Tension-CompressionDamageModelwithConsistentCrack ...downloads.hindawi.com/journals/ace/2019/2810108.pdf · tension-compression load reversal. e classical strain- driven damage models

Research ArticleTension-Compression Damage Model with Consistent CrackBandwidths for Concrete Materials

Wei He 1 Ying Xu1 Yu Cheng 2 Peng-Fei Jia 2 and Ting-Ting Fu 3

1School of Civil Engineering and Architecture Anhui University of Science amp Technology Huainan China2Jiangsu Transportation Institute Nanjing China3General Station of Construction Project Quality amp Safety for Administration and Supervision Jinhua China

Correspondence should be addressed to Wei He wheausteducn

Received 29 August 2018 Revised 7 November 2018 Accepted 23 December 2018 Published 7 February 2019

Academic Editor Giovanni Garcea

Copyright copy 2019Wei He et alis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper proposes a tension-compression damage model for concrete materials formulated within the framework of ther-modynamics of irreversible processes e aim of this work is to solve the following problems the premature divergence ofnumerical solutions under general loading conditions due to the conflict of tensile and compressive damage bounding surfaceswhich is a result of the application of the spectral decomposition method to distinguish tension and compression and theunsatisfactory reproduction of distinct tension-compression behaviors of concrete by strain-driven damage modelse former issolved by the sign of the volumetric deformation while the latter is solved via two separated dissipationmechanisms Moreover ofspecific interest is an improved solution to the problem of mesh-size dependency using consistent crack bandwidths which takesinto account situations with irregular meshes and arbitrary crack directions in the context of the crack band approach eperformance of themodel is validated by the well-documented experimental datae simplicity and the explicit integration of theconstitutive equations render the model well suitable for large-scale computations

1 Introduction

Concrete is still one of the most widely used constructionmaterials in civil engineering applications [1] ereforedeveloping an efficient constitutive model for concrete is ofgreat importance since it contributes to engineering designsfailure analyses and life-cycle assessments by reducing costsand time [2] Microscopically the mechanical behavior ofconcrete materials is mainly governed by the nucleation andgrowth of microcracks and their coalescence into macrocracks[3] which may result in the failure of a structure Physically itis an irreversible thermodynamic process accompanying en-ergy dissipation and changes of the microstructure [4] Such aprogressive degradation of material properties can be phe-nomenologically described by models [5ndash17] based on con-tinuum damage mechanics (CDM) which is generallyaccepted by the scientific and engineering communities [18]

Despite the achievements of CDM over a wide range ofmaterials (eg [19ndash22]) the rational and accurate CDM

modeling of concrete materials still represents a challengingtask ree modeling challenges have been identified in finiteelement (FE) analyses e first challenge is the criteriondistinguishing tension and compression which is used foractivatinginactivating damage evolutions in tension andcompression e existing models usually utilize the spectraldecomposition of a stress or strain tensor into tensile (pos-itive) and compressive (negative) eigenvalues (eg [23 24])in order to describe different behaviors of concrete undergeneral loading conditions However in the use of thismethod the following problem would arise divergence ofnumerical solutions will occur under biaxial or reverseloading conditions due to the conflict of tensile and com-pressive damage bounding surfaces induced by Poissonrsquos ratioeffect [25] A better solution to this problem is the in-troduction of a weighted damage parameter (eg [5 25])Moreover the sign of the volumetric strain ([26]) is used todistinguish what is tension and what is compression How-ever it should be noted that the aforementioned method is

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 2810108 16 pageshttpsdoiorg10115520192810108

not applicable to incompressible materials which have a valueof Poissonrsquos ratio around 05 (eg fresh concrete andconcrete under confinement) since the volumetric strain isnull erefore the present model is limited to the com-pressible materials characterized by Poissonrsquos ratio lowerthan 05

e second challenge is the modeling of tension-compression behaviors of concrete materials which ex-hibit large differences in their peak strengths and postpeakstages as well as the unilateral effect characterized by a totalor partial recovery of the degraded stiffness during thetension-compression load reversal e classical strain-driven damage models incorporating one damage crite-rion alone may be incapable of predicting such distinctbehaviors Among them (eg [27ndash32]) one case can bementioned as an example the predicted strength ofconcrete under biaxial compression is seriously lower thanthat under uniaxial compression To solve the problemthis paper proposes two separate damage criteria de-scribing the state of damage under tension and compres-sion respectively whereby the distinct behaviors can bereproduced in a physically realistic fashion Due to thescope of this paper the unilateral effect will not bediscussed

e third challenge is to avoid the lack of meshobjectivity of numerical results when the modeling ofstrain-softening is taken into account At the stage ofstrain-softening the concrete does not lose its entire load-carrying capacity but exhibits a progressively decreasingresidual strengtherefore capturing the strain-softeningis important because it is responsible for predicting thebehavior of overloaded structures However damagemodels based on conventional continuum theories usuallyproduce mesh-dependent results in FE analyses [6] emesh dependency can be further classified into mesh-size[33] and mesh-bias [34] dependencies e mesh-sizedependency which originates from the loss of ellipticityof the governing partial differential equations for staticproblems [35] which means that both the computed en-ergy dissipation due to strain-softening and the corre-sponding force-displacement response are governed by thefineness of the FE mesh [36] It also reflects a physicallyunrealistic fact that the infinitesimal element size leads to avanishing energy dissipation during the structural failure[37] e mesh-bias dependency which is a result of theinadequacy of the spatial discretization procedure [38 39]means that the numerical result depends on the orientationof the FE mesh in the sense that the computed direction ofthe strain localization band tends to propagate alongcontinuous mesh lines [40] which may give rise to spu-rious crack trajectories Although the problem of mesh-bias dependency is important for damage modeling ofconcrete materials this study focuses only on the solutionto the mesh-size dependency

To solve the problem of mesh-size dependency in theframework of continuum mechanics various types of regu-larization methods have been developed that can be distin-guished by the crack band approach (or fracture energyregularization) [33 41 42] nonlocal models including the

integral-type [6 43 44] and the gradient-type [13 45]micropolar method [46] and viscous or rate-dependent [11]methods On one hand the nonlocal method seems to bemore recent (eg [28 30 47]) However the nonlocal modelshave their own difficulties such as (i) incorrect crack initia-tion ahead of the crack tip [48] and (ii) incorrect shieldingeffect with nonzero nonlocal interactions across a cracksurface [49] Furthermore nonlocal methods have no po-tential for large-scale computations of concrete structuressince they require small enough elements eg their sizes areat least one-third of the characteristic length of the material[50] to subdivide strain localization bands of the structurewhich are usually a priori unknown On the other hand thecrack band approach is widely used inmost of the commercialFE packages and is still an important analytical tool forpractical applications (eg [51 52]) One goal of the presentwork is to propose a constitutive model which needs to ensurea high algorithmic efficiency and an easy implementationtherefore a fracture energy regularization based on the crackband approach [33] is adopted

One key parameter in the crack band approach is thecrack bandwidth through which the fracture energy(regarded to be a material property) is related to the vol-umetric (or specific) fracture energy and it therefore ensuresan objectivity of the energy dissipation regardless of themesh refinement e existing estimations of the crackbandwidth can be divided into three categories (i) methodsbased on the square root of the element area (for two-dimensional elements) or the cubic root of the elementvolume (for three-dimensional elements) (eg [53ndash55]) (ii)the notable methods proposed by Oliver [56] and (iii)projection methods (eg [57ndash60]) e methods in the firstcategory which estimate the crack bandwidth as a constantduring the numerical analysis process are less applicable tosituations with irregular meshes or arbitrary crack directions[56] e methods in the last two categories in which crackbandwidths are varied with respect to crack directions andspatial positions solve the aforementioned problemsHowever they may encounter difficulties such as excessivelylarge estimation [59] or misprediction of the crack band-width is paper therefore develops an improved estima-tion of the crack bandwidth to circumvent the abovedifficulties Moreover two different crack bandwidths (onefor the tensile strain-softening and the other for the com-pressive strain-softening) are taken into account in thepresent study because of which concrete exhibits quitedifferent localization behaviors in tension and compression[61] It should also be noted that the crack band approachcan only be regarded as a partial regularization method formesh-size dependency in the sense that the objectivity of theenergy dissipation is ensured [62] and thus the globalstructural response can be captured correctly [56] but themathematical description of the rate boundary valueproblem (BVP) still remains ill-posed [63]

e main purpose of this study is not to describe all themechanical behaviors of concrete but rather to give meth-odological efforts to solve the aforementioned difficultieserefore the plastic dissipation is not considered and theisotropic damage is assumed for simplification Accordingly

