Importante Per Displacement-based Damage Model II

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Damage analysis of concrete structures using polynomial wavelets

C.M. Silva 1 , L.M.S.S. Castro

Departamento de Engenharia Civil e Arquitectura, Instituto Superior Tcnico, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

Article history:Received 7 January 2012

Accepted 13 February 2012Available online 17 March 2012

Keywords:Continuum damage mechanicsNon-local integral modelsPolynomial waveletsHybrid-mixed stress modelsFinite elementsConcrete structures

a b s t r a c t

This paper presents and discusses a hybrid-mixed stress nite element model based on the use of poly-nomial wavelets for the physically non-linear analysis of concrete structures. The effective stress and the

displacement elds in the domain of each element and the displacements on the static boundary areindependently approximated. As none of the fundamental equations is locally enforced a priori, thehybrid-mixed stress formulation enables the use of a wide range of functions. In the numerical modelreported here, all approximations are dened using complete sets of polynomial wavelets. These basespresent some important features. In one hand, the functions are orthogonal, which is an important issuewhen implementing hybrid-mixed stress elements as it ensures high levels of sparsity. On the other hand,the polynomial wavelet basis is dened through linear combinations of Legendre polynomials. This factenables the use of closed-form solutions for the computation of the integrations involved in the denitionof all linear structural operators. A simple isotropic damage model is adopted and a non-local integral for-mulation where the strain energy release rate is taken as the non-local variable is considered. The numer-ical model is both incremental and iterative and is solved with a modied NewtonRaphson method thatuses the secant matrix. Classical benchmark tests are chosen to illustrate the use of the model under dis-cussion and to assess its numerical performance.

2012 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction

The main goal of this paper is to present and discuss a hybrid-mixed stress nite element model based on the use of polynomialwavelets for the physically non-linear analysis of concrete struc-tures. The hybrid-mixed model used in this work was rst devel-oped by Freitas et al. [1] during the 1990s. In recent years, thisnon-conventional nite element formulation has been extendedto non-linear analysis using isotropic damage models [25] .

In [2,6] the hybrid-mixed stress model based on the use of orthonormal Legendre polynomials [7] is used. The stress and thedisplacement eldsin thedomain of each element andthe displace-ments on the static boundary are independently approximated.None of the fundamental relations is enforced a priori and all eldequations are enforced in a weighted residual form, ensuring thatthe discrete numerical model embodies all the relevant propertiesof the continuum it represents. The Mazars isotropic model [8] isadopted and a non-local integral formulation where the damagevariable is taken as the non-local variable is considered.

In [9,3] an improved hybrid-mixed stress model is presentedand discussed. The approximation of the stress eld in the domain

is here replaced by the approximation of the effective stress eld.The isotropic damage models presented by Comi and Perego[10,11] are now adopted using a non-local integral model. An alter-native technique based on the denition of an explicit enhancedgradient model has also been tested [4] .

The objective of this paper is to use complete sets of polynomialwavelets to dene the approximation bases for the effective stresseld required by the hybrid-mixed stress model presented in [9,3] .This orthogonal wavelet basis, introduced by Frolich and Uhlmann[12] , is dened using a linear combination of Legendre polynomialswhere the expansion coefcients are taken as roots of theChebyshev polynomials of the second kind. For the approximationof the displacement elds, both in the domain and on the staticboundary, complete sets of orthonormal Legendre polynomialsare adopted.

This wavelet system has been selected due to the properties itpresents. In one hand, the functions are orthogonal, which is animportant issue whenimplementing hybrid-mixed stress elementsbecause it ensures high levels of sparsity in the global governingsystem. On the other hand, the polynomial wavelet basis is denedthrough linear combinations of Legendre polynomials. This fact en-ables the use of closed-form solutions for the computation of theintegrations involved in the denition of all linear structural oper-ators by following the techniques discussed in [13] . Numericalintegration schemes can be fully avoided, with clear advantagesboth in terms of accuracy and numerical performance.

