Simple shear in nonlinear Cosserat micropolar elasticity: Existence of minimizers, numerical simulations, and occurrence of microstructure (original) (raw)
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Mechanics Research Communications, 2013
In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.
Planar Cosserat Elasticity of Materials With Holes and Intrusions
Applied Mechanics Reviews, 1995
Recently, Cherkaev, Lurie, and Milton (I 992) established an invariance of stress field in planar linear anisotropic elasticity under a specific shift in bulk and shear moduli; this is now known as the CLM theorem. Motivated by the importance of micropolar models in mechanics of media with micropolar structures, Ostaja-Starzewski and Jasiuk (1995) generalized the CLM theorem to planar micropolar elastic materials and considered inhomogeneous simply-connected materials.
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We study the reduced parameter dependence in linear plane Cosserat elasticity with eigenstrains and eigencurvatures. The focus is on singly connected inhomogeneous materials. We find conditions on the eigenstrains and eigencurvatures for the planar stress field to be invariant under a shift in area Cosserat compliances. The analysis can be extended to multiply connected inhomogeneous or multiphase materials. The special case is linear planar uncoupled micropolar thermoelasticity where eigenstrains represent the product of thermal expansion and temperature change.
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011
The stress-invariance problem for a chiral (non-centrosymmetric) micropolar material model is explored in two different planar problems: the in-plane and the anti-plane problems. This material model grasps direct coupling between the Cauchy-type and Cosserat-type (or micropolar) effects in Hooke's law. An identical strategy of invariance is set for both problems, leading to a remarkable similarity in their results. For both problems, the planar components of stress and couple-stress undergo strong invariance, while their out-of-plane counterparts can only attain weak invariance, which restricts all compliance moduli transformations to a linear type. As an application, when heterogeneous (composite) materials are subjected to weak invariance, their effective (volume-averaged) compliance moduli undergo the same linear shift as that of the moduli of the local phases forming the material, independently of the microstructure, geometry and phase distribution. These analytical results ...
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Modeling of Plastic Deformation Based on the Theory of an Orthotropic Cosserat Continuum
Physical Mesomechanics, 2020
In the paper, the plastic deformation of heterogeneous materials is analyzed by direct numerical simulation based on the theory of an elastic-plastic orthotropic Cosserat continuum, with the plasticity condition taking into account both the shear and rotational mode of irreversible deformation. With the assumption of a block structure of a material with elastic blocks interacting through compliant plastic interlayers, this condition imposes constraints on the shear components of the asymmetric stress tensor, which characterize shear, and on the couple stresses, which irreversibly change the curvature characteristics of the deformed state of the continuum upon reaching critical values. The equations of translational and rotational motion together with the governing equations of the model are formulated as a variational inequality, which correctly describes both the state of elastic-plastic deformation under applied load and the state of elastic unloading. The numerical implementation of the mathematical model is performed using a parallel computing algorithm and an original software for cluster multiprocessor systems. The developed approach is applied to solve the problem of compressing a rectangular brick-patterned blocky rock mass by a rough nondeformable plate rotating with constant acceleration. The effect of the yield stress of the compliant interlayers on the stress-strain state of the rock mass in shear and bending is studied. The field of plastic energy dissipation in the rock mass is analyzed along with the fields of displacements, stresses, couple stresses, and rotation angle of structural elements. The obtained results can help to validate the hypothesis about the predominant effect of curvature on plastic strain localization at the mesolevel in microstructural materials.
International Journal for Numerical Methods in Engineering, 2012
A computational model for micropolar hyperelastic-based finite elastoplasticity that incorporates isotropic hardening is developed. The basic concepts of the non-linear micropolar kinematic framework are reviewed, and a thermodynamically consistent constitutive model that features Neo-Hooke-type elasticity and generalized von Mises plasticity is described. The integration of the constitutive initial value problem is carried out by means of an elastic-predictor/plastic-corrector algorithm, which retains plastic incompressibility. The solution procedure is developed carefully and described in detail. The consistent material tangent is derived. The micropolar constitutive model is implemented in an implicit finite element framework. The numerical example of a notched cylindrical bar subjected to large axial displacements and large twist angles is presented. The results of the finite element simulations demonstrate (i) that the methodology is capable of capturing the size effect in three-dimensional elastoplastic solids in the finite strain regime, (ii) that the formulation possesses a regularizing effect in the presence of strain localization, and (iii) that asymptotically quadratic convergence rates of the Newton-Raphson procedure are achieved. Throughout this paper, effort is made to present the developments as a direct extension of standard finite deformation computational plasticity.