2 Advances in Civil Engineering

the objectives of this paper are fourfold (i) to define thecriterion distinguishing tension and compression by the sign ofthe volumetric deformation (ii) to propose amodel to describedistinct tension-compression behaviors of concrete materialsunder a framework similar to that of Cervera et al [11] andComi and Perego [26] (iii) to develop an improved estimationof the crack bandwidth using the projectionmethod for solvingthe problem of mesh-size dependency and (iv) to validate theproposed model by the selected benchmark tests

2 Tension-Compression Damage Model withConsistent Crack Bandwidths

21 Helmholtz Free Energy Potential e thermodynamicsof irreversible processes provides a general framework forformulating constitutive equations In other words the state ofthematerial can be described by the thermodynamic potentialConsequently the stresses and other associated variables canbe derived from this potential [64] Because of the scope of thispaper the present model is developed based on the followingassumptions (i) damage evolutions are assumed to be iso-tropic and to occur in small strains and (ii) the rate-dependentbehavior and the thermal effect are not considered To takeinto account distinct tension-compression behaviors of con-crete the postulated thermodynamic potential in terms ofHelmholtz free energy ψ is decomposed into volumetric anddeviatoric parts ψv and ψd in a similar way as in the study ofCervera et al [11] and Comi and Perego [26] as follows

ρψ ε dt dc( 1113857 ρψv εv dt dc( 1113857 + ρψd εd dt dc( 1113857

ρψ+v ε+

v dt dc( 1113857 + ρψminusv εminusv dt dc( 1113857

+ ρψd εd dt dc( 1113857

(1)

ρψ+v ε+

v dt dc( 1113857 32

1minus dt( 1113857 1minusdc( 1113857K0ε+v ε+

v

ρψminusv εminusv dt dc( 1113857 32

1minus dc( 1113857K0εminusv εminusv

(2)

ρψd εd dt dc( 1113857 1minusH(trε)dt1113858 1113859 1minusH(minustrε)dc1113858 1113859G0εd εd

(3)

where ρ is the mass density of the material ε is the second-order strain tensor dt and dc are the damage internal variablesthat represent the level of material degradation in tension andcompression K0 and G0 are the initial bulk and shear moduliof the undamagedmaterial respectivelyH(bull) is theHeavisidefunction defined by H(x) 1 if xge 0 and H(x) 0if xlt 0 and tr(bull) is the trace of a tensor e tensile andcompressive damage internal variables are monotonouslyincreasing scalars both of which grow from zero (undamagedmaterial with the intact stiffness) to one (completely damagedmaterial) εv and εd are second-order volumetric and devia-toric strain tensors defined by the following equation

εv Iv ε (4)

εd Id ε (5)

Iv 13Iotimes I (6)

Id Is minus Iv (7)

where Iv and Id are the fourth-order volumetric anddeviatoric projections respectively and I and Is are thesecond-order identity tensor and the fourth-order sym-metric identity tensor respectively

ε+v and εminusv are positive and negative volumetric strain

tensors defined according to

ε+v H(trε)εv (8)

εminusv H(minustrε)εv (9)

22DamageCriteria To describe the damage due to tensionor compression it first necessitates a criterion that definestensilecompressive states As mentioned in Introductionthe method of spectral decomposition may lead to thepremature convergence problem in numerical solutions[25] erefore the sign of the volumetric deformation(relative volume change) which is calculated by trε is usedto distinguish tension and compression in this paper esimilar application can be found in the work of Comi andPerego [26] In the following the tensile and compressivestrain states are thus defined by trεge 0 and trεlt 0respectively

e damage criterion characterizes the state of damageand represents a bounding surface whose boundary corre-sponds to the states at which the damage will grow To clearlyspecify the damage due to tension or compression twodamage criteria are introduced as follows

Fi εeqi κi( 1113857 εeqi (ε)minus κi le 0 (i t or c) (10)

where Ft and Fc are the tensile and compressive damagecriteria εeqt and εeqc are the equivalent tensile and com-pressive strains and κt and κc are the current tensile andcompressive damage thresholds respectively It is note-worthy that the damage does not develop if Fi lt 0 and growsif Fi 0

e damage threshold represents the largest value of theequivalent strain ever reached by the material during theloading history [65ndash67] erefore κt and κc can be furtherexpressed as follows

κi(t) maxτlet

εeqi (τ)( 1113857 (i t or c) (11)

Similar to the work of Fichant et al [68] the equivalentstrains in equation (10) are assumed to be as follows

εeqi 1

E0αiI1σ +βi

J2σ

1113968( 1113857 (i t if trεge0 i c otherwise)

(12)

with

Advances in Civil Engineering 3

I1σ trσe

J2σ 12σe σe minus

16I21σ

σe C0 ε

(13)

αt 12

1minusft

fc1113888 1113889

βt

3

radic

21 +

ft

fc1113888 1113889

αc 1minusfc

fbc

βc 3

radic2minus

fc

fbc1113888 1113889

(14)

where E0 is the initial Youngrsquos modulus of the undamagedmaterial I1σ is the first invariant of the effective stress tensorJ2σ is the second invariant of the deviatoric effective stresstensor σe is the effective stress tensor C0 is the fourth-orderelastic stiffness tensor of the undamagedmaterial αi and βi arethe material constants ft is the uniaxial tensile strength andfc and fbc are the uniaxial and biaxial compressive strengthsof the material respectively Following the experimental resultfrom the study of Kupfer et al [69] the typical value offbcfc 116 for concrete is adopted throughout this paper

23 Evolution Laws for Internal Variables From themacroscopic viewpoint within the framework of thermody-namics the damage variable can be introduced as an internalstate variable to quantify the average material degradation [22]In order to capture distinct tension-compression behaviors ofconcrete two types of damage evolution laws which relate thecorresponding damage internal variable to the associateddamage threshold are defined in the following general form

di gi κi( 1113857 (i t or c) (15)

In this paper explicit damage evolution laws are furtherpostulated as follows

dt gt κt( 1113857

0 if κt le κ0t

1minusκ0tκt

κut minus κtκut minus κ0t

1113888 1113889

At

if κ0t lt κt lt κut

1 if κt ge κut

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

dc gc κc( 1113857

1minus1

1 +(cminus 1) κcκ0c( 1113857(ccminus1)

if κc le κ0c

1minusκ0ccκc

1minusκc minus κ0cκuc minus κ0c

1113888 1113889

2⎡⎣ ⎤⎦ if κ0c lt κc lt κuc

1 if κc ge κuc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

where κ0t ftE0 and κ0c cfcE0 are the initial tensile andcompressive damage thresholds respectively with c being thematerial parameter At is the nonnegative material parameterthat controls the shape of the tensile stress-strain relation asshown in Figure 1 It is noteworthy that if At 1 then thedamage evolution law in tension represents a linear softeningκut and κuc are the ultimate tensile and compressive damagethresholds respectively According to the crack band ap-proach [33] the ultimate damage threshold depends on thefracture energy and the crack bandwidth It will be furtherdiscussed later

At any material point damage can evolve only if thecurrent state reaches the boundary of the elastic domaincharacterized by Fi(ε

eqi κi) 0 To reflect the loading

unloading and reloading conditions more generally theevolution of damage internal variables is governed by thefollowing KuhnndashTucker relations

Fi εeqi κi( 1113857le 0 di

bull

ge 0

Fi εeqi κi( 1113857di

bull

0 (i t or c)

(18)

24 Mechanical Dissipation and Constitutive Laws efailure of concrete is related to the degradation of physicalproperties of the material During this damage process theenergy dissipation is always nonnegative and accompanied byan increase of entropy [66] It hence implies an irreversibleprocess of thermodynamics erefore the mechanical dis-sipation ϕd under the isothermal condition needs to satisfy thesecond principle of thermodynamics which can be expressedby the ClausiusndashDuhem inequality as follows

ϕdbull

σ εbullminus ρψ

bullge 0 (19)

where σ is the second-order stress tensorSubstitution of equation (1) into equation (19) yields the

following equation

ϕd

bull

σminus ρzψzε

1113888 1113889 εbull

+ Ytdt

bull

+ Ycdc

bull

ge 0 (20)

with the tensile and compressive damage energy release ratesdefined as follows

Yi minusρzψzdi

minus12ε

zC

zdi

ε (i t or c) (21)

where C is the fourth-order secant stiffness tensor of thedamaged material

Because of the fact that the inequality of equation (20)must be satisfied by the arbitrary change of the strain tensorε the coefficient of the first term of the inequality thereforemust be zero is requirement yields the following con-stitutive laws

σ ρzψzε

(22)

C dt dc( 1113857 zσzε

ρz2ψzεzε

(23)


Using equation (22) the ClausiusndashDuhem inequalityexpressed by equation (20) is eventually written in thefollowing form