0965-9978/$ - see front matter 2012 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.doi: 10.1016/j.advengsoft.2012.02.009

Corresponding author. Tel.: +351 218418253.E-mail addresses: [email protected] (C.M. Silva), [email protected]

(L.M.S.S. Castro).1 Tel.: +351 218418356.

Advances in Engineering Software 50 (2012) 6981

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Because the hybrid-mixed stress model used here is built on anaturally hierarchical basis, it can be implemented using coarsemeshes of macro-elements, where the renement is achieved byincreasing the degree of the approximation.

The nonlinear behavior of the structural material is modeledassuming an isotropic continuumdamage model. Continuum dam-age mechanics is an important tool that describes the evolution of

the mechanical properties of the continuum as micro crackingdevelops [1417] . In this present paper, the simple isotropic con-tinuum damage model with only one damage variable introducedby Comi and Perego [10,18] is considered.

A strainsoftening materiallaw is consideredin order to beable tomodel a structural softening behavior. However, strain softening iswell known to produce strain localization with consequent depen-dence on the data of the nite element model, as for instance themesh and the degree of the approximation functions adopted[19,20] . To overcome thisproblem, severalregularization techniquesare proposed in the literature, in particular nonlocal integral [21,22]and gradient-enhanced damage formulations [23,24] . Following[10] , the present work adopts a nonlocal integral model where thestrain energy release rate is adopted as the nonlocal variable.

Only static and monotonic loads are considered and smallstrains and rotations are assumed. The numerical models are bothincremental and iterative and are solved with a modied NewtonRaphson method that uses the secant matrix.

This paper is organized as follows: the formulation of the prob-lem and the adopted damage model are presented in Sections 2and 3 . The non-conventional nite element formulation and theapproximation functions are described in Sections 4 and 5 . Thenumerical examples are shown in Section 6 and nally. Section 7summarizes the main conclusions and indicates future researchwork in this eld

2. Fundamental relations

Consider a domain V limited by the boundary C , referred to acartesian coordinate system. The static boundary C r (or Neumannboundary) and the kinematic boundary C u (or Dirichlet boundary)are complementary sub-regions of the boundary C , whereon trac-tion-resultants and displacements are respectively prescribed.

The body under analysis is assumed to be homogeneous andisotropic. The model is geometrically linear and only static andmonotonic loads are considered. No viscous, thermal or othernon-mechanical dissipative effects are taken into account.

The fundamental equilibrium equations may be written in amatrix form as follows:

Dr b 0 in V ; 1Nr tc on Cr ;where D is the differential equilibrium operator. The matrix N con-

tains the components of the unit outward normal vector to the sta-tic boundary C r . The vector r lists the independent components of the stress tensor. The vector b represents the body force vector inthe domain V and t c corresponds to the tractions vector on thestaticboundary C r .

The compatibility equations may be written in the followingformat:

e Du in V ; 2u u on Cu ;where D is the differential compatibility operator, adjoint of thedifferential equilibrium operator D since the model is geometricallylinear. The vector e collects the independent components of thestrain tensor and the vector u lists the independent components

of the displacement eld. The vector u denotes the prescribed dis-placements on the boundary C u.

The constitutive relation depends on the damage modeladopted, as detailed in Section 3.

3. Non-local damage model

The mechanical behavior of quasi-brittle materials such as con-crete, is characterised by the development of micro-cracks and

subsequent evolution to localized macro cracking. The ContinuumDamage Mechanics models describe the evolution of the mechan-ical properties of the continuum as cracking develops. This type of constitutive models are able to describe, with a continuum ap-proach, some of the material properties observed in experiments,such as global softening, stiffness degradation, anisotropy anddevelopment of inelastic deformations [25,16,15,26,17] .

