A coupled damage–plasticity model for concrete based on thermodynamic principles: Part II:...

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

A coupled damage–plasticity model for concrete based onthermodynamic principles: Part II: non-local regularization and

numerical 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.

SUMMARY

Non-local regularization is applied to a new coupled damage–plasticity 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 damage–plasticity 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 constitutivemodels 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 strainsoftening 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]‡Postdoctoral Fellow (formerly Research Student, University of Oxford).§Professor of Civil Engineering.

Copyright q 2007 John Wiley & Sons, Ltd.

G. D. NGUYEN AND G. T. HOULSBY

behaviour. The use of damage mechanics, in combination with plasticity theory, enables us toderive 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-fracturingprocesses in concrete. Therefore, macroscopic variables such as damage indicator and plastic strainare definitely not true representatives of those complicated fracturing processes. Nevertheless, tosome extent, they could be considered appropriate in capturing some essential features of thematerial behaviour at macroscopic level, such as the stiffness and strength reductions and residualstrain, although the underlying microscopic processes are much more complicated. The damage andplasticity zones in coupled damage–plasticity models therefore could be viewed as macroscopicrepresentation of the underlying micro-cracking and frictional slips in a non-zero volume fractureprocess 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 structuralanalysis. In the literature, non-local regularization techniques have been found to be appropriate forthe 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 damage–plasticity 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 calibrationof parameters for the non-local model is addressed, with the procedures proposed in Nguyen andHoulsby [7] being used.

Numerical implementation plays an important part in the development of constitutive models forengineering materials. The implementation here comprises a method for the solution of the partialdifferential equations in solid mechanics, and the incorporation of the constitutive models into thissystem of governing differential equations. In this study, the finite element method is employed forsolving boundary value problems in continuum mechanics. However, as concrete exhibits highlynonlinear behaviour after peak stress, the incorporation of the non-local constitutive relationshipsinto the system of equations is not straightforward. A modified backward integration scheme, basedon 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 algebraicequations in finite element analysis, the arc-length method in combination with local constraintequations employing dominant displacements [9] is implemented, and proves its reliability in thisstudy. Numerical examples showing the stability of the integration scheme, and the capability ofthe arc-length control procedure to handle highly nonlinear behaviour in structural analysis, willbe 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 proceduresfor calibration of parameters for this non-local coupled damage–plasticity model. The numericalimplementation of the non-local model is presented in Section 5, followed by numerical examplessimulating real structural problems.

2. THE LOCAL MODEL

The constitutive equations governing the local behaviour of a coupled damage–plasticity model [1]consist of a stress–strain 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. (2007)DOI: 10.1002/nag

A COUPLED DAMAGE–PLASTICITY MODEL FOR CONCRETE: PART II

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

�i j = (1 + �)�i j − ��kk�i jE(1 − �td)(1 − �cd)

+ �i j (1)

y∗p = ��kk + �′

i j�′i j

2− k = 0 (2)

ytd = (1 + pt)�+i j�

+i j − pt〈�kk〉〈�ll〉

2E(1 − �td)2

− F t1(�

td, �

cd) = 0 (3)

ycd = (1 + pc)�−i j�

−i j − pc〈−�kk〉〈−�ll〉

2E(1 − �cd)2

− Fc1 (�cd) = 0 (4)

where 〈〉 denotes the Macaulay bracket (〈x〉= 0, x<0; 〈x〉= x, x�0), �i j the Kronecker delta, kand � two parameters of the yield criterion, pt and pc two parameters of the damage criteria, andF t1 and Fc

1 denote two functions governing the evolutions of damage variables �td and �cd [10].The following evolution rules for plastic strain �i j and damage variables �td and �cd are implicitlyembedded in the above constitutive equations:

�i j = �p�yp��i j

= �p(�r�i j + �′i j ) = �p(�r�i j + �′

i j ) (5)

�td = �td�ytd��td

= �td (6)

�cd = �cd�ycd��cd

= �cd (7)

where �p, �cd and �td are corresponding non-negative scalar multipliers; �td and �cd two generalizedstresses associated with �td and �cd, respectively [1]; and

yp = �[r�kk + (1 − r)�kk] + �′i j�

′i j

2− k = 0 (8)

is the yield function written in the generalized stress space �i j [1, 11]. It is noted here that yp(in Equation (8)) is only used to define a non-associated flow rule, through the use of the ratio rgoverning the deviation of the plastic strain rate from the normal to the yield surface y∗

p in truestress space �i j (see [10, 12] for details). For computational aspects, the yield surface y∗

p (seeEquation (2)), obtained by substituting �i j = �i j into yp, is used [1]. The use of both y∗

p and yphere can be considered similar to the use of a yield criterion and a plastic potential, respectively,for the definition of a non-associated flow rule in conventional plasticity.

