Mesoscopic modelling of concrete under impact

R.R. Pedersen, A. Simone, M. Stroeven & L.J. SluysDelft University of Technology, Faculty of Civil Engineering and Geosciences, Delft, The Netherlands

ABSTRACT: We present a mesoscopic model for the analysis of tensile failure of concrete under impact load-ing. Rate dependency is incorporated through viscosity terms in the constitutive formulation. The model isapplied to mesoscopic analyses of three-phase mesoscopic descriptions of concrete consisting of elastic aggre-gates, inelastic bulk material and interfacial transition zones (ITZs). The tensile strength of the ITZ influencesthe global strength and the shape of the final failure zone. Different aggregate distributions affect the diffusefailure pattern but the change of the overall strength is less significant. The numerical responses are comparedwith experimental results from a gravity driven Split Hopkinson bar test setup.

1 INTRODUCTIONExperimental observations have provided an insightinto the complicated fracture process of concrete instatics and dynamics (Reinhardt and Weerheijm 1991;Rossi et al. 1992). Concrete is a heterogeneous brittlematerial that fractures via the formation, growth andcoalescence of microcracks. Furthermore, the failureprocess is dependent on the loading rate. Micro in-ertia of the material adjacent to propagating microcracks and moisture in the capillary pores characterisethe rate dependent behaviour of concrete. A represen-tative computational model of concrete in dynamicsmust include these features.

A more realistic description of fracture processesranging from diffuse failure to localisation and finaldiscrete failure can be described by means of an ex-plicit model of the mesostructure. In dynamics themodified inertia effect and the geometrical dispersivecharacteristic are other consequences of the meso-scopic model. For the constituents in the mesostruc-ture we apply a rate dependent constitutive model.

Realistic homogeneous macroscopic models forheterogeneous materials incorporate informationfrom the lower scales. A successful constitutivetheory is the microplane theory (Bazant and Prat1988; Carol and Bazant 1997). This approach is ex-tended including damage and plasticity formulationsby D’Addetta et al. (2002) and Kuhl et al. (2000).Also enhanced continuum theories implicitly incor-porate lower scale effects by introducing a lengthscale parameter. An example of these theories is theCosserat continuum (Muhlhaus 1989) in which ro-tational degrees of freedom are included to account

for the non-local effects due to the motion of the mi-crostructures. Discrete models exist which are basedon Voronoi cell representations of the heterogeneousstructure. In these models, a beam lattice network isused to model the cohesive and compressive forcesbetween the neighbouring cells (Bolander and Suku-mar 2005; Nagai et al. 2004; Ibrahimbegovic and De-laplace 2003; Schlangen and van Mier 1992). The co-hesive crack methodology (Camacho and Ortiz 1996)is also applied at a mesoscopic level (Maiti et al. 2005;Tijssens et al. 2001; Carol et al. 2001).

The mesoscopic problem is solved here using anembedded multiple scale approach. Therefore, thedifferent constituents are modelled explicitly andwe apply an idealised geometrical model for themesoscale consisting of matrix and aggregates sur-rounded by an interfacial transition zone. The con-stitutive model used for the constituents is based onthermodynamics of chemically reactive porous me-dia (Lackner et al. 2002; Sercombe et al. 2000; Ulmand Coussy 1995). A viscoelastic model, coupled to aviscoplastic damage model (Simone and Sluys 2004),accounts for the strengthening effect associated withthe viscous phenomenon due to moisture. The vis-coplastic part contributes to an additional rate effectin the failure zone which is linked to inertia effects.

In this contribution, we focus on the influence ofaggregate distribution and strength of the interfacialtransition zone. A gravity driven Split Hopkinson Bartest setup is employed to determine the moisture de-pendent tensile strength (Vegt et al. 2006). A compari-son is made between experimental results and numer-ical estimations for different values of the parameter

governing the viscosity in the bulk material.

