Boundary line integrals

In some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary. Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals. 

The last teim in the right-hand side of Equation (4.143) represents boundary line integrals. These result from the application of Green s theorem to... 

After the evaluation of the definite integrals in the coefficient matrix and the boundary line terms in the right-hand side, Equation (2.58) gives... 

In the relationships above, 8 is the angle that the boundary normal makes with a fixed direction in the plane of the specimen. Because the curvature is the rate of change of the boundary normal as the line integral is carried out, k = dO/ds. Also, 6 varies between 0 and 27r in the integration, because the normal rotates by 2/T as the boundary is traversed. Therefore, independent of the shape of the grain, Eq. 15.30 becomes... 

Step 9 To plot a variable along a line, choose Postprocessing/Cross-Section Plot Parameters. Click on the Line/Extrusion tab and choose the two points (r, z) = (0, 0) and (0.5,0). Change the function plotted to be the z-velocity and click OK. The same picture arises. Figure 10.3. This method can be used to integrate over any line, not necessarily a boundary line. 

The path of integration of the line integral in Eq. (13-114) is specified by the assumption of a Henderson boundary. The Henderson boundary requires the concentration of component i in the boundary to be given as a function of x by the expression... 

Gauss theorem [24] allows to convert an area integral over an area A into a line integral along its boundary For the integral in question, we thus get... 

More detailed results are possible if the bodies in contact are isotropic and such that plane strain, or stress, conditions apply and body forces are zero. Let us suppose, as in the last section, that outside of the contact region, only surface tractions (as opposed to displacements) are prescribed on the boundary and further that any line integral of the stresses around a closed contour T, covering any portion of either or both bodies, is zero ... 

Line integration boundary 2 Expression chds.ndflux c... 

---