In the damage criteria (3) and (4), the total stress tensor �i j is decomposed into two parts�+i j and �−

i j corresponding to tension and compression, respectively (see [13, 14]). The damage-induced softening processes are governed by two functions F t

1 and Fc1 , the forms and parameter



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

F t1 = (1 − �cd)

f ′2t

2E

[E + Ept(1 − �td)

nt

E(1 − �td) + Ept(1 − �td)nt

]2(9)

and

Fc1 = f ′2

c

2E

{E + Epc(1 − �cd)

nc[ln(1 + �cd)]mc

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

}2(10)

Besides material properties such as Young modulus E , Poisson’s ratio �, uniaxial tensile andcompressive strengths f ′

t and f ′c, the following parameters are used in the above functions: Ept

and nt for tensile damage, and Epc, mc and nc for compressive damage. The physical meaning ofthese 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 = fcy fty3

(11)

� = fcy − fty3

(12)

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

fcy = ( fc0 + Hc�cp)(1 − �cd) (13)

fty = ( ft0 + Ht�tp)(1 − �td)(1 − �cd) (14)

In the above expressions, the two initial yield thresholds are denoted as fc0 = 0.3 f ′c and ft0 = f ′

tfor uniaxial compression and tension, respectively. Linear hardening rules are assumed here, withHt>0 and Hc>0 being the hardening moduli and �tp and �cp being corresponding accumulatedplastic strains in tension and compression [1]. We rewrite the definitions of �tp and �cp here:

�tp = cF t4(�kk)

√�i j �i j (15)

�cp = cFc4 (�kk)

√�i j �i j (16)

where c= √2/3 and

F t4 =min

(1,

⟨�kk + fcyfcy + fty

⟩)(17)

Fc4 =min

(1,

⟨fty − �kkfcy + fty

⟩)= 1 − F t

4 (18)

The determination of all parameters of the proposed coupled damage–plasticity model, based onfracture properties of the material and widths of the fracture zones, has been presented in Part Iof this paper [1] in which further details on the model can be found.



3. NON-LOCAL REGULARIZATION

To deal with softening-related problems, the local constitutive model described in the precedingsection needs to be regularized. The regularization can be based on any of several methods rangingfrom 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 byJirasek 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 damageapproaches 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, withdamage-induced softening in the constitutive model, the two damage criteria (3) and (4) shouldbe treated as non-local criteria. Therefore, the regularization in this case is realized through thenon-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 thenon-local softening processes. The thermomechanical aspects of this non-local regularization havebeen 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(‖y − x‖) (1 + pt)�+i j�


2E(1 − �td)2

dV − F t1(�

td, �

cd) = 0 (19)

ycd = 1

G(x)

∫Vd

g(‖y − x‖) (1 + pc)�−i j�

−i j − pc〈−�kk〉〈−�ll〉

2E(1 − �cd)2

dV − Fc1 (�cd) = 0 (20)

where all the stress terms in Equations (19) and (20) are evaluated at point y which lies inside asphere (or circle in two dimensions) of volume Vd, centre x and radius R;G(x)= ∫

Vdg(‖y − x‖) dV

is used to normalize the weighting scheme applied to the energy-like terms in (19) and (20); andg(‖y − x‖) is a bell-shaped weighting function defined by

g(r) = g(‖y − x‖) =

⎧⎪⎨⎪⎩0 if r>R(1 − r2

R2

)2

if r�R(21)

