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A 3D Lattice Modelling Study of Drying Shrinkage Damage in Concrete Repair Systems Mladena Luković *, Branko Šavija, Erik Schlangen, Guang Ye and Klaas van Breugel Faculty of Civil Engineering and Geosciences, Delft 2628 CN, The Netherlands; [email protected] (B.Š.); [email protected] (E.S.); [email protected] (G.Y.); [email protected] (K.v.B.) * Correspondence: [email protected]; Tel.: +31-624391694 Academic Editor: Jorge de Brito Received: 12 June 2016; Accepted: 8 July 2016; Published: 14 July 2016

Abstract: Differential shrinkage between repair material and concrete substrate is considered to be the main cause of premature failure of repair systems. The magnitude of induced stresses depends on many factors, for example the degree of restraint, moisture gradients caused by curing and drying conditions, type of repair material, etc. Numerical simulations combined with experimental observations can be of great use when determining the influence of these parameters on the performance of repair systems. In this work, a lattice type model was used to simulate first the moisture transport inside a repair system and then the resulting damage as a function of time. 3D simulations were performed, and damage patterns were qualitatively verified with experimental results and cracking tendencies in different brittle and ductile materials. The influence of substrate surface preparation, bond strength between the two materials, and thickness of the repair material were investigated. Benefits of using a specially tailored fibre reinforced material, namely strain hardening cementitious composite (SHCC), for controlling the damage development due to drying shrinkage in concrete repairs was also examined. Keywords: concrete repair; drying shrinkage; lattice model; strain hardening cementitious composites (SHCC)

1. Introduction Concrete repair implies integration of the new (repair) material with the old concrete substrate in order to form a composite system capable of enduring exposure to mechanical loads and varying environmental conditions. What makes this difficult is the mismatch in age, properties, and performance of the two materials, often leading to premature failure of the multi-layered system [1,2]. Differential shrinkage, primarily drying shrinkage, is considered to be the major cause of failure [3,4]. After casting, the repair material is usually covered or coated in order to enable good curing and prevent plastic shrinkage cracking. After such curing, the repair system is uncovered and exposed to ambient temperature and relative humidity. Due to environmental drying, ongoing hydration, and moisture absorption by the concrete substrate, the repair material loses water and tends to shrink. Its deformation is, however, restrained by the concrete substrate. As a consequence, stresses build up in the repair system and cracking and/or debonding follows. Cracking or debonding of the overlay reduces the load-carrying capacity of the system and allows water, together with hazardous substances such as chloride, to penetrate into concrete and further speed up the deterioration process. In order to make some practical design recommendations, a number of analytical models for bonded overlays subjected to differential shrinkage have been developed [4–8]. Also, a number of two-dimensional continuum models were used [3,9–13]. One way of coupling moisture transport and fracture simulations, while taking into account the influence of drying on cracking mechanism, is through discrete lattice type modelling [14,15]. The main disadvantage of all these numerical Materials 2016, 9, 575; doi:10.3390/ma9070575

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Materials 2016, 9, 575

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models is that restrained shrinkage deformation is studied mostly in 2D while the experiments are always 3D. In previous work with lattice approach [15], although a 3D mesh was used, the thickness of the simulated specimen was small in order to limit computational costs. Applied boundary conditions resulted in deformation of the material being restrained in one direction only, which does not completely agree with the behaviour in the experiments. Therefore, in order to further verify the applied modelling approach, cracking due to drying shrinkage in concrete repair systems needs to be simulated in 3D. The aim of this paper is to understand parameters that influence shrinkage induced cracking by modelling moisture transport and fracture processes during drying in concrete repair systems. 3D mesoscale lattice models are used for simulating moisture distribution and resulting damage development in repair systems. Benefits of using lattice models for fracture simulations are that they can mimic physical structure and processes, so that realistic crack patterns can be achieved [16–18]. Such models have been used for modelling fracture of quasi-brittle (i.e., concrete, mortar or cement paste), but also ductile (i.e., fibre reinforced) materials [19,20]. In the presented models, the material structure is explicitly represented; in the lattice transport model it is captured through assigning different diffusivity (conductivity) properties [21–23], and in the lattice fracture model through assigning different mechanical properties [16] to lattice elements that represent a certain material phase. The influence of different parameters—for example substrate surface preparation (i.e., roughness), bond strength between the two materials, and thickness of the repair material on damage caused by drying shrinkage in concrete repair—were investigated. The damage in the repair system can be reduced if the repair material is reinforced with fibres, as in case of strain-hardening cementitious composite—SHCC [24]. Fibres bridge crack faces and prevent large crack openings. In SHCC these cracks successively form and are limited to 100 micrometres. With successive microcracking in SHCC, load that can be transferred increases, implying that strain hardening behaviour, commonly observed in metals, is achieved. For applications as a repair material, multiple cracks of SHCC are beneficial for relief of stresses induced by deformational incompatibility between the repair material and the substrate [24,25]. Furthermore, dense microcracking and local debonding lead to high ductility of SHCC in repair systems, which is beneficial in suppressing crack localization due to flexural loading [26,27]. In this paper, implications of using SHCC as a repair material on damage caused by restrained drying shrinkage are addressed. Simulated 3D damage patterns in repair systems with different types of repair materials, their thickness, and bond strength are compared with experimental observations and cracking tendency in various brittle and ductile materials. 2. Modelling Approach Lattice models have been widely used to simulate fracture, moisture transport, and chloride diffusion in cement-based materials [14,16,21,23]. In the transport lattice approach, concrete is treated as an assembly of one-dimensional “pipes”, through which the flow takes place. In the mechanical lattice approach, concrete is discretized as a set of truss or beam elements that transfer forces. The output from the moisture transport model is used as an input for simulating fracture. This is a staggered flow-stress analysis: moisture transport induces strain which might result in cracking, but there is no influence of the (micro)cracking on the moisture transport. This is a reasonable assumption as drying shrinkage microcracks seem not to have a significant effect on drying rate of mortars and concrete [28]. With wetting and drying cycles, however, cracks may have a significant influence, and transport and fracture simulations would probably need to be fully coupled. 2.1. Spatial Discretization The approach proposed here uses the same lattice network for both the moisture and the fracture simulations. For the spatial discretization of the specimen in three dimensions, the basis is the prismatic domain [21]. Discretization of the domain was performed according to the following procedure:


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the prismatic domain [21]. Discretization of the domain was performed according to the following 3 of 19 procedure:


