This paper proposes a micromechanics algorithm utilising the finite element method (FEM) for the analysis of heterogeneous matter. The characterisation procedure takes the material properties of the constituents, axons and extracellular matrix (ECM) as input data. The material properties of both the axons and the matrix are assumed to have linear viscoelastic behaviour with a perfect bonding between them. The results of the modelling have been validated with experimental data with material white input from brainstem by considering the morphology of brainstem in which most axons are oriented in longitudinal direction in the form of a uniaxial fibrous composite material. The method is then employed to examine the undulations of axons within different subregions of white matter and to study the impact due to axon/matrix volume fractions. For such purposes, different unit cells composed of wavy geometries and with various volume factions have been exposed to the six possible loading scenarios. The results will clearly demonstrate the undulation and axon volume fraction impacts. In this respect, undulation affects the material stiffness heavily in the axon longitudinal direction, whereas the axons' volume fraction has a much greater impact on the mechanical properties of the white matter in general. Also the results show that the created stresses and strains in the axons and matrix under loading will be impacted by undulation change. With increase in undulation the matrix suffers higher stresses when subjected to tension, whereas axons suffer higher stresses in shear. The axons always exhibit higher stresses whereas the matrix exhibits higher strains. The evaluated time-dependent local stress and strain concentrations within a repeating unit cell of the material model are indicative of the mechanical behaviour of the white tissue under different loading scenarios.

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An efficient computational algorithm is proposed to evaluate the viscoelastic properties of fibrous composites. A repeating unit cell (RUC) based on a pre-determined fiber packing is assumed to represent the microstructure of the composite. Finite element analysis of this unit cell under six specified loading conditions is carried out to study and quantify the time-dependent composite viscoelastic property of the unit cell. Volume averaging scheme is implemented to determine the averaged response as function of time in terms of stresses and strains. This time history analysis constitutes the data for characterization of the composite and determination of the viscoelastic parameters and coefficients. The individual viscoelastic properties of the constituents' materials as well as the composite are assumed to follow the Prony series definitions. To verify the algorithm certain examples are included. In these examples, the constituting fiber materials are assumed to be elastic and the matrix is assumed viscoelastic. Three different types of fiber packings; hexagonal (HEX), square (SQR) and bidirectional crossed fibers (BCF), will be used and each packing will be examined for different fiber volume (V f/V) fractions. The accuracy and verification of the results have been examined with known circumstances. The methodology is to be accurate and efficient so far the periodicity of the composite material rules. This micromechanics tool could make a powerful viable algorithm for determination of many linear as well as nonlinear properties in continuum mechanics material characterization and analysis.

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