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A Problem in Nature Of XFEM Approximation

There is a basic problem in NATURE of approximation of discontinuity in element using XFEM.I tried to illustrate that below:
for example, If you consider an element with strong discontinuity( like fracture or contact) which usually we use heaviside function as enrichment function. we expect two parts of element deform independently because of nature of problem.

In equation for approximation of this element we have two main parts. First  is 'regular' or 'standard' part which uses the standard shape functions of element and Second is the enriched part.

In the 'regular' part we use standard shape functions which are continuous over the interface. it means because of 'regular' part in approximation, displacements of nodes which are on two different sides of enriched element are dependent, and it is a paradox the nature of problem.

for example if we enrich a 4-noded quadratic element using heaviside enrichment function with an interface cross the middle, by investigating the stiffness matrix (assume nodes 1 and 3 are at two different sides of interface) components (1,5),(1,6),(2,5),(2,6) are NOT zero( which is because of regural part in our approximation) which 
means there is an internal bonding between two parts of element in two sides of interface.

How can we omit this internal bonding?



I suppose what you do have with heaviside function as enrichment is that your displacements corresponding to your enrichment are independent not the first part of your solution.
I mean the second term of the solution is independent because if you omit it you will get the same results as ordinary finite element. Therefore you have dependant solution for the first term but independent solution for the second term.

Furthermore, what one can surely say is that with heaviside function you have strongly discontinuous solutuion with both different derivative and value at one point andthis is what we mean by independent. In the cases when you need to compromise for the parasitic terms in blending elements you impose dependecy on the solution by using lagrange mutiplier.

I am not an expert in XFEM but I have had a course in this field so every comment is appreciated.

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