In this paper, the simple isotropic continuum damage modelwith only one damage variable used e.g. in the works of [14,10,18] is considered. The main characteristics of this damagemodel are summarized in Table 1 .

The strain softening behavior is well known to produce strainlocalization with consequent dependence on the data of the niteelement model, as for instance the dependence on the mesh and on

the degrees of the approximation functions [19,20] . To overcomethis problem, several regularisation techniques are proposedin the literature, in particular non-local integral [21,22] andgradient-enhanced damage formulations [23,24] . FollowingPijaudier-Cabot and Baz ant [21] , the present work assumes anon-local approach, where strain energy release rate is adoptedas the non-local variable.

As dened by Pijaudier-Cabot and Baz ant in [21] , a generic non-local variable v is computed considering the following weightedaverage over the whole domain:

v x Z V W x; sv sds;where v corresponds to the original local variable and W ( x,s) is aweight function taken here as the normalized Gauss function:

W x; s 1

W 0 xexp

k x sk2

2l2 !;W 0 x Z V exp k x sk22l2 !ds:The length l in the previous equation is a geometric length, usuallydenoted as characteristic length . It worksas a localization limiter and

Table 1

Continuum damage model with one scalar variable.

Model 1 [14,10,18]

Damage variable dHelmholtz free energy density a W 12 1 de

t Ee Win nwith

Win n k1 nPni0 n !i! ln i c 1n State equations b r @ W@ e 1 dEe

v @ W@ n W0in n

Y @ W@ d 12

e t EeActivation function f Y v Y v 12 e

t Ee vEvolution laws c _d @ f @ Y _c _c

_n @ f @ v _c _cKuhn Tucker conditions f 6 0; _d P 0; _df 0Non-local variable Y 12 e

t Ee

a n is a scalar internal variable of kinematic nature and variables k, n and c arematerial parameters.

b e , n and d are the state variables and r , v and Y represent the correspondingassociated variables.

c For this particular damage model, the internal variable n coincides with thedamage variable d.

70 C.M. Silva, L.M.S.S. Castro / Advances in Engineering Software 50 (2012) 6981


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regularises the mathematical problem. According to [27] , thislength may also be interpreted as a material-dependent parameterrelated to the width of the fracture process zone. A normalizedweight function is chosen because the non-local model should beable to reproduce correctly local uniform elds.

The damage model adopted in this work presents the limitationof considering the same behavior for the material in prevailing ten-sion and compression states, which is not realistic for most of thematerials. To overcome this limitation, it is assumed that damagemay only appear and develop if the strain tensor trace is positive,tr e > 0. The constitutive model with the referred assumption issuitable for studying structures subjected mainly to tension stres-ses and it is competitive due to its simplicity.

4. Hybrid-mixed stress formulation

The hybrid-mixed stress (HMS) formulation adopted in thiswork was for the rst time described by Silva in [3] . Comparedto the original version of the hybrid-mixed stress formulation [1]the particularity of the new model is that the approximation of the stress eld r is replaced by the approximation of the effectivestress eld ~r , dened e.g. by Lemaitre in [14] . The approximationsmay be expressed as:

~r Sv

eX in V ; 3

u Uv q v in V ;u Ucqc on Cr ;

where the matrices Sv , Uv and Uc collect the approximationfunctions and the vectors eX ; q v and q c list the associated weights(generalized variables). Since the three elds are approximatedindependently, it is possible to adopt different degrees of approxi-mation for each one.

Due to the properties presented by polynomial wavelets, it ispossible to increase the degree of the approximations without hav-ing any problems in terms of numerical stability. This fact enablesthe implementation of highly efcient p-renement procedures, asit is possible to dene high degree approximations without deteri-orating the condition number of the global governing system.

While the concrete is linear elastic, the model proposed coin-cides with the one described by Freitas et al. in [1] . When damageappears, the models are different since the effective stress and thestress eld are no longer coincident. In the context of a non-linearanalysis with softening, the main advantage of the proposed ap-proach when compared to the one described by Freitas et al. [1]is that the effective stress eld is directly related to the evolutionof damage, since it is comparable to the strain eld, while thestress eld is not.

