# Static and Fatigue Behaviour of Double-Lap Adhesive Joints and Notched Metal Samples Reinforced by Composite Overlays

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Samples

_{1}= 4 mm. The chemical composition and mechanical properties of this steel are given in Table 1 and Table 2, respectively. Different kinds of materials were applied for reinforcing overlays of the investigated samples. These were made from the following materials:

- steel S355J2+N (U.S. Steel Košice, Košice, Slovak Republic),
- R-glass/epoxy composite HEXCEL TVR 380 with stacking sequence [+45°/−45°]
_{4}(HEXCEL Corporation, Stamford, CT, USA), - E-glass woven roving/Epidian 601 (R&G Faserverbundwerkstoffe GmbH, Waldenbuch, Germany),
- carbon S&P C-Laminate 150/2000 (S&P Clever Reinforcement Company AG, Seewen, Switzerland).

_{0}= 1 mm.

_{2}= 4 mm (samples 1–10—i.e., Figure 3a), and S&P C-Laminate 150/2000 with thickness g

_{2}= 1.4 mm (samples 11–15—i.e., Figure 3b). The detailed list of the samples and applied loading conditions are provided in Table 3, and photographs of samples with double-lap joints are presented in Figure 3.

_{4}(Figure 4b), E-glass woven roving (0°/90°) (Figure 4c), and S&P C-Laminate 150/2000 (0°) (Figure 4d). The second overlay (sample no. 22) had a rectangular shape with 15 × 180 mm size and was bonded on both sides of the notch (Figure 2c). In this case, overlays were made of S&P C-Laminate 150/2000 (0°) (Figure 4e). In all cases, overlays were bonded on both sides of the steel surfaces.

#### 2.2. Theoretical Solution of Double-Lap Joint

- Shear stresses are constant through the thickness of the adhesive joint;
- Linear and elastic material model;
- Deformation of adhesive is caused only by shear stress;
- Deformation of adherends is caused only by tension;
- Bending moments and peel stresses are neglected.

_{i}, Young moduli E

_{i}and Poisson’s ratios ν

_{i}, where i = 1 is valid for inner adherend and i = 2 refers to outer adherend (overlays). The adhesive material is defined by its thickness g

_{0}and shear modulus G

_{0}. The length of one side of the joint is equal to L, and the space between the inner part is equal to L

_{sp}. The width of both adherends and adhesive joint is equal to b, which is also the width of the tested samples.

_{0}(2), and by taking into account stiffness and forces in both adherends (3):

_{1}, N

_{2}, and F, and after mathematical transformations, the following equation is obtained

_{0}is equal to

_{1}and A

_{2}can be determined from the boundary conditions:

- For x = 0: N
_{2}= 0; - For x = L: N
_{2}= F/2;

#### 2.3. FEM Models

## 3. Results

#### 3.1. Double-Lap Joints

#### 3.1.1. Static Tensile Test of Double-Lap Joint

_{x}and major strain ε

_{x}on parts of the adherends surfaces and overlay determined by DIC are presented in Figure 11a,b, respectively. Obviously, the highest strain is observed in the adhesive layer.

_{x}/ε

_{x}

_{,over,cent}in the middle inspection section (parallel to the x-axis—see Figure 11b) determined by DIC is compared with the FEM solution in Figure 12. Here, ε

_{x}

_{,over,cent}is the strain in the middle cross-section of the overlap (perpendicular to x-axis), for x = L + 0.5 L

_{sp}= 15 mm—more details are given in Figure 6 and Figure 11. The most interesting part is for x in the range from 0 to 30, which presents strain on the overlay surface. The ranges x > 30 and x < 0 show distributions of strain in upper and bottom steel adherends, respectively. Similar to Figure 11 high local peaks appear at the adhesive at the end of the overlay. The shown overlay has been torn off from the upper adherend (x = 30 mm) when the highest strain occurred.

_{x}displacement in the middle vertical section of the double-lap adhesive joint is presented in the same way as in Figure 12. The rapid changes in displacements are observed at both ends of the overlay (for x = 0 and x = 30). It is caused by shear deformation γ of the adhesive and leads to the formation of the shear stress in the adhesive (see Equations (2) and (3)).

#### 3.1.2. FEM and Analytical Calculations of Double-Lap Joint

#### 3.1.3. Fatigue Tensile Tests of Double-Lap Joint

^{5}–10

^{6}cycles.

