩ : (¯ ε − εINEQ) (16) where εINEQ = DMΩ−1·(fΩ·DΩ·TΩ : SΩ+fM·DM−fΩ·DMΩ·TΩ : SΩ) : εIN (18)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Individual phase and composite equations

σM = DM : εMel = DM : (εM − εIN) (13) σΩ = DΩ : TΩ : (εM − SΩ : εIN) (14) Substitution and rearranged total strain equation εM = (fΩ · TΩ + fM)−1 : (¯ ε + fΩ · TΩ · SΩ : εIN) (15) Constitutive equation ¯ σ = DMΩ : (¯ ε − εINEQ − εa) (17) where εINEQ = DMΩ−1·(fΩ·DΩ·TΩ : SΩ+fM·DM−fΩ·DMΩ·TΩ : SΩ) : εIN (18)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Penny-shaped micro-cracks in the matrix

σM = DM : (εM − εIN − εF RM) (19) σΩ = DΩ : εΩ = DΩ : (εo +εc) (20)

Consistency equation DΩ : (εo + εc) = (1 − ω)DM : (εo + εc − ετ) (21) Constrained strain εc = SΩ : (ετ − εIN) (23)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Penny-shaped micro-cracks in the matrix

σM = DM : (εM − εIN − εF RM) (19) σΩ = DΩ : εΩ = DΩ : (εo +εc) (20)

Consistency equation DΩ : (εo + εc) = DM : (εo + εc − ετ − εFRMΩ) (22) Constrained strain εc = SΩ : (ετ − εIN) (23)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Penny-shaped micro-cracks in the matrix

σM = DM : (εM − εIN − εF RM) (19) σΩ = DΩ : εΩ = DΩ : (εo +εc) (20)

Consistency equation DΩ : (εo + εc) = DM : (εo + εc − ετ − εFRMΩ) (22) Constrained strain εc = SΩ : (ετ − εIN) (23) Transformation eigenstrain ετ = AΩ : (εo−SΩ : εIN)−BΩ : εFRMΩ (24) where BΩ = [(DΩ − DM : SΩ + DM]−1·DM (25)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Local model of fracture strain

eFRM = CLM : sM (26a) sM = N · σM (26b) eFRM = CLM : N · σM (26c) εFRM =

  • NT · eFRM · ds

(26d)

εF RM = (I2s + Cadd)−1 · Cadd : DM : (εM − εIN) (27a) εF RMΩ = (I2s + Cadd : DM : VΩ)−1 · Cadd : DM : UΩ : (εM − SΩ : εIN) (27b) where Cadd = 1 2π

  • 2π
  • π

2

Nε : CLM : N · ω(θ, ψ) 1 − ω(θ, ψ) sin(ψ)dψdθ (28a) UΩ = I2s + (SΩ − I2s) : AΩ, (28b) VΩ = I2s + (SΩ − I2s) : BΩ (28c)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Inelastic strain and micro-cracks in the matrix

Constitutive equation ¯ σ = DMΩFR : (¯ ε − εINFREQ) (29) where

DMΩF R = F : H (30a) εINF REQ = [(DMΩF R

−1 : G) − J] : εIN

(30b) F = fΩ · DM UΩ − fΩ · DM : VΩ · (I2s + Cadd : DM : VΩ)−1 · Cadd : DM : UΩ + fM · DM − fM · DM Cadd · (I2s + Cadd : DM )−1 · DM (30c) H = [fΩ · D−1

Ω

: DM : UΩ + fM · I2s −fΩ·D−1

Ω

: DM : VΩ · (I2s + Cadd : DM : VΩ)−1 · Cadd : DM : UΩ]−1 (30d) G = fΩ · DM UΩ : SΩ − fΩ · DM : VΩ · (I2s + Cadd : DM : VΩ)−1 · Cadd : DM :UΩ : SΩ + fM · DM − fM · DM Cadd · (I2s + Cadd : DM )−1 · DM (30e) J = [fΩ · D−1

Ω

: DM : UΩ − fΩ · D−1

Ω

: DM :VΩ · (I2s + Cadd : DM : VΩ)−1 · Cadd : DM : UΩ] : SΩ (30f) (30g)

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Free shrinkage with inelastic strain in matrix only with no micro-cracking

  • 0.0016
  • 0.0014
  • 0.0012
  • 0.001
  • 0.0008
  • 0.0006
  • 0.0004
  • 0.0002

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Strain in Inclusion xx Strain in Matrix xx Total Strain xx Inelastic Strain in Matrix xx Elastic Strain in Matrix xx

  • 20
  • 15
  • 10
  • 5

5 10 15 5 10 15 20 25 30 35 40 Stress σx-x (N/mm2) Time (Days) Stress in Inclusion xx Stress in Matrix xx Total Stress xx

