# Classification of Multiaxial Behaviour of Fine-Grained Concrete for the Calibration of a Microplane Plasticity Model

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## Abstract

**:**

## 1. Introduction

## 2. Microplane Drucker–Prager Cap Plasticity

#### 2.1. Formulation of the Material Model

#### 2.2. A Discussion of Material Parameter Fitting for the Numerical Model

## 3. Experimental Investigations

#### 3.1. Background

#### 3.2. Experimental Methodology

#### 3.3. Material

^{®}” (formerly TUDALIT) for use in the strengthening of ferro-concrete components with carbon concrete [30]. As defined by the standard [31], this product is a dry mortar with a maximum grain size of $1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$. As with other mortars, all the dry components are already packaged in bags, so that only water needs to be added on site. The concrete guarantees a compressive strength of at least $80\phantom{\rule{0.166667em}{0ex}}\mathrm{N}/{\mathrm{mm}}^{2}$ after 28 days and a modulus of elasticity above 25,000 $\mathrm{N}/{\mathrm{mm}}^{2}$. The actual concrete properties of the specimens tested are given in the following description of the test results.

#### 3.4. Test Setup

#### 3.5. Biaxial Compressive Strength

#### 3.6. Compressive Strength with Transversal Tensile Stress

## 4. Discussion

#### 4.1. Experimental Results

#### 4.2. Parameter Fitting

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the directions of plastic flow with the Drucker–Prager cap model as presented in [19] (

**left**) and with the new model (

**right**) plotted over the respective yield function.

**Figure 3.**Yield function set into relation with macroscopic yield stresses (

**left**) and exemplary stress–strain plot with these macroscopic yield stresses for uniaxial compression (blue), biaxial compression (orange), triaxial compression (grey) and uniaxial tension (green) (

**right**, not true to scale).

**Figure 5.**Test setup (

**left**), arrangement of measuring equipment (

**middle**) and specimen placed between the load brushes (

**right**).

**Figure 7.**Stress-strain curves of the cubic specimens (batch 2) compared with curves by Hampel [15].

**Figure 8.**Comparison of a typical fracture pattern with uniaxial (

**left**) and biaxial (

**right**) loading.

**Figure 10.**Comparison of a typical fracture pattern with different combined compressive and tensile load paths.

**Figure 11.**Stress-strain curves of the cubic specimens compared with curves from Hampel [15].

**Figure 13.**Comparison of the numerical results with the experimental results for biaxial compression tests of cube batch 2.

Biaxial Compressive Loads | Compressive-Tensile Loads | |||||
---|---|---|---|---|---|---|

−1/0 | −1/−0.5 | −1/−1 | −1/0.05 | −1/0.5 | 0/1 | |

Cubes $(100\times 100\times 100\phantom{\rule{0.166667em}{0ex}}\mathrm{mm})$ | ||||||

Batch 1 (tested after 28 days) | 4 | 4 | 4 | - | - | - |

Batch 2 (tested after 43 days) | 4 | 4 | 4 | - | - | - |

Discs $(200\times 200\times 40\phantom{\rule{0.166667em}{0ex}}\mathrm{mm})$ | ||||||

Batch 1 (tested after 56 days) | - | - | - | 2 | 2 | 2 |

Batch 2 (tested after 63 days) | - | - | - | 2 | 2 | 2 |

Compressive Strength [N/mm ^{2}] | Flexural Strength [N/mm ^{2}] | Young’s Modulus [N/mm ^{2}] | Poisson’s Ratio [-] | |
---|---|---|---|---|

Mean value prisms | 105.9 | 7.86 | 39,250 | - |

(acc. to EN 196-1 [34]) | ||||

Batch 1 | 102.9 | 6.9 | 38,750 | - |

Batch 2 | 108.9 | 8.9 | 39,750 | - |

Mean value cubes | 96.7 | - | 41,750 | 0.26 |

(uniaxial compressive test) | ||||

Batch 1 | 99.6 | - | 41,250 | 0.27 |

Batch 2 | 93.7 | - | 42,250 | 0.24 |

Compressive Strength [N/mm ^{2}] | Flexural Strength [N/mm ^{2}] | Tensile Strength [N/mm ^{2}] | Young’s Modulus [N/mm ^{2}] | Poisson’s Ratio [-] | |
---|---|---|---|---|---|

Mean value prisms | 100.7 | 7.76 | 4.06 | - | - |

(acc. to EN 196-1 [34]) | - | - | - | - | - |

Batch 1 | 101.5 | 6.84 | 3.79 | ||

Batch 2 | 99.8 | 8.68 | 4.33 | ||

Mean value discs | 87.6 | - | 2.79 | 35,000 | 0.26 |

(uniaxial tensile tests) | (calc. 0.87 ${f}_{c,pr}^{\prime}$) | (tensile) | (tensile) | ||

Batch 1 | 88.3 (calc.) | - | 2.37 | ||

Batch 2 | 86.8 (calc.) | - | 3.22 |

$\mathit{\alpha}$ [-] | ${\mathit{\sigma}}_{0}$ [N/mm ^{2}] | ${\mathit{\sigma}}_{\mathit{V}}^{\mathit{T}}$ [N/mm ^{2}] | ${\mathit{\sigma}}_{\mathit{V}}^{\mathit{C}}$ [N/mm ^{2}] | h [N/mm ^{2}] |
---|---|---|---|---|

0.221 | 55.716 | −25 | −55.5 | 5000 |

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**MDPI and ACS Style**

Betz, P.; Curosu, V.; Loehnert, S.; Marx, S.; Curbach, M.
Classification of Multiaxial Behaviour of Fine-Grained Concrete for the Calibration of a Microplane Plasticity Model. *Buildings* **2023**, *13*, 2704.
https://doi.org/10.3390/buildings13112704

**AMA Style**

Betz P, Curosu V, Loehnert S, Marx S, Curbach M.
Classification of Multiaxial Behaviour of Fine-Grained Concrete for the Calibration of a Microplane Plasticity Model. *Buildings*. 2023; 13(11):2704.
https://doi.org/10.3390/buildings13112704

**Chicago/Turabian Style**

Betz, Peter, Verena Curosu, Stefan Loehnert, Steffen Marx, and Manfred Curbach.
2023. "Classification of Multiaxial Behaviour of Fine-Grained Concrete for the Calibration of a Microplane Plasticity Model" *Buildings* 13, no. 11: 2704.
https://doi.org/10.3390/buildings13112704