Mesomechanic Analysis of Time Dependent Concrete Behavior

R. Lorefice & G. EtseUniversidad Nacional de Tucuman, CONICET, ArgentinaC.M Lopez & I. CarolETSECCPB-UPC, E-08034, Barcelona, Spain

ABSTRACT: This paper presents a mesomechanic approach for the analysis of time dependent problemsin concrete structures by using a consistent viscoplastic formulation at meso level of observation. Creepand relaxation phenomena are addressed under the unified theoretical framework provided by Perzynasviscoplasticity theory.

1 INTRODUCTION

It is widely accepted that heterogeneous materi-als like concrete require different levels of observa-tions to fully understand the mechanism govern-ing their response behaviors when they are sub-jected to complex loading cases that activate non-linear responses. This is particularly the case whentime dependent effects are involved as the pre-cise calibration of traditional macroscopic mod-els based on continuous or smeared-crack conceptneed observations at meso and, moreover, microlevels to accurately evaluate and distinguish therate sensitivity of the different constituents as wellas their influences in the overall behavior. Severalauthors have already recognized the importance ofmesostructure evaluations of heterogeneous mate-rial responses with three main features: it includesa non-regular array of particles representing thelargest aggregates, a homogeneous matrix mod-elling the behavior of mortar plus small aggregates,and the interfaces between the two phases. Indeed,the mesomechanic approach provides a consistentextension of the Fictitious Crack Model conceptinto realistic finite element analyses of discrete fail-ure processes of brittle materials.

After reviewing the main features of theconsistent viscoplastic interface model for rate-dependent mesomechanic analyses by Lorefice,Etse and Carol (2005) the predictions at the con-stitutive and mesostructure levels of observationof creep and relaxation processes are evaluated.Thereby, two different approaches are consideredfor the matrix in case of the meso-level analyses

of time-dependent effects in concrete: on the onehand, an aging Maxwell chain assigned to the con-tinuum elements conforming the matrix phase asproposed by Lopez et al. (2001),(2003). On theother hand, a discrete approach in which the rate-dependent non-linear response of the matrix ismodelled by means of the interface elements in-serted in-between the continuum elements of thematrix phase. Numerical analyses of relaxationand creep processes are performed to evaluate dif-ferent aspects of the concrete rate-dependent re-sponse behavior. The results demonstrate the ca-pability of the mesoscopic analysis based on theproposed viscoplastic interface model formulationto reproduce the main features of concrete rate-dependency including stress relaxation under con-stant strain and basic creep (without humiditychanges) under constant stress.

2 TIME DEPENDENCY OF CONCRETE BE-HAVIOR

Under sustained loads/strains, the influence ofcreep and/or relaxation effects becomes relevantto the evaluation of the life-time and durabilityof concrete structures. The creep mechanism hasbeen addressed by several authors, see a.o. Bazantand Wittman (1982), Bazant and Panula (1997),Domone (1974), Neville (1996), Westman (1994),Granger (1994), Granger and Bazant (1995), Ben-boudjema (2002). At meso-level of observation, ba-sic creep has been investigated by Lopez et al.(2001),(2003) and Ciancio et al. (2003). When aconcrete specimen is subjected to sustained stress

it experiences a gradual increase in deformationwith time as shown in Figure 1.

DELASTICDCREEP PERMANENT

DCREEP RECOVERY

DELASTIC

(a) (b) (c) (d)

t0 tu time

De

(a) (b) (c) (d)

t0 tu time

s

s0

(a) (b) (c) (d)

DELASTIC DELASTIC

DCREEP

DCREEP PERMANENT

DCREEP RECOVERY

DELASTIC

Figure 1: Creep deformation definitions: (a) origi-nal length, (b) elastic deformation, (c) creep load-ing, and (d) permanent creep after loading (Weiss1999)

sRELAXATION

(a) (b) (c) (d)

t0 tu time

(a) (b) (c) (d)

t0 tu time

e

e0

(a) (b) (c) (d)