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A cubical grid was chosen (square for 2D lattice) and the domain was divided into a number of A cubical grid was chosen (square for 2D lattice) and the domain was divided into a number of cubical cells (Figures 1a,b and 2b). In all simulations presented herein, linear dimension of the cubical cells (Figures 1a,b and 2b). In all simulations presented herein, linear dimension of the cubical cell A (marked in Figure 1a) was 1 mm. cubical cell A (marked in Figure 1a) was 1 mm. In each cell of size A, a sub cell with linear dimension of s was defined (marked in Figure 1a). InThe each cell of size was A, a randomly sub cell with linear dimension of sub s was defined (marked in ratio Figures/A 1a). lattice node positioned inside this cell (Figure 1a). The is The lattice node was randomly positioned inside this sub cell (Figure 1a). The ratio s/A is defined as defined as randomness of a lattice. When randomness is 0, the node is located in the centre of randomness of a lattice. is 0,Ifthe node is located in centre of cell and regular cell and regular latticeWhen meshrandomness is generated. the randomness is the higher than 0, beams have lattice mesh is generated. If the randomness is higher than 0, beams have different lengths and different lengths and material disorder is built through the geometry of lattice mesh. With material disorder built through the geometry of lattice mesh.degree With randomness 1, therefore, randomness of 1,is therefore, materials have the maximum of disorder.ofThe choice of materials have the maximum degree of disorder. The choice of randomness affects the randomness affects the simulated fracture of materials to a certain extent, assimulated different fracture of materials to a certain extent, as different orientation of meshes can affect the crack orientation of meshes can affect the crack shape [29]. In order to include benefits of random shape [29]. In order to include benefits of random mesh, the randomness here was set to be 0.5. mesh, the randomness here was set to be 0.5. Voronoi prismatic domain domainwith withrespect respecttoto specified of nodes Voronoitessellation tessellation of of the the prismatic thethe specified set set of nodes was was performed. Nodes with adjacent Voronoi cells were connected by lattice elements performed. Nodes with adjacent Voronoi cells were connected by lattice elements (Figures 1a,b (Figures 1a,b and 2b). Since diagrams Voronoi diagrams arewith dual with Delaunay tessellation, approachis and 2b). Since Voronoi are dual Delaunay tessellation, thisthis approach isequivalent equivalenttotoperforming performingaaDelaunay Delaunaytessellation tessellationofofthe theset setofofnodes nodes[21]. [21]. InInorder to take material heterogeneity into account, either a computer-generated order to take material heterogeneity into account, either a computer-generatedmaterial material structure, obtained by bymicro-CT micro-CTscanning scanning[17,30] [17,30]can can used. Here, structure,or or aa material material structure obtained bebe used. Here, the the concrete mesostructure was simulated using the Anm material model originally developed concrete mesostructure was simulated using the Anm material model originally developed by by Garboczi [31] andimplemented implementedinina a3D 3Dpacking packingalgorithm algorithmby by Qian Qian et al. [32]. Garboczi [31] and [32]. It It isisbased basedon on placing placingmultiple multipleirregular irregularshape shapeparticles particlesseparated separatedinto intoseveral severalsieve sieveranges rangesinto intoaapredefined predefined empty emptycontainer container(Figure (Figure2a). 2a).Aggregates Aggregatessmaller smaller than than44mm mmin inthe thesubstrate substrateare arenot notexplicitly explicitly modelled together with cement paste, are considered to be part of the mortar phase. Similarly, modelledand, and, together with cement paste, are considered to be part of the mortar phase. inSimilarly, the repairinmaterial, thematerial, largest sand are around 200 µm and, as200 such, also the repair the (filler) largestparticles sand (filler) particles are around μmare and, as not explicitly simulated. such, are also not explicitly simulated. Material Materialoverlay overlayprocedure procedure(schematically (schematically shown shown in in Figures Figures1a,b 1a,band and2b) 2b)was wasemployed: employed:the the beams that belong to each phase were identified by overlapping the material mesostructure beams that belong to each phase were identified by overlapping the material mesostructure (i.e., (i.e.,substrate substratemortar, mortar,repair repairmaterial materialmortar, mortar,and andaggregates) aggregates)on ontop topofofthe thelattice. lattice.Interface Interface elements elementsbetween betweenthe therepair repairmaterial materialand andthe thesubstrate substratefor forsmooth smoothand andrough roughsubstrate substratesurface surface were between substrate nodes andand repair material nodes (Interface MS/RM, FigureFigure 1a,b, weregenerated generated between substrate nodes repair material nodes (Interface MS/RM, respectively). Aggregate-mortar interface (ITZ) elements were generated between mortar mortar nodes 1a,b, respectively). Aggregate-mortar interface (ITZ) elements were generated between and aggregates (Figure 2b).(Figure By applying thisapplying overlay procedure, the geometrical of the nodes and aggregates 2b). By this overlay procedure, roughness the geometrical substrate was explicitlywas modelled (Figure 3a,b). roughness of also the substrate also explicitly modelled (Figure 3a,b).

(a)

(b)

Figure1.1.Two-dimensional Two-dimensional overlay procedure generation of lattice the lattice model the interface Figure overlay procedure for for generation of the model in thein interface zone zone between the fibre reinforced repair material and (a) a smooth; and (b) a rough surface substrate. between the fibre reinforced repair material and (a) a smooth; and (b) a rough surface substrate.


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(a) (b) (a) (b) Figure 2. Including material mesostructure of the concrete substrate (a) aggregates in the concrete Figure 2. Including material mesostructure of the concrete substrate (a) aggregates in the concrete substrate Anm model [32]; (b) two-dimensional overlay procedure for generation of the Figure 2. generated Including by material mesostructure of the concrete substrate (a) aggregates in the concrete substrate generated by Anm model [32]; (b) two-dimensional overlay procedure for generation of the lattice model in the interface zone between and substrate (ITZ).for generation of the substrate generated by Anm model [32]; (b) aggregate two-dimensional overlaymortar procedure lattice model in the interface zone between aggregate and substrate mortar (ITZ). lattice model in the interface zone between aggregate and substrate mortar (ITZ).

(a) (a)

(b) (b)

Figure 3. Generated lattice model for (a) a smooth; and (b) a “sandblasted” surface substrate. Figure 3. Generated lattice model for (a) a smooth; and (b) a “sandblasted” surface substrate.

Figure 3. Generated lattice model for (a) a smooth; and (b) a “sandblasted” surface substrate. Different transport and mechanical properties were assigned to different phases (Tables 1 and 2, respectively). Interfaces, as used in the present model, are always considered one row of Different transport and mechanical properties were assigned to different phases as (Tables 1 and beam elements.and As such, they do properties not coincide withare thealways ofconsidered real interfaces. In reality, 2, respectively). Interfaces, as used in exactly the present as(Tables one row1 of Different transport mechanical weremodel, assigned tosize different phases and 2, interface thickness in used athey range tens of micrometres, while elements the present beam elements. As is such, doof not exactly coincide with the interface sizeconsidered of real interfaces. In reality, respectively). Interfaces, as in the present model, are always as in one row of beam model takethey up is ado piece ofexactly aggregate (ormicrometres, repair material) and a piece of mortar.Inin Therefore, the interface thickness in not a range of tens of while elements the present elements. Asalso such, coincide with the size ofinterface real interfaces. reality, interface actual also size take of the interface in the model depends on theand characteristic elementTherefore, size and,the in model up a piece of aggregate (or repair material) a piece of mortar. thickness is in a range of tens of micrometres, while interface elements in the present model also presented is around actual sizesimulations, of the interface in the1 mm. model depends on the characteristic element size and, in take up a piece of aggregate (or repair material) and a piece of mortar. Therefore, the actual presented simulations, is around 1 mm.

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size of the interface in parameters the modelused depends on themoisture characteristic and, in presented Table 1. Diffusivity in the lattice transportelement model to size obtain diffusion 2 simulations, is around 1 mm. coefficient, D (mm /day) (Equation (2)). are adjusted such that the moisture in Table 1. Diffusivity parameters used in Parameters the lattice moisture transport model to obtainprofiles diffusion 2/day) the repair system 1 day,(Equation 10 days and days areare theadjusted same as such thosethat obtained by Wittmann coefficient, D (mmat (2)). 110 Parameters the moisture profilesand in Tablethe 1. repair Diffusivity parameters usedand in 110 the days lattice transport modelbytoWittmann obtain diffusion Martinola [12]. system at 1 day, 10 days aremoisture the same as those obtained and 2 coefficient, D (mm Martinola [12]. /day) (Equation (2)). Parameters are adjusted such that the moisture profiles in Diffusivity Repair Mortar the repair system at 1 day, 10 days andInterface 110 days are the same as thoseITZ obtained by Wittmann and Aggregates Parameters Mortar Substrate Diffusivity Repair Mortar Martinola [12]. Interface ITZ Aggregates β 0.022 0.022 0.022 0.066 0.00022 Parameters Mortar Substrate 7.5 7 4 4 0 βγ 0.022 0.022 0.022 0.066 0.00022 Diffusivity Repair Mortar Interface ITZ Aggregates γ 7.5 7 4 4 0 Parameters Mortar Substrate

β γ

0.022 7.5

0.022 7

0.022 4

0.066 4

0.00022 0


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Table 2. Input values for lattice fracture model [27], E stands for the modulus of elasticity, ft stands for the tensile strength and fc for the compressive strength of lattice elements. Element

E (GPa)

ft (MPa)

fc (MPa)

Matrix (repair mortar-RM)

20

3.5

35


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Table 2. Input values for lattice fracture model [27], E stands for the modulus of elasticity, ft stands Fibre 40 7380 (7380) for the tensile strength and fc for the compressive strength of lattice elements.

