1-s2.0-S0013794403000237-main

Engineering Fracture Mechanics 71 (2004) 681–698

www.elsevier.com/locate/engfracmech

CDM based modelling of damage and fracture mechanismsin concrete under tension and compression q

Halina Kuna-Ciskał *, Jacek J. Skrzypek

Institute of Mechanics and Machine Design, Cracow University of Technology, 31-864 Krak�oow, Poland

Received 30 October 2002; accepted 4 November 2002

Abstract

Anisotropic damage evolution and crack propagation in the elastic–brittle materials is analysed by the concepts of

continuum damage mechanics (CDM) and finite element method (FEM). The modified Murakami–Kamiya (MMK)

model of elastic-damage material is used to describe damage anisotropy in concrete. The Helmholtz free energy rep-

resentation is discussed. The unilateral crack opening/closure effect is incorporated in such a way that the continuity

requirement during unloading holds. The incremental form of the stress–strain equations is developed. The general

failure criterion is proposed by checking the positive definiteness of the Hessian matrix of the free energy function. The

local approach to fracture (LAF) by FEM is applied to the pre-critical damage evolution that precedes the crack

initiation, and the post-critical damage/fracture interaction. Crack is modelled as the assembly of failed finite elements

in the mesh, the stiffness of which is reduced to zero when the critical points at stress–strain curves are reached. A

concrete specimen with the pre-load, inclined crack is analysed in order to simulate different fracture mechanisms in

tension or compression. The constitutive model is capable of predicting the kinked-type crack under tension and the

wing-type crack under compression.

� 2003 Published by Elsevier Ltd.

Keywords: Anisotropic damage; Local approach to fracture; Unilateral response; Crack modelling

1. Introduction

The increasing demands for a high strength concrete require the adequate constitutive and damage

growth modelling, as well as the appropriate predictions of the overall failure mechanisms under monotonic

or cyclic loads. Concrete belongs to engineering materials commonly classified as brittle (polycrystalline

rocks, concrete, ceramics, and cast iron). The microstructure of concrete exhibits a large number of pre-load micro-cracks, voids, inclusions and other non-homogeneities. The nucleation, growth and interaction

of these micro-defects under external loads result in a deterioration process of the material on the micro-

scale and, as a consequence, change of the constitutive properties of the material. This micro-scale damage

qThis paper is the extended version of [29].* Corresponding author.

E-mail address: [email protected] (H. Kuna-Ciskał).

0013-7944/$ - see front matter � 2003 Published by Elsevier Ltd.

doi:10.1016/S0013-7944(03)00023-7

mail to: [email protected]

682 H. Kuna-Ciskał, J.J. Skrzypek / Engineering Fracture Mechanics 71 (2004) 681–698

response causes a fracture process, which yields a structural failure of the element and loss of its carrying

capacity. Deformation process in the damaged material is characterised by a non-linear stress–strain curve

that precedes the local failure. Proper formulation of the local failure criterion is one of the goals of the

present study. The experimental evidence reveals that in brittle-damaged materials the stress–strain non-linearity is mainly caused by a degradation of the elastic properties induced by a progressive deterioration

of the material with increasing loads. The strain softening effect on the energy dissipation due to inelastic

(visco-plastic) deformation may usually be considered as insignificant.

In general, the models used to describe a non-linear damage response of concrete and other brittle

materials have to be capable of capturing essential features of this class of materials. These are: a degra-

dation of the elastic properties and the strain softening due to the development of micro-defects, an ac-

quired anisotropy of the material even though in a virgin state it is considered as initially isotropic, and the

unilateral damage response of concrete in tension or compression. When subjected to the predominant axialcompression a concrete specimen fails by the axial splitting. In contrast, when subjected to the axial tension

it splits perpendicularly to the axial direction. Hence, ultimate failure mechanisms predicted by these

theories must also exhibit essential difference in an overall structural failure in case of compression or

tension.

Depending on the scale, two different approaches may be used in order to describe an overall structural

response of a concrete structure on the macro-scale. In general, micro-mechanical damage (MD) models

relate the macro-properties and the macro-response of a structure to its microstructure. In such approach

the damage is a discrete and stochastic phenomenon induced by a number of weakly or strongly interactingmicro-defects that influence the overall structural response (cf. [1]). The MD models have the advantage of

being able to sustain heterogeneous structural details on the micro-scale and meso-scale, and to allow a

micro-mechanical formulation of the damage evolution equations based on the accurate micro-crack

growth processes involved (cf. [2]). The simplified models of a kinked micro-crack under biaxial tension,

motivated by experimental evidence for concrete and rocks, were developed on the micro-mechanical basis

by Fanella and Krajcinovic [1], as well as Basista and Gross [2–4]. An attempt to the phenomenological

damage description in the framework of irreversible thermodynamics with internal variables, relating

random and heterogeneous microstructure of a material to the macro-response of a specimen, is due toBasista [5]. At the present stage of development these models occur somewhat computationally inefficient in

practical applications, and can be applied to the limited classes of materials and damage-to-failure

mechanisms (cf. also [6,7]).

