Nguyen Et Al-2008-International Journal for Numerical and Analytical Methods in Geomechanics

7/23/2019 Nguyen Et Al-2008-International Journal for Numerical and Analytical Methods in Geomechanics

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2008; 32:391413Published online 9 August 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.649

A coupled damageplasticity model for concrete based on

thermodynamic principles: Part II: non-local regularization andnumerical implementation

Giang D. Nguyen1,,, and Guy T. Houlsby2,

1Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, U.S.A.2Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, U.K.

SUMMARYNon-local regularization is applied to a new coupled damageplasticity model (Int. J. Numer. Anal.

Meth. Geomech.2007; DOI: 10.1002/nag.627), turning it into a non-local model. This procedure resolvessoftening-related problems encountered in local constitutive models when dealing with softening materials.The parameter identification of the new non-local coupled damageplasticity model is addressed, withall parameters being shown to be obtainable from the experimental data on concrete. Because of theappearance of non-local spatial integrals in the constitutive equations, a new implementation scheme isdeveloped for the integration of the non-local incremental constitutive equations in nonlinear finite elementanalysis. The performance of the non-local model is assessed against a range of two-dimensional structuraltests on concrete, illustrating the stability of the stress update procedure and the lack of mesh dependencyof the model. Copyright q 2007 John Wiley & Sons, Ltd.

Received 15 January 2007; Revised 21 April 2007; Accepted 1 June 2007

KEY WORDS: concrete; damage; plasticity; non-local; numerical implementation

1. INTRODUCTION

The resolving of softening-related problems plays a crucial role in the development of constitutive

models for strain softening materials in general, and for concrete in particular. In the consti-

tutive modelling of concrete, localization due to softening is of great importance, because strain

softening and strength reduction are two of the most important features of the macroscopic material

Correspondence to: Giang D. Nguyen, Department of Mathematics and Statistics, University of New Mexico,Albuquerque, NM 87131, U.S.A.

E-mail: [email protected], [email protected] Fellow (formerly Research Student, University of Oxford).Professor of Civil Engineering.

Copyright q 2007 John Wiley & Sons, Ltd.


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392 G. D. NGUYEN AND G. T. HOULSBY

behaviour. The use of damage mechanics, in combination with plasticity theory, enables us to

derive appropriate constitutive models for the material[1]. However, it should be noted here thatin this study a macroscopic approach is used to model complicated underlying micro-fracturing

processes in concrete. Therefore, macroscopic variables such as damage indicator and plastic strain

are definitely not true representatives of those complicated fracturing processes. Nevertheless, to

some extent, they could be considered appropriate in capturing some essential features of the

material behaviour at macroscopic level, such as the stiffness and strength reductions and residual

strain, although the underlying microscopic processes are much more complicated. The damage and

plasticity zones in coupled damageplasticity models therefore could be viewed as macroscopic

representation of the underlying micro-cracking and frictional slips in a non-zero volume fracture

process zone (FPZ).

As the material exhibits significant post-peak softening, appropriate treatments, called regu-

larization techniques, need to be applied to the constitutive modelling as well as the structural

analysis. In the literature, non-local regularization techniques have been found to be appropriate for

the modelling of softening materials[2 5], and help to avoid pathological problems encounteredin the constitutive modelling of these materials[6]. In this study, these techniques are applied toa coupled damageplasticity models recently developed[1]. In particular, spatial integrals nowappear in the two damage criteria, with the physical interpretation of redistributing the energiesassociated with the damage processes. The yield criterion remains in its local form. The calibration

of parameters for the non-local model is addressed, with the procedures proposed in Nguyen and

Houlsby[7] being used.Numerical implementation plays an important part in the development of constitutive models for

engineering materials. The implementation here comprises a method for the solution of the partial

differential equations in solid mechanics, and the incorporation of the constitutive models into this

system of governing differential equations. In this study, the finite element method is employed for

solving boundary value problems in continuum mechanics. However, as concrete exhibits highly

nonlinear behaviour after peak stress, the incorporation of the non-local constitutive relationships

into the system of equations is not straightforward. A modified backward integration scheme, based

on the algorithm proposed by Crisfield [8 (Chapter 6, vol. 1)] is employed for the integration ofthe non-local incremental constitutive equations. For the solution of systems of nonlinear algebraic

equations in finite element analysis, the arc-length method in combination with local constraint

equations employing dominant displacements[9] is implemented, and proves its reliability in thisstudy. Numerical examples showing the stability of the integration scheme, and the capability of

the arc-length control procedure to handle highly nonlinear behaviour in structural analysis, will

be presented.

In the next sections, a summary of the model in Nguyen and Houlsby[1] and the incorporationof non-locality into this model are presented. This is followed by a summary of the procedures

for calibration of parameters for this non-local coupled damageplasticity model. The numerical

implementation of the non-local model is presented in Section 5, followed by numerical examples

simulating real structural problems.