2 MATERIAL MODELThe bulk material and the material in the ITZ obeya viscoelastic viscoplastic damage model which isbriefly described next. We assume a strain decompo-sition of the type

ǫ = ǫe + ǫ

ve + ǫvp (1)

where the strain tensor is split into the elastic, theviscoelastic and the viscoplastic parts. Here we fol-low Ju (1989) in coupling plasticity and damage andby making use of the effective stress concept and thehypothesis of strain equivalence. This approach doesnot pose any restriction on the nature of the plas-tic moduli, apart from the requirement of formulat-ing the plastic moduli in the effective stress space.Therefore, rate-dependent plasticity models, whichpreserve well-posedness of the governing equationsin the softening regime, can be coupled to damage. Aquantity in the effective stress space will be denotedby a superimposed tilde.

In the context of a numerical procedure, the stressupdate relation at the end of a time step for the com-bined model reads

σn+1 = (1− ωn+1) σn+1 (2)

whereσ is the Cauchy stress tensor andω (0≤ ω ≤ 1)a damage variable which is updated through

ωn+1 = α(

1− e−βκn+1)

. (3)

α and β are parameters regulating the asymptoticvalue of damage and the slope of the damage evolu-tion law, respectively, whileκn+1 is a measure of thedeformation cumulated in the plastic regime.

The viscoelastic strain contribution is expressed ina rate form as

ǫve = χb, (4)

whereχ is the equivalent viscoelastic strain andb isdefined as

b = s/ |s| , (5)

with s the deviatoric stress tensor. The equivalentviscoelastic strain obeys the relation Sercombe et al.(2000, eq. (56))

η1

∂χ

∂t= |s| −E1χ, (6)

whereE1 is the viscoelastic spring stiffness, andη1

is the viscosity in the Kelvin element. The right-handside of (6) can be interpreted as the macroscopic rep-resentation of the viscous effect due to moisture innano- and micro-pores. It is similar to the hardening

force in plasticity, which represents micro crackingcontrolled by the equivalent plastic strain.

The viscoplastic strain contribution is expressedaccording to the formulation proposed by Perzyna(1966). When the yield criterion is violated in the ef-fective stress space, i.e. whenf ≥ 0 with f the yieldfunction, the viscoplastic strain rate is expressed inthe associative form as

ǫvp =1

τ2

φ fσ, (7)

whereτ2 = η2/E2 is the viscoplastic relaxation time,η2 is the corresponding viscosity of the damper ele-ment, fσ = ∂f/∂σ and the overstress functionφ isgiven by the following power-law form

φ(

f)

=

(

f

ft

)N

, (8)

with ft the initial tensile strength andN (N ≥ 1) areal number.

Another important component of our model is thehardening forceq which defines the elastic domain ofthe material and is made a function of both the equiv-alent plastic and equivalent viscoelastic strains as

q (κ,χ) = ft g (κ)h (χ) . (9)

The functiong is defined here as an exponential soft-ening curve function of the equivalent plastic strainκin the effective space:

g (κ) = (1 + a) exp (−bκ)− a exp (−2bκ) , (10)

with the parametersa and b governing the residualand the slope. The second function is defined by

h (χ) =

{

1 + χa log

(

χ

χs

)}

, (11)

where, by using thelog function, small values of theequivalent viscoelastic strain result in an increase ofthe tensile strength. This is particularly useful in dy-namics where the viscous strain decreases with in-creasing loading rate. From a physical standpoint, thisrepresents the retardation effect in the nano-pores dueto the Stefan effect – the Stefan effect is an increaseof tensile strength due to adhesive forces betweenmoisture and skeleton.χs is the maximum value ofthe equivalent viscoelastic strain when the Stefan ef-fect is activated. When the equivalent measure ofthe viscoelastic strains exceeds this value (no Ste-fan effect) thelog term is negative and the strengthis multiplied with a term lower than one and plas-ticity is activated earlier. Here, we add an additionalrate effect to the viscosity terms. The two viscosity

F (t)

5050 100 100

x

100 50 50

74

SG 2SG 1

Es/2020ρs ρs

EsEs/10 Es

ρs 10ρsEs/10 Es/20

20ρs10ρs

y

Figure 1: Model of the Split Hopkinson bar test setup (di-mensions in mm).

