Elastic-Plastic Finite Element Algorithm Study of Heterogeneous Cosserat Bodies Based on Parametric Variational Principle
|School||Dalian University of Technology|
|Keywords||PVP PQPM Cosserat model VCFEM elastic-plastic analysis|
Due to the high efficiency and flexibility, the classic continuum theory based finiteelement method (FEM) has been applied in a wide range of engineering areas such asaerospace, astronauics, mechanism, civil and nuclear engineerings. However, it will facemany challenges when the classical continuum theory based finite element method is used tosolve the mechanical problems of heterogenous materials such as granular materials, porousmedia and metal matrix composite materials, becasue of the strong heterogeneity of thesematerials. The same problem will exist in the numerical simulation of some special physicalphenomena such as strain localization in material deformation, even the incorrect results maybe obtained. Therefore, it is necessary to improve the classic FEM and develop some newFEMs for efficient computation of heterogeneous materials and strain localization problems.In order to study the effects of micro structures on the macro equavilent mechanicsbehavior of heterogeneous materials, Ghosh and coworkers proposed a material basedVoronoi cell finite element method (VCFEM) for the numerical analysis of the elastic-plasticmechanics properties of heterogeneous materials. In this method, each cell which has asecond phase with arbitrariness in shape, size and distribution is treated as a finite element forthe simulation of the micro structure of actual materials. As a result, the finite element meshgeneration is simplified, the number of finite elements is reduced and the correspondingcomputational efficiency is increased.In attempt to simulate the strain localization and size effect in materials, the Cosseratmodel was proposed by Cosserat brothers in 1909. The kinematics of a Cosserat model ischaracterized by an independant rotation degree of freedom, besides the displacement degreesof freedom in the element. By introducing the internal length scale in the constitutiveformulation, the regularization of the problem is realized. The strain localization problem canbe solved correctly by the Cosserat model and the results turn out to be mesh-independent.Traditionally, the incremental iteration method is widely adopted in solution of thenon-linear problems. Nevertheless, it would result in the problems of low convergent speed ornon-convergence. When the stiffness matrix changes, modification and inversion of thestiffness matrix have to be done and this will prolong the computation time and lower theefficiency. To overcome the limitations of the classic methods, Zhong and coworkersproposed the parametric variational principle (PVP) and the parametric quadratic programming method (PQPM) in which the variational method is generalized to adapt to theproblems where the classical variational principle can not be used and the iteration procedurescan be avoided. For the elastic-plastic problem, this method can avoid the limitation of theDrucker hypothesis. It can also be applied to the non-associated plastic constitutive model,non-normal sliding and strain softening problems.Based on the PVP, a new elastic-plastic FEM is developed in the area of the Cosserattheory for the solution of the stress concentration problem and the strain localization problem.A new PQPM based on the VCFEM is proposed to apply to the macro elastic-plastic analysisof heterogeneous materials. Moreover, this paper makes an effort in incorporating theCosserat theory in the VCFEM for the elastic-plastic analysis of heterogeneous materialsbased on the PVP. The chapters are divided as follows:In Chapter 1, the author summarizes the study on the equivalent mechanics properties ofheterogeneous materials (including the development of the universal relationship which isirrelevant to the micro structure, the discussion of the limitations of the effective moduli, thecomputation of the effective properties of heterogeneous materials with the micromechanicsmethod and the FEM, and the equivalence of the heterogeneous micropolar materials) and thepresentation, the evolution, the corresponding numerical studies of the Cosserat model and itsapplications in engineering.In Chapter 2, the PVP and PQPM are described. At first, the limitations of the classicvariational principle are discussed and the basic idea of the PVP is given. Then, theparametric minimum potential energy principle and the parametric minimum complementaryenergy principle are developed. By transferring the corresponding elastic-plastic problem tothe linear complementary problem, the parametric quadratic programming model isconstructed.In Chapter 3, three quadrilateral plane isoparametric elements based on the linearCosserat theory are constructed. The patch tests for the verification of the elements aredescribed. The stress concentration problem around a circular hole is solved to validate thecharacteristics of the Cosserat model.In Chapter 4, based on the PVP, a new algorithm is developed for the elastic-plasticanalysis of the Cosserat continuum. The parametric minimum potential energy principle andthe corresponding PQPM of the Cosserat continuum are verified and constructed, separately.Strain localization problems are computed numerically with the new method and meshindependent results can be obtained.In Chapter 5, PVP based VCFEM is developed for the computation of the effectivemechanics properties of heterogeneous materials. The finite element formulations of theVoronoi elements with and without inclusion are deduced. The corresponding parametric minimum complementary energy principle of Voronoi element with inclusion is verified. ThePQPM for the Voronoi elements with and without inlusion are presented and the influence ofmicrostructure on the overall mechanics properties of heterogeneous materials is studied.In Chapter 6, the elastic-plastic analysis of heterogeneous Cosserat materials is carriedout with the VCFEM. The parametric complementary energy principle of the Cosserat theoryis developed and the PQPM for the VCFEM are established. Based on the new method,influence of microscopic heterogeneities on the overall mechanical responses ofheterogeneous materials is studied.Finally, the main contributions of the dissertation are concluded and some possiblefurther research work are suggested.The research of the dissertation is supported by the National Natural Science Foundationof China (50679013, 10421202), the Program for Changjiang Scholars and InnovativeResearch Team in University of China (PCSIRT) and the National Key Basic ResearchSpecial Foundation of China (2005CB321704).