# Influence of Boundary Conditions on Numerical Homogenization of High Performance Concrete

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Methodology

#### 2.2. Recipes for Modelled HPC

#### 2.3. Microstructure of Modeled HPC

**Mixture I**(Figure 2a):

- fine quartz sand (dark blue): E = 48,200 MPa, $\nu $ = 0.20, 37.4 wt%
- steel micro fibres (black): E = 210,000 MPa, $\nu $ = 0.30, yield stress = 2100 MPa, 8.8 wt%
- air voids (yellow): empty space (no finite elements), 4 wt%.

**Mixture II**(Figure 2b):

- fine quartz sand (dark blue): E = 48,200 MPa, $\nu $ = 0.20, 22.6 wt%
- thick quartz sand (sky blue): E = 73,200 MPa, $\nu $ = 0.20, 12.3 wt%
- steel micro fibres (black): E = 210,000 MPa, $\nu $ = 0.30, yield stress = 2100 MPa, 8.6 wt%
- steel fibres (orange): E = 210,000 MPa, $\nu $ = 0.30, yield stress = 1100 MPa, 4.3 wt%
- air voids (yellow): empty space (no finite elements), 4 wt%.

#### 2.4. Parameters of Concrete Damage Plasticity (CDP) Model

- $\beta $—the internal friction angle of concrete. In the CDP model, $\beta $ is defined as the inclination angle of the Drucker–Prager surface asymptote to hydrostatic axis of the meridional plane;
- m—eccentricity of the surface of the plastic potential. This is the distance measured along the hydrostatic axis between the apex of the Drucker–Prager hyperbola and the intersection of the asymptote of this hyperbola, calculated in practice as a ratio of tensile strength to strength for compression;
- ${f}_{b0}/{f}_{c0}$—number specifying the compressive strength ratio in a two-axis state for the strength in a single-axis state;
- ${K}_{c}$—parameter defining the shape of the surface of the plastic potential on a deviatoric plane;
- $\eta $—viscoplasticity parameter, used to regularize the concrete constitutive equations.

#### 2.5. Numerical Homogenization and Boundary Conditions

#### 2.5.1. Linear Displacement Boundary Conditions (DBC)

#### 2.5.2. Uniform Traction Boundary Conditions (TBC)

#### 2.5.3. Periodic Boundary Conditions (PBC)

## 3. Results and Discussion

#### 3.1. Test Example: Homogeneous and Linear Elastic RVE with Hole

#### 3.2. Compression Test for Mixtures I and II

#### 3.3. Tensile Test for Mixture I and II

#### 3.4. Shear Test for Mixture I and II

## 4. Concluding Remarks

- The periodic boundary conditions (PBC) lead to stable results for the full range of deformations up to failure in all performed numerical calculations.
- In the compression and tensile tests, the upper estimate of values of macro-parameters is reached for mixture I by imposing the PBC whereas for mixture II by imposing the applied displacement boundary conditions (DBC).
- Use of the DBC provides the upper estimate of values of macro-parameters in the shear test for both mixtures I and II.
- Application of the traction boundary conditions (TBC) leads in all analyzed cases to a lower estimate of values of the macro-parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Representative volume element of high-performance concrete (HPC): (

**a**) microstructure [58] and (

**b**) an exemplary finite element model of representative volume element (RVE) with the given number of finite elements and wt% of components.

**Figure 2.**RVE for (

**a**) mixture I and (

**b**) mixture II. The RVE (10 × 10 mm) is divided into 2500 finite elements, each of dimensions 0.2 × 0.2 mm.

**Figure 7.**Shear test for homogeneous linear elastic RVE with a hole, demonstrating the influence of the kind of boundary condition (periodic boundary condition (PBC), displacement boundary condition (DBC), and traction boundary conditions (TBC)) on the solution.

**Figure 8.**Compression test for mixture I, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Figure 9.**Compression test for mixture II, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Figure 10.**Tensile test for mixture I, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Figure 11.**Tensile test for mixture II, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Figure 12.**Distribution of damage parameter: (

**a**) compression of mixture I with PBC; (

**b**) tension of mixture I with PBC; (

**c**) compression of mixture II with DBC; and (

**d**) tension of mixture II with DBC. The white spots represent pores.

**Figure 13.**Shear test for mixture I, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Figure 14.**Shear test for mixture II, demonstrating the influence of the type of boundary condition (PBC, DBC, and TBC) on the solution.