^{5}cycles) was achieved for fatigue stresses below 12 MPa. The fatigue life of the specimen subjected to the maximal fatigue stress 11.96 MPa (corresponding to 48.3% of static tensile strength) was 489,873 cycles. The sample subjected to the maximal fatigue stress of 10.4 MPa (42% of static tensile strength) did not fail at 1.1 × 10

^{6}cycles.

^{5}—10

^{6}cycles. The samples subjected to the maximal fatigue stress 7.51 MPa and 6.01 MPa (30% and 24% of static tensile strength, respectively) did not fail at 1.0 million cycles. For higher fatigue loads (maximal fatigue stress 10.36 MPa and 11.03 MPa) the fatigue life was reduced to 3.83 × 10

^{5}cycles and 3.71 × 10

^{5}cycles, respectively. Such loadings correspond to 42% and 45% of static tensile strength of samples with composite S&P C-Laminate 150/2000 overlays.

^{5}cycles) was that the approximate 55%–60% reduction of the fatigue strength with respect to the tensile strength was obtained. In the presented study such results were used for the determination and verification of the size of the overlays for notched metal samples discussed in Section 3.2.

#### 3.2. Notched Steel Samples Reinforced by Composite Overlays

#### 3.2.1. Static Tensile Test

_{TOT}= 0.48% (see Figure 19). It can be seen that on the composite overlay the zone with high major strain has been increased (Figure 21b) in comparison with the unreinforced sample (Figure 21a). This confirms the possibility of increasing the load-carrying capacity on a notched object by reinforcing overlays. The highest strains are observed in the adhesive in the corners of the overlays (Figure 21b). Such stress concentrations caused by shear stress and peel stress in adhesive finally lead to steel/adhesive interfacial failure. A more detailed study of this problem can be found in [9].

#### 3.2.2. Fatigue Tensile Tests of Notched Samples Reinforced by Composite Overlays

_{max}= 44.1 kN were applied during the fatigue tests. The experimental tests were performed for non-reinforced notched sample no. 17 (geometry in Figure 2a) and notched samples no. 19–22 with composite overlays (geometries in Figure 2b,c). The main aim of this experimental study was a preliminary assessment of the possibilities of increasing the fatigue life by the application of composite overlays. The fatigue life was determined as the rupture of the adherend (core) part of the sample. The initial size of the overlays (Figure 2b) was assumed on the basis of the static tensile test (Figure 19) and fatigue tests carried out for samples with double-lap joints (Section 3.1.3).

_{t}= 2.5 [9,15], it can be concluded that the maximal applied stresses at the notches were much higher than the yield limit of the steel core. The existence of such high stresses (theoretical stress at the notch is 919 MPa, while the minimum guaranteed yield limit is 355 MPa; the determined yield limit from the tensile test was 372 MPa [9]) resulted in the short fatigue life of non-reinforced sample no. 17, namely N

_{f}= 34,303 cycles to rupture (see Figure 22 and Table 4). Failure forms of the investigated samples are presented in Figure 23.

_{1}= 85–115 mm on the outer surface of the overlay—Figure 25, which indicates the proper load transfer from the core). During the fatigue test, the first damage in the adhesive layer was caused by peel stresses at both ends of the overlays (see Figure 24b). Further degradation results in steel/adhesive interfacial failure and further cracks forming and propagating in adherend.

## 4. Discussion

_{t}—see Table 5).

_{t}= 2.183—Table 5), the maximal adhesive shear stress may achieve 14 MPa (for the maximal load of 54.48 kN) which corresponds to the low fatigue regime. The experimental fatigue tests were carried out for lower loadings; however, the concentrations of the shear stresses in the adhesive were still at a high level.

_{t}) is observed after the application of two long reinforcing paths (size 180 × 15 mm) made from composite material.

