Kelvin Notation for Stabilizing Elastic-Constant Inversion

Dellinger, Joe; Vasicek, D.; Sondergeld, Carl H.

doi:10.2516/ogst:1998063

Cited by 46 publications

(24 citation statements)

References 18 publications

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“…The 2D elastic wave solver used in this study is based on a Kelvin notation method. 42,43 Using this notation, the three eigenvectors of the elastic constants tensor ͑in 2D͒ correspond to three eigenstress/eigenstrain vectors. These vectors represent directions where applied stress and response strain are in the same direction.…”

Section: ͑11͒mentioning

confidence: 99%

A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy

Goursolle

Callé

Santos

et al. 2007

The Journal of the Acoustical Society of America

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One way to characterize metallic materials in the presence of defects like dislocation networks is to measure their large dynamic nonlinear elastic response. In this numerical study, a new method combining the nonlinear elastic wave spectroscopy (NEWS) method with a time reversal (TR) process is proposed. This method, called NEWS-TR, uses nonlinear analysis as a pretreatment of time reversal and then consists of retrofocusing only nonlinear components on the defect position. A two-dimensional pseudospectral time domain algorithm is developed here to validate the NEWS-TR method as a potential technique for damage location. Hysteretic nonlinear behavior of the materials being studied is introduced using the Preisach-Mayergoyz model. Moreover, in order to extend this solver in two dimensions, the Kelvin notation is used to modify the elastic coefficient tensor. Simulations performed on a metallic sample show the feasibility and value of the NEWS-TR methodology for microdamage imaging. Retrofocusing quality depends on different parameters such as the filtering method used to keep only nonlinear components and the nonlinear effect measured. In harmonic generation, pulse inversion filtering seems to be a more appropriate filtering method than classical harmonic filtering for most defect positions, mainly because of its ability to filter all fundamental components.

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Section: ͑11͒mentioning

confidence: 99%

A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy

Goursolle

Callé

Santos

et al. 2007

The Journal of the Acoustical Society of America

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“…Major topics of interest in which the concept has been used are: the use of six eigenstiffnesses and orthogonal eigenstates for a better understanding of material behaviour (Rychlewski 1984;Annin and Ostrosablin 2008); different aspects of a spectral decomposition of the stiffness tensor (Theocaris and Philippidis 1991;Theocaris 2000;Bolcu et al 2010); the investigation of material symmetries and preferred deformation modes of anisotropic media, e.g., composite materials (Mehrabadi and Cowin 1990;Bóna et al 2007) including the relationship to fabric tensors (Moesen et al 2012) and deformation-induced anisotropy (Cowin 2011); the transformation of the properties of one anisotropic medium to the closest effective medium from a differing symmetry group (Norris 2006;Diner et al 2011;Kochetov and Slawinski 2009;Moakher and Norris 2006); wave attenuation and elastic constant inversion from wave traveltime data (Carcione et al 1998;Dellinger et al 1998). The inversion of Hooke's law in the case of incompressible or slightly compressible materials was studied by Itskov and Aksel (2002), while the use of the spectral decomposition of the stiffness tensor in a constitutive formulation for finite hyperelasticity in a finite element context was described in Dłuzewski and Rodzik (1998).…”

Section: The Kelvin Mappingmentioning

confidence: 99%

On Advantages of the Kelvin Mapping in Finite Element Implementations of Deformation Processes

Nagel¹,

Görke²,

Kolditz³

et al. 2016

Preprint

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Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix-vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin's method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress-or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost.

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“…It should be mentioned that there are many choices for these 18 independent elastic material constants. For example, some researchers [Cowin and Mehrabadi, 1992;Rychlewski, 1995;Dellinger et al, 1998;Kowalczyk-Gajewska and OstrowskaMaciejewska, 2009] had pointed out that the 21 independent elastic constants in Kelvin notation are consisted of 6 Kelvin moduli, 12 stiffness distributors and 3 orientation angles. The 6 Kelvin moduli are the eigenvalues of the compliance matrix and do not vary with the coordinate system.…”

Section: ∂Smentioning

confidence: 99%

“…They also found the third vanishing component through a specific rotation of coordinate system. It has also been pointed out that the 21 independent elastic constants are consisted of 6 Kelvin moduli, 12 stiffness distributors and 3 orientation angles [Cowin and Mehrabadi, 1992;Rychlewski, 1995;Dellinger et al, 1998;Kowalczyk-Gajewska and Ostrowska-Maciejewska, 2009]. Hence, 18 independent elastic constants are proved, which however has not been clearly presented to students through most textbooks due to the involvement of tensor computation.…”

Section: Introductionmentioning

confidence: 99%

Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient

Zhao

Song

Liu

2016

Int. J. Appl. Mechanics

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The compliance matrix for a general anisotropic material is usually expressed in an arbitrarily chosen coordinate system, which brings some confusion or inconvenience in identifying independent elastic material constants and comparing elastic properties between different materials. In this paper, a unique stiffest orientation-based standardized compliance matrix is established, and 18 independent elastic material constants are clearly shown. During the searching process for the stiffest orientation, it is interesting to find from our theoretical analysis and an example that a material with isotropic tensile stiffness does not definitely possess isotropic elasticity. Therefore, the ratio between the maximum and minimum tensile stiffnesses, although widely used, is not a correct measure of anisotropy degree. Alternatively, a simple and correct measure of anisotropy degree based on the maximum shear-extension coupling coefficient in all orientations is proposed. However, for a two-dimensional constitutive relation, both the stiffness ratio and the shear-extension coupling coefficient can be adopted as proper measures of anisotropy degree.

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Kelvin Notation for Stabilizing Elastic-Constant Inversion

Cited by 46 publications

References 18 publications

A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy

A two-dimensional pseudospectral model for time reversal and nonlinear elastic wave spectroscopy

On Advantages of the Kelvin Mapping in Finite Element Implementations of Deformation Processes

Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient

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