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Interface (Matrix/Fibre) Element Mortar substrate Matrix (repair mortar-RM) Aggregate Fibre Interface ITZ (Matrix/Fibre) Mortar substrate Matrix (Repair mortar-RM) Aggregate Interface (Matrix/fibre) ITZ Weak Matrix (Repair mortar-RM) Interface (MS/RM) Interface (Matrix/fibre) Strong Weak Interface (MS/RM) Strong

20 E (GPa) 25 20 70 40 15 20 20 25 70 20 15 15 20 15 20 15 15

90 ft (MPa) 4 3.5 8 7380 90 2.5 4 3.5 8 90 2.5 1 3.5

90 3 1 3

900 fc (MPa) 40 35 80 (7380) 90025 4035 80 900 25 3510

90030 10 30

For fracture simulations, fibre elements were added in the repair material with a design volume  For fracture simulations, were added the repair material of with design volume content (0.5%), fibre length (8fibre mm)elements and diameter (0.04 in mm). The location thea first node of each content (0.5%), fibre length (8 mm) and diameter (0.04 mm). The location of the first node of fibre was chosen randomly in the specified volume and a random direction was defined which each fibre was chosen randomly in the specified volume and a random direction was defined determined the position of the second node (Figure 4a). If the second node was outside the mesh which determined the position of the second node (Figure 4a). If the second node was outside boundary, then the fibre was automatically cut off. the mesh boundary, then the fibre was automatically cut off. Extra nodes inside the fibres were generated wherethe thefibre fibrecrosses crosses  Extra nodes inside the fibres were generatedatateach eachlocation location where thethe gridgrid (Figure(Figure 1a,b). 1a,b).

(a)

(b)

Figure 4. Periodic boundary conditions in horizontal directions for (a) fibres (grey colour) in the

Figure 4. Periodic boundary conditions in horizontal directions for (a) fibres (grey colour) in the SHCC SHCC repair material (b) aggregates (light grey colour) in the concrete substrate. repair material (b) aggregates (light grey colour) in the concrete substrate.

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Fibre/matrix interface elements were generated between fibre nodes and the matrix nodes in the

neighbouring cell.elements Also, the end nodes of the fibres werefibre connected interface to the Fibre/matrix interface were generated between nodeswith andan the matrixelement nodes in the matrix node in the where theof fibre waswere located (Figure 1a,b). neighbouring cell. Also, thecell end nodes theend fibres connected with an interface element to  matrix Note node that ininorder to reduce time, 0.5% (Figure instead of 2% of fibres, as commonly the the cell wherethe thecomputational fibre end was located 1a,b). used in SHCC, was simulated. In order to obtain SHCC fracture behaviour in simulations with Note that in order to reduce the computational time, 0.5% instead of 2% of fibres, as commonly a lower percentage of implemented fibres, fibre/matrix interface strength and fibre tensile used instrength SHCC,from was Table simulated. In order to obtain fracture measured behaviourvalues in simulations 2 are higher compared to SHCC experimentally which canwith be a lower percentage of implemented fibres, fibre/matrix interface strength and fibre tensile strength found in [27]. from 2 are higher compared experimentally whichthat canone be side found  Table Samples were simulated with to periodic boundary measured conditions.values This means of in the[27]. specimen was connected to periodic the other boundary end. Therefore, aggregates andmeans fibres were in the Samples were simulated with conditions. This that distributed one side of such a way that they continue through boundaries and periodically repeat. The example for the specimen was connected to the other end. Therefore, aggregates and fibres were distributed in boundary conditions of the boundaries fibres is givenand in Figure 4a and for aggregates in Figurefor 4b. the such aperiodic way that they continue through periodically repeat. The example periodic boundary conditions of the fibres is given in Figure 4a and for aggregates in Figure 4b.


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Fibre elements and fibre/matrix interface elements were assumed not to take part in the moisture transport and therefore are not modelled in the transport simulations. In order to have a representative volume, the length of the repair system was chosen to be at least 5 times larger than the maximum aggregate size and fibre length in the repair material.

2.2. Lattice Moisture Transport Model A random lattice (s/A = 0.5) was used first to model moisture transport caused by water exchange in a repair system. The model treats concrete as an assembly of one-dimensional, linear, “pipe” elements, through which moisture transport takes place. A governing partial differential equation for moisture transport in 1D is: BH B BH “ pD pHq q (1) Bt Bx Bx where H is the relative humidity and D(H) is a humidity dependent diffusion coefficient which can be given as: H D pHq “ βeγ (2) Here β and γ are parameters that can be determined by calibration with experimental results [33]. In the model presented herein, although moisture transport is in reality a multi-scale problem, no distinction was made between the different flow mechanisms that take place at the micro-scale [34]. If Equation (1) is discretized using the standard Galerkin method [35], the following set of linear equations arises (in matrix form): . M H ` KH “ f (3) where M = mass matrix; K = diffusion matrix and f = force vector. Vector H is the vector of unknown quantities (in this case relative humidity) and dot over H indicates the time derivative. Element matrices, Me and Ke , have the following forms: Aij lij Me “ 6ω

«

2 1

1 2

ff

D pHq Aij Ke “ lij

«

1 ´1

ff ´1 1

(4)

where lij is the length of the element between nodes i and j, Aij is its cross sectional area, and D(H) its diffusion coefficient. Cross sectional areas of individual elements were assigned using the so-called Voronoi scaling method [36]. All matrices were equivalent to those of regular one-dimensional linear elements [35], except the correction parameter ω in the mass matrix (Equation (4)). This parameter was used to convert the volume of all lattice elements to the volume of the specimen, due to overlap of volume of adjacent lattice elements (Figure 5). Therefore, ω corresponds to the ratio between the total area of Voronoi facets through which moisture transport takes place and the volume represented by lattice mesh, and can be determined as [37]: řm Al ω “ i “1 i i (5) V where m is the total number of elements, Ai and li cross sectional area and length of each lattice element, k element number, and V the total volume of the specimen. In this way, the assemblage of lattice elements in the mesh provided 3D moisture simulations by using 1D moisture flow in lattice elements. When flux, qs , occurs between the material boundary and the atmosphere, it is necessary to account for convective boundary conditions: qs “ C f pHs ´ Ha q

(6)

Here, Cf is the film coefficient, qs is the moisture flux across the boundary, Hs and Ha are the relative humidities at the material surface and surrounding atmosphere, respectively. In the lattice model, the evaporation rate was implemented through force vector in the element fe :


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qs Aij fe “ ϑ

«

=

1 0

ff

C f Aij “ ϑ

1

«

Hs,i ´ Ha 0 ,

=

ff

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(7)

−

(7)

where Hs,i is the relative humidity of the surface0 node (node at 0 the surface exposed to drying) and ϑ is the correction which is determined where Hs,ifactor is the relative humidity of theas: surface node (node at the surface exposed to drying) and ϑ is the correction factor which is determined as: řn As ϑ “ ∑i“1 (8) A (8) = Here,Here, n is the number of of elements tothe thesurface surface nodes, Asthe is the n is total the total number elementscorresponding corresponding to nodes, As is areaarea of of Voronoi facetsfacets corresponding to these elements and and A isAthe areaarea of the exposed to drying. Voronoi corresponding to these elements is total the total of surface the surface exposed to The concept similar asisfor determination of the correction factor ωfactor (see ωEquation (5)). (5)). drying. is The concept similar as for determination of the correction (see Equation

Figure 5. Definition of overlap area for determination of parameter ω (adapted from [34]).

Figure 5. Definition of overlap area for determination of parameter ω (adapted from [34]).