The volume Vd here is defined by the non-local interaction radius R. For an interpretation of thelength 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 canbe 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). Itshould be noted here that the non-locality of damage in (19) and (20) also helps to prevent the



localization of both damage and plastic strain into zones of infinitesimal volumes, as encounteredin local models. Due to this non-locality, damage at a material point, once activated, will alsotrigger damage at neighbouring material points, reducing the yield thresholds at those neighbouringpoints (Equations (13) and (14)). Plasticity at the neighbouring damaged points can therefore takesplace at the same time as damage (e.g. in uniaxial tension), or can be activated once reduced yieldthresholds are met. With the focus here on the constitutive behaviour of the model at this stage ofthe development, an analytical proof that the non-local model achieves these regularization featuresis 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 localstress–strain behaviour of the model strongly depends on the imaginary widths wt and wc of theFPZs in tension and compression, respectively. Unlike in smeared crack models, where wt and wccan 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 stress–strain behaviour of the model. It is therefore difficult, even impossible,to determine analytically the relationship between wt (or wc) and the non-local interaction radiusR in non-local models, except in some simple cases with a linear softening law (e.g. see [23–25]).For that reason, a condition on the equivalence between the energy dissipated by a non-local

model with imaginary crack bandwidth wt (or wc in compression) and that specified by the fractureenergy GF (or Gc) is used in this study. Non-local numerical analyses of a simple bar under uniaxialtension (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 calibrationof 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 morecomputationally expensive) calibration methods (e.g. those based on numerical inverse analysisand 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 materialcharacteristic length.

A summary of the models and parameters used in the non-local model described above can befound in Nguyen [10]. Those parameters are not independent, but interact closely. They can beclassified 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 oncalibration 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



constitutive models can be derived from the full version by dropping out appropriate terms inthe constitutive equations and setting appropriate values for some model parameters, e.g. settingHt = 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 nowpresented. A new integration scheme is proposed for the integration of the non-local incrementalconstitutive equations derived from Sections 2 and 3. Local arc-length control based on thealgorithm 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 integrationscheme for non-local incremental constitutive equations is presented here; the implementation ofthe local arc-length control is given by Nguyen [10].

Integrating the incremental constitutive equations is a vital part of the numerical implementationof 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 integrationmethod, 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 numericalmethods. The analysis is, however, more complex than in models where the evolution laws ofdamage are explicitly defined (e.g. in models by Peerlings et al. [30], Peerlings [31], Jirasek andPatzak [32] and Jirasek et 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 valueproblem. The first discretization is required for the numerical solution of the partial differentialequations, 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, asthe integration of the constitutive equations can be carried out pointwise. For the implementationused here, the same discretization scheme is used for the solution of governing partial differentialequations and the integration of non-local constitutive equations. This results in the non-localevaluation of energy-like quantities in the two damage criteria over the Gauss points used in thefinite element discretization.

For the numerical analysis using finite elements, the integrals in (19) and (20) are transformedto summations over Gauss points. Denoting by the energy-like quantity to be averaged, we canexpress its corresponding non-local counterpart as [10]:

(x) =∑n

e=1∑me

i=1 wei g(‖yei − x‖) det Jei (yei )∑n

e=1∑me

i=1 wei g(‖yei − x‖) det Jei

=n∑

w (22)

in which e denotes the element index and n the total number of elements inside the interactionvolume Vd defined by a sphere at centre x and of radius R; i is the i th Gauss point of elemente; me is the number of Gauss points of this element inside the interaction volume; we

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



number of Gauss points inside the interaction volume at point x; w and are, respectively, theweight and energy-like quantity associated with the i th Gauss point of element e, in which w isdefined by

w = wei g(‖yei − x‖) det Jei∑n

e=1∑me

i=1 wei g(‖yei − x‖) det Jei

(23)

An implicit Euler integration scheme is adopted for the integration of the incremental constitutiveequations. However, due to the presence of the spatial integrals in the damage criteria, the stressupdate procedure cannot be carried out pointwise as is normal in local models. Non-locality inthis case turns the pointwise-defined stress–strain constitutive equations to a system of non-localcoupled equations, relating the stresses, strains and internal variables at several integration pointsin the failure region. This coupling makes the stress update routine more complicated, requiringconsiderable effort in the formulation and implementation as well as a time cost in the numericalcomputation.

