Problems with numerical integration of discontinuous functions

Hi everybody,

 I am a beginner in doing research :-) and my topic is about "Micro Indentation Analysis using Continuum Dislocation Theory". I am applying high-order finite element method for this nonlinear problem.

 My plan is first writing a subroutine for the element. However, when I intend to compute the internal force  by using Gauss Integration, I see a problem with the integrand function of some components of the internal force vector. This integrand is discontious function. It is therefore, I cannot get a good approximation with the standard Gauss integration.

 Does anybody have idea on how to deal with this problem? I attached two images: one of the integrand function before and one of the integrand function after using standard Gauss integration in order to clarify my question. Thank you very much.

Best Regards,

Minh