Continuum damage mechanics (CDM) approach provides the constitutive and damage evolution

equations in the framework of thermodynamics of irreversible processes. When the CDM approach is used

a concept of the effective quasi-continuum is applied. The material heterogeneity (on the micro- and meso-

scale) is smeared out over the representative volume element (RVE) of the piece-wise discontinuous ma-

terial. The true state of damage within RVE, represented by the topology, size, orientation and number ofmicro-defects, is mapped to a material point of the pseudo-undamaged quasi-continuum, Fig. 1. The true

distribution of micro-defects within the RVE, and the correlation between them are measured by the

change of the effective constitutive modules, stiffness eKKðDÞ or compliance eKK�1ðDÞ. The material deterio-

ration is defined by the set of internal state variables D ¼ fD;Da;D2;D4; . . .g of the scalar, vectorial or

tensorial nature (cf. [8–12]). The constitutive tensors for the damaged material are defined by the use of

fourth-rank damage effect tensors MðDÞ that map state variables from the physical damaged ðr; eeÞ to the

fictitious pseudo-undamaged ð~rr;~eeeÞ configurations. Representation of the damage effect tensors depend on

the equivalence principle between the damaged and pseudo-undamaged states of the material. Among themwe mention: the principle of strain equivalence [13], the principle of stress equivalence [14], the principle of

elastic energy equivalence [15], the principle of total energy equivalence [16,17].

The anisotropic CDM-based elastic damage model of the high strength concrete, capable of predicting the

fatigue life under compressive stress of the order up to 120 MPa, has recently been developed by Al-Gadhib

Fig. 1. (a) Virgin-undamaged, (b) physical-damaged and (c) pseudo-undamaged continuum in CDM; the equivalence principles are

used in order to smear the true micro-defects distribution over the RVE to yield the effective constitutive module for damaged materials

(after [12]).

H. Kuna-Ciskał, J.J. Skrzypek / Engineering Fracture Mechanics 71 (2004) 681–698 683

et al. [18]. The model is based on the concept of damage effect tensorM used to define the effective compliance

matrix eCC in the elastic damage constitutive law e ¼ eCC : r. Loading surface concept in the space of thermo-

dynamic-force conjugates Ri is used to develop damage growth equations for principal damage components

xi. The critical energy release rate Rc is used for the bounding surface (limit fracture). A capability of the

model is limited to the proportional loading conditions and the special case of uniaxial compression.

In the present analysis a more general phenomenological model, based on the irreversible thermody-

namics, originate from Murakami and Kamiya (MK) [19], is used to describe the elastic damage material

by the total stress–strain constitutive equation. The Helmholtz free energy is assumed as the state potential

where two internal variables D and b are used as arguments. They are responsible for the anisotropicdamage and the isotropic hardening of the damage threshold, respectively. The stress–strain elasticity law

r ¼ eKKs : e is derived from the state potential. The effective secant stiffness matrix eKKs changes following the

damage evolution.

In what follows the extended incremental form of the elastic damage constitutive equation is developed,

where the effective tangent stiffness matrix eKKt is introduced. The dissipation potential is defined in the space

of thermodynamic-force conjugates ðY;�BÞ of damage variables ðD; bÞ, and the damage evolution of _DD;and _bb are established from the normality rule. The modified Murakami–Kamiya (MMK) model is capable

of predicting a fully anisotropic damage growth in concrete with no limitation to the proportional loading

induced. Loss of the positive definiteness of the Hessian matrix of the free energy function is used as thelocal failure criterion. Admitting for the consecutive macro-crack growth in a material when the failure

criterion is reached does an essential extension of the damage growth stage. Crack is modelled as the as-

sembly of failed elements in the FE mesh, the stiffness of which has been reduced to zero when the failure

criterion is reached. Local approach to fracture (LAF) is the framework of fracture analysis (cf. e.g. [20]).

2. Total formulation of the Murakami–Kamiya MK model of the elastic damage material

The general thermodynamically based theory of the constitutive and evolution equations of elastic–brittle damaged materials in a total stress–strain formulation [19] is the key for a further extension. The


Helmholtz free energy is a function of the elastic strain tensor ee, the second-rank damage tensor D, and

another scalar damage variable b, D ¼ DðD; bÞ. The following Helmholtz free energy decomposition into

the elastic and the damage terms is postulated as the state potential:

qwðee;D; bÞ ¼ qweðee;DÞ þ qwdðbÞ ð1Þ

According to the representation theory of non-linear algebra, the most general form of a scalar function of

the second-rank tensors weðee;DÞ can be expressed as the combination of 10 basic invariants of the tensorsee, D (cf. [21–23]):

we ¼ wefTr ee;TrðeeÞ2;TrðeeÞ3;TrD;TrðDÞ2;TrðDÞ3;TrðeeDÞ;Tr½ðeeÞ2D;Tr½eeðDÞ2;Tr½ðeeÞ2ðDÞ2gð2Þ

However, at the initial elastic state, the elastic behaviour is isotropic linear, hence the Helmholtz free energyqweðee;DÞ is quadratic with respect to ee. Additionally, since qweðee;DÞ will decrease with damage growth, it

is supposed to be linear in D (cf. [19,24]). Eventually, the simplified representation of the function qweðee;DÞcan be used, which depends on the five basic invariants of ee and D, as a linear combination of the following

terms:

ðTr eeÞ2;TrðeeÞ2; ðTr eeÞ2TrD;TrðeeÞ2TrD;Tr eeTrðeeDÞ;Tr½ðeeÞ2D

In order to properly describe the unilateral damage response in tension or compression a modified elastic

strain tensor ee in the principal strain co-ordinate system is defined

eI ¼ heIi � fh�eIi ¼ kðeIÞeI ; f 2 h0; 1i;kðeIÞ ¼ kI ¼ HðeIÞ þ fHð�eIÞ; I ¼ 1; 2; 3

ð3Þ

Symbol h i denotes the Macauley bracket, Hð Þ is the Heaviside step function, eI ðI ¼ 1; 2; 3Þ are principal

values of ee, and f is an additional material constant responsible for the unilateral damage response effect

under tension or compression (cf. [25]). For f ¼ 1 the modified strain tensor ee is identical to ee and the

unilateral damage (crack opening/closure) effect is not accounted for. In contrast, for f ¼ 0, the strain

tensor ee is modified in such a way that negative principal strain components are replaced by zeros, whereas

positive ones remain unchanged.In the general co-ordinate system the modified strain tensor is expressed as follows (cf. [24]):

eij ¼X3

I¼1

eI QIiQIj ¼X3

I¼1

kðeIÞeIQIiQIj ¼ Bijklekl ð4Þ

where

Bijkl ¼X3

I¼1

kðeIÞQIiQIjQIkQIl ð5Þ

is the fourth-rank tensor built of direction cosines between the principal strain axes and the current spatial

system.