2. THE LOCAL MODEL

The constitutive equations governing the local behaviour of a coupled damageplasticity model[1]consist of a stressstrain relationship, a plasticity yield criterion y p along with two damage criteria

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2008; 32:391413

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A COUPLED DAMAGEPLASTICITY MODEL FOR CONCRETE: PART II 393

ytd and ycd for tensile and compressive damage, respectively. They all are written as follows:

i j=(1+)i jkki jE(1td)(1cd)

+i j (1)

yp= kk+i ji j

2k= 0 (2)

ytd=(1+ pt)+i j+i j ptkkll

2E(1td)2 Ft1(td, cd)= 0 (3)

ycd=(1+ pc)i ji j pckkll

2E(1cd)2 Fc1 (cd)= 0 (4)

where denotes the Macaulay bracket (x= 0,x


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identification of which have been detailed in Nguyen and Houlsby[1]. They are rewritten here asfollows:

Ft1= (1cd)f2t2E

E+Ept(1td)ntE(1

td)

+Ept(1

td)

nt 2

(9)

and

Fc1=f2c2E

E+Epc(1cd)nc[ln(1+cd)]mc

E(1cd)+Epc(1cd)nc[ln(1+cd)]mc2

(10)

Besides material properties such as Young modulus E, Poissons ratio , uniaxial tensile and

compressive strengths ft and fc , the following parameters are used in the above functions: Ept

and n t for tensile damage, and Epc, m c and n c for compressive damage. The physical meaning of

these parameters is explored in Part I of this paper[1], and is summarized in the next section.The two parameters k and in yield criterion (2) are defined as

k=

fcyfty

3(11)

= fcy fty3

(12)

in which fty and fcy are yield stresses in uniaxial tension and compression, defined as

fcy= (fc0+Hccp)(1cd) (13)

fty= (ft0+Http)(1td)(1cd) (14)

In the above expressions, the two initial yield thresholds are denoted as fc0= 0.3fc and ft0= ftfor uniaxial compression and tension, respectively. Linear hardening rules are assumed here, with

Ht>0 and Hc>0 being the hardening moduli and tp and cp being corresponding accumulated

plastic strains in tension and compression[1]. We rewrite the definitions of tp and cp here:

tp= cFt4(kk)i ji j (15)

cp= cFc4 (kk)i ji j (16)

where c =2/3 and

Ft4= min

1,

kk+ fcyfcy+ fty

(17)

Fc4= min1, ftykkfcy+ fty = 1 Ft4 (18)The determination of all parameters of the proposed coupled damageplasticity model, based on

fracture properties of the material and widths of the fracture zones, has been presented in Part I

of this paper[1] in which further details on the model can be found.


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3. NON-LOCAL REGULARIZATION

To deal with softening-related problems, the local constitutive model described in the preceding

section needs to be regularized. The regularization can be based on any of several methods ranging

from simple (e.g. fracture energy regularization by Bazant and Oh[

15]) to more complicated

(non-local or gradient approaches). A detailed classification of regularization methods is given by

Jirasek and Patzak[16]. Here full regularization of the constitutive equations based on non-localtheories is used.

Following Pijaudier-Cabot and Bazant[2], it is necessary to apply non-local treatment onlyto variables or quantities directly controlling the softening process. Various non-local damage

approaches using different non-local quantities has been analysed at length in Jirasek[17]. Recently,Jirasek and Marfia[4] developed a non-local averaging scheme using displacement with someadvantages over other averaging schemes (e.g. using strain or stress averaging). In this study, with

damage-induced softening in the constitutive model, the two damage criteria (3) and (4) should

be treated as non-local criteria. Therefore, the regularization in this case is realized through the

non-locality of the two energy-like terms associated with the damage processes (see Equations (3)

and (4)). This is done by placing these energy-like terms in (3) and (4) under two non-local spatialintegrals. A physical interpretation is that the energy redistribution due to micro-crack interactions

[18, 19] is accounted for in the non-local model. The damage energies defined by the first termsof Equations (3) and (4) and arising from all material points inside the volume Vd govern the

non-local softening processes. The thermomechanical aspects of this non-local regularization have

been discussed by Nguyen[10]and are also subjects of an on-going study[20]. From (3) and (4),we obtain two non-local damage criteria as follows:

ytd=1

G(x)

Vd

g(yx)(1+ pt)+i j+i j ptkkll

2E(1td)2 dV Ft1(td, cd)= 0 (19)

yc

d=1

G(x) Vd g(yx)(1+ pc)i ji j pckkll

2E(1cd)2 dV

Fc

1

(c

d

)=

0 (20)

where all the stress terms in Equations (19) and (20) are evaluated at point y which lies inside a

sphere (or circle in two dimensions) of volumeVd, centre x and radiusR;G(x)=

Vdg(yx) dV

is used to normalize the weighting scheme applied to the energy-like terms in (19) and (20); and

g(yx) is a bell-shaped weighting function defined by

g(r)= g(yx)=

0 ifr>R

1 r2

R2

2ifrR

(21)

The volume Vd here is defined by the non-local interaction radius R. For an interpretation of the

length parameter R, recent results[21] in bridging micro-scale and macro-scale could be used fora relationship between the internal length R and the size of the representative volume element.

