# A 3D Lattice Modelling Study of Drying Shrinkage Damage in Concrete Repair Systems

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## Abstract

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## 1. Introduction

## 2. Modelling Approach

#### 2.1. Spatial Discretization

- In each cell of size A, a sub cell with linear dimension of s was defined (marked in Figure 1a). The lattice node was randomly positioned inside this sub cell (Figure 1a). The ratio s/A is defined as randomness of a lattice. When randomness is 0, the node is located in the centre of cell and regular lattice mesh is generated. If the randomness is higher than 0, beams have different lengths and material disorder is built through the geometry of lattice mesh. With randomness of 1, therefore, materials have the maximum degree of disorder. The choice of randomness affects the simulated fracture of materials to a certain extent, as different orientation of meshes can affect the crack shape [29]. In order to include benefits of random mesh, the randomness here was set to be 0.5.
- Voronoi tessellation of the prismatic domain with respect to the specified set of nodes was performed. Nodes with adjacent Voronoi cells were connected by lattice elements (Figure 1a,b and Figure 2b). Since Voronoi diagrams are dual with Delaunay tessellation, this approach is equivalent to performing a Delaunay tessellation of the set of nodes [21].
- In order to take material heterogeneity into account, either a computer-generated material structure, or a material structure obtained by micro-CT scanning [17,30] can be used. Here, the concrete mesostructure was simulated using the Anm material model originally developed by Garboczi [31] and implemented in a 3D packing algorithm by Qian et al. [32]. It is based on placing multiple irregular shape particles separated into several sieve ranges into a predefined empty container (Figure 2a). Aggregates smaller than 4 mm in the substrate are not explicitly modelled and, together with cement paste, are considered to be part of the mortar phase. Similarly, in the repair material, the largest sand (filler) particles are around 200 μm and, as such, are also not explicitly simulated.
- Material overlay procedure (schematically shown in Figure 1a,b and Figure 2b) was employed: the beams that belong to each phase were identified by overlapping the material mesostructure (i.e., substrate mortar, repair material mortar, and aggregates) on top of the lattice. Interface elements between the repair material and the substrate for smooth and rough substrate surface were generated between substrate nodes and repair material nodes (Interface MS/RM, Figure 1a,b, respectively). Aggregate-mortar interface (ITZ) elements were generated between mortar nodes and aggregates (Figure 2b). By applying this overlay procedure, the geometrical roughness of the substrate was also explicitly modelled (Figure 3a,b).
- Different transport and mechanical properties were assigned to different phases (Table 1 and Table 2, respectively). Interfaces, as used in the present model, are always considered as one row of beam elements. As such, they do not exactly coincide with the size of real interfaces. In reality, interface thickness is in a range of tens of micrometres, while interface elements in the present model also take up a piece of aggregate (or repair material) and a piece of mortar. Therefore, the actual size of the interface in the model depends on the characteristic element size and, in presented simulations, is around 1 mm.
- For fracture simulations, fibre elements were added in the repair material with a design volume content (0.5%), fibre length (8 mm) and diameter (0.04 mm). The location of the first node of each fibre was chosen randomly in the specified volume and a random direction was defined which determined the position of the second node (Figure 4a). If the second node was outside the mesh boundary, then the fibre was automatically cut off.
- Extra nodes inside the fibres were generated at each location where the fibre crosses the grid (Figure 1a,b).
- Fibre/matrix interface elements were generated between fibre nodes and the matrix nodes in the neighbouring cell. Also, the end nodes of the fibres were connected with an interface element to the matrix node in the cell where the fibre end was located (Figure 1a,b).
- Note that in order to reduce the computational time, 0.5% instead of 2% of fibres, as commonly used in SHCC, was simulated. In order to obtain SHCC fracture behaviour in simulations with a lower percentage of implemented fibres, fibre/matrix interface strength and fibre tensile strength from Table 2 are higher compared to experimentally measured values which can be found in [27].
- Samples were simulated with periodic boundary conditions. This means that one side of the specimen was connected to the other end. Therefore, aggregates and fibres were distributed in such a way that they continue through boundaries and periodically repeat. The example for the periodic boundary conditions of the fibres is given in Figure 4a and for aggregates in Figure 4b.
- 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

_{e}and K

_{e}, have the following forms:

_{ij}is the length of the element between nodes i and j, A

_{ij}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]:

_{i}and l

_{i}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.

_{s}, occurs between the material boundary and the atmosphere, it is necessary to account for convective boundary conditions:

_{f}is the film coefficient, q

_{s}is the moisture flux across the boundary, H

_{s}and H

_{a}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 f

_{e}:

_{s,i}is the relative humidity of the surface node (node at the surface exposed to drying) and ϑ is the correction factor which is determined as:

_{s}is the area of Voronoi facets corresponding to these elements and A is the total area of the surface exposed to drying. The concept is similar as for determination of the correction factor ω (see Equation (5)).

#### Transport Properties of Lattice Elements

_{a}= 0.5) and film coefficient of the surface was 0.7 mm/day. The terms in the diffusivity Equation (2) and Table 1 were set to obtain humidity profiles similar to those obtained by Martinola and Wittmann [12]. The concrete substrate was pre-saturated, which means that the top layer had the same initial relative humidity as the repair material (H = 1).

#### 2.3. Lattice Fracture Model

#### Mechanical Properties of Lattice Elements

#### 2.4. Coupling of the Lattice Moisture Model with the Lattice Fracture Model

_{sh}is the unrestrained shrinkage strain, ΔH is the change in relative humidity and α

_{sh}hygral coefficient of shrinkage which can be measured from drying tests at different relative humidities. Hygral coefficients were taken from experimental measurements of Martinola and Wittmann [12]: for repair material α

_{sh,RM}= 0.0048; for interface between repair material and substrate α

_{sh,INT}= 0.0028; and for substrate α

_{sh,SUB}= 0.0013. Note that, for different mixtures of repair material and concrete substrate, hygral coefficients will differ and should be measured experimentally.

