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Dear Omid and Sepehr

I think a non-zero entry in the stiffness matrix doesn't necessarily imply dependence of ith DOF on the jth DOF. This is the case also for standard FEM (without any enrichments). For example for a quadrilateral 2D element the x component of displacement has interaction with y component at each node, but as we know, two independent values of translation in x and y directions can

be added to the solution. May be the rank deficiency instead of non-zero entry, would be a more convenient criteria for examining dependencies. For standard FEM, the 8*8 stiffness matrix has 3 equations which can be constructed by the linear combinations of the other 5 equations. These 3 equations add no bonding on the unknowns, i.e. we have 8 unknowns and 5 independent equations. These three relate to two translations and one rotation in 2D. When this element is enriched, the 16*16 stiffness matrix has 6 dependent equations which are related to the translation and rotation of the two pieces produced by the crack (or translation and rotation of the primary element before defining the crack and that of the crack faces with respect to each other).

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