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Mechanics of Building MaterialsConcrete Plasticity

F. Wittel


Concrete plasticity

Challenges:

Non-linear stress-strain relation in multi-axial stress states.

Stain softening and anisotropic elastic degradation.

Progressing crack formation by tensile stress, or strain.

Pullout of reinforcement, aggregate interlocking.

Time dependent behavior like creep or shrinkage.

Concrete behavior and failure

hardening models for concrete

softening models for concrete

damage models for concrete


Mechanical concrete behavior - uniaxial

• Linear region, inelastic region and localization.• Tensile strength/ ultimate tensile strain approx. 10% of compressive values.• Non-linear, elastic behavior under tension t=0.015% .

Growth of a dominant crack brittle failure.• Non-linear growth under compression c=0.25%.

Simultaneous growth of multiple cracks Dilatancy• Unloading along linear-elastic paths.• MOE depends on damage progress.• Reloading at identical MOE, damage starts to evolve, when «yield stress»

is reached.

1 2 3v


Tensile strain Compressive strain Dilatation Contraction Tensile strain Compressive strain Tensile strain Compressive strain

Mechanical behavior of concrete - biaxial

Ductility for biaxial compressive-compressive failure higher than uniaxial one. Aggregate detachment is retarded by compressive stress. Reduced dilatancy by biaxial compressive stress, since cracks are compressed. Compression-tension combination reduces the strength and «ductility». Tensile-tensile uni- and biaxial ultimate strength approx. 0.08%

Compressive-compressive Compressive-tensile Tensile-tensile

(Kupfer etal 1969, Nelissen 1972)


Mechanical behavior of concrete - triaxial

Pressure has significant influence on deformation behavior. Under hydrostatic stress the aggregate detachment is strongly reduced. Failure is no longer dominated by the growth of single cracks.

Ductility increases with increasing hydrostatic load. Ultimate strain and strength increase significantly under hydrostatic load.

(Palaniswamy und Shah 1974,Balmer 1949)

Strain

Long

itudi

nal s

tres

s [M

Pa]


Failure criteria for concrete - triaxial

1 2 3 1 2 3( , , ) 0 ( , , ) 0 ; ( , , ) 0f f I J J f

Failure envelopes are functions of the three principal stresses, stress invariants or in HAIGH-WESTERGAARD coordinates and mainly path dependent.

Failure modes depend on the combination of stress components. Biaxial compressive-compressive failure is higher than uniaxial compressive failure (up to 25%) Compressive-tensile combination reduces the strength with increasing tensile stress. Tension-tension is identical to uniaxial tensile failure.Non-affine intersections in the deviatoric plane as function of Merges from triangular to circular shape with increasing volumetric stress.Non-linear function in the meridional plane,

(Kupfer etal 1969, Nelissen 1972)

2

.tf

c o n s

2 11 0 .8t cf f

2

1 2 1

23 .6 5 0

c c c

c

f f f

f


1 2 2 2 1

2

111 23

111 23 3

( , , ) 1 0

( , , ) 3 1 02 2

c o s c o s ( c o s 3 ) fü r c o s 3 0

c o s c o s ( c o s 3 ) fü r c o s 3 0

f I J a J J b I

af b

k k

k k

OTTOSEN criterion (4P):

Tests:1. Uniaxial compressive test fc (=60°)2. Uniaxial tensile test ft=0.1fc (=0°)3. Biaxial compressive test 1= 2=-1.16fc4. Triaxial stress state at (fc, fc)=(-5,4) on the compressive meridian

(=60°).

Example: (a=1.2759,b=3.1962, k1=11,7365, k2=0,9801)

Valid for large span width of stress combinations but complicated expression for


Compressive meridian

Tensile meridian

Biaxialenvelope

Deviatoric plane


Tests:1. Uniaxial compression fc (=60°)2. Uniaxial tension ft=0.1fc (=0°)3. Biaxial compression 1= 2=1.15fc4. Triaxial stress state at (oct/fc, oct/fc)=(-1.95,1.6) on the compressive

meridian (=60°).

Example: (a=2.0108, b=0.9714, c=9.1412, d=0.2312;A=a/2=1.0054, B=(2/3)1/2c=7.4638, C=b/21/2=0.6869, D=31/2(d+B/61/2)=5.678

Acceptable predictions, however for high I1 rather conservative. Not continuous differentiable!