terms reflect microscale processes as the Stefan effectand distributed microcracking. These micromechan-ical effects are rate dependent. The idea is to incor-porate the following rate dependent viscosity relationin the model, similar to what is done in mantle rhe-ology (Becker 2006) and in rate dependent frictionalsliding problems (Coker et al. 2005):

ηi = η0

i

(

1 + (ǫeq)1−n

n

)

. (12)

In the above relation,ǫeq is the equivalent measureof the strain tensor andη0

i (i = 1,2) are the viscosi-ties in the elastic and plastic part in case of staticloading conditions. In Equation (12)n is a real value(0 < n ≤ 1). The mesoscopic representation com-bined with the rate dependent constitutive model isapplied to reproduce the results from a Split Hopkin-son Bar test.

3 SPLIT HOPKINSON BAR TESTThe Split Hopkinson bar apparatus consists of an in-cident bar (∼ 5 m), the test specimen (0.1 m), and theoutput bar (∼ 5 m). In the computational model thelength of the incident and output bars is 250 mm.We apply material based absorbing boundary con-ditions, see Figure 1. SG 1 refers to the positionfor strain measurement of the incoming tensile wavewhile SG 2 indicates the points where the transmit-ted load pulse is measured. The values forEs andρs are adjusted to obtain comparable results at SG 1with respect to the experiments (Es = 4572 MPa, ρs =182910−11N s2/mm4). Identical acoustic impedances√

Eρ of the materials is necessary to avoid spuri-ous waves in the response. A detailed description ofthe experiments can be found in (Rossi et al. 1992;Vegt et al. 2006). For a mesoscopic analysis, anaccurate and realistic geometrical model is neces-sary: the shape, size and distribution of the aggre-gate particles must resemble real concrete in a sta-tistical sense. There exist specific algorithms for gen-eration of random aggregate structures taking into ac-count the size, shape and spatial distributions (Hafneret al. 2006; Wriggers and Moftah 2006). Here, weuse HADES (HADES 2006), a discrete element pack-age that allows for the simulation of granular ma-terials. The individual particles are generated in anon-overlapping way in a region. This region can be

tablenameTable 1: Model parameters.

E2 38500 MPaν 0.2ρ 2200 10−12 N s2/mm4

ft 3 MPa (matrix)τ2 0.08 sb 500a 0α 1β 1800E1 60000 MPaη1 10000 MPa sχa 0.2χs 0.00001n 0.3N 1∆t 10−8 s

defined by periodic boundaries, rigid boundaries orpartly periodic and partly rigid boundaries. Each par-ticle is given an random initial linear and angularvelocity. The particles are iteratively displaced to aposition that is obtained by integrating the velocityover a small time period. Similarly, the velocity ofa particle at the next iteration is calculated by inte-grating the force or torque that acts on each particle.Gravitational forces, paste friction and contact forcesbetween particles have been implemented. Next, theboundaries can be moved according to user-definedfunctions. In this way, a number of experiments canbe reproduced. Dense packings can be obtained inthis way, but it is also possible to move the periodicor rigid walls of the container increasing the volumedensity of the mixture. In this way arbitrary densitiescan be obtained up to the maximum density. The out-put of a two-dimensional simulation can be used tocreate a distribution of aggregates as shown in Fig-ure 2 and 3.

There are clear differences in the spatial distribu-tions of the aggregates. Three realisations are shownin Figure 2. For realisation (2) the aggregates are con-centrated in the middle of the specimen. In the otherdistributions (1,3), several aggregates are placed closeto the free boundaries, and create therefore, poten-tial positions for damage initiation. In realisation (1)particles to the left are closely arranged in a columnand failure can develop and propagate in this weakzone. In average, 28000 linear triangular elements areused for the discretisation and a Courant number ofapproximately0.75 is ensured with a time step size∆t = 10−7s.

1

2

3Figure 2: Different realisations of the test specimen. Parti-cle density is 30%.

