# On the Calibration of a Numerical Model for Concrete-to-Concrete Interface

^{*}

## Abstract

**:**

## 1. Introduction

`USDFLD`user subroutine coded in FORTRAN, which facilitates the correct prediction of the interface strength subjected to compressive normal forces. The issue of fracture energy in two modes of fracture (tensile and shear) was discussed and implemented in the analyses. The results of experimental tests found in the literature (i.e., in references [21,25,26]), as well as the results obtained during own pilot studies, were used to validate the proposed strategy. The validation part was divided into four case studies (CS). The following types of concrete-to-concrete interface laboratory tests were examined: three-point bending of a beam with a notch, splitting of cubic specimens, and slant-shear tests on prisms.

## 2. Strength Test Used for Calibration

#### 2.1. Notched Beam by Chen et al. (2021)

#### 2.2. Own Splitting and Shear-Slant Tests

#### 2.3. Splitting and Shear Slant Tests by Santos and Julio (2009, 2011)

## 3. Numerical Modelling Strategy

**elastic regime**(before damage initiation), the traction–separation law neglecting normal-shear states coupling has the following form:

**damage initiation criterion**. Among the few available ones, the quadratic nominal stress criterion was selected. It can be represented as:

`USDFLD`to make the failure envelope pressure-dependent:

`USDFLD`subroutine allowed us to modify the parameters of models implemented in the Abaqus code by default at the beginning of each increment of the Newton scheme. Consequently, it is important to set a small value of increment load size when one uses such an approach. On the other hand, the problem is highly non-linear, so small increments are necessary for the convergence of the incremental-iterative procedure. The proposed algorithm reads the normal traction and calculates the admissible shear stress according to Carol’s formula. All the discussed criteria are shown in Figure 5.

**post-cracking behaviour**. After reaching the damage-initiation criterion, tractions are calculated taking into account the damage factor D:

**concrete**region was modelled with C3D8 continuum elements with selective integration [15]. The concrete damage plasticity (CDP) was chosen as a constitutive model for concrete. The version without the scalar damage parameter was assumed since only monotonic loads were analysed. The theoretical background of this model was presented in references [50,51]. The model has gained significant popularity and was discussed in detail in many references, e.g., [52,53,54,55], so its thorough description could be omitted in this study.

## 4. The Validation of the Proposed Strategy

#### 4.1. CS1—Tests with One Cohesive Element

- displacement-driven tension (pure mode I), i.e., ${\delta}_{n}>0$, ${\delta}_{s}={\delta}_{t}=0$,
- displacement-driven pure shear (pure mode II), i.e., ${\delta}_{n}={\delta}_{t}=0$, ${\delta}_{s}>0$,
- at the first step, the pressure that induces compressive normal traction; at second-step displacement—driven shear, i.e., ${\delta}_{n}={\delta}_{t}=0$, ${\delta}_{s}>0$.

#### 4.2. CS2—Simulation of Three-Point Bending of a Notched Beam

#### 4.3. CS3—Simulation of Own Tests

#### 4.4. CS4—Simulation of Santos and Julio Tests

## 5. Discussion

`USDFLD`user procedure, which enabled us to introduce the strength envelope, which is dependent on a normal traction value. The model took into account the different fracture energy values for modes of fractures I and II. The proposed approach was verified and validated with four case studies concerning: one element test, simulation of three-point bending of the bi-material notched beam [26], simulation of two series of splitting, and slant-shear tests—our own pilot research and the research made by Santos and Julio [21,25]. The results of the performed analyses are summarised in Table 10. The ratio of the ultimate force predicted by the FEM model and obtained as an outcome of a laboratory test was calculated for CS2-CS4. Its mean value was 1.01, and the CoV was 5%, which proves the the accuracy of the proposed modelling strategy.

`UMAT`or

`UEL`user procedure [57].

## 6. Conclusions

`USDFLD`user subroutine coded in FORTRAN, which enabled us to relate a shear strength with tractions normal to the interface. Different values of fracture energy in fracture modes I and II were also included. The results presented in the article can be used in the analyses of complex composite structures made of concrete layers cast at different times as well as in the assessment of various methods of repairing deteriorated structures. The presented modelling approach can also be useful in the simulation of masonry structures—the interface model can be easily adjusted to the behaviour of mortar [41].