As the constitutive relationships in this case contain coupled equations relating the stress andstrain increments at several integration points, finding the intersection points between the stresspath and the yield/damage surface is almost impossible. To avoid the need for this, an elasticpredictor–damage–plastic corrector integrating scheme is adopted here for the integration of theequations. This is based on the algorithm proposed by Crisfield [8 (Chapter 6, vol. 1)] and canbe considered as a form of the backward-Euler integration scheme [8 (Chapter 6, vol. 1)]. Thisalgorithm uses the normal at the elastic trial point and hence avoids the necessity of computing theintersection between the elastically assumed incremental stress vector (using secant elastic–damagebehaviour) and the yield–damage surfaces. Furthermore, the method enforces yield (2) and damagecriteria (19), (20) at any stage of the loading process, thus removing the inaccuracies encounteredwhen 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 takingthe first-order Taylor expansion of the yield and damage functions about the elastic trial pointB (Figure 1, with y in the figure representing either the yield or compressive/tensile damagesurface). 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 andHoulsby [1]) on the composite yield–damage surface is automatically taken into account. Bydropping appropriate terms, we can straightforwardly treat simpler situations, in which only one ortwo 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)]).



governing the constitutive behaviour of the model can be written:

d�i j = (1 + �)d�i j − � d�kk�i jE(1 − �td)(1 − �cd)

+ d�i j

+ (1 + �)�i j − ��kk�i jE(1 − �td)

2(1 − �cd)d�td + (1 + �)�i j − ��kk�i j

E(1 − �td)(1 − �cd)2d�cd (24)

y∗p = y∗

p |B + �y∗p

��i jd�i j + �y∗

p

��tdd�td + �y∗

p

��cdd�cd + �y∗

p

��tpd�tp + �y∗

p

��cpd�cp (25)

ytd|C = ytd|B+∑

w

⎡⎢⎢⎢⎢⎣(1+pt)�

+i j d�i j − pt〈�kk〉�i j d�i j

E(1 − �td)2

+ (1+pt)�+i j�


E(1 − �td)3

d�td

⎤⎥⎥⎥⎥⎦−�F t1

��tdd�td−

�F t1

��cdd�cd = 0 (26)

ycd|C = ycd|B +∑

w

⎡⎢⎢⎢⎢⎣(1 + pc)�

−i j d�i j + pc〈−�kk〉�i j d�i j

E(1 − �cd)2

+ (1 + pc)�−i j�

−i j − pc〈−�kk〉〈−�ll〉

E(1 − �cd)3

d�cd

⎤⎥⎥⎥⎥⎦− �Fc1

��cdd�cd = 0 (27)

The spatial integrals in Equations (26) and (27) obtained from two damage criteria have beenreplaced by summations over integration points. The terms y∗

p |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 herethat all the derivatives and terms in Equations (25)–(27) are evaluated at this stress point. Due tothe appearance of the summations in (26) and (27), it should also be noted that B for these twodamage 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 addingelastic 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

d�i j = �p�yp��i j

(28)

and the definitions of d�tp and d�cp in (15)–(18), the plastic strain rate d�i j is obtained as

d�i j = �p�yp��i j

= −

⎡⎢⎢⎢⎢⎢⎣y∗p |B + �y∗

p

��kld�kl + �y∗

p


p

��cdd�cd

c

√�yp��mn

�yp��mn

(F t4(�kk)

�y∗p

��tp+ Fc

4 (�kk)�y∗

p

��cp

)⎤⎥⎥⎥⎥⎥⎦

�yp��i j

(29)

Secondly, back substituting the above plastic strain increment into (24), some manipulationleads to the relationship between the stress, strain and internal variables, written in incremental



form as follows:

d�mn =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

d�i j +

(y∗p |B + �y∗

p


p

��cdd�cd

)�yp��i j

c

√�yp��kl

�yp��kl

(F t4

�y∗p

��tp+ Fc

4

�y∗p

��cp

)

−[(1 + �)�i j − ��kk�i j ]d�tdE(1 − �td)

2(1 − �cd)− [(1 + �)�i j − ��kk�i j ]d�cd

E(1 − �td)(1 − �cd)2

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[Dst

i jmn]−1 (30)

where Dsti jmn is the constitutive matrix, which is tangent with respect to plasticity and secant with