Following Murakami and Kamiya [19] assumptions, both terms of the free energy (2) are represented as:

qweðee;DÞ ¼ 1

2kðTr eeÞ2 þ lTrðeeÞ2 þ g1ðTr eeÞ2TrDþ g2TrðeeÞ2TrD

þ g3Tr eeTrðeeDÞ þ g4Tr½ðeeÞ2D; qwdðbÞ ¼ 1

2Kdb

2 ð6Þ


where k and l are Lam�ee constants for undamaged materials, g1, g2, g3, g4, and Kd are material constants.

Note that the modified strain tensor ee is applied in the last term of we exclusively, which ensures the

continuous transition of the stress–strain response from crack opening to closure (cf. [26,27]). In general, it

may be shown that the unilateral crack opening/closure effect can influence only the diagonal componentsof the stiffness matrix Ks

ii, if the constitutive law is written in the principal damage directions (cf. [26]). The

unilateral damage takes place during unloading when e ¼ 0 or r ¼ 0. In compression the initial stiffness is

partially recovered, depending on the parameter f (3). It may be shown that in the case considered, when

only the last term in (6) is affected by the unilateral damage, the non-diagonal terms Ksijði 6¼ jÞ are not

modified by the active/passive unilateral condition (cf. [28]).

The following constitutive equation of anisotropic elasticity coupled with damage is furnished from (6)

according to the conventional procedure of the thermodynamic formalism

Fig. 2.

force c

r ¼ oðqwÞoee

¼ ½kðTr eeÞ þ 2g1ðTr eeÞðTrDÞ þ g3TrðeeDÞIþ 2½l þ g2ðTrDÞee

þ g3TrðeeÞDþ g4ee :

oee

oeeD

�þD

oee

oee

�ð7Þ

where oee=oee is the fourth-rank tensor derived from (4) and (5).

The thermodynamic-force conjugates of D and b are also derived from (6)

Y ¼ �qowoD

¼ �½g1ðTreeÞ2 þ g2ðTrðeeÞ2I� g3ðTreeÞee � g4eeee;

B ¼ qowd

ob¼ Kdb

ð8Þ

The damage dissipation potential in the space of force conjugates fY;�Bg is assumed in the form

F ðY;BÞ ¼ Yeq � ðB0 þ BÞ ¼ 0; Yeq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12Y : L : Y

q; Lijkl ¼ 1

2ðdikdjl þ dildjkÞ ð9Þ

where B0 and B stand for the initial damage threshold and the subsequent damage force conjugates of b,respectively. Hence, experimentally motivated by Murakami and Kamiya [19] the isotropic hardening of

damage dissipation surface (9) in the fY;Bg space is introduced, as sketched in Fig. 2. Note that inside the

damage dissipation surface the damage evolution does not occur. A damage threshold hardening effect

Initial and subsequent damage dissipation surfaces and illustration of the normality rule in the fY;Bg space of thermodynamic-

onjugates of fD; bg.


results from the micro-mechanical response in concrete. The pre-load micro-cracks initially activated under

applied loading, might further be arrested when the obstacles are met.

The damage evolution equations are finally established from the normality rule

Fig. 3.

after M

_DD ¼ _kkd

oFoY

; _bb ¼ _kkd

oFoð�BÞ ¼

_kkd ð10Þ

and the consistency condition _FF ¼ 0 ¼ oFoY

: _YYþ oFoB

_BB is used to eliminate _kkd from (10)

_kkd ¼aoFoY

: _YY

oBob

¼ aL : Y

2KdYeq

: _YY ð11Þ

A factor a ¼ 1 or a ¼ 0 is used for the active or passive damage growth, respectively.

When exposed to tension or compression, the MK constitutive phenomenological model is capable of

capturing unilateral damage response. Under the uniaxial tension condition the damage component D11 is

dominant, whereas the other two components D22 and D33 are negligible. By contrast, under the uniaxialcompression the transverse damage components D22 ¼ D33 become predominant, but a non-negligible axial

component D11 is also visible (cf. Fig. 3). In contrast to other models that are time-dependent (cf. e.g. [9]),

damage evolution described by the MK model results from the stress and strain growth, but it is time-

independent and does not occur under the constant stress condition. Note also that in the MK model below

the damage threshold A0ðr0; e0Þ none damage nucleation or growth occurs. On the non-linear pre-peak

stress–strain curve the damage evolution cause the elastic modulus in the constitutive equation to drop,

until the critical point in a sense of the material instability in the r–e curve Afðrf ; efÞ is met and, hence, the

local failure criterion is satisfied.

ε11ε11

ε11

ε0 ε0 εr

εr

εr εrε0

σ 11[M

Pa]

σ 11[M

Pa]

D11

D =D22 33

0 0-5 -10 -15 -20 -25

-10

-20

-30

-40

-50

00

2

0.05

2

2

4

0.10

4

4

6

0.15

6

6

10-4

10-4

10-410-4

8

0.20

10

0.25

xx

x

AfAf

A0

A0

(a) (b)

ε11

ε0

D11

D =D22 33

-50

0.04

0.08

0.12

0.16

0.20

0.24

-10 -15 -20 -25x

σ

σ

σ

σ

Stress–strain curves and damage growth in a concrete specimen under the uniaxial tension or the uniaxial compression tests,

urakami and Kamiya [19].