2

1 1

1 2 1 2 2 1 1

( , , ) ( c o s ) 1 0

m it c o s ( 3 2 6 )

( , , ) 1 0

f A B C D

I

f I J a J b J c d I

HSIEH-TING-CHEN (HTC) criterion (4P):

! All stresses are normalized by fc !

(Launay et al 1970)



Tests:1. Uniaxial cylinder compression strength2. Uniaxial tensile strength

for the strength ratio3. Uniform biaxial compressive strength

for the strength ratio4. High compressive stress

in the tensile meridian5. High compressive stress

on the compressive meridianAdditionally the parabolas have to intersect at the apex at the hydrostatic axis:

a0=0.1025; a1=-0.8403; a2=-0.0910; b1=-0.4507; b2=-0.1018

WILLAM-WARNKE criterion (1974) (5P):

! All stresses normalize by fc !Intersection of meridians on hydrostatic axis a0=b0

20 1 2

20 1 2

m t t

m c c

a a a

b b b

1 / 3m I

( 6 0 , 0 )c cf f ( 0 )tf

t t cf f f( 0 , 0 )b c b cf f

b c b c cf f f

2 2, ,m c m cf f

1 1, ,m c m cf f 1( 0 , 0 )

2( 6 0 , 0 )

00 0 0( ) ( ) 0 a t 0m

t ccf



WILLAM-WARNKE criterion (1974) (5P):

122 2 2 2 2

2 2 2 2

( ) 4 ( ) c o s 2 ( 2 ( ) c o s

2 4 ( ) c o s 5 4 )

c t c t c c t

c t c c t t t c

Valid for all stress combinations and gives in practically relevant regimes good agreement with reality.

Contains for different parameter sets all other models (VM, DP, 3P WW) Failure envelope opens up to negative hydrostatic axis No failure for

very high



Comparison:



Concrete plasticity: hardening behavior

Basic concepts for concrete plasticity:

1. Definition of an initial yield surface and if a failure surface in stress space limitation of the elastic domain and the hardening domain.

2. Hardening ruleDevelopment of a loading function as well as of hardening parameters.

3. Flow rule incremental stress-strain relation.


Concrete Plasticity: Hardening behavior

Initial yield surface:

Early approaches:Scaling of failure surface with 0.3fc.

Problem: Plastic volume change under hydrostatic load is ignored. Behavior at different stress states is from a mechanical point

of view not identicalIsotropic scaling must be wrong


Hardening rule:

Subsequent yield surfaces:

Non-uniform, plastic hardening model (Han, Chen 1985): Only one hardening parameter k0

Model with multiple plastic hardening (Ohtani, Chen 1988): Three hardening parameters c,bc,t

Functions of the effective plastic strains

Different modes like (Vermeer, Deborst 1984) via MC.

1 2( , , , . . . , ) 0i j nf



Flow rules:

Only non-associated flow rules due to dilatation

i scalar parameter (i)

Non-uniform, plastic hardening model (Han, Chen 1985): DRUCKER-PRAGER type of potential surface

Other models (Onate 1988, Vermeer 1984) used potential surfaces of MOHR-COULOMB-type:

Dilatational angle

Modell with multiple plastic hardening (Ohtani, Chen 1988): we will have a closer look later...

1 2( , , , . . . , ) 0i j ng g

1 1 2( , )i jg I J

22i jp

ij i ji j

sgd d d

J

3p pV iid d d

1 11 23 3

( , ) s in c o s s in s ini jg I J



Concrete plasticity: Non-uniform plastic hardening model

Non-uniform plastic hardening model (Han, Chen 1987): Closed initial yield surface, Subsequent yield surfaces determined by hardening parameter k0

Failure surface limits all subsequent surfaces.