ϕdbull Ytdt

bull+ Ycdc

bullge 0 (24)

erefore the second principle of thermodynamics issatised as long as the damage variables dt and dc aremonotonically increasing since Yt and Yc are strictly non-negative because of their quadratic forms which involvezCzdi le 0

25 Fracture Energy Regularization with Consistent CrackBandwidths To solve the problem of mesh-size dependencya fracture energy regularization based on the crack bandapproach [33] is adopted in the present study e generalidea of the crack band approach is to rescale the softeningbranch of the stress-strain curve in order to impose the sameenergy dissipation per unit area for dierent element sizes[59] is adjustment consists in relating the fracture energy(regarded to be a material property) to the volumetric (orspecic) fracture energy (Figures 2(c) and 2(d)) which de-pends on the crack bandwidth To achieve this goal of reg-ularization it is nally required to relate the ultimate damagethreshold to the fracture energy and the crack bandwidth

e crack band approach assumes that the failure ofmaterial occurs within a damage zone having a limited widthwhereas the other parts are unloaded [59] during the process offailure (Figure 2) According to Bazant andOh [33] and Jirasekand Mara [43] the same energy dissipation per unit area canbe ensured using following relations

Gft l

(P)t gft l

(P)t int

κut

0σnnt κt( ) dκt (25)

Gfc l

(P)c gfc l

(P)c int

κuc

κ0cσnnc κc( ) dκc (26)

where Gft and G

fc are the Mode-I and compressive fracture

energies dened as the energy dissipation per unit area of a

surface due to cracking or crushing l(P)t and l(P)c are the tensileand compressive crack bandwidths gft and g

fc are the tensile

and compressive volumetric fracture energies dened as theenergy dissipation per unit volume and σnnt and σnnc are thetensile and compressive stresses normal and tangential to thesurface of cracking respectively It is noteworthy that Gf

t andgft can be represented by the area under the traction-separation (tensile stress normal to the crack versus crackopening displacement) curve and under the stress-straincurve respectively as shown in Figures 2(c) and 2(d)

Using equation (16) substitution of equation (22) intoequation (25) yields the following equation

κut 1 + At( )Gf

t

l(P)t ft+1minusAt

2κ0t (27)

It is noteworthy that equation (27) is derived under thecondition of uniaxial loading Nevertheless the previousstudies (eg [20 70 71]) have revealed that it is also anacceptable approximation under general loading conditions

Similarly the compressive damage threshold κuc can beexpressed as follows

κuc 32

Gfc

l(P)c fc+ κ0c (28)

Generally the crack bandwidth can be regarded as an FEdiscretization parameter which depends on the chosen ele-ment shape (eg quadrilaterals or triangles) element sizeelement orientation (or mesh alignment) with respect to thedirection of the strain localization band interpolationfunction or order of the displacement approximation(eg linear or quadratic) and numerical integration scheme(number of integration points) [53 59 60] For the sake ofsimplicity only four-node quadrilateral and three-node tri-angular elements are used throughout the paper It shouldalso be noted that only low-order elements (ie elements withlinear or multi-linear interpolation functions) are adopted inthis study since the stress elds obtained from high-orderelements (ie elements with quadratic interpolation func-tions) are highly disturbed for crack band simulations [59]

Inspired by the works of Cervenka et al [57] and Jirasekand Bauer [59] this paper develops an improved estimationof crack bandwidths in tension and compression by pro-jecting the element onto the directions normal and tan-gential to the crack direction respectively Mathematicallythey can be written as follows

l(P)i lmaxi minus l

mini

∣∣∣∣∣∣∣∣∣∣ (i t or c) (29)

with

lmaxt max

N1NmminusnxN middot n(x)[ ]

lmint min


lmaxc max

N1NmminusnxN middot t(x)[ ]

(30)

lminc min

N1NmminusnxN middot t(x)[ ] (31)

0

ft

At = 02At = 06At = 10

At = 20At = 60

κt0 ε

σ

Figure 1 Inuence ofAt on uniaxial stress-strain relation in tension


where l(P)t and l(P)c are the tensile and compressive crackbandwidths lmax

i and lmini (i t or c) are the maximum and

minimum projection values xN is the midpoint of the elementedge (for four-node quadrilateral elements) or the nodal point(for three-node triangular elements) Nmminusn is the number ofelement midpoints (for four-node quadrilateral elements) ornodes (for three-node triangular elements) x represents thepoint (eg Gaussian point) in the structure where the crackbandwidth is to be evaluated and n(x) and t(x) are the unitvectors with the direction normal and tangential to the crackdirection at the point x respectivelye geometrical meaningof equation (29) is illustrated in Figure 3

For comparison the existing estimation formulas of thecrack bandwidth mentioned in Introduction are recalled asfollows

e tensile crack bandwidth l(R)t which is estimated byRots [53] belonging to the rst category can be written asfollows

l(R)t A

radic for quadratic two-dimensional elements

2A

radic for linear two-dimensional elements

(32)

with A being the area of the elemente tensile crack bandwidth l(O)t which is estimated

according to the rst method of Oliver [56] can be expressedas follows

l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with

ϕi 1 if xi minus xc( ) middot n(x) ge 00 if xi minus xc( ) middot n(x) lt 0

(34)

whereNc is the number of corner nodes of the elementNi isthe standard shape function for the corner node i belongingto an element with Nc corner nodes ϕi is the nodal value atthe corner node i and xi and xc are the corner node and thecenter of the element respectively

e tensile crack bandwidth l(G)t which is estimated byGovindjee et al [58] belonging to the third category can begiven as follows

l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with

τi xi minus xc( ) middot n(x) minus τmin

τmax minus τmin

τmax maxi1Nc

xi minus xc( ) middot n(x)[ ](36)

τmin mini1Nc

xi minus xc( ) middot n(x)[ ] (37)

e geometrical meanings of the aforementioned tensilecrack bandwidths and their dierences with respect to thedirection of n(x) are illustrated in Figures 4 and 5 where θ isthe angle between the normal to the crack and the global axisx at the point x e crack bandwidths denoted by l(O)t(according to equation (33)) become very large (Figures 4(b)and 5(b)) under certain circumstances and they also show adiscontinuity (Figures 4(c) and 5(c)) while the crackbandwidths denoted by l(P)t and l(G)t (according to equations(29) and (35) respectively) are free from such problemsediscontinuous crack bandwidth obtained from equation (33)is caused by the discontinuous nature of ϕi whose value iseither 1 or 0 as dened by equation (34) It is also noteworthythat the present method (labelled as l(P)t ) and the method ofGovindjee et al [58] (labelled as l(G)t ) give exactly the same

+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone

w (crack opening displacement)

Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f

Figure 2 Damage zone and fracture energy regularization (a) damage zone in the concrete specimen under tensile loading (b) stress-strainresponse outside the damage zone (c) traction-separation law in the damage zone (d) general stress-strain relation for the concretespecimen


estimation of the tensile crack bandwidth for the lineartriangular element as shown in Figures 5(a) and 5(c) sinceboth of them use the method of projecting corner nodes ofthe element onto the direction normal to the crack directione further comparative study of these estimation methodswill be given in the next section

3 Model Verification

To establish the validity of the proposed constitutive modelthe computed results were either veried theoretically orcompared with measured ones from experiments ese arepresented in the following sections

31 Analysis of Biaxial Tests of Concrete To examine therationality of the adopted criterion distinguishing tensionand compression and to validate the present model inreproducing the distinct strength features of concrete underbiaxial stresses a series of plain concrete specimens sub-jected to dierent stress ratios (denoted by α) which weretested by Kupfer et al [69] were chosen for the numericalanalysis e material properties used for the computationwere Youngrsquos modulus E0 32 times 104 MPa Poissonrsquos ratio]0 02 concrete uniaxial tensile strength ft 328MPaAt 10 uniaxial compressive strength fc 328MPac 23 Mode-I fracture energy Gf

t 01Nmm and com-pressive fracture energy Gf

c 150Nmm A one-element

Gaussian pointCrack

Node of elementMidpoint of element edge

n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x

Figure 3 Proposed estimation of crack bandwidths

Node of elementMidpoint of element edgeGaussian pointCrack

20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)

Figure 4 Comparison of estimations of crack bandwidth for a rectangular quadrilateral element (a) geometrical meanings of crackbandwidths estimated by the present model Oliver [56] and Govindjee et al [58] (b) large crack bandwidth estimated by the rst method ofOliver [56] (c) relations of tensile crack bandwidth and angle θ


mesh (Mesh A was 2000mm long times 2000mm high) with afour-node plane stress quadrilateral element using a 2 times 2Gaussian integration scheme [44] was used to represent theplain concrete specimen (Figure 6)