DELASTICDCREEP PERMANENT

DCREEP RECOVERY

ePERMANENT

s

sELASTIC

Figure 2: Creep/relaxation definitions: (a) origi-nal length, (b) elastic deformation, (c) creep load-ing, and (d) permanent creep strain after loading(Weiss 1999)

Def

orm

atio

n

Time

TotalCreepDeformation

DryingCreepDeformation

BasicCreepDeformation

DryingShrinkageDeformation

ElasticDeformation

Figure 3: . Time-dependent deformations in con-crete subjected to sustained load (after Neville1996)

This type of time dependent deformation isknown as creep. Another form of concrete time-

dependency is the stress relaxation that takesplaces when a restrained concrete specimen is sub-jected to a constant deformation. This effect, thatis produced by the same causes as creep, leads toa progressive reduction of the stress with time asshown in Figure 2.

Actually, the mechanisms that result in creep ofconcrete are complex. It is generally accepted thatmoisture movement within concrete is the mainmechanism responsible for creep, see Mehta andMonteiro (1993). When a hydrated cement pasteis subjected to sustained stress, depending on themagnitude of stress, it looses a significant amountof the water physically adsorbed between the layersof calcium silicate hydrate (CSH) resulting in creepdeformation (Mehta and Monteiro 1993). Bazant(1982) has proposed that creep of concrete oc-curs when interlayer water migrates from stressedgel (micropores) to stress-free zones (larger pores).Since the bonds and the contacts between the lay-ers of cement gel are highly disordered and un-stable, this water migration helps the solid parti-cles to move out of the loaded regions. The mi-grations of interlayer water and of solid particlesin cement gel are responsible for creep. Recently,Rossi (1988),(1994), proposed a new hypothesisconcerning the physical mechanism of basic creep.According to this hypothesis when a concrete spec-imen is subjected to loading, microcracks developthroughout the volume of the specimen. This re-sults in a gradient of water molecule concentration,which leads to a migration of water vapor fromcapillaries to the microcracks. The local movementof water causes significant drying of the capillar-ies. Precisely, this post-cracking local drying of thecapillaries is the main cause of basic creep.

3 VISCOPLASTICITY THEORY

Among the different proposed constitutive theo-ries for rate dependent materials, see a.o. Perzyna(1963),(1966), Duvaut and Lions (1972), Bodnerand Partom (1975), the most widely accepted isthe viscoplastic formulation by Perzyna. The fea-tures of the Perzynas model have been addressedby several authors, see a.o. Simo (1989), Sluys(1992), Wang (1997), Simo and Hughes, (1998),Etse and Willam (1999), Carosio et al. (2000), Etseand Carosio (2002). Restricting our analysis to thesmall-strain case, the total strain rate can be de-composed into an elastic and a viscoplastic partwhich accounts for both irreversible and viscousdeformation,

= e+ vp (1)

The stress rate is related to the strain rate by theconstitutive relation

= E : e (2)whereby, E is the fourth-order material tangent op-erator. The evolution of the viscoplastic strain ratein the Perzyna model is defined as

vp = G(,F,) =1

(F )m (3)

with the viscosity parameter,m the viscoplasticpotential gradient defined as m =A1 : n, beingn the gradient tensor to the yield surface F andAthe fourth order transformation tensor, and (F ) adimensionless monotonically increasing over-stressfunction defined as

(F ) =

[F (,q)

F0

]N(4)

F0 is a normalizing factor, usually chosen equal tothe initial yield limit and N a constant defining theorder of the Perzynas viscoplatic formulation.The evolution law for the set of harden-ing/softening variables q is defined as

q =1

(F )H :m, (5)

being H a suitable tensorial function of the statevariables. Like in classical elastoplasticity, the vis-coplastic flow rule takes the form

vp = m (6)

Combining equations (3) and (6), the viscoplasticmultiplier in the Perzynas theory can be writtenas