( , , ) ( , ) 0 ; 6 0m f mf General form of failure surface:

22 J with

OTTOSEN 4P model: 212( , ) 2 2 8 ( 3 1)f m ma a b

HSIEH-TING-CHEN 4P model: 212( , ) ( c o s ) ( c o s ) 4 ( 3 1)f m mA B C B C A D

WILLAM-WARNKE 5P model: 1( , ) ( )f m v s t 2 2( , ) 2 ( ) c o sm c c ts s ( , ) 2m c t ct t u

2 2 2 2( , ) 4 ( ) c o s 5 4m c t t t cu u 22 2 2( , ) 4 ( ) c o s 2m c t c tv v

2 21 1 2 0 1 1 2 0

2 2

1 14 ( ) ; 4 ( )2 2c m t mb b b b a a b a ab a

Parabolic meridians:


Initial yield - and subsequent yield surfaces:(Meridians)

1. T-T zone mt) Yield surface identical to failure surface Brittle failure.2. C-T zone tmc): Plastic hardening zone slowly develops.3. C zone with small hydrostatic pressure cmk): Meridians are proportionally reduced failure surfaces.4. C-C zone mk): Yield surface gradually closes at the hydrostatic axis large hardening domain emerges.

12 0

3IJ 1

2 03

IJ

12 10 u n d 0

3IJ I 1

2 10 u n d 03

IJ I

0( , , ) ( , ) ( , ) 0m m f mf k k Subsequent surfaces: with 0 1yk k



Incremental stress-strain relations:- General relations

( )e pij ijk l k l ijk l k l k ld C d C d d

e pk l k l k ld d d

2

1 2i jk l ik jl il jk ij k lC G

0i j pij p

f fd f d d

Subsequent yield surface withConsistency condition.

pH( ) 0p p

ijk l k l k l pij

f fd f C d d H d

:pd d

:pij

ij

gd d

Flow rule1

p q k l k lp q

fd C d

h

pm n p q

m n p q

f g fh C H

with

pij ijk l ijk l k ld C C d

*1pijk l ij i jC H H

h

*ij i jm n

m n

ij p q k lp q

gH C

fH C

Constitutive law:



Incremental stress-strain relation:- Constitutive equation with associated flow rule

f g

0 1 2i j i j i ji j i j

g fB B s B t

2 2 3 2 20 1 2 1 2 2 2 2

1 22 3 2 6

1 2 3p f

h G B B J B B J B J H

*

0 1 2

12

1 2i j i j i j i j i jH H G B B s B t

1p

ijk l i j k lC H Hh

Plastic stiffness tensor

223 233 ;i j ij i j jk k i ij ijt s s s s J t t J with

- Constitutive equation with non-associated flow rule g=Drucker-Prager function

20 1 2 3

2

312 3

1 2 2pB f

h G B B J J HJ

*

2

1 12

1 2 2i j i j i jH G s

J

f g

0 1 2

12

1 2i j i j i j i jH G B B s B t



Effective stress – effective strain dp

2 3 ; 3;m f c

2 3 0 3 2c cf k k

d

pp ij i j i j p

ij

gd W d d d

Uniaxial compression test (0,0)

1ij

ij

g

dp:

• Associated flow:0 1 1 2 2 32 ( 2 3 )

3 c

B I B J B J

k

• Non-associated flow:

1 22 ( )

3 c

I J

k

:f

0ij

ij

f fd f d d

3 3d d

3 3

2

3 33c c

m m

df f k d k

d d

3 3

3 3

3 3

2 3

3m

with



Parameters and model prediction 4-5 strength tests for material constants, depending on failure surface. Uniaxial compression test for hardening behavior. Knowledge on deformation behavior for modification parameters N(m,). Definition of dilatation factor for flow behavior.



Concrete plasticity: Multiple hardening plasticity model

Definition of segments, that can evolve independent from each other.

First version of independent hardening models by Murray (2D)

3 independent hardening parameters (1compressive yield stress, 2 tensile yield stresses).

3D Extension by Ohtani and Chen

3 hardening parameters with physical meaning adjustment to experimental data

Associated flow is assumed.


General form of failure surfaces:

1 2( , , , . . . , ) 0 ;ij Nf ( )M M M

1( , , . . . , )M M M ij N Td d M-th hardening parameter

p pp p ij i jd d d 1 2( , , , . . . , , )M M ij N pd d

Damage parameter:

1 21 2

. . . 0i j Nij N

f f f fd f d d d d

Consistency condition:

M MM M M p

M M

d dd d dd d p

ij ij

g gd

0i jk l k l i ji j k l i j i j

f g g gd f C d d d d

1 2

1 21 1 2 2

. . . NN

N N

f f f

with

1i jk l k l

i j

fd C d

h

with em n s t

m n s t m n m n

f g g gh C



Initial yield surfaces and subsequent loading surfaces:

Ohtani and Chen model

( , , , ) 0 ;i j c b c tf

c Momentary yield stress in uniaxial compressionValues in between initial yield stress and strength fc< c< f’c

bc Momentary equal biaxial compressive yield stress fbc< bc< f’bc

t Momentary uniaxial tensile yield stress ft< t< f’t

C-C zone:21

2 13( ) 0i j c cf J A I 2 2

2

b c cc

b c c

A

2 ( 2 )

3 ( 2 )c b c c b c

cb c c

with

T-T and T-C zones:2 21 1

2 1 16 3( ) 0i j t tf J I A I 12 ct tA 2 1

6t c t with

( , , , ) 0 ;i j c b c tf f f f

( , , , ) 0 ;i j c b c tf f f f ( , , , ) 0 ;i j c b c tf f f f

Before hardening:

Hardening:

Failure:



Evolution of hardening parameters

Hardening parameter: ( )

( )

( )

c c p c

b c b c p b c

t t p t

1

2

3

( , , , )

( , , , )

( , , , )

p c p c ij c b c t p

p b c p b c ij c b c t p

p t p t i j c b c t p

d d

d d

d d

Effective strain:

Conditions (for In pure states i = 1. The resulting yield surface has to be convex. Plastic strains in C-C- zone have no influence of tensile hardening and vice

versa. Plastic strain in T-C and C-T zone result in tension and compression

hardening.

1 2 31; 0 1 2 30 ; 1 1 11 2 3;t t

c t c t

I I

C-C C-T; T-C T-T



Incremental stress-strain relation:

*1( ) ; m it p p

ij i jk l ijk l k l i jk l i j i jd C C d C H Hh

*i j i j i jk l

k l

fH H C

Associated flow:

With loading functions:

2 20 0 2 23 2 ; 9 4 3 2i j i j i jH K B G s h K B G J J 0 1

1

1

3

fB A n I

I

with

13

, 0

,c

t

A A n

A A n

C-C::T-T / T-C:1 1 2 2 3 3

p p pc b c tQ H Q H Q H

; ; ;p p pc b c tc b c t

p c p b c p t

d d dH H H

d d d

C-C:2 2

1 1

2 22 12

3

1 1( 4 )( 2 )

3 ( 2 )

2 1( 4 )( 2 )

3 ( 2 )

0

c c b c b c b cc b c c

c c b c b c b cb c b c c

t

fQ I

fQ I

fQ

T-T / T-C

1 1

2

3 1

1( 2 )

6

0

1( 2 )

6

tc

b c

ct

fQ I

fQ

fQ I



Parameter and model prediction

Experimental stress-strain curve for the 3 loadings: uniaxial tension and compression, biaxial compression

T-C curves exhibit large discrepancy with experiments for 2,3 due to strong slope of the yield curve in this stress region in combination with associated flow.

T-T gives almost brittle failure for tension hardening.

C-C describes well the trend of dilatation, however non-linearity is more pronounced in the experiment.

, , , , ,c c b c b c t tf f f f f f , ,p p pc b c tH H H

1 .1 5; 0 .0 9 1; 0 .6 ; 0 .4 5;

0 .5; 9 9 0 .0 ; 0 .2

b c c t c c c b c b c

t t c

f f f f f f f f

f f E f



Concrete plasticity: Softening behavior

Conditions for tests:

1. Homogeneous sample

2. Homogeneous stress state

3. No significant change of specimen geometry.

Strain softening:

1. Transition from continuum to a structure with significant cross sectional change.

2. Behavior is not a real material behavior but structural property size effects.

3. Strain localization formation of a shear band.

4. Shear band width is independent on sample size.

5. Similar behavior under tension.

6. Insertion of localization zone (crack band model / fictitious crack model) or smeared over sample length (plasticity model).


Formulation of plasticity in strain space.

Generalization of the uniaxial behavior to multi-axial stress states.

Problem: When formulations are in stress space, inwards directed stress increments only result in elastic strain increments, while outwards pointing ones result in hardening. Strain hardening is not part of the theory.

For strain hardening also inwards directed increments must result in plastic strain increments.

Formulation of the loading surface F in strain space:

Hardening and softening can be simultaneously studied.