The generalized strains, e , body forces, Q v , and tractions, Q c aredened by

e Z St v dV ; Q v Z Ut v b dV ; Q c Z Ut ct c dCr ; 4in order to ensure the inner product invariance between thepairs of

dual discrete variables eX ; e; qv ; Q v , and ( q c,Q c) and the contin-uum elds they represent.

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

00 (x)

01 (x)

Scaling functions

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 12

1.5

1

0.5

0

0.5

1

wavelet (j=0)

00 (x)

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 13

2.5

2

1.5

1

0.5

0

0.5

1

1.5

wavelets (j=1)

10 (x)

11 (x)

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 16

5

4

3

2

1

0

1

2

wavelets (j=2)

20 (x)

21 (x)

22 (x)

23 (x)

Fig. 1. 1D Polynomial wavelets.

C.M. Silva, L.M.S.S. Castro / Advances in Engineering Software 50 (2012) 6981 71


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As demonstrated in [9,3,4] , using the denition of the effectivestress in the form r ~r 1 d[14] and enforcing the fundamentalequations (Section 2) on average, in the sense of Galerkin, one ob-tains the following equilibrium equations for the discrete model:

At v Mv eX Q v in V ; 5 At c Mc eX Q c on Cr ;where the matrices M v , Mc and Av , Ac are dened as follows:Mv R DUv t Sv ddV R NUv t Sv ddC ; Mc R Ut cNSv ddCr ; Av R DSv t Uv dV ; Ac R NSv t Uc dCr : 6

The compatibility condition in the discrete model (Eq. (7) ) may beobtained integrating by parts the average enforcement of the com-

patibility equation in thedomain and then replacing in theresultingexpression the approximations for the displacements (Eq. (3) ) [9,3] :

e Av q v Acq c e ; with e Z NSv t u dCu : 7The relation between the independent components of the effectivestress tensor and the strain components can be expressed as [3] :

F~r ; 8

where F is the symmetric non-singular matrix of elastic constantscharacterizing a linear reciprocal elastic law.

Introducing the constitutive relation (Section 3) and the gener-alized strains (Eq. (4) ) in Eq. (7) , we obtain Eq. (9) that encom-passes the compatibility and the constitutive relations of thediscrete model:

F

eX Av q v Acqc e ; with F Z S

t v FSv dV : 9

10.5 0

0.51

10.5

0

0.5

1

1.5

2

00 (x) 00 (y)

10.5

00.5

1

10.5

00.5

10.5

0

0.5

1

1.5

2

00 (x)01 (y)

10.5

00.5

1

10.5

00.5

13

2

1

0

1

2

00 (x) 00 (y)

10.5 0

0.51

0.5

0

0.5

1

1.5

2

01 (x) 00 (y)

10.5

00.5

1

10.5

00.5

10.5

0

0.5

1

1.5

2

01 (x) 01 (y)

1 0.50 0.5

1

10.5

00.5

12.5

21.5

1

0.50

0.51

1.5

01 (x) 00 (y)

10.5 0

0.51

10.5

00.5

13

2

10

1

2

00 (x) 00 (y)

1 0.50 0.5

1

10.5

00.5

12.5

21.5

10.5

0

0.51

1.5

00 (x) 01 (y)

10.5

00.5

1

10.5

00.5

12

1

01

2

3

00 (x) 00 (y)

10.5

00.5

1

10.5

00.5

1

Fig. 2. 2D Polynomial wavelets.



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Combining Eqs. (5) and (9) , one obtains the following solving sys-tem for each nite element:

F Av Ac At v Mv 0 0

At c Mc 0 0264

375e

X q v q c

8>:

9>=>;

e

Q v Q c

8>:

9>=>;

: 10

The governing system of the nite element mesh is assembled bydirect allocation of the contribution of the elementary systems [1] .