Fig. 4. Stress–strain loops in the elastic damage material under cyclic uniaxial loading conditions; upper index ‘‘i’’ denotes the number

of loading cycle, i ¼ 1; 2; 3; . . ., lower index ‘‘0’’ corresponds to the damage threshold.


To illustrate this behaviour the numerical simulation of the damage response under the uniaxial tensionand compression was performed by Murakami and Kamiya [19]. The model was calibrated for a high

strength concrete to yield the following set of material constants

E ¼ 21:4 ½GPa; v ¼ 0:2; g1 ¼ �400 ½MPa; g2 ¼ �900 ½MPa; g3 ¼ 100 ½MPa;g4 ¼ �23500 ½MPa; f ¼ 0:1; Kd ¼ 0:04 ½MPa; B0 ¼ 0:0026 ½MPa

ð12Þ

Under the cyclic loading condition a cycle-by-cycle softening of the material follows the damage evo-

lution. It is accompanied by the damage threshold increase from cycle to cycle ri0, and a simultaneous

critical stresses rimax drop until, after a number of cycles, the failure criterion is met at the critical point

Afðrf ; efÞ (Fig. 4). In the case of uniaxial stress state considered the critical point (failure) corresponds to the

maximum at the stress–strain curve.

In the original paper by Murakami and Kamiya [19] it is assumed that the failure criterion corresponds

to the critical eigenvalue of damage Dimax ¼ Dcrit which has been established at the level of Dcrit ¼ 0:4.However, in a general case, the critical damage value at the critical point changes with the loading tra-jectory and, hence, the concept of constant critical damage Dcrit cannot be used as the failure criterion. In a

multiaxial case the more general failure criterion has to be formulated that requires the Hessian matrix of

the function w is positive definite (see Section 3). To this end the incremental form of the constitutive

equation of the elastic damage material has to be developed.

3. Incremental formulation of the modified Murakami–Kamiya MMK model and failure criterion

When the total formulation is used the constitutive equation (7) represents stress–strain relations by the

use of the secant elastic damage stiffness eKKsðDÞ that suffers from damage evolution in a material

r ¼ eKKs : ee or rij ¼ eKKsijkl : eekl ð13Þ

The incremental form of the stress–strain relations may be established from (13) as follows:

dr ¼ eKKs : dee þ ee :oeKKs

oD: dD

!ð14Þ


The damage increment dD on dee is obtained from (10) and (11). Eventually, the following incremental

state equation is derived (cf. [28,29])

dr ¼ eKKs þ aee :oeKKs

oD:

YY :oY

oee

4KdY 2eq

0B@1CA

264375 : dee ð15Þ

The square bracket in (15) represents the effective tangent stiffness eKKtðee;DÞ and factor a equal 0 or 1 is used

for passive or active damage process, respectively. By the use of Voigt notation the state equations, total

(13) and incremental (15), may easily be rewritten in a more convenient matrix forms:

frg ¼ ½eKKsðDÞfeeg or ri ¼ eKKsije

ej ð16Þ

fdrg ¼ ½eKKtðee;DÞfdeeg orri ¼ eKKtijdeej ð17Þ

Note that superscript lower case ‘‘t’’ is used for the tangent stiffness matrix, and should not be confused

with the transposition operator ‘‘T’’. In a general 3D case the following matrix representation of the total

constitutive equation holds

r11

r22

r33

r23

r31

r12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

eKKs11

eKKs12

eKKs13

eKKs14

eKKs15

eKKs16eKKs

22eKKs

23eKKs

24eKKs

25eKKs

26eKKs33

eKKs34

eKKs35

eKKs36eKKs

44eKKs

45eKKs

46

sym: eKKs55

eKKs56eKKs66

2666666664

3777777775

e11e22e33c23c31c12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð18Þ

The ‘‘full’’ 6� 6 symmetric secant stiffness matrix is furnished as follows:
eKKs11 ¼ k þ 2l þ 2ðg1 þ g2ÞTrDþ 2ðg3 þ g4ÞD11eKKs22 ¼ k þ 2l þ 2ðg1 þ g2ÞTrDþ 2ðg3 þ g4ÞD22eKKs33 ¼ k þ 2l þ 2ðg1 þ g2ÞTrDþ 2ðg3 þ g4ÞD33eKKs12 ¼ k þ 2g1TrDþ g3ðD11 þ D22Þ; eKKs
13 ¼ k þ 2g1TrDþ g3ðD11 þ D33ÞeKKs23 ¼ k þ 2g1TrDþ g3ðD22 þ D33ÞeKKs44 ¼ 0:5½2l þ 2g2TrDþ g4ðD33 þ D22Þ; eKKs

