A variational approach to statistically nonhomogeneous fields

Classical variational principles of elasticity are cast in a form amenable to the treatment of certain problems concerned with statistically nonhomogeneous fields. The variational principles are utilized to obtain bounds directly upon the ensemble average values of physical quantities which in some sense characterize their problems. General formulations and applications are given for randomly heterogeneous elastic media and elastically supported beams with random axial variation of sectional rigidity and foundation stiffness. Bounds are obtained for the average displacement at the inner surface of a hollow sphere under pressure, the average elongation of a cylinder in uniform tension, and the average displacement of a beam directly under a point load. General methods are given for the selection of the deterministic admissible functions which provide the best bounds that can be obtained from classical principles for this restricted class of admissible functions. The closeness of the bounds is of the same order as the elementary bounds on effective material properties, which are of quantitative value only for small dispersion of the random coefficients. In a special statistically homogeneous case, previously obtained bounds for an effective Young’s modulus are regained. Usually, however, the bounded quantities do not possess the generality of effective properties but are more closely associated with the results of particular problems. The developments of the present paper are considered an initial approach to the investigation of statistically nonhomogeneous problems.