The tirne scale in concrete fracture: A model based on partitions of unity

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1 Fracture Mechanics of Concrete Structures, e Borst et al (es) 21 Swets & Zeitlinger, Usse, SBN The tirne scale in concrete fracture: A moel base on partitions of unity G.P.A.G.van Zijl Faculty of Architecture, Delft University of Technology, The Netherlans/Civil Engineering Department, University of Stellenbosch, South Africa G.N.Wells Koiter nstitute Delft, Delft University of Technology, The Netherlans ABSTRACT: ntense research efforts have been put in the formulation of theories for crack initiation an propagation in concrete. Yet, little attention has been pai to the time-epenence of fracture, espite evience of its significance. This paper reports a finite element moel which captures the time scale in concrete fracture. Visco-elasticity is employe to capture bulk creep. n the fracture process zone a ifferent time scale acts. Therefore, a rate-epenent cracking resistance is moelle. A recently evelope finite element metho for moelling cohesive cracks is employe. t is base on partitions of unity, by which means isplacement jumps are introuce inepenently of the mesh structure. This avois the requirement of ense meshes by regularise continuum approaches to moel localisation, an a priori knowlege of where cracks occur for stanar iscrete cracking approaches via interfaces. NTRODUCTON The behaviour of concrete is highly time- an rateepenent. The significant increase in structural resistance upon loaing rate increase in the ynamic range is well-ocumente. However, more recently experimental evience of rate epenence in the quasi-static range, i.e. where inertia an wave effects are negligible, has been prouce by, for example Bafant an Gettu (1992), Zhou (1992) an Bafant an Xiang (1997). The time-epenent crack growth can cause elaye structural collapse, renering the incorporation of the time scale in computational moels imperative. The mechanisms of the time epenence are not yet fully unerstoo. The micro-structural processes in concrete, which hol the key to the rate effects, are governe by an intricate hygro-thermalmechanical inter-epenence. This has inspire couple approaches to moel concrete behaviour (Biot 1955; Coussy 1955; Lewis an Schrefter 1998). Such an approach is computationally costly, even if a macroscopic approach is followe. This is worsene by the requirement of ense finite element meshes in localisation zones (Askes et al. 1998; e Borst et al. 21). The alternative cohesive zone concept for moelling cracking in concrete, as traitionally applie in terms of interface elements by (Rots 1988; Schellekens an De Borst 1993), suffers from the requirement of a ptiori knowlege of crack paths. A promising solution lies in a iscontinuous moel base on the partitions of unity concept (Wells an Sluys 21). Such an approach is followe in this paper. n a step towars formulating a fully couple hygro-thermo-rnechanical moel for concrete fracture, which employs a iscontinuous moel formulation, the mechanical part is formulate in this paper. To capture the time epenence, a phenomenological approach is followe, by moelling bulk creep with linear visco-elasticity, while the finite cracking rate is capture with a rate-epenent cohesive behaviour in the iscontinuity. To verify the moel, the creep failure experiments by Zhou (1992) are analyse. 2 PARTTONS OF UNTY BASED CRACK MODELLNG The isplacement fiel in a boy crosse by a iscontinuity, figure 1, can be expresse as (Wells an Sluys 21) U=fi+ f. ii (1) where u an ii are continuous functions on Q,!-. is the Heavisie jump centere at the iscontinuity surface r. For finite element implementation, equation (1) is iscretise to rea u=na+ f. Nb (2) where N is a matrix containing the usual element shape functions, a are the regular noal egrees of 31