=1

(F ) (7)

From eqs. (4) and (7) a viscoplastic constrain con-dition can be defined as (Ponthot (1995), Etse andWillam (1999))

F = F 1(

)= 0 (8)

Equation (8) can be viewed as a generalizationof the inviscid yield condition F = 0 for rate-dependent Perzyna type materials. The name con-tinuous formulation is due to the fact that the con-dition = 0 (without viscosity effect) leads to theelastoplastic yield condition F = 0. Moreover, fromequation (7) follows that when 0 the consis-tency parameter remains finite and positive since

also the over-stress goes to zero. The other extremecase, , leads to the inequality F < 0 for everypossible stress state, indicating that only elasticresponse may be activated. The constraint condi-tion defined by equation (8) allows a generalizationof the Kuhn-Tucker conditions which may be nowwritten as

F = 0 , 0, F 0. (9)Finally, the viscoplastic consistency condition ex-pands into

F = n : + rq + s = 0 (10)

where

r =F

q=

(F

q

1( )q

)h (11)

and

s=1()

(12)

4 RATE-DEPENDENT INTERFACE MODEL

In this section the rate-dependent extension of theinterface model by Carol et al. (1997) is summa-rized. The viscoplastic yield condition of the inter-face constitutive model can be expressed as

F = 2 (c tan)2 + (c tan)2 ()(13)being and the normal and tangential stresscomponents at the interface with the tractionstrength (vertex of hyperbola), c the apparentcohesion (shear strength) and the friction an-gle.The evolution of the fracture process is drivenby the cracking parameters and c, which dependon a single parameter: the work spent on the ratedependent fracture process during crack formation,qvcr, see Lorefice et al. (2005) that is defined as

qvcr = uvcr + vvcr if 0 (14)or

qvcr = vvcr(1 -

tan) if < 0 (15)

whereby uvcr and vvcr are the normal and tan-gential (critical) rate-dependent rupture displace-ments, respectively. As defined by Lorefice et al.(2005) the viscoplastic flow of the rate dependentinterface model is fully associated in tension whilenon-associated in compression, with the gradient

vectors m and n to the viscoplastic potential andyield surface, respectively, related to each other bymeans of the transformation matrix A as m = An.The continuum viscoplastic form of the rate de-pendent interface constitutive model is defined bythe following set of equations:

u = uel + uvcr (16)

uel = E1 (17) = E(u uvcr) (18)

where T = (, ) is the stress vector, uT = (u, v) isthe rate of the relative displacement vector which isdecomposed into the elastic and viscoplastic com-ponents, uel and uvcr, respectively, and E the 2x2elastic stiffness matrix defined as

E =(

EN 00 ET

)(19)

The viscoplastic consistency condition in equation

(10) takes now the form F = nt + rq + s = 0,with q = q(c,) representing the evolution of theinternal variable during the cracking process andr and s defined as:

r =

(dF

dc

dc

dqvcr+dF

d

d

dqvcr

)(qvcr

uvcr

)Tm (20)

The non-linear system of equations is solved us-ing a Newton-Raphson iterative procedure in theframework of the Closest Point Projection Method(CPPM) starting from the expansion of a Taylorsseries truncated at the first term

iF =i1 F +i1(

dF

d

)id = 0 (21)

from equation (21), the differential change in theelasto-viscoplastic multiplier is derived as

id = i1F[i1(

dF

d

)]1(22)

Assuming the hypothesis: d = d/t, see Pon-thot (1995), Wang (1997), Carosio et al. (2000),the derivative of the viscoplastic yield function re-spect to takes the form

dF

d= nt

d

d+ r

t(23)

Replacing in eq.(23) d/d=Emm, with Em =(E1 +M)1 = [E1 +m/]1 the mod-ified elastic matrix and M the Hessian matrixfor the interface model as given in Lorefice et al.(2005), the expression for d finally results

id = i1F

i1[ntEmm+ r t]

(24)

from where the increments of the stress vector andstate variables can be obtained.

