Issue 51

R. Massabò et alii, Frattura ed Integrità Strutturale, 51 (2020) 275-287; DOI: 10.3221/IGF-ESIS.51.22 276 cohesive zone approach to capture the displacement discontinuities and an in-plane discretization to follow the evolution of the cracks. The main drawback of these approaches is that the number of displacement variables is related to the number of mathematical layers and to the kinematics assumed to describe the layers. To model a n -layered beam with perfectly bonded layers and transverse inextensibility using Timoshenko beam theory in each layer leads to a total of ( n +2) displacement unknowns (namely (2 n +1) initial displacements minus ( n -1) displacement continuity conditions at the interfaces). Delamination damage evolution in the beam can be studied by releasing the displacement continuity at the layer interfaces and introducing cohesive interfacial tractions, normal and tangential, and cohesive traction laws which relate the tractions to the relative sliding and opening displacements [2–4]. This approach requires 3 n unknowns for each beam region (or (2 n +1) unknowns for mode II dominant problems, where crack opening is neglected and a single displacement variable describes the transverse displacements of the layers). An effective approach for modeling multilayered structures was proposed in the early works in [5,6] and is known in the community as “zigzag approach”. The basic idea of the global-local zigzag approach is to enrich the displacement field of an equivalent single layer theory (ELST) by local zigzag functions [5,6, and subsequent works] in order to account for the effects of the multilayer structure (zigzag). This adds a large number of local kinematic variables to the global variables, which are then removed from the problem by imposing continuities conditions on interfacial tractions at the layer interfaces [6]. Variational techniques are then used to derive the equilibrium equations, which depend on the global variables only. Other refined approaches have been proposed, e.g. [7,8], which introduce additional global variables to try and overcome some drawbacks of the original theories. The presence of thin interlayers and interfacial damage has been modelled following two different techniques. The first, known as “compliant layer concept”, is based on a description of thin interlayers or interfacial damaged regions as regular layers of the stack, with elastic constants apt to describe their response; the method then essentially follows the procedure described above for fully bonded structures [9,10]. The second technique, known as “spring-layer approach”, introduces zero thickness interfaces and interfacial traction laws which describe the response of the interlayer/damaged regions [11– 14]. In addition to the zigzag functions, the displacement field of the global equivalent single layer theory is then enriched by interfacial jumps. Considering the n -layered plate in Fig. 1a, the displacements ( ) k v  (  =1,2,3) in an arbitrary layer k , with upper and lower coordinates 1 3 k x  and 3 k x , are described by the following equations: 1 1 ( ) 1 1 2 3 01 1 2 3 1 1 2 1 1 2 3 3 1 1 2 1 1 ˆ ( , , , ) ( , , ) ( , , ) ( , )( ) ( , , ) k k k i i i i i v x x x t v x x t x x x t x x x x v x x t              1 1 ( ) 2 1 2 3 02 1 2 3 2 1 2 2 1 2 3 3 2 1 2 1 1 ˆ ( , , , ) ( , , ) ( , , ) ( , )( ) ( , , ) k k k i i i i i v x x x t v x x t x x x t x x x x v x x t              (1) 1 1 ( ) 3 1 2 3 0 1 2 3 3 1 2 3 1 2 3 3 3 1 2 1 1 ˆ ( , , , ) ( , , ) ( , , ) ( , )( ) ( , , ) k k k i i i i i v x x x t w x x t x x x t x x x x v x x t              where 01 02 0 1 2 3 , , , , , v v w    are the primary variables of the equivalent single layer model (here a first order shear and normal deformation plate theory), 1 2 3 3 ( , )( ) i i x x x x    for 1, 2, 3   and 1,..., 1 i n   are piecewise linear zigzag functions and ( 1) ( ) 1 2 3 3 3 3 ˆ ( , , ) ( ) ( ) i k k k k v x x t v x x v x x         for 1, 2, 3   and 1,..., 1 i n   are displacement jumps at the layer interfaces [14]. Higher order equivalent single layer models can be used which increase the number of the global variables and better describe the transverse stresses in fully bonded problems [1]. Zigzag functions and displacement jumps are then defined by imposing continuity of transverse shear and normal tractions at the layer interfaces and the interfacial constitutive laws to relate tractions and jumps. This yields the macro-scale displacement field and homogenized equilibrium equations which depend on the global variables only. Using this second approach care must be put on properly accounting for the strain energy absorbed during deformation within the interface [15] in the variational equations used for the derivation of the equations of motion. This term was neglected in the early models in [11–13] and corrected only later in [14]. Zigzag models have been formulated to study interfacial damage evolution using the compliant-layer concept and continuum damage approaches in [9,16,17]. Recently, the model in [14], which uses a cohesive zone approach and the zigzag kinematic approximation in Eqn. (1), has been applied to simulate single and multiple delamination growth under mode II dominant conditions in wide plates within the framework of fracture mechanics [18].