## 5. Conclusions

- −
- The static tensile strengths of double-lap joints with the same adhesive area were similar for samples with overlays made of different materials and thicknesses; however, the fatigue strength of adhesive in double-lap joint strongly depends on the stiffness of the adherend–overlays arrangement,
- −
- The analytical formulation used for the calculations of the shear stresses in double-lap joints shows good agreement with the numerical solution,
- −
- The application of the DIC system reveals good convergence with the FEM solution and enables the determination of the strain concentrations at the notches,
- −
- The application of the overlays increases load-carrying capacity under static tensile loading conditions,
- −
- The fatigue strength of the notched samples can be significantly increased by the application of the overlays. However, the weakest point of such a joint is the steel/adhesive connection,
- −
- The fatigue strength of the adhesive joint can be increased by increasing the adhesively bonded area. However, additional technological treatments (i.e., chamfering of the overlays) are necessary to reduce peel stresses at the ends of the joint.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

CFRP | carbon-fibre-reinforced plastics, |

DIC | digital image correlation, |

DLJ | double-lap joint, |

FEM | finite element method, |

GFRP | glass-fibre-reinforced plastics, |

Variables | |

A_{0}, A_{1}, A_{2} | constants of theoretical solution of DLJ, |

b | width of sample, |

C_{1}, C_{2} | stiffness of adherend and overlap, respectively, |

E | Young stiffness modulus, |

F | applied tensile force, |

F_{max} | maximal applied value of tensile force during fatigue test, |

f_{u} | ultimate tensile strength, |

v | tensile speed, |

G | shear modulus, |

g_{0} | adhesive thickness, |

g_{1} | adherend thickness, |

g_{2} | overlap thickness, |

i | number of samples |

K_{t} | stress concentration factor, |

L | length of one side of DLJ joint, |

L_{sp} | space between inner parts of DLJ, |

N_{1}, N_{2} | forces in adherend and overlaps in particular cross-section, respectively, |

N_{f} | number of cycles to failure, |

R | stress ratio, |

u_{x} | elongation of sample in tension direction, |

Y_{eH} | Yield limit, |

ε_{TOT} | total strain of sample in tension direction, |

ε_{x} | major strain in tension direction, |

ε_{x}_{,over,cent} | strain in middle cross-section of overlap, |

ν | Poisson’s ratio, |

δ_{inc} | ratio of increase of fatigue life of reinforced sample in relation to not reinforced one, |

θ | fibre angle orientation in layers with respect to tension direction, |

σ | peel stress in adhesive, |

τ | shear stress in adhesive, |

τ_{adh} | adhesive shear strength, |

τ_{avg} | average shear stress in adhesive, |

τ_{avg,FAT} | maximal applied value of average adhesive shear stress during fatigue test, |

γ | shear deformation. |

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**Figure 2.**Geometry of tested samples with square notches with rounded corners: (

**a**) without composite overlays—samples 16 and 17; (

**b**) with two 45 × 45 mm composite overlays with hole at centre on both sides—samples 18–21; (

**c**) with four rectangular 15 × 180 composite overlays on both sides—sample no. 22.

**Figure 3.**Samples with double-lap joint: (

**a**) with metal overlay; (

**b**) with composite S&P C-Laminate 150/2000 overlays; (

**c**) magnification of speckle pattern on sample with composite overlay.

**Figure 4.**Notched samples with composite overlays: (

**a**) sample no. 16 without overlays; (

**b**) sample no. 19—Hexcel TVR 380 [+45°/−45°]

_{4}; (

**c**) sample no. 20—E-glass woven roving; (

**d**) sample no. 21—S&P C-Laminate 150/2000—45 × 45 mm; (

**e**) sample no. 22—S&P C-Laminate 150/2000—15 × 180 mm.

**Figure 6.**Double-lap joint: (

**a**) geometry, dimensions and material parameters; (

**b**) internal forces in investigated joint and direction of x-axis.

**Figure 7.**FEM model: (

**a**) boundary conditions and part of FEM model with visible half-part of bonded joint and marked materials: light blue—steel adherend, red—adhesive, purple—overlays; (

**b**) part of FEM model with finite element discretization.

**Figure 9.**Failure forms of adhesive joints: (

**a**) adhesive; (

**b**) adhesive caused by peel stress; (

**c**) cohesive; (

**d**) cohesive caused by peel stress; (

**e**) mixed mode.

**Figure 10.**Failure forms of adhesive joints: (

**a**) sample no. 1—steel overlays; (

**b**) sample no. 11—S&P C-Laminate 150/2000 overlays.

**Figure 11.**Results from DIC analyses for double-lap joint with S&P C-Laminate 150/2000 overlays: (

**a**) vertical displacement u

_{x}; (

**b**) major strain ε

_{x}.

**Figure 12.**Distribution of normalized surface major strain in middle section of double-lap adhesively joint with S&P C-Laminate 150/2000 overlays—DIC and FEM results.