Figure 6 shows the comparison between the experi-mental and computed results of the strength envelope whichcorresponds to the envelope of the maximum stress everreached in the material during the loading in a biaxialprincipal stress space e results indicate that the predictedbiaxial strengths of concrete are very close to the experi-mental ones either in the biaxial tensile domain or in thebiaxial compressive domain or even in the tensile-compressive domain e numerical results in Figure 6also demonstrate the rationality of the adopted criteriondistinguishing tension and compression Figure 7 shows thecomparison of stress-strain curves of numerical and ex-perimental results [69] under biaxial compression (σ3 0)according to the following stress ratios (α σ2σ1) α 0α 052 and α 1 It shows that the proposed modelcan predict well both the strength enhancement of concreteand the corresponding peak strain in biaxial compressivestates

32 Analysis of a Double-Edge Notched Concrete Specimenunder Direct Tension To validate the proposed model incapturing the strain-softening a double-edge notched(DEN) concrete specimen under direct tension which wasinvestigated experimentally byHordijk [72] was used for thenumerical analysis e test specimen shown in Figure 8which had dimensions 125mm height times 60mm width times50mm thickness with two symmetrical notches of 5times 5mm2

in the middle cross-section was glued to the loading platens

and subjected to an axial tensile load under the deformationcontrol e average deformation δ which denotes theelongation of the specimen was measured experimentally byusing four pairs of extensometers located at the front andrear faces of the specimen e gauge length of extensom-eters was 35mm (Figure 8) e average stress σavg in theexperiment was dened as the vertical load divided by thenet cross-sectional area (50times 50mm2)

e material properties of the test specimen usedfor the numerical analysis were Youngrsquos modulusE0 18 times 104 MPa Poissonrsquos ratio ]0 02 concreteuniaxial tensile strength ft 35MPa At 60 uniaxial

Node of elementGaussian pointCrack

lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)

Figure 5 Comparison of estimations of crack bandwidth for a right triangular element (a) geometrical meanings of crack bandwidthsestimated by the present model Oliver [56] and Govindjee et al [58] (b) large crack bandwidth estimated by the rst method of Oliver [56](c) relations of tensile crack bandwidth and angle θ

Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02

ndash14 ndash12 ndash10 ndash08 ndash06 ndash04 ndash02 00 02

Experimental (Kupfer)Proposed model

σ1| fc|

σ2|fc|

Figure 6 Biaxial strength envelope of concrete


compressive strength fc 340MPa and Mode-I fractureenergy Gf

t 007Nmme DEN concrete specimen was modeled by a 298-

element mesh with four-node plane stress quadrilateralelements using a 2 times 2 Gaussian integration scheme asshown in Figure 9 According to the boundary conditions of

the experiment all FE nodes at the lower boundary wereassumed to be xed (Figure 9) Figure 10 presents thecomputed average stress-deformation curve in tension enumerical result shows a satisfactory agreement with theexperimental one including both the load-carrying capacityand the postpeak behavior

33 Analysis of a Bar Subjected to Uniaxial Tension UsingElementswithaConstantStrainField To prove the ecopyciencyof the proposed estimation of the crack bandwidth withrespect to the FE mesh irregularity a bar subjected touniaxial tension was used for the numerical analysis byfollowing the work of Oliver [56] Figure 11 shows thegeometry boundary and loading conditions of the bar witha thickness of 50mm

e material properties used for the numerical analysiswere Youngrsquos modulus E0 20 times 104 MPa Poissonrsquos ratio]0 00 concrete uniaxial tensile strength ft 20MPaAt 10 uniaxial compressive strength fc 200MPa andMode-I fracture energy Gf

t 01NmmAccording to Oliver [56] the localization of the bar due

to strain-softening was induced articially by the prescribedimperfection (Figure 11) with a small reduction of theuniaxial tensile strength in shaded elements as shown inFigure 12

e bar was discretized by the four-node quadrilateral orthree-node triangular plane stress elements both of which

00

02

04

06

08

10

12

14

ndash40ndash30ndash20ndash1000

Experimental α = 0Experimental α = 052Experimental α = 1

Numerical α = 0Numerical α = 052Numerical α = 1

Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c

Figure 7 Computed stress-strain curves of concrete under biaxialcompression

125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen

Figure 8 DEN concrete specimen geometry boundary andloading conditions (dimensions in mm)

Figure 9 FE mesh for the DEN concrete specimen


adopted a linear interpolation function (implying a constantstrain eld) and a 1-point Gaussian integration schemeFigure 12 shows the meshes used for the analysis withdierent mesh irregularities

Due to the fact that the projection methods (eg thepresent method and that of [58]) have solved the dicopyculty ofthe excessively large estimation of the crack bandwidtharising in the rst method of Oliver [56] shown in theprevious section only the element area-based Govindjeeet al [58]-based and the present methods of estimatingthe crack bandwidth are compared in this section

Figure 13(c) shows that the load-displacement responsesobtained by the present method are almost coincident withthe theoretical solution and are hardly detectable in the plotswhile the corresponding responses which were obtained bythe crack bandwidth estimated by

2A

radicand by Govindjee

et al [58] using equation (35) show an unacceptable meshdependency with respect to the FE mesh irregularityConsequently the present model provides an accurateestimation of the crack bandwidth that yields mesh-size independent results with respect to the meshirregularity

00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

ExperimentalNumerical

Average deformation δ (times10ndash6 m)

Ave

rage

stre

ss σ

avg (

MPa

)

Figure 10 Average stress-deformation curves for the DEN concrete specimen

F

100mm

25mm

Prescribed imperfection

Figure 11 A bar subjected to uniaxial tension

(a) (b)

(c) (d)

Figure 12 FE meshes used for the study of mesh irregularity (after the study [56]) (a) Mesh A (b) Mesh B (c) Mesh C (d) Mesh D


34 Analysis of a Single-Edge Notched Concrete Beam Sub-jected to Four-Point Shear Loading To demonstrate themesh-size objectivity and the capability of the proposedmodel in predicting the structural response in shear loadinga single-edge notched (SEN) concrete beam of Arrea andIngraea [73] was used for the numerical verication egeometry of the SEN concrete beam boundary and four-point shear loading conditions which were used to induce amixed tensileshear failure are shown in Figure 14 e loadwas applied using a steel beam through which the load wastransmitted to the SEN concrete beam at points A and B(Figure 14)

e material properties of the SEN beam used in theanalysis were Youngrsquos modulus E0 248 times 104 MPa Pois-sonrsquos ratio ]0 018 concrete uniaxial tensile strengthft 35MPa At 10 uniaxial compressive strengthfc 350MPa c 22 Mode-I fracture energyGft 0075Nmm and compressive fracture energy

Gfc 150Nmm e steel plates used as load distributors

were assumed to be a linear elastic material with Youngrsquosmodulus Es 210GPa and Poissonrsquos ratio ]s 03

e SEN concrete beam and the steel plates were dis-cretized by four-node quadrilateral (with 2 times 2 Gaussian

00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D

eoretical and Mesh A

Forc

e F (k

N)

Displacement δ (mm)

(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

eoretical Mesh A and Mesh D

(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

eoretical Mesh A Mesh BMesh C and Mesh D

(c)

Figure 13 Force-displacement responses obtained by dierent estimations of crack bandwidth with dierent mesh irregularities crackbandwidth estimated by (a)

2A

radic (b) Govindjee et al [58] and (c) the present model

306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate

Figure 14 SEN concrete beam geometry boundary and loadingconditions (dimensions in mm)


integration) and three-node triangular plane stress elementsin the FE analysis respectively ree dierent mesh sizeswere used for the mesh-size dependency study coarsemedium and ne meshes which had a total of 480 810 and1176 elements respectively (Figure 15) e correspondingminimum element sizes of these three meshes were 191mm128mm and 96mm respectively

Figure 16 presents the predicted structural responses interms of the load P at point B versus the crack mouth slidingdisplacement (CMSD) Figure 17 shows the simulated naltensile damage distribution which is denoted by the darkcolor corresponding to a high damage value close to oneFigure 18 shows the simulated crack patterns at the nalcomputation stage

According to the numerical results it is clear that (i) theglobal structural responses are well predicted compared tothe experimental results (Figure 16) (ii) the proposed model

provides an acceptable prediction of the curved macroscopiccrack as compared with the experimentally observed cracktrajectory (Figures 17 and 18) and (iii) the simulated cracktrajectories obtained with the medium and ne meshes areidentical which indicates that the predicted paths of thecrack propagation are almost mesh-size independent

e predicted thicknesses of the macroscopic crack areinconsistent when using dierent mesh sizes as shown inFigures 17 and 18 As more clearly shown in Figure 18 thesimulated localization band tends to occur in one row ofelements is is due to the inherent deciency of the crackband approach since the rate BVP still remains ill-posed[63] e similar conclusion to simulations with low-orderelements can be found in the study of Jirasek and Bauer [59]who also pointed out that the minimum size of the locali-zation band is the size of one row of integration points(ie subelement domains) but not the size of one row ofelements in simulations with high-order elements