2 t Figure 1: Boy Q crosse by a isplacement iscontinuity r. freeom an bare 'enhance' noal egrees of freeom. The isplacement jump at a iscontinuity is given by Nb, x E r' Formally, the interpolation can be interprete as an interpolation base on the partition of unity concept (Babuska an Melenk 1997; Duarte an Oen 1996; Wells an Sluys 21). The infinitesimal strain fiel is obtaine by taking the symmetric graient of the isplacement fiel: r (3) where B is the usual matrix containing spatial erivatives of the element shape functions. The superscripts inicates that only the symmetric part of the isplacement fiel graient is taken. For convenience, the superscript will be left out of the subsequent elaborations. The starting point for the finite element formulation is the virtual work equation, which reas without boy forces: (4) where 1} are amissible isplacement variations, ecompose as (5) a is the stress fiel an i are external traction forces (see figure 1). The graient of amissible isplacement variations is given by Vfl = Vf; + [. Vfj +Dr (fj n), (6) with 8r the Dirac-elta function, centere at the iscontinu1ty an n is the normal vector to the iscontinuity, pointing to Q+. nserting equations (5) an (6) into the virtual work equation (4) leas to: f Vf;:a Q+ { Vij:a Q lo lo+ where t ( = an) are the traction forces acting at the surface r. The Dirac-elta term has been eliminate using the well-known integration property of the istribution. Since equation (7) must hol for all f; (fj = ), as well as for all ii (f; = ), inserting the iscretise expressions from equations (2) an (3) into equation (7) leas to two iscrete weak governing equations (Wells an Sluys 21): The stress rate in a visco-elastic continuum can be expresse as (van Zijl et al. 21) (J = nve (na+ycf.bb) +:E (9) where nve is an equivalent, time-epenent stiffness moulus an:e is a viscous stress term which accounts for the history. The traction rate at a iscontinuity can be expresse as t=tnb (1) where T relates the instantaneous traction an isplacement jump rates. nserting the stress an traction rate expressions into equation (8) an employing a linear time integration scheme gives K {Lla} = Llb {fe,a}-{fi,a} (11) f e,b fi,b where the stiffness matrix K has the form: fobtnvebq fo+btnvebq l [ f BTnveBQ f ntnvebq+ f Jo+ Jo+ Jr NTTNr (12) Note that a semi-analytical time integration is performe (van Zijl et al. 2lb) for an accurate solution of the history. Thereby, the history term is assume to be constant uring a time increment an oes not appear in the stiffness matrix. The internal an external force vectors are equal to: f e,a = r T- N t r ru (13a) (13b) f. = { BTaQ i,a lo (13c) 32

3 See Wells an Sluys (21) for a etaile iscussion of implementation aspects. 3 CONSTTUTVE MODEL The stress rate equation (9) can be integrate with a linear scheme, which prouces the stress increment uring the time increment /).t where [ Eo(t*) + (1-e - ~~) En(t*)] D n=l /).t/sn is assume in the iscontinuity, where K is the historically largest crack normal opening isplacement, G 1 is the fracture energy an k, is a constant, reference crack opening velocity, which, together with the moel parameters k an k 1 are obtaine by inverse analysis of experiments at various (quasi-static) loaing rates. The shearing traction ts is given by: (18) where ks is inepenent of the normal opening to preserve symmetry of the global stiffness matrix. For unloaing, the secant stiffness is use. The loaing an unloaing behaviour of a boy containing the escribe iscontinuity is illustrate in figure 2. F 1 ~-~~-~-~--~-~ (15) n this moel an aging Maxwell chain can be ientifie, with time epenent element stiffnesses E 11 (t) an viscosities 'Yin = E 11 s 11, Sn being the relaxation time of chain element n=l,2,...,n. The stress vector 1 a 11 contains the stress components in chain element n at the en of the previous time step, i.e. at time t. Note that the parameters are assume to be constant in each time interval an are evaluate at a time t S t* S t + /).t. D is the matrix representation of the fourth orer tensor D.. kt = _1_ [_v_s.. s.k+! (s.ks.,+8.18 k)] ) 1 + V 1-2V ) J 2 J J (16) which is epenent on Poisson's ratio v. A iscontinuity is extene when the maximum principal stress at any integration point in the element ahea of the iscontinuity tip excees the tensile strength ft. A rate-epenent cracking normal traction (Wu an Bafant 1993; van Zijl et al. 21) of the form -.2 ~-~--~-~--~ u Figure 2: Uniaxial loaing an unloaing behaviour. 4 CASESTUDY To stuy the time scale in concrete fracture, the case of failure uner sustaine loa is investigate. The three-point bening experiments by Zhou (1992) are analyse. The specimens were 8mm long concrete beams with section loomm x loomm an a cen- /\\ ~ q\\ n '--- k. V1 // ~ \\ h \ /f \ t ~" ~' J '~......,~~ Displacement (111111) Figure 3: Three-point bening test experimental results (Zhou 1992)..8 33