5 CONSTITUTIVE ANALYSIS

In this section numerical examples are presentedat constitutive level to evaluate model perfor-mance when simulating time dependent failureprocesses in concrete. Two different types of loadcases are examined: a) A constant normal displace-ment/strain field is applied, see Figure 2, that acti-vates plastic material behavior as the correspond-ing stress state exceeds the yield surface. In thiscase, the evolution of the stress state in time do-main is of interest and is known as relaxation pro-cess. (b) A constant load/stress field is applied tothe interface model that exceeds the yield limit.Our interest in this case is to study the evolutionof the displacements in time domain or creep.The analysis in this section will explore the re-

laxation test for the following set of material pa-rameters: EN = 1.E7MPa/m, = 2.0MPa, G

fI =

0.00003MPa.m, GfII = 10GfI . The yield limit in

pure traction for the considered parameters is= 2.0MPa. A displacement of 3.E 7m was ap-plied that corresponds to a uniform stress state of3.0MPa. The results of the elastoplastic and theelasto-viscoplastic computations for several valuesof the viscosity parameter are plotted in Figure 4.

s

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Time(Hours)

(MPa)Viscosity=1.E7

Viscosity=1.E6

Viscosity=1.E4

Elastoplasticstrength

Figure 4: Relaxation test - constitutive level

It is important to note that in the relaxation test ofelastoplastic materials, the stresses cannot exceed

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Time(Hours)

(MPa) Perfect ViscoplasticitySoftening

Hardening

ViscoelasticMaxwell

ElastoplasticStrength

s

Figure 5: Relaxation test - comparison of differentmodels for t = 1

the yield limit, while in case of viscoplastic mate-rials they are allowed to exceed this limit duringtransient stage regime. Figure 5 compares the vis-coelastic solution of a Maxwell chain with the vis-coplastic one for a relaxation time t = /E = 1.In the numerical analysis three different harden-ing/softening behaviors were considered in the vis-coplastic interface model: a perfect viscoplastic, ahardening and a softening behavior. From Figure 5can be observed that while for perfect viscoplastic-ity the stress state relaxes up to the elastoplasticstrength limit, the relaxation process in the vis-coelastic model has no limit and causes a stress re-duction up to zero value in the final stage. On theother hand we observe that contrarily to the per-fect viscoplasticity case, the hardening and soften-ing viscoplastic behaviors lead to final stress stagesthat do not agree with the strength limit.

The results in Figure 4 illustrate clearly the con-trolling effect of the viscosity on the rate of thestress relaxation process.

From a physical standpoint, it is important torealize that the controlling factor in the relaxationprocess is the relative time t/t. The absolute timet [0; 1) is regarded to be short or long only whencompared to the natural relaxation time t = /E.We explore now the creep test for the same setof material parameters and for the softening be-havior of the viscoplastic interface model. At timet = 0, a constant load is applied to impose a ten-sile stress state just above the elastoplastic yieldlimit, see Figure 1. The evolution of the normaldisplacements uN against time is plotted in Figure6 for several values of the viscosity parameter .Once the stress state reaches the limit strength,the softening behavior is activated and the creepprocess takes place.

This is the reason for the increasing slope of thedisplacement rate in Figure 6, which inversely de-pends on the viscosity , i.e. as the viscosity tends

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

2.0E-04

0 50 100 150 200 250 300 350 400

Time (Hours)

uN (cm)

Viscosity = 1.E6

Viscosity = 5.E5

Viscosity = 1.E5

Figure 6: Creep curves - constitutive level

to zero the slope of the creep displacement ratetends to be vertical.

