Micropolar Continuum Theory
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Fiber reinforced composites are attractive as lightweight materials, especially in the aerospace industry. To model their behavior at the global scale (macro-scale), continuum models are often preferred for their computational efficiency over micromechanics approaches. However, classical (Cauchy) continuum models often disregard the local structural mechanisms, such as local bending and rotations of the constituent fibers at the microscale. Similarly, in cellular materials, the inherent bending of the cell walls at the micro-scale during loading contribute to the overall deformation at the continuum-scale.
Higher order micropolar continuum theory introduces these local effects with the generalization of the kinematic degrees of freedom. In addition to the displacement field, there is also an additional independent rotational field introduced into the continuum formulation. As a consequence, there is couple-stress (moment stresses) tensor in addition to the classical force-stress tensor. These manifest from the local rotations and moments at the micro-scale of the material. These aspects of micropolar theory are appropriate for representing the local mechanics of fibrous composites and cellular materials.
Micropolar continuum model becomes especially critical when the wavelength of the deformation is comparable to the characteristic size of the microstructure (i.e. stress gradients around notches, localization phenomenon). This is because of the intricate patterns and complex phenomena exhibited by the morphology of the material at the micro and nano-scales transcend scales and affect the macroscopic behavior.
Localization Phenomenon in Composites (Fiber Kinking)
Composite materials under compression can fail by fiber kinking instability due to the interaction of shear nonlinearity and imperfections in fiber orientation. This is characterized by the formation of a kink band in the material, with a characteristic width approximately equal to the fiber diameter. As a result of large deformation gradients inside the kink band, microstructural deformation modes of the fibers (bending and twisting) become significant, affecting the overall deformation of the material. To overcome the limitations of micromechanics, which requires explicit modeling of each individual fiber and is computationally expensive, this phenomenon can be represented as a localization in the continuum through micropolar theory.
Classical continuum theories fail to capture localization phenomena due to the loss of ellipticity in the governing equations. However, micropolar theory introduces an intrinsic length scale into the continuum formulation through the moment-curvature relation, which increases the order of the differentials in the governing equations and prevents the ill-posedness associated with localization. Rather than introducing material softening into the constitutive relation, micropolar theory shows this to be unnecessary. Instead, localization is induced through the coupling of geometric and material nonlinearity, along with the misalignment of the principal material direction with respect to the dominant loading directions, which introduces axial-shear coupling. As a result, micropolar theory can predict the snap-back behavior in the macroscopic stress-strain response, along with the localization associated with fiber kinking.
Hemitropic Micropolar Properties of 3D Woven Textile Composites
Continuum theories rely on the principle of separation of scales between the micro-scale (d), the intermediate meso-scale (l), and the structural macro-scale (L) (d << l << L). However, due to the complex architecture of 3D woven textile composites (3DWTC) and the relatively large tow size, this separation of length scales is not always evident (i.e., d << l < L). This can result in the material exhibiting complex macroscopic deformations that classical continuum theory cannot effectively predict. The aim of this work is to propose an alternative methodology by replacing the heterogeneous 3DWTC medium with a generalized micropolar continuum. Instead of modeling each feature of the meso-structure in detail, this approach offers computational efficiency while retaining numerical accuracy by representing the 3DWTC material as a homogeneous micropolar continuum.
At the structural scale, the material can be modeled as a micropolar continuum with additional material properties. A generalized Aero-Kuvshinski model is used to represent the constitutive relation of a 3D woven composite (3DWC). Anisotropic micropolar properties and additional hemitropic material constants are determined numerically by analyzing a representative volume element (RVE). In the generalized homogenization scheme, a displacement load, represented by a polynomial field, is applied on the boundary of the RVE. The order of the polynomial dictates the macroscopic degrees of freedom of the continuum: first-order terms correspond to classical strain deformation modes of the RVE, second-order terms produce local bending and twisting, and third-order components correspond to higher-order shear deformation modes.
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