Issue 18

S. Marfia et alii, Frattura ed Integrità Strutturale, 18 (2011) 23-33; DOI: 10.3221/IGF-ESIS.18.03 26 2 2 1 2 eq         (11) with 1  and 2  the local principal strains. Moreover, the following condition is introduced: t c D D    (12) in order to prescribe that the damage in tension should not be lower than the damage in compression. Interface damage model without coupling A phenomenological interface model based on the micromechanical idea, developed in [13] and [14], is proposed. The displacement fields of the two joined bodies are denoted as 1 u and 2 u , while the relative displacement at the typical point  x on the interface  is defined as       1 2       s x u x u x . At the micromechanical level, the representative area at the point  x is considered; in the representative area microcracks could be present, so that it can be modeled in a simplified form splitting the representative area in two parts: the undamaged and damaged part. The damage parameter D  is introduced as the ratio between the damaged area with respect to the reference area; it can vary from zero to one: 0 D   corresponds to the undamaged state (no microcarcks are present in the representative area), while 1 D   corresponds to the completely damaged state (the representative area is completely cracked). The stress-relative displacement relationship is formulated: ( ) D              σ K s c p (13) where  K is the interface stiffness matrix,  c is the unilateral contact vector and  p is the sliding friction vector. A local coordinate system on the interface   , T N x x , where the indices N and T indicate the normal and the tangential directions of the interface, respectively, is introduced. In this coordinate system, the stiffness matrix, the unilateral contact vector and the sliding friction vector are represented as: 0 0 ( ) 0 0 N N N T T K s H s K p                         K c p (14) In order to define the evolution of the inelastic slip relative displacement, the stress given in Eq. (13) is rewritten in the following form:   (1 ) d D           σ σ K c p (15) defining the contact-frictional stress d  σ as:     d D          σ K s c p (16) It is assumed that the stress d  σ governs the evolution of the inelastic slip relative displacement. In particular, the classical Coulomb yield function is introduced:   d d N dT d N dT             σ (17) where  is the friction coefficient and the symbol d N   denotes the negative part of the contact-frictional stress. The following non-associated flow rule is considered for the evolution of the components of the vector  p :