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A decomposition of Laplace stretch with applications in inelasticity

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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.

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Fig. 1

Notes

  1. In the previous papers on this topic [6,7,8,9], 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.

  2. 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.

  3. 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.

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

  5. 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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Acknowledgements

Constructive discussions took place over the course of this work between ADF and Prof. Charis Harley from the University of the Witwatersrand, Johannesburg, South Africa, with Dr. John Clayton from the Army Research Laboratory at Aberdeen, MD, with our colleague Prof. Arun Srinivasa, and with ADF’s student Shahla Zamani.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843, USA

    Alan D. Freed & K. R. Rajagopal

  2. Impact Physics Branch, U.S. Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD, 21005, USA

    Alan D. Freed

  3. Department of Aerospace Engineering, Texas A&M University, College Station, TX, 77843, USA

    Jean-Briac le Graverend

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  1. Alan D. Freed
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Correspondence to Alan D. Freed.

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Freed, A.D., le Graverend, JB. & Rajagopal, K.R. A decomposition of Laplace stretch with applications in inelasticity. Acta Mech 230, 3423–3429 (2019). https://doi.org/10.1007/s00707-019-02462-3

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