6 MESO-LEVEL ANALYSIS OF TIME-DEPENDENT CONCRETE FRACTURE

In this section the creep and relaxation processesare evaluated at the mesostructure level. To thisend, the plane finite element mesh showed in Fig-ure 7 is considered as composed by polygonal ag-gregates with randomly generated shapes and lo-cations, according to Stankowski (1990) and Lopez(1999). The aggregates are embedded on a ma-trix phase which represents the mortar plus smallaggregates. A number of zero-thickness interfaceelements are inserted along the aggregate-matrixinterfaces, and also across the mortar matrix inorder to capture the most relevant features of itsfailure mechanism. In the present study, the tri-angular finite elements representing the aggregateand mortar phases follow an elastic constitutivelaw, while the rheological non-linear behavior isattributed to the zero-thickness interfaces accord-ing to the previously described elasto-viscoplasticrate-dependent model.Firstly, we analyze the stress relaxation case

under a uniaxial tensile stress state consideringtwo meshes: a coarse mesh with a 4x4 arrangementof aggregates and a fine mesh with a 6x6 arrange-ment, see Lopez (1999). In this load case, theevolution of the stress relaxation process dependson the applied stress level. Two situations aredistinguished, namely: a) the applied stress levelexceeds the strength of the mortar-aggregatesinterfaces, representing the weakest interfaces,and b) the stress level exceeds the inviscidstrength of the mortar-mortar interfaces. Theconsidered material parameters are as follows:En = Et = 10

9MPa/m, = 2MPa, c = 7MPa,tan = 0.8, GIf = 0.03N/mm, G

IIf = 10G

If and

dil = 40MPa (the remaining parameters are set

Figure 7: Finite Element arrangement 6x6 Mesh a)mortar elements, b) aggregate elements, c) inter-face locations, d) discretization detail

to zero). For the mortar-mortar joints, the sameparameters, except that = 4MPa, c = 14MPaand GIf = 0.06N/mm. For the continuum ele-ments Em = 25000MPa and Eag = 70000MPa. Inboth cases, =0.2, while the viscosity parameter = 106MPa/s and the time step t = 0.014days. For case a), the numerical analysis isperformed imposing a deformation in such away that the reached stress stage exceeds by50% the strength limit of the mortar-aggregatesinterfaces, i.e. (/ = 1.5). In case b) the applieddisplacement leads to an overstress that exceedsthe inviscid strength of the mortar-mortar jointsby 50%. The numerical responses for both casesare illustrated in Figure 8. In the two consid-ered discretizations the case a) results in a flatrelaxation curve indicating a very reduced timedependent behavior, while the case b) leads to avery pronounced relaxation process, according toan exponential descending curve. The reason forthis behavior is that the stress level in case b)activates the viscoplasticity formulation of bothmortar-aggregates and mortar-mortar interfaces.Thus, a strong rate dependent process initiatesduring which the viscoplastic overstress relaxes tothe inviscid strength of 3.0MPa according to theadopted material parameters. In the case a) thisdoes not occur as only few mortar-aggregate jointsare activated by the applied stress level, while allthe mortar-mortar interfaces remain inactive.

We analyze now the creep test by means ofthe 6x6 aggregate FE-mesh. To appropriately re-produce the boundary conditions of the test (the

stresses are prescribed while the vertical displace-ments are almost uniform due to the high stiffnessof the loading plate) the discretization in this nu-merical analysis includes also the upper plate aswell as the interface between the steel plate andthe concrete specimen. The computational evalu-ation was repeated for different stress levels thatare applied and kept constant on the 28 days oldconcrete specimen. Figure 9 illustrates the nomi-nal stress-strain evolution in the form of the well-known isochronous curves. The curve on the leftextreme side of this figure from which evolve allthe others, represents the instantaneous responseobtained for a very low strain rate. The differ-ent curves in Figure 9 follow the instantaneous re-sponse up to a certain stress level from which thestress is kept constant. The isochronous curves areobtained by connecting the points that correspondto the same time (amount of days) in the differentanalyses.