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

Dsti jmn = Di jmn

(1 − �td)(1 − �cd)−

�yp��i j

�y∗p

��mn

c

√�yp��kl

�yp��kl

(F t4

�y∗p

��tp+ Fc

4

�y∗p

��cp

) (31)

where Di jmn is the elastic compliance matrix.Finally, substituting (30) into (26) and (27), we obtain a system of linear equations in d�td and

d�cd. This system is written for all integration points in the FPZ, each pair (tension and compression)of which has the following form:

ytd = ∑

wQt − �F t1

��tdd�td − �F t

1

��cdd�cd + ytd

∣∣∣∣∣B

= 0 (32)

ycd = ∑

wQc − �Fc1

��cdd�cd + ycd

∣∣∣∣∣B

= 0 (33)

where Qt and Qc are defined as follows:

Qt =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(1 + pt)�+kl − pt〈�qq〉�kl

E(1 − �td)2

[Dsti jkl ]−1

(y∗p |B + �y∗

p


p

��cdd�cd

)�yp��i j

c

√�yp��mn

�yp��mn

(F t4

�y∗p

��tp+ Fc

4

�y∗p

��cp

)

− (1 + pt)�+kl − pt〈�qq〉�kl

E(1 − �td)2

[Dsti jkl ]−1 [(1 + �)�i j − ��qq�i j ]d�td

E(1 − �td)2(1 − �cd)

− (1 + pt)�+kl − pt〈�qq〉�kl

E(1 − �td)2

[Dsti jkl ]−1 [(1 + �)�i j − ��qq�i j ]d�cd

E(1 − �td)(1 − �cd)2

+ (1 + pt)�+i j�


E(1 − �td)3

d�td

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(34)



and

Qc =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(1 + pc)�−kl + pc〈−�pp〉�kl

E(1 − �cd)2

[Dsti jkl ]−1

(y∗p |B + �y∗

p


p

��cdd�cd

)�yp��i j

c

√�yp��mn

�yp��mn

(F t4

�y∗p

��tp+ Fc

4

�y∗p

��cp

)

− (1 + pc)�−kl + pc〈−�pp〉�kl

E(1 − �cd)2

[Dsti jkl ]−1 [(1 + �)�i j − ��pp�i j ]d�td

E(1 − �td)2(1 − �cd)

− (1 + pc)�−kl + pc〈−�pp〉�kl

E(1 − �cd)2

[Dsti jkl ]−1 [(1 + �)�i j − ��pp�i j ]d�cd

E(1 − �td)(1 − �cd)2

+ (1 + pc)�−i j�

−i j − pc〈−�kk〉〈−�ll〉

E(1 − �cd)3

d�cd

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(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 elastictrial point B [8 (Chapter 6, vol. 1)]. Solving the above system of equations (32)–(33) will givethe damage increments at all points in the damaged zone of the structure. Back substituting thedamage increments into (30), noting that the strain increments have been applied in the predictorstep (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 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(y∗p |B + �y∗

p

��td��td + �y∗

p

��cd��cd

)�yp��i j

c

√�yp��kl

�yp��kl

(F t4

�y∗p

��tp+ Fc

4

�y∗p

��cp

)

−[(1 + �)�i j − ��kk�i j ]��tdE(1 − �td)

2(1 − �cd)− [(1 + �)�i j − ��kk�i j ]��cd

E(1 − �td)(1 − �cd)2

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[Dst

i jmn]−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 processbut 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 loadingsurfaces. Mathematically, this is an iterative Newton–Raphson process to return the stress at Bto loading surfaces. This simple but rather efficient stress return algorithm is advocated here due



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 thecombination of sub-incrementation and the repetition of the above predictor–corrector processes,all of which have been implemented in the finite element code OXFEM. Furthermore, to reducethe errors in updating the stresses, in this study the integration scheme above was carried out basedon 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 damage–plasticity model.Analyses of example structures are carried out to show the features of the model. These testsrange from simple to complex loading cases, and hence require appropriate choice of constitutivemodels.