Figure 3: Detail of the finite element discretisation num-ber 3. Light grey is the matrix, the aggregates in dark greysurrounded by the white interfacial transition zones.

ft = 1.0 MPa

ft = 2.0 MPa

ft = 3.0 MPa

Figure 4: Fracture planes corresponding to a differentstrength of the interfacial transition zone. Configuration3is analysed. Black corresponds to damage parameterω = 1.

4 APPLICATIONS

We test the influence of the tensile strength of theinterfacial transition zone and the consequences ofdifferent distributions on the overall dynamic tensilestrength. Furthermore, the rate dependent propertiesof the bulk material are changed to examine the influ-ence on failure patterns and the global strength.

4.1 Strength of ITZ

In concrete a gradient in water cement ratio devel-ops around the aggregate particles during casting, re-sulting in a different microstructure of the surround-ing hydrated cement paste. Therefore, the ITZ con-tains a gradient of porosity and a gradient of proper-ties which is here neglected. Instead we restrict our-self to variations (1.0,2.0 and3.0 MPa) of the tensilestrength of the ITZ. When the matrix and ITZ haveidentical strength the global strength is obviouslyhigher compared to situations with lower strength ofthe ITZ, see Figure 5. However, it is important to

experiment normal

Time [ms]

µǫ

ft = 3MPa

ft = 2MPa

ft = 1MPa

0.80.60.40.20

100

80

60

40

20

0

Figure 5: Time-strain curves at the upper bar (SG2). Trans-mitted pulse and a measure of the strength of the material.The fracture planes are shown in Figure 4.

note the relatively high impact of the ITZ strength onthe global strength comparing the number of ITZ ele-ments to the total number of elements in the simula-tions. Therefore, the most critical points are the ITZsand the mutual connection between ITZs. The exper-imental result (normal saturation level) is reported inFigure 5. In the mesoscopic model the aggregates areelastic and therefore no direct contribution to the rateeffect is present. The fracture planes, represented bythe damage parameterω are shown in Figure 4.

4.2 Different distributionsIn Figure 7 the final fracture patterns are shown fordifferent distributions of the aggregates. The tensilestrength of the ITZ is1.0 MPa. In the three distribu-tions the final failure consists of 1-2 fracture planes.For realisation (2) damage initiates at the notches,whereas for (1) and (3) failure is initiated betweenaggregates and the free boundary. These possiblefracture planes can be compared to experimental re-sults (Vegt et al. 2006) for the fracture zone. Fig-ure 8 shows a picture from an optical microscope,where the specimen (impregnated with a fluorescentepoxy) is studied to examine possible microcracks.Figure 6 shows that there is no significant variation ofthe global tensile strength for different distributions ofthe particles. The overall tensile strength is obtainedby multiplying the strain measurement, see Figure 5,with the Young’s modulus of the upper bar.

4.3 Influence of viscosity in bulk materialThe previous analysis proved that the initial strengthof the interfacial transition zone is an important pa-rameter for the global response of the model. The keyparameters in our model are the two viscosity param-eters. Next, we fix the material properties for the ITZand focus on the behaviour of the bulk material bychanging viscosity parameterη1. This parameter re-

Tensile strength ITZ [MPa]

Ten

sile

sten

gth

[MP

a]

43210

6

5

4

3

2

Figure 6: Tensile strength of the concrete specimen as afunction of the tensile strength of the ITZ.

flects the Stefan effect and is related to the microstruc-ture via the moisture content and the pore size distri-bution. The influence ofη1 for the bulk material onthe global strength is not significant as shown in Fig-ure 9. We use a wide range of the viscosity parame-ter to reproduce the different moisture contents fromthe experiments. An explanation for the less signifi-cant influence of the viscoelastic contribution is that

1

2

3Figure 7: Fracture planes for different realisations. Tensilestrengthft = 1 MPa. Black corresponds to damage param-eterω = 1.