Because the hybrid-mixed stress nite element model adoptedin this work use macro-element meshes, it is not possible to con-trol the length of the nonlinear strain localization band throughthe nite element mesh, as usually happens in a traditional dis-placement formulation. Consequently, a more rened mesh mustbe chosen to implement the nonlocal integral model. In this work,the Lobatto points mesh is used for this purpose. Since the hybrid-mixed stress model requires the knowledge of the damage evolu-tion on the boundary, the Lobatto quadrature rule is used insteadof the usual Gauss quadrature rule. In order to capture the strainlocalization band, it is necessary to ensure that a convenient num-

ber of Lobatto control points lie inside the process zone, so thenumber of control points must be large.

The algorithm used in the solution of the non-linear governingsystem follows a secant NewtonRaphson method. At load step jthe iterative algorithm can be described by the following steps:

(1) Initialize the variables by setting v0 = v( j1)

(2) Error = 10 tol and iter = 1(3) while Error > tol

(a) iter = iter + 1(b) computation of the non-local variable at each Lobatto

point;(c) validation of the Kuhn-Tucker conditions (see Table 1 ) in

order to dene the new values for the damage variable;

(d) computation of the secant matrix, A;(e) computation of the residual vector, R ;

bar 20mm

500

250 250

110 250 230

pk No.4b=100mm

210 20 20

0 0 5

0 5 2

0 5 2

0 0 2

0 8 2

pk No.1

22 226

100

pk No.3

i f u g

A e r

0 2

pk No.2

dimensions in [mm]

figure A

Fig. 3. L-shaped plate: experimental device [28] .

Fig. 4. L-shaped plate: nite element meshes.

Table 2

Discretizations used in the analysis of the L-shaped plate.

Disc. nelem j pL ndof

A 3 1 3 389B 3 2 7 1449C 10 1 3 1262D 10 2 7 4094



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(a) Discretization A (b) Discretization B

(c) Discretization C (d) Discretization D

Fig. 5. Damage distribution obtained for u 0 :75 mm.

(a) Discretization A (b) Discretization B

(c) Discretization C (d) Discretization D

Fig. 6. ~r yy effective stress distribution obtained for u 0:75 mm.



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(f) solution of the system AD sol = R ;(g) update the value for the generalized variables,

soliter = soliter 1 + D sol;(h) computation of the new value for the controlling

parameter, Error;(4) store the nal value for the generalized variables,

v( j) = viter .

The secant operator A corresponds to the matrix presented in

Eq. (10) . The solution vector v collects the generalized effectivestress parameters, eX , and the generalized domain and static

boundary displacement variables, q v and q c. The residual vectorR is dened according to Eq. (10) .

5. Polynomial wavelets

The orthogonal polynomial wavelet systems are based on thedenition of linear combinations of Legendre polynomials andwere introduced by Frolich and Uhlmann [12] . The roots of the

Chebyshev polynomials of the second kind are used to obtainthe corresponding expansion coefcients. The details concerning

(a) Comparison with experimental data

(b) Comparison with other numerical tools

(c) Solutions obtained with all tested discretizationsFig. 7. Reaction (N)prescribed displacement (mm) diagrams.

(a) Comparison with experimental data

(b) Comparison with other numerical tools

(c) Solutions obtained with all tested discretizationsFig. 8. Reaction (N)horizontal displacement (mm) diagrams.



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Fig. 9. Damage evolution.

Fig. 10. ~r yy Effective stress distribution evolution.



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the construction of these systems of wavelets are presented in[12] .

