Issue 29

R Massabò, Frattura ed Integrità Strutturale, 29(2014) 230-240; DOI: 10.3221/IGF-ESIS.29.20 232 assumptions above, the displacement components then simplify in 1 0  v , 2 2 2 3  ( , ) v v x x and 3 2  ( ) v w x . Following the classical assumption of lower-order plate theories, in the constitutive relationships for the generic layer k the normal stresses, 33  ( ) k for k =1..n-1, are assumed to be negligibly small compared to the other components. This yields: 22 22 22    ( ) ( ) ( ) k k k C and 23 55 23 2    ( ) ( ) ( ) k k k C (1a)  22 22 22    ( ) ( ) ( ) k k k A and 23 55 23 2    ( ) ( ) ( ) k k k A (1b) with   22 22 23 32 33   ( ) ( ) / k k C C C C C and   22 22 21 12 11 ( ) ( ) / k k A A A A A    , where the ( ) k ij C and ( ) k ij A are the coefficients of the 6×6 stiffness and compliance matrices (engineering notation). Transverse normal tractions will then be derived a posteriori from local equilibrium. The interfaces are described by interfacial traction laws which relate the interfacial shear tractions, acting along the surface of the layer k at the interface with unit positive normal vector,  S ( ) k , 23 2 23 2 3 3       ( ) ˆ ˆ ( ) ( , ) k k k k S x x x x (2) to the interface relative sliding displacement:         1 2 2 2 2 2 3 3 2 2 3 3       ˆ ˆ ( ) , , k k k k k k v v x v x x x v x x x (3) The interfacial traction law is generally nonlinear to represent different physical mechanisms, which may include the elastic response of thin interfacial layers, cohesive/bridging mechanisms developed by trans-laminar reinforcements or other means, material rupture, elastic contact along the delamination surfaces [3-7,20,21]. The law can be approximated as a piecewise linear function so that the arbitrary branch i is described by an affine function of the relative displacement: 23 2 ˆ ˆ ˆ i k i k i k k i k S S S K v t      (4a)   2    ˆ ˆ k i k i k i k S S S v B t (4b) where i k S K and i k S B are the interface tangential stiffness and compliance and i k S t is a constant interfacial traction which is assumed to act when 2 0 ˆ k v  , and is typically positive/negative for positive/negative 2 ˆ k v , i.e. 2 2        ˆ ˆ ( ) H( ) i k k k i k S S t H v v t with H the Heaviside step function. A purely elastic interface is described by a single branch with 0  k S t , perfectly bonded interfaces are defined by 0  k S t and  k S B 0, which yields 2  ˆ k v 0 , and fully debonded interfaces by 0  k S t and  k S K 0 , which yields   ˆ k S 0 . For 0  k S t , the affine law of Eq. (4) could describe the bridging mechanisms developed by a through-thickness reinforcement, e.g. stitching, applied to a laminated composite [21]. If the relationship (4) is used to represent branches of cohesive traction laws, different linear functions may be needed to represent processes inducing loading and unloading of the delaminations and the associated sliding and reverse-sliding mechanisms. In this paper, equilibrium equations will be derived for plates with interfaces described by the arbitrary branch i of Eq. (4) and, for the sake of simplicity, the superscript i will be removed. Two length-scales displacement field The displacement field is assumed to be given by the superposition of a global field and local perturbation terms (or enrichments). The nonzero components of the displacement vector at an arbitrary point   1 2 3  x , , T x x x are: 1 1 2 2 3 02 2 3 2 3 3 2 1 1              ˆ ( , ) ( ) n n k k k k k k k v x x v x x x H v H (5a) 3 2 2 0 ( ) ( ) v x w x w   (5b) where 3 3   ( ) k k H H x x  3 3 0   , ; k x x  3 3 1  , k x x and the terms on the right hand side of Eq. (5a) denote different contributions in the displacement representation: 02 02 2  ( ) v v x , 0 0 2  ( ) w w x and 2 2 2 ( ) x    define standard first order shear deformation theory terms, which are continuous with continuous derivatives in the thickness direction, 1 3 C , and,