Onset of shear band localization by a local generalized eigenvalue analysis

Abstract

Shear bands are material instabilities associated with highly localized intense plastic deformation zones which can form in materials undergoing high strain rates. Determining the onset of shear band localization is a difficult task and past work reported in the literature attempt to detect this instability by computing the eigenvalues of the acoustic tensor or by studying the linear stability of the perturbed governing equations. However, both methods have their limitations and are not suited for general rate dependent materials in multidimensions.In this work we propose a novel approach to determine the onset of shear band localization and alleviate the limitations of the above mentioned methods.Owing to the implicit mixed finite elements discretization employed in this work, we propose to cast the instability analysis as a generalized eigenvalue problem by employing a particular decomposition of the element Jacobian matrix. We show that this approach is attractive, as it is applicable to general rate dependent multidimensional cases where no special simplifying assumptions ought to be made.To verify the accuracy of the proposed eigenvalue analysis, we first extend an analytical criterion by applying linear perturbation techniques to the continuous PDE model, considering an elastoplastic material with thermal diffusion and a nonlinear Taylor-Quinney coefficient. While this extension is novel on its own, it requires strenuous derivations and is not easily extended to general multidimension applications. Hence, herein it is only used for verification purposes in 1D.Numerical results on one-dimensional problems show that the eigenvalue analysis exactly recovers the instability point predicted by the analytical criterion with non-linear Taylor-Quinney coefficient. In addition, the proposed generalized eigenvalue analysis is applied on two-dimensional problems where propagation of the instability can be easily determined.

Instability analysis of shear bands using the instantaneous growth-rate method

“…In section 3.1 the local analytical instability criterion is recalled, evidentiating the correspondence between this result and experimental observations. It is also shown why this analytical condition, commonly used in the literature, is equivalent to the spectral stability analysis in Arriaga et al [6]. The presentation of these topics is essential for validation of the proposed criterion.…”

“…The inner blocks in M and K are presented in Arriaga et al [6]. The subscripts on each block indicate the fields which are coupled, thus the pair (i,j) represents the variation in the j direction of the residual associated with field i.…”

“…It has been shown in Arriaga et al [6] that the local instability can be computed by doing an eigenvalue analysis of the element stiffness matrix. This was more expensive than the computation of the analytical local instability criterion but was much easier to implement in more complex situations.…”

“…The local stability criteria proposed by Bai [8], Ling and Belytschko [38] and others, are usually derived from a perturbation analysis, which under the same assumptions is equivalent to a local eigenvalue analysis [6]. These works have shown that this instability condition predicts well the localized region and, for a certain special class of problems, it will mark the point after which the solution will rapidly localize.…”

“…It has been shown that the local spectral analysis based on element eigenvalues recover the analytical local instability criterion [6]. Albeit considerably more expensive, the local spectral analysis posses the same implementation simplicity that the instantaneous growth rate and the Rayleigh Quotient have, i.e.…”