# Thoughts on the Importance of Similitude and Multi-Axial Loads When Assessing the Durability and Damage Tolerance of Adhesively-Bonded Doublers and Repairs

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{p},

_{thr})/(1 − K

_{max}/A)

^{1/2}.

_{max}in Equation (2) is the maximum value of the stress intensity factor seen in a load cycle, ∆K = (K

_{max}− K

_{min}), K

_{min}is the minimum value of the stress intensity factor seen in the cycle, K

_{max}is the maximum value of stress intensity factor seen in the cycle and A is the cyclic fracture toughness. The term ∆K

_{thr}in Equation (2) represents the “fatigue threshold” below which a crack will not grow, i.e., the value of ΔK for which da/dN = 0.

^{−10}and p = 2, are taken from [27,28,29,36]. Furthermore, as per [27,28,29,36] when computing the durability of an AA7050-T7451 component the threshold term ΔK

_{thr}was taken to be 0.1 MPa√m. Examples of how this approach has been shown to be able to compute the durability of a range of other materials are given in [27,28,29,30,31,32,33,34,36,37].

_{thr})/(1 − √G

_{max}/√A′)

^{1/2}.

_{max}in Equation (3) is the maximum value of energy release rate seen in a fatigue (load) cycle and $\u2206\sqrt{G}$ is defined as:

_{min}is the minimum value of the energy release rate in the load cycle, and A’ is the cyclic fracture toughness. The term Δ√G

_{thr}in Equation (3) represents the “fatigue threshold” below which a crack will not grow, i.e., the value of Δ√G for which da/dN = 0.

_{max}, the term Δκ′ is a valid similitude parameter for representing/modelling the growth of cohesive cracking in adhesive joints.

## 3. The Importance of Fatigue Testing under the True Operational Multi-Axial Stress State

#### 3.1. Introduction

_{1}and σ

_{2}, see Figure 5. Let us further assume that the panel contains a small crack of length l emanating from one edge of the hole, see Figure 5. The stress state σ(r), at an arbitrary point along the line of the crack, in the direction perpendicular to the line of the crack in the uncracked panel is given in [44] by the expression:

_{1}(2 + a

^{2}/r

^{2}+ 3a

^{4}/r

^{4})/2 + σ

_{2}(a

^{2}/r

^{2}− 3a

^{4}/r

^{4})/2.

_{2}, and the stress state perpendicular to the crack, σ

_{1}.

#### 3.2. A Simple Worked Example

_{2}has on crack growth let us consider the case of a centrally located 6 mm diameter hole in an AA7050-T7451 aluminum-alloy panel subjected to maximum remote stresses of σ

_{1}= 150 MPa, σ

_{2}= −150 MPa and R = 0.1. The stress state of σ

_{2}= −σ

_{1}has been chosen so as to mimic that seen by cracks that arose from the fuel-drain hole in Mirage III aircraft in service with the RAAF [11]. For simplicity, the panel was assumed to have an initial crack length of l = 0.01 mm emanating from one side of the central hole, see Figure 5. This crack length was chosen as it is the length recommended in [45] for performing a durability analysis of AA7050-T7451 structural components.

^{−10}and p = 2, were used to study the effect of load biaxiality on the panel shown in Figure 5. (In this analysis the stress intensity factor associated with a given crack length was computed using the stress field given in Equation (6) and the weight function solution for this class of problems that is given in [46]). As in previous studies in [27,28,29,34] involving the growth of small cracks in metallic structures, the term ΔK

_{thr}in Equation (2) was set to a small value, i.e., ΔK

_{thr}= 0.1 MPa√m. Furthermore, as per [27], the cyclic fracture toughness term was set to be A = 39 MPa√m. The resultant computed crack growth histories for the case of biaxial loading (σ

_{1}= 150 MPa, σ

_{2}= −150 MPa and R = 0.1), and the case when only the uniaxial load (σ

_{1}= 150 MPa and R = 0.1) was considered and are shown in Figure 6.