Further details on non-local formulations and the non-local damage criteria described above can

be found in Bazant and Jirasek[3] and Nguyen[10], respectively.The set of non-local constitutive equations now comprises Equations (1)(2) and (19)(20). It

should be noted here that the non-locality of damage in (19) and (20) also helps to prevent the


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localization of both damage and plastic strain into zones of infinitesimal volumes, as encountered

in local models. Due to this non-locality, damage at a material point, once activated, will also

trigger damage at neighbouring material points, reducing the yield thresholds at those neighbouring

points (Equations (13) and (14)). Plasticity at the neighbouring damaged points can therefore takes

place at the same time as damage (e.g. in uniaxial tension), or can be activated once reduced yield

thresholds are met. With the focus here on the constitutive behaviour of the model at this stage of

the development, an analytical proof that the non-local model achieves these regularization features

is not given, but numerical examples (Section 6) demonstrate its effectiveness.

4. DETERMINATION OF MODEL PARAMETERS

As can be seen in Part I of this paper[1], the determination of parameters controlling the localstressstrain behaviour of the model strongly depends on the imaginary widths wt and wc of the

FPZs in tension and compression, respectively. Unlike in smeared crack models, wherewt andwccan be directly determined from the finite element size[15, 22], these two widths in the non-localmodel are related to the non-local interaction radius R, and also depend on other parameterscontrolling the local stressstrain behaviour of the model. It is therefore difficult, even impossible,

to determine analytically the relationship betweenwt (or wc) and the non-local interaction radius

R in non-local models, except in some simple cases with a linear softening law (e.g. see[2325]).For that reason, a condition on the equivalence between the energy dissipated by a non-local

model with imaginary crack bandwidth wt(or wcin compression) and that specified by the fracture

energyGF(or Gc)is used in this study. Non-local numerical analyses of a simple bar under uniaxial

tension (or compression) will be carried out for the determination of a relationship between wt(or wc) and the length parameter R of the non-local model. Once this relationship is established,

a value of R to give a numerical crack band width wt close to that proposed in the literature

(e.g. wt 3dmax in[15, 26]), and satisfying the condition of fracture energy balance, is readilydetermined.

The determination of the ratios wt/R and wc/R in non-local constitutive models has beenpresented at length in Nguyen[10], and by Nguyen and Houlsby[7]. In this study, we use theprocedure proposed in Nguyen and Houlsby[7] to determine wt and wc from a local constitutivemodel, experimental material properties, and a specified value of non-local radius R . The calibration

of non-local model parameters using the procedures in Nguyen and Houlsby [7] can only giveconsistent responses of the model on the basis of fracture energy. More advanced (and more

computationally expensive) calibration methods (e.g. those based on numerical inverse analysis

and optimization by Carmelliet[27] and Le Bellego et al.[28]) for parameters of non-localmodels can be used to obtain a supposedly unique model response corresponding to the material

characteristic length.

A summary of the models and parameters used in the non-local model described above can be

found in Nguyen

[10

]. Those parameters are not independent, but interact closely. They can be

classified as local parameters and a spatial parameter, which is the non-local radius R in this case.The local parameters govern the pointwise behaviour of the model and are determined based on

calibration procedures in Nguyen and Houlsby[1, 7]. Following those procedures, the widths wtand wc of the FPZs are obtained based on a specified value of non-local radius R (see[7]), andthen used to compute these local parameters (see[1]). In those procedures, the mutual effects ofall parameters of the model (spatial and local) on each other are accounted for. Different simplified


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constitutive models can be derived from the full version by dropping out appropriate terms in

the constitutive equations and setting appropriate values for some model parameters, e.g. setting

Ht=Hc=, and ft0= fc0= to exclude plasticity from the full model (see[1] for details).This feature has been incorporated into the finite element code OXFEM used in this study.

5. NUMERICAL IMPLEMENTATION

The finite element implementation of the non-local constitutive model described above is now

presented. A new integration scheme is proposed for the integration of the non-local incremental

constitutive equations derived from Sections 2 and 3. Local arc-length control based on the

algorithm by May and Duan[9] is used to handle snap-through and snap-back responses whentracing the equilibrium paths for structures made of softening materials. Only the new integration

scheme for non-local incremental constitutive equations is presented here; the implementation of

the local arc-length control is given by Nguyen[10].Integrating the incremental constitutive equations is a vital part of the numerical implementation

of the model, as it directly affects the stability of the numerical solutions. The integration schememust also be able to deal with non-locality in the constitutive equations. An implicit integration

method, based on the integration scheme proposed by several researchers [8 (Chapter 6, vol. 1); 29]

is therefore adopted and modified here.

The responses of every material point in the structure must satisfy entirely the system of con-

stitutive equations (1)(2) and (19)(20), which in general can only be solved using numerical

methods. The analysis is, however, more complex than in models where the evolution laws of

damage are explicitly defined (e.g. in models by Peerlings et al.[30], Peerlings[31], Jirasek andPatzak[32] and Jiraseket al.[33]). Because of the appearance of the spatial integral in Equations(19) and (20), two spatial discretization schemes are necessary for solving the boundary value

problem. The first discretization is required for the numerical solution of the partial differential

equations, as is employed in conventional finite element analysis. The second is an inner dis-

cretization, which deals with the integration of the rate constitutive equations along a loading path.Normally, for models based on local theory, only the outer discretization scheme is needed, as

the integration of the constitutive equations can be carried out pointwise. For the implementation

used here, the same discretization scheme is used for the solution of governing partial differential

equations and the integration of non-local constitutive equations. This results in the non-local

evaluation of energy-like quantities in the two damage criteria over the Gauss points used in the

finite element discretization.