- In the case of small specimens, the mesh density of 5 mm was sufficient to obtain satisfactory accuracy of the FEA results.
- The numerical model was able to cover the chipping of sharp edges for slant-shear specimens (see Figure 17).
- The value of fracture energy has a noticeable influence on the ultimate force predicted by the model (see Figure 18), so this issue should be experimentally studied in more depth due to the small amount of the experimental data.
- The traction stress distribution is not homogenous along the interface during the whole loading history in the case of slant-shear specimens made of concrete, which are of different classes (see Figure 22 and reference [21]). Consequently, the interface strength characteristics determined according to Equations (3) should be corrected in order to take into account stress concentrations present in the prism specimens.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples of concrete-to-concrete interfaces in structures: (

**a**) a composite floor on hybrid beams, (

**b**) a slab-column connection.

**Figure 3.**(

**a**) The test stand after a splitting test. (

**b**) Examples of surfaces after slant-shear tests. (

**c**) Examples of surfaces after splitting tests.

**Figure 4.**Cohesive element—local coordinate system and separations. The initial configuration is shown with black lines and the deformed one with grey lines.

**Figure 6.**The plot of implemented traction–separation law. $t=\sqrt{{\langle {t}_{n}\rangle}^{2}+{t}_{s}^{2}+{t}_{t}^{2}}$.

**Figure 7.**The relationship between the bond efficiency coefficient and a decrease in the fracture energy.

**Figure 8.**One element tests: (

**a**) the numerical model; (

**b**) the results for three different loading histories.

**Figure 10.**Meshes adopted in the analyses. Mean mesh size in the vicinity of the notch (

**left**to

**right**): 10 mm, 5 mm, and 2 mm.

**Figure 12.**(

**a**) Normal traction distribution along with the interface for different load levels. (

**b**) Deformed model.

**Figure 13.**(

**a**) Model of the splitting test. (

**b**) Meshes adopted in the analyses (from the top): 10 mm, 5 mm, and 2 mm.

**Figure 14.**(

**a**) Model of slant-shear test. (

**b**) Meshes adopted in the analyses (respectively): 10 mm, 5 mm, and 2 mm.

**Figure 15.**(

**a**) Results of the splitting test simulation. (

**b**) Results of the slant-shear test simulation.

**Figure 16.**(

**a**) The splitting test—deformed mesh. (

**b**) The splitting test—a map of the equivalent plastic strain in tension PEEQT.

**Figure 17.**(

**a**) The slant-shear test—deformed mesh. (

**b**) The slant-shear test—map of the equivalent plastic strain in compression PEEQ.

**Figure 19.**(

**a**) Model of splitting test. (

**b**) Mesh adopted in the analysis—element size in the vicinity of the interface—5 mm.

**Figure 20.**(

**a**) Results of the splitting test simulation. (

**b**) Results of the slant-shear test simulation.

**Figure 21.**Results of the splitting test simulation—evolution of the degradation parameter in the interface.

**Table 1.**The main laboratory tests outcomes from reference [26]. n/a—not applicable, n/t—not tested, n/g—not given.

Tested Item | ${f}_{cm,cube}$ | CoV | ${f}_{ctm,split}$ | CoV | ${f}_{ctm,flex}$ | CoV | ${G}_{f}$ | CoV |
---|---|---|---|---|---|---|---|---|

(MPa) | (%) | (MPa) | (%) | (MPa) | (%) | (N/m) | (%) | |

Low-strength concrete (LSC) | 68.3 | n/g | 3.7 | n/g | 3.99 | 6.3 | 225.8 | 11.3 |

High-strength concrete (HSC) | 34.4 | n/g | 2.7 | n/g | 6.40 | 6.2 | 271.2 | 5.4 |

LSC-to-HSC interface | n/a | n/a | n/t | n/t | 1.84 | 7.6 | 57.4 | 25.6 |

Mix Composition | Concrete A | Mix Composition | Concrete B |
---|---|---|---|

Compound | (kg/m${}^{3}$) | Compound | (kg/m${}^{3}$) |

Cement I 42.5 R | 350 | Cement I 42.5 R | 506 |

Vistula sand 0/2 mm | 687 | Fly ash | 83 |

Gravel 2/8 mm | 465 | Vistula sand | 620 |

Gravel 8/16 mm | 706 | Amphibole 2/8 mm | 889 |

Water | 163 | Amphibole 2/8 mm | 279 |

Superplasticizer | 3.25 | Water | 159 |

Superplasticizer | 9.3 |

Tested Concrete | ${f}_{cm,cube}$ | STD | CoV | ${f}_{ctm,split}$ | STD | CoV |
---|---|---|---|---|---|---|