The scaling functions, / ji( x), and the wavelets, w ji( x), are denedby:

/ ji x C U ji X

2 j

k0

U k y2 j 1

i ffiffiffiffiffiffiffiffiffiffiffiffik 12r Lk x; 11 with j = 0,1, . . . , i = 0, . . . ,2 j, and by:

w ji x C W ji X

2 j 1

k2 j 1

U k y2 ji ffiffiffiffiffiffiffiffiffiffiffiffik 12r Lk x; 12

with j = 0,1, . . . , i = 0, . . . ,2 j1.In Eqs. (11) and (12) , Lk( x) and U k( x) represent the orthogonal

Legendre polynomials and the Chebyshev polynomials of the sec-

ond kind, respectively. The Legendre polynomials can be given bythe recursive expression:

Lk 1 x 2k 1k 1

xLk x k

k 1Lk1 x; 13

with L0( x) = 1 and L1( x) = x.The Chebyshev polynomials of second kind are given by:

U k 1 x 2 xU k x U k1 x; 14

with U 0( x) = 1 and U 1( x) = 2 x.The support of both scaling functions and wavelets is given

by:

supp / ji x supp w ji x 1 ; 1 : 15

The parameters yni used in Eqs. (11) and (12) correspond to thezeros of the nth order Chebyshev polynomial of the second kindand are given by:

yni cos

i 1pn 1

; i 0 ; . . . ; n 1 16

(a) (b)

Fig. 11. Hassanzadehs test [30] : (a) geometry and (b) adopted nite element meshes.

Table 3

Discretizations used in the analysis of the Hassanzadeh test.

Disc. nelem j pL ndof

A 7 1 3 877B 7 2 7 2853C 13 1 3 1631D 13 2 7 5303

(a) Solutions obtained with all tested discretizations

(b) Comparison with other numerical tools andexperimental data

Fig. 12. Hassanzadehs test: reaction (N)prescribed displacement (mm) diagrams.



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The expansion coefcients C U ji and C W ji are dened by:

C U ji ffiffiffiffiffiffiffiffiffiffiffiffiffi22 j 2s sin i 1p2 j 2 17 C W ji ffiffiffiffiffiffiffiffiffiffiffiffiffi22 j 1s sin i 1p2 j 1 18 An unconditional orthogonal basis for L2([1,1]) is dened by thefollowing set of functions:f / 00 x; / 01 x; f w ji xg j0 ; 1 ; ... ; jmax i0 ; 2 j1g 19

According to (19) , 2 j wavelets are dened at a given level of resolu-tion, j. It is not difcult to demonstrate that the total number of functions in the basis dened by (19) is given by n f 2 jmax 1 1.Polynomials with a degree up to 2 jmax 1 are exactly representedusing that basis.

The scaling functions / 00 ( x) and / 01 ( x) and the wavelets withrenement levels ranging from j =0 to j = 2 are represented inFig. 1 .

Tensor products involving one-dimensional bases in each coor-dinate direction lead to the construction of polynomial waveletbases for 2D domains. Fig. 2 presents the nine 2D functions denedat renement level j = 0.

6. Numerical tests

6.1. Analysis of an L-shaped plate

Let us consider the L-shaped concrete plate presented in Fig. 3 .The thickness of the plate is 100 mm and an upward vertical dis-placement, u , at the lowest right corner is prescribed. The experi-mental results and the solutions obtained with several numericalsimulations are presented in [28,29] .

The available experimental data are the Young modulusE = 25,850 MPa, the Poisson coefcient, m= 0.18 and the maximumstrength in tension, f t = 2.70 MPa. The remaining material parame-

ters are dened in order to minimize the differences between theexperimental and numerical load-prescribed displacement dia-grams. According to [29] , the following values have been assumed:n = 9.5, k = 1.1 10 11 MPa, c = 270 and l = 11 mm.

