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Acta Mechanica

Erschienen in:

15.07.2019 | Note

A decomposition of Laplace stretch with applications in inelasticity

verfasst von: Alan D. Freed, Jean-Briac le Graverend, K. R. Rajagopal

Erschienen in: Acta Mechanica | Ausgabe 9/2019

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Abstract

The kinematics of an inelastic solid established in terms of Laplace stretch and its rate are decomposed into one stretch that describes an elastic response, another stretch that describes an inelastic response, and their rates. These kinematics are a direct consequence of Laplace stretch belonging to the group of all real, \(3 \times 3\), upper-triangular matrices with positive diagonal elements. The Laplace stretch follows from a Gram–Schmidt decomposition of the deformation gradient.

Vorheriger Artikel Generalized strain space formulations and eigenprojection tensors

Nächster Artikel Correction to: A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

In the previous papers on this topic [6‐9], ADF et al. adopted a notation introduced by Srinivasa [10] to denote the upper-triangular representation of deformation that, viz. \(\tilde{{\mathbf {\mathsf{{F}}}}}\), they called distortion. McLellan [4, 5] denoted this upper-triangular representation of deformation as \({\mathbf {\mathsf{{H}}}}\), but he did not name this field. Here, we call it the Laplace stretch and we represent it as \(\varvec{\mathcal {U}}\), using a script-like calligraphic font to denote it and its associated fields in an attempt to simplify the notation and make it more intuitive.

The set of all real, \(3 \times 3\), upper-triangular matrices with positive diagonal elements constitutes a mathematical group under matrix multiplication. In stark contrast, the set of all real, \(3 \times 3\), symmetric matrices with positive eigenvalues does not constitute a group under matrix multiplication, because there is no closure under such a multiplication.

Eckart [16] was the first to propose such a decomposition, within the context of a much more general, differential, geometric consideration. Specifically, he considered stress to be some function of a deformation gradient \({\mathbf {\mathsf{{F}}}}\) whose reference configuration varied.

See also Rajagopal’s discussion on the status of natural configurations [17].

With rapid developments in the experimental technologies of digital image correlation (DIC) and electron backscatter diffraction (EBSD), it is now possible to apply these techniques, collaboratively, to measure changes in displacement and crystallographic angle, respectively, whereby components of the deformation gradient can be quantified within individual grains of heterogeneously deformed samples [23]. Such technologies could be used to compare our decomposition \({\mathbf {\mathsf{{F}}}} = {\mathbf {\mathsf{{Q}}}} \varvec{\mathcal {U}}^e \varvec{\mathcal {U}}^p\) against Lee’s [1] decomposition \({\mathbf {\mathsf{{F}}}} = {\mathbf {\mathsf{{F}}}}^e {\mathbf {\mathsf{{F}}}}^p\).

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Titel: A decomposition of Laplace stretch with applications in inelasticity
verfasst von: Alan D. Freed
Jean-Briac le Graverend
K. R. Rajagopal
Publikationsdatum: 15.07.2019
Verlag: Springer Vienna
Erschienen in: Acta Mechanica / Ausgabe 9/2019
Print ISSN: 0001-5970
Elektronische ISSN: 1619-6937
DOI: https://doi.org/10.1007/s00707-019-02462-3

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