Figure 9 also shows the isochronous curves byLopez et al (2001) for the same problem that wereobtained with a mesomechanic formulation includ-ing a elastoplastic behavior for the interface ele-ments while a linear viscoelastic behavior is as-signed to the continuum elements of the matrixby means of a Maxwell chain model. As the de-layed strains obtained with the viscoplastic modelare irreversible once the interfaces are inelasticallyactivated at the stress level /f

c = 0.4 then the

corresponding isochronous curves are shown start-ing at this stress level.

The numerical results in Figure 9 reproduce rel-evant aspects that were experimentally observedsuch as the non-linear stress-strain behavior whichprogressively increases with the applied stresslevel. Figures 10 and 11 illustrate the crack pat-tern in terms of the released energy obtained withthe viscoplastic interface model for the stress level/f c = 0.95 and for t t=100 and t t=10000days, respectively. In the same way, Figures 12 and13 show similar results obtained by Lopez et al.(2001) using the viscoelastic Maxwell chain for thematrix continuum elements while the interfaces re-main inviscid. We observe that under sustainedload both models predict that the microcrackingincreases with time. However, the internal mecha-nism developed in both cases is completely differ-ent. In the approach by Lopez et al. (2001),(2003)the viscoelastic behavior of the matrix continuumelements generates an internal stress redistributionand a stress transference from the matrix to theaggregates that are considered elastic. This is re-sponsible for the progressive crack evolution in thematrix-aggregate interfaces under constant exter-nal loading. Contrarily, in the elasto-viscoplastic

interface approach, time evolution is basically re-sponsible for the increasing cracking. The elasticbehavior turns into the inviscid elastoplastic oneas time tends to infinity, providing the activationof the inelastic response.In both approaches the crack evolution is more

severe as the applied constant stress level increases.This is the reason for the increasing nonlinearityshown by the isochronous curves.The different internal responses obtained with

the considered approaches for the simulation ofthe time-dependence effect in the mesomechanicanalyses in this work, see Figure 9 infers the im-portance to consider a combination between bothtime-dependent formulations to realistically pre-dict concrete creep behavior at the mesomechaniclevel of observation.

s

2.0

3.0

4.0

5.0

6.0

0 50 100 150 200

Time(days)

(MPa) Casea)ConcreteMesostructure-4x4Mesh

Casea)ConcreteMesostructure-6x6Mesh

Caseb)ConcreteMesostructure-4x4Mesh

Caseb)ConcreteMesostructure-6x6Mesh

Figure 8: Relaxation test - concrete mesostructure

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.001 0.002 0.003

Instantaneoust-t'=18-Viscoelasticityt-t'=100-Viscoelasticityt-t'=1000-Viscoelasticityt-t'=10000-Viscoelasticityt-t'=18-Viscoplasticityt-t'=100-Viscoplasticityt-t'=1000-Viscoplasticityt-t'=10000-Viscoplasticity

/fcs/fc

e

Figure 9: Creep test - isochrone curves

7 CONCLUSIONS

From the obtained results we conclude thatthe Perzynas based rate-dependent interface

Figure 10: Energy dissipation t-t=100 days - vis-coplasticity

Figure 11: Energy dissipation t-t=10000 days -viscoplasticity

Figure 12: Energy dissipation t-t=100 days (Lopezet al. (2001))

model is able to capture the main featuresof time/rate-dependent concrete failure undersustained loads/strains, like basic creep, stressrelaxation and increasing non-linearity of theisochronous curves for increasing stress levels.The mesomechanic level of observation combinedwith a viscoplastic theory allows to numerically

Figure 13: Energy dissipation t-t=10000 days(Lopez et al. (2001))

evaluate the influence of the composite mesostruc-ture including the coupling between creep andfracture processes in a realistic manner. However,the inclusion of the viscoelastic Maxwell chainmodel for the continuum matrix elements asconsidered by Lopez et al (2001), (2003) seems tobe necessary to fully capture the time dependenteffects of concrete at the mesostructure level ofobservation.

8 REFERENCES

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