The structural tests used here comprise those exhibiting important features in the behaviour ofquasi-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 cyclicloading. The material models used can be pure tensile damage or tensile–compressive damagewith plastic deformation being accounted for whenever the experimental cyclic loading data areavailable. The calibration procedures for parameters of non-local models, proposed in Nguyen andHoulsby [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 incrementalanalysis, and Newton–Raphson method for the iterative technique. Finite element meshes of six-node triangular elements are used in all examples. The convergence tolerance parameter is 10−4

for the norm of the out-of-balance force vector in the Newton–Raphson iterative process. The sametolerance is used in the stress update routine to gauge the errors occurring in returning the stresses tothe 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 thelocal 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 andpost-peak stages) is used, in which the controlling minimum and maximum numbers of iterationsfor the last two stages are normally 12–18 and 18–27, respectively.

To achieve convergence in severe cases with snap-back observed in the equilibrium paths, theconstraint 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 istotally due to damage, the local secant stiffness matrix is used. For the Newton–Raphson iterativemethod, the stiffness matrix is only recomputed at the beginning of the load increment, and keptunchanged 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 onhow to obtain these model parameters can be found in Nguyen [10].



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) 18.0f ′t (N/mm2) 2.0 (or 2.86) 3.33 3.0 3.55 4.08f ′c (N/mm2) 38.0fc0 (N/mm2) 11.0R (mm) 12.0 16.0 6.0 25.0 2.0wt/R 2.02 (1.98) 1.96 2.32 1.86 2.0Ept (N/mm2) 3090.0 (9081.0) 6899.0 6381.0 14 129.0 1988.0nt 0.26 (0.36) 0.32 0.33 0.37 0.28Ht (N/mm2) 33 800.0wc/R 2.2Epc (N/mm2) 6000.0mc 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-nodetriangular finite elements (Figure 2(c)), with prescribed vertical displacements on the top edge ofthe specimen.

The numerical results are depicted in Figure 3, showing the agreement in the load–displacementcurves obtained from different finite element meshes, thus proving the lack of mesh dependence



0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2Prescribed displacement (mm)

Lo

ad (

kN)

ExperimentalNumerical, mesh 1, ft=2.0MPaNumerical, mesh 2, ft=2.0MPaNumerical, mesh 3, ft=2.0MPaNumerical, mesh 1, ft=2.86MPa

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 specimen—numerical results: (a) load–displacement curves and (b) damagepattern (mesh 3, prescribed displacement of 0.19mm).

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 byits two numerical counterparts (0.94 and 1.26 kN) corresponding to two used values of the tensilestrength. In addition, the overall shape of the numerical load–displacement curves is consistentwith the experiment. The failure of the specimen can be seen in Figure 3(b). No clear macro-crackcan 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 interactionradius could be used, at higher computational cost, if the observation of crack propagation isimportant (see [10]).

6.2. Three-point bending test—notched 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 propertiesare taken from the experimental test of Petersson [35], with the effect of the beam’s 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 formof load–deflection curve and damage pattern, are shown in Figure 6. The damage process zone



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

Lo

ad (

N)

Deflection (mm)

Exp., GF=115N/mExp., GF=137N/mNum., mesh a, GF=124N/mNum., mesh b, GF=124N/m

Figure 6. Load–deflection 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 experimentalone in Figure 4. The numerical load–deflection curves obtained from the two meshes are almostidentical, 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 test—notched beam—cyclic loading

In this example, the four-point bending test experimentally performed by Hordijk [36] is simulatedusing the coupled damage–plasticity model. The geometry of the specimen and finite elementmeshes of a half-beam model are shown in Figure 7.

The numerical load–deflection curve overestimates the experimental one in the post-peak regionnear peak load (Figure 8(a)), while it underestimates the tail behaviour (Figure 8(b)). However, the



50

10030

P

500mm

P

150 150 150

Crack path

Figure 7. Four-point bending test—geometry and finite element meshes.

numerical peak load is rather close to the value given by experiment (Figure 8). The permanentdeformations at zero stress state are clearly seen in Figure 8, showing the capability of the modelin 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 independenceof the obtained numerical results.

In Figure 8(c), we can see the zones of damage and plastic strain along with their incrementsat 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 damagemechanisms. In other words, the reduction of the yield thresholds due to the non-locality of damagewill also result in the activation of plasticity at neighbouring points of the material point underconsideration. Therefore, the accumulated plastic strain defined in (15) also spreads out, realizingthrough 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 schemeand solution methods in dealing with highly nonlinear structural response. The geometrical datafor 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 = 210GPa, � = 0.3) (Figure 10). The analyses were carriedout with the applied load under indirect control, based on the relative displacements between nodalpoints of elements in the fracture zone (local control). The arc-length method was used for theincremental analysis and the Newton–Raphson method for the iterative equilibrium. This helps tocapture effectively the snap back of the load–displacement curve.

