Hi all,

I'm interested in problems include cracks using XFEM module. I have implied high-order terms of crack tip enrichment into the OOFEM code. The code is been tested and give correct solutions for different orders of XFEM. Now I want to calculate the crack open displacement, but face problem when I use high order formulation.

The additional terms for crack tip are:
first order:      { r^0.5*sin(θ/2), r^0.5*cos(θ/2), r^0.5*sin(θ/2)*sin(θ), r^0.5*cos(θ/2)*sin(θ) }
second order:  { r^1*sin(θ),       r^1*cos(θ) }
third order:     { r^1.5*sin(θ/2), r^1.5*cos(θ/2), r^1.5*sin(θ/2)*sin(θ), r^1.5*cos(θ/2)*sin(θ) }
forth order:     { r^2,                 r^2*cos(2θ),     r^2*sin(2θ) }

After the deformation for the body, the crack might open. Assuming the displacement is u1(r, +π) above the crack, which is u2(r, +π) below the crack. And then I want to know the open displacement of the crack Δu = u1 - u2.

Since the crack tip enrichments of first order XFEM are continuities except the r^0.5*sin(θ/2) item, the crack open displacement (assuming very close to the crack tip and have no heaviside enrichment) is:
    Δu = sum( N_i(r)*r^0.5*(sin(θ/2)-sin(θ/2))*c_i ) = 2*sum( N_i(r)*r^0.5*c_i ).
Here N_i(r) is the shape function, and c_i is the associated degree of freedom. This part of code I have tested and work well.

The formulation of crack open displacement for second order XFEM is the same as what for the first order XFEM because the second order enrichments have no significant. Similarly, formulations of crack open displacement for third and forth order XFEM are same because significant terms are  r^0.5*sin(θ/2) and r^1.5*sin(θ/2). They are:
    Δu = sum( N_i(r)*r^0.5*(sin(θ/2)-sin(θ/2))*c_i ) + sum( N_i(r)*r^1.5*(sin(θ/2)-sin(θ/2))*d_i ).
         = 2*sum( N_i(r)*r^0.5*c_i ) + 2*sum( N_i(r)*r^1.5*d_i ).
Here c_i and d_i are associated degrees of freedom. But the answer is wrong, even through the hole displacement field in the VTK file is correct.

Could any one give me some suggestion?

Thanks
Zhang Ji
20160-05-28