A Continuum-Microscopic Algorithm for Modeling Fibrous, Heterogeneous Media with Dynamic Microstructures

Many materials undergo reconfiguration of microscopic structure in response to applied stress. Computing the mechanical behavior of such materials at the continuum level requires a locally valid stress-strain relation. Due to the dynamic microstructure reconfiguration, such relations are difficult to obtain analytically. Numerical simulation of the microscopic dynamics is an alternative, albeit one that is computationally expensive. Continuum-microscopic (CM) interaction algorithms seek to reduce computational cost by microscopic simulation over some small fraction of the continuum time step of interest, enough to determine the locally valid stress-strain relationship, assumed to hold over the entire continuum time step. One difficulty with this approach is the problem of recreating a valid microscopic configuration at the start of the next continuum time step. In most previous CM algorithms, the microscopic structure at the beginning of a new continuum time step is assumed to obey some predefined statistical distribution. This paper extends current CM methods by applying a probability distribution function (PDF) estimation procedure to describe local microscopic states within each continuum computational cell. The estimated PDFs are extrapolated forward over a continuum time step to recreate new microscopic configurations. This procedure captures local variability in the microscopic structure. The algorithm is applied to a generic fibrous material with randomly oriented, cross-linked fibers. Numerical results show that the procedure furnishes continuum stress and strain values to within 5-10% of those obtained from averaging a full microscopic simulation. The computational time is reduced by a factor of two in serial computation and by an order of magnitude when the PDF estimation procedures are computed in parallel.