For the numerical analysis using finite elements, the integrals in (19) and (20) are transformed

to summations over Gauss points. Denoting by the energy-like quantity to be averaged, we can

express its corresponding non-local counterpart as[10]:

(x)= ne

=1

mei

=1w

eig(

yei

x

) det Jei (y

ei)ne=1mei=1weig(yei x) det Jei =

n

w (22)in which e denotes the element index and n the total number of elements inside the interaction

volume Vd defined by a sphere at centre x and of radius R; i is the i th Gauss point of element

e; me is the number of Gauss points of this element inside the interaction volume; wei and J

ei

are, respectively, the weight and Jacobian matrix at Gauss point i of element e; n is the total


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number of Gauss points inside the interaction volume at point x; w and are, respectively, the

weight and energy-like quantity associated with the i th Gauss point of element e , in which w is

defined by

w=weig(

yei

x

) det Jeine=1mei=1weig(yei x) det Jei (23)

An implicit Euler integration scheme is adopted for the integration of the incremental constitutive

equations. However, due to the presence of the spatial integrals in the damage criteria, the stress

update procedure cannot be carried out pointwise as is normal in local models. Non-locality in

this case turns the pointwise-defined stressstrain constitutive equations to a system of non-local

coupled equations, relating the stresses, strains and internal variables at several integration points

in the failure region. This coupling makes the stress update routine more complicated, requiring

considerable effort in the formulation and implementation as well as a time cost in the numerical

computation.

As the constitutive relationships in this case contain coupled equations relating the stress and

strain increments at several integration points, finding the intersection points between the stress

path and the yield/damage surface is almost impossible. To avoid the need for this, an elasticpredictordamageplastic corrector integrating scheme is adopted here for the integration of the

equations. This is based on the algorithm proposed by Crisfield [8 (Chapter 6, vol. 1)] and can

be considered as a form of the backward-Euler integration scheme [8 (Chapter 6, vol. 1)]. This

algorithm uses the normal at the elastic trial point and hence avoids the necessity of computing the

intersection between the elastically assumed incremental stress vector (using secant elasticdamage

behaviour) and the yielddamage surfaces. Furthermore, the method enforces yield (2) and damage

criteria (19), (20) at any stage of the loading process, thus removing the inaccuracies encountered

when the incremental consistency conditions of the yield and damage functions are used[29].At the starting point, the system of equations are rewritten in the incremental form by taking

the first-order Taylor expansion of the yield and damage functions about the elastic trial point

B (Figure 1, with y in the figure representing either the yield or compressive/tensile damage

surface). In this coupling case, it is assumed here that yielding and both mechanisms of damagetake place at the same instant. Therefore, treatment of corners (see Figure 6 of Nguyen and

Houlsby[1]) on the composite yielddamage surface is automatically taken into account. Bydropping appropriate terms, we can straightforwardly treat simpler situations, in which only one or

two failure mechanisms are activated. From (1), (2), (19), (20), a system of incremental equations

X

B (Elastic trial point)

C

y = 0 (elastic limit)

y =yC

> 0

y =yB> 0

Figure 1. Pictorial presentation of the integration scheme (after de Borst[29] andCrisfield [8 (Chapter 6, vol. 1)]).


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governing the constitutive behaviour of the model can be written:

di j=(1+)di j dkki j

E(1td)(1cd)+di j

+ (1+)i jkki jE(1td)2(1cd)

dtd+ (1+)i jkki j

E(1td)(1cd)2dcd (24)

yp=yp |B+ypi j

di j+yptd

dtd+ypcd

dcd+yptp

dtp+ypcp

dcp (25)

ytd|C=y td|B+w

(1+pt)+i j di j ptkki j di j

E(1td)2

+(1+pt)+i j+i j ptkkll

E(1td)3 dtd

Ft1

tddtd

Ft1

cddcd= 0 (26)

ycd|C=ycd|B+w

(1+ pc)i j di j+ pckki j di j

E(1cd)2

+(1+ pc)i ji j pckkll

E(1cd)3 dcd

Fc1

cddcd= 0 (27)

The spatial integrals in Equations (26) and (27) obtained from two damage criteria have been

replaced by summations over integration points. The terms yp |B , ytd|B and ycd|B in (25)(27) arethe values of the loading functions at the elastic trial point B. Furthermore, it is implied here

that all the derivatives and terms in Equations (25)(27) are evaluated at this stress point. Due to

the appearance of the summations in (26) and (27), it should also be noted that B for these two

damage functions denotes several elastic trial points at different physical points, from which ytdand ycd are evaluated, rather than a single point in the original scheme.