As also can be observed in Figures 17 and 18 thepredicted crack trajectories are less curved than its experi-mental counterpart due to the approximation error of thespatial discretization in the standard nite element methodIt can be explained by the fact that the localized damage zonecannot be resolved with sucopycient accuracy because of theminimum element size (191mm) of the coarse mesh beingtoo large as compared with the thickness of the notch(5mm)

4 Conclusions

is paper presents a tension-compression damage modelwith consistent crack bandwidths developed for the mac-roscopic description of concrete behaviors e model isformulated within the framework of the internal variabletheory of thermodynamics so that the constitutive relationsare derived without ad hoc assumptions Based on thenumerical and experimental results the following conclu-sions can be drawn

(i) e essential features of the proposedmodel includethe identication of tensilecompressive strain statesby the sign of the volumetric deformation the

(a) (b)

(c)

Figure 15 Dierent FE mesh sizes for the SEN concrete beam in the mesh-size dependency study (a) coarse mesh (b) medium mesh (c)ne mesh

0

20

40

60

80

100

120

140

160

000 005 010 015

ExperimentalNumerical(coarse mesh)

Numerical(medium mesh)Numerical(fine mesh)

CMSD (mm)

Load

P (k

N)

Figure 16 Load-CMSD curves for SEN concrete beam obtainedwith dierent FE mesh sizes


rational definition of two equivalent strains twointernal variables and two separated damage cri-teria describing distinct damage mechanics undertension and compression Such features enable thepresent model to provide a better prediction of thedistinct tension-compression behaviors of concretematerials Furthermore the adopted method usingthe sign of the volumetric deformation avoids thepremature convergence difficult in the numericalanalysis which is often encountered in existingmodels using the spectral decomposition of a stressor strain tensor

(ii) e main contribution of this paper is that itprovides an improved estimation of the crackbandwidth which is developed for two-dimensionallinear elements by taking into account the elementshape element size and element orientation with

respect to crack directions and spatial positions Itscapability in numerical analyses of irregular meshesand arbitrary crack directions has been validated bythe comparative study of existing estimations of thecrack bandwidth at the element level e proposedmodel incorporating this enhanced description ofthe crack bandwidth has also been proven to bemesh-size independent at the structural level epresent model is able to provide objective simula-tion results with respect to the finite element meshrefinements that are very important for the com-putation of the real engineering structures

(iii) e proposed model has been validated by the well-documented experimental data of concrete egood agreement between the predicted and exper-imental results consequently demonstrates that thepresent model provides an effective tool for

Damage zone(computed)

Crack envelope(experimental)

B B B

(a) (b) (c)

P P P

Figure 17 Final tensile damage profiles in the central part of the SEN concrete beam (a) coarse mesh (b) medium mesh (c) finemesh

P

(a)

P

(b)

P

(c)

Figure 18 Computed final crack patterns in the central part of the SEN concrete beam (a) coarse mesh (b) medium mesh (c) fine mesh


modeling concrete behaviors in tension andcompression

(iv) e present model has a limited number of materialparameters each of which has a clear physicalmeaning and can be identified by standard labo-ratory tests Moreover the simplicity and the ex-plicit integration of the constitutive equations makethe model well suited for large-scale computations

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e work described in this paper was supported by theNational Natural Science Foundation of China (51208005)the Transportation Science Research Project of JiangsuProvince (0213Y25) the Natural Science Foundation ofAnhui Province of China (1208085QE80) the EducationalDepartment of Anhui Province of China (KJ2012A095) andthe Doctoral Fund of Anhui University of Science ampTechnology (11084)

References

[1] L K Turner and F G Collins ldquoCarbon dioxide equivalent(CO2-e) emissions a comparison between geopolymer andOPC cement concreterdquo Construction and Building Materialsvol 43 pp 125ndash130 2013

[2] Z-D Xu and ZWu ldquoEnergy damage detection strategy basedon acceleration responses for long-span bridge structuresrdquoEngineering Structures vol 29 no 4 pp 609ndash617 2007

[3] S Yazdani ldquoOn a class of continuum damage mechanicstheoriesrdquo International Journal of Damage Mechanics vol 2no 2 pp 162ndash176 2016

[4] C Goidescu H Welemane O Pantale M Karama andD Kondo ldquoAnisotropic unilateral damage with initialorthotropy a micromechanics-based approachrdquo In-ternational Journal of Damage Mechanics vol 24 no 3pp 313ndash337 2014

[5] J Mazars ldquoA description of micro- and macroscale damage ofconcrete structuresrdquo Engineering Fracture Mechanics vol 25no 5-6 pp 729ndash737 1986

[6] G Pijaudier-Cabot and Z P Bazant ldquoNonlocal damagetheoryrdquo Journal of Engineering Mechanics vol 113pp 1512ndash1533 1987

[7] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal ofSolids and Structures vol 23 no 7 pp 821ndash840 1987

[8] D Krajcinovic ldquoDamage mechanicsrdquoMechanics of Materialsvol 8 no 2-3 pp 117ndash197 1989

[9] G Z Voyiadjis and T M Abu-Lebdeh ldquoDamage model forconcrete using bounding surface conceptrdquo Journal of Engi-neering Mechanics vol 119 no 9 pp 1865ndash1885 1993

[10] J L Chaboche P M Lesne and J F Maire ldquoContinuumdamage mechanics anisotropy and damage deactivation for

brittle materials like concrete and ceramic compositesrdquo In-ternational Journal of Damage Mechanics vol 4 no 1pp 5ndash22 2016

[11] M Cervera J Oliver and O Manzoli ldquoA rate-dependentisotropic damage model for the seismic analysis of concretedamsrdquo Earthquake Engineering amp Structural Dynamicsvol 25 no 9 pp 987ndash1010 1996

[12] S Murakami and K Kamiya ldquoConstitutive and damageevolution equations of elastic-brittle materials based on ir-reversible thermodynamicsrdquo International Journal of Me-chanical Sciences vol 39 no 4 pp 473ndash486 1997

[13] R H J Peerlings R de Borst W A M Brekelmans andM G D Geers ldquoGradient-enhanced damage modelling ofconcrete fracturerdquoMechanics of Cohesive-frictional Materialsvol 3 no 4 pp 323ndash342 1998

[14] N Xie Q-Z Zhu J-F Shao and L-H Xu ldquoMicromechanicalanalysis of damage in saturated quasi brittle materialsrdquo In-ternational Journal of Solids and Structures vol 49 no 6pp 919ndash928 2012

[15] J J C Pituba and G R Fernandes ldquoAnisotropic damagemodel for concreterdquo Journal of Engineering Mechanicsvol 137 no 9 pp 610ndash624 2011

[16] S Zhu and C Cai ldquoStress intensity factors evaluation forthrough-transverse crack in slab track system under vehicledynamic loadrdquo Engineering Failure Analysis vol 46pp 219ndash237 2014

[17] S Zhu Q Fu C Cai and P D Spanos ldquoDamage evolutionand dynamic response of cement asphalt mortar layer of slabtrack under vehicle dynamic loadrdquo Science China Techno-logical Sciences vol 57 no 10 pp 1883ndash1894 2014

[18] B Valentini and G Hofstetter ldquoReview and enhancement of3D concrete models for large-scale numerical simulations ofconcrete structuresrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 37 no 3 pp 221ndash246 2013

[19] C Wen and S Yazdani ldquoAnisotropic damage model forwoven fabric composites during tension-tension fatiguerdquoComposite Structures vol 82 no 1 pp 127ndash131 2008

[20] L Pela M Cervera and P Roca ldquoAn orthotropic damagemodel for the analysis of masonry structuresrdquo Constructionand Building Materials vol 41 pp 957ndash967 2013

[21] A Khennane M Khelifa L Bleron and J Viguier ldquoNu-merical modelling of ductile damage evolution in tensile andbending tests of timber structuresrdquo Mechanics of Materialsvol 68 pp 228ndash236 2014

[22] G Z Voyiadjis and P I Kattan ldquoMechanics of small damagein fiber-reinforced composite materialsrdquo Composite Struc-tures vol 92 no 9 pp 2187ndash2193 2010

[23] M Ganjiani R Naghdabadi and M Asghari ldquoAnalysis ofconcrete pressure vessels in the framework of continuumdamage mechanicsrdquo International Journal of Damage Me-chanics vol 21 no 6 pp 843ndash870 2011