Using the Crank-Nicholson procedure [35], the system of linear equations was then discretized

Using the Crank-Nicholson procedure [35], the system of linear equations was then discretized in in time, and the following equation resulted: time, and the following equation resulted: (9) ( ) ) ( + 0.5∆

´

=

¯

− 0.5∆

´

+∆

¯

n´1for each 1 (∆t) This equation wasM then solved timen´ step contents were (9) ` 0.5tK H n “ discrete M ´ 0.5tK H n´1and ` t moisture f obtained. Since parameter D and, therefore, K is dependent on relative humidity H, an iterative procedure was avoided by calculating the relative humidity in each step (n) based on values of This equation was then solved for each discrete time step (∆t) and moisture contents were hydraulic diffusivity determined from the previous step (n − 1). Although this implies a certain obtained. Since parameter D and, therefore, K is dependent on relative humidity H, an iterative amount of error, it significantly shortens the simulation time and the error is small as long as the procedure was∆tavoided by calculating relative humidity in each stepis (n) values of time step is kept small. A benefit ofthe using a lattice type moisture model thatbased furtheron fracture hydraulic diffusivity determined from thesimulations previous step ´ 1). Although this implies aand certain simulations are coupled with moisture whilst(nthe same material mesostructure amount of are error, it significantly shortens the simulation time and the error is small as long as the mesh used.

time step ∆t is kept small. A benefit of using a lattice type moisture model is that further fracture Transport of Lattice Elements simulations areProperties coupled with moisture simulations whilst the same material mesostructure and mesh are used. For the analysis of the time-dependent drying shrinkage, a repair system was simulated. A similar problem was studied numerically by Wittmann and Martinola [12], Sadouki and van Mier

Transport Properties ofand Lattice Elements [11], and Bolander Berton [14]. Therefore, the same input parameters were used: the system was exposed to environment with 50% relative humidity (Ha = 0.5) and film coefficient of the surface was

For the analysis of the time-dependent drying shrinkage, a repair system was simulated. A similar 0.7 mm/day. The terms in the diffusivity Equation (2) and Table 1 were set to obtain humidity problem was studied numerically by Wittmann and Martinola [12], Sadouki and van Mier [11], and profiles similar to those obtained by Martinola and Wittmann [12]. The concrete substrate was Bolander and Berton [14]. means Therefore, the top same input wererelative used: the systemaswas pre-saturated, which that the layer hadparameters the same initial humidity theexposed repair to environment with 50% relative humidity (H = 0.5) and film coefficient of the surface was 0.7 mm/day. a material (H = 1). The termsConcrete in the diffusivity Equation (2) and Table 1 were set to obtain humidity profiles similar substrate mesostructure was included. The volume percentage of the crushed stone in to simulated substrate was 30%. This approximately correspond to the amount which of those the obtained by concrete Martinola and Wittmann [12].should The concrete substrate was pre-saturated, aggregates in the while the sand andhumidity cement paste wererepair considered to form meanscoarse that the top layer hadconcrete, the same initial relative as the material (H =a mortar 1). phase. Sixty-seven (67%) of coarse in the sieve range of (8,16) mm and 33% in in Concrete substratepercent mesostructure was aggregates included. were The volume percentage of the crushed stone the sieve range of (4,8) mm. In the present model, aggregates were assumed to be impermeable (as the simulated concrete substrate was 30%. This should approximately correspond to the amount of coarse aggregates in the concrete, while the sand and cement paste were considered to form a mortar


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phase. Sixty-seven percent (67%) of coarse aggregates were in the sieve range of (8,16) mm and 33% in the sieve range of (4,8) mm. In the present model, aggregates were assumed to be impermeable (as similarly done by Wang and Ueda when modelling capillary absorption of concrete in [23]) and were ascribed very low diffusivity properties (Table 1). It has to be highlighted however, that for different type of aggregates, and especially for lightweight aggregates, this assumption is not valid. The surface nodes of aggregates, which are in contact with surrounding matrix, have the same initial relative humidity as the surrounding matrix. As a zone with higher porosity, ITZ has higher diffusivity properties than the bulk matrix. In current simulations, 3 times higher diffusivity compared to the bulk matrix diffusivity is ascribed to ITZ. This is in line with ratio used by Šavija et al. [21] for chloride diffusivity. In their study, effective diffusion rate for chloride penetration in various concrete mixtures was calculated, and 2.5–7 times higher diffusion coefficient for ITZ elements compared to bulk matrix was obtained. Since both chloride diffusivity and water movement are directly related to the porosity, the same ratio was assumed to be valid in this study. In this way, effective diffusivity for the ITZ elements (which depends on the chosen mesh size) was fitted from experiments. It might, however, be argued whether the same ratio for moisture transport and chloride penetration can be used in these simulations. Wang and Ueda [23], for example, used 10 times higher diffusivity of ITZ elements when modelling moisture absorption of the concrete. Note that in their simulations, however, the thickness of ITZ was around 50 micrometres, while interface simulated herein was much thicker (around 1 mm). Therefore, the assumed ratio of 3 is probably a reasonable starting point but should be calibrated with experimental results for the specific mixture. To check the accuracy of the model, non-linear solution procedure, and implementation of convective boundary conditions, simulation results from the lattice moisture model were previously compared with those from a commercial finite element model FEMMASSE MLS [38] and published in [15]. The lattice model showed performance comparable to the commercial FE model. Note that, for different water-to-cement ratios and aggregate content of the materials, diffusion parameters (as given in Table 1) would be different. In order to obtain moisture diffusivity experimentally, concrete substrate and repair mortar should be exposed to different environmental conditions, and an inverse numerical technique should be applied on processing the experimental data [12]. Therefore, for different repair materials and substrates, moisture transport properties need to be measured and used as input for the simulation. 2.3. Lattice Fracture Model In lattice fracture models, the material is discretized as a network of beam elements connected at the ends. All individual elements have linear elastic behaviour. In each loading step, an element that exceeds the limit stress capacity is removed from the mesh. The analysis procedure is then repeated until a pre-determined failure criterion is reached. For the fracture criterion, only axial forces are taken into account to determine the stress in the beams. An element in the lattice fracture model can fail either in tension or in compression, when the stress exceeds its strength. Mechanical Properties of Lattice Elements In fracture simulations, different mechanical properties were ascribed to elements of a certain phase (Table 2). Two different mixtures of repair material were simulated. First, repair material was simulated as a repair mortar, with homogenous local properties. In the other mixture, PVA (polyvinyl alcohol) fibres were added. These fibres enable crack bridging and a strain hardening behaviour of a material, without a significant change in tensile strength. Such a strain-hardening cementitious composite (SHCC) can be used as a repair material. SHCC was first tailored through a single fibre pull-out test and then tested in a direct tension test. More details about the procedure can be found in [27]. Mechanical properties of interface elements, which connected the repair material and mortar substrate used in the simulations, were assumed values. As the interface is usually the weakest zone


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in the system, lower properties were ascribed to elements that characterize this zone. To investigate the influence of interface strength on the fracture behaviour of the repair system, the tensile strength of the interface is varied from 1 to 3 MPa. For simplicity, it was assumed that the concrete substrate strength is similar to the repair material strength. Although in real repair systems this is not the case, here it was done in order to avoid Materials 2016, 9, 575 9 of 19 additional influence of this parameter on the response of the system. Properties of ITZ and mortar in thethe concrete were, therefore, adjusted so that of thethetensile oftensile the substrate influencesubstrate of interface strength on the fracture behaviour repair strength system, the strength was similar to the strength of thefrom repair of the interface is varied 1 tomaterial. 3 MPa. It has to be highlighted, however, that assumed substrate strength will the balance betweenthat debonding and cracking repair ismaterial substrate, For affect simplicity, it was assumed the concrete substrate in strength similar and/or to the repair material in strength. Although in real repair systems this is not the case, here it was done in order to as discussed Chapter 7 in [39]. avoid additional influence of this parameter on the response of the system. Properties of ITZ and

2.4. Coupling theconcrete Lattice substrate Moisture were, Modeltherefore, with the Lattice Fracture Model mortar inofthe adjusted so that the tensile strength of the substrate was similar to the strength of the repair material. It has to be highlighted, however, that assumed

For coupling moisture and fracture analysis, it can be assumed that the moisture distribution substrate strength will affect the balance between debonding and cracking in repair material and/or produces a shrinkage fieldinaccording [3]: substrate, as discussed Chapter 7 to in equation [39]. αsh HFracture Model 2.4. Coupling of the Lattice Moisture Model withε sh the“Lattice

