# 4. Some continuum mechanics and related geometric problems

The following paper was mentioned above under 2. in connection with the p-harmonic equation:

*On p-harmonic functions, convex duality and an asymptotic formula for injection mould filling*, Euro. Jnl. of Appl. Math. 7(1996), 417-437.

It lead to an asymptotic solution formula for the filling flow in injection moulding of plastics, the so-called {\sl distance model}. The distance in question is a weighted distance within the mould geometry, and the weight is the reciprocal of the gap width (which is not assumed constant). The distance model predicts some nice geometric properties of the mould filling flow. Further, a simple computational algorithm has been developed and implemented by students in Linköping, so that the filling process can be nicely visualized.

The basic theory and a number of solved (= simulated) examples can be found in the departmental report

*Mold-Filling Patterns, Studied by the Distance Model*, LiTH-MAT-R-2000-6, Linköping University.

Further geometric results and the algorithm are explained in the paper

*Five Geometric Principles for Injection Molding*, Intern. Polymer Processing XVIII(2003), 91-94. Hanser Publishers, Munich.

The properties of the weighted distance in a polygonal geometry were studied in the report

*On a weighted distance function*, LiTH-MAT-R-2001-02, Linköping University.

A deformation problem for a so-called power-law fluid (a kind of non-newtonian fluid) was presented in the report

*Asymptotic solution of a compression moulding problem. Sketch of a new approach*, LiTH-MAT-R-95-01, Linköping University.

It leads to an interesting moving boundary problem involving non-local effects. The theory was further developed in the paper

*An asymptotic Model for Compression Molding*, by G.A. and C.Evans, Indiana Univ. Math. J., 51:1(2002), 1-36.

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Last updated: 2019-11-29