A note on Eringen’s moment balances

The present study provides a comparison of Eringen’s [Eringen, A.C. (1970). Balance laws of micromorphic mechanics. International Journal of Engineering Science, 8, 819–828] general moment balances for micromorphic continua with Germain’s [Germain, P. (1973). The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal on Applied Mathematics, 25, 556–575] momentum balances based on virtual work principles, and with those derived in the present paper by a two-scale Fourier analysis of heterogeneous media. It has not been possible to establish a clear-cut correspondence between Eringen’s balances and either of the latter, partly because Eringen’s balances involve a mixture of surface and volume averages over microdomains.There is disagreement between the last two methods, arising from the fact that Germain’s treatment involves spatial gradients not occurring in the elementary two-scale Fourier analysis. A brief discussion is given of the possible extension of the latter to achieve agreement with the former.As a separate matter, a construction of path-moments of density fields serves to establish a source-flux duality in continuum balances, which inter alia establishes a fairly direct connection between Newton’s and Cauchy’s laws and provides an expression for stress suggested by the statistical mechanics of point-particles.