(10)

coupling moisture and fracture analysis, it can be assumed that the moisture distribution where εsh For is the unrestrained shrinkage strain, ∆H is the change in relative humidity and αsh hygral produces a shrinkage field according to equation [3]: coefficient of shrinkage which can be measured from drying tests at different relative humidities. (10) [12]: = measurements ∆ Hygral coefficients were taken from experimental of Martinola and Wittmann for repair for interface and substrate = 0.0028; wherematerial εsh is theαunrestrained shrinkage strain, between ΔH is therepair changematerial in relative humidity andαsh,INT αsh hygral sh,RM = 0.0048; and for substrate αsh,SUB = which 0.0013.can Note that, for different mixtures repairrelative material and concrete coefficient of shrinkage be measured from drying tests at of different humidities. Hygral coefficients were taken from experimental measurements of Martinola and Wittmann [12]: substrate, hygral coefficients will differ and should be measured experimentally. forarepair material αsh,RM = 0.0048; for to interface between repair material and substrate sh,INT = 0.0028; If material is completely free deform, shrinkage deformation due to αmoisture change andonly for substrate αsh,SUB = 0.0013. Note that, forand different mixtures of repair materialinternal and concrete results in volume change of the material no stress occurs. However, restraints substrate, hygral coefficients will differ and should be measured experimentally. in material (i.e., aggregates, stiff inclusions) and external restraints (i.e., bond with the substrate) result If a material is completely free to deform, shrinkage deformation due to moisture change results in eigenstresses and local stress concentrations that might exceed the material strength. Once the only in volume change of the material and no stress occurs. However, internal restraints in material strength exceeded, cracking in the material occurs. The lattice this behaviour. Due to (i.e.,isaggregates, stiff inclusions) and external restraints (i.e., mesh bond mimicked with the substrate) result in the relative humidity change (∆H), every element tended to change its length (Figure 6). However, eigenstresses and local stress concentrations that might exceed the material strength. Once the the deformation restrained by element’s connectivity with other elements in the ThisDue resulted strength was is exceeded, cracking in the material occurs. The lattice mesh mimicked thismesh. behaviour. in generation of tensile stresses. Each (ΔH), element, according its cross local elastic to the relative humidity change every elementtotended to section changeA, itsitslength (Figuremodulus 6). However, the deformation was restrained by element’s connectivity E, and the resulting moisture change, was axially loaded with a force: with other elements in the mesh. This resulted in generation of tensile stresses. Each element, according to its cross section A, its local elastic modulus E, and the resulting moisture change, Nsh “ ε sh E A was axially loaded with a force: (11) =

(11)

Once an element broke, it was removed from the mesh. Stresses in that element were released and Once an element broke, it was removed from the mesh. Stresses in that element were released other elements connected to the broken element were less restrained. During further analysis, new and other elements connected to the broken element were less restrained. During further analysis, restrain levels and moisture gradients led to stress generation and redistribution among the surviving new restrain levels and moisture gradients led to stress generation and redistribution among the beams, until the strength of the next element was reached and that element also broke. This is analogue surviving beams, until the strength of the next element was reached and that element also broke. to material theshrinkage—once material cracks,the thematerial magnitude ofthe restraints in the material in is reduced This is shrinkage—once analogue to material cracks, magnitude of restraints the and the stresses are partially released. Lower restraint levels led to higher deformability of the system material is reduced and the stresses are partially released. Lower restraint levels led to higher and lower stresses.of the system and lower stresses. deformability

Figure 6. Coupling lattice moistureand andlattice lattice fracture time (H-relative humidity). Figure 6. Coupling lattice moisture fracturemodel modelinin time (H-relative humidity).

3. Numerical Results and Discussion In order to verify the proposed modelling approach, crack development and distribution due to restrained shrinkage was examined and compared to damage propagation in experiments. Periodic


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3. Numerical Results and Discussion In order to verify the proposed modelling approach, crack development and distribution due to restrained shrinkage was examined and compared to damage propagation in experiments. Periodic Materials 2016, 9, 575 10 of 19 boundary conditions were applied in two (horizontal) directions (Figure 4). All parameters for moisture transport andconditions fracture properties wereintaken as given in directions Tables 1 and 2. 4). All parameters for boundary were applied two (horizontal) (Figure 2 with the total height of 30 mm. Due to Dimensions of the repair system were 40 ˆ 40 mm moisture transport and fracture properties were taken as given in Tables 1 and 2. high computational specimen werethe limited; for the simulation with Dimensions ofdemands, the repair system weredimensions 40 × 40 mm2 with total height of 30 mm. Due to highSHCC, computational specimen (Repair dimensions were limited; for the simulation with SHCC, there were in totaldemands, 639,255 elements material, RM = 103,752, Mortar Substrate, MSthere = 144,639, were in total 639,255 (Repair material, RM = = 103,752, Substrate, MS = 144,639, Aggregates = 62,414, ITZelements = 43,832, Interface (MS/RM) 11,731,Mortar Interface (Matrix/Fibre) = 183,011, Aggregates 62,414, ITZ =nodes 43,832, Interface (MS/RM) 11,731, system Interfacewithout (Matrix/Fibre) = 183,011, Fibre = 89,876) =and 154,045 (48,000 nodes for the= repair fibres and the rest are Fibre = 89,876) and 154,045 nodes (48,000 nodes for the repair system without fibres and the rest are additional nodes for fibre elements). Repair material thickness was either 6 or 10 mm. The substrate additional nodes for fibre elements). Repair material thickness was either 6 or 10 mm. The substrate contained 30% of coarse aggregate particles (by volume), generated and packed using the Anm material contained 30% of coarse aggregate particles (by volume), generated and packed using the Anm model [32]. Top 10 mm of the substrate was fully saturated, while the bottom had an initial relative material model [32]. Top 10 mm of the substrate was fully saturated, while the bottom had an initial humidity 90%. Wet surface the substrate came either from from the wetting of the substrate prior to relativeofhumidity of 90%. Wetofsurface of the substrate came either the wetting of the substrate application of the repair material, or from or capillary absorbed waterwater fromfrom the repair material. Smooth prior to application of the repair material, from capillary absorbed the repair material. andSmooth sandblasted surface imitated of the concrete (Figure 3a,b). and sandblasted surfaceroughness imitated roughness of the substrate concrete substrate (FigureSandblasted 3a,b). Sandblasted surface was with 3 mm of aggregates the surface (Figure 7a). After thesurface surface was mimicked withmimicked 3 mm of aggregates at the surfaceat(Figure 7a). After the substrate substrate surface was “prepared”, the repair material was “cast” on top of it (Figure 7b). Duesubstrate to was “prepared”, the repair material was “cast” on top of it (Figure 7b). Due to drying and drying and substrate absorption, moisture transport in repair system took place (Figure 7c). absorption, moisture transport in repair system took place (Figure 7c).

(a)

(b)

(c)

Figure 7. Lattice preparation procedure and moisture simulations (a) Imitating sandblasted surface

Figure 7. Lattice preparation procedure and moisture simulations (a) Imitating sandblasted surface (top 3 mm of aggregates is exposed); (b) “Casting” repair mortar with 10 mm thickness; (c) Moisture (top 3 mm of aggregates is exposed); (b) “Casting” repair mortar with 10 mm thickness; (c) Moisture distribution in the repair system at 110 days of drying. distribution in the repair system at 110 days of drying.

First, non-reinforced repair mortar was used as a repair material. Influence of the interface strength and substrate roughness on the simulated pattern at 110Influence days of drying shown in First, non-reinforced repair mortar was used as acrack repair material. of the is interface strength Figure 8. Crack width in the lattice model can be calculated for each analysis step. It is defined as Figure an and substrate roughness on the simulated crack pattern at 110 days of drying is shown in 8. increase in length of the damaged element at a certain analysis step compared to the unloaded state. Crack width in the lattice model can be calculated for each analysis step. It is defined as an increase in Debonding in the model (interface crack) is defined as the increase in length of the damaged length of the damaged element at a certain analysis step compared to the unloaded state. Debonding in interface element. In all following images where the damage is shown, deformation is scaled 100×. the model (interface crack) is defined as the increase in length of the damaged interface element. In all Maximum crack width and debonding (0.12 mm and 0.056 mm, respectively) are observed when the following images where the damage shown, deformation scaled 100ˆ. width and substrate surface is smooth and theis interface strength is 1 is MPa (Figure 8a).Maximum Cracks in crack the repair debonding (0.12 mm and 0.056 mm, respectively) are observed when the substrate surface is smooth mortar form at close to a 90° angle. A rough surface with the same interface strength enables a andsimilar the interface strength is 1 MPa (Figure 8a). Cracks in the repair mortar form at close to a 90˝ angle. polygonal pattern with almost straight cracks (Figure 8b). However, due to more restraint and higher at same the interface, the higher roughness profile, more cracks with A rough surfacefriction with the interfaceprovided strength by enables a similar polygonal pattern with almost straight smaller crack spacing and smaller (maximum crackfriction width of mm and debonding cracks (Figure 8b). However, due tocrack morewidths restraint and higher at 0.090 the interface, provided by the of 0.044 mm) form. the bond strength at the crack interface increases, more cracks observed higher roughness profile,If more cracks with smaller spacing and smaller crack are widths (maximum (maximum crack width is 0.077 mm and debonding is 0.03 mm) and cracks are not that straight crack width of 0.090 mm and debonding of 0.044 mm) form. If the bond strength at the interface increases, (Figure 8c). Apart from T-junctions (cracks meeting at the angle of 90°), there is also a Y-junction more cracks are observed (maximum crack width is 0.077 mm and debonding is 0.03 mm) and cracks are (triplet junction with cracks meeting at the angle of 120°). ˝