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Free shrinkage with inelastic strain in matrix with micro-cracking

  • 0.0016
  • 0.0014
  • 0.0012
  • 0.001
  • 0.0008
  • 0.0006
  • 0.0004
  • 0.0002

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Strain in Inclusion xx Strain in Matrix xx Total Strain xx Inelastic Strain in Matrix xx Elastic Strain in Matrix xx

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 5 10 15 20 25 30 35 40 Stress σx-x (N/mm2) Time (Days) Stress in Inclusion xx Stress in Matrix xx Total Stress xx

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Free shrinkage with inelastic strain in matrix with micro-cracking

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Free shrinkage comparison with elastic constraint method (early results)

  • 0.0016
  • 0.0014
  • 0.0012
  • 0.001
  • 0.0008
  • 0.0006
  • 0.0004
  • 0.0002

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Strain in Inclusion xx Strain in Matrix xx Total Strain xx Inelastic Strain in Matrix xx Elastic Strain in Matrix xx

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 5 10 15 20 25 30 35 40 Stress σx-x (N/mm2) Time (Days) Stress in Inclusion xx Stress in Matrix xx Total Stress xx

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Uni-axial tension strain path with inelastic strain in matrix

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 0.0002

0.0002 0.0004 0.0006 0.0008 Stress σx-x (N/mm2) Strain εx-x Inelastic Strain in Matrix

  • 0.0015
  • 0.001
  • 0.0005

0.0005 0.001 0.0015 0.002 0.0025 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Total Strain Inelastic Strain in Matrix (Total Strain - Inelastic Strain in Matrix )

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Uni-axial tension strain path with inelastic strain in matrix

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 0.0002

0.0002 0.0004 0.0006 0.0008 Stress σx-x (N/mm2) Strain εx-x Inelastic Strain in Matrix 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 Damage ωi Time (Days) Damage

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Uni-axial tension strain path with inelastic strain in matrix

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 0.0002

0.0002 0.0004 0.0006 0.0008 Stress σx-x (N/mm2) Strain εx-x Inelastic Strain in Matrix 0.99 0.992 0.994 0.996 0.998 1 5 10 15 20 25 30 35 40 Damage ωi Time (Days) Damage

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Uni-axial tension strain path no inelastic strain in matrix

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

  • 0.0002

0.0002 0.0004 0.0006 0.0008 Stress σx-x (N/mm2) Strain εx-x Inelastic Strain in Matrix No Inelastic Strain in Matrix 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Total Strain Inelastic Strain in Matrix (Total Strain - Inelastic Strain in Matrix )

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Uni-axial tension strain path comparison with Elastic Constraint Method (early results)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

  • 0.0002

0.0002 0.0004 0.0006 0.0008 Stress σx-x (N/mm2) Strain εx-x Inelastic Strain in Matrix No Inelastic Strain in Matrix Elastic Contraint Method

  • 0.0015
  • 0.001
  • 0.0005

0.0005 0.001 0.0015 0.002 0.0025 0.003 5 10 15 20 25 30 35 40 Strain εx-x Time (Days) Total Strain Inelastic Strain in Matrix (Total Strain - Inelastic Strain in Matrix )

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Content Outline

1

Elastic two-phase composite

2

Non-linear behaviour within the composite

3

A two-phase composite with inelastic strain in the matrix only

4

Micro-cracks in the matrix

5

Damage predictions

6

Concluding remarks

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Concluding remarks

A new way of introducing inelastic strains into the matrix Introducing micro-cracking into the matrix Models predict an expected inelastic and micro-cracking response Basic framework for simulating real behaviour of cementitious materials Future work Inelastic creep strain using solidification theory Integration of new thermo-hygro model Modelling self-healing processes Experimental work for model validation

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

Thank you for your attention. I now welcome your questions.

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Elastic Composite Non-linear Inelastic matrix Micro-cracks Damage Conclusion

First principle derivation

1 One material ellipsoid undergoes a transformation

σM = DM : (εo+εc) σMi = DM : (εo+εc−εi) where εc = SΩ : εi

2 Standard two material problem

σΩ = DΩ : (εo + εc) = DM : (εo + εc − ετ) where εc = SΩ : ετ

3 Standard two material problem with εi in the Ω

DΩ : (εo+εc−εi) = DM : (εo+εc−ετ−εi) where εc = SΩ : (ετ+εi)

4 One material problem with M undergoing εi

σM = DM : (εo+εc) σMi = DM : (εo+εc−εi) where εc = SΩ : εi

transformation strain on outside (added to both sides) σM = DM : (εo + εc + εi) σMi = DM : (εo + εc − εi + εi) where εc = SΩ : εi Ω undergoing -ve strain to start with (removed to both sides) σM = DM : (εo + εc − εi) σMi = DM : (εo + εc + εi − εi) where εc = −SΩ : εi