As mentioned above, the stresses at the elastic trial point B (Figure 1) are obtained by adding

elastic incremental stresses to the stresses at point X. Our aim is to compute the stress increment

i j |BC, which is needed in going from B to C(Figure 1), from system (24)(27). At first, from(25), using the flow rule

di j=pyp

i j(28)

and the definitions ofdtp and dcp in (15)(18), the plastic strain rate di j is obtained as

di j=p

ypi j

= yp

|B

+

ypkl

dkl

+

yp

t

d

dtd

+

yp

c

d

dcd

c

yp

mn

yp

mn

Ft4(kk)

yptp

+Fc4 (kk)ypcp

ypi j

(29)

Secondly, back substituting the above plastic strain increment into (24), some manipulation

leads to the relationship between the stress, strain and internal variables, written in incremental


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form as follows:

dmn=

di j

+

yp |B+

yptd

dtd+ypcd

dcd

yp

i j

cypkl

yp

kl

Ft4 yptp

+Fc4 ypcp

[(1+)i jkki j ]d

td

E(1td)2(1cd) [(1+)i jkki j ]d

cd

E(1td)(1cd)2

[Dsti j m n]1 (30)

where Dsti j m n is the constitutive matrix, which is tangent with respect to plasticity and secant with

respect to damage (see[10] for details):

Dsti j m n=Di j m n

(1

td)(1

cd)

yp

i j

ypmn

cypkl

yp

klFt4 yp

tp+Fc4

ypcp

(31)

where Di j m n is the elastic compliance matrix.

Finally, substituting (30) into (26) and (27), we obtain a system of linear equations in dtd and

dcd. This system is written for all integration points in the FPZ, each pair (tension and compression)

of which has the following form:

ytd=wQt

Ft1

tddtd

Ft1

cddcd+y td

B

= 0 (32)

yc

d= wQcFc1

cddc

d+y c

dB = 0 (33)where Qt and Qc are defined as follows:

Qt=

(1+ pt)+kl ptqq klE(1td)2

[Dsti j k l ]1

yp |B+

yptd

dtd+ypcd

dcd

yp

i j

c

yp

mn

yp

mn

Ft4

yptp

+Fc4ypcp

(1+ pt)+kl ptqq kl

E(1

td)

2 [Dsti j k l ]1

[(1+)i jqq i j ]dtdE(1

td)

2(1

cd)

(1+ pt)+kl ptqq kl

E(1td)2 [Dsti j k l ]1

[(1+)i jqq i j ]dcdE(1td)(1cd)2

+(1+ pt)+i j+i j ptkkll

E(1td)3 dtd

(34)


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and

Qc=

(1+ pc)kl+ pcppklE(1

cd)2 [

Dsti j k l]1

yp |B+

yptd

dtd+ypcd

dcd

yp

i j

c yp

mn

yp

mn

Ft4

yptp

+ Fc4ypcp

(1+ pc)

kl+ pcppkl

E(1cd)2 [Dsti j k l ]1

[(1+)i jppi j ]dtdE(1td)2(1cd)

(1+ pc)kl+ pcppkl

E(1cd)2 [Dsti j k l ]1

[(1+)i jppi j ]dcdE(1td)(1cd)2

+(1+ pc)i ji j pckkll

E(1cd)3 dcd

(35)

As can be seen, the strain increments do not appear in (34) and (35) when substituting (30)into Equations (26) and (27), as they have been applied in moving from point X to the elastic

trial point B [8 (Chapter 6, vol. 1)]. Solving the above system of equations (32)(33) will give

the damage increments at all points in the damaged zone of the structure. Back substituting the

damage increments into (30), noting that the strain increments have been applied in the predictor

step (from X to B) and now must be removed from that expression, we obtain the stress increment

i j |BC in going from B to C. Finally, the stress at point C (see Figure 1) is updated using[8 (Chapter 6, vol. 1)]:

i j |C=i j |B+ i j |BC (36)

in which i j |BC is computed by

mn |BC=

yp |B+

yptd

td+ypcd

cd

yp

i j

c

yp

kl

yp

kl

Ft4

yptp

+ Fc4ypcp

[(1+)i jkki j ]td

E(1td)2(1cd) [(1+)i jkki j ]

cd

E(1td)(1cd)2

[Dsti j m n]1 (37)

In the above expression, all damage increments have been computed by solving systems (32)(33),

and all derivatives are evaluated at the elastic trial point B. Normally, due to the linearization,

the updated stress points do not lie on the yield/damage surfaces (Figure 1) and relevant tech-niques should be applied to return them to the loading surfaces. Repetition of the same process

but with point B replaced with C, noting that there are no elastic stress increments in the sub-

sequent steps, would be an appropriate and simple way to bring the stresses back to the loading

surfaces. Mathematically, this is an iterative NewtonRaphson process to return the stress at B

to loading surfaces. This simple but rather efficient stress return algorithm is advocated here due


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to the complexity of the system of incremental constitutive equations (24)(27). Alternatively,

further enhancement to the exactness of the integration scheme can be obtained through the

combination of sub-incrementation and the repetition of the above predictorcorrector processes,

all of which have been implemented in the finite element code OXFEM. Furthermore, to reduce

the errors in updating the stresses, in this study the integration scheme above was carried out based

on the incremental instead of iterative strains (see [8 (Chapter 6, vol. 1)] for details).

6. NUMERICAL EXAMPLES

This section examines the numerical validation of the non-local coupled damageplasticity model.

Analyses of example structures are carried out to show the features of the model. These tests

range from simple to complex loading cases, and hence require appropriate choice of constitutive

models.