45 ¼ g4D12eKKs55 ¼ 0:5½2l þ 2g2TrDþ g4ðD11 þ D33Þ; eKKs

46 ¼ g4D13eKKs66 ¼ 0:5½2l þ 2g2TrDþ g4ðD11 þ D22Þ; eKKs

56 ¼ g4D23eKKs14 ¼ g3D23; eKKs

24 ¼ eKKs34 ¼ ðg3 þ g4ÞD23eKKs

25 ¼ g3D13; eKKs15 ¼ eKKs

35 ¼ ðg3 þ g4ÞD13eKKs36 ¼ g3D12; eKKs

16 ¼ eKKs26 ¼ ðg3 þ g4ÞD12

ð19Þ

Let us mention that the above general formulae are derived for a special case f ¼ 1, where ee � ee (the

unilateral damage effect is ignored in (19)). Representation of the effective tangent stiffness matrix in a

general 3D state is rather cumbersome hence, in what follows, we confine ourselves to the simpler plane

stress state.In the case if incremental form of the constitutive law is used (17) the (6� 6) matrix representation can

be written as follows:


dr11

dr22

dr33

dr23

dr31

dr12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

o2woe211

o2woe11oe22

o2w

oe11oe33o2w

oe11oc23

o2woe11oc31

o2woe11oc12

o2woe222

o2woe22oe33

o2woe22oc23

o2woe22oc31

o2woe22oc12

o2woe233

o2woe33oc23

o2woe33oc31

o2woe33oc12

o2woc223

o2woc23oc31

o2woc23oc12

sym:o2woc231

o2woc31oc12o2woc212

2666666666666666666666664

3777777777777777777777775

de11de22de33dc23dc31dc12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð20Þ

The (6� 6) matrix of the components of the fourth-rank tensor Hijkl ¼ o2w=oeijoekl is known as the Hessian

matrix of the function w. In order to introduce the general failure criterion the Drucker�s material stability

postulate is adopted:

drijdeij > 0 ð21Þ

Substituting for drij formula (20) into the stability criterion (21) we obtain

o2woeijoekl

deijdekl ¼ Hijkldeijdekl > 0 ð22Þ

The quadratic form ðo2w=oeijoeklÞdeijdekl must be positive definite for arbitrary values of the components

deij, hence, eventually the condition (22) requires that Hessian matrix ½H be positive definite (cf. [30]).

According to the Sylvester criterion the symmetric matrix ½H of the nth order is positive definite if and only

if

det½Hk > 0 ðk ¼ 1; 2; . . . ; nÞ ð23Þ

where ½Hk is the (k � k) minor of the matrix ½H (cf. [31]).The local tangent stiffness matrix is used for the quasi-Newton algorithm for first iteration step of solving

the non-linear equation (17) as long as the local failure criterion (23) holds. The stiffness of the element in

the FE mesh that has come to failure is next reduced to zero. As a consequence, the failed element is

completely released from stress and the appropriate stress redistribution occurs in the neighbouring ele-

ments to ensure the global equilibrium. Note that the above failure criterion (23) assumes the brittle failure

mechanism. However, when broader class of materials is considered, a post-peak softening regime can also

be admitted, that would result in strain localisation and a smooth stiffness drop in elements that come to

failure. Neglecting the effect of visco-plastic deformation on the energy dissipation in the MMK modelconsidered might be inconsistent with the post-peak analysis and, hence, the additional brittle fracture

criterion is necessary to use.

By the use of local approach to fracture LAF, based on FEM and CDM, the crack is modelled as the

assembly of failed elements in the mesh. Subsequent elements in the FE mesh that have come to failure

(along the crack) are released from stresses and the appropriate redefinition of the global stiffness of the

structure occurs. The procedure is continued as long as the overall fracture mechanism of the structure is

reached. The described method of LAF is capable of predicting the fracture initiation and the ultimate

fracture pattern, as well as the limit load that corresponds to the considered fracture mechanism in astructure. The crack growth stage and the ultimate crack pattern are, in general, mesh-dependent. To


mitigate the effect of element size and shape the additional regularisation procedures are required. To this

end, the thermodynamic-force conjugate Y can be subjected to a non-local treatment bYY by applying the

formula analogous to that described in Skrzypek et al. [32] where the non-local and limited stress variable

was defined for the Litewka model. The regularisation applied in [32] allowed to avoid mesh effect andobtain the convergent solution with respect to the element size and shape. In present case the following

formula for the non-local variable bYY can be used:

bYYðxÞ ¼ RXdYðnÞuðx; nÞdXdRXd

uðx; nÞdXd

; uðx; nÞ ¼ exp

"� dðx; nÞ

d

� �2#

ð24Þ

Additionally, to avoid the singularity of Y at the crack tip when the mesh size tends to zero, the cut-off

procedure should be used in the neighbourhood of a crack tip according to the scheme

Y ¼ kY; k ¼ 1 if Yeq 6 Yu

Yu=Yeq if Yeq > Yu

ð25Þ

where the cut-off factor is determined from (9) as follows:

k ¼ Yu

Yeq

¼ B0 þ BYeq

ð26Þ

In the examples presented below the regularisation has not been used. The results obtained for crack

prediction exhibit the mesh effect, so that a non-local treatment has to be involved in order to ensure

convergence.

4. Matrix constitutive equations in plane stress state

The effective application of (4) to the constitutive equation (7) requires the explicit formula for the

derivative oee=oee which accounts for the unilateral opening/closure effect. To this end the unilateral

transformation matrix Bijkl (5) must explicitly be expressed. In the case of plane stress considered the

transformation reduces to the plain rotation by the angle a

a ¼ 1

2arctg

2e12e11 � e22

ð27Þ

such that the derivative oee=oee must be calculated according to the procedure for the complex function

oeijoekl

¼ Bijkl þoBijpq

oaoaoekl

epq ð28Þ

The above scheme is cumbersome in the numerical applications. In a particular case of plane stress state,

when the rotation angle a is small, Eq. (5) may be reduced to the simplified form

e11e22e12e33

8>><>>:9>>=>>; ¼

a 0 b 0

0 c b 0

0:5b 0:5b d 0

0 0 0 1

26643775

e11e22e12e33

8>><>>:9>>=>>; ð29Þ

where

a ¼ 1

2ðk1 þ k2Þ þ

1

2ðk1 � k2Þ cos 2a; b ¼ 1

2ðk1 � k2Þ sin 2a

c ¼ 1

2ðk1 þ k2Þ �

1

2ðk1 � k2Þ cos 2a; d ¼ 1

2ðk1 þ k2Þ

ð30Þ


When the assumption of plane stress is used r33 ¼ 0 the total form of the matrix constitutive equations