(MPa) | (MPa) | (%) | (MPa) | (MPa) | (%) | |

Concrete A | 46.6 | 2.8 | 6.0 | 3.20 | 0.07 | 2.3 |

Concrete B | 79.4 | 1.6 | 2.0 | 8.10 | 0.12 | 1.5 |

Test | Results | |||||
---|---|---|---|---|---|---|

Test | ${\mathit{f}}_{\mathit{ctm},\mathit{split}}$(MPa) | STD (MPa) | CoV (%) | Failure Mode | ${\mathit{\alpha}}_{\mathit{int}}$ | |

Splitting | 2.8 | 0.3 | 10.0 | Mixed | 0.88 | |

${\mathit{F}}_{\mathit{ult}}$(kN) | STD (kN) | CoV (%) | ${\mathit{\sigma}}_{\mathit{ult}}$(MPa) | ${\mathit{\tau}}_{\mathit{ult}}$(MPa) | Failure Mode | |

Slant-shear | 193.3 | 33.1 | 17.1 | 4.83 | 8.37 | Adhesive |

**Table 5.**The main laboratory tests’ outcomes for the used concretes in reference [25].

Tested item | ${f}_{cm,cube}$ (MPa) | STD (MPa) | CoV (%) |
---|---|---|---|

Substrate | 79.3 | 4.8 | 6.0 |

Added | 66.4 | 2.1 | 3.2 |

**Table 6.**The main laboratory tests’ outcomes for the interface in reference [25]. n/a—not applicable, n/t—not tested.

Test | Results | |||||
---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{ctm},\mathit{split}}$(MPa) | STD (MPa) | CoV (%) | Failure Mode | ${\mathit{\alpha}}_{\mathit{int}}$ | ||

Splitting | 1.8 | 0.3 | 15.4 | Adhesive | 0.46 | |

${\mathit{F}}_{\mathit{ult}}$(kN) | STD (kN) | CoV (%) | ${\mathit{\sigma}}_{\mathit{ult}}$(MPa) | ${\mathit{\tau}}_{\mathit{ult}}$(MPa) | Failure Mode | |

Slant-shear | 621.5 | 97.7 | 15.7 | 6.9 | 12.0 | Adhesive |

**Table 7.**The calibration of a traction–separation model for a concrete-to-concrete interface—main assumptions.

Symbol | Parameter | Assumed Value/Formula | Reference |
---|---|---|---|

Elasticity | |||

${K}_{nn}$ | Stiffness in normal direction | ${E}_{weak}$ | [13] |

${K}_{ss}$, ${K}_{tt}$ | Stiffness in tangential directions | ${K}_{ss}={K}_{tt}={G}_{weak}$ | [13] |

t | Interface thickness | $t=0.005\phantom{\rule{4pt}{0ex}}a$ | [13] |

Damage Initiation Criterion | |||

${f}_{t}$ | Tensile strength (mode I) | ${f}_{t}\approx 0.9\phantom{\rule{4pt}{0ex}}{f}_{ctm,split}$ | [49] |

${f}_{sh}$ | Shear strength (mode II) | ${f}_{sh}\left({t}_{n}\right)$ acc. to Carol’s Formula (9), USDFLD | [42] |

Damage Evolution Rule and Viscous Regularisation | |||

Softening type | Exponential shape | ||

$\alpha $ | Parameter of exponential function | ≈7 | [44] |

${\delta}_{m}^{fail,I}$ | Failure separation in mode I | Sequentially acc. to: (16), (17), (15) | [36] |

${\delta}_{m}^{fail,II}$ | Failure separation in mode II | Sequentially acc. to: (16), (18), (15) | [46] |

$\mu $ | Viscosity parameter | 0.0001 |

**Table 8.**The calibration of the CDP model—main assumptions. (*)—strength substituted in MPa and dimensions in mm.

Symbol | Parameter | Assumed Value/Formula | Reference |
---|---|---|---|

Elasticity | |||

E | Young’s Modulus | ${E}_{cm}=22,000{\left({f}_{cm}\right)}^{0.3}$ (*) | EN 1992-1-1 [56] |

$\nu $ | Poisson’s ratio | 0.2 | EN 1992-1-1 [56] |

Ultimate surface | |||

${f}_{cm}$ | Uniaxial compressive strength | ${f}_{cm}\approx 0.8\phantom{\rule{4pt}{0ex}}{f}_{cm,cube}$ | EN 1992-1-1 [56] |

