I study boundary value problems which may mathematically model the development and growth of damage in composite materials. Damage in such materials may consist of cracks, voids, and microcracks. In attempting to develop realistic models of damage growth in composite materials, it is necessary first to identify the important dimensionless parameters governing the fracture process. This is accomplished by consideration of idealized problems which model the most significant physical aspects of the process, but which still are amenable to detailed analysis. This work includes techniques from complex variable analysis, numerical analysis, approximation and expansions of functions, integral equations, and partial and ordinary differential equations. Mathematical modeling uses a wide range of mathematical skills from a wide variety of different mathematical fields to create and analyze mathematical models.
Current Projects: Current studies include the propagation of an accelerating crack in the viscoelastic epoxy of a composite, the effect of a crack along a fiber on the overall physical response of the material, and the effect of the inhomogeneity caused by damage on crack propagation in the material. Analytic solutions are constructed for the stresses and displacements of these models by forming Riemann-Hilbert problems which are solved using complex variable techniques. These solutions typically must be numerically approximated to determine how the dimensionless parameters affect the fracture process.
J.R. Walton and J.M. Herrmann, [1992], A new method for solving dynamically accelerating crack problems: Part 1. The case of a semi-infinite mode III crack in elastic material revisited, Q. Appl. Math., Volume L, No.2 pp. 373-387.
J.M. Herrmann and J.R. Walton, [1994], On the energy release rate for dynamic transient mode I crack propagation in a general linear viscoelastic body, to appear in Q. Appl. Math., Volume LII, No. 2, pp. 201-228.
J.M. Herrmann and L. Schovanec, [1994], Dynamic steady state mode III fracture in a nonhomogeneous viscoelastic body, Acta Mechanica, Volume 106, pp. 41-54.
J.M. Herrmann, [1994], The displacement field due to an interface crack along an elastic inclusion in a differing elastic matrix, Acta Mechanica, Volume 105, pp. 207-226.