The structural tests used here comprise those exhibiting important features in the behaviour of

quasi-brittle materials in general and concrete materials in particular. They can be classified astension tests, bending tests, shear tests and compression-related tests, under monotonic or cyclic

loading. The material models used can be pure tensile damage or tensilecompressive damage

with plastic deformation being accounted for whenever the experimental cyclic loading data are

available. The calibration procedures for parameters of non-local models, proposed in Nguyen and

Houlsby[1, 7] and summarized in Section 4, are used throughout the numerical examples.All the numerical analyses are carried out using the local arc-length control [9] for the incremental

analysis, and NewtonRaphson method for the iterative technique. Finite element meshes of six-

node triangular elements are used in all examples. The convergence tolerance parameter is 104for the norm of the out-of-balance force vector in the NewtonRaphson iterative process. The same

tolerance is used in the stress update routine to gauge the errors occurring in returning the stresses to

the loading surfaces. Automatically chosen numbers of increments (see [8 (Chapter 9, vol. 1); 9]),

controlled by the number of iterations required for each load increment, are used throughout theexamples. Due to the complexity of the non-local formulation, a local stiffness matrix based on the

local constitutive matrix in Equation (31) (see[10] for details) is used in all numerical examples.A loading scheme consisting of three load stages (1st: fully elastic behaviour, 2nd and 3rd: peak and

post-peak stages) is used, in which the controlling minimum and maximum numbers of iterations

for the last two stages are normally 1218 and 1827, respectively.

To achieve convergence in severe cases with snap-back observed in the equilibrium paths, the

constraint equation of the arc-length control is based on relative nodal displacements in the FPZ,

although crack mouth opening displacements (CMODs) or crack mouth sliding displacements

(CMSDs) can also be used without any difficulty (see[9, 10, 29] for details). The local stiffnessmatrix, which is tangent with respect to plasticity and secant with respect to damage (Equation (31);

see also

[10

]), is used throughout the numerical examples in this section. When the dissipation is

totally due to damage, the local secant stiffness matrix is used. For the NewtonRaphson iterativemethod, the stiffness matrix is only recomputed at the beginning of the load increment, and kept

unchanged throughout the iterations[10]. Furthermore, the stress update is based on the incrementalinstead of iterative strains[8].

Table I summarizes model parameters used in numerical examples in this section. Details on

how to obtain these model parameters can be found in Nguyen[10].


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Table I. Model parameters used in numerical examples.

Examples and models

6.1 6.2 6.3 6.4 6.5

Tensile damage Tensile andParameters Pure damage Pure damage and plasticity Pure damage compressive damage

E (N/mm2) 2.4 104 3.0 104 3.8 104 2.48 104 3.1 104 0.2 0.2 0.2 0.18 0.2GF (N/mm) 0.059 0.124 0.125 0.1085 0.072Gc (N/mm) 18.0

ft (N/mm2) 2.0 (or 2.86) 3.33 3.0 3.55 4.08fc (N/mm2) 38.0fc0 (N/mm

2) 11.0R (mm) 12.0 16.0 6.0 25.0 2.0wt/R 2.02 (1.98) 1.96 2.32 1.86 2.0

Ept (N/mm2) 3090.0 (9081.0) 6899.0 6381.0 14 129.0 1988.0

nt 0.26 (0.36) 0.32 0.33 0.37 0.28

Ht (N/mm2) 33 800.0wc/R 2.2

Epc (N/mm2) 6000.0

mc 6.0nc 0.24

(c)(b)(a)60mm

60 105

2

60

Figure 2. (a) Double edge notched specimen (10-mm thick)geometry; (b) experimentalcrack pattern[34]; and (c) FE meshes.

6.1. Double-edge notched specimen under tension

In this example, simulations of a double edge notched specimen under tension

[34

]are presented

(Figure 2(a)). In the analyses, the specimen is fixed in both directions at the bottom edge, and inhorizontal direction at the top edge. The analyses were carried out using three meshes of six-node

triangular finite elements (Figure 2(c)), with prescribed vertical displacements on the top edge of

the specimen.

The numerical results are depicted in Figure 3, showing the agreement in the loaddisplacement

curves obtained from different finite element meshes, thus proving the lack of mesh dependence


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0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2

Prescribed displacement (mm)

Load

(kN)

Experimental

Numerical, mesh 1, ft=2.0MPa




0.5

0.7

0.9

1.1

1.3

0 0.005 0.01 0.015 0.02 0.025

(a) (b)

Figure 3. Double edge notched specimennumerical results: (a) loaddisplacement curves and (b) damagepattern (mesh 3, prescribed displacement of 0.19 mm).

P/2

L /250

200100

P

2000

Crack path

Figure 4. Geometrical data and half-beam model used in the numerical analysis.

of the model. The experimental peak load (1.13 kN) in the figure can be seen to be bounded by

its two numerical counterparts (0.94 and 1.26 kN) corresponding to two used values of the tensile

strength. In addition, the overall shape of the numerical loaddisplacement curves is consistent

with the experiment. The failure of the specimen can be seen in Figure 3(b). No clear macro-crack

can be observed, as attention here is paid to the structural response of the specimen under loading,

rather than the crack propagation and interaction. A finer mesh and smaller non-local interaction

radius could be used, at higher computational cost, if the observation of crack propagation is

important (see[10]).

6.2. Three-point bending testnotched beam

Analyses are carried out for a notched beam in a three-point bending test (Figure 4), aiming atinvestigating Mode I fracture and crack propagation. The geometrical data and material properties

are taken from the experimental test of Petersson[35], with the effect of the beams self-weightbeing eliminated from the measured fracture energy (see[35]).