_{2}(as was conducted in the coupon tests performed by Lockheed to assess the DADT of the Lockheed C-130J wing [20]) results in an erroneous estimate of the crack growth history. Figure 6 also suggests that to ensure that uniaxial coupon tests yield a crack growth history that is consistent with that seen under a multi-axial stress state representative of an operational aircraft, it would be necessary to adjust the applied loads, in the uniaxial test, so that at each crack length the similitude parameter, Δκ, corresponded to that under the multi-axial stress state. To achieve this objective, it requires a valid similitude parameter. As discussed in Section 2, Δκ is one parameter that could be used to achieve this goal.

#### 3.3. Adhesively-Bonded Joints

## 4. Cohesive Crack Growth in Adhesively-Bonded Double Lap Joints under Uniaxial Loads

^{®}finite element computer code, see Appendix A for more details. A comparison between the computed, by Zencack using Equation (5) using the values of D, p, A′ and $\u2206\sqrt{{G}_{thr}}$ given in Table 1, and measured crack growth histories is shown in Figure 8 where we see excellent agreement. The initial crack size in this analysis is approximately 0.2 mm.

## 5. Cohesive Crack Growth in Adhesively-Bonded Doublers under Uniaxial and Biaxial Loads

‘How does the growth of sub mm cracks in an adhesive bond subjected to uniaxial loads relate to growth under multi-axial loads?’

- (i)
- Uniaxial constant amplitude fatigue stresses with the peak stresses in the spectrum being σ
_{1}= 193 MPa and σ_{2}= 0. - (ii)
- Biaxial constant amplitude fatigue stresses with the peak stresses in the spectrum being σ
_{1}= 193 MPa and σ_{2}= σ_{1}/3. - (iii)
- Biaxial constant amplitude fatigue stresses with the peak stresses in the spectrum being σ
_{1}= 193 MPa and σ_{2}= −σ_{1}/3.

_{1}= 193 MPa was chosen to coincide with that in Section 3. As previously stated, the governing equation for the FM73 adhesive was taken to be as given by Equation (5), with the values of the constants as given in Table 2.

_{1}= 193 MPa and σ

_{2}= σ

_{1}/3 and less severe than the biaxial load case (iii), i.e., σ

_{1}= 193 MPa and σ

_{2}= −σ

_{1}/3.

## 6. Cohesive Crack Growth in Adhesively-Bonded Joints under a Combat Aircraft Flight Load Spectrum

- (i)
- The peak stresses in the spectrum are σ
_{1}= 193 MPa and σ_{2}= 0. - (ii)
- The peak stresses in the spectrum are σ
_{1}= 193 MPa and σ_{2}= σ_{1}/3. - (iii)
- The peak stresses in the spectrum are σ
_{1}= 193 MPa and σ2 = −σ_{1}/3.

- (i)
- The effect of biaxial loads on crack growth under operational flight loads can be significant.
- (ii)
- When comparing the results shown in Figure 12 and Figure 13, we see that the relative effect of the biaxial stresses on crack growth is similar regardless of whether the fatigue load spectra is a constant amplitude spectrum, as in Figure 12, or a FALSTAFF flight load spectrum, as in Figure 13. By this we mean that case (iii) had faster crack growth than the uniaxial load case, i.e., case (i), and that case (ii) had slower crack growth than the uniaxial load case. This observation reflects the way in which the various stress states interact with the disbond, i.e., with the crack in the adhesive.

## 7. Conclusions

- (i)
- Firstly, it should be noted that an essential requirement of any such laboratory test program that is performed so as to determine inspection intervals/durability of an adhesively bonded repair/doubler. Regardless of whether uniaxial or biaxial laboratory tests are performed, it is necessary to establish that at each stage in the testing regime the LEFM similitude parameter in the test specimen corresponds to that in the (operational) airframe.
- (ii)
- Secondly, the results of the present study have led to the hypothesis that: For uniaxial coupon tests to yield a crack growth history that is consistent with that seen under a multi-axial stress state representative of an operational aircraft, it may be necessary to adjust the magnitude of the applied loads as the crack grows such that at each crack length the similitude parameter, i.e., Δκ′, in the uniaxial test corresponds to that present under the true multi-axial stress state. The scientific community is challenged to evaluate the potential/validity of this hypothesis.
- (iii)
- Thirdly, a simpler, but possibly less desirable, approach, that is consistent with the building block approach to certification delineated in MIL-STD-1530D and JSSG-2006 and with the approaches outlined in [10,12,15], is to first establish that you can compute both the uniaxial and the multi-axial test crack histories in specimens, albeit with geometries and support conditions that may not be truly representative of the operational structure, using the same input parameters in both cases. (By this it is meant that the crack growth equation used in both studies would be identical, and that there would be no disposable parameters that could be “tweaked” to improve the fit to the experimental data.) Having thus validated the analysis methodology, the engineer would then run the analysis using a finite element model that has the actual geometry, boundary conditions, operational flight loads, etc., and compute the in service performance, i.e., the remaining life, inspection intervals, etc.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