Snap-back behaviour can be clearly seen on the load–displacement curves in Figure 11(b), withthe vertical displacement taken at point A on the upper side of the beam and under the steel loaddistributor (Figure 9). The difference in the load-displacement curves from the analyses using twomeshes comes mainly from the difference in size of the modelled load distributor. This is onlya local effect, and the overall structural responses in Figure 11(a) are almost identical and notsignificantly affected by the size of the load distributors.



0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2 0.25

Lo

ad (

kN)

0

0.5

1

1.5

2

2.5

3

Lo

ad (

kN)

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

Numerical, mesh 1

Numerical, mesh 2Experimental Numerical, mesh 1

Numerical, mesh 2Experimental

0 0.1 0.2 0.3 0.4 0.5

X

(a) (b)

(c)

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

There is however a poor match between the experimental and numerical load–CMSD curves;and this has also been found in other research dealing with this mixed-mode test [38, 39]. It canbe 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 testmeans 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 thanits experimental counterpart.



P

306

156

306

82

203 203396 61 61 396

0.13PObserved crack pathA

Figure 9. Four-point shear test—geometrical 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)

Lo

ad P

(kN

)

Lo

ad P

(kN

)

Mesh 1Mesh 2

Mesh 1Mesh 2

Experimental, lowerExperimental, upper

0

20

40

60

80

100

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

Figure 11. (a) Load–CMSD responses and (b) load–displacement curves.



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

Figure 13. Four-point shear test: crack pattern at CMSD of 0.26mm (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 tensilestrength 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 = 75mm, B = 50mm and thefollowing widths of the load bearing strip: b1 = 0.08D= 6mm (numerical test 1; named STP75-8in [40]) and b2 = 0.16D= 12mm (numerical test 2; STP75-16 in [40]).

In both tests, the results obtained from the two finite element meshes are almost identical, provingthe mesh independence of the proposed non-local model. However, the peak loads obtained inneither of the numerical tests agree well with the experimental data. In addition, only the numericalload–displacement 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 andthe peak load only changes insignificantly. For the second test (b/D= 0.16), oscillation in theload–displacement response can be observed, which has also been found in other numerical resultsfor this kind of splitting test [43, 44].

The paper by Rocco et al. [40] does not provide any results on the load–displacement curvesor the crack pattern of the specimen. In this study, the damage contours obtained from numericalanalysis are presented to illustrate the failure processes in the specimen at different loading



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)

Lo

ad (

kN)

0

5

10

15

20

25

30

Lo

ad (

kN)

Num., Comi & Perego (2001)Num., Mesh 1, Gc=18N/mmNum., Mesh 2, Gc=18N/mmNum., Mesh 2, Gc=10N/mmExp. peak load, b/D=0.08

B

A

Num., Mesh 1Num., Mesh 2Num., 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. Load–displacement 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 andcompressive damage mechanisms, which are coupled in this example. The damage zones in bothtests are depicted in Figures 16 and 17, where the splitting effect can be clearly seen in Figure 17



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 atthe 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 damage–plasticity modelemploys two independent length parameters in tension and compression. All parameters of thismodel are shown to be identifiable and can be calibrated based on the experimental data onconcrete, 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 nonlinearfinite element analysis. This implicit integration scheme for non-local rate constitutive equationswas shown to be stable through the simulation of real structural tests in Section 6. This stabilityis 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 damage–plasticity 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 computationalaspects of the proposed non-local model were only briefly considered in this study, with theproposal of an integration scheme for the stress update routine. In addition, use of a local stiffnessmatrix, instead of a non-local stiffness matrix consistent with the stress update algorithm, in the



numerical analysis significantly increased the computational costs. Further research will focus onthe thermodynamic, regularization and computational aspects of non-local constitutive modelling,and the incorporation of anisotropic behaviour to appropriately predict failure mode and orientationof 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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A coupled damage–plasticity model for concrete based on thermodynamic principles: Part II:...

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