Student Research Abstracts

Numerical Computation for Singular Nonlinear Solid Mechanics Problems: the Modified Picard Method

Kelly Dickson 

  While a Runge-Kutta order-four system solver has been a predominant numerical solution technique for differential equations, a shooting algorithm using the Picard method can potentially produce better accuracy in less time near a singularity.  A theory for a more efficient Picard approach developed by Parker and Sochacki (JMU, Dept. of Math. and Stat.) requires the right hand side of an ODE to be converted into polynomial form, which is possible for nearly all systems/equations arising from continuum physical principles.  This produces an effective numerical shooting technique for boundary-value problems (BVPs) and allows the Taylor polynomial to be formed at each step with each term of the series generated essentially by a single function evaluation. The algorithm allows for a smaller number of steps as the solution marches toward the singularity and provides a simple manner in which to increase (or decrease if desired) the order of the algorithm during the computation, resulting in general in more accurate solutions nearby and at singularities.  The algorithm also routinely “pulls off” the coefficients of the Taylor series to store them for later use, producing high efficiency.  This modified Picard shooting method is first developed theoretically and is then demonstrated for a particular singular test problem and is compared against a standard Runge-Kutta procedure.  Finally, singular nonlinear BVPs associated with modeling both concentrated loads and cavitation (void formation in solids) in finite elasticity are examined numerically via the Modified Picard and Runge-Kutta techniques.

Asymptotic analysis of an inward point load acting on a half-space:  a nonlinearly elastic near-load model.

 Mary Lee

  Concentrated load problems have a long history as a subject of considerable interest in both the linear and nonlinear theories of elasticity, and have applications in a host of physical contexts.  The problem of a compressive point load acting on an elastic half-space was solved by Boussinesq in 1885 within the framework of linear elasticity.  His solution is still in use today for a variety of geotechnical and engineering applications, for example, soil/rock mechanics, soil-structure interaction, earthquake prediction, soil compaction by heavy machinery and its effect on agricultural crops.  While the most interesting and crucial material behavior takes place under and nearby the point load, it is precisely here where Boussinesq’s solution breaks down, predicting physically unreasonable behavior and violating the basic premise of the linear theory in which it was derived.  This work studies this important problem in the context of incompressible nonlinear elasticity.  An asymptotic analysis of the exact, nonlinear, governing, partial differential equations of equilibrium and boundary conditions is carried out and we are able to determine and apply a series of simple tests to decide whether an elastic material may sustain a finite deflection under a point load.