(18) reduces to

r11

r22

0

r12

8>><>>:9>>=>>; ¼

eKKs11

eKKs12

eKKs13

eKKs16eKKs

12eKKs

22eKKs

23eKKs

26eKKs13

eKKs23

eKKs33

eKKs36eKKs

16eKKs

26eKKs

36eKKs

66

2666437775

e11e22e33c12

8>><>>:9>>=>>; ð31Þ

where the symmetric 4� 4 effective secant stiffness matrix, which depends on D, is furnished as follows:

eKKs11 ¼ k þ 2l þ 2ðg1 þ g2ÞTrDþ 2g3D11 þ g4 D11 2a2

� þ 1

2b2

�þ 1

2D22b2 þ 2abD12

!eKKs

22 ¼ k þ 2l þ 2ðg1 þ g2ÞTrDþ 2g3D22 þ g4

1

2b2D11

þ D22 2c2

�þ 1

2b2

�þ 2bcD12

!eKKs

33 ¼ l þ g2TrDþ 12g4fðD11 þ D22Þðb2 þ d2Þ þ 4D12bdgeKKs

12 ¼ eKKs21 ¼ k þ 2g1TrDþ g3ðD11 þ D22Þ þ g4

12b2ðD11

"þ D22Þ þ D12bðaþ cÞ

#eKKs

13 ¼ eKKs31 ¼ g3D12 þ 1

2g4fD11bð2aþ dÞ þ bdD22 þ 2D12ðb2 þ adÞgeKKs

23 ¼ eKKs32 ¼ g3D12 þ 1

2g4fbdD11 þ D22bð2cþ dÞ þ 2D12ðb2 þ cdÞgeKKs

16 ¼ eKKs61 ¼ g3D12 þ 1

2g4½D11bð2aþ dÞ þ bdD22 þ 2D12ðb2 þ adÞeKKs

26 ¼ eKKs62 ¼ g3D12 þ 1

2g4½bdD11 þ D22bð2cþ dÞ þ 2D12ðb2 þ cdÞeKKs

36 ¼ eKKs63 ¼ g3D12eKKs

66 ¼ l þ g2TrDþ 12g4½ðD11 þ D22Þðb2 þ d2Þ þ 4D12bd

ð32Þ

Thermodynamic-force conjugates Y and B are defined by (8) as follows:

Y11

Y22

Y33

Y23

Y31

Y12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

�g1ðTr eeÞ2 � g2TrðeeÞ2 � g3ðTr eeÞe11 � g4½ðae11 þ be12Þ2 þ ð0:5bðe11 þ e22Þ þ de12Þ2�g1ðTr eeÞ2 � g2TrðeeÞ2 � g3ðTr eeÞe22 � g4½ðce22 þ be12Þ2 þ ð0:5bðe11 þ e22Þ þ de12Þ2

�g1ðTr eeÞ2 � g2TrðeeÞ2 � g3ðTr eeÞe33 � g4k23e

233

0

0�g3ðTr eeÞe12 � g4ð0:5bðe11 þ e22Þ þ de12Þðae11 þ 2be12 þ ce22Þ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð33Þ

B ¼ Kdb ð34Þ

where

Tr ee ¼ ðe11 þ e22Þ½2l þ 2g2TrDþ ðg3 þ 2g4k23ÞD33 � g3ðD11e11 þ D22e22 þ 2D12e12Þ

k þ 2l þ 2ðg1 þ g2ÞTrDþ 2ðg3 þ g4k23ÞD33

ð35Þ

Note that the 4� 4 secant stiffness matrix eKKsðDÞ may further be reduced to 3� 3 one by eliminating e33from (31).

When the general incremental form of the constitutive equations (17) is applied to plane stress dr33 ¼ 0,

the following 4� 4 tangent stiffness matrix representation eKKtijðee;DÞ is furnished in terms of the corre-

sponding secant matrix components and the additional terms depending on damage and strain tensors (15)


dr11

dr22

0dr12

8>><>>:9>>=>>; ¼

eKKt11

eKKt12

eKKt13

eKKt16eKKt

12eKKt

22eKKt

23eKKt

26eKKt13

eKKt23

eKKt33

eKKt36eKKt

16eKKt

26eKKt

36eKKt

66

2666437775

de11de22de33dc12

8>><>>:9>>=>>; ð36Þ

For instance, when the first element of the matrix (36) is concerned, the following formula holds:

eKKt11 ¼ eKKs

11 þ e11 2ðg1

þ g2Þ

oD11

oe11

�þ oD22

oe11þ oD33

oe11

�þ 2g3

oD11

oe11þ g4

oD11

oe11ð2a2

$þ 0:5b2Þ

þ 0:5b2 oD22

oe11þ 2ab

oD12

oe11

%!þ e22 2g1

oD11

oe11

� þ oD22

oe11þ oD33

oe11

�þ g3

oD11

oe11

�þ oD22

oe11

�þ g4 0:5b2 oD11

oe11

�$þ oD22

oe11

�þ bðaþ cÞ oD12

oe11

%!þ e33 2g1

oD11

oe11

� þ oD22

oe11þ oD33

oe11

�þ g3

oD11

oe11

�þ oD33

oe11

�!þ c12 g3

oD11

oe11

þ 0:5g4 bð2a

$þ dÞ oD11

oe11þ bd

oD22

oe11þ 2ðb2 þ adÞ oD12

oe11

%!
ð37Þ
Other elements of the matrix (36) can be established in a similar fashion, when the general equation (15) is

used. Also in this case de33 can be eliminated from (36) to reduce the tangent stiffness matrix to the 3� 3

one.