not that straight (Figure 8c). Apart from T-junctions (cracks meeting at the angle of 90 ), there is also a Y-junction (triplet junction with cracks meeting at the angle of 120˝ ).


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Side view

Top view

Top view (crack widths)

(a)

(b)

(c)

Figure 8. Influence of bond strength and substrate roughness on cracking after 110 days of drying,

Figure 8. Influence of bond strength and substrate roughness on cracking after 110 days of drying, side side and top view: white—repair material, blue—substrate, black—cracks, orange—aggregates, and top view: white—repair material, blue—substrate, black—cracks, orange—aggregates, red—ITZ; red—ITZ; top view (crack widths): colours as indicated in the legend (a) Smooth surface, interface top view (crack widths): colours as indicated in the legend (a) Smooth surface, interface strength 1 MPa; strength 1 MPa; (b) Rough surface, interface strength 1 MPa; (c) Rough surface, interface strength 3 (b) Rough MPa. surface, interface strength 1 MPa; (c) Rough surface, interface strength 3 MPa.

Experimentally, Groisman Kaplan found transition T-junction to Y-junction Experimentally, Groisman andand Kaplan [40][40] found transition fromfrom T-junction to Y-junction cracking cracking when the thickness of drying material was reduced. They reported it to be a consequence of when the thickness of drying material was reduced. They reported it to be a consequence of crack crack nucleation from defects and inhomogeneities in the material when the layer thickness is nucleation from defects and inhomogeneities in the material when the layer thickness is sufficiently sufficiently small. Shorlin et al. [41] also reported a few triplet Y-junctions from experiments when small. Shorlin et al. [41] also reported a few triplet Y-junctions from experiments when sand particles sand particles were sprinkled evenly in drying slurry. In presented simulations, sand particles in the were repair sprinkled evenly in not drying slurry. In presented simulations, sand particles repair materials materials were explicitly simulated, but inhomogeneity originates fromin thethe rough surface wereof not explicitly simulated, but inhomogeneity originates from the rough surface of the substrate. the substrate. The influence of substrate roughness on crack patterns observed at the surface of the The influence of substrate crack patterns at the surface of the repair material repair material increasesroughness with higheron bond strength and observed smaller repair material thickness. Therefore, further analyses were performed different repair mortar thicknesses and Therefore, bond strengths. increases with higher bond strengthfor and smaller repair material thickness. further analyses In Figurefor 9a,different crack propagation in thethicknesses repair mortar with 10 mm thickness is shown. Interface were performed repair mortar and bond strengths. strength between the repair material and the concrete substrate is 1 thickness MPa, which leads to Interface low In Figure 9a, crack propagation in the repair mortar with 10 mm is shown. restraint at the interface. First, a single crack forms. This crack is not affected by successive cracking. strength between the repair material and the concrete substrate is 1 MPa, which leads to low restraint at the interface. First, a single crack forms. This crack is not affected by successive cracking. However, it determines the boundary conditions for the following crack. The following crack meets the existing crack at close to 90˝ angle. This is due to the fact that a crack tends to produce the maximum relief


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However, it determines the boundary conditions for the following crack. The following crack meets the existing crack at close to 90° angle. This is due to the fact that a crack tends to produce the of stresses. Therefore, it propagates perpendicular to the direction wheretothethe stresses are the highest. maximum relief of stresses. Therefore, it propagates perpendicular direction where the Both theare second and theBoth thirdthe crack initiate the concave edgefrom of the crack.edge Thisofwas stresses the highest. second and from the third crack initiate thefirst concave thealso first observed bywas Shorlin al. [41] when they explained why the they peak explained of crack distribution is at of slightly crack. This alsoetobserved by Shorlin et al. [41] when why the peak crack ˝ . In this crack development, cracks form successively one after another and, therefore, more than 90is distribution at slightly more than 90°. In this crack development, cracks form successively one subsequent cracks the existing cracks at roughly 90˝existing . Cracking sequence in the simulation after another and, meet therefore, subsequent cracks meet the cracks at roughly 90°. Cracking resembles thethe experimentally investigated drying shrinkage cracks [41].drying shrinkage cracks [41]. sequence in simulation resembles the experimentally investigated In repair material materialthickness thicknesswas wasreduced reducedtoto6 6mm. mm.Interface Interface strength was also In Figure Figure 9b, the repair strength was also set set 1 MPa. It can observedthat thatimmediately immediatelya acouple coupleof of Y-junctions Y-junctions form, mostly above to 1toMPa. It can bebeobserved above the the exposed thethe concrete substrate (orange zoneszones in Figure 9b). These are formed exposedaggregates aggregatesinin concrete substrate (orange in Figure 9b). Y-junctions These Y-junctions are almost and laterand they meet. Bohn et Bohn al. observed [42] that[42] “starlike cracks formedsimultaneously almost simultaneously later theySimilarly, meet. Similarly, et al. observed that “starlike ˝ with relative angles of 120 of occur at material defects in a thininlayer and thatand there is neither cracks with relative angles 120°mainly occur mainly at material defects a thin layer that there is aneither temporal succession of the cracks norcracks a localnor geometrical hierarchy”. hierarchy”. Maximum crack widths are a temporal succession of the a local geometrical Maximum crack smaller (in the repair material 0.049 mm, debonding 0.026 mm) compared to those in thicker layer widths are smaller (in the repair material 0.049 mm, debonding 0.026 mm) compared to those in of repairlayer material. In addition, crack patterns from 9a,bfrom confirms that9a,b crack spacingthat is simply thicker of repair material. In addition, crackFigure patterns Figure confirms crack proportional to the thickness of thetodrying layer as was previously and Kaplan spacing is simply proportional the thickness of the drying stated layer by as Groisman was previously stated[40]. by In their tests, was tested. A similar trendwas is also observed for trend drying Groisman anddrying Kaplancoffee-water [40]. In theirmixture tests, drying coffee-water mixture tested. A similar is damage in other [18,41]. in other materials [18,41]. also observed formaterials drying damage

Figure 9. Cont.

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Figure Figure 9. Damage development development as as aa function function of of time, time, rough rough substrate substrate surface, surface, from from top top to to bottom, bottom, at at Figure 9. 9. Damage 5000, 10,000, 20,000 damaged elements, final crack widths at 110 days of drying (a) interface strength 5000, 10,000, 20,000 damaged elements, final crack widths at 110 days of drying (a) interface strength 11 MPa MPa and mm thick repair material; (b) strength MPa and 66 mm repair material MPaand and1010 10mm mm thick repair material; (b) interface interface strength MPa andrepair mm repair thickness material thick repair material; (b) interface strength 1 MPa11and 6 mm material thickness and interface 66 mm thick repair thickness and (c) (c)strength interface3strength strength MPa and mm thick repair mortar mortar and (c) interface MPa and33 6MPa mmand thick repair mortar.