A plane stress behavior is considered and the vertical displace-ment at edge A is prescribed. The two nite element meshes pre-sented in Fig. 4 have been considered. For each mesh, two

different discretizations have been adopted. The rst uses polyno-mial wavelets with resolution level j = 1 to dene the effectivestress eld approximation in the domain of each nite elementand Legendre polynomials of degree pL = 3 to approximate the dis-placement elds, both in the domain and on the static boundary.The second discretization considers polynomial wavelets with res-olution level given by j = 2 to approximate the effectivestress eldsand Legendre polynomials up to degree pL = 7 to approximate alldisplacement elds.

Table 2 lists the main characteristics of these different discreti-zations, namely the number of element in the nite element mesh,nelem , and the total number of degrees of freedom, ndof . In all cases,a (20 20) Lobatto mesh points is used.

Fig. 5 presents the damage distribution obtained by each testeddiscretization for a prescribed vertical displacement given byu 0:75 mm. In the rst {second} row, the results obtained withthe three {10} element mesh are plotted. The results obtained withresolution level j = 1{ j = 2} are presented in the rst{second} col-umn. It is possible to observe that solutions obtained with the low-est levels of resolution (Discretizations A and C) are associatedwith less accurate damage distributions. In these cases, damageappears in regions where it is not supposed to exist, namely alongthe boundary between elements 2 and 3 (Discretization A) andalong the plate boundary (Discretization B). The same conclusioncan be extracted from the analysis of the ~r yy effective stressdistributions presented in Fig. 6 .

In Fig. 7 a the reactionprescribed displacement diagram ob-tained with Discretization D is compared with the set of experimental curves reported in the literature and referred in

Fig. 13. Final damage distribution obtained for u 0:04 mm.



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[29] . It can be veried that the numerical solution is able to capturethe observed global behavior of the structure. Fig. 7 b compares thesolution obtained with Discretization D with the numerical solu-tion provided by the use of a hybrid-mixed stress model basedon the use of orthonormal Legendre polynomials as approximationfunctions [29] . The slight difference existing between both curvesmay be explained by the fact that in the analysis reported in thispaper the vertical displacement is prescribed along the completeright edge of the plate while in the simulation presented in [29]

a point prescribed displacement load has been considered. Thereactionprescribed displacement diagrams obtained with alltested discretizations are presented in Fig. 7 c. The solutions areconsistently rened by increasing the number of elements in themesh or by increasing the renement level of the functions usedto dene the approximation bases.

Fig. 8 presents the evolution for the horizontal displacement atthe upper left corner of the structure. The numerical response isstiffer than the experimental behavior at the beginning of the load-ing procedure. This type of behavior is also observed in othernumerical simulations and can be justied by the fact that thenumerical simulation does not take into account the rotationalstiffness of the steel device that embraces the L-shaped concretestructure.

Fig. 9 presents the damage distribution obtained with Discreti-zation D for the following loading steps: u 0 :125 mm ,

u 0 :25 mm , u 0 :50 mm and u 1 :00 mm. The corresponding~r yy effective stress distributions are presented in Fig. 10 .

6.2. Hassanzadeh test

The second test presented in this paper corresponds to thenumerical simulation of the Hassanzadehs experiment [30] , illus-trated in Fig. 11 a. This numerical test has been used by severalauthors [31,11,10] to assess the behavior of concrete in prevailing

tension mechanisms. Due to the geometry and to the applied load,only the tension mechanism is activated through the loading his-tory. In this paper, the numerical results obtained with the pro-posed model are compared with the experimental results of Hassanzadeh [30] and with the numerical results presented in ref-erence [10] .

Following Comi and Perego [10] , a two-dimensional analysis isperformed and the total vertical reaction is computed in order totake into account the three dimensionality of the problem. Thestructure is analyzed as a strain plane problem using both meshesshown in Fig. 11 b. The symmetry of the problem is not consideredin order to conrm if the model simulates this property correctly.The material parameters of the damage model are n = 12,k = 5.8 10 14 MPa, c = 405, l = 1.6 mm, E = 36GPa and m= 0.15.