## 3. Numerical Results and Discussion

^{2}with the total height of 30 mm. Due to high computational demands, specimen dimensions were limited; for the simulation with SHCC, there were in total 639,255 elements (Repair material, RM = 103,752, Mortar Substrate, MS = 144,639, Aggregates = 62,414, ITZ = 43,832, Interface (MS/RM) = 11,731, Interface (Matrix/Fibre) = 183,011, 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 contained 30% of coarse aggregate particles (by volume), generated and packed using the Anm material model [32]. Top 10 mm of the substrate was fully saturated, while the bottom had an initial relative humidity of 90%. Wet surface of the substrate came either from the wetting of the substrate prior to application of the repair material, or from capillary absorbed water from the repair material. Smooth and sandblasted surface imitated roughness of the concrete substrate (Figure 3a,b). Sandblasted surface was mimicked with 3 mm of aggregates at the surface (Figure 7a). After the substrate surface was “prepared”, the repair material was “cast” on top of it (Figure 7b). Due to drying and substrate absorption, moisture transport in repair system took place (Figure 7c).

## 4. Conclusions

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

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Two-dimensional overlay procedure for generation of the lattice model in the interface zone between the fibre reinforced repair material and (

**a**) a smooth; and (

**b**) a rough surface substrate.

**Figure 2.**Including 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 aggregate and substrate mortar (ITZ).

**Figure 4.**Periodic boundary conditions in horizontal directions for (

**a**) fibres (

**grey**colour) in the SHCC repair material (

**b**) aggregates (

**light grey**colour) in the concrete substrate.

**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 distribution in the repair system at 110 days of drying.

**Figure 8.**Influence of bond strength and substrate roughness on cracking after 110 days of drying, side and top view: white—repair material, blue—substrate, black—cracks, orange—aggregates, red—ITZ; top view (crack widths): colours as indicated in the legend (

**a**) Smooth surface, interface strength 1 MPa; (

**b**) Rough surface, interface strength 1 MPa; (

**c**) Rough surface, interface strength 3 MPa.

**Figure 9.**Damage development as a function of time, rough substrate surface, from top to bottom, at 5000, 10,000, 20,000 damaged elements, final crack widths at 110 days of drying (

**a**) interface strength 1 MPa and 10 mm thick repair material; (

**b**) interface strength 1 MPa and 6 mm repair material thickness and (

**c**) interface strength 3 MPa and 6 mm thick repair mortar.

**Figure 10.**Crack development in the 10 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 (

**c**) final crack width at the age of 110 days.

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

**a**) repair mortar and (

**b**) SHCC.

**Figure 12.**Experiments: Crack patterns in a commercial repair material at (

**a**) low and (

**c**) high magnification and in strain hardening cementitious composite (SHCC) repair material at (

**b**) low and (

**d**) high magnification (more results can be found in Chapter 6 in [39]).

**Figure 13.**Cracks in concrete (

**a**) T-junctions; (

**b**) mostly Y-junctions (photos of existing structures).

**Table 1.**Diffusivity parameters used in the lattice moisture transport model to obtain diffusion coefficient, D (mm

^{2}/day) (Equation (2)). Parameters are adjusted such that the moisture profiles in the repair system at 1 day, 10 days and 110 days are the same as those obtained by Wittmann and Martinola [12].

Diffusivity Parameters | Repair Mortar | Interface | Mortar Substrate | ITZ | Aggregates |
---|---|---|---|---|---|

β | 0.022 | 0.022 | 0.022 | 0.066 | 0.00022 |

γ | 7.5 | 7 | 4 | 4 | 0 |

**Table 2.**Input values for lattice fracture model [27], E stands for the modulus of elasticity, f

_{t}stands for the tensile strength and f

_{c}for the compressive strength of lattice elements.

Element | E (GPa) | f_{t} (MPa) | f_{c} (MPa) | |
---|---|---|---|---|

Matrix (repair mortar-RM) | 20 | 3.5 | 35 | |

Fibre | 40 | 7380 | (7380) | |

Interface (Matrix/Fibre) | 20 | 90 | 900 | |

Mortar substrate | 25 | 4 | 40 | |

Aggregate | 70 | 8 | 80 | |

ITZ | 15 | 2.5 | 25 | |

Matrix (Repair mortar-RM) | 20 | 3.5 | 35 | |

Interface (Matrix/fibre) | 20 | 90 | 900 | |

Interface (MS/RM) | Weak | 15 | 1 | 10 |

Strong | 15 | 3 | 30 |

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**MDPI and ACS Style**

Luković, M.; Šavija, B.; Schlangen, E.; Ye, G.; Van Breugel, K. A 3D Lattice Modelling Study of Drying Shrinkage Damage in Concrete Repair Systems. *Materials* **2016**, *9*, 575.
https://doi.org/10.3390/ma9070575

**AMA Style**

Luković M, Šavija B, Schlangen E, Ye G, Van Breugel K. A 3D Lattice Modelling Study of Drying Shrinkage Damage in Concrete Repair Systems. *Materials*. 2016; 9(7):575.
https://doi.org/10.3390/ma9070575

**Chicago/Turabian Style**

Luković, Mladena, Branko Šavija, Erik Schlangen, Guang Ye, and Klaas Van Breugel. 2016. "A 3D Lattice Modelling Study of Drying Shrinkage Damage in Concrete Repair Systems" *Materials* 9, no. 7: 575.
https://doi.org/10.3390/ma9070575