DRUCKER’s stability postulates render invalid.

Weak stability criteria that allow for instable behavior.



Basic equations:Previously:

e pij ij ij

Now:e p

ij ij ij eij ijk l k lC

(1 ) (1 2 )

( )2 (1 )

ijk l ij k l

ik jl il jk

EC

E

p pij ijk l k lC

eij ijk l k lD 1

( )2i jk l ij k l ik jl i l jkD

E E

e p pij ij i j ijk l k l ij

e p pij ij ij ijk l k l ij

C

E

Loading surface / relaxation surface: ( , , ) 0p

ij ijF k

0i ji j

Fd

Unloading(elastic)

Neutral loading

Loading(plastic)

0i ji j

Fd

0i ji j

Fd

0pijd 0p

ijd 0pijd



Flow rule

0pij i jd W d d

p

iji j

Fd d

1

k lk l

F Fd d

h h

( , , ) ( , , )p pij ij ij ij ij

k lk l

F F d d k F d k

Fd

1pij k l

ij k l

F Fd d

h

Il-yushin’s postulate:

Normality condition (flow rule):

Stress relaxation increment:



Incremental constitutive relation:1p

ij k lij k l

F Fd d

h

1i j ijk l k l

ij k l

F Fd C d

h

e pij i j i jd d d

1e p pijk l ijk l ijk l ijk l

ij k l

F FC C C C

h

Concrete plasticity: Softening behavioreij ijk l k lC

m n ijm n

FD

Compliance tensor:

1i j ijk l k l

ij k l

F Fd C d

h

p q k l k lp q

m n rsm n rs

FD d

dF F

h D

e p pij i j ij ijk l k l i jtu tud d d D d D d

ijtu p q k ltu p q

ij i jk l k l

m n rsm n rs

F FD D

d D dF F

h D

1

i j m n ij k l m n ij i jk l m n ij k lm n m n m n ij k l

F F F F Fd D d D C D d

h

d h d 1

p

iji j

Fd d


Consistency condition:0p p

ij ij ijp pij ij ij

F F F kd F d d d

k

1

i j

i j

d Fd

h

m n p q m n p qp pm n p q m n p q

F F F k Fh D D

k


p pk l ij m n ijd d D

p

iji j

Fd d

0

i j m n ijp pij i j i j i j

F F F k Fd d D

k

/ m n

F


Concrete plasticity: Formulation with plastic failure Ideal plastic body, if unloading and reloading paths are parallel to initial tangent

For concrete stiffness decreases with increasing plastic strain damage development

Model for progressively damaged body (Dougill) considers damage but no remaining plastic strain.

Combination = Model for plastic failure (Bazant, Kim) Plastic deformation via plastic yield theory Stiffness degradation via fracture theory by Dougill.

Stress and strain increment Plastic damage work increment

Dougill 1975


Basic relation:e p f

ij ij ij ijd d d d

ijk l k lC d pijk l k lC d f

ijk l k lC d

e p fij i j ij ijd d d d

ijk l k lD d

Relaxation surface and flow rule: ( , , ) 0p p fij i jF W

0i ji j

Fd

Unloading (elastic) Neutral loading Loading (plastic)

0i ji j

Fd

0i ji j

Fd

0p fi jd 0p f

i jd 0p fi jd

p f p fij i j i jd d d 0p f

ij i jd W d d p f

ijij

Fd d

Normality conditionIl’yushins postulate

Concrete plasticity: Formulation with plastic failure

elastic plastic failure


Energy dissipation rate and decomposition

1 12 2( )p f p e e e p f

ij ij ijk l ij k l i j ij ijD d W d d C d d Plastic deformation Stiffness reduction

( )p fijk l ijk lC C W ijk l

ijk l p f

d CC

d W Stiffness reduction rate:

12( )i jk l p f e p f

ijk l ijk l m n m n m np f

d Cd C d W C d d

d W

12( ) f e e e p f f p f

i j i jk l k l i jk l k l m n m n m n ijk l k ld d C C d d T d Transformation tensor :

fijk l i jm n m n k lT M N 1 1

2e e

ijm n im jn ijp q p q m nM C e e

m n k l m n p q p q k lN C p p p f

ij ijk l k ld T d Transformation tensor

p p fijk l ik jl ijk lT T

Constitutive relation1

i jij

Fd d

h

12( )p e p f

ijm n m n k l m n m n k l m n k lp p fij k l k l

F F F Fh D T T T

W

1e p f p fij ij ij ijk l k l ij ijk l k l

ij k l

F Fd d d C d d C d

h

Concrete plasticity: Formulation with plastic failure


Development of the CDMBasic assumptions of CDMThermo dynamic basics

Strain basedStress based

Scalar damage modelScalar damage model for concreteUnilateral, elastic damage model for concrete

