PSI - Issue 28

M.Z. Sadeghi et al. / Procedia Structural Integrity 28 (2020) 1601–1620 M.Z. Sadeghi et al./ Structural Integrity Procedia 00 (2019) 000–000

1605

� � 12β � � 3α � �β√3α 36β � � 3α L

(4)

in which L is the half span length, α and β are correction parameters and defined as followed: α � 1 � m m β � �� � � ��� �� � � ���2��� a 0 is the initial crack length, h is the thickness of one adherent and half of the adhesive layer, E and G are the elastic and shear modulus respectively and m is the Mixed Mode Ratio (MMR). χ is the correcting factor for the crack length and can be calculated as follows (Reeder, 2003): � � � 11 E G �3 � 2 � 1 � Γ Γ � � � Γ � 1�1� G E m � � �� � � �� �� . For measuring the fracture energy of the adhesive under different mixed modes based on the J-integral approach, one needs to measure angular changes of the load introduction points. For example, in MMB tests since the load introduction points are four points, the angular rotation of the load introduction points was measured with Inclinometers. 2.1. Determination of fracture energy In this work, two methods were applied for the determination of fracture energy to ensure consistency between the obtained fracture toughness G c . The first method is J-integral which is based on the angular changes of the load introduction points. Another method is the equivalent crack length method in which fracture energy is calculated based on an equivalent modulus and an equivalent crack length during the crack propagation. In the following, both methods are discussed for DCB and MMB tests. J-integral Rice (Rice, 1968) proposed a parameter for fracture characterising of materials with non-linear behaviour on the crack tip. According to Rice, the energy release rate is equivalent to the following integration around any arbitrary path: � � �� ��� � � � ∂ ∂ u x ��� � (5) where, w is the strain energy density, T is the traction vector perpendicular to the integration path Γ and u is the displacement vector. Paris and Paris (Paris A. and Paris P., 1988) later by considering the traction forces on the load introduction points offered a simple closed-form relation for J-integral in a double cantilever beam (DCB): � � 2P b θ (6) 2.1.1