Impact of heterogeneity of the substrate roughness on the damage development in the repair Impact Impact of of heterogeneity heterogeneity of of the the substrate substrate roughness roughness on on the the damage damage development development in in the the repair repair material is enhanced by increased bond strength (Figure 9c). Triple Y-junction cracks form, and, due material strength (Figure 9c).9c). Triple Y-junction cracks form, and, and, due material is is enhanced enhancedby byincreased increasedbond bond strength (Figure Triple Y-junction cracks form, to more restraint at the interface, crack spacing and maximum crack widths are lower. This is in to more restraint at the interface, crack spacing and maximum crack widths are lower. This is due to more restraint at the interface, crack spacing and maximum crack widths are lower. This in is accordance with the experimental results [40,41], where the friction between drying material and accordance with the experimental results [40,41], where the friction between drying material and in accordance with the experimental results [40,41], where the friction between drying material and substrate was varied (by coating with Vaseline, using Teflon, or drying on aa rough substrate wasvaried varied(by (by coating Vaseline, or on drying on sandpaper rough sandpaper sandpaper substrate was coating withwith Vaseline, usingusing Teflon,Teflon, or drying a rough substrate). substrate). In these experiments, bottom friction with the substrate critically influenced the final substrate). In these experiments, bottom friction with the substrate critically influenced final In these experiments, bottom friction with the substrate critically influenced the final crackthe spacing crack spacing and width. Compared to the simulated repair system, there is an analogy between crack spacing and width. Compared the simulated repair system, there is anbond analogy between and width. Compared to the simulatedtorepair system, there is an analogy between strength and bond strength and friction with the substrate. bond strength and friction with the substrate. friction with the substrate. In simulations, shift in angle between cracks was also observed for the smooth surface case, In In simulations, simulations, shift shift in in angle angle between between cracks cracks was was also also observed observed for for the the smooth smooth surface surface case, case, when only the bond strength changes. This means that the higher restraint level itself when only only the the bond bond strength strength changes. changes. This itself (without (without when This means means that that the the higher higher restraint restraint level level itself (without heterogeneities in the material or interface) will also contribute change in the heterogeneities contribute to to the cracking cracking pattern pattern heterogeneities in in the the material material or or interface) interface) will will also also contribute to change change in in the cracking pattern (Figure 10). With smooth surface substrate and high bond strength, stresses in the repair mortar are (Figure 10). 10). With (Figure With smooth smooth surface surface substrate substrate and and high high bond bond strength, strength, stresses stresses in in the the repair repair mortar mortar are are more uniform. Y-cracks are then preferable considering the “ratio of elastic energy relief to energy more more uniform. uniform. Y-cracks Y-cracks are are then then preferable preferable considering considering the the “ratio “ratio of of elastic elastic energy energy relief relief to to energy energy created by created by cracking cracking surface” surface” [40]. [40]. created by cracking surface” [40].

(a) (a)

(b) (b)

(c) (c)

Figure 10. development in the 10 mm thick Figure 10. Crack Crack thick repair repair material material with with smooth smooth surface surface of of substrate substrate Figure 10. Crack development development in in the the 10 10 mm mm thick repair material with smooth surface of substrate and high interface strength (3 MPa) at (a) 5000 damaged elements; (b) 10000 damaged elements and and high interface strength (3 MPa) at (a) 5000 damaged elements; (b) 10000 damaged elements and high interface strength (3 MPa) at (a) 5000 damaged elements; (b) 10000 damaged elements and and (c) final crack width at the age of 110 days. (c) final crack crack width width at at the the age age of of 110 110 days. days. (c) final

As As the the thickness thickness of of the the drying drying layer layer decreases, decreases, more more Y-cracks Y-cracks over over T-cracks T-cracks can can be be observed observed in in As the thickness of the drying layer decreases, more Y-cracks over T-cracks can be observed in an an an unconfined unconfined hardened hardened cement cement paste paste sample sample exposed exposed to to drying drying shrinkage shrinkage by by Bisschop Bisschop and and Wittel Wittel unconfined hardened cement paste sample exposed to drying shrinkage by Bisschop and Wittel [43]. [43]. [43]. In In their their study, study, however, however, no no aggregates aggregates were were added added to to the the mixture mixture and and the the tested tested thicknesses thicknesses In their study, however, no aggregates were added to the mixture and the tested thicknesses were 3, were were 3, 3, 44 and and 66 mm. mm. All All of of these these thicknesses thicknesses are are at at least least 30 30 times times larger larger than than the the size size of of aa cement cement 4 and 6 mm. All of these thicknesses are at least 30 times larger than the size of a cement particle and, particle particle and, and, therefore, therefore, crack crack patterns patterns should should not not be be influenced influenced by by the the ratio ratio between between maximum maximum therefore, crack patterns should not be influenced by the ratio between maximum inhomogeneity and inhomogeneity inhomogeneity and and thickness thickness of of the the sample. sample. However, However, friction friction at at the the bottom bottom and and therefore therefore higher higher

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restraints provided by the reduced thickness are probably sufficient to provide the change of angles in the crack pattern. thickness of the sample. However, friction at the bottom and therefore higher restraints provided by With SHCC as a are repair material, once the arethe activated, they arrest the reduced thickness probably sufficient to fibres provide change of angles inthe thecrack crack(Figure pattern.11b). Simulation results a regular repaironce mortar (without reinforcement) are given in With SHCC as aofrepair material, the fibres are fibre activated, they arrestand the SHCC crack (Figure 11b). Figure 11a,b respectively. Bothrepair systems have (without the same fibre transport and mechanical properties. Similar Simulation results of a regular mortar reinforcement) and SHCC are given in crack 11a,b patterns are obtained in drying shrinkage experiments, on specimens containing Figure respectively. Both systems have the same transport and mechanical properties. Similar alkali-resistant glass fibresin [44]. Fibreshrinkage alignment and orientation enhancescontaining the heterogeneity and the crack patterns are obtained drying experiments, on specimens alkali-resistant levelfibres of restraint in the repair material (similar enhances to sand grains). As a result, more form glass [44]. Fibre alignment and orientation the heterogeneity and the Y-junctions level of restraint and (similar later these cracks meet. This in accordance with findings of Horing et al. and [45]: insimultaneously, the repair material to sand grains). As a is result, more Y-junctions form simultaneously, theythese concluded strong disorder, the system “breaks through coalescence of initially later cracks that, meet.for This is in accordance with findings of Horing et al.the [45]: they concluded that, independent pointthe defects”. the contrary, forcoalescence the less disordered it “breaks through for strong disorder, system On “breaks through the of initiallymaterial independent point defects”. crack Apart from introducing higher heterogeneity, fibrespropagation”. also prevent Apart cracksfrom from On the propagation”. contrary, for the less disordered material it “breaks through crack widening. Therefore, the maximum crack andcracks crackfrom spacing in SHCC is smaller introducing higher heterogeneity, fibres alsowidth prevent widening. Therefore, thecompared maximumto repair mortar mm compared 0.077compared mm). This is in mortar accordance the authors’ crack width and (0.0546 crack spacing in SHCC is to smaller to repair (0.0546with mm compared to experiments (Figure 12), in which due experiments to drying shrinkage was in twodue repair 0.077 mm). This is in accordance withdamage the authors’ (Figure 12), in studied which damage to systems (Chapterwas 6 instudied [39]). In 12a,b, crack (Chapter patterns 6due to restrained in patterns a system drying shrinkage in Figure two repair systems in [39]). In Figureshrinkage 12a,b, crack withtocommercial repair mortar and SHCC, respectively,repair are presented. InSHCC, Figure respectively, 12c,d maximum due restrained shrinkage in a system with commercial mortar and are crack widths in two12c,d systems are given. presented. In Figure maximum crack widths in two systems are given. In essence, essence, aggregate and local restraint in the In and fibres, fibres, or orany anysort sortofofheterogeneity, heterogeneity,introduce introduce local restraint in material. As As thethe level of ofheterogeneity irregular. the material. level heterogeneityincreases, increases,cracks cracksbecome become more more tortuous tortuous and irregular. Y-junctionsin in model simultaneously and subsequently On the contrary, Y-junctions thethe model form form simultaneously and subsequently coalesce. coalesce. On the contrary, T-junctions T-junctions form successively when a new crack intersects the existing one. Both types of cracks can form successively when a new crack intersects the existing one. Both types of cracks can be found beconcrete found in concrete 13).crack Simulated crack patterns closely resemble in other in (Figure 13). (Figure Simulated patterns closely resemble those found inthose otherfound homogenous homogenous and heterogeneous materials.using Furthermore, using the proposed additional and heterogeneous materials. Furthermore, the proposed model, additionalmodel, phenomena can phenomena can be captured [18,40,41]: crack spacing is proportional to the layer thickness and as the be captured [18,40,41]: crack spacing is proportional to the layer thickness and as the friction with friction with substrate increases bondbetween strength) spacing between decreases. Once substrate increases (or bond strength)(or spacing cracks decreases. Oncecracks quantitatively verified quantitatively verified with experimental model can be reliably used to investigate with experimental observations, the model observations, can be reliablythe used to investigate the influence of critical the influence critical for the repair system performance. parameters forof the repairparameters system performance.

