13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- where x, y, z are coordinates, subscript S and M denote the slave and master nodes respectively, superscript i is the number of rigid beams. Similarly, the relation of nodal forces between master and slave nodes can be expressed as: S Fi M i i F = t F (3) where M i F and S i F is the master and slave nodal force of ith rigid beam. Fi t is the nodal force transformation matrix. Furthermore, the relation of the two transformation matrices can be obtained according to the principal of virtual work: T 1 δi Fi − − t = t (4) The relations (1) and (3) for all rigid beams in an interface element are brought together as: N N N N S δ M S F M and = u =Tu F T F (5) where δ1 δ2 1 T δ F δ δ8 0 and = ... 0 − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ t t T T T t (6) The displacement jumps of internal cohesive element, δ, can be obtained as: N S δ=Bψu (7) where 24 48 1 0 0 0 0 0 , 0 1 0 0 0 0 ... 0 0 1 0 0 0 × ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ γ γ ψ= γ = γ is used to pass translational movements of the slave nodes to cohesive element. B transforms the nodal displacements of the cohesive element to local displacement jumps. The constitutive law for internal cohesive element can be defined as: τ =Dδ (8) where τ is stress vector, D is damaged elasticity matrix. With the use of virtual work principle, we get the stiffness matrix of the interface element: T T T δ δ d Γ Γ ∫ K =T ψ B DB ψT (9) The nodal force vector of the element is: T T T δ = d Γ Γ ∫ F T ψ B τ (10) It is seen that the nodal force vector and stiffness matrix of the new interface element can be simply obtained from cohesive element as follows: T T cohesive δ cohesive ( ) ( ) and =( ) δ δ K = ψT K ψT F ψT F (11) 2.3. Bilinear constitutive law A bilinear constitutive damage model under pure Mode I loading is adopted for the new interface element, see Fig.3. The interfacial damage is initiated after the normal traction attains the interfacial tensile strength. After that, the stiffness is gradually reduced to zero. The onset displacement is obtained as: o 3 = / N K δ , where N is the interfacial tensile strength and K is the interfacial stiffness. The area under the traction-displacement jump curve is the Mode I fracture toughness ICG :
RkJQdWJsaXNoZXIy MjM0NDE=