Mathematical modeling of a Cosserat method in finite-strain holonomic plasticity (original) (raw)
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Meccanica, 2019
This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.
On rotation deformation zones for finite-strain Cosserat plasticity
Acta Mechanica, 2015
In this article, a numerical solution method for the finite-strain rate-independent Cosserat theory of crystal-plasticity is developed. Based on a time-incremental minimization problem of the mechanical energy, a limited-memory Broyden-Fletcher-Goldfarb quasi-Newton method applied to a finite-difference discretization is proposed. First benchmark tests study the convergence to an analytic solution. Further simulations focus on the investigation of rotation localization zones, the bending of a rod and a torsion experiment.
Computational Implementation of Cosserat Based Strain Gradient Plasticity Theories
The current trend in microelectronics towards miniaturization has pushed for an interest in theories intended to explain the behavior of materials at small scales. In particular, an increase in yield strength with decreasing size has been experimentally observed in several materials and under different loading conditions. A class of non-classical continuum mechanics theories has been recently employed in order to explain the wide range of observed size dependent phenomena. The theories are non-classical in the sense that they bring about additional kinematic variables. In the numerical treatment of such theories two issues are clearly identified. First, in a displacement based finite element approach the need appears for higher orders of continuity in the interpolation functions or else alternative formulations must be used. Second, if nonlinear-inelastic material response is expected the theories should be recast in rate form and the corresponding integration algorithms should complement the implementation. In this article we address both problems for the particular case of a Cosserat couple stress theory. We describe alternatives for the numerical treatment and then we extend the framework to the case of a rate independent inelastic-non-linear material behavior. The equations are presented in its flow theory form together with integration algorithms.
The current trend towards miniaturization in the microelectronics industry has pushed for the development of theories intended to explain the behavior of materials at small scales. In the particular case of metals, a class of available non-classical continuum mechanics theories has been recently employed in order to explain the wide range of observed behavior at the micron scale. The practical use of the proposed theories remains limited due to issues in its numerical implementation. First, in displacement-based finite element formulations the need appears for higher orders of continuity in the interpolation shape functions in order to maintain the convergence rate upon mesh refinement. This limitation places strong restrictions in the geometries of the available elements. Second, the available inelastic constitutive models for small scale applications have been cast into deformation theory formulations limiting the set of problems to those exhibiting proportional loading only. In this article two contributions are made for the particular case of a Cosserat couple stress continuum. First it describes a numerical scheme based on a penalty function/reduced integration approach that allows for the proper treatment of the higher order terms present in Cosserat like theories. This scheme results in 1 PhD in computational mechanics, jgomezc1@eafit.edu.co, associate professor, applied mechanics group, EAFIT University, Medellín-Colombia.
Finite element formulation of slender structures with shear deformation based on the Cosserat theory
International Journal of Solids and Structures, 2007
This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.
Finite Elements in Analysis and Design, 2015
The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these theories and the associated non-classical boundary conditions, special types of elements in the finite element method maybe necessary. This makes the numerical implementation of this higher-order theory not straightforward. In this paper, a robust but straightforward numerical implementation of higher-order gradient-dependent plasticity theories is presented. The novelty of this paper is in (1) the application of the meshless methods, particularly the moving weighted least square method, combined with the finite element method for the numerical computation of plastic strain gradients, and (2) the numerical implementation of different types of higher-order microscopic boundary conditions at internal/external surfaces, interfaces, and moving elastic-plastic boundaries. The proposed numerical implementation algorithms can be easily adapted in the implementation of any form of higher-order gradient-dependent constitutive models. Examples of applying the current numerical approach is demonstrated for capturing mesh-objective shear band formation and size effect and boundary layer formation in thin films on elastic substrates and metal matrix composites with embedded elastic inclusions through the consideration of stiff, intermediate, and soft interfaces.
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, 2005
In this paper an algorithm for large strain elasto-plasticity with isotropic hyperelasticity based on the multiplicative decomposition is formulated. The algorithm includes a (possible) constitutive equation for the plastic spin and mixed hardening in which the principal stress and principal backstress directions are not necessarily preserved. It is shown that if the principal trial stress directions are preserved during the plastic flow (as assumed in some algorithms) a plastic spin is inadvertently introduced for the kinematic/mixed hardening case. If the formulation is performed in the principal stress space, a rotation of the backstress is inadvertently introduced as well. The consistent linearization of the algorithm is also addressed in detail. of hypoelastic constitutive relations with a Jaumann formulation produces dissipation under elastic closed cycles [5], which is inconsistent with the definition of an elastic deformation path. It has also been found that the use of the additive decomposition of the strain tensor in finite deformation inelasticity may produce dependence of incremental elastic deformations on the deformation history . Nonetheless, new corrected formulations using similar approaches have been recently advocated due to their simplicity and broad application [7, 8] and reported numerical simulations using isotropic plasticity seem to produce results similar to multiplicative plasticity. The approaches given in Reference [8] are based on an additive decomposition of total (instead of incremental) logarithmic strains and, going back to the work of Green and Nagdhi, the plastic logarithmic strain is defined in terms of the plastic metric tensor, considered as an internal variable.
Multisurface plasticity for Cosserat materials: Plate element implementation and validation
International Journal for Numerical Methods in Engineering, 2016
The macroscopic behaviour of materials is affected by their inner micro-structure. Elementary considerations based on the arrangement, and the physical and mechanical features of the micro-structure may lead to the formulation of elastoplastic constitutive laws, involving hardening/softening mechanisms and nonassociative properties. In order to model the non-linear behaviour of micro-structured materials, the classical theory of time-independent multisurface plasticity is herein extended to Cosserat continua. The account for plastic relative strains and curvatures is made by means of a robust quadratic-convergent projection algorithm, specifically formulated for non-associative and hardening/softening plasticity. Some important limitations of the classical implementation of the algorithm for multisurface plasticity prevent its application for any plastic surfaces and loading conditions. These limitations are addressed in this paper, and a robust solution strategy based on the Singular Value Decomposition technique is proposed. The projection algorithm is then implemented into a finite element formulation for Cosserat continua. A specific finite element is considered, developed for micropolar plates. The element is validated through illustrative examples and applications, showing able performance.
Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains
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.