Introduction to Continuum Damage Mechanics (CDM)

A

AD

P

P

dAn

dAD


Basic assumptions of the CDM

Elastic-plastic body

Progressively damaged body

Plastic-damagedbody

Plastic deformations Progressive damage both

Plasticity theory Damage mechanics CDM

Description:

Source for non-linearity:

Approach:

Stereotypes of material behavior:


Dusan Krajcinovic, "It is often argued that the ultimate task of engineering research is to provide not so much a better insight into the examined phenomenon but to supply a rational predictive tool applicable in design”


Typical approach of CDM:1. Representation of the damage state by internal state variables.2. Description of the mechanical behavior of the damaged material.3. Damage progress via initiation and evolution laws for state variables.

Ductile damage (Lemaitre, Voyiadjis, Kattan, Duffaily)Fatigue (Lemaitre, Chaboch)Creep (Leckie, Hayhurst, Hult, Lemaitre, Chaboche)

Micro scale: Scale inherent to mechanisms of strain and damage:Atoms - Displacements - Inclusions - Micro cracks

Macro scale: Scale on which RVEs are valid (~. 0.1mm metal, 1mm polymers, 10mm wood, 100mm concrete). A macro crack is that large, that the phenomenological constitutive equations dominate the behavior on the scale.

Structural scale: Scale of mechanical components, on which the crack is several mm or cm long.

Basic assumptions of the CDM


Basic assumptions of the CDM – damage variablesContinuum mechanical variables phenomenologic variables

2

* 23 (1 ) 3(1 2 ) H

e qe q

32 ( ) ( )e q ij H ij i j H ij

13H k k

Equivalent and hydrostatic stress

230

t p pij i jp

0

tp pij i jw d t

• Equivalent damage stress

• Accumulated plastic strain (vonMises)

• Plastic deformation energy

vP

V

• Porosity= rel. Pore volume c a vr• Pore radius

( )n

S

S

• Real micro crack surface and intersection of pores in arbitrary planes (normal vector n)

( )n

A A

A

( )

( )

0

1

n

n

Undamaged

Fully damaged


Elimination of cracks and pores

Hypothesis of strain equivalence:

(1 )e E E

Decomposition into distinct mechanisms:Elimination of pores

Elimination of cracks

Elimination of cracks

Elimination of pores

(1 ) (1 )c v

1 1 1v c

c v v c

Basic assumptions of the CDM – damage variables


Thermo dynamic Basics (strain based)

( , )i j D Free energy potential:Scalar damage:

12 (1 ) i jk l i j k lD C

0i j i j Clausius-Duhem inequality:

0i j i j i ji j

DD

0 o r

0

i j i ji j i j

DD

Conditions:

Ydissipative flux

2 2

i j k li j k l ij

d d d DD

Stress-strain relation:

Thermodynamic force


CDM- damage evolution (strain based)

Damage evolution law:• In principle each component of the damage variable is function of the present value of all state

variables and internal variables.• Analogy to plasticity via potential surface normality condition

0Y D Clausius-Duhem inequality:

0pij i j

Drucker’s stability postulate:

( , , ) 0i jf f D Y Dissipation potential surface:

State variables= parameters

fD

Y

Damage rate vector

• Damage evolution law or dissipative potential function correlates directly to the microstructural changes in the material.