Figure 8: Microscopic picture of the fracture plane fromexperiment (Vegt et al.2006).

experiment wetexperiment dryexperiment normal

Time [ms]

µǫ

η1 = 106MPas

η1 = 104MPas

η1 = 103MPas

0.80.60.40.20

100

80

60

40

20

0

Figure 9: Time-strain curves at the upper bar (SG2) for dif-ferent viscositiesη1.

only the viscosity for the matrix material is varied.However, the failure initiates in the weak ITZ ele-ments, where the same viscosity parameter is used forall analyses. Therefore, the weak zones and paths arethe same as in the previous analysis and the globalstrength is not clearly affected by only changing thesurrounding bulk material. However, an increase ofviscosity in the bulk material reduces the diffusion ofdamage around the aggregates as a consequence ofthe increased strength of the material, see Figure 10.

4.4 Influence of viscosity in bulk material and ITZ

We now change the viscosity in the viscoelastic partfor both the bulk material and the ITZ. The time strainresponse is reported in Figure 11. We can comparethese results to Figure 9. A clear difference is the in-creased dependency of the global tensile strength onthe viscosity parameterη1. Therefore, it is importantto properly change the material properties of the ITZelements as they are the first active elements in thelocalisation process while the surrounding bulk ma-terial is predominantly unloading. However, the over-all tendency in the failure process does not change ifwe compare the final fracture planes in Figure 10 andFigure 12.

η1 = 103 MPa s

η1 = 104 MPa s

η1 = 106 MPa s

Figure 10: Fracture planes corresponding to different val-ues of the viscosity. Here only the bulk material is changed.Black corresponds to damage parameterω = 1.

experiment wetexperiment dryexperiment normal

Time [ms]

µǫ

η1 = 106MPas

η1 = 104MPas

η1 = 103MPas

0.80.60.40.20

100

80

60

40

20

0

Figure 11: Time-strain curves at the upper bar (SG2) fordifferent viscositiesη1.

η1 = 103 MPa s

η1 = 104 MPa s

η1 = 106 MPa s

Figure 12: Fracture planes corresponding to different val-ues for the viscosity. Both the material for the bulk and ITZis changed. Black corresponds to damage parameterω = 1.

5 CONCLUSIONSThe influence of the heterogeneous mesostructure ofconcrete on the impact response is analysed. The ge-ometrical model consists of aggregates, interfacialtransition zones and a matrix. With this lower levelmodel a more in-depth study on failure mechanismsand inertia effects is possible. The ITZ between ag-gregate particles and bulk paste plays a major rolein the material behaviour of cementitious compos-ites. We observed that the strength of the interfacialtransition zone significantly influences the overall ten-sile strength. Different fracture planes are observedas a consequence of changing the distribution of theaggregates. A small increase of the overall tensilestrength was found when changing the rate dependentproperties of the bulk material alone. However, whenwe change the viscosity both for the ITZ and the bulkmaterial a more significant increase of the global ten-sile strength was observed. Therefore, the rate effectof the material is strongly related to the weakest partof the material, where failure is initiated.

REFERENCESBazant, Z. P. and P. Prat (1988). Microplane model for

brittle-plastic material: I. theory.Journal of Engi-neering Mechanics 114, 1672–1688.

Becker, T. W. (2006). On the effect of temperatureand strain-rate dependent viscosity on global man-tle flow, net rotation, and plate driving forces.Geo-physical Journal International 167(2), 943–957.

Bolander, J. E. and N. Sukumar (2005). Irregular lat-tice model for quasistatic crack propagation.Physi-cal Review B 71, 1–12.

Camacho, G. T. and M. Ortiz (1996). Computationalmodelling of impact damage in brittle materials.In-ternational Journal of Solids and Structures 33(20-22), 2899–2938.

Carol, I. and Z. P. Bazant (1997). Damage and plastic-ity in microplane theory.International Journal ofSolids and Structures 34, 3807–3835.

Carol, I., C. Lopez, and O. Roa1 (2001). Micromechani-cal analysis of quasi-brittle materials using fracture-based interface elements.International Journal forNumerical Methods in Engineering 52, 193–215.

Coker, D., G. Lykotrafitis, A. Needleman, andA. Rosakis (2005). Strain localization and bifurca-tion in a nonlocal continuum.Journal of the Me-chanics and Physics of Solids 53, 884–922.