[24] A Sellier G Casaux-Ginestet L Buffo-Lacarriere andX Bourbon ldquoOrthotropic damage coupled with localizedcrack reclosure processing Part I constitutive lawsrdquo Engi-neering Fracture Mechanics vol 97 pp 148ndash167 2013

[25] X Tao and D V Phillips ldquoA simplified isotropic damagemodel for concrete under bi-axial stress statesrdquo Cement andConcrete Composites vol 27 no 6 pp 716ndash726 2005

[26] C Comi and U Perego ldquoFracture energy based bi-dissipativedamage model for concreterdquo International Journal of Solidsand Structures vol 38 no 36-37 pp 6427ndash6454 2001

[27] J H P de Vree W A M Brekelmans and M A J van GilsldquoComparison of nonlocal approaches in continuum damage


mechanicsrdquo Computers and Structures vol 55 no 4pp 581ndash588 1995

[28] A Pros P Dıez and C Molins ldquoNumerical modeling of thedouble punch test for plain concreterdquo International Journal ofSolids and Structures vol 48 no 7-8 pp 1229ndash1238 2011

[29] P E Bernard N Moes and N Chevaugeon ldquoDamage growthmodeling using the thick level set (TLS) approach efficientdiscretization for quasi-static loadingsrdquo Computer Methods inApplied Mechanics and Engineering vol 233-236 pp 11ndash272012

[30] S Y Alam P Kotronis and A Loukili ldquoCrack propagationand size effect in concrete using a non-local damage modelrdquoEngineering Fracture Mechanics vol 109 pp 246ndash261 2013

[31] S Saroukhani R Vafadari and A Simone ldquoA simplifiedimplementation of a gradient-enhanced damage model withtransient length scale effectsrdquo Computational Mechanicsvol 51 no 6 pp 899ndash909 2012

[32] G Sciume F Benboudjema C De Sa F PesaventoY Berthaud and B A Schrefler ldquoA multiphysics model forconcrete at early age applied to repairs problemsrdquo EngineeringStructures vol 57 pp 374ndash387 2013

[33] Z P Bazant and B H Oh ldquoCrack band theory for fracture ofconcreterdquoMaterials and Structures vol 16 no 3 pp 155ndash1771983

[34] M Cervera L Pela R Clemente and P Roca ldquoA crack-tracking technique for localized damage in quasi-brittlematerialsrdquo Engineering Fracture Mechanics vol 77 no 13pp 2431ndash2450 2010

[35] R de Borst L J Sluys H B Muhlhaus and J PaminldquoFundamental issues in finite element analyses of localizationof deformationrdquo Engineering Computations vol 10 no 2pp 99ndash121 1993

[36] J Oliver A E Huespe and I F Dias ldquoStrain localizationstrong discontinuities and material fracture matches andmismatchesrdquo Computer Methods in Applied Mechanics andEngineering vol 241ndash244 pp 323ndash336 2012

[37] Z P Bazant and M Jirasek ldquoNonlocal integral formulationsof plasticity and damage survey of progressrdquo Journal ofEngineering Mechanics vol 128 no 11 pp 1119ndash1149 2002

[38] M Cervera M Chiumenti and R Codina ldquoMixed stabilizedfinite element methods in nonlinear solid mechanics Part Iformulationrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 37ndash40 pp 2559ndash2570 2010

[39] M Cervera M Chiumenti and R Codina ldquoMixed stabi-lized finite element methods in nonlinear solid mechanicsPart II strain localizationrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 37ndash40 pp 2571ndash2589 2010

[40] M Cervera and M Chiumenti ldquoSmeared crack approachback to the original trackrdquo International Journal for Nu-merical and Analytical Methods in Geomechanics vol 30no 12 pp 1173ndash1199 2006

[41] W He Y-F Wu K M Liew and Z Wu ldquoA 2D total strainbased constitutive model for predicting the behaviors ofconcrete structuresrdquo International Journal of EngineeringScience vol 44 no 18-19 pp 1280ndash1303 2006

[42] L Pela M Cervera S Oller and M Chiumenti ldquoA localizedmapped damage model for orthotropic materialsrdquo Engi-neering Fracture Mechanics vol 124-125 pp 196ndash216 2014

[43] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[44] W He Y-F Wu Y Xu and T-T Fu ldquoA thermodynamicallyconsistent nonlocal damage model for concrete materials with

unilateral effectsrdquo Computer Methods in Applied Mechanicsand Engineering vol 297 pp 371ndash391 2015

[45] R K Abu Al-Rub and G Z Voyiadjis ldquoGradient-enhancedcoupled plasticity-anisotropic damage model for concretefracture computational aspects and applicationsrdquo In-ternational Journal of Damage Mechanics vol 18 no 2pp 115ndash154 2009

[46] R de Borst and L J Sluys ldquoLocalisation in a Cosserat con-tinuum under static and dynamic loading conditionsrdquoComputer Methods in Applied Mechanics and Engineeringvol 90 no 1ndash3 pp 805ndash827 1991

[47] C Giry C Oliver-Leblond F Dufour and F RagueneauldquoCracking analysis of reinforced concrete structuresrdquo Euro-pean Journal of Environmental and Civil Engineering vol 18no 7 pp 724ndash737 2014

[48] A Simone H Askes and L J Sluys ldquoIncorrect initiation andpropagation of failure in non-local and gradient-enhancedmediardquo International Journal of Solids and Structures vol 41no 2 pp 351ndash363 2004

[49] M Jirasek S Rolshoven and P Grassl ldquoSize effect on fractureenergy induced by non-localityrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28no 78 pp 653ndash670 2004

[50] K Haidar J F Dube and G Pijaudier-Cabot ldquoModellingcrack propagation in concrete structures with a two scaleapproachrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 27 no 13 pp 1187ndash12052003

[51] J A Paredes A H Barbat and S Oller ldquoA compression-tension concrete damage model applied to a wind turbinereinforced concrete towerrdquo Engineering Structures vol 33no 12 pp 3559ndash3569 2011

[52] M Alembagheri and M Ghaemian ldquoIncremental dynamicanalysis of concrete gravity dams including base and liftjointsrdquo Earthquake Engineering and Engineering Vibrationvol 12 no 1 pp 119ndash134 2013

[53] J G Rots Computational modeling of concrete fracture PhDthesis Delft University of Technology Delft the Netherlands1988

[54] P H Feenstra and R de Borst ldquoConstitutive model forreinforced concreterdquo Journal of Engineering Mechanicsvol 121 no 5 pp 587ndash595 1995

[55] R Scotta R Vitaliani A Saetta E Ontildeate and A Hanganu ldquoAscalar damage model with a shear retention factor for theanalysis of reinforced concrete structures theory and vali-dationrdquo Computers and structures vol 79 no 7 pp 737ndash7552001

[56] J Oliver ldquoA consistent characteristic length for smearedcracking modelsrdquo International Journal for NumericalMethods in Engineering vol 28 no 2 pp 461ndash474 1989

[57] V Cervenka R Pukl J Ozbold and R Eligehausen ldquoMeshsensitivity effects in smeared finite element analysis of con-crete fracturerdquo in Fracture Mechanics of Concrete StructuresF H Wittmann Ed pp 1387ndash1396 Aedificatio PublishersFreiburg Germany 1995

[58] S Govindjee G J Kay and J C Simo ldquoAnisotropic mod-elling and numerical simulation of brittle damage in con-creterdquo International Journal for Numerical Methods inEngineering vol 38 no 21 pp 3611ndash3633 1995

[59] M Jirasek andM Bauer ldquoNumerical aspects of the crack bandapproachrdquo Computers and Structures vol 110-111 pp 60ndash782012

[60] A T Slobbe M A N Hendriks and J G Rots ldquoSystematicassessment of directional mesh bias with periodic boundary


conditions applied to the crack band modelrdquo EngineeringFracture Mechanics vol 109 pp 186ndash208 2013

[61] C Comi ldquoA non-local model with tension and compressiondamage mechanismsrdquo European Journal of Mechanics-ASolids vol 20 no 1 pp 1ndash22 2001

[62] M Jirasek and P Grassl ldquoEvaluation of directional mesh biasin concrete fracture simulations using continuum damagemodelsrdquo Engineering Fracture Mechanics vol 75 no 8pp 1921ndash1943 2008

[63] R de Borst ldquoSome recent developments in computationalmodelling of concrete fracturerdquo International Journal ofFracture vol 86 no 1-2 pp 5ndash36 1997

[64] I Carol and K Willam ldquoSpurious energy dissipationgeneration in stiffness recovery models for elastic degrada-tion and damagerdquo International Journal of Solids andStructures vol 33 no 20ndash22 pp 2939ndash2957 1996