**Table 1.**Concrete mix proportions [53].

Component | Mixture I [kg/m${}^{3}$] | wt% | Mixture II [kg/m${}^{3}$] | wt% |
---|---|---|---|---|

Cement CEM I 42.5R | 905 | 34.2 | 905 | 33.2 |

Silica fume | 230 | 8.7 | 230 | 8.4 |

Quartz sand 0.063–0.4 mm OS 36 | 702 | 26.6 | 330 | 12.1 |

Quartz sand 0.04–0.125 mm OS 38 | 285 | 10.8 | 285 | 10.5 |

Quartz sand 0.2–0.8 mm OS 30 | - | - | 335 | 12.3 |

Water | 260 | 9.8 | 260 | 9.5 |

Superplasticizer Woerment FM 787 BASF | 29.6 | 1.1 | 29.6 | 1.1 |

Micro steel fibres DM 6/0.17 KrampeHarex | 233 | 8.8 | 233 | 8.6 |

Steel fibres DW 38/1.0 N KrampeHarex | - | - | 117 | 4.3 |

Density | 2645 | - | 2725 | - |

$\mathit{\beta}$ | m | ${\mathit{f}}_{\mathit{b}0}/{\mathit{f}}_{\mathit{c}0}$ | ${\mathit{K}}_{\mathit{c}}$ | $\mathit{\eta}$ |
---|---|---|---|---|

${36}^{\circ}$ | 0.1 | 1.16 | 0.667 | 0 |

**Table 3.**The values of the coefficients in Equation (1).

A | B | C | D | |
---|---|---|---|---|

Compression | 0.00002 | −0.01003 | 3.25030 | −59.29370 |

Tension | 0.00002 | 0.02236 | −651.59869 | 1885015.6 |

**Table 4.**Extremal macro-stresses and corresponding macro-strains in mixtures I and II (plastic range).

Mixture_BC | Compression | Tensile | Shear | |||
---|---|---|---|---|---|---|

$\overline{\mathit{\sigma}}$ [MPa] | $\overline{\mathit{\epsilon}}$ [-] | $\overline{\mathit{\sigma}}$ [MPa] | $\overline{\mathit{\epsilon}}$ [-] | $\overline{\mathit{\sigma}}$ [MPa] | $\overline{\mathit{\epsilon}}$ [-] | |

M1_PBC | 110.91 | $3.20\times {10}^{-3}$ | 13.50 | $4.22\times {10}^{-4}$ | 13.40 | $3.16\times {10}^{-3}$ |

M1_DBC | 107.18 | $3.00\times {10}^{-3}$ | 13.30 | $3.98\times {10}^{-4}$ | 14.9 | $3.29\times {10}^{-3}$ |

M1_TBC | 45.55 | $1.20\times {10}^{-3}$ | 10.40 | $3.21\times {10}^{-4}$ | 4.49 | $6.14\times {10}^{-4}$ |

M2_PBC | 137.77 | $3.40\times {10}^{-3}$ | 18.00 | $5.29\times {10}^{-4}$ | 12.90 | $2.57\times {10}^{-3}$ |

M2_DBC | 151.21 | $3.70\times {10}^{-3}$ | 19.50 | $5.52\times {10}^{-4}$ | 15.60 | $3.45\times {10}^{-3}$ |

M2_TBC | 96.61 | $2.20\times {10}^{-3}$ | 13.20 | $3.70\times {10}^{-4}$ | 8.43 | $1.42\times {10}^{-3}$ |

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**MDPI and ACS Style**

Denisiewicz, A.; Kuczma, M.; Kula, K.; Socha, T.
Influence of Boundary Conditions on Numerical Homogenization of High Performance Concrete. *Materials* **2021**, *14*, 1009.
https://doi.org/10.3390/ma14041009

**AMA Style**

Denisiewicz A, Kuczma M, Kula K, Socha T.
Influence of Boundary Conditions on Numerical Homogenization of High Performance Concrete. *Materials*. 2021; 14(4):1009.
https://doi.org/10.3390/ma14041009

**Chicago/Turabian Style**

Denisiewicz, Arkadiusz, Mieczysław Kuczma, Krzysztof Kula, and Tomasz Socha.
2021. "Influence of Boundary Conditions on Numerical Homogenization of High Performance Concrete" *Materials* 14, no. 4: 1009.
https://doi.org/10.3390/ma14041009