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Laplace stretch: Eulerian and Lagrangian formulations

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Abstract

Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work-conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting for the purpose of constitutive development, though it has been introduced before in a purely kinematic setting.

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Notes

  1. Regarding Lagrangian stretches with triangular elements, McLellan [1, 5] was the first to propose an upper-triangular decomposition of the deformation gradient. Later, Souchet [6] constructed a stretch tensor with lower-triangular components. We use Srinivasa’s [7] approach for populating an upper-triangular stretch because, of these three Lagrangian approaches, this is the simplest framework to apply.

  2. The upper-left \(2 \! \times \! 2\) matrix in Eq. (6) is equivalent to Eq. (9.3) in Boulanger and Hayes [16].

  3. The Gram factorization of a square matrix results in an orthogonal matrix and an upper-triangular matrix. Here we apply the same strategy, but we secure a different orthogonal matrix and a lower-triangular matrix; hence, the terminology ‘Gram like.’

  4. See Ref. [9] for one way to extend this approach to anisotropic materials.

  5. Curiously, \(\varvec{\mathcal {U}}^{-1} \dot{\varvec{\mathcal {U}}}\) has components akin to Eq. (36), except its components are upper triangular instead of lower triangular, and are expressed in terms of the Lagrangian stretch attributes instead of their Eulerian counterparts.

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Acknowledgements

This research was inspired by a conversation that ADF had with Prof. Michael Sacks at the 2019 Annual Meeting of the Society for Engineering Science held at Washington University in St. Louis, where he encouraged the author to develop of an Eulerian constitutive framework suitable for biomechanics. The authors also thank anonymous reviewers for their constructive comments.

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Freed, A.D., Zamani, S., Szabó, L. et al. Laplace stretch: Eulerian and Lagrangian formulations. Z. Angew. Math. Phys. 71, 157 (2020). https://doi.org/10.1007/s00033-020-01388-4

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  • DOI: https://doi.org/10.1007/s00033-020-01388-4

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