PSI - Issue 18

Roberta Massabò et al. / Procedia Structural Integrity 18 (2019) 484–489 Author name / Structural Integrity Procedia 00 (2019) 000–000

485

2

is nonlinear. A novel approach has been formulated in (Massabò, 2014; Massabò and Darban, 2019) which extends to delaminated beams the multiscale strategy in (Massabò and Campi, 2014, 2015; Pelassa and Massabò, 2015) for layered plates with imperfect continuous interfaces and the original zigzag theory in (Di Sciuva, 1986). Brittle delamination fracture under Mode II dominant conditions in unidirectional composite, laminated and layered beams and wide plates is studied. The model captures the unstable propagation of cracks, snap-back and snap-through instabilities, the effects of the interaction of multiple cracks on the macrostructural response and of the layered structure on the energy release rate. The layered structure and the delaminations are described by introducing local enrichments, in the form of zigzag functions and cohesive interfaces, to a classical first-order shear deformation plate theory. The model applies to layers with principal material directions parallel to the geometrical axes, depends on only three displacement variables and the solution of specific problems requires only in-plane discretization, for any numbers of layers and delaminations. Closed form solutions have been derived for the energy release rate in bi-material beams and applications have been presented to homogeneous, bi-material and layered, simply supported and cantilever, bend beams, with one and two delaminations in (Massabò and Darban, 2019). The model overcomes some inconsistencies which have been recently observed (Flores et al, 2018; Darban and Massabò, 2018) in modeling approaches based on the refined zigzag theory (Tessler et al, 2009) and the compliant layer concept, such as the model in (Eijo et al, 2013). The model in (Massabò and Darban, 2019) will be briefly recalled in Section 2 and applications to single and multiple delamination problems will be presented in Section 3. The comparison with accurate 2D solutions and experiments highlights the accuracy of the approach. Some limitations of the model, which are a consequence of the homogenization technique, are discussed. Applications to cross-ply laminated beams will be presented at the meeting. 2. Model The model refers to the schematic shown in Fig. 1 which describes a beam or wide-plate (deforming in cylindrical bending) with n layers and multiple delaminations subjected to transverse load. The delaminations are assumed to be under mode II dominant conditions and in constrained contact. The layers are joined by cohesive interfaces and the interfacial traction law, which relates the interfacial tractions to the relative sliding displacement of the layers at their interfaces, is shown in Fig. 1b; The law is piecewise linear in order to describe the intact portions of the plate, where 1/ 0 S K  , and the delaminated portions, where 0 S K  . The plate is studied using the multiscale strategy originally developed in (Massabò and Campi, 2014, 2015) which assumes a small-scale displacement field where the displacements of a classical first order shear deformation plate theory are enriched by local zig-zag functions, to account for the multilayered structures, and interfacial relative sliding displacements, to account for the presence of the delaminations. Homogenization and a variational approach are then used to derive the macro-scale displacement field, equilibrium equations and boundary conditions which describe the homogenized beam in Fig. 1a. The homogenization imposes continuity conditions on the transverse shear tractions at the layer interfaces and the equivalence of the transverse shear traction and the cohesive interfaces. The equilibrium equations are:

0

0 C w S

S

,

(  

) , 

,

0

A v

22 B C





22 02 22

2 22

0 222

(1)

0 B C v  ( S

1 D C C ( 2    S

2

1 C C w A w    2 0 222 ( ) , ( , S S

) ,

) , 

) 0

S

  

22

02 22

22

2 22

44 0 2 2

0 C v S

1 C C S

2

2 C w A w   0 2222 , ( , S

S

,

(  

) , 

, )    f 

0

02 222

2 222

44 0 22 2 2

3

and depend on three global displacement variables, which coincide with those of first order shear deformation theory, 02 2 ( ) v x , 2 2 ( ) x  , 0 2 ( ) w x , and define the in-plane displacement, bending rotation and transverse displacement of the reference surface, at 3 0 x  , when this falls into the lowest layer. The coefficients in Eq. (1) depend on the layup and the status of the interfaces and differ in the different regions of the homogenized beam; they can be calculated a priori and are given in (Massabo and Darban, 2019). The boundary conditions are defined in the same paper. The form of Eq. (1) is similar to that of first order shear deformation theory which is recovered when the coefficients which depend on the zigzag functions and the relative displacements at the interfaces are set to zero (coefficients with upper-script S). The formulation assumes the layers to be homogeneous and orthotropic, with principal material directions parallel to the geometrical axes, and neglects transverse compressibility and normal stresses.