4 tral notch 4mm wie an 5mm eep. The specimens were seale to avoi rying. The results of two isplacement-controlle experiments at a constant, central eflection rate of 5 µm.s- 1, as well as creep experiments at sustaine loas of 92%, 85%, 8% an 76% of the peak loa are shown in figure 3. The experiments were analyse recently with a rate-epenent continuum plasticity moel by van Zijl et al. (21). They are re-analyse to verify the cun-ent moel, before turning to cases where the superiority of the partition of unity base moel is exploite. The material parameters were obtaine by van Zijl et al. (21) from separate experiments by Zhou (1992): Young's moulus 3 kn.mm- 2, tensile strength ft=2.66 N.mm- 2 an fracture energy G 1 =.35 N.mm- 1. For this particular case the response is insensitive to the shear stiffness coefficient ks. A 1-element Maxwell chain was fitte to relaxation test ata (figure 4). same as employe by van Zijl et al. (21), while the reference cracking rate is obtaine from the reference strain rate by multiplication with the crack ban with lb=4 mm. A weaker rate enhancement can be seen in figure 5 for the cun-ent moel. This is ue to the if- 8 ~ 6 \ 2 & 4 2 Exp. (Zhou 1992) Num. (current moel) ~-- Num. (van Zijl et al. 21) ~\ \ ~~ ~.,. "'::.-:;~... "~- '----~~~-~~-=,_,,,. ) Displacement (mm) Figure 6: Three-point bening results at fixe loaing rate. e Experimental (Zhou 1992) +--tmaxwell moel fit 15 ~~~~~~~~~-'--'-~~~ 1 2 Time(s) 3 4 Figure 4: Relaxation moulus employe. Furthermore, Zhou performe three-point bening tests at various loaing rates on notche beams, from which the cracking rate moel parameters coul be obtaine by inverse analysis. The parameters obtaine in this way are k =.5,k 1 =.1, Kr= 4 x 1-7 mm.s- 1 The rate epenence of the peak strength is shown in figure 5. The first two parameters are the 1 _ ll Experimental (Zhou 1992) Numerical (current moel) o--o Numerical: (van Zijl et al. 21) 8 ~ 6 2 & 4 -- Coarse mesh ---- Refine mesh ' Deflection rate (µm.s- 1 ) Figure 5: Loaing rate effect on peak strength Displacement (111111) Figure 7: Loa-isplacement response for three-point bening analysis. 34

5 ferent constitutive moel employe here. No attempt has been mae to improve the paramaters to obtain a better agreement with the experimental results. Firstly, the isplacement-controlle case (central eflection rate 5 µm.s- 1 ) is analyse. Goo agreement is obtaine with the experimental results, as well as with the continuum moel results (figure 6). To stuy the mesh epenence, a coarse mesh an a refine mesh are employe for the analysis. n figure 7 it can be seen that the response is insensitive to the mesh. n the subsequent analyses, the coarse mesh is employe. Next, the creep experiments are analyse. The numerical responses are shown in figure 8. The experimental observation (figure 3) that the constant velocity response forms an envelope for creep failure is confirme by the numerical results. n the creep analyses by van Zijl et al. (21), isplacement control was resume once the point of failure uner the sustaine loa was approache, to ensure that failure, or the inability of the beams to resist the sustaine central loa, was inee imminent. n the current stuy a negative iagonal term in the stiffness matrix is assume to inicate failure uner the sustaine loa. t is most important that the time scale of fracture is capture accurately. n figure 9 the times between the instant the full sustaine loa is applie an fail- 1 /\ 8 85% 8% 76% 6 ~ <1l ~ µ ~pl. control (Sµm.s- 1 ) 2 ~Cr~,, ure are shown. Goo agreement is foun with the experiments. The iscrepancy with the numerical results by van Zijl et al. (21) is ascribe to the ifferent constitutive moel, which also le to the lower peak loa enhancement with loaing rate, seen in figure 5. Also shown in figure 9 are the times to failure if the finite cracking rate is not consiere, in which case the times to failure are overestimate by several orers of magnitue. 5 CONCLUSONS The extension of a cohesive zone moel to incorporate the time scale in concrete fracture, has been presente. The moel is base on partitions of unity. Displacement iscontinuities can traverse arbitrarily through a finite element mesh avoiing a priori crack path knowlege require by iscontinuity moelling via intetfaces. Also, there is no requirement for ense meshes in fracture process zones, as in the case of regularise continuum approaches to crack moelling. To capture the time scale of concrete fracture, a phenomenological approach has been followe. Two sources of time epenence have been inclue in the mechanical constitutive behaviour, namely linear visco-elasticity in the continuum to capture bulk creep, an a rate-epenent normal traction in the isplacement iscontinuity, to simulate the finite crack rate. This simplifie approach has been shown to introuce the time scale accurately, through the analyses of creep failure experiments. ACKNOWLEDGEMENTS This research is supporte by the Technology Founation STW, The Netherlans. _J _._ Displacement (111111) Figure 8: Numerical preiction of failure uner sustaine loa..7 G Experimental (Zhou 1992) " Numerical (current moel) o----- Numerical (van Zijl et al. 21) o- - -o Numerical (no crack rule epenence) Time to failure t 1 (s) Figure 9: Time to failure uner sustaine loa. REFERENCES Askes, H., Boe, L. an Sluys, L.J. (1998). ALE analyses of localisation in wave propagation problems. Mechanics of Cohesive-Frictional Materials 3, Babuska,. an J.M. Melenk (1997). The Partition of Unity Metho. lntemational Journal for Numerical Methos in Engineering 4(4), Bafant, Z.P. an Gettu, R. (1992). Rate effects an loa relaxation in static fracture of concrete. AC! Materials Joumal 89(5), 1-2. Bafant, Z.P. an Xiang, Y. (1997). Crack growth an lifetime of concrete uner long term loaing. Joumal of Engineering Mechanics 123(4),

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