13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- in which ( ) ( ) m k λ g represent the unknown stress and displacement fields due to the mechanical load and ( ) t k c λ+ u are the known initial displacement components from the uniform temperature variation TΔ . The left subscripts c λ+ represent the summation of singular and non-singular components of the variables. 0Π represents the total potential, given as an initial value, associated with the displacements and stresses under thermal load. Matrix n contains 3 2× components of the unit outward normal to boundary kΓ . ( )k u are displacements under thermo-mechanical loads. Introducing the coordinate system transformation matrices () ( ) = g g g Z and =u Uβ, σ Pβ = , ( ) s k = %u Lq , where matrix L is the linear interpolation function, and vector sq is the nodal displacement on the boundary segment kΓ of the super corner element, then we have s 0 1 2 m T m m T t m T Π Π =− − + + β H β β f β Gq (2) where 2 ( ) ( ) T ( ) ( ) ( )T ( ) ( ) ( ) 1 1 d 2 k k T k T k k k k T k k u u k S σ σ Γ = ⎡ ⎤ = + ⎣ ⎦ ∑ ∫ H PZnZU UZnZP , 2 ( ) ( ) 1 d k k T k T k S σ Γ = =∑∫ G P Z nL 2 ( ) ( ) ( ) T ( ) ( ) 1 k t k k T k T k t k u c p k dS σ λ Γ + = =∑∫ f P Z n Z u Setting 0 δΠ= and noting that 0 0 δΠ = , we determine that: { } 1 s m t − = − β H Rq f (3) and thus 1 1 1 s s s 0 1 1 2 2 T T T T t t T t Π Π − − − = − + + q G H Gq q G H f f H f (4) From Eq.(4), we have the following matrices: 1 T s − = K G H G, 1 t T t − = F G H f (5) where sK is the stiffness matrix of the super corner element and t F represents the nodal force due to thermal load on the boundary segment kΓ . This ad hoc element is used to model the near-field region and is combined with the conventional standard four-node hybrid-stress elements in the far-field region.
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