PSI - Issue 64

Lingzhen Li et al. / Procedia Structural Integrity 64 (2024) 1318–1325 Author name / Structural Integrity Procedia 00 (2019) 000–000

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length longer than an effective bond length can be considered sufficiently long. A second assumption pertains to that the stress-strain behavior of the adherent monotonically increases, which aligns with the behavior of the majority of engineering materials. This allows for a later representation of the supplementary strain energy. Fig. 1 (b) schematically illustrates an adherent infinitesimal element, whose equilibrium can be expressed using Equation (1). = (1) where and are the tensile and shear stresses, respectively, applied on the adherent infinitesimal element; and are the adherent strip thickness and coordinate in the longitudinal (loading) direction, respectively. The substrate deformation is ignored in the current model derivation, as the adherent strip deformation is significantly more pronounced. As a result, the engineering strain of the adherent strip can be computed via Equation (2). = (2) where refers to the relative displacement (slip) between the adherent strip and substrate; stands for the tensile strain of the adherent strip. The multiplication of Equations (1) and (2) leads to Equation (3). ⋅ = ⋅ (3) An integral of Equation (3) from the free end ( =0 ) to an arbitrary position ( = ) along the bond line in Fig. 1 (a) is expressed as Equation (4). � ⋅ 0 = � ⋅ 0 (4) Given the assumed sufficient bond length, the slip is zero at the free end, i.e., | =0 =0 . In addition, the tensile stress there should be zero, due to the free surface, i.e., | =0 =0 . Therefore, by replacing the integral target in Equation (4) from to on the left-hand side and from to on the right-hand side, one obtains Equation (5). � ⋅ 0 = � 0 (5) In the context of a monotonically increasing stress-strain curve, the left-hand side of Equation (5) corresponds to the supplementary strain energy, which is depicted in Fig. 2 (a). The right-hand side of Equation (5) can be further replaced by , / , where , is the partial fracture energy as expressed in Equation (6). , = � 0 (6) where , refers to the fracture energy ( ) when the slip ( ) exceeds the ultimate slip ( ) of the bond-slip behavior; , forms a part of the fracture energy when the slip ( ) is less than the ultimate slip ( ). When a random location in the bond line ( in Fig. 1 (a)) experiences a complete damage, the fracture energy at this point fully dissipated. As a result, , represents the (total) fracture energy, while the tensile stress held in the adherent strip ( ) corresponds to the bond capacity.