Because of symmetry, only half of the beam was modelled (Figure 5). Results, in the form

of loaddeflection curve and damage pattern, are shown in Figure 6. The damage process zone


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Figure 5. Finite element meshes: (a) full beam and (b) zoomed-in at centre.

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

Load

(N)

Deflection (mm)

Exp., GF=115N/m

Exp., GF=137N/m

Num., mesh a, GF=124N/m

Num., mesh b, GF=124N/m

Figure 6. Loaddeflection curve and damage pattern at very late stage of the numerical analysis (finer mesh,zoomed-in at centre part of the half-beam).

can be clearly seen in this figure and the numerical crack path agrees well with the experimental

one in Figure 4. The numerical loaddeflection curves obtained from the two meshes are almost

identical, again demonstrating the lack of mesh dependency of the proposed model. In addition,

they also match the experimental curves quite well.

6.3. Four-point bending testnotched beamcyclic loading

In this example, the four-point bending test experimentally performed by Hordijk[36]is simulatedusing the coupled damageplasticity model. The geometry of the specimen and finite element

meshes of a half-beam model are shown in Figure 7.

The numerical loaddeflection curve overestimates the experimental one in the post-peak region

near peak load (Figure 8(a)), while it underestimates the tail behaviour (Figure 8(b)). However, the


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50

100

30

P

500mm

P

150 150 150

Crack path

Figure 7. Four-point bending testgeometry and finite element meshes.

numerical peak load is rather close to the value given by experiment (Figure 8). The permanent

deformations at zero stress state are clearly seen in Figure 8, showing the capability of the model

in capturing residual strains in tension. On the other hand, it can also be seen that there is almostno difference in the numerical results using different FE meshes, proving the mesh independence

of the obtained numerical results.

In Figure 8(c), we can see the zones of damage and plastic strain along with their increments

at a specific moment of the failure process. Although plasticity is local in the proposed model,

its activation and development in the structure is however induced by the non-local damage

mechanisms. In other words, the reduction of the yield thresholds due to the non-locality of damage

will also result in the activation of plasticity at neighbouring points of the material point under

consideration. Therefore, the accumulated plastic strain defined in (15) also spreads out, realizing

through the sizes of the zones of accumulated plastic strain and its increment in Figure 8(c).

6.4. Four-point shear test

The four-point shear test of Arrea and Ingraffea[37]is selected here to demonstrate the capabilityof the model in capturing the structural responses in shear loading. The snap back of the load

displacement curve helps to demonstrate the stability of the proposed numerical integration scheme

and solution methods in dealing with highly nonlinear structural response. The geometrical data

for structure are shown in Figure 9.

To avoid local failure, the load distributors were also modelled in the two finite element meshes,

and assumed to be made of steel (E= 210 GPa, = 0.3) (Figure 10). The analyses were carriedout with the applied load under indirect control, based on the relative displacements between nodal

points of elements in the fracture zone (local control). The arc-length method was used for the

incremental analysis and the NewtonRaphson method for the iterative equilibrium. This helps to

capture effectively the snap back of the loaddisplacement curve.

Snap-back behaviour can be clearly seen on the loaddisplacement curves in Figure 11(b), withthe vertical displacement taken at point A on the upper side of the beam and under the steel load

distributor (Figure 9). The difference in the load-displacement curves from the analyses using two

meshes comes mainly from the difference in size of the modelled load distributor. This is only

a local effect, and the overall structural responses in Figure 11(a) are almost identical and not

significantly affected by the size of the load distributors.


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0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2 0.25

Load(k

N)

0

0.5

1

1.5

2

2.5

3

Load(k

N)

Mid-span deflection (mm)Mid-span deflection (mm)

Numerical, mesh 1

Numerical, mesh 2

Experimental Numerical, mesh 1

Numerical, mesh 2

Experimental

0 0.1 0.2 0.3 0.4 0.5

X

(a) (b)

(c)

Figure 8. Four-point bending testloaddeflection curves: (a) peak and (b) tail behaviour. (c) Contourlines of internal variables and their increments (mesh 2), at point X(b) on the loaddeflection curve.

There is however a poor match between the experimental and numerical loadCMSD curves;

and this has also been found in other research dealing with this mixed-mode test

[38, 39

]. It can

be supposed that the observed mismatch comes from the irrelevant use of pure Mode I fractureenergy in a mixed-mode analysis. However, lack of relevant material properties from the real test

means that this mismatch cannot be fully explained.

The numerical crack pattern can be seen in Figures 12 and 13, and compared with the experi-

mentally observed crack path in Figure 9. The numerical crack path seems to be less curved than

its experimental counterpart.


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P

306

156

306

82

203 203396 61 61 396

0.13PObserved crack path

A

Figure 9. Four-point shear testgeometrical data (dimensions in mm).

Figure 10. Finite element meshes.

0

20

40

60

80

100

120

140

160

0 0.05 0.1 0.15 0 0.05 0.1 0.15 0.2

CMSD (mm)

Load

P(

kN)

Load

P(

kN)

Mesh 1Mesh 2

Mesh 1

Mesh 2

Experimental, lowerExperimental, upper

0

20

40

60

80

100

Displacement at load point (mm)(a) (b)

Figure 11. (a) LoadCMSD responses and (b) loaddisplacement curves.


