Extracting the left and right critical eigenvectors from the LDU-decomposed non-symmetric Jacobian matrix in stability problems

Fujii, Fumio; Yamakawa, Yuki; Noguchi, Hirohisa

doi:10.1007/s00466-009-0412-5

Cited by 4 publications

(1 citation statement)

References 40 publications

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“…The eigenvector is necessary in the judgment of whether the critical point is a bifurcation point or a load limit point, and it is used as a perturbation to access to the bifurcated path. An efficient numerical method, L D U ‐mode extraction method, proposed by Fujii et al is used to compute the critical eigenvector at the bifurcation point. By this method, we can obtain critical left and right eigenvectors of the nonsymmetric matrix separately in an effective approximating way avoiding computationally costly direct eigenanalysis.…”

Section: Numerical Bifurcation Analysis Proceduresmentioning

confidence: 99%

Diffuse bifurcations engraving diverse shear bands in granular materials

Yamakawa

Ikeda

Saiki

et al. 2017

Num Anal Meth Geomechanics

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SummaryShear bands with characteristic spatial patterns observed in an experiment for a cubic or parallelepiped specimen of dry dense sand were simulated by numerical bifurcation analysis using the Cam-clay plasticity model. By incorporating the subloading surface concept into the plasticity model, the model became capable of reproducing hardening/softening and contractive/dilative behavior observed in the experiment. The model was reformulated to be compatible with the multiplicative hyperelasto-plasticity for finite strains. This enhanced constitutive model was implemented into a finite-element code reinforced by a stress updating algorithm based on the return-mapping scheme, and by an efficient numerical procedure to compute critical eigenvectors of elastoplastic tangent stiffness matrix at bifurcation points. The emergence of diamond-and column-like diffuse bifurcation modes breaking uniformity of the materials, followed by the evolution of shear bands through strain localization, was observed in the analysis. In the bifurcation analysis of plane strain compression test, unexpected bifurcation modes, which broke out-of-plane uniformity and led to 3-dimensional diamond-like patterns, were detected. Diffuse bifurcations, which were difficult to observe by experiments, have thus been found as a catalyst creating diverse shear band patterns.

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Section: Numerical Bifurcation Analysis Proceduresmentioning

confidence: 99%

Diffuse bifurcations engraving diverse shear bands in granular materials

Yamakawa

Ikeda

Saiki

et al. 2017

Num Anal Meth Geomechanics

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Interpreting the Critical Left Eigenvector of a Non-Symmetric Singular Matrix in Mechanics

et al. 2013

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In mathematical problems and mechanical engineering, there are a number of examples, in which a non-symmetric Jacobian matrix is involved in solution of simultaneous linear equations. The left and right eigenvectors of such non-symmetric matrices are in general complex and must be well discerned to each other. Their properties and practical meaning are, however, hardly discussed in engineering applications. Especially when the non-symmetric matrix is singular, the critical left eigenvector corresponding to null eigenvalues is of increased significance in the examination of the solvability of the problem. The present paper describes and interprets the substantial role of the critical left eigenvector of the non-symmetric singular matrix in mechanics. Model examples in applied mathematics, solids, structures and rigid bodies will illustrate the meaning of the critical left eigenvector, when the singularity is unavoidable in the problem to be solved. The discussion will be then extended to the critical left singular vector of a rectangular matrix.

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Extended subloading surface model based on multiplicative finite strain elastoplasticity framework: constitutive formulation and fully implicit return-mapping scheme

Iguchi

Yamakawa

Hashiguchi

et al. 2017

Transactions of the JSME (In Japanese)

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This paper presents a finite strain elastoplastic constitutive model incorporating the extended subloading surface concept within the unconventional plasticity framework for cyclic loadings. This is a reformulated and extended version of the small strain model [Iguchi et al., Trans. JSME (in Japanese), Vol. 82 (2016), No. 841 p. 16-00197]. The constitutive formulation is underpinned by the multiplicative decomposition of the deformation gradient tensor, which is the wellestablished modern kinematical framework in geometrically nonlinear elastoplasticity. In addition to the conventional multiplicative decomposition into elastic and plastic parts, we further introduce two kinds of multiplicative decompositions of the plastic deformation gradient tensor. One decomposition is related to kinematic hardening, and the other an evolution of the elastic-core tensor, i.e. a key internal variable in the extended subloading surface model, which stands for a stress state where the material exhibits most elastic responses. In each decomposition, the plastic deformation gradient tensor is split into an energetic part and a dissipative part, and the former is related to the back-stress tensor for kinematic hardening or the elastic-core tensor defined on a pertinent intermediate configuration via a hyperelastic format. Therefore, the whole constitutive theory can be formulated in terms of deformation-like tensorial variables without resort to any objective or co-rotational rates of stress-like variables such as the back-stress, fulfilling the principle of material frame indifference. We then focus on the numerical stress update scheme for the proposed material model based on the fully implicit return-mapping. Basic properties of the proposed model, as well as the capability of the developed numerical scheme, are examined and verified through numerical examples.

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Extracting the left and right critical eigenvectors from the LDU-decomposed non-symmetric Jacobian matrix in stability problems

Cited by 4 publications

References 40 publications

Diffuse bifurcations engraving diverse shear bands in granular materials

Diffuse bifurcations engraving diverse shear bands in granular materials

Interpreting the Critical Left Eigenvector of a Non-Symmetric Singular Matrix in Mechanics

Extended subloading surface model based on multiplicative finite strain elastoplasticity framework: constitutive formulation and fully implicit return-mapping scheme

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