**Figure 13.**Distribution of surface vertical displacements u

_{x}(DIC) in middle section of double-lap adhesively joint with S&P C-Laminate 150/2000 overlays.

**Figure 14.**Distribution of normalized adhesive shear stresses in adhesive in double-lap joints calculated by analytical and numerical approaches: (

**a**) sample no. 1 with steel overlays; (

**b**) sample no. 11 with composite overlays.

**Figure 17.**Failure modes of samples with steel overlays with visible steel/adhesive interfacial failure: (

**a**) τ

_{avg,FAT}= 18.35 MPa; (

**b**) τ

_{avg,FAT}= 13.48 MPa; (

**c**) τ

_{avg,FAT}= 11.96 MPa.

**Figure 18.**Failure modes of samples with composite overlays with visible steel/adhesive interfacial failure: (

**a**) τ

_{avg,FAT}= 11.03 MPa; (

**b**) τ

_{avg,FAT}= 10.36 MPa.

**Figure 19.**Comparison of tensile curves for samples without (no. 16) and with composite overlays (no. 18).

**Figure 20.**Failure form of notched samples: (

**a**) no. 16 without overlays; (

**b**) no. 18 with composite Hexcel TVR 380 [+45°/−45°]

_{4}overlays. Samples were covered by speckle patterns for DIC analyses.

**Figure 21.**Surface strains determined from DIC analyses (total strain ε

_{TOT}= 0.48%): (

**a**) sample no. 16 without overlays; (

**b**) sample no. 18 with composite Hexcel TVR 380 [+45°/−45°]

_{4}overlays.

**Figure 23.**Failure modes of samples with composite overlays: (

**a**) no. 19—Hexcel TVR 380 [+45°/−45°]

_{4}; (

**b**) no. 20—E-glass woven roving; (

**c**) no. 21—S&P C-Laminate 150/2000—45 × 45 mm; (

**d**) no. 22—S&P C-Laminate 150/2000—15 × 180 mm.

**Figure 24.**Distribution of surface strain (DIC) for sample no. 22 for maximal load 44.1 kN at 0.25 cycle: (

**a**) around notch; (

**b**) strain concentrations in adhesive at ends of overlays.

Chemical Components of S355J2+N Steel (in Weight %) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Material | C | Si | Mn | P | S | Cu | Al | Cr | Fe |

S355J2+N (tested material) [9] | 0.19 | 0.20 | 0.99 | 0.012 | 0.01 | 0.03 | 0.04 | 0.02 | res. |

S355J2, Standards [49] | 0.20–0.22 | 0.55 | 1.60 | 0.025 | 0.025 | 0.55 | - | - | res. |

Steel | |||||||
---|---|---|---|---|---|---|---|

Material | E [GPa] | ν | Y_{eH}[MPa] | f_{u}[MPa] | |||

S355J2+N | 210 | 0.3 | Min 355 | 470–630 | |||

Composites | |||||||

Material | E_{1}[GPa] | E_{2}[GPa] | G_{12}[GPa] | G_{23}[GPa] | ν_{1} | ν_{2} | f_{u}[MPa] |

HEXCEL TVR 380 M12/26%/R-glass/epoxy [+45°/−45°] _{4} [9,12] | 46.43 | 14.92 | 5.23 | 9.15 | 0.269 | 0.3 | 141.8 |

E-glass woven roving/Epidian 601 [50] | 16.8 | 16.8 | 3.4 | 3.4 | 0.14 | 0.14 | 220 |

S&P C-Laminate 150/2000 [51] | 165 | 10 | 5 | 5 | 0.3 | 0.3 | 2800 |

Adhesive | |||||||

Material | E[GPa] | ν | τ_{adh}[MPa] | Adhesion steel to steel(tensile strength) [MPa] | |||

S&P Resin 220 Epoxy Adhesive [52] | 7.1 | 0.35 | 26 | 14 |

Sample Number i | Geometry | Type of Tension Load | Overlay Material | Overlay Thickness g _{2} in mm | Loading Conditions ^{1} |
---|---|---|---|---|---|

First series of experimental tests—double lap-joint (DLJ) samples | |||||

1 | Figure 1 | Static | S355J2+N | 4 | v = 0.5 mm/min |

2–10 | Figure 1 | Fatigue | S355J2+N | 4 | R = 0.1, 9 samples tested, τ _{avg,FAT =} 18.4, 14.9, 13.6, 13.6, 13.5, 13.4, 12.9, 12.0, 10.4 MPa |