[65] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7pp 803ndash833 1989

[66] R Faria J Oliver and M Cervera ldquoA strain-based plasticviscous-damage model for massive concrete structuresrdquo In-ternational Journal of Solids and Structures vol 35 no 14pp 1533ndash1558 1998

[67] J Y Wu J Li and R Faria ldquoAn energy release rate-basedplastic-damage model for concreterdquo International Journal ofSolids and Structures vol 43 no 3-4 pp 583ndash612 2006

[68] S Fichant C La Borderie and G Pijaudier-Cabot ldquoIsotropicand anisotropic descriptions of damage in concrete struc-turesrdquoMechanics of Cohesive-frictional Materials vol 4 no 4pp 339ndash359 1999

[69] H Kupfer H K Hilsdorf and H Rusch ldquoBehavior ofconcrete under biaxial stressesrdquo ACI Journal vol 66 no 8pp 656ndash666 1969

[70] R K Abu Al-Rub and S-M Kim ldquoComputational applica-tions of a coupled plasticity-damage constitutive model forsimulating plain concrete fracturerdquo Engineering FractureMechanics vol 77 no 10 pp 1577ndash1603 2010

[71] G Z Voyiadjis Z N Taqieddin and P I Kattan ldquoeoreticalformulation of a coupled elastic-plastic anisotropic damagemodel for concrete using the strain energy equivalenceconceptrdquo International Journal of Damage Mechanics vol 18no 7 pp 603ndash638 2008

[72] D A Hordijk Local approach to fatigue of concrete PhDthesis Delft University of Technology DelfteNetherlands1991

[73] M Arrea and A R IngraffeaMixed-Mode Crack Propagationin Mortar and Concrete Department of Structural Engi-neering Cornell University Ithaca NY USA 1982


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not applicable to incompressible materials which have a valueof Poissonrsquos ratio around 05 (eg fresh concrete andconcrete under confinement) since the volumetric strain isnull erefore the present model is limited to the com-pressible materials characterized by Poissonrsquos ratio lowerthan 05

e second challenge is the modeling of tension-compression behaviors of concrete materials which ex-hibit large differences in their peak strengths and postpeakstages as well as the unilateral effect characterized by a totalor partial recovery of the degraded stiffness during thetension-compression load reversal e classical strain-driven damage models incorporating one damage crite-rion alone may be incapable of predicting such distinctbehaviors Among them (eg [27ndash32]) one case can bementioned as an example the predicted strength ofconcrete under biaxial compression is seriously lower thanthat under uniaxial compression To solve the problemthis paper proposes two separate damage criteria de-scribing the state of damage under tension and compres-sion respectively whereby the distinct behaviors can bereproduced in a physically realistic fashion Due to thescope of this paper the unilateral effect will not bediscussed

e third challenge is to avoid the lack of meshobjectivity of numerical results when the modeling ofstrain-softening is taken into account At the stage ofstrain-softening the concrete does not lose its entire load-carrying capacity but exhibits a progressively decreasingresidual strengtherefore capturing the strain-softeningis important because it is responsible for predicting thebehavior of overloaded structures However damagemodels based on conventional continuum theories usuallyproduce mesh-dependent results in FE analyses [6] emesh dependency can be further classified into mesh-size[33] and mesh-bias [34] dependencies e mesh-sizedependency which originates from the loss of ellipticityof the governing partial differential equations for staticproblems [35] which means that both the computed en-ergy dissipation due to strain-softening and the corre-sponding force-displacement response are governed by thefineness of the FE mesh [36] It also reflects a physicallyunrealistic fact that the infinitesimal element size leads to avanishing energy dissipation during the structural failure[37] e mesh-bias dependency which is a result of theinadequacy of the spatial discretization procedure [38 39]means that the numerical result depends on the orientationof the FE mesh in the sense that the computed direction ofthe strain localization band tends to propagate alongcontinuous mesh lines [40] which may give rise to spu-rious crack trajectories Although the problem of mesh-bias dependency is important for damage modeling ofconcrete materials this study focuses only on the solutionto the mesh-size dependency

To solve the problem of mesh-size dependency in theframework of continuum mechanics various types of regu-larization methods have been developed that can be distin-guished by the crack band approach (or fracture energyregularization) [33 41 42] nonlocal models including the

integral-type [6 43 44] and the gradient-type [13 45]micropolar method [46] and viscous or rate-dependent [11]methods On one hand the nonlocal method seems to bemore recent (eg [28 30 47]) However the nonlocal modelshave their own difficulties such as (i) incorrect crack initia-tion ahead of the crack tip [48] and (ii) incorrect shieldingeffect with nonzero nonlocal interactions across a cracksurface [49] Furthermore nonlocal methods have no po-tential for large-scale computations of concrete structuressince they require small enough elements eg their sizes areat least one-third of the characteristic length of the material[50] to subdivide strain localization bands of the structurewhich are usually a priori unknown On the other hand thecrack band approach is widely used inmost of the commercialFE packages and is still an important analytical tool forpractical applications (eg [51 52]) One goal of the presentwork is to propose a constitutive model which needs to ensurea high algorithmic efficiency and an easy implementationtherefore a fracture energy regularization based on the crackband approach [33] is adopted

One key parameter in the crack band approach is thecrack bandwidth through which the fracture energy(regarded to be a material property) is related to the vol-umetric (or specific) fracture energy and it therefore ensuresan objectivity of the energy dissipation regardless of themesh refinement e existing estimations of the crackbandwidth can be divided into three categories (i) methodsbased on the square root of the element area (for two-dimensional elements) or the cubic root of the elementvolume (for three-dimensional elements) (eg [53ndash55]) (ii)the notable methods proposed by Oliver [56] and (iii)projection methods (eg [57ndash60]) e methods in the firstcategory which estimate the crack bandwidth as a constantduring the numerical analysis process are less applicable tosituations with irregular meshes or arbitrary crack directions[56] e methods in the last two categories in which crackbandwidths are varied with respect to crack directions andspatial positions solve the aforementioned problemsHowever they may encounter difficulties such as excessivelylarge estimation [59] or misprediction of the crack band-width is paper therefore develops an improved estima-tion of the crack bandwidth to circumvent the abovedifficulties Moreover two different crack bandwidths (onefor the tensile strain-softening and the other for the com-pressive strain-softening) are taken into account in thepresent study because of which concrete exhibits quitedifferent localization behaviors in tension and compression[61] It should also be noted that the crack band approachcan only be regarded as a partial regularization method formesh-size dependency in the sense that the objectivity of theenergy dissipation is ensured [62] and thus the globalstructural response can be captured correctly [56] but themathematical description of the rate boundary valueproblem (BVP) still remains ill-posed [63]

e main purpose of this study is not to describe all themechanical behaviors of concrete but rather to give meth-odological efforts to solve the aforementioned difficultieserefore the plastic dissipation is not considered and theisotropic damage is assumed for simplification Accordingly






ρψ+v ε+


+ ρψd εd dt dc( 1113857

(1)

ρψ+v ε+

v dt dc( 1113857 32


v



(2)


(3)


εv Iv ε (4)

εd Id ε (5)

Iv 13Iotimes I (6)

Id Is minus Iv (7)




ε+v H(trε)εv (8)







κi(t) maxτlet

εeqi (τ)( 1113857 (i t or c) (11)


εeqi 1

E0αiI1σ +βi

J2σ


(12)

with


I1σ trσe


16I21σ

σe C0 ε

(13)

αt 12

1minusft

fc1113888 1113889

βt

3

radic

21 +

ft

fc1113888 1113889

αc 1minusfc

fbc

βc 3

radic2minus

fc

fbc1113888 1113889

(14)



di gi κi( 1113857 (i t or c) (15)


dt gt κt( 1113857

0 if κt le κ0t

1minusκ0tκt


1113888 1113889

At


1 if κt ge κut

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

dc gc κc( 1113857

1minus1


if κc le κ0c

1minusκ0ccκc


1113888 1113889


1 if κc ge κuc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)






bull

ge 0


bull

0 (i t or c)

(18)


ϕdbull

σ εbullminus ρψ

bullge 0 (19)


following equation

ϕd

bull

σminus ρzψzε

1113888 1113889 εbull

+ Ytdt

bull

+ Ycdc

bull

ge 0 (20)


Yi minusρzψzdi

minus12ε

zC

zdi

ε (i t or c) (21)



σ ρzψzε

(22)


ρz2ψzεzε

(23)



ϕdbull Ytdt

bull+ Ycdc

bullge 0 (24)




Gft l

(P)t gft l

(P)t int

κut


Gfc l

(P)c gfc l

(P)c int

κuc


where Gft and G




fc are the tensile




κut 1 + At( )Gf

t

l(P)t ft+1minusAt

2κ0t (27)