CDM- damage evolution (stress based)

Stress based formulation:

0i j i j Clausius-Duhem inequality:

1( , )

2 (1 )i j i jk l i j k lD DD

i ji j

0DD

Y

2

1 22 2

1 1 (1 2 ) (1 )

61 1i jk l i j k lY D I J

E ED D

2 2

i j k li j k l i j

d d d DD

Stress-strain relation


CDM – Scalar damage model12 (1 ) i jk l i j k lD C Free energy: Undamaged damaged0 1D

(1 )i j i jk l k lij

D Cd

0 0Y D D Clausius-Duhem inequality:

12 i jk l ij k lY C

d D

Damage criterion:2

ii

0

0 0i i

ii

fo r

fo r

( ) ( ) 0F D K D

Equivalent Strain:

0( 0 )K KInitial condition:

0 0 0 0

( ) 0 0

fo r f a n d f o r fD

F fo r f a n d f

0

( ) ( ) ( )M

M MD F d F


10 D

A

A

AreationalsecCrossTotal

DamageofAreaTotalD D

D = 0: free of damageD = 1: failure

n

nD

dA

dAD nx,

A

AD

P

P

dAn

dAD x – locationn – normal vector

CDM – Scalar damage model


Damage variables Dt and Dc ( )

( )t t

c c

D F

D F

Compression:

Tension:

t t c cD D D c, t Parameter that depend on the stress state

i j i j i j Positive negativePrincipal stresses

Partitioniung of the stress tensor:

1 i i i i i iI

i j i j t i j c Partitioning of the strain tensor:

1

1

i j t i j i i

i j c ij i i

E E

E E

2

2

( )

( )

ti t i c it i

i

c i t i c ic i

i

H

H

i ti c i 1 0

0 0i

ii

fo rH

fo r

CDM – Scalar damage model for concrete


Damage evolution Dt and Dc

00

00

(1 )( ) 1 e x p [ ( ) ]

(1 )( ) 1 e x p [ ( ) ]

tt t t

cc c c

K AD A B K

K AD A B K

Compression:

Tension:

Mazars 1986loading Initial surface

K0 Limit value for damageAt Bt material parameter from bending testAc Bc material parameter from compression test


||Institute for Building MaterialsMazars 1986

Only for uniformly distributed cracks

NO large single cracks!



CDM – Unilateral, elastic damage model

( ) ( ) ( )i j i j i j

1 1( )

2 1

1 1( )

2 1

i j i jk l i j k lt

i j i jk l i j k lc

DD

DD

i ji j ij

1 1

(1 ) (1 )

11

(1 )

11

(1 )

i j i jk l k l i jk l k lt c

ij i j k k ijt

i j k k ijc

D DD D

D E

D E

Complementary free energy:

Stress-strain relation:

21 22

21 22

1 (1 2 ) (1 )

(1 ) 6

1 (1 2 ) (1 )

(1 ) 6

tt t

cc c

Y I JD D E E

Y I JD D E E

Thermodynamic force:

0t t c cY D Y D 0 a n d 0t cD D Clausius-Duhem inequality:


Damage criterion:

( ) 0

( ) 0t t t t

c c c c

f Y K D

f Y K D

Initial value: 0 0( 0 ) u n d ( 0 ) t t c cK Y K Y

0 2 0 20 0( ) ( )

u n d 2 2

t ct cY Y

E E

From uniaxial test:

Loading condition:Unloading:

Loading:

0 o r 0 a n d 0 , th e n 0

0 o r 0 a n d 0 , th e n 0

t t t t

c c c c

f f f D

f f f D

0 a n d 0 , th e n ( )

0 a n d 0 , th e n ( )

t t t t t t

c c c c c c

f f D F Y Y

f f D F Y Y

Damage evolution law:0

3 / 2 0

0

3 / 2 0

(1 ),

2 ( ) 2 e x p [ ( ) ]

(1 )

2 ( ) 2 e x p [ ( ) ]

t t t tt

t t t t t

c c c cc

c c c c c

Y a a bF

Y Y b Y Y

Y a a bF

Y Y b Y Y


||Institute for Building MaterialsMazars 1986

0 3

0 3

1

2 1

3 1

0 .2

1 5 0 /

5 .1 6 7 /

0 .8 ; 0 .1 6

1 .5 6 5; 0 .5 1 0

t

c

t t

c c

E G P a

Y N m m

Y N m m

a b P a

a b P a



Summary on Concrete Plasticity

Concrete behavior and failure envelopes

Hardening models for concrete

uniform plastic hardening

multiple plastic hardening

Softening models for concrete

plasticity formulations in strain space

damage plasticity models

Damage models for concrete (CDM)

Scalar damage model

Unilateral elastic damage model


Thank you for your attention.

09.09.2013 55

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