D’Addetta, G. A., F. Kun, and E. Ramm (2002). On theapplication of a discrete model to the fracture pro-cess of cohesive granular materials.Granular Mat-ter 4, 77–90.

HADES (2006). Habanera’s discrete element simulator.http://www.habanera.nl/.

Hafner, S., S. Eckardt, T. Luther, and C. Konke (2006).Mesoscale modeling of concrete: Geometry and nu-merics.Computers and Structures 84, 450–461.

Ibrahimbegovic, A. and A. Delaplace (2003). Mi-croscale and mesoscale discrete models for dy-namic fracture of structures built of brittle material.Computers and Structures 81, 1255–1265.

Ju, J. W. (1989). On energy-based coupled elastoplasticdamage theories: constitutive modeling and compu-tational aspects.International Journal of Solids andStructures 25(7), 803–833.

Kuhl, E., E. Ramm, and K. Willam (2000). Failure anal-ysis of elasto-plastic material models on dierent lev-els of observation.International Journal of Solidsand Structures 37, 7259–7280.

Lackner, R., C. Hellmich, and H. A. Mang (2002). Con-stitutive modeling of cementitious materials in theframework of chemoplasticity.International Jour-nal for Numerical Methods in Engineering 53,2357–2388.

Maiti, S., K. Rangaswamy, and P. H. Geubelle (2005).Mesoscale analysis of dynamic fragmentation of ce-ramics under tension.Acta Materialia 53, 823–834.

Muhlhaus, H.-B. (1989). Application of Cosserat the-ory in numerical solutions of limit load prob-lems. Archive of Applied Mechanics (IngenieurArchiv) 59, 124–137.

Nagai, K., Y. Sato, and T. Ueda (2004). Mesoscopicsimulation of failure of mortar and concrete by 2drbsm.Journal of Advanced Concrete Technology 2,359–374.

Perzyna, P. (1966). Fundamental problems in viscoplas-ticity. In Advances in Applied Mechanics, Volume 9,pp. 243–377. New York: Academic Press.

Reinhardt, H. and J. Weerheijm (1991). Tensile frac-ture of concrete at high loading rates taking accountof inertia and crack velocity effects.InternationalJournal of Fracture 51, 31–42.

Rossi, P., J. G. M. van Mier, C. Boulay, and F. L.Maou (1992). The dynamic behaviour of con-crete: influence of free water.Materials and Struc-tures/Materiaux et Constructions 25, 509–514.

Schlangen, E. and J. G. M. van Mier (1992). Sim-ple lattice model for numerical simulation of frac-ture of concrete materials and structures.Materialsand Structures/Materiaux et Constructions 25, 534–542.

Sercombe, J., F.-J. Ulm, and H.-A. Mang. (2000). Con-sistent return mapping algorithm for chemoplasticconstitutive laws with internal couplings.Interna-tional Journal for Numerical Methods in Engineer-ing 47, 75–100.

Simone, A. and L. J. Sluys (2004). The use of displace-ment discontinuities in a rate-dependent medium.Computer Methods in Applied Mechanics and En-gineering 193, 3015–3033.

Tijssens, M. G. A., L. J. Sluys, and E. van derGiessen (2001). Simulation of fracture of cemen-titious composites with explicit modeling of mi-crostructural features.Engineering Fracture Me-chanics 68, 1254–1263.

Ulm, F.-J. and O. Coussy (1995). Modelling of ther-mochemomechanical couplings of concrete at earlyages.Journal of Engineering Mechanics 121, 785–794.

Vegt, I., J. Weerheijm, R. R. Pedersen, and L. J.Sluys (2006, 27–30 March). Modelling of im-pact behaviour of concrete - an experimental ap-proach. In G. Meschke, R. de Borst, H. A.Mang, and N. Bicanic (Eds.),Computational Mod-elling of Concrete Structures – EURO-C 2006,Mayrhofen, Tyrol, Austria. Taylor & Francis, Lon-don/Leiden/New York/Philadelphia/Singapore.

Wriggers, P. and S. Moftah (2006). Mesoscale mod-els for concrete: Homogenisation and damage be-haviour.Finite Elements in Analysis and Design 42,623–636.