5. Numerical simulation of damage and fracture in concrete

5.1. Micro-mechanical observations

Concrete is a strongly heterogeneous engineering material where initial (pre-load) micro-cracks, which

are most frequently located at the cement matrix–aggregate interfaces, are randomly oriented with respect

to the load direction (cf. e.g. [1]).

When the tensile load is applied to the concrete specimen the micro-cracks that are oriented per-pendicularly to the load axis start to open and grow until the adjacent micro-cracks bridge is formed.

Eventually, when the fracture toughness of the matrix is exceeded, micro-cracks coalesce to form the

macro-crack (see also, [5]). Preferable orientation of the macro-crack is also, roughly speaking, perpendicular

to the tensile load axis. If the obstacle does not arrest it, a final transverse splitting of the specimen is met.

Note, however, that not only perpendicular micro-cracks come to grow under tension. Also micro-cracks

that are inclined at a certain angle to the tensile direction may grow to form a kinked-type micro-crack as

shown in Fig. 5(a). Corresponding stress–strain curve in tension exhibits three stages that correspond to

linear elastic, hardening and softening regime.Damage response of concrete under compression exhibits: the gradual degradation of elastic module

after the damage threshold from linear elastic behaviour is reached, the positive dilatancy due to consec-

utive opening of micro-cracks, the anisotropy of micro-crack evolution, and the existence of softening

regime. The final failure that corresponds to macro-crack formation in compression occurs at the stress

much higher than that at which the individual micro-crack can be activated. Micro-mechanical mechanism

of damage and fracture in compression is much more complex than in tension, and the effect of micro-

cracks interaction is much stronger. Sliding micro-crack mechanism plays here the essential role. Consider a

single pre-load micro-crack inclined to the compressive direction at a certain angle, Fig. 5(b) (cf. [4], also[1]). During the first phase a frictional sliding along the faces of the pre-load micro-crack occurs with no

Fig. 5. (a) Simplified macro-crack growth under tension (after [3]) and (b) sliding crack mechanism with the actual and the simplified

geometry of tensile wing cracks (after [2]).


length change of it, when the actual shear stress at a point of micro-crack face exceeds the resistance of

friction. When the elastic energy release rate G at the crack tip reaches the critical value, the second phase

occurs when a secondary curvilinear tensile-type wing micro-crack starts to open at each crack-tip. Next,following a path for which the stress intensity factor KI reaches a maximal value (or KII ¼ 0) the wing

micro-crack grows, gradually changing its initial direction, such that eventually it coincides with the

compressive load axis. Final failure mechanism in compression results from a coalescence and interaction

of the wing micro-cracks to finally form an axial splitting of the specimen or a shear fracture at small

confinement or a pseudo-ductile failure at large confinement (cf. Basista [5]). Motivated by the above

micro-mechanical evidence of damage in concrete, Basista and Gross [3] developed the simplified models of

the kinked micro-crack growth under biaxial tension as well as a simplified model of the sliding micro-crack

under compression [4]. Finally the non-linear macroscopic stress–strain relationships were obtained, wherethe Rice internal variable theory, based on the micro-to-macro transition, was used as a constitutive

framework.

In what follows, damage and failure mechanisms in concrete are studied by the use of the continuum

damage mechanics (CDM), modified Murakami–Kamiya (MMK) constitutive model, and the local ap-

proach to fracture (LAF). The material constants used to the model calibration for the high strength

concrete are given by (12). All the numerical calculations were performed by the use of the ABAQUS

system.

5.2. Failure mechanism in a 2D structure with the pre-load crack in tension

Consider the plane-stress concrete structure with the pre-load crack of the infinitely small width, inclined

at the angle of 45� to the tension direction. Structure is subject to the uniform tension at top and bottomedges, and is free to move at all sides, Fig. 6. The monotonically increasing tensile load q causes the pre-load

crack of the zero width in a pre-load state, to open. Stress concentration at the zones neighbouring crack

tips is accompanied by the cumulation of the D11 damage component in the element where the secondary

kinked crack starts to open when the failure criterion (23) is locally met. The magnitudes and directions of

Fig. 6. A plane stress concrete structure with the pre-load inclined crack, subjected to tension in the direction (1): (a) the geometry and

mesh, (b) maximal principal stress distribution in the surrounding of the pre-load crack tip; the arrows represent the actual principal

stresses.


the principal stress components in elements neighbouring crack tip are sketched in Fig. 6 just before the

instant of the secondary kinked crack opening. The magnified square zone neighbouring the pre-load crack

tip is shown in Fig. 6(b), where the magnifications factor for displacements equal 100 is used for figure

clarity.

With the increasing tensile load the consecutive elements come to failure, to ultimately form a ‘‘kinked’’

crack that splits the concrete specimen approximately perpendicularly to the tension direction, as shown in

Fig. 7(a)–(c). Note releasing the elements neighbouring the crack face from stresses following the kinked

crack growth. Anisotropic damage growth and fracture mechanism in a specimen causes on a final stage theoverall stiffness of the structure to drastically drop. The predicted critical load for crack initiation qf

1 differs

from that of ultimate failure qfu by the amount of approximately 15%.

