PSI - Issue 68

Lukas Dominik Geisel et al. / Procedia Structural Integrity 68 (2025) 1273–1279 L. D. Geisel and S. Marzi / Structural Integrity Procedia 00 (2024) 000–000

1275

3

Using Euler-Bernoulli beam theory (without any consideration of the e ff ects of the adhesive or the shear forces that act to the left and the right of the seesaw), the overall beam deflections within the relevant interval x ∈ [ a , l + l s ] (i.e., behind the crack tip but still between the two load introduction points, where the bending moment is constant) of both the short and the long beam is governed by ∂ 2 ∂ x 2 w ( F , x ) = − M ( F ) 2 EI = − 2 H ˜ ε ( F ) . (3) However, this simplified depiction is lacking, when it comes to the calculation of cohesive laws. The adhesive does not exist in this simplified description of the specimen, since the crack is merely thought of as a jump within in the beamsti ff ness. This jump is a consequence of the fact, that the adhesively bonded section consists of two adherends instead of one. Within the fracture process zone, these two adherends do not follow the same deflections, since the adhesive cannot couple both beams rigidly. Therefore, both adherends need to be considered on their own within this process zone by ∂ 2 ∂ x 2 w long ( F , x ) = − 2 H ˜ ε long ( F , x ) and (4) ∂ 2 ∂ x 2 w short ( F , x ) = − 2 H ˜ ε short ( F , x ) . (5) These relationships allow the calculation of the individual beam deflections from backface strain measurements by twofold numerical integration. Furthermore, the crack opening displacement (COD) of the 4-OSLB specimen can now be computed along the fracture process zone by calculating the di ff erence between the deflections of the short and long beam, i.e. w COD ( F , x ) = w short ( F , x ) − w long ( F , x ) . (6) Substituting the angles in Eq. (2) with their respective algebraic expressions (which can be obtained via beam theory), J ext can be expressed in terms of the curvature and thus in terms of the backface strains by

w ( F , x )

EI 4 h −

2

F 2 (2 L − l 64 EIh

∂ 2 ∂ x 2

s )

J OFS ( F , x ) =

(7)

.

=

Eq. (7) assignes each point located within x ∈ [ a , l + l s ] an ERR value. However, this quantity in itself does not characterize the ERR within the adhesive layer since it only takes the overall beam curvature into account and not the curvature of the individual adherends. The deformation of the adhesive layer however, is not caused by the curvature of the overall beam (i.e. − d 2 w / d x 2 ), but by the di ff erence between the curvatures of two adherends. The correct equation of the ERR thus takes on the form

EI 4 h

w COD ( F , x ) 2 .

∂ 2 ∂ x 2

J OFS ( F , x ) =

(8)

Since both Eqs. (6) and (8) can be expressed in terms of the curvature, which is in turn obtainable using backface strain measurements, traction-separation laws can be gathered using only these backface strain measurements.

3. Experimental setup

All experiments were executed in a displacement-controlled manner, using the servo-hydraulic testing machine MTS Landmark ® 370.02 (MTS Systems Corporation, Eden Prairie, USA) which has a load capacity of 15 kN. The im plemented setup is depicted in Fig. 2. Fiber optic strain gauges were applied to the bottom surface of the specimens for the measurements of the backface strains. To measure these strains with glass fibers, the optical distributed sensor interrogator ODiSI 6101 (LUNA Inc., Blacksburg, USA) was used. This device records the strains along the optical fiber with a spatial resolution of 0.65 mm whilst achieving a measurement accuracy of ± 25 µε . The measurement frequency was set to 1 Hz.