${f}_{ctm}$ | Uniaxial tensile strength | ${f}_{ctm}\approx 0.9\phantom{\rule{4pt}{0ex}}{f}_{ctm,split}$ | [49] |

$\frac{{f}_{b0}}{{f}_{c0}}$ | Ratio of biaxial to uniaxial compressive strength | 1.16 | [15] |

${K}_{c}$ | Rarameter controlling the shape of deviatoric section | 0.667 | [15] |

Hardening/Softening Rule | |||

${f}_{cy}\left({\u03f5}^{pl}\right)$ | Hardening rule in compression | EC2 parabola | EN 1992-1-1 [56] |

${G}_{f}$ | Fracture energy in tension | ${G}_{f}=10{\left({d}_{max}\right)}^{0.33}{\left({f}_{cm}\right)}^{0.33}$ (*) | JSCE [36] |

Plastic Potential and Viscoplastic Regularisation | |||

$\psi $ | Dilatancy angle | ${30}^{\circ}$ | [52] |

${\u03f5}_{0}$ | Eccentricity | 0.1 | [15] |

$\mu $ | Viscosity parameter | 0.0001 |

CS1 & CS4 | CS2 | CS3 | |
---|---|---|---|

CDP—weaker concrete | |||

E (GPa) | 36.3 | 29.8 | 32.6 |

$\nu $ (1) | 0.2 | 0.15 | 0.2 |

0 ${f}_{cm}$ (MPa) | 53.1 | 27.5 | 37.3 |

${f}_{ctm}$ (MPa) | 3.9 | 2.43 | 2.86 |

${G}_{f}$ (N/m) | 98 | 225 | 140 |

CDP—stronger concrete | |||

E (GPa) | 38.3 | 36.6 | 38.3 |

$\nu $ (1) | 0.2 | 0.15 | 0.2 |

${f}_{cm}$ (MPa) | 63.5 | 54.6 | 63.5 |

${f}_{ctm}$ (MPa) | 4.23 | 3.3 | 7.3 |

${G}_{f}$ (N/m) | 104 | 270 | 154 |

Interface | |||

${K}_{nn}$ (GPa) | 36.3 | 29.8 | 32.6 |

${K}_{tt}$ or ${K}_{ss}$ (GPa) | 15.1 | 13 | 13.6 |

t (m) | 0.001 | 0.001 | 0.001 |

${f}_{ctm}$ (MPa) | 1.6 | 0.95 | 2.55 |

c (MPa) | 4.5 | 1.9 | 6.6 |

$fi$ (deg) | 50 | 40 | 40 |

${G}_{f,I}$ (N/m) | 22.4 | 57 | 37 |

${\delta}_{m}^{fail,I}$ (m) | $9.86\xb7{10}^{-5}$ | 0.000423 | 0.000102 |

${G}_{f,II}$ (N/m) | 539 | 1425 | 925 |

${\delta}_{m}^{fail,II}$ (m) | 0.001035 | 0.006477 | 0.001283 |

Case Study No. | ${F}_{ult,exp}$ | ${F}_{ult,FEM}$ | $\frac{{F}_{ult,FEM}}{{F}_{ult,exp}}$ |
---|---|---|---|

CS2 | 4.19 | 4.418 | 1.05 |

CS3—splitting | 100 | 95.8 | 0.96 |

CS3—slant-shear | 193.3 | 208.1 | 1.08 |

CS4—splitting | 62.9 | 62.6 | 1.00 |

CS4—slant-shear | 621.5 | 613.9 | 0.99 |

mean | 1.01 | ||

STD | 0.05 | ||

CoV | 0.05 |

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**MDPI and ACS Style**

Dudziak, S.; Jackiewicz-Rek, W.; Kozyra, Z.
On the Calibration of a Numerical Model for Concrete-to-Concrete Interface. *Materials* **2021**, *14*, 7204.
https://doi.org/10.3390/ma14237204

**AMA Style**

Dudziak S, Jackiewicz-Rek W, Kozyra Z.
On the Calibration of a Numerical Model for Concrete-to-Concrete Interface. *Materials*. 2021; 14(23):7204.
https://doi.org/10.3390/ma14237204

**Chicago/Turabian Style**

Dudziak, Sławomir, Wioletta Jackiewicz-Rek, and Zofia Kozyra.
2021. "On the Calibration of a Numerical Model for Concrete-to-Concrete Interface" *Materials* 14, no. 23: 7204.
https://doi.org/10.3390/ma14237204