A qualitatively similar kinked crack in a brittle rock-like specimen was predicted by Basista [5], where

the micro-mechanical damage model (MD) was used on the micro-level.

Fig. 7. Formation of the ‘‘kinked-type’’ crack under tension; maximal principal stress distribution on different stages of crack

propagation.


5.3. Failure mechanism in a 2D structure with the pre-load crack in compression

To illustrate the failure mechanism under compression, consider the plane concrete structure with the

pre-load crack inclined at the angle of about 70� to the compression direction. The uniform compression isapplied at top and bottom edges, and the structure is free to move at all sides. No confinement is applied at

specimen sidewalls (Fig. 8) and the frictionless pre-load crack is assumed for simplicity.

Contact at the pre-load crack edges was defined by identifying and pairing potential contact surfaces. To

define a sliding interface between two surfaces, one of the surfaces (the ‘‘slave’’ surface) is covered with 3-

node contact elements. The second surface (the ‘‘master’’ surface) is defined by a series of nodes ordered in

the segments of the slide line. For each node on the slave surface the closest point on the master surface is

looked for, where the master surface�s normal passes through the node on the slave surface. The interaction

is then discretized between the point on the master surface and the slave node (cf. [33]).FEM analysis, performed at the end of first phase when sliding occurs on pre-load crack faces with no

crack length change, shows the maximum tensile stress vectors at both crack tips. The maximum tensile

stress causes the secondary wing crack to open (Fig. 9(a)), when the failure criterion (23) is reached. Next

the stiffness (and stress) in the failed element is reduced to zero. During next loading steps subsequent

elements in FE mesh come to failure to eventually form two straight wing cracks growing in the com-

pressive load direction (Fig. 9(b)).

After a number of loading steps a slight change in the crack path was observed indicating a mixed type

tensile-shear mechanism of the crack growth (Fig. 9(c)). However, the mesh effect may also be moresignificant there. When the secondary crack splits the structure, the zones neighbouring the crack faces

are gradually released from stresses, whereas the front of the maximum tensile stress propagates outwards

with the wing crack length increase (Fig. 10(a)–(c)). The predicted wing cracks in a concrete specimen

under compression do not exhibit curvilinear shape, as shown in Fig. 5, but they start from the pre-

load crack tips in a straight line manner growing approximately in a direction parallel to the loading axis

(cf. [1]).

Fig. 8. A plane stress concrete structure with the pre-load inclined crack, subjected to compression in the direction (1): (a) the geometry

and mesh and (b) maximal principal stress distribution in the surrounding of the pre-load crack tip; frictionless sliding is assumed on

the pre-load crack surface, the maximum tensile stress initiates the tensile ‘‘wing-type’’ secondary crack starting from the pre-load

crack tip.

Fig. 9. Formation of the ‘‘wing-type’’ crack under compression; subsequent stages of the macro-crack growth.

Fig. 10. Maximal principal stress distribution on several stages of the ‘‘wing-type’’ crack growth under compression.


6. Final remarks

1. Phenomenological CDM based MK model is capable of predicting anisotropic damage growth in a con-

crete specimen under monotonic or cyclic loading conditions.

2. When solving the non-linear elastic damage problem the incremental stress–strain matrix constitutive

equation of the modified Murakami–Kamiya MMK model is necessary to use to ensure the convergence.

Loss of the positive definiteness of the tangent stiffness matrix is used as the failure criterion. To this end,

the set of sub-determinants of the Hessian matrix of the free energy function is checked at every integra-tion point, at each step of loading. This criterion is more general than the criterion proposed in [19].

3. By the use of local approach to fracture LAF and FEM the crack growth in a concrete specimen may

successfully be simulated until the ultimate fracture mechanism is achieved. Crack is modelled as the as-

sembly of failed elements in the FE mesh the stiffness of which and stress are reduced to zero.

4. Crack opening/closure effect is included in the model by introducing a modified strain tensor to the con-

stitutive law. The modification of the diagonal components of the elastic stiffness matrix by the active-

passive unilateral damage satisfies the continuity requirement during unloading, when e ¼ 0 or r ¼ 0.


5. A concrete specimen with the pre-load crack fails in a different way under tension or compression. When

loaded by the axial tension, a ‘‘kinked-type’’ crack is formed in the direction perpendicular to the tension

axis. In contrast, when loaded by the axial compression, a slip on the primary crack faces results in a

‘‘wing-type’’ crack which starts from the primary crack tips and grows in the direction of axial compres-sion.

6. To properly model the sliding-type crack it is necessary to define the contact on the sliding surfaces. To

this end, the appropriate value of the sliding friction coefficient should be implemented. In the example

presented in this paper the frictionless sliding on the pre-load crack surfaces was assumed for simplicity.

7. The similarity between the fracture mechanisms in tension and compression on the micro-scale and the

corresponding fracture mechanisms on the macro-scale is shown. However, the mechanisms simulated

on the macro-scale are obtained for a single pre-load crack without the interaction effect between the

cracks considered. The procedure described in this paper may also be capable of predicting the final frac-ture pattern in the case of larger number of pre-load cracks.

8. Simulation of the crack propagation in the plane stress under tension or compression by the use of local

approach to fracture is capable of predicting crack pattern in a qualitative sense. The solution is, in gen-

eral, the mesh-dependent. In order to reduce mesh-dependence an additional regularisation treatment is

necessary. It may be done by the introduction of the non-local definition of the thermodynamic-force

conjugate Y and an additional cut-off procedure, in a similar fashion as it was done in [32].

Acknowledgements

The Grant 7T07A 03819 from the State Committee for Scientific Research KBN Poland is gratefully

acknowledged.

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1-s2.0-S0013794403000237-main

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