11 | Figure 1 | Static | S&P C-Laminate 150/2000 | 1.4 | v = 0.5 mm/min |

12–15 | Figure 1 | Fatigue | S&P C-Laminate 150/2000 | 1.4 | R = 0.1, 4 samples tested, τ _{avg,FAT =} 11.0, 10.4, 7.5, 6.0 MPa |

Second series of experimental tests—notched steel samples reinforced by composite overlays | |||||

16 | Figure 2a | Static | without | - | v = 0.5 mm/min |

17 | Figure 2a | Fatigue | without | - | F_{max} = 44.1 kN, R = 0.1 |

18 | Figure 2b | Static | HEXCEL TVR 380 [+45°/−45°]_{4} | 2.1 | v = 0.5 mm/min |

19 | Figure 2b | Fatigue | HEXCEL TVR 380 [+45°/−45°]_{4} | 2.1 | F_{max} = 44.1 kN, R = 0.1 |

20 | Figure 2b | Fatigue | E-glass woven roving | 2.1 | F_{max} = 44.1 kN, R = 0.1 |

21 | Figure 2b | Fatigue | S&P C-Laminate 150/2000 | 1.4 | F_{max} = 44.1 kN, R = 0.1 |

22 | Figure 2c | Fatigue | S&P C-Laminate 150/2000 | 1.4 | F_{max} = 44.1 kN, R = 0.1 |

^{1}v—tensile speed in mm/min, τ

_{avg,FAT}—maximal applied value of average adhesive shear stress during fatigue test in MPa, F

_{max}—maximal value of the applied fatigue force in kN.

**Table 4.**Results of experimental fatigue tests of notched metal samples; maximal tensile force F

_{max}= 44.1 kN, stress ratio R = 0.1.

Sample Number i | Overlay Material | Fatigue Life N _{f} (in Cycles) | ${\mathit{\delta}}_{\mathit{i}\mathit{n}\mathit{c}}^{\mathit{i}}=\raisebox{1ex}{${\mathit{N}}_{\mathit{f}}^{\mathit{i}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{N}}_{\mathit{f}}^{17}$}\right.$ | Failure Form |
---|---|---|---|---|

17 | without | 34,303 | 1 | - |

19 | HEXCEL TVR 380 [+45°/−45°]_{4} | 95,377 | 2.7 | Slight fibre/matrix debonding around notches and adhesive failure of bonded joint—Figure 23a |

20 | E-glass woven roving | 61,910 | 1.8 | Adhesive failure of bonded joint—Figure 23b |

21 | S&P C-Laminate 150/2000 | 66,250 | 1.9 | Overlay failure and adhesive failure of bonded joint—Figure 23c |

22 | S&P C-Laminate 150/2000 | 242,500 | 7.1 | Adhesive failure of bonded joint—Figure 23d |

**Table 5.**Stress concentration factor at notch in adherend, θ—fibre angle orientation in layers with respect to tension direction.

Overlay Dimensions in (mm) | K_{t} (−) | θ | Corresponding Sample Number i |
---|---|---|---|

Without Overlay | |||

- | 2.508 | - | 17 |

Rectangular Patch | |||

Size (45 × 45) | 2.183 | [+45°/−45°]_{4} | 19 |

Size (45 × 45) | 2.014 | [0°]_{8} | 21 |

Size (180 × 15) | 1.366 | [0°]_{8} | 22 |

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**MDPI and ACS Style**

Romanowicz, P.J.; Szybiński, B.; Wygoda, M.
Static and Fatigue Behaviour of Double-Lap Adhesive Joints and Notched Metal Samples Reinforced by Composite Overlays. *Materials* **2022**, *15*, 3233.
https://doi.org/10.3390/ma15093233

**AMA Style**

Romanowicz PJ, Szybiński B, Wygoda M.
Static and Fatigue Behaviour of Double-Lap Adhesive Joints and Notched Metal Samples Reinforced by Composite Overlays. *Materials*. 2022; 15(9):3233.
https://doi.org/10.3390/ma15093233

**Chicago/Turabian Style**

Romanowicz, Paweł J., Bogdan Szybiński, and Mateusz Wygoda.
2022. "Static and Fatigue Behaviour of Double-Lap Adhesive Joints and Notched Metal Samples Reinforced by Composite Overlays" *Materials* 15, no. 9: 3233.
https://doi.org/10.3390/ma15093233