A Wiener Path Integral Solution Treatment and Effective Material Properties of a Class of One-Dimensional Stochastic Mechanics Problems

A Wiener path integral (WPI)-based approximate technique is developed for determining the joint response probability density function (PDF) of a class of one-dimensional stochastic mechanics problems. Specifically, employing a variational formulation and relying on the concept of the most probable path, the original stochastic problem is recast into a number of deterministic boundary value problems (BVPs) to be solved numerically. Next, it is shown that by employing a Rayleigh-Ritz solution scheme for the BVP, and potentially combining with an appropriate expansion for the joint response PDF, the total computational cost of the technique is kept at a minimal level. In particular, the system response PDF at a specific point in space is determined by solving only very few deterministic systems of algebraic equations. The developed WPI technique can account for nonhomogeneous stochastic fields modeling the material properties as well. Further, it can be construed as a local technique due to the fact that the response PDF at a specific point of interest can be directly determined without the need for obtaining the global solution first. This feature renders the technique particularly well-suited for determining efficiently statistics of effective material properties in homogenization problems, which, in many cases, require knowledge of response statistics at few points/regions only. Finally, the stochastic beam bending problem is considered in detail as an illustrative example. In this regard, a specific case is included as well where, notably, the exact beam joint response PDF coincides with the WPI approximate solution. Comparisons with pertinent Monte Carlo simulation data demonstrate the reliability of the technique.