Table 3 lists the main characteristics of the discretizations con-sidered in the analysis of the Hassanzadeh problem, namely the

Fig. 14. Hassanzadehs test: damage distribution evolution.



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number of element in the nite element mesh, nelem , and the totalnumber of degrees of freedom, ndof . In all cases, a (20 20) Lobattomesh points is used. As before, two different discretizations havebeen adopted for each mesh. The rst uses polynomial waveletswith resolution level j = 1 to dene the effective stress eldapproximation in the domain and Legendre polynomials up to de-gree pL = 3 to approximate the displacement elds, both in the do-

main and on the static boundary. The second discretizationconsiders polynomial wavelets with resolution level given by j = 2 to approximate the effective stress elds and Legendre poly-nomials up to degree pL = 7 to approximate all displacement elds.

Fig. 12 a presents the reactiondisplacement diagrams obtainedwith the different tested discretizations. Except for the case of Dis-cretization A, all the remaining solutions are quite similar. This factillustrates the objectivity of the numerical solutions obtained andproves that the non-local integral regularization scheme beingadopted is working properly.

Fig. 12 b compares the reactiondisplacement diagram obtainedwith Discretization D with the experimental results obtained byHassanzadeh [30] . It can be easily veried that both the numericaland the experimental solutions are similar. The bump experimen-tally observed in the softening branch is due to rotational instabil-ity and therefore is not present in the numerical simulations. Thesame gure presents the reactiondisplacement curves obtainedby Comi and Perego using classical nite elements [10] and by Sil-va [29] using a hybrid-mixed stress nite element model based onthe use of Legendre polynomials as approximation functions. Inthis last case, a total of 1153 have been considered in the analysis.It is possible to state that all numerical models provide quite sim-ilar and accurate results.

The nal damage distribution obtained with each tested dis-cretization is presented in Fig. 13 . With Discretization A it is notpossible to obtain an accurate nal damage distribution. This factindicates that with this number of nite elements the renementlevel considered is not able to ensure the computation of adequatesolutions, especially at the nal stages of the loading procedure.This can be overcome by increasing the number of elements inthe mesh or by increasing the renement level. Both renementapproaches proved to work as expected, as Discretizations B andC provided accurate nal damage distributions.

For the solution obtained with Discretization D, Fig. 14 presentsthe damage distribution evolution for different values of the pre-scribed displacement. As expected, the damage rst appears nearthe re-entrant corners and then evolves localizing the damagearound the fracture zone.

7. Conclusions

The hybrid-mixed stress model based on the use of polynomial

wavelets proved to be a stable and robust numerical technique forthe physically non-linear analysis of concrete structures using con-tinuum damage mechanics. From the numerical tests reported inthis paper, it is possible to conclude that when using polynomialwavelets to dene the approximation bases for the effective stresselds the use of the lowest level of resolution, j = 1, is notrecommended.

In all performed numerical tests, the quality of the solutionsdoes not depend on the nite element mesh orientation ( meshbias ). This behavior results mainly from the use of macroelementmeshes associated with the implementation of highly effective p-renement procedures. As discussed in [32] , this type of phenom-ena may inuence the quality of the results provided by the classi-cal FEM computations.

The potential associated with the use of polynomial waveletbasis is not completely explored as only uniform renement

was taken into account. This means that all wavelets at all levelsof resolution with j 6 jmax are considered in the basis. To fullyexploit the properties of wavelet systems, adaptive algorithmsbased on non-uniform renement procedures should beimplemented. In these cases, only the wavelets located nearthe regions where the detail is important are necessary toinclude in the approximation basis. The implementation of

such adaptive algorithms is one of the main objectives for thefuture.

Acknowledgements

This research work corresponds to part of the activities of theMechanics, Modeling and Analysis of Structures Group of Institutode Engenharia de Estruturas, Territrio e Construo, ICIST. It hasbeen supported by Fundao para a Cincia e Tecnologia as partof research Program PTDC/ECM/71519/2006.

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