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Figure 12. Deformed structure at CMSD of 0.26 mm (mesh 2, magnification of 100).

Figure 13. Four-point shear test: crack pattern at CMSD of 0.26 mm (mesh 2).

6.5. Splitting test on a concrete prism

The splitting test (Brazilian test) here serves as an indirect testing method to measure the tensile

strength of the material, helping to resolve disadvantages in the implementation of direct tensiletest[40]. The splitting strength can then be used to calculate the uniaxial tensile strength of thematerial[41, 42]. The testing arrangement was quite simple, as shown in Figure 14, and test on theprism specimen was adopted for the numerical simulation, with D= 75 mm, B= 50 mm and thefollowing widths of the load bearing strip:b1= 0.08D= 6 mm (numerical test 1; named STP75-8in[40]) and b2= 0.16D= 12 mm (numerical test 2; STP75-16 in[40]).

In both tests, the results obtained from the two finite element meshes are almost identical, proving

the mesh independence of the proposed non-local model. However, the peak loads obtained in

neither of the numerical tests agree well with the experimental data. In addition, only the numerical

loaddisplacement curve for the first test (b/D= 0.08) shows a close resemblance to the numericalreference curve[43]. Analyses have also been carried out using a lower value of the compressivefracture energy (dashed dot curve, Figure 15(a)). However, only the tail response is affected and

the peak load only changes insignificantly. For the second test (b/D= 0.16), oscillation in theloaddisplacement response can be observed, which has also been found in other numerical results

for this kind of splitting test[43, 44].The paper by Rocco et al.[40] does not provide any results on the loaddisplacement curves

or the crack pattern of the specimen. In this study, the damage contours obtained from numerical

analysis are presented to illustrate the failure processes in the specimen at different loading


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D

P

D

P B

b

Figure 14. Splitting test: geometry and finite element model.

0

5

10

15

20

0 0.02 0.04 0.06

Prescribed displacement (mm)

Load

(kN)

0

5

10

15

20

25

30

Load

(kN)

Num., Comi & Perego (2001)

Num., Mesh 1, Gc=18N/mm



Exp. peak load, b/D=0.08

B

A

Num., Mesh 1

Num., Mesh 2

Num., Comi & Perego( 2001)

Exp. peak load, b/D=0.16

C

DE

F

(a)

0 0.02 0.04 0.06

Prescribed displacement (mm)(b)

Figure 15. Loaddisplacement curves: (a) b/D= 0.08 and (b) b/D= 0.16.

At A (peak) At B

Figure 16. Tensile damage (left) and compressive damage (right) for b/D= 0.08.

stages. This is useful as it helps to show the complex failure process involving both tensile and

compressive damage mechanisms, which are coupled in this example. The damage zones in both

tests are depicted in Figures 16 and 17, where the splitting effect can be clearly seen in Figure 17


DOI: 10.1002/nag


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At C (just after peak)

At D At F

At E

Figure 17. Tensile damage (left) and compressive damage (right) for b/D= 0.16.

for b/D= 0.16. Tensile damage in this case (b/D= 0.16) occurs at the centre of the specimenand quickly develops through its height, while failure due to compressive damage just happens at

the corner of the load bearing strip. This is different from the first test ( b/D= 0.08) where failuredue to both mechanisms of damage localizes underneath the load bearing strip (Figure 16).

7. CONCLUSIONS

Enhancements using non-local theory to a newly developed constitutive model[1] to cope withsoftening-related problems are presented in this paper. The regularized damageplasticity modelemploys two independent length parameters in tension and compression. All parameters of this

model are shown to be identifiable and can be calibrated based on the experimental data on

concrete, using the procedures developed by Nguyen and Houlsby[1, 7]. This is an importantfeature which makes the model readily applicable in practice.

An integration scheme, based on that by de Borst[29] and Crisfield [8 (Chapter 6, vol. 1)],was proposed for updating stress during the incremental-iterative solution procedure in nonlinear

finite element analysis. This implicit integration scheme for non-local rate constitutive equations

was shown to be stable through the simulation of real structural tests in Section 6. This stability

is enabled by the use of a local arc-length control[9] in the incremental analysis. The principalstructural responses were captured in numerical examples using the proposed non-local approach,

and the lack of mesh dependency of the non-local constitutive model was established.

The formulation of a non-local coupled damageplasticity in this study (Part I: Nguyen andHoulsby[1] and Part II: this paper) is considered as an initial step in the development of athermomechanical approach for the non-local constitutive modelling of concrete. The computational

aspects of the proposed non-local model were only briefly considered in this study, with the

proposal of an integration scheme for the stress update routine. In addition, use of a local stiffness

matrix, instead of a non-local stiffness matrix consistent with the stress update algorithm, in the


DOI: 10.1002/nag


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numerical analysis significantly increased the computational costs. Further research will focus on

the thermodynamic, regularization and computational aspects of non-local constitutive modelling,

and the incorporation of anisotropic behaviour to appropriately predict failure mode and orientation

of failure planes, all of which have been discussed by Nguyen[10].

ACKNOWLEDGEMENT

Financial support from the Jenks family (in the U.S.A.) through the Peter Jenks Vietnam scholarship tothe first author, while he was working in Oxford, is gratefully acknowledged.

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DOI: 10.1002/nag

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