Figure 11. Cont.

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Figure 11. Damage development in repair systems with high interface strength (3 MPa) and a rough Figure 11. 11. Damage Damage development in in repair repair systems systems with high high interface strength strength (3 (3 MPa) MPa) and and aa rough rough Figure substrate surface asdevelopment a function of time, from top with to bottom,interface at 5000, 10,000 and 20,000 damaged substrate surface as a function of time, from top to bottom, at 5000, 10,000 and 20,000 damaged substrate surface as a widths functionatof110 time, from to bottom, at 5000, andthick 20,000 elements, elements, final crack days of top drying, repaired with 10,000 a 10 mm (a)damaged repair mortar and elements, final crack widths atof110 days repaired of drying, repaired with a 10(a) mm thickmortar (a) repair mortar and final crack widths at 110 days drying, with a 10 mm thick repair and (b) SHCC. (b) SHCC. (b) SHCC.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 12. Experiments: low and (c) high Figure Experiments: Crack Crackpatterns patternsin in aaa commercial commercial repair repair material material at at (a) (a) Figure 12. 12. Experiments: Crack patterns in commercial repair material at (a) low low and and (c) (c) high high magnification and in strain hardening cementitious composite (SHCC) repair material at (b) low and magnification and in strain hardening cementitious composite (SHCC) repair material at (b) low magnification and in strain hardening cementitious composite (SHCC) repair material at (b) low and and (d) high magnification magnification (more results results can be be found in in Chapter 66 in in [39]). (d) (d) high high magnification(more (more resultscan can befound found inChapter Chapter 6 in[39]). [39]).


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(a)

(b)

Figure (b) mostly mostly Y-junctions Y-junctions (photos (photos of of existing existing structures). structures). Figure 13. 13. Cracks Cracks in in concrete concrete (a) (a) T-junctions T-junctions;; (b)

4. Conclusions 4. Conclusions In the the present present work, work, aa lattice-based lattice-based model model is is proposed proposed to to simulate simulate moisture moisture transport transport and and In drying shrinkage-induced cracking in the repair systems. The model was compared to experimental drying shrinkage-induced cracking in the repair systems. The model was compared to experimental observations totoevaluate evaluate its ability to mimic the processes Different influencing parameters observations its ability to mimic the processes involved. involved. Different parameters influencing cracking in repair systems caused shrinkage, by drying shrinkage, as strength, the bond strength, repair cracking in repair systems caused by drying such as thesuch bond repair material material thickness, and type of repair material, have been simulated and evaluated. Based on above thickness, and type of repair material, have been simulated and evaluated. Based on above findings, findings, the following conclusions can be drawn: the following conclusions can be drawn:  ‚  ‚

‚

‚

The influence of repair layer thickness, amount of restraint, and heterogeneity of the repair The influence of repair layer thickness, amount of restraint, and heterogeneity of the repair material on crack pattern (crack spacing, geometry, and angles) due to drying shrinkage can be material on crack pattern (crack spacing, geometry, and angles) due to drying shrinkage can be captured with the presented modelling approach. captured with the presented modelling approach. If there is no continuous delamination, with the same bond strength and drying conditions, If therecracks is no continuous same bond strength and drying conditions, more more (with lowerdelamination, crack spacing)with are the observed in thinner overlays. The number of cracks cracks (with lower crack spacing) are observed in thinner overlays. The number of cracks increases increases with an increase in bond strength or substrate roughness. This is in accordance with with an increaseobservations in bond strength or substrate roughness. is in accordance experimental in drying shrinkage tests This of different brittle with and experimental quasi-brittle observations in drying shrinkage tests of different brittle and quasi-brittle materials. materials. Y-junctions Y-junctions in in the the lattice lattice model model form form simultaneously simultaneously as as aa consequence consequence of of higher higher heterogeneities heterogeneities (defects or inclusions, surface roughness of the substrate) and higher restraint by (defects or inclusions, surface roughness of the substrate) and higher restraint by the the substrate substrate (smaller thickness, high bond strength). On the contrary, T-junctions are formed successively, (smaller thickness, high bond strength). On the contrary, T-junctions are formed successively, when when one one crack crack intersect intersect an anexisting existingcrack crack(bigger (biggerthickness, thickness,lower lowerbond bondstrength). strength). With adequate bonding, SHCC performs better in simulated drying shrinkage With adequate bonding, SHCC performs better in simulated drying shrinkage tests. tests. It It exhibits exhibits small crack widths which is beneficial for durability properties of repair systems. Repair small crack widths which is beneficial for durability properties of repair systems.material Repair should strain-hardening behaviour in order in to order enabletomultiple crack formation and to materialhave should have strain-hardening behaviour enable multiple crack formation prevent wide opening of the crack. and to prevent wide opening of the crack.

The failure failuremode modeofof repair system to drying shrinkage very sensitive to substrate The thethe repair system duedue to drying shrinkage is veryissensitive to substrate surface surface roughness, thickness of the overlay and moisture gradients to drying. Simulation roughness, thickness of the overlay and moisture gradients due to due drying. Simulation resultsresults show show that, in general, performance of the repair system due to drying shrinkage can be that, in general, performance of the repair system due to drying shrinkage can be well imitated. Inwell the imitated.however, In the models, a number of assumptions areinput made regarding Diffusivity the input models, a numberhowever, of assumptions are made regarding the parameters. parameters.and Diffusivity parametersused andas hygral coefficients useddifferent as inputs in reality, differentand for parameters hygral coefficients inputs are, in reality, forare, every repair material every repair material and concrete substrate. Therefore, for they should be measured for every the mixture, concrete substrate. Therefore, they should be measured every mixture, as they influence state as stresses they influence the system. state of Due stresses in the repair system. Due to difficulties experimental of in the repair to difficulties in experimental quantification of theindiffusivity and quantification of the diffusivity and fracture properties of interfaces, values used in the model are fracture properties of interfaces, values used in the model are only reasonable assumptions. only Edge reasonable assumptions. boundary conditions were not included in the model. However, it was considered that, after Edgeboundary boundaryconditions conditionsaround were not in the model. However, it was considered that, cracking, thisincluded crack represent edge effect in a repair system. after cracking, boundary conditions around this crack represent edge effect in a repair system.


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Time dependency was not taken into account in simulations. In reality, material properties (diffusion coefficient, modulus of elasticity, strength) are time dependent. In this research only shrinkage strains were considered as a function of time, and they were fitted such that they correspond to experimentally measured values. No change in diffusion coefficient, modulus of elasticity, or strength was considered. In future research, time (hydration) dependency, should also be included. Due to stress relaxation in the repair material, the magnitude of stresses due to internal and external restraints will be lower and, therefore, crack initiation will be postponed. Because, in case of a repair system, relaxation has a beneficial influence, neglecting its influence is a conservative approach when determining the performance of repair systems. For a more accurate prediction and quantification of the repair system performance, both the relaxation of the repair material and the creep of the substrate need to be considered. As long as the same fracture behaviour (crack widths, spacing, debonding) of repair systems in experiments and simulations is obtained as a function of time, maybe it can be (roughly) considered that these relaxation and creep are implicitly taken into account. Acknowledgments: Financial support by the Dutch Technology Foundation (STW) for the project 10981—“Durable Repair and Radical Protection of Concrete Structures in View of Sustainable Construction” is gratefully acknowledged. Author Contributions: Mladena Luković, Branko Šavija, Erik Schlangen, Guang Ye and Klaas van Breugle performed the analyses and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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