^{®}[49,50,51,52,53,54,55] software interface version 9.3-1 to the ABAQUS

^{®}(version 9.3.1) finite element code. Zencrack is a fracture mechanics-based 3D crack propagation simulation software which is interfaced to a number of commercial finite-element analysis (FEA) codes. It allows calculation of fracture mechanics parameters such as energy release rate and stress intensity factors via automatic generation of focused cracked meshes from uncracked finite element models.

^{®}process is shown in the flowchart in Figure A1. This iterative process continues to advance the crack until certain criteria are satisfied, see [49,50,51] for details.

^{®}is able to perform this process with minimal requirements placed on the meshing within the uncracked mesh. Indeed, the initial crack definition is geometry based and, as such, is mesh independent. Updates to loads and boundary conditions in the re-meshed region are carried out as necessary.

**Figure A1.**Crack propagation fracture mechanics software interaction with the FEA software version 9,3-1.

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**Figure 5.**Schematic representation of a small, sub-mm, crack that emanates from one side of a small hole in a large plate subjected to arbitrary remote biaxial stresses.

**Figure 8.**The computed (from the present study) and measured [39] crack growth histories.

**Figure 9.**Plan view of the specimen which is a plate with adhesively bonded doublers showing the location of the edge cracks (disbonds) in the adhesive bonds. (The lower and upper doublers bonded to the “inner” AA2024-T3 plate both contain identical disbonds at each of their corners.).

**Figure 10.**Cross-section showing a quarter of the specimen, which is a plate with adhesively bonded doublers, see Figure 9.

**Table 1.**Values of the Hartman-Schijve constants used for the ‘FM73’ adhesive, from [39].

D (m/cycle) | p | A′ (J/m^{2}) | $\u2206\sqrt{{\mathit{G}}_{\mathit{t}\mathit{h}\mathit{r}}}$ (√(J/m^{2})) |
---|---|---|---|

1.9 × 10^{−10} | 2.7 | 2000 | 7.1 |

AA2024-T3 | FM73 | |
---|---|---|

Young’s modulus, E, in MPa | 72,000 | 2295 |

Poisson’s ratio, υ | 0.33 | 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Jones, R.; Chandwani, R.; Timbrell, C.; Kinloch, A.J.; Peng, D.
Thoughts on the Importance of Similitude and Multi-Axial Loads When Assessing the Durability and Damage Tolerance of Adhesively-Bonded Doublers and Repairs. *Aerospace* **2023**, *10*, 946.
https://doi.org/10.3390/aerospace10110946

**AMA Style**

Jones R, Chandwani R, Timbrell C, Kinloch AJ, Peng D.
Thoughts on the Importance of Similitude and Multi-Axial Loads When Assessing the Durability and Damage Tolerance of Adhesively-Bonded Doublers and Repairs. *Aerospace*. 2023; 10(11):946.
https://doi.org/10.3390/aerospace10110946

**Chicago/Turabian Style**

Jones, Rhys, Ramesh Chandwani, Chris Timbrell, Anthony J. Kinloch, and Daren Peng.
2023. "Thoughts on the Importance of Similitude and Multi-Axial Loads When Assessing the Durability and Damage Tolerance of Adhesively-Bonded Doublers and Repairs" *Aerospace* 10, no. 11: 946.
https://doi.org/10.3390/aerospace10110946