κuc 32

Gfc

l(P)c fc+ κ0c (28)



l(P)i lmaxi minus l

mini

∣∣∣∣∣∣∣∣∣∣ (i t or c) (29)

with

lmaxt max


lmint min


lmaxc max


(30)

lminc min


0

ft

At = 02At = 06At = 10

At = 20At = 60

κt0 ε

σ








l(R)t A


2A


(32)



l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with


(34)



l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with


τmax minus τmin

τmax maxi1Nc


τmin mini1Nc



+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone


Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f









Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






ρψ+v ε+


+ ρψd εd dt dc( 1113857

(1)

ρψ+v ε+

v dt dc( 1113857 32


v



(2)


(3)


εv Iv ε (4)

εd Id ε (5)

Iv 13Iotimes I (6)

Id Is minus Iv (7)




ε+v H(trε)εv (8)







κi(t) maxτlet

εeqi (τ)( 1113857 (i t or c) (11)


εeqi 1

E0αiI1σ +βi

J2σ


(12)

with


I1σ trσe


16I21σ

σe C0 ε

(13)

αt 12

1minusft

fc1113888 1113889

βt

3

radic

21 +

ft

fc1113888 1113889

αc 1minusfc

fbc

βc 3

radic2minus

fc

fbc1113888 1113889

(14)



di gi κi( 1113857 (i t or c) (15)


dt gt κt( 1113857

0 if κt le κ0t

1minusκ0tκt


1113888 1113889

At


1 if κt ge κut

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

dc gc κc( 1113857

1minus1


if κc le κ0c

1minusκ0ccκc


1113888 1113889


1 if κc ge κuc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)






bull

ge 0


bull

0 (i t or c)

(18)


ϕdbull

σ εbullminus ρψ

bullge 0 (19)


following equation

ϕd

bull

σminus ρzψzε

1113888 1113889 εbull

+ Ytdt

bull

+ Ycdc

bull

ge 0 (20)


Yi minusρzψzdi

minus12ε

zC

zdi

ε (i t or c) (21)



σ ρzψzε

(22)


ρz2ψzεzε

(23)



ϕdbull Ytdt

bull+ Ycdc

bullge 0 (24)




Gft l

(P)t gft l

(P)t int

κut


Gfc l

(P)c gfc l

(P)c int

κuc


where Gft and G




fc are the tensile




κut 1 + At( )Gf

t

l(P)t ft+1minusAt

2κ0t (27)



κuc 32

Gfc

l(P)c fc+ κ0c (28)



l(P)i lmaxi minus l

mini

∣∣∣∣∣∣∣∣∣∣ (i t or c) (29)

with

lmaxt max


lmint min


lmaxc max


(30)

lminc min


0

ft

At = 02At = 06At = 10

At = 20At = 60

κt0 ε

σ








l(R)t A


2A


(32)



l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with


(34)



l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with


τmax minus τmin

τmax maxi1Nc


τmin mini1Nc



+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone


Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f









Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


I1σ trσe


16I21σ

σe C0 ε

(13)

αt 12

1minusft

fc1113888 1113889

βt

3

radic

21 +

ft

fc1113888 1113889

αc 1minusfc

fbc

βc 3

radic2minus

fc

fbc1113888 1113889

(14)



di gi κi( 1113857 (i t or c) (15)


dt gt κt( 1113857

0 if κt le κ0t

1minusκ0tκt


1113888 1113889

At


1 if κt ge κut

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

dc gc κc( 1113857

1minus1


if κc le κ0c

1minusκ0ccκc


1113888 1113889


1 if κc ge κuc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)






bull

ge 0


bull

0 (i t or c)

(18)


ϕdbull

σ εbullminus ρψ

bullge 0 (19)


following equation

ϕd

bull

σminus ρzψzε

1113888 1113889 εbull

+ Ytdt

bull

+ Ycdc

bull

ge 0 (20)


Yi minusρzψzdi

minus12ε

zC

zdi

ε (i t or c) (21)



σ ρzψzε

(22)


ρz2ψzεzε

(23)



ϕdbull Ytdt

bull+ Ycdc

bullge 0 (24)




Gft l

(P)t gft l

(P)t int

κut


Gfc l

(P)c gfc l

(P)c int

κuc


where Gft and G




fc are the tensile




κut 1 + At( )Gf

t

l(P)t ft+1minusAt

2κ0t (27)



κuc 32

Gfc

l(P)c fc+ κ0c (28)



l(P)i lmaxi minus l

mini

∣∣∣∣∣∣∣∣∣∣ (i t or c) (29)

with

lmaxt max


lmint min


lmaxc max


(30)

lminc min


0

ft

At = 02At = 06At = 10

At = 20At = 60

κt0 ε

σ








l(R)t A


2A


(32)



l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with


(34)



l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with


τmax minus τmin

τmax maxi1Nc


τmin mini1Nc



+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone


Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f









Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ϕdbull Ytdt

bull+ Ycdc

bullge 0 (24)




Gft l

(P)t gft l

(P)t int

κut


Gfc l

(P)c gfc l

(P)c int

κuc


where Gft and G




fc are the tensile




κut 1 + At( )Gf

t

l(P)t ft+1minusAt

2κ0t (27)



κuc 32

Gfc

l(P)c fc+ κ0c (28)



l(P)i lmaxi minus l

mini

∣∣∣∣∣∣∣∣∣∣ (i t or c) (29)

with

lmaxt max


lmint min


lmaxc max


(30)

lminc min


0

ft

At = 02At = 06At = 10

At = 20At = 60

κt0 ε

σ








l(R)t A


2A


(32)



l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with


(34)



l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with


τmax minus τmin

τmax maxi1Nc


τmin mini1Nc



+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone


Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f









Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







l(R)t A


2A


(32)



l(O)t sumNc

i1ϕizNi(x)

zx middot n(x)

minus1

(33)

with


(34)



l(G)t sumNc

i1τizNi(x)

zx middot n(x)

minus1

(35)

with


τmax minus τmin

τmax maxi1Nc


τmin mini1Nc



+ =

(b) (c) (d)

Stre

ss

Strain

Stre

ss

Strain

(a)

Dam

age

zone


Trac

tion

Separation

Gtf

lt(P)gtf = Gt

f









Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








Gaussian pointCrack


n(x)t(x)

lt(P)

lc(P)

x

y

x

n(x)t(x)

lt(P)

lc(P)

x



20

lt(O)lt(G)

lt(P)

10

n(x)

x

yϕ1 = 0 ϕ2 = 1

ϕ3 = 1ϕ4 = 0

xθ

(a)


lt(O) 20

ϕ4 = 1 ϕ3 = 1

ϕ2 = 0ϕ1 = 0

n(x)

10x

y

x

θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degree)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)










lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









lt(O)

lt(P) lt(G)

10

20

n(x)

x

y

ϕ3 = 1

ϕ2 = 1ϕ1 = 0

xθ

(a)


ϕ3 = 1

ϕ2 = 0ϕ1 = 0

lt(O)

10

20n(x)

x

y

x θ

(b)

00

05

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100 120 140 160 180Angle θ (degrees)

Tens

ile cr

ack

band

wid

th

lt(P)lt(G)lt(O)

A

2A

(c)


Mesh A

σ1

σ2 = ασ1

ndash14

ndash12

ndash10

ndash08

ndash06

ndash04

ndash02

00

02



σ1| fc|

σ2|fc|












00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











00

02

04

06

08

10

12

14




Mesh A

σ1

σ2 = ασ1

ε1 (times10ndash3)

σ 1f c


125125 1111 13

125

5

35

P

5

60

Concrete specimen

Strain gauge

Loading platen







2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





2A



00

05

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100



Ave

rage

stre

ss σ

avg (

MPa

)


F

100mm

25mm



(a) (b)

(c) (d)








00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C

Mesh D


Forc

e F (k

N)


(a)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015

Mesh B

Mesh C


(b)

Forc

e F (k

N)


00

05

10

15

20

25

30

000 005 010 015


(c)


2A


306

156

203 2036161 397397

82

A B

013P P

113PCSteel beam

Concrete beam

Steel plate









4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








4 Conclusions



(a) (b)

(c)


0

20

40

60

80

100

120

140

160

000 005 010 015



CMSD (mm)

Load

P (k

N)









B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








B B B

(a) (b) (c)

P P P


P

(a)

P

(b)

P

(c)





Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Data Availability




Acknowledgments


References



















































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Tension-CompressionDamageModelwithConsistentCrack ...downloads.hindawi.com/journals/ace/2019